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bibcode stringlengths 19 19	abstract stringlengths 113 4.11k	date stringclasses 121 values	identifier stringlengths 70 194	keyword stringlengths 19 518	title stringlengths 4 265	year int64 2.01k 2.02k	read_count float64 108 235	cite_read_boost float64 0 0.98	esources stringclasses 5 values	citation_count float64 0 10.7k	arXiv_PDF_Link stringlengths 35 36	splitted_sections stringlengths 2 1.43M
2024ApJ...976..223W	This paper selected eight totally eclipsing contact binaries for photometric and spectroscopic studies. Spectral data were analyzed by University of Lyon Spectroscopic analysis Software and photometric data were analyzed using PHOEBE through Markov Chain Monte Carlo MCMC sampling. We used two methods to calculate the initial values for running MCMC one method is a new approach proposed by ourselves to model light curves without spots while the other method is the genetic algorithm which can determine physical parameters with spots. The results imply that these eight targets are all contact binary stars with a small mass ratio below 0.25. There are four systems exhibiting the OConnell effect. By adding a dark spot on the primary component the ideal fitting can be obtained. Meanwhile it was found that two systems are shallow contact binaries while the remaining six are moderate contact binaries. An O C analysis of the eight eclipsing binary stars revealed that seven of them exhibit longterm changes. Four of them display a longterm decreasing trend in orbital period while the other three show a longterm increasing trend and two targets exhibit periodic variations. A decrease in period may be caused by the transfer of matter from the more massive component to the less massive component while an increase in period may be caused by transfer in the opposite way. The absolute physical parameters orbital angular momentum initial masses and ages of these eight systems were calculated. Additionally their massluminosity and massradius distributions were analyzed.	2024-12-01T00:00:00Z	['10.48550/arXiv.2409.09743', '2024ApJ...976..223W', '10.3847/1538-4357/ad7f4b', 'arXiv:2409.09743', '2024arXiv240909743W']	['Eclipsing binary stars', 'Fundamental parameters of stars', 'Contact binary stars', '444', '555', '297', 'Astrophysics - Solar and Stellar Astrophysics']	Photometric and Spectroscopic Analysis of Eight Totally Eclipsing Contact Binaries with Small Mass Ratios	2,024	188	0.55	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.09743.pdf	{'Photometric and Spectroscopic analysis of eight totally eclipsing contact binaries with small mass ratios': 'L i -H eng W ang , 1 K ai L i , 1 Y a -N i G uo , 1 J ing -Y i W ang , 1 X iang G ao , 1 X ing G ao , 2 and G uo -Y ou S un 3 \n1 Shandong Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and Physics, Institute of Space Sciences, Shandong University, Weihai, Shandong 264209, China \n2 Xinjiang Astronomical Observatory, 150 Science 1-Street, Urumqi 830011, China \n3 Xingming Observatory, Urumqi, Xinjiang, China', 'ABSTRACT': "This paper selected eight totally eclipsing contact binaries for photometric and spectroscopic studies, spectral data were analyzed by ULySS, and photometric data were analyzed using PHOEBE through MCMC sampling. Weused two methods to calculate the initial values for running MCMC: one method is a new approach proposed by ourselves to model light curves without spots, while the other method is the genetic algorithm (GA) which can determine physical parameters with spot. Due to the results, these eight targets are all small mass ratio contact binary stars with a mass ratio below 0.25. There are four systems exhibiting O'Connell e ff ect. By adding a dark spot on the primary component, the ideal fitting can be obtained. Meanwhile, it was found that two systems are shallow contact binaries, while the remaining six are moderate contact binaries. An O-C analysis of the eight eclipsing binary stars revealed that seven of them exhibit long-term changes. Four of them display a long-term decreasing trend, while the other three show a long-term increasing trend, and two targets exhibit periodic variations. The decrease in period may be caused by the transfer of matter from the more massive component to the less massive component, while the increase in period may be caused by the transfer of matter from the less massive component to the more massive component. The absolute physical parameters, orbital angular momentum, initial masses, and ages of these eight systems were calculated. Additionally, their mass-luminosity and mass-radius distributions were analyzed. \nKeywords: Eclipsing binary stars; Fundamental parameters of stars; Contact binary stars; Stellar evolution", '1. INTRODUCTION': "Binary stars are extremely common star systems in the universe. A thorough understanding of the evolution of binary stars will be of great help in studying the evolution of the universe and the existence of interesting astronomical phenomena. For example, a binary star merger event: two short-period binary stars merge into a rapidly rotating single star due to loss of angular momentum (Qian et al. 2017; Bradstreet & Guinan 1994; Tylenda et al. 2011). According to the filling factor of the two Roche lobes of binary stars, they are divided into detached binary (not filled), semidetached binary (only one Roche lobe is filled), and contact binary (two Roche lobe overfilling). Contact binaries are in the late stage of binary star evolution and serve as a connection between single stars and binary stars, making them of great research value. There are still many unresolved issues in the current research on contact binary, such as short period end (Jiang et al. 2012; Li et al. 2019; Zhang et al. 2023), the O'Connell e ff ect (e.g., O'Connell 1951; Liu & Yang 2003; Qian et al. 2014), and the minimum mass ratio limit (Rasio 1995; Arbutina 2007; Jiang et al. 2010), etc. Physical parameters of a large number of contact binaries are expected as a basis to better solve related issues. \nRucinski (1992) first noticed a short period cuto ff of 0.22 days by using contact binary data in the General Catalogue of Variable Stars (GCVS). Recently, Li et al. (2019) combined with the Variable Star Index (VSX), GCVS and photometric surveys from around the world to determine the period distribution of contact binary stars and found that there is still a significant decline around 0.22 days. Many researchers have tried to solve this problem in history. Rucinski (1992) claimed that the short-period cuto ff could possibly be explained by the fully convective limit. Stepien (2006) suggested that the time scale of the angular momentum loss (AML) is too long, and even at the age of the universe, short period contact binaries below the short period end \nshould not be discovered. Jiang et al. (2012) suggested that the main reason for the short period limit may be the unstable mass transfer when the initial low mass primary component fills its inner Roche lobe. \nContact binary often exhibits unequal heights at the two maxima of the light curve. This phenomenon is called the O'Connell e ff ect (O'Connell 1951). Wilsey & Beaky (2009) outlined three possible explanations for the O'Connell e ff ect: spots on one or both components, the material accretion between two components (Shaw 1994), or the circumstellar material surrounding the binary (Liu & Yang 2003). The most common explanation is star-spot due to magnetic activity (Knote et al. 2022). \nIn past studies, the theoretically determined lower limit of the mass ratio of contact binary was about 0.05-0.09 (e.g., Rasio 1995; Li & Zhang 2006; Wadhwa et al. 2021), but observations in recent years have continuously broken this lower limit, such as: V857 Her ( q ∼ 0.065; Qian et al. 2005), ASAS J083241 + 2332.4 ( q ∼ 0.068; Sriram et al. 2016), V1187 Her ( q ∼ 0.044; Caton et al. 2019), VSX J082700.8 + 462850 ( q ∼ 0.055; Li et al. 2021), TYC 4002-2628-1 ( q ∼ 0.0482; Guo et al. 2022), CRTS J224827.6 + 341351 ( q ∼ 0.0791; Liu et al. 2023) and WISE J185503.7 + 592234 ( q ∼ 0.0514; Guo et al. 2023). Therefore, physical parameters of a large number low-mass ratio contact binaries are needed to determine the true lower limit of the mass ratio and the evolutionary fate of the contact binary. \nDue to the statistical work on contact binaries with both spectroscopic and photometric observations by Pribulla et al. (2003) and Li et al. (2021) and the numerical simulations by Terrell & Wilson (2005), it was found that the mass ratios of total eclipsing contact bianries should be reliable even without spectroscopic observations. In this paper, we selected eight contact binary stars with total eclipses from All-Sky Automated Survey for SuperNovae (ASAS-SN; Shappee et al. 2014; Jayasinghe et al. 2018). The relevant information of the selected targets is shown in Table 1. We analyzed the light curves and orbital period changes of these eight targets.", '2.1. Photometric Observations': "From 2019 to 2021, we used the 60 cm Ningbo Bureau of Education and Xinjiang Observatory Telescope (NEXT) in China to observe the eight binary stars. This telescope is equipped with an FLI PL23042 CCD and has a field of view of 22 ' × 22 ' . During the observations, standard Johnson-Cousins filter and Sloan filter were used. The observation details are shown in Table 2 , and the data reduction is using the IRAF 1 package. First, bias subtraction and flat correction were performed, followed by aperture photometry to extract the instrumental magnitudes. Stars with constant brightness were selected as comparison and check stars, and di ff erential photometry method was used to obtain the light curve of the target. The data of the light curves can be found in ChinaVO (https: // nadc.china-vo.org / res / r101411 / ).", '2.2. Spectroscopic Observations': 'From 2021 to 2022, we conducted spectroscopic observations of these targets using the Xinglong Observatory 2.16 m telescope. During our observation, we used Beijing Faint Object Spectrograph and Camera (BFOSC) and employed G4 mode, with a single pixel spectral resolution of 2.97 (Å) and a wavelength coverage of 3850-7000 (Å). The observation time, exposure time, and signal-to-noise ratio of these eight targets are shown in Table 3. These spectral data were processed using IRAF. Bias subtraction and flat correction were performed, followed by the elimination of cosmic rays. Subsequently, the processed spectra were obtained through spectral extraction, wavelength calibration, and flux normalization. Then, the University of Lyon Spectroscopic analysis Software (ULySS; Koleva et al. 2009) was used to fit the normalized spectral data, and the atmospheric parameters were obtained, such as the temperature, [Fe / H], and log g, the corresponding parameters are recorded in Table 3. The fitted image of the spectrum is shown in Figure 1.', '3.1. data preparation': "First, the method of Kwee & van Woerden (1956) was used to calculate the eclipsing times of the light curves of these eight contact binaries. Then we used the primary eclipsing times as the zero point and the equation: HJD = HJD 0 + P × E , to convert the data from time to phase. Next, obvious outliers were manually removed to ensure data quality. Finally, the magnitudes were converted into flux, which were then normalized to obtain the final light curves. \nVersion 2.4 of PHOEBE (Prˇsa & Zwitter 2005; Prˇsa et al. 2016; Horvat et al. 2018; Conroy et al. 2020; Jones et al. 2020) was used to determine the physical parameters of the eight contact binaries. The contact model was chosen for calculations. First, \nFigure 1. Using ULySS to fit the spectral data. The black line represents the original data and the red line represents the fitted data. \n<!-- image --> \nTable 1. The information of the eight Targets \nwe set the e ff ective temperature obtained from the spectral analysis as the temperature of the primary component. The gravity darkening and bolometric albedo coe ffi cients of all stars are set to g 1 , 2 = 0.32 and A 1 , 2 = 0.5 respectively due to Lucy (1967) and \nTable 2. Photometric observation Details of the eight TargetsTable 3. Spectral observation information and the atmospheric parameters \nRuci'nski (1973). We choose the atmospheric model of Castelli & Kurucz (2004), and limb-darkening coe ffi cients were derived by a logarithmic law.", '3.2. Calculation of initial values of PHOEBE': "According to PHOEBE's introduction, an accurate prior can help the model converge faster. Therefore, we would like to use accurate values as priors for Markov Chain Monte Carlo (MCMC) sampling. The physical parameters that need to be calculated for contact binary stars are mass ratio ( q ), the e ff ective temperature of the secondary component ( T 2), orbital inclination ( i ), luminosity of the primary component ( l 1), and degree of contact ( f ). If the target exhibits the O'Connell e ff ect, we add a spot on the primary component. To quickly obtain the spot parameters, θ (the latitude of the spot) is fixed at 90 · , and other parameters, such as λ (the longitude of the spot), rs (the radius of the spot) and Ts (the relative temperature of the spot), are calculated accordingly. When calculating the initial values of these parameters, we set PHOEBE's pblum mode to dataset -scaled to eliminate the need to calculate the luminosity of the primary component. \nTwo methods are used to obtain the initial values of the parameters for PHOEBE MCMC sampling. One method o ff ers faster calculation speed but cannot model parameters with spots, while the other method can model parameters with spots but takes longer calculation time. Since there is a possibility that q is greater than 1, we need to set the parameter phaseshift to 0.5 and recalculate the parameters using the same method. By comparing with the Residual Sum of Squares (RSS) of the fitting residuals, the parameters with the smallest RSS are selected as the final results. \nThe first method is a new algorithm proposed by ourselves. For the light curve without O'Connell e ff ect, where only four parameters ( q , T 2, i , f ) need to be determined, the parameter space can be divided into equally space intervals. Subsequently, computational traversals can be executed to pinpoint the optimal fitting parameters. To enhance e ffi ciency, a two-step calculation approach is employed. Initially, exploration occurs within a parameter space featuring wide intervals. For example, in this step, the parameter interval for q is 0.1, and for i is 2 · . Subsequently, a narrow interval is applied for parameter search in proximity to the optimal solution from the previous step, thereby enhancing both speed and accuracy. And in this step, the parameter \nTable 4. Search range and precision for the physical parameters \ninterval for q is 0.02, and for i is 1 · . Of course, we can include other parameters, such as the parameters of the spot, in this calculation if we want, but this may cause the computation time to grow exponentially, which is undesirable for us. Utilizing a computer equipped with 80 cores to process 400 data points yields the desired initial values within a mere 10 minutes. Compared to the second method, this method demonstrates superior speed and result stability. This method's precision is listed in Table 4. However, it is unsuitable for light curve with O'Connell e ff ect, presenting a notable limitation in its applicability. \nTherefore, we employed the second method to compute the light curve with O'Connell e ff ect. The second method is Genetic algorithm (GA; Holland 1992). GA is a computational method inspired by natural evolution, rooted in Darwin's theory. It can e ffi ciently explores parameter spaces, seeking optimal solutions without predefined rules, and autonomously adapts and refines search strategies, exhibiting robust exploratory capabilities. The procedure of a GA involves evaluating the fitness of each individual based on predefined criteria, selecting the fittest individuals from the population, and applying genetic operations like crossover and mutation to generate a new population. This process iterates until termination criteria are satisfied. GA excels in exploring parameter spaces devoid of directional cues, enabling the search for globally optimal solutions. In the actual operation, python package scikit-opt 2 was used to implement GA, setting the population size size pop = 500, the mutation rate prob mut = 0 . 01, and the number of iterations max iter = 100. Since it is necessary to obtain an initial value close to the real value, the space range of each parameter is divided into a finite number of equidistant data points, and this addresses the issue of potentially excessive computation time caused by overly high precision. For example, we set the minimum interval for q to 0.02, Table 4 gives the precision for each parameter. \nAfter obtaining the values of q , i , T 2, f , λ , rs and Ts , the value of primary luminosity can be automatically determined, and these initial values are input as the priors of the MCMC calculation. \nIt should be noted that we additionally obtain a set of parameters which contributes to the third light ( l 3) in our calculation. Therefore, we end up with two distinct sets of parameters; one includes l 3, whereas the other does not. We have incorporated the third light into both algorithms mentioned above, with specific search ranges and interval settings as detailed in Table 4.", '3.3. MCMC calculation': 'Before performing MCMC, the prior distribution of the parameters was chosen to follow a Gaussian distribution, with the previously calculated parameters used as the mean value of the Gaussian distribution, and set σ for di ff erent parameters as shown in Table 4. When dealing with parameters containing spots, a higher number of nwalkers is needed. Consequently, walker is set to 32 for parameters with spots, while the remaining uses 24. The initial iteration number is set as 2000. In order to confirm the convergence of the MCMC calculation, it is ensured that the number of iterations for each parameter is 10 times of the autocorrelation time according to Conroy et al. (2020). During the operation, it can be seen that certain nwalker chains may converge to local values. However, this does not a ff ect the ultimate result. To uphold calculation rigor, we opt to discard convergence chains with lower lnprobability (the cost function) branch and only retain those with higher lnprobability (indicating closer proximity to the real value) branch, and resample all walkers from the higher branch and continue iterations from there. \nRegarding results that do not include l 3, the posterior distribution of two targets (with spot and without spot, respectively) are shown in Figures 2 and 3 for example. Figure 4 shows the observed data, fitted curves, and O-C residuals. And physical \nTable 5. The physical parameters of the eight Targets \nTable 6. Luminosity ratio of the primary component for eight targets \nparameters of the eight systems are summarized in Tables 5 and 6. It should be mentioned that, the errors of these parameters are all underestimated (Prˇsa & Zwitter 2005). We combined the error of T 1 due to spectral fitting with the error of T 2 determined by MCMC sampling, resulting in the final error for T 2. And we show the result with l 3, the fitted curves and one of the posterior distribution are shown in Figures 5 and 6, and the physical parameters of the eight systems are shown in Tables 7 and 8. \nWe found that by introducing the l 3 parameter, there are significant shifts in the q values for all targets, which we attribute to the strong correlation between q and l 3. This high degree of correlation makes it challenging to ascertain the true value of q . To investigate whether our observations are influenced by other stars, we cross-matched our targets with 5 arcsec radius in the Gaia catalog (Gaia Collaboration et al. 2016, 2021) . The results revealed that, apart from the target V0678 Peg, no additional stars were detected in the vicinity of the targets. Concerning V0678 Peg, we found an adjacent star, however, the brightness of V0678 Peg far surpasses that of this neighboring star by a factor of 489, meaning that our observations were not a ff ected by nearby stars. Thus, we proposed that the results without third-light are more reliable. This viewpoint aligns with the stance presented by Liu (2021), which posits that l 3 has a significant impact on q , yet is very faint, it is better to set it to zero. In light of these considerations, we have adopted the results excluding third-light contributions for all subsequent analyses and computations.', '4. O-C ANALYSIS': 'The O-C analysis is a powerful tool to study the dynamical evolution of contact binaries and search for additional companions (Li et al. 2014, 2016). Therefore, we seek to gather as many eclipsing times as possible from global photometric surveys to analyze the O-C diagrams of the eight targets, the sources for computing eclipsing minima include: ASAS-SN,the Zwicky Transient Facility (ZTF) survey (Bellm et al. 2019; Masci et al. 2019), O-C gateway 3 , AAVSO 4 , the Transiting Exoplanet Survey Satellite \nFigure 2. Probability distributions of q , T 2, i , f , L 1 g / LTg , L 1 r / LTr and L 1 i / LTi determined by the MCMC modeling of J005148. \n<!-- image --> \n(TESS; Ricker et al. 2015), and Wide Angle Search for Planets (SuperWASP; Butters et al. 2010). Because the observations, such as ASAS-SN and ZTF, were very dispersed, we use the period shift method proposed by Li et al. (2020) to obtain the eclipsing times. We need to divide the data into groups and use the equation: HJD = HJD 0 + P × E , where HJD is the observation time, HJD 0 is the reference time, and P is the period, to shift the data into one period. Then, we can calculate the eclipsing times using the Kwee & van Woerden (1956) method. The eclipsing times for other observations, such as TESS-2min, 10min, and SuperWASP, can be calculated directly. Due to the time of TESS data is BJD, and that of the others is HJD, so we converted HJD \nFigure 3. Probability distributions of q , T 2, i , f , L 1 g / LTg , L 1 r / LTr , L 1 i / LTi , Ts , rs and λ determined by the MCMC modeling of NSVS 503993. \n<!-- image --> \nto BJD using an online tool 5 . Then, the O-C values of each system are calculated through the following equation, \nBJD = BJD 0 + P × E , (1) \nwhere BJD is the observational eclipsing times, BJD 0 is mentioned in Table 1, and P is the orbital period obtained from ASASSN. The O-C values of these eight systems are displayed in Table 9, and the corresponding O-C diagrams are shown in Figure 7. It is evident from this figure that the periods of most targets undergo long-term changes. \nFigure 4. These figures show the fitting results of each target. The scatter points are the real observation data, the curves are the fitting results, and the fitting residuals are shown in the lower panel. \n<!-- image --> \nTable 7. The physical parameters of the eight Targets after incorporating l 3 \nTable 8. The luminosity ratio of the primary component and the third light ratio for eight targets after incorporating l 3 \nFor J055741, only linear correction was used. For the other stars (except NSVS 6133550 and V0678 Peg), the following equation, \nO -C = ∆ T 0 + ∆ P 0 × E + β 2 × E 2 , (2) \nwas used to fit the O-C diagram. For NSVS 6133550 and V0678 Peg, there is a periodic variation except the secular change, we use the following equation to fit their O-C diagrams. \nO -C = ∆ T 0 + ∆ P 0 × E + β 2 × E 2 + A × sin ( 2 π P 3 × E + φ ) , (3) \nThe periodic variation in the equation may be caused by the magnetic activity cycle or the light travel time e ff ect of a third body, and we will discuss this in detail in Section 5. The results are shown in Table 10. We can see that there are some outlier points in the O-C curves . According to our analysis, this may be due to insu ffi cient accuracy when calculating eclipsing times or magnetic activity in our targets. In order to honor the integrity of the original data, we have chosen to retain the existence of this portion. \nThe results illustrate that three stars exhibit an upward trend, indicating a long-term increase orbital periods. In contrast, the other four stars display a downward trend, suggesting a long-term decrease in their orbital periods. It is worth noting that due to \nFigure 5. Probability distributions of q , T 2, i , f , L 1 g / LTg , L 1 r / LTr , L 1 i / LTi , L 3 g / LTg , L 3 r / LTr and L 3 i / LTi determined by the MCMC modeling of J005148. \n<!-- image --> \nthe limitations imposed by the observational time span and the density of observations across di ff erent phases. Therefore, we are currently restricted to perform some qualitative analyses, and it will still require more observations in the future to assist us in making quantitative calculations.', '5. DISCUSSIONS AND CONCLUSIONS': "Utilizing PHOEBE, we derived the physical parameters of eight totally eclipsing contact binaries. Because they are all totally eclipsing contact binaries, the photometric physical parameters are reliable (Pribulla et al. 2003; Terrell & Wilson 2005). Our analysis revealed that all of them are low mass ratio (q < = 0.25) contact binaries. The light curves of NSVS 503993, V0394 Cam, \nFigure 6. These figures show the fitting results of each target after incorporating l 3. The scatter points are the real observation data, the curves are the fitting results, and the fitting residuals are shown in the lower panel. \n<!-- image --> \nFigure 7. The O-C diagrams for our eight targets. \n<!-- image --> \nTable 9. Eclipse Timings for the Eight Targets \nNote. (1) ASAS-SN; (2) This paper; (3) TESS; (4) O-C gateway; (5) ZTF; (6) SuperWASP; (7) AAVSO \nHere we only show a portion of the table, the table is available in its entirety from ChinaVO (https: // nadc.china-vo.org / res / r101411 / ). \nTable 10. O-C fitting coe ffi cients and errors \nJ055741, and V0737 Cep exhibit noticeable asymmetry, which is attributed to dark spot on the primary component. Additionally, two systems display shallow contact configuration (f < = 0.25), while the remaining six systems feature moderate contact configuration. O-C analysis of all available eclipsing times revealed a long-term decrease in the orbital periods of three systems, whereas four systems show a long-term increase in their orbital periods. \nTo investigate the reasons for the changes in orbital periods, as well as the evolutionary status, initial masses, and ages of the eight contact binaries, we need to derive their absolute parameters. Due to the lack of radial velocity curve, it is di ffi cult to obtain them. Nevertheless, for contact binary stars with a temperature less than 10000K, there is a relation between the orbital period \nTable 11. Absolute parameters of the eight Targets \n( P ) and the semi-major axis ( a ) (Li et al. 2022), \na = 0 . 501( ± 0 . 063) + 5 . 621( ± 0 . 138) × P . (4) \nCombining this equation with the period of the eight binaries, their absolute physical parameters can be obtained as shown in Table 11. \nThe long-term orbital period increase is caused by the transfer of matter from the less massive component to the more massive component. And the long-term orbital period decrease generally caused by angular momentum loss (AML) or the transfer of matter from the more massive component to the less massive component. Then, the following equation (Kwee 1958), \n˙ P P = -3 ˙ M ( 1 M 1 -1 M 2 ) , (5) \nwas used to calculate material transfer rates of these stars, the results are shown in Table 10. For the three targets with long-term decreasing period, the negative sign indicates that the more massive primary is losing mass. We use the equation τ th = GM 2 1 R 1 L 1 to calculate the thermal timescale, and τ th = 1 . 26 × 10 7 yr for J005148, τ th = 1 . 82 × 10 7 yr for V0394 Cam, τ th = 1 . 34 × 10 7 yr for NSVS 2561806 and τ th = 5 . 48 × 10 7 yr for V0737 Cep. The thermal timescale mass transfer rate is determined to be M 1 /τ th = 1 . 13 × 10 -7 M ⊙ yr -1 for J005148, M 1 /τ th = 8 . 08 × 10 -8 M ⊙ yr -1 for V0394 Cam, M 1 /τ th = 1 . 09 × 10 -7 M ⊙ yr -1 for NSVS 2561806 and M 1 /τ th = 2 . 45 × 10 -8 M ⊙ yr -1 for V0737 Cep. For all of them, the thermal mass transfer rates are similar to those listed in Table 10, suggesting that the long-term decrease in orbital period may be caused by mass transfer. For the two targets with periodic variations, the periodic changes may be due to the magnetic activity cycle or the light travel time e ff ect of a third body. We do not provide detailed discussions, mainly because the time span of the eclipsing times is short and the coverage is insu ffi cient, so these results can only be considered preliminary. \nNSVS 6133550 and V0678 Peg showed periodic variations in their O-C diagrams, such variations could potentially be explained by the Applegate mechanism or the light travel time e ff ect caused by a third body (Zhou et al. 2016). \nThe Applegate mechanism (Applegate 1992) suggested that variations in the quadrupole moment of solar-type stars, driven by magnetic activity, can account for cyclic orbital period oscillations. Using the equations (Lanza & Rodon'o 2002; RovithisLivaniou et al. 2000), \n∆ P P = -9 ∆ Q Ma 2 , (6) \n∆ P = p [1 -cos (2 π P / P 3)] × A , (7) \nwe derived the required quadrupole moments. For NSVS 6133550, the primary component has ∆ Q 1 = 1 . 1 × 10 47 g cm 2 and the secondary component has ∆ Q 2 = 1 . 9 × 10 46 g cm 2 . For V0678 Peg, the primary component has ∆ Q 1 = 3 . 8 × 10 47 g cm 2 and the secondary component has ∆ Q 2 = 9 . 5 × 10 46 g cm 2 . However, the typical values of the required quadrupole moments for active close binaries range from 10 51 to 10 52 g · cm 2 . Therefore, it seems unsuitable to explain the cyclic variations using the Applegate mechanism for the two targets. \nAnother explanation is the light travel time e ff ect due to the presence of a third body. Using the mass function, \nf ( m ) = ( m 3sin i ) 3 ( m 1 + m 2 + m 3) 2 = 4 π GP 2 3 × ( a 12sin i ) 3 , (8) \nwhere a 12sin i = A × c (where c is the speed of light). We obtained f ( m ) = 2 . 67( ± 1 . 31) × 10 -3 M ⊙ for NSVS 6133550 and f ( m ) = 10 . 63( ± 2 . 67) × 10 -3 M ⊙ for V0678 Peg. And we assumed the orbital inclination of the third body is i = 90 · , the minimum mass of the tertiary companion of NSVS 6133550 was determined to be m 3 = 1 . 99( ± 0 . 59) × 10 -1 M ⊙ with a separation of 5 . 05( ± 1 . 71) AU , while the minimum mass of the tertiary companion V0678 Peg was determined to be m 3 = 3 . 63( ± 0 . 58) × 10 -1 M ⊙ with a separation of 4 . 30( ± 0 . 77) AU . We assumed the third body to be a main-sequence star and estimated the luminosity by interpolating from the Table 5 provided by Pecaut & Mamajek (2013). We obtained the luminosity of the third body for the two targets to be 0 . 004 L ⊙ and 0 . 016 L ⊙ . The third light contribution for these two targets was calculated to be 0 . 36% and 0 . 43%, respectively. Since the contribution of the third light was very small, we did not consider its e ff ect in the analysis of the light curve (Liu 2021). \nBased on the absolute physical parameters, we plotted the mass-luminosity ( M -L ) and the mass-radius ( M -R ) distributions of the eight binaries as shown in Figure 8. The solid and dotted lines correspond to the zero-age main-sequence (ZAMS) and terminal-age main-sequence (TAMS) provided by Hurley et al. (2002), respectively. Circles and triangles denote the more massive and less massive components of the contact binary stars. It is obvious that the more massive primary components are around the ZAMS, indicating that they are either non-evolved or only little-evolved main-sequence stars. The less massive secondary components, on the other hand, are above the TAMS, indicating that they have evolved away from the main sequence. This phenomenon may be attributed to the transfer of mass and energy from the more massive star to the less massive star. \n<!-- image --> \nFigure 8. M-L and M-R diagrams. the solid line is the ZAMS and the dotted line is the TAMS. The circles represent the primary components and the triangles represent the secondary components. \n<!-- image --> \nThe orbital angular momentum of the contact binary stars can be calculated by the following equation proposed by Eker et al. (2006), \nJorb = 1 . 24 × 10 52 × M 5 / 3 T × P 1 / 3 × q × (1 + q ) -2 , (9) \nwhere MT is the total mass of the binaries, P is the period, and q is the mass ratio. The relationship between logJorb and logMT is shown in Figure 9. The detached binaries from Eker et al. (2006) and the contact binaries from Li et al. (2021) are displayed in Figure 9 for comparison. The boundary line between detached and contact binary stars obtained by Eker et al. (2006) is also marked in Figure 9. It was found that the orbital angular momentum of our eight targets is within the contact binary region. For detached binaries and contact binaries with the same total mass, contact binaries have smaller orbital angular momentum. This may be due to the loss of orbital angular momentum during the formation and evolution of the contact binary stars. \nFor contact binary stars, their initial masses are crucial to their evolution. Therefore, we used the method suggested by Yildiz &Do˘gan (2013) to calculate the initial masses of the two components. First, the following equation is used to calculate the initial mass of the secondary component: \nM 2 i = M 2 + ∆ M = M 2 + 2 . 50( ML -M 2 -0 . 07) 0 . 64 , (10) \nwhere M 2 i is the initial mass of the secondary component, M 2 is the mass of the secondary component at present, and we can calculate ML by mass-luminosity relation: ML = ( L 2 / 1 . 49) 1 / 4 . 216 . Then we can calculate the initial mass of the primary \nFigure 9. The relationship between log MT and log Jorb . The crosses represent detached binaries, the solid circles display contact binaries, and the open circles denote our ten targets. The dashed line refers to the boundary between detached and contact binaries derived by Eker et al. (2006). \n<!-- image --> \ncomponent using the following equation, \nM 1 i = M 1 -( ∆ M -Mlost ) = M 1 -∆ M (1 -γ ) , (11) \nwhere M 1 i is the initial mass of the primary component, M 1 is the mass of the primary component at present, Mlost is the mass lost by the system during the evolution, γ is the ratio of Mlost to ∆ M . We set it to 0.664, the same value as used in Yildiz & Do˘gan (2013), and the results are shown in Table 11. At last, we calculated the age of the targets using the equation proposed by Yıldız (2014), \nt ≈ tMS ( M 2 i ) + tMS ( M 2) , (12) \ntMS = 10 ( M / M ⊙ ) 4 . 05 × (5 . 60 × 10 -3 ( M M ⊙ + 3 . 993) 3 . 16 + 0 . 042) , (13) \nwhere M 2 = ( M 2 i + ML ) / 2. We tabulated the results in Table 11 and we can see that six of them, except V0737 Cep and NSVS 6133550, are similar to the results obtained by Yildiz & Do˘gan (2013) and Yıldız (2014). The masses of V0737 Cep and NSVS 6133550, derived from the empirical relationship, may be inaccurate, and radial velocity curves are required to calculate more accurate absolute parameters in the future. \nThese eight stars have all been studied by Li et al. (2024). In addition, V0737 Cep has been studied by Michaels (2018), and V0678 Peg has been studied by Sun et al. (2020). In order to compare the physical parameters of these binaries, we listed our determined parameters alongside those from Li et al. (2024) in Table 12. Notable di ff erences between the results can be observed. For V0737 Cep, Michaels (2018) use Wilson-Devinney (W-D) code (Wilson & Devinney 1971; Wilson 1979, 1990, 1994) get q = 2 . 5. To find out the true paraments, we use PHOEBE and W-D to model the data from Michaels (2018). We obtained q = 0 . 18 and q = 0 . 16 respectively using PHOEBE and W-D, with setting the phaseshift to 0.5. For V0678 Peg, Sun et al. (2020) get q = 0 . 25, which is equal to our result. Regarding other targets, due to our higher precision of our observational data and the fact that all were observed in three bands, we consider our results to be more reliable. \nIn conclusion, we have observed eight totally eclipsing contact binaries, and their light curves are analyzed using PHOEBE. O-C investigation was performed with all available eclipsing times. We studied their evolutionary states, initial masses, and ages. Future observations of radial velocities are required to obtain more accurate physical parameters, which will help in understanding their formation and evolution. Due to the short time span of the observations, further observations are required to confirm the results of orbital period variations. \nTable 12. Comparison with the physical parameters obtained by Li et al. (2024)", 'ACKNOWLEDGEMENTS': "We extend our heartfelt gratitude to the anonymous reviewer for the insightful comments and constructive suggestions, which have significantly enhanced the quality and clarity of our manuscript. This work was supported by National Natural Science Foundation of China (NSFC) (No. 12273018), and the Joint Research Fund in Astronomy (No. U1931103) under cooperative agreement between NSFC and Chinese Academy of Sciences (CAS), and by the Qilu Young Researcher Project of Shandong University, and by Young Data Scientist Project of the National Astronomical Data Center and by the Cultivation Project for LAMOST Scientific Payo ff and Research Achievement of CAMSCAS. The calculations in this work were carried out at Supercomputing Center of Shandong University, Weihai. \nThis paper makes use of data from ASAS-SN. ASAS-SN isfunded in part by the Gordon and Betty Moore Foundation through grant numbers GBMF5490 and GBMF10501 to the Ohio State University, and also funded in part by the Alfred P. Sloan Foundation grant number G-2021-14192. \nThis work includes data collected by the TESS mission. Funding for the TESS mission is provided by NASA Science Mission Directorate. We acknowledge the TESS team for its support of this work. \nThis paper makes use of observation from the Two Micron All Sky Survey (MASS), a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center / California Institute of Technology. Funding of MASS is provided by the National Aeronautics and Space Administration and the National Science Foundation. \nThis paper makes use of data from the DR1 of the WASP data (Butters et al 2010) as provided by the WASP consortium, and computational resources supplied by the project 'e-Infrastruktura CZ' (e-INFRA CZ LM2018140) supported by the Ministry of Education, Youth and Sports of the Czech Republic.", 'REFERENCES': 'Applegate, J. H. 1992, ApJ, 385, 621, doi: 10.1086 / 170967 Arbutina, B. 2007, MNRAS, 377, 1635, doi: 10.1111 / j.1365-2966.2007.11723.x Bellm, E. C., Kulkarni, S. R., Barlow, T., et al. 2019, PASP, 131, 068003, doi: 10.1088 / 1538-3873 / ab0c2a Bradstreet, D. H., & Guinan, E. F. 1994, in Astronomical Society of the Pacific Conference Series, Vol. 56, Interacting Binary Stars, ed. A. W. Shafter, 228 Butters, O. W., West, R. G., Anderson, D. R., et al. 2010, A&A, 520, L10, doi: 10.1051 / 0004-6361 / 201015655 Castelli, F., & Kurucz, R. 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2023ApJ...953L..29L	We report the discovery of an accreting supermassive black hole at z 8.679. This galaxy denoted here as CEERS1019 was previously discovered as a Lybreak galaxy by Hubble with a Ly redshift from Keck. As part of the Cosmic Evolution Early Release Science CEERS survey we have observed this source with JWSTNIRSpec MIRI NIRCam and NIRCamWFSS and uncovered a plethora of emission lines. The H line is best fit by a narrow plus a broad component where the latter is measured at 2.5 with an FWHM 1200 km sSUP1SUP. We conclude this originates in the broadline region of an active galactic nucleus AGN. This is supported by the presence of weak highionization lines N V N IV and C III as well as a spatial pointsource component. The implied mass of the black hole BH is log M SUBBHSUBM SUBSUB 6.95 0.37 and we estimate that it is accreting at 1.2 0.5 times the Eddington limit. The 18 m photometric spectral energy distribution shows a continuum dominated by starlight and constrains the host galaxy to be massive log MMSUBSUB 9.5 and highly starforming star formation rate or SFR 30 MSUBSUB yrSUP1SUP log sSFR 7.9 yrSUP1SUP. The line ratios show that the gas is metalpoor ZZ SUBSUB 0.1 dense n SUB e SUB 10SUP3SUP cmSUP3SUP and highly ionized log U 2.1. We use this present highestredshift AGN discovery to place constraints on BH seeding models and find that a combination of either superEddington accretion from stellar seeds or Eddington accretion from very massive BH seeds is required to form this object.	2023-08-01T00:00:00Z	['10.48550/arXiv.2303.08918', 'arXiv:2303.08918', '2023arXiv230308918L', '10.3847/2041-8213/ace619', '2023ApJ...953L..29L']	['AGN host galaxies', 'Black holes', 'High-redshift galaxies', 'Galaxies', 'Infrared spectroscopy', 'Spectroscopy', 'Observational astronomy', '2017', '162', '734', '573', '2285', '1558', '1145', 'Astrophysics - Astrophysics of Galaxies']	A CEERS Discovery of an Accreting Supermassive Black Hole 570 Myr after the Big Bang Identifying a Progenitor of Massive z gt 6 Quasars	2,023	189	0.71	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	257	https://arxiv.org/pdf/2303.08918.pdf	{'No Header': '<!-- image -->', 'A CEERS Discovery of an Accreting Supermassive Black Hole 570Myr after the Big Bang: Identifying a Progenitor of Massive z > 6 Quasars': "Rebecca L. Larson 1,40 , Steven L. Finkelstein 1 , Dale D. Kocevski 2 , Taylor A. Hutchison 3,41 , Jonathan R. Trump 4 , Pablo Arrabal Haro 5 , Volker Bromm 6 , Nikko J. Cleri 7,8 , Mark Dickinson 5 , Seiji Fujimoto 1 , Jeyhan S. Kartaltepe 9 , Anton M. Koekemoer 10 , Casey Papovich 7,8 , Nor Pirzkal 11 , Sandro Tacchella 12,13 , Jorge A. Zavala 14 , Micaela Bagley 1 , Peter Behroozi 15,16 , Jaclyn B. Champagne 17 , Justin W. Cole 7,8 , Intae Jung 10 , Alexa M. Morales 18 , Guang Yang 19,20 , Haowen Zhang 15 , Adi Zitrin 21 , Ricardo O. Amorín 22,23 , Denis Burgarella 24 , Caitlin M. Casey 25,26 , Óscar A. Chávez Ortiz 6 , Isabella G. Cox 9 , Katherine Chworowsky 6,40 , Adriano Fontana 27 , Eric Gawiser 28 , Andrea Grazian 29 , Norman A. Grogin 10 , Santosh Harish 9 , Nimish P. Hathi 30 , Michaela Hirschmann 31 , Benne W. Holwerda 32 , Stéphanie Juneau 33 , Gene C. K. Leung 34 , Ray A. Lucas 10 , Elizabeth J. McGrath 2 , Pablo G. Pérez-González 35 , Jane R. Rigby 3 , Lise-Marie Seillé 24 , Raymond C. Simons 4 , Alexander de la Vega 36 , Benjamin J. Weiner 37 , Stephen M. Wilkins 38,39 , and L. Y. Aaron Yung 3,41 \nand The CEERS Team 1 The University of Texas at Austin, Department of Astronomy, Austin, TX, USA; [email protected] 2 Department of Physics and Astronomy, Colby College, Waterville, ME 04901, USA 3 Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA 4 Department of Physics, 196 Auditorium Road, Unit 3046, University of Connecticut, Storrs, CT 06269, USA 5 NSF ' s National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA 6 Department of Astronomy, The University of Texas at Austin, Austin, TX, USA 7 Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242 USA 8 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242 USA 9 Laboratory for Multiwavelength Astrophysics, School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623, USA 10 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 11 ESA / AURA Space Telescope Science Institute, /uniF0A0 Baltimore, MD 21218, USA 12 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK 13 Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK 14 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 15 Department of Astronomy and Steward Observatory, University of Arizona, Tucson, AZ 85721, USA 16 Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 17 Steward Observatory, University of Arizona, 933 N. Cherry Ave, Tucson, AZ 85719, USA 18 Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Austin, TX 78712, USA 19 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands 20 SRON Netherlands Institute for Space Research, Postbus 800, 9700 AV Groningen, The Netherlands 21 Physics Department, Ben-Gurion University of the Negev, P.O. Box 653, Be ' er-Sheva 84105, Israel 22 Instituto de Investigación Multidisciplinar en Ciencia y Tecnología, Universidad de La Serena, Raul Bitrán 1305, La Serena 2204000, Chile 23 Departamento de Astronomía, Universidad de La Serena, Av. Juan Cisternas 1200 Norte, La Serena 1720236, Chile 24 Aix Marseille Univ, CNRS, CNES, LAM Marseille, France 25 The University of Texas at Austin, 2515 Speedway Blvd Stop C1400, Austin, TX 78712, USA 26 Cosmic Dawn Center ( DAWN ) , Denmark 27 INAF -Osservatorio Astronomico di Roma, via di Frascati 33, I-00078 Monte Porzio Catone, Italy 28 Department of Physics and Astronomy, Rutgers, the State University of New Jersey, Piscataway, NJ 08854, USA 29 INAF -Osservatorio Astronomico di Padova, Vicolo dell ' Osservatorio 5, I-35122, Padova, Italy 30 Space Telescope Science Institute, Baltimore, MD, USA 31 Institute of Physics, Laboratory of Galaxy Evolution, Ecole Polytechnique Fdrale de Lausanne ( EPFL ) , Observatoire de Sauverny, 1290 Versoix, Switzerland 32 Physics & Astronomy Department, University of Louisville, Louisville, KY 40292, USA 33 NSF ' s NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA 34 Department of Astronomy, The University of Texas at Austin USA 35 Centro de Astrobiología ( CAB ) , CSIC-INTA, Ctra. de Ajalvir km 4, Torrejón de Ardoz, E-28850, Madrid, Spain 36 Department of Physics and Astronomy, University of California, 900 University Ave, Riverside, CA 92521, USA 37 MMT / Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA 38 Astronomy Centre, University of Sussex, Falmer, Brighton, BN1 9QH, UK 39 Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta Received 2023 March 15; revised 2023 June 1; accepted 2023 June 1; published 2023 August 22", 'Abstract': 'We report the discovery of an accreting supermassive black hole at z = 8.679. This galaxy, denoted here as CEERS\\_1019, was previously discovered as a Ly α -break galaxy by Hubble with a Ly α redshift from Keck. As \n<!-- image --> \nOriginal content from this work may be used under the terms \nof the Creative Commons Attribution 4.0 licence. Any further \ndistribution of this work must maintain attribution to the author ( s ) and the title of the work, journal citation and DOI. \npart of the Cosmic Evolution Early Release Science ( CEERS ) survey, we have observed this source with JWST / NIRSpec, MIRI, NIRCam, and NIRCam / WFSS and uncovered a plethora of emission lines. The H β line is best /uniFB01 t by a narrow plus a broad component, where the latter is measured at 2.5 σ with an FWHM ∼ 1200 km s -1 .We conclude this originates in the broadline region of an active galactic nucleus ( AGN ) . This is supported by the presence of weak high-ionization lines ( NV,NIV ] , and C III ] ) , as well as a spatial point-source component. The implied mass of the black hole ( BH ) is log ( M BH / M e ) = 6.95 ± 0.37, and we estimate that it is accreting at 1.2 ± 0.5 times the Eddington limit. The 1 -8 /uni03BC m photometric spectral energy distribution shows a continuum dominated by starlight and constrains the host galaxy to be massive ( log M / M e ∼ 9.5 ) and highly star-forming ( star formation rate, or SFR ∼ 30 M e yr -1 ; log sSFR ∼-7.9 yr -1 ) . The line ratios show that the gas is metal-poor ( Z / Z e ∼ 0.1 ) , dense ( n e ∼ 10 3 cm -3 ) , and highly ionized ( log U ∼-2.1 ) . We use this present highest-redshift AGN discovery to place constraints on BH seeding models and /uniFB01 nd that a combination of either super-Eddington accretion from stellar seeds or Eddington accretion from very massive BH seeds is required to form this object. \nUni /uniFB01 ed Astronomy Thesaurus concepts: AGN host galaxies ( 2017 ) ; Black holes ( 162 ) ; High-redshift galaxies ( 734 ) ; Galaxies ( 573 ) ; Infrared spectroscopy ( 2285 ) ; Spectroscopy ( 1558 ) ; Observational astronomy ( 1145 )', '1. Introduction': "One of the most consequential periods in cosmic history is the Epoch of Reionization ( EoR ) , where the material between galaxies underwent a signi /uniFB01 cant transition from neutral to ionized hydrogen. Satisfactory explanations for the sources of radiation that contributed to this process, when it started, and how long it lasted are all presently wanting, primarily from the lack of necessary observatories and instruments tuned to the early Universe -until now. With the launch of JWST, we are on the cusp of an exciting new era in astronomy, as detailed studies of galaxies in the /uniFB01 rst billion years are /uniFB01 nally possible. \nOne of the key questions JWST was designed to answer was when and how the /uniFB01 rst black holes ( BHs ) formed. De /uniFB01 ning this epoch will help constrain the role these sources played in reionization alongside ionizing photons from massive stars. Supermassive black holes ( SMBHs ) exist at the centers of massive galaxies in the present-day Universe and exhibit a wellstudied correlation with the velocity dispersions ( and stellar masses ) of galaxy bulge components ( see the review by Kormendy & Ho 2013 ) . These massive objects have > 13 Gyr of cosmic time to grow to their present-day mass, which is possible via standard accretion scenarios with stellar-mass BH seeds ( ∼ 1 -10 M e ; possibly up to 100 M e if a Population III star remnant ) . However, the surprising discovery of z > 6 quasars with the Sloan Digital Sky Survey ( SDSS ) with BH masses of log ( M BH / M e ) > 9 ( e.g., Fan et al. 2006; Bañados et al. 2018;Wang et al. 2021; Farina et al. 2022 ) challenges models of BH growth. Such objects require very early ( z ∼ 25 -30 ) stellar-mass seeds with near-unity duty cycles of Eddington-limited accretion and / or super-Eddington accretion to grow to such a mass from a stellarmass seed in /uniF088 1Gyr ( e.g., Volonteri et al. 2021 and references therein ) . None of these scenarios are adequately predicted by simulations ( e.g., Volonteri et al. 2021;Fontanotetal.2023 ) .This tension is further exacerbated by the similarly massive quasars now being discovered at z > 7 ( e.g., Mortlock et al. 2011; Bañados et al. 2018;seeFanetal.2022 for a recent review ) , with the current highest-redshift known active galactic nucleus ( AGN ) being a bright quasar at z = 7.64 ( Wang et al. 2021 ) . \nThese surprisingly massive BHs in the /uniFB01 rst Gyr of cosmic history have led to an alternative seeding theory: direct collapse black holes ( DCBHs; Bromm & Loeb 2003 ) .Inthismodel, minihalos irradiated by Lyman -Werner photons ( 11.2 -13.6 eV ) cannot form molecular gas and thus do not form Population III stars. As these halos grow, they eventually cross the atomic cooling regime, at which point the gas will begin to cool rapidly via H I cooling ( primarily Ly α ) . Further fragmentation-inducing \ncooling is avoided due to the H2-suppressing UV background, leading to collapse into massive BHs in the range of 10 4 -6 M e , preceded by a brief phase as a supermassive star ( see Smith & Bromm 2019; Woods et al. 2019 and references therein for a detailed explanation of this process ) . A similarly massive BH seed could be formed as the remnant of a supermassive star powered by WIMP-like dark matter annihilation ( so-called ' dark stars ' ;Ilieetal.2012; Freese et al. 2016 ) . Such massive BH seeds, forming at z ∼ 10 -15 ( by necessity after the /uniFB01 rst generation of stars ) , could more feasibly grow into the observed z ∼ 6 -7 quasar population ( e.g., Madau et al. 2014;Natarajan et al. 2017; Regan et al. 2019; Latif et al. 2021;Pacucci& Loeb 2022;Massonneauetal.2023; Trinca et al. 2023 ) .These massive quasars must represent only the extreme cases -there likely exists a much larger population of lower-mass BHs and / or obscured BHs waiting to be discovered. \nWhile DCBHs could alleviate the tension between observed BH masses and our theories of BH growth, such objects have yet to be observed. One clear pathway to understanding BH growth is to observe more SMBHs in the EoR. Identifying modest-sized BHs at earlier cosmic times could provide further evidence as to whether DCBHs are a necessary pathway. Additionally, the discovery of such a population would both better explain how the observed z ∼ 6 quasar population originally formed and inform on the potential contribution of AGNs to reionization both through X-ray heating ( e.g., Jeon et al. 2014 ) and through ionizing photon contributions ( e.g., Finkelstein et al. 2019; Giallongo et al. 2019; Grazian et al. 2020, 2022; Yung et al. 2021 ) . \nPrior to JWST, only the most massive SMBHs at high redshifts could be identi /uniFB01 ed. However, the spectroscopic capabilities of JWST now enable the search for signs of AGN activity from less luminous sources, particularly those embedded in galaxies whose stellar emission dominates the total galaxy luminosity ( e.g., Endsley et al. 2022;Kocevskietal.2023 ) and / or where the bulk of the accretion emission is obscured ( e.g., Fujimoto et al. 2022; seealsoFurtaketal.2023 ) . \nHere we report the discovery of the /uniFB01 rst known AGN at z > 8. The galaxy harboring this AGN was /uniFB01 rst identi /uniFB01 ed as a candidate z ∼ 8 galaxy by Roberts-Borsani et al. ( 2016a; named EGSY-2008532660 ) . Its spectroscopic redshift was measured via Ly α emission via Keck / MOSFIRE to be zLy = a 8.683 0.004 0.001 -+ from Zitrin et al. ( 2015; as EGSY8p7 ) ; at the time, and for several years, it was the farthest known Ly α emitter. It later gained a potential N V detection at zsys = 8.667, from Mainali et al. ( 2018 ) , from additional Keck / MOSFIRE data, \nand it was also the brightest z > 8.5 galaxy found in the CANDELS survey ( Finkelstein et al. 2022a; EGS\\_z910\\_6811 ) . Here we report the results from JWST with the Cosmic Evolution Early Release Science ( CEERS ) Survey ( Finkelstein et al. 2022b ) data set. The NIRSpec spectroscopy ( Arrabal Haro et al. 2023, in preparation ) of this source has an ID from the micro shutter array ( MSA ) of 1019; thus, and hereafter, we refer to it as CEERS\\_1019. \nIn Section 2, we present the data from four different JWST observational modes from the CEERS Survey. In Section 3,we describe our method of emission-line /uniFB01 tting, while we explore the detected emission lines in detail in Section 4. In Section 5, we analyze the properties of this galaxy from the available imaging data and from our spectroscopic data in Section 6.We discuss the implications of these results in Section 7 and present our conclusions in Section 8. Throughout this paper we assume a /uniFB02 at Planck cosmology, with H 0 = 67.36 km s -1 Mpc -1 , /uni03A9 m = 0.3153, and /uni03A9Λ = 0.6847 ( Planck Collaboration et al. 2020 ) . All magnitudes are in the AB system, and all restframe wavelengths are vacuum.", '2. Data': 'Data presented in this work were taken as part of the CEERS Survey ( ERS 1345; PI: S. Finkelstein; Bagley et al. 2023; Finkelstein et al. 2022b ) in the CANDELS ( Grogin et al. 2011; Koekemoer et al. 2011 ) Extended Groth Strip ( EGS ) /uniFB01 eld. The complete details of the CEERS program will be presented in Finkelstein et al. ( 2023, in preparation ) , and the program data can be found at doi:10.17909 / z7p0-8481 and at https: // archive.stsci.edu / hlsp / ceers. This source is one of the /uniFB01 rst to be observed and published with four JWST ( Gardner et al. 2023 ) observing modes: NIRSpec ( Böker et al. 2023 ) , NIRCam ( Rieke et al. 2023 ) , MIRI ( Wright et al. 2023 ) , and NIRCam / Wide-Field Slitless Spectrograph, or WFSS ( Greene et al. 2017 ) . We describe these observations below and provide a summary of information about this source in Table 1. \nAdditional imaging data for this source were obtained in the rest-frame infrared from Spitzer / MIPS at 24 /uni03BC m and Herschel / PACS at 100 /uni03BC m, as well as SCUBA-2 at 850 /uni03BC m and the JVLA at 3 GHz, which are shown in Appendix A. X-ray imaging from the Chandra Space Observatory for this source is also discussed in Appendix B.', '2.1. NIRSpec Observations': "The source presented in this work is included in the JWST / NIRSpec ( Jakobsen et al. 2022 ) multi-object shutter ( MOS ) con /uniFB01 gurations taken with the MSA ( Ferruit et al. 2022 ) during the CEERS Epoch 2 observations ( 2022 December ) . These NIRSpec observations are split into six different MSA pointings, each of them observed with the G140M / F100LP, G235M / F170LP, and G395M / F290LP medium-resolution ( R ≈ 1000; here denoted by ' M ' ) gratings plus the prism ( R ≈ 30 -300 ) , fully covering the ∼ 1 -5 /uni03BC m wavelength range. The MSA was con /uniFB01 gured to use three-shutter slitlets, enabling a three-point nodding pattern, shifting the pointing by a shutter length plus the size of the bar between the shutters in each direction for background subtraction. The total exposure time per disperser is 3107 s, distributed as three integrations ( one per nod ) of 14 groups each in the NRSIRS2 readout mode. \nJWST / NIRSpec 2D + 1D spectra for this source in each of the M gratings ( G140M, G235M, and G395M ) are shown in \nTable 1 Source Information for CEERS\\_1019 \nNote. Column ( 1 ) : right ascension. Column ( 2 ) : declination. Column ( 3 ) : observed AB magnitude in the F160W /uniFB01 lter from Finkelstein et al. ( 2022a ) . Column ( 4 ) : observed AB magnitude in the F356W /uniFB01 lter ( Section 2.2 ) . Column ( 5 ) : photometric redshift measured with HST ( prior to JWST photometry; Finkelstein et al. 2022a ) . Column ( 6 ) : photometric redshift measured including JWST photometry from CEERS ( Section 5 ) . Column ( 7 ) : spectroscopic redshift measured from Keck / MOSFIRE spectroscopy via Ly α emission-line detection from Zitrin et al. ( 2015 ) . Column ( 8 ) : spectroscopic redshift measured from JWST / NIRSpec spectroscopy via Ly α . Column ( 9 ) : spectroscopic redshift measured from JWST / NIRSpec spectroscopy via [ O III ] emission-line detection ( Section 4.1 ) . Column ( 10 ) : stellar mass from Prospector SED /uniFB01 t ( Section 5 ) . Column ( 11 ) : speci /uniFB01 c star formation rate ( Section 5 ) . Column ( 12 ) : BH mass ( Section 6.1 ) . Column ( 13 ) : electron temperature ( Section 6.2 ) . Column ( 14 ) : Te -based metallicity ( Section 6.2 ) . Column ( 15 ) : metallicity ( Section 6 ) . Column ( 16 ) : electron density ( Section 6.2 ) . Column ( 17 ) : UV spectral slope ( Section 5.1 ) . Column ( 18 ) : dust attenuation ( Sections 5 and 6.4 ) . \nFigure 1, with the location of the source marked by a horizontal red line in each 2D spectrum ( see Appendix C for details of the prism observations that were corrupted ) . Details of the extraction method from 2D to 1D are detailed in Section 2.1.1 below.", '2.1.1. NIRSpec Data Reduction': "The details of the CEERS NIRSpec data processing are presented in Arrabal Haro et al. ( 2023 ) . We summarize the main steps of the reduction here. The NIRSpec data are processed with the Space Telescope Science Institute Calibration Pipeline 42 ( version 1.8.5; Bushouse et al. 2022a ) and the Calibration Reference Data System ( CRDS ) context jwst \\_ 1027.pmap . We use the calwebb \\_ detector1 pipeline module to subtract the bias and the dark current, correct the 1 / f noise, and generate count-rate maps ( CRMs ) from the uncalibrated images. At this stage, the parameters of the jump step are modi /uniFB01 ed for an improved correction of the ' snowball ' events 43 associated with high-energy cosmic rays. \nFigure 1. 2D and 1D spectra of CEERS\\_1019 from three JWST / NIRSpec M gratings: G140M ( top ) , G235M ( middle ) , and G395M ( bottom ) . The horizontal red dashed line identi /uniFB01 es the central location of the source in the 2D spectrum and is the extraction center for our 1D spectra. Measured emission lines are indicated with purple dotted lines and discussed in Section 4. A description of the CEERS NIRSpec observations for this source is given in Section 2.1, and the data reduction process is described in Section 2.1.1. \n<!-- image --> \nThe resulting CRMs are then processed with the calwebb \\_ spec2 pipeline module, which creates two-dimensional ( 2D ) cutouts of the slitlets, performs the background subtraction making use of the three-nod pattern, corrects the /uniFB02 at /uniFB01 elds, implements the wavelength and photometric calibrations, and resamples the 2D spectra to correct the distortion of the spectral trace. The pathloss step accounting for the slitloss correction is turned off at this stage of the reduction process ( see Figure 2 ) . Instead, we introduce slit-loss corrections based on the morphology of the sources in the NIRCam bands and the location of the slitlet hosting them. \nThe one-dimensional ( 1D ) spectra of the sources were obtained via an optimal extraction ( Horne 1986 ) with a spatial weight pro /uniFB01 le from the trace of the source, such that the pixels near the peak of the trace are maximally weighted. To create the extraction pro /uniFB01 le, the 2D signal-to-noise ratio ( S / N ) spectrum was collapsed in the spectral direction for each grating independently, taking the median value at each spectral pixel and /uniFB01 tting a Gaussian to the positive trace. This source is effectively unresolved at all wavelengths ( the intrinsic size FWHM is less than the JWST point-spread function, or PSF, FWHM ) , so the central trace is unaffected by upper and lower negative traces. A signi /uniFB01 cant trace was measured in both the G140M and G395M gratings, but was not as apparent in the G235M grating such that it could be /uniFB01 tted with a similar Gaussian pro /uniFB01 le. We thus used the pro /uniFB01 le from the G140M grating for the extraction from 2D to 1D in the G235M grating. \nWe evaluate the scaling factor for the slit-loss correction compared to the NIRCam photometry for this source, again \nusing the reduction without the path-loss correction applied. In all NIRCam /uniFB01 lters, we repeatedly calculate the /uniFB02 uxes enclosed within the 0 2 × 0 46 rectangle aperture, using the source and MSA shutter positions, and estimate the scaling factors to match them with the total /uniFB02 ux measurements. The scaling factors range from approximately 2.0 -2.5 among the NIRCam /uniFB01 lters. We correct the spectrum with the scaling factor of the /uniFB01 lter whose central wavelength is closest to that of the observed wavelength. This self-consistently applies an aperture correction accounting for the variable PSF across the observed wavelength range. We assess the uncertainty in these slit losses by comparing them to another method based solely on the F444W /uniFB01 lter for multiple sources in the CEERS observations. We /uniFB01 nd differences in the correction factor on the order of 10% -40% ( Fujimoto et al. 2023 ) on average for all sources. We note that the slit-loss correction is multiplicative, such that any systematic uncertainty in this correction does not affect the signi /uniFB01 cance measurements for emission lines. \nWe test the accuracy of the error spectrum computed by the pipeline by comparing the normalized median absolute deviation of the science spectrum ( masking out emission lines and removing a smoothed continuum ) to the median of the error spectrum in each grating individually. The real data show /uniFB02 uctuations ∼ 1.5 -2 times larger than the typical error value. Thus, we measure and scale the error spectrum up by this scale factor in each grating. \nOnce the errors are corrected, and the spectra are scaled to the NIRCam photometry, the three M gratings are combined into a single spectrum, resampling to a common wavelength \nFigure 2. Comparison of the JWST / NIRSpec combined M grating spectrum when using the native pipeline pathloss correction ( black ) vs. without this step of the pipeline, but rather scaling the spectra to the measured JWST / NIRCam photometry ( blue ) , as described in Fujimoto et al. ( 2023 ) and in Section 2.1.1. Our scaled spectrum ( purple ) highlights how the default pipeline pathloss correction underpredicts the total slit-loss correction for this source and that corrections to the NIRSpec spectrum are required for the /uniFB02 ux calibration of resolved sources. \n<!-- image --> \narray at the overlapping wavelengths and adopting the mean /uniFB02 ux at each pixel, weighted by the /uniFB02 ux errors. This combined spectrum is then used for the remainder of the analysis in this paper.", '2.2. NIRCam Imaging': "The galaxy discussed in this work was observed in the CEERS JWST / NIRCam ( Rieke et al. 2003, 2005; Beichman et al. 2012; Rieke et al. 2023 ) imaging taken in 2022 December ( in CEERS NIRCam Field 8 ) ; these imaging data, including the detailed reduction steps, are described in Bagley et al. ( 2023 ) . 44 Photometry of this source was measured using Source Extractor ( SE; Bertin & Arnouts 1996 ) in dual-image mode. The photometry procedure is broadly similar to that described in full by Finkelstein et al. ( 2022b ) , though with a few differences we detail here. First, to improve color accuracy, especially in the Hubble Space Telescope ( HST ) / WFC3 bands, we do a two-step PSF correction. For any /uniFB01 lters with a PSF FWHM smaller than the NIRCam F277W ( which includes HST / Hubble ' s Advanced Camera for Surveys ( ACS ) , F606W and F814W and NIRCam F115W, F150W, and F200W ) ,we convolve the images with a kernel designed to match these images ' PSFs to that in F277W. For images with larger PSFs ( WFC3 F105W, F125W, F140W, and F160W and NIRCam F356W, F410M, and F444W ) , we derive a correction factor. This is done by convolving the F277W image with a kernel designed to match the PSF to the larger PSF image, with a correction factor derived as the ratio of the /uniFB02 ux in the broadened image to that in the original F277W image. In this way, we correct for light not captured in our default aperture in the bands with larger PSFs, without needing to smooth all bands to that with the largest PSF ( which would be WFC3 F160W ) . \nWe tested this process using source injection simulations to con /uniFB01 rm that accurate colors were recovered. Finally, following Finkelstein et al. ( 2022b ) , we derived accurate total /uniFB02 uxes, /uniFB01 rst by correcting the /uniFB02 uxes in all bands by an aperture correction derived in F277W ( as the ratio between the /uniFB02 ux in our small Kron aperture used to measure colors and in the default larger ' FLUX\\_AUTO ' Kron aperture ) , and then correcting this by a \nfactor of 8% -10%, to account for missing light from the wings of the PSF, with this correction factor derived via the source injection simulations ( see Finkelstein et al. 2022b for more details ) . \nThis /uniFB01 nal photometry catalog includes measurements over the full CEERS NIRCam wavelength range in the F115W, F150W, F200W, F277W, F356W, F410M, and F444W /uniFB01 lters, which have exposures of ∼ 3000 s per /uniFB01 lter ( ∼ 6000 s for F115W ) , as well as in the existing HST / CANDELS ACS and WFC3 F606W, F814W, F105W, F125W, F140W, and F160W bands. We include the F098M ( nondetection ) data for this source from Finkelstein et al. ( 2022a ) . \nThe measured photometric redshift for this galaxy before JWST was ( ) z HST 8.84 phot 0.25 0.12 = -+ ( Finkelstein et al. 2022a ) . With the addition of the JWST / NIRCam imaging ( and by removing the blended Spitzer / IRAC data ) , we measure a photometric redshift with EAZY ( using the same process as Finkelstein et al. 2022b ) nearly equal to the spectroscopic redshift ( z spec = 8.679 ) of ( ) z HST JWST 8.68 phot 0.15 0.09 + = -+ .", '2.3. MIRI Imaging': 'This source was also observed with JWST / MIRI in CEERS Epoch 1 with /uniFB01 lters F560W and F770W. The photometry is obtained from the CEERS team DR0.5 data release for the MIRI 3 and MIRI 6 /uniFB01 elds ( G. Yang et al. 2023, in preparation ) . 45 The MIRI data reduction process is described by Yang et al. ( 2021 ) , 46 and the photometric measurements for this source have previously been reported by Papovich et al. ( 2023 ) . We present the JWST / NIRCam and MIRI photometry for this source in Table 2. With the addition of the MIRI photometry, the photometric redshift is slightly modi /uniFB01 ed to ( ) z HST JWST 8.72 phot 0.06 0.04 + = -+ .', '2.4. NIRCam Grism Observations': "This source was also observed with the JWST / NIRCam WFSS in December 2022 as part of the CEERS Epoch 2 data. These data were obtained using the F356W /uniFB01 lter with the orthogonal column ' C ' and row ' R ' grisms with a resolution R ∼ 1600. Grism spectra for this source are shown in Figure 3; \nTable 2 Photometric Measurements for CEERS\\_1019, with Fluxes in nJy \nNote. The HST photometry is updated from Finkelstein et al. ( 2022a ) using the NIRCam-selected apertures as described in Finkelstein et al. ( 2022b ) . JWST / NIRCam photometry from CEERS Epoch 2 imaging measured in a similar way as Finkelstein et al. ( 2022b ) , described in Section 2.2. JWST / MIRI photometry from Papovich et al. ( 2022 ) using CEERS Epoch 1 imaging is described in Section 2.3. \nthe data reduction methodology will be presented in N. Pirzkal et al. ( 2023, in preparation ) . 47 We observe several emission lines detected in the combined grism spectrum. These are consistent with [ OII ] , [ Ne III ] ,H ò , and H δ if a wavelength shift ( of ∼ 40 Å ) is applied from the NIRCam to NIRSpec spectra. This is likely due to as yet improving wavelength calibrations for both instruments, though given the agreement of our observed Ly α wavelength with ground-based measurements, the offset is likely dominated by the NIRCam WFSS spectra. While upcoming improved calibrations will provide the necessary adjustments to compare better the resulting line /uniFB02 uxes and detections from JWST, these detected lines show the utility of this NIRCam WFSS mode for early galaxy spectroscopy. For the remainder of this paper, we use all measurements from the higher-S / N NIRSpec spectra.", '3. Methods: Emission-line Search': "To search for emission-line features in our 1D spectrum, we utilize an automated line/uniFB01 nding code /uniFB01 rst published in Larson et al. ( 2018 ) and outlined here. This code uses a Monte Carlo Markov Chain ( MCMC ) routine to /uniFB01 t a model that consists of a Gaussian line plus a continuum constant to a given wavelength range, with four free parameters: the continuum level, line central wavelength, line FWHM, and integrated line /uniFB02 ux. To run the MCMC, we use an IDL implementation of the af /uniFB01 neinvariant sampler ( Goodman & Weare 2010 ) to sample the posterior similar to that used in Finkelstein et al. ( 2019 ) , which is similar to the Python emcee package ( Foreman-Mackey et al. 2013 ) . \nAs a /uniFB01 rst pass-through for emission-line features, we search the entire spectrum with our automated code. At each wavelength pixel, we do an initial S / N check ( /uniFB02 ux / error ) .If \nthis is greater than unity, we run our /uniFB01 tting routine ( this ' precheck ' is not required, but helps with ef /uniFB01 ciency, as no detectable emission feature would have S / N < 1 at the line center ) . We use a /uniFB01 tting range of 100 times the pixel scale in the G140M grating on either side of the center pixel, which equates to 630 Å across the full spectrum. This ensures we /uniFB01 t the same wavelength range for every line, regardless of pixel scale. We /uniFB01 t a single Gaussian feature to the spectrum: \n( ) ( ) ( ) /uni239C /uni239F /uni239B /uni239D /uni239E /uni23A0 f f f exp 1 2 . 1 c 0 0 2 2 l l l s = + -- \nWe also impose noninformative priors on each free parameter, designed to increase the ef /uniFB01 ciency of the line detections while removing the chance for the MCMC chain to exit a range of realistic parameters. For the continuum constant ( fc ) , we let it vary between -3 to 10 times the average /uniFB02 ux across the /uniFB01 tting range. These values are larger than the 1 σ noise level and broadly encompass the typical continuum values for our source. We restrict the peak wavelength ( λ 0 ) to be the wavelength at that pixel ± one pixel, such that we /uniFB01 ta Gaussian within each pixel. The actual value in Å varies, as the pixel scale ( /uni0394 λ ) for NIRSpec is wavelength-dependent, with /uni0394 λ G140M = 6.4 Å , /uni0394 λ G235M = 10.7 Å , and /uni0394 λ G395M = 17.9 Å . We limit the FWHM ( FWHM () 2 2ln 2 2.355 s s = = to ± 30 km s -1 of the observed FWHM of the [ OIII ] 5008 line, the brightest feature in the spectrum ( see Figure 4 and Table 3 ) . Tying the FWHM to [ OIII ] 5008 in velocity units rather than pixels is critical, as the wavelength dispersion changes across the observed wavelength range. The line /uniFB02 ux prior requires the line /uniFB02 ux ( f 0 ) to be greater than the average of the /uniFB02 ux over the plotting range after a > 1 σ clipping ( estimate of the continuum at that location ) and less than 100 times the average /uniFB02 ux, well outside the expected range of line /uniFB02 uxes. \nTo de /uniFB01 ne the starting point for our MCMC routine, we used the IDL routine mpfit , a Levenberg -Marquardt least-squares /uniFB01 tting routine that /uniFB01 ts a Gaussian with the above parameters ( Markwardt 2009 ) . We then run our MCMC /uniFB01 tting code with 10,000 iterations and 100 walkers on each pixel and determine the best line /uniFB01 t results as outlined in the following sections. We use the median of the last 10,000 steps of our MCMC chain for our /uniFB01 t parameters. To measure the MCMC error on our parameters, we use the robust sigma calculation: using the median absolute deviation as the initial estimate, then weighting points using Tukey ' s biweight ( Equation ( 9 ) from Beers et al. 1990 ) , assuming a Gaussian posterior distribution. As our MCMC /uniFB01 ts marginalize over the line /uniFB02 ux and the continuum /uniFB02 ux, the errors on both are included in our /uniFB01 nal line /uniFB01 t error. \nWe place several constraints on the resulting Gaussian /uniFB01 ts when determining a successful line detection, including the wavelength and comparison to neighboring pixels. First, we mask out the edges of the wavelength range for the three M Gratings ( within 0.97 -5.1 /uni03BC m ) , where the grating transmission curve falls below 70%. To determine the correct peak pixel for the line, we select the /uniFB01 t at the pixel that has the highest peak /uniFB02 ux ( the /uniFB02 ux at the peak of the Gaussian /uniFB01 t ) within the FWHM of the line. We calculate an integrated S / N from the MCMC Gaussian /uniFB01 t as the median line /uniFB02 ux divided by the robust sigma of the line /uniFB02 ux as described above. We also measure a peak S / N value, calculated by taking the maximum /uniFB02 ux of the Gaussian /uniFB01 t and calculating the ratio of this to the noise per pixel within our /uniFB01 tting range ( similar to Larson et al. 2022 ) . For \nFigure 3. The spectrum of CEERS\\_1019 obtained with JWST / NIRCam ' s wide/uniFB01 eld slitless spectroscopy mode. This consists of spectra with both the column ( ' C ' ; blue ) and row ( ' R ' ; purple ) grisms, both taken with the F356W blocking /uniFB01 lter, with a combined spectrum shown in black. While these data have a higher noise level than the NIRSpec spectra, we observe detections of the same [ O II ] 3727 + 3729, [ Ne III ] 3870, and bluer Balmer lines that we see in NIRSpec ( Figure 5 ) . While we do not use the grism measurements in this paper, these data highlight the utility of this mode for obtaining slitless measurements of modestly faint emission lines out to high redshifts. \n<!-- image --> \nFigure 4. Combined JWST / NIRSpec spectrum from G140M + G235M + G395M of CEERS\\_1019, plotted in F λ [ 10 -19 erg s -1 cm -2 Å -1 ] vs. observed wavelength [ /uni03BC m ] . Plotted in green is the /uniFB01 t to the continuum in the spectrum ( see Section 3.1 ) plus the detected emission lines, as discussed in Section 4. We require an S / N > 2.3 ( with our simulation-estimated noise as described in Section 3.2 ) for a line to be considered detected. \n<!-- image --> \nour initial search through the spectrum for emission lines, we only consider lines found at > 5 σ in both integrated and peak S / N. This initial pass is run for masking out signi /uniFB01 cant features in the spectrum to /uniFB01 t and remove the detected continuum, as described in Section 3.1, and for identifying potentially unexpected emission-line detections in sources at this redshift. In Section 3.2, we discuss how these initial S / N measurements overestimated the signi /uniFB01 cance of our emission-line detections and how we determine the /uniFB01 nal line /uniFB02 ux errors and S / N values for our detected emission lines.", '3.1. Measuring Continuum in NIRSpec': 'To /uniFB01 t the continuum to the NIRSpec spectra, we /uniFB01 rst mask out all the emission lines ( the peak of the line ± FHWM ) and interpolate over those pixels with an average of the three notmasked pixels on either side of the line region. We then smooth the entire spectrum using a boxcar /uniFB01 lter with a width of 60 pixels. As each M grating has a different pixel scale, we then smooth each section of the spectrum with a larger boxcar /uniFB01 lter at the blue end compared to the red end. We use 120 pixels where λ < 1.75 /uni03BC m ( the G140M grating ) , 60 pixels from 1.75 < λ < 2.9 /uni03BC m ( the G235M grating ) , and 30 pixels where λ > 2.9 /uni03BC m ( the G395 grating ) . To remove any discontinuous jumps between the gratings, we smooth the whole spectrum again with the 60 pixel boxcar /uniFB01 lter. This measured continuum \nis plotted in green over the combined NIRSpec spectrum in Figure 4, with notable detection in the G140M grating but nonsigni /uniFB01 cant detection in the redder gratings. This estimated continuum is consistent with the observed photometric spectral energy distribution ( SED ) , as expected due to our slit-loss correction methodology described in Section 2.1.1.', '3.2. Determining the Signi /uniFB01 cance of the Emission Lines': "Upon inspection of the initially identi /uniFB01 ed lines described above, it was found that this methodology tended to overestimate the S / N, especially for faint lines. We thus empirically derive line /uniFB02 ux errors and appropriate S / N measurements for each line that passes the above selection cuts in the following way. Using the line-masked spectrum as above, we /uniFB01 nd the 20 closest ' blank ' pixels, free of any other nearby detected emission lines, on either side of the emission line ( line center + FWHM, such that we are not overlapping the emission line at all ) . At each of these blank pixels, we insert a fake emission line with the same parameters as our detected emission line ( F ( λ ) ,FHWM,and fc ) and a line center ( λ 0 ) at the wavelength of that pixel. We then run the same line/uniFB01 tting routine at each pixel and record the recovered line /uniFB02 ux from the MCMC ( F ( λ ) out ) for each of our 40 simulated lines. Our reported line /uniFB02 ux error ( /uni0394 F ( λ )) is then the median absolute deviation of these recovered line /uniFB02 uxes, as reported \nThe Astrophysical Journal Letters, \n953:L29 \n] \n1 \n- \n) \ns \n10 \n( \nkm \n[ \n[Å] \n) \n9 \n( \n332.94 \n± \n218.61 \n0.93 \n± \n10.67 \n85.20 \n± \n522.45 \n- \n2.30 \n± \n3.90 \nv \n/uni0394 \nEW \nRest \nobs \nλ \nLine \nBlue \n/ \nCentral \n[Å] \nrest \nλ \n) \n2 \n( \n) \n1 \n( \nNV \n( \n26pp \n183.26 \n± \n16.15 \n1.47 \n± \n7.96 \n34.71 \n± \n11.43 \n5.80 \n± \n34.79 \n) \n, 2023 August 20 \nLarson et al. \n19.64 \n± \n8.51 \n7.07 \n± \n44.27 \n77.78 \n± \n71.95 \n3.87 \n± \n8.43 \n30.11 \n± \n23.34 \n- \n6.07 \n± \n26.53 \n46.36 \n± \n53.32 \n5.50 \n38.12 \n± \n27.63 \n- \n8.48 \n70.68 \n± \n19.06 \n6.45 \n32.06 \n± \n31.55 \n- \n18.96 \n13.64 \n± \n1.00 \n27.51 \n124.41 \n± \n0.59 \n- \n82.2 \n74.18 \n± \n144.74 \n9.90 \n15.64 ± 39.36 ± 17.78 ± 128.94 ± 235.35 ± 725.59 ± 6.04 ± 11.56 ± \nin \nline \nblue \nthe \nts, \n/uniFB01 \nsingle-Gaussian \nfor \n) \n1 \ndual-component \nthe \nin \ncomponent \nbroad \n1 \n- \nSection \n( \ndual-component \nthe \nin \ncomponent \nbroad \n. \n[Å] \nin \nNIRSpec \nin \nline \nthe \nof \nwavelength \nColumn \nCEERS\\_1019; \nin \nvalues \nemission-line \ned \n/uniFB01 \nIdenti \nNote. \nthe \nr \no \nt \n/uniFB01 \n) \n3.4.2 \nSection \n( \ndoublet \nthe \nin \nline \n8.679; \n= \n] \nIII \nO \n[ \nz \n( \nCEERS\\_1019 \nof \nredshift \n; \nÅ \nframe, \nrest \nthe \nin \nÅ \n1908 \nand \n1906 \nat \npeaks \nsigni \nlow \na \nat \ndetected \nLines \n. \n) \n1 \n- \nnk \nms \n;i \n4.1 \nSection \nS / N Line Flux Total Single [ 10 -18 cgs ] [ km ( 3 ) ( 4 ) ( 14.4 5.41 ± 0.38 209.77 2.0 1.74 ± 0.88 358.93 7.6 3.36 ± 0.44 363.99 11.9 2.94 ± 0.25 357.55 12.84 3.5 ± 0.27 358.44 2.3 0.66 ± 0.29 357.13 5.5 2.04 ± 0.37 347.65 3.1 1.14 ± 0.37 358.56 6.6 2.60 ± 0.39 360.56 3.0 1.16 ± 0.38 361.11 10.2 7.32 ± 0.72 354.80 26.3 13.08 ± 0.50 358.52 47.2 39.58 ± 0.84 358.66 1.63 1.70 ± 1.05 642.90 1.18 1.62 ± 1.37 358.76 see Figure 5 for individual plots. ) : total integrated line /uniFB02 ux in F λ ( units 3.4.1 ) /uniFB01 ts, or the narrow component asymmetric ( Section 3.4.1 ) /uniFB01 ts, or the narrow 3.4.3 ) /uniFB01 t ( in 10 -18 erg s -1 cm -2 Å rest-frame EW ( Rest EW ) of the line ( in /uniFB01 cance are reported below the horizontal which uses a single-Gaussian pro \n6.58 \n± \n11775.10 \n1215 \nα \nLy \n3.39 \n± \n8.77 \n± \n11969.40 \n4.09 \n± \n14357.60 \n1242 \n+ \n2.29 \n± \n36074.80 \n1486 \n+ \n1483 \n] \nIV \nN \n9.73 \n± \n37456.30 \n3729 \n+ \n3869 \n] \nIII \nNe \n[ \n3727 \n] \nII \nO \n[ \n37656.80 \n3890 \nη \nH \n+ \n3889 \nI \nHe \n1238 \n3.75 \n± \n6.07 \n± \n38407.90 \n5.25 \n± \n39717.90 \n3971 \nò \nH \n+ \n3968 \n] \nIII \nNe \n[ \n42018.20 \n4102 \nδ \nH \n4341 \nγ \nH \n9.91 \n± \n4.91 \n± \n42244.90 \n1.87 \n± \n47059.80 \n4364 \n] \nIII \nO \n[ \n1.14 \n± \n48009.60 \n48473.50 \n4960 \n] \nIII \nO \n[ \n5008 \n] \nIII \nO \n[ \n8 \n9.22 \n± \n18493.30 \n8 \n* \n190 \n] \nIII \nC \n6.67 \n± \n27077.90 \n2803 \n+ \n2796 \nI \nI \nMg \nts, \n/uniFB01 \nsingle-Gaussian \nfor \n) \n1 \n- \nÅ \n2 \n- \ncm \n1 \n- \ns \nin \nSection \n( \nasymmetric \nand \n) \n3.4.2 \nSection \n( \ndoublet \nthe \n: \n) \n9 \n( \nColumn \n. \n) \n1 \n- \ns \nkm \nin \n( \nt \n/uniFB01 \n) \n3.4.3 \nSection \n( \n/uniFB01 \nt, \n/uniFB01 \nbest \nthe \nshow \nand \ndoublet \nthis \nresolve \nnot \ndo \nwe \nand \n) \n3.4.2 \nSection \n( \ndoublet \nthe \nin \nline \nblue \nthe \n4862 \nβ \nH \n149.63 \n± \n309.51 \n3.85 \n4 \n( \nColumn \n. \n) \n3.2 \nSection \n( \nline \nthe \nf \no \nN \n/ \n:S \n) \n3 \n( \nColumn \nin column 4 of Table 3.TheS / N of our emission lines is taken as the measured line /uniFB02 ux from our line /uniFB01 t divided by this error, S / N = F ( λ ) / /uni0394 f ( λ ) , and is reported in column 3 of Table 3. To derive a consistent threshold in S / Nwherewe consider a line to be detected, we examine the results of the automated line/uniFB01 nding routine, searching for lines with restframe wavelengths that do not match a known line ( using a large line list ) . The point below which we detect no ' unknown ' ( spurious ) emission lines is S / N = 2.3, which we use as our detection threshold for emission lines in CEERS\\_1019, unless otherwise noted below.", '3.3. Rest-frame Equivalent Width Measurements': 'To measure rest-frame equivalent widths ( EWs ) for our emission lines ( EW ( ) ( ) F f z rest 1 c = l + ) , we require a measurement of the continuum emission in the region near a given emission line. While, in principle, this could come from the spectrum itself, the continuum is only detected at a high signi /uniFB01 cance in the G140M ( blue ) grating. Thus, to enable a uniform procedure for all emission lines, we use the best/uniFB01 t Prospector model ( see Section 5 and Figure 7 ) for our rest-frame EW measurements. We note that this is based on the same NIRCam photometry used to derive our slit-loss corrections, and we also /uniFB01 nd that the agreement with the spectroscopic continuum in G140M is excellent. To determine the continuum value of the Prospector model, we /uniFB01 rst mask out any emission lines and smooth the spectrum following the same process described above. Our EW continuum value is the median of the Prospector continuum over the width of the line. The error is taken as the median absolute deviation of the Prospector model over this same range.', '3.4. Non-Gaussian Emission-line Fits': 'In our automated line search above, we /uniFB01 t a single Gaussian to each emission line, which is not physical for many of the lines in our spectrum. We thus implement three additional line pro /uniFB01 les: a dual-component Gaussian, a doublet Gaussian, and an asymmetric Gaussian /uniFB01 t as described below. The /uniFB01 ts are done using the same MCMC routine as above, and any parameters not mentioned below as having been changed or added are the same as those in the single-Gaussian /uniFB01 ts.', '3.4.1. Asymmetric Gaussian Fit': "Given the detection of Ly α in our NIRSpec spectra, we want to /uniFB01 t an appropriate asymmetric Gaussian pro /uniFB01 le to the data to account for intergalactic medium ( IGM ) absorption on the blue side of the line. For this /uniFB01 t, we add an extra FWHM parameter, such that we now have ' blue ' and ' red ' FWHMs of the line. The blue FWHM can vary between 0, as the blue side of the line may be fully attenuated, and FWHM [ O III ] + 30 km s -1 , which is the same maximum as the single-Gaussian /uniFB01 t. We start the MCMC chain for this parameter at FWHM [ O III ] . The red FWHM is less restricted, as it can vary between FWHM [ O III ] and 2000 km s -1 , and the MCMC chain is started at FWHM [ O III ] + 30 km s -1 , the maximum value for the blue FWHM. Following the same method as Jung et al. ( 2020 ) , the \nequation for the asymmetric Gaussian is: \n( ) ( ) ( ) ( ) ( ) ( ) /uni23A7 /uni23A8 /uni23AA /uni23A9 /uni23AA f f f exp for , exp for . 2 c Asym 0 1 2 0 1 2 0 0 2 blue 2 0 2 red 2 l l l l l = + ' --> l l s l l s -- \nThis returns the continuum /uniFB02 ux ( f c ) , peak line /uniFB02 ux value ( f 0 ) , a peak wavelength ( λ 0 ) , and the blue- and red-side line widths ( σ blue and σ red ) . To measure our line /uniFB02 ux, we integrate this emission-line pro /uniFB01 le and report the integrated line /uniFB02 ux in column 4 of Table 3", '3.4.2. Doublet Gaussian Fits': 'Several of the UV and optical emission lines are not a single emission-line feature, but rather a doublet. To accurately /uniFB01 t these lines, we implement a doublet Gaussian line pro /uniFB01 le to our MCMC routine: \n( ) ( ) ( ( )) ( ) /uni239C /uni239F /uni239C /uni239F /uni239B /uni239D /uni239E /uni23A0 /uni239B /uni239D /uni239E /uni23A0 f f f f exp 1 2 exp 1 2 . 3 c Doublet 1 1 2 2 2 1 2 2 l l l s l l dl s = + --+ --+ \nFor this /uniFB01 t, we employ the continuum /uniFB02 ux ( fc ) plus two Gaussians with the same FWHM ( σ ) and individual line /uniFB02 uxes for each line ( f 1 and f 2, respectively ) , with the blue peak ( λ 1 ) /uniFB01 xed to ± 6 pixels from the expected location and the red peak /uniFB01 xed to the redshifted separation between the two doublet lines ( δλ = δλ rest ( 1 + z )) . This returns a total combined line /uniFB02 ux for the doublet ( f ( λ )) as reported in column 4 in Table 3, as well as individual line /uniFB02 uxes for each component, as reported in columns 6 and 7 of that same table.', '3.4.3. Dual-component Gaussian Fit': 'Upon inspection of the initial emission lines, the H β line had a noticeable second broad component feature, as discussed in Section 4.2 below. In order to accurately measure this, we employ a dual-component Gaussian pro /uniFB01 le: \n( ) ( ) ( ) ( ) /uni239C /uni239F /uni239C /uni239F /uni239B /uni239D /uni239E /uni23A0 /uni239B /uni239D /uni239E /uni23A0 f f f f exp 1 2 exp 1 2 . 4 c Dual nar 0 2 narrow 2 broad 0 2 broad 2 l l l s l l s = + --+ -- \nThis /uniFB01 t utilizes the continuum /uniFB02 ux ( f c ) plus two Gaussians with the same line center ( λ 0 ) , individual line /uniFB02 uxes for each line component ( f nar and f broad, respectively ) , and separate FWHMs for each ( σ nar and σ broad ) . The narrow FWHM has the same constraints as the single-Gaussian /uniFB01 t ( ± FHWM [ O III ] ) , but the broad component is not allowed to be smaller than the narrow component ( > FHWM [ O III ] + 30 km s -1 ) and can extend to the unrestricted FWHM ( < 2000 km s -1 ) . We start the MCMC chain for the narrow FWHM at FHWM [ O III ] and the FWHM of the broad component at three times this value. This returns a total combined line /uniFB02 ux for the line ( F ( λ )) as reported in column 4 in Table 3, as well as individual line /uniFB02 uxes for each component, as reported in columns 6 and 7 of that same table.', '4. Emission-line Measurements': 'We /uniFB01 nd abundant emission lines in CEERS\\_1019 and present our measured line /uniFB02 uxes in Table 3. A plot indicating the detected emission lines is shown in Figure 4, and /uniFB01 ts to each line are shown in Figure 5. In the following sections, we brie /uniFB02 y discuss the /uniFB01 ts to different lines and provide a detailed description ( where necessary ) of the detected emission lines. We note that H β has a signi /uniFB01 cant broad component, which we discuss in detail in Section 4.2.', '4.1. Redshift Con /uniFB01 rmation': 'We determine the spectroscopic redshift using the /uniFB01 t to the [ OIII ] 5008 emission line, the brightest in our spectrum. This line is detected with a peak wavelength of λ = 48473.50 ± 1.14 Å , yielding a spectroscopic redshift for this source of z [ O III ] = 8.6788 ± 0.0002. When comparing this to the redshift we measure from the line centers of the other strong [ OIII ] 4960 line, we get a consistent measurement within /uni0394 z / z /uni0084 10 -4 . We use this value as the systemic redshift for CEERS\\_1019.', '4.2. Broad H β Feature': "The observed pro /uniFB01 le of the H β emission line shows a strong narrow component, similar to [ OIII ] , but also a weaker broad component. To explore the signi /uniFB01 cance of this broad component, we perform a dual Gaussian /uniFB01 t to this line, as described in Section 3.4.3. Uncertainties on both components were derived via similar simulations, as described in Section 3.2. From this /uniFB01 t, we /uniFB01 nd a distinct broad component to the H β emission line, with an FWHM = 1196 ± 349 km s -1 . We show this /uniFB01 t ( green ) in Figure 6. The /uniFB02 ux in this broad component is comparable to the /uniFB02 ux in the narrow component, with a broadto-narrow /uniFB02 ux ratio of 0.93. \nWe consider multiple measures of the signi /uniFB01 cance of this broad feature. The /uniFB01 rst is the S / N of the broadline /uniFB02 ux from this /uniFB01 t, which is 2.5. However, this does not consider how good the /uniFB01 t is without the broadline component. We next try a narrowline-only /uniFB01 t, with the FWHM allowed to be free ( i.e., not tied to [ OIII ] ) , shown as the blue line in Figure 6.We /uniFB01 nd a single-line FWHM of 500 ± 52 km s -1 , signi /uniFB01 cantly broader than the /uniFB01 ducial narrowline-only FWHM of 367 ± 17 km s -1 ( the purple line in Figure 6 ) , making it clear that the broad component affects even a single-line /uniFB01 t. \nFinally, we calculate the /uniF0A0 Bayesian Information Criterion ( BIC ) between the dual-component and narrowline-only /uniFB01 ts. The BIC is a method of comparing the goodness-of/uniFB01 t between two models, accounting for differing degrees of freedom. It is de /uniFB01 ned as BIC = ( ) klnN 2 c + , where k is the number of free parameters and N is the number of data points ( the number of spectral pixels, in this case; Liddle 2004 ) . Our narrowline-only /uniFB01 t with four free parameters ( f c , f 0, λ 0, and σ -see Section 3 ) has a BIC of 21.4 when the FWHM is tied to [ OIII ] and 21.8 when the FWHM is free. The narrow + broad /uniFB01 t with six free parameters ( f c , λ 0, f nar, f broad, σ nar, and σ broad -see Section 3.4.3 ) has a BIC of 18.7. The /uni0394 BIC = 2.7 -3.1 gives ' positive ' evidence ( using the scale of Jeffreys 1961, where /uni0394 BIC > 6is ' strong ' evidence ) that a broadline component is necessary to /uniFB01 t the H β line successfully. \nThis signi /uniFB01 cant broad component of H β thus indicates the presence of high-velocity gas, which we interpret as emitting from the broadline region of an AGN. Large-scale out /uniFB02 ows can \nsimilarly produce broad velocity widths, but would be apparent in all emission lines ( e.g., Amorín et al. 2012; Hogarth et al. 2020 ) . We do not observe any broad features in lines with much higher S / N, like [ OIII ] λ 5008. Instead, [ OIII ] and other forbidden lines have widths that are inconsistent with the out /uniFB02 ow scenario and are instead consistent with the narrow H β component. Broadline AGNs typically exhibit broad components for permitted lines ( like H β ) and narrow widths of forbidden lines ( like [ OIII ] ; e.g., Schmidt et al. 2016; Vanden Berk et al. 2001 ) , as observed in our spectrum. We discuss further evidence for the AGN nature of this source in Section 4.4 below.", '4.3. Ly a': 'The Ly α line was /uniFB01 rst detected by Zitrin et al. ( 2015 ) at zLy α = 8.683 with Keck / MOSFIRE. We detect an emission line at this same wavelength, which we /uniFB01 t with an asymmetric Gaussian pro /uniFB01 le, as described in Section 3.4.1, and measure a line /uniFB02 ux f Ly α = 5.41 ± 0.38 × 10 -18 erg s -1 cm -2 . Using our measured central wavelength and the vacuum wavelength for Ly α of 1215.67 Å , we measure a Ly α -based redshift of 8.6854 ± 0.0045, 218 ± 333 km s -1 higher than that of [ OIII ] . This is lower, although consistent within ∼ 1 σ , of the measured Ly α velocity offset ( relative to N V ) of + 362 km s -1 from Mainali et al. ( 2018 ) . \nWe /uniFB01 nd a clear asymmetric line pro /uniFB01 le with an extended redside tail ( FWHMred > 1000 km s -1 ) . Although it becomes more dif /uniFB01 cult to detect Ly α at this high redshift, due to the increasingly neutral IGM, one can expect the escape of Ly α that is scattered and redshifted enough to avoid resonant scattering ( Dijkstra et al. 2004 ) , as seen in the extended red tail in our Ly α spectra. Additionally, a sharp cutoff at the blue edge of the line pro /uniFB01 le may indicate a signi /uniFB01 cant contribution of resonant scattering, which can be caused by a proximate optically thick medium ( Mason & Gronke 2020 ) . Particularly, reionization simulations predict the sharp blue-side edge of the red peak, due to the infall motion of neutral gas around a galaxy, as seen in our Ly α spectrum. The cutoff location is found at ∼ 200 km s -1 in our spectra, which is comparable to the predicted velocity of the infalling gas around an M UV = -22 galaxy at this redshift ( Park et al. 2021 ) . \nOur Ly α line /uniFB02 ux is ∼ seven times fainter than the line /uniFB02 ux measured from MOSFIRE. This could be due, in part, to the MOSFIRE slits being wider than the NIRSpec micro shutters ( 0 7 versus 0 2 ) . Our NIRSpec slit-loss correction would not correct for extended Ly α emission in this source. The MOSFIRE observations of this source were taken ∼ 8 -9yr prior to the JWST / NIRSpec data, ∼ 1 yr in the rest frame, and it is possible that the variability common in AGNs might contribute to the difference in the measured line /uniFB02 uxes. AGN variability is typically ∼ 20% ( e.g., MacLeod et al. 2012 ) and is unlikely to fully explain this large difference. \nThe N V1243 line was detected from this source with Keck / MOSFIRE by Mainali et al. ( 2018 ) at 12019.5 Å , with the other line in the doublet, N V1238, obscured by a skyline at 11981 Å . In our JWST / NIRSpec data, we do not detect a line at either of these wavelengths, but there is a 1.8 σ peak at 12007 Å . This would correspond to a rest-frame wavelength of 1240 Å between the peaks of the N V doublet. The velocity offset of the N V line from the systemic redshift z [ O III ] = 8.679, if indeed the λ 1238 line is ∼ + 442 ± 85 km s -1 . This is consistent ( within the error ) with the offset we measure for the \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 5. Fits to the emission lines identi /uniFB01 ed in the JWST / NIRSpec spectrum of CEERS\\_1019. Each panel shows an individual emission-line /uniFB01 t, with the type of line pro /uniFB01 le as described in Section 3. The panels are plotted in F λ ( 10 -19 erg s -1 cm -2 Å -1 ) vs. observed wavelength ( /uni03BC m ) , are presented in order of increasing wavelength, and are scaled vertically to show the extent of the highlighted emission line. Emission-line values are tabulated in Table 3 and discussions of speci /uniFB01 c lines are given in Section 4. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 6. Left: dual-component Gaussian /uniFB01 t to the H β emission line ( green ) , yielding FWHMbroad = 1196.3 ± 349.1 km s -1 and FWHMnar = 354.8 ± 19.6 km s -1 . Alternative single-Gaussian line pro /uniFB01 le /uniFB01 ts are shown where the FWHM is restricted to FWHM [ O III ] ± 30 km s -1 ( blue ) or left unrestricted ( purple ) . In both cases, the /uniFB01 t to the H β emission line is worse than the dual-component /uniFB01 ts, as indicated by the higher Bayesian Information Criterion ( BIC ) measurement ( Section 4.2 ) . Right: scaled-down /uniFB01 ts of the same dual-component /uniFB01 t from H β to the peak of the H δ ( top ) and H γ ( bottom ) emission lines. In both cases, the broad component is not distinguishable from the noise, indicating that a broad component could be present in these lines below our noise threshold. \n<!-- image --> \nCIII ] of ∼ + 310 ± 150 km s -1 . If it is the λ 1243 line, the velocity offset is ∼-521 ± 85 km s -1 .', '4.4. Other Potential Broad Emission Lines': "As noted above, broadline ( Type 1 ) AGNs typically exhibit broad permitted lines and narrow forbidden lines, interpreted as high-velocity dense gas in a ' broadline region ' near the BH ( and dominated by its gravitational in /uniFB02 uence ) and lowervelocity gas in a ' narrowline region ' that is more distant from the BH and dominated by the galaxy kinematics ( e.g., Netzer 2015 ) . Unfortunately, the H α line is redshifted beyond the wavelength range of NIRSpec for our target ( though it is observable by MIRI spectroscopy ) . The other lines in the Balmer series ( H γ ,H δ , etc. ) are too weak for strong constraints on the presence of broad components ( see Figure 6 ) . In AGNs with a broad H β line, broad features are also typically observed in the permitted UV lines Ly α ,CIV,CIII ] , and Mg II with similar or stronger /uniFB02 uxes to the broad H β line ( e.g., Vanden Berk et al. 2001 ) . \nWe detect a signi /uniFB01 cant broad width in the observed Ly α line of our target. The blue-side absorption of the Ly α forest makes it dif /uniFB01 cult to measure the line width precisely, but we note that our asymmetric Gaussian /uniFB01 t ( described in Section 3.3 ) /uniFB01 nds a red-side width of FWHM = 1360.6 ± 479.9 km s -1 : comparable to the best/uniFB01 t width of the broad H β component. \nThe best/uniFB01 t Gaussian to the C III ] line also indicates a broadline width of FWHM = 643 ± 157 km s -1 . The observed CIII ] feature is not well /uniFB01 t by two narrow Gaussians, with widths constrained to be the same as the [ OIII ] λ 5008 line for each of the doublet lines. This could indicate that the doublet is unresolved in a noisy part of our data, leading to a single line with a broader width. Alternatively, the broader FHWM of this line compared to [ OIII ] may have some physical implications for this galaxy. The line width of the best/uniFB01 t broad C III ] pro /uniFB01 le is somewhat narrower than would be expected for a strati /uniFB01 ed broadline region, in which the higher-ionization C III ] gas orbits closer to the BH and consequently has a broader velocity width \nthan H β . The line center of the best/uniFB01 t broad C III ] pro /uniFB01 le is redshifted by + 310 km s -1 with respect to the systemic redshift ( determined from the narrow [ OIII ] λ 5008 line as described in Section 4.1 ) . The shift in line center and narrower-thanexpected width may indicate a blueshifted absorption component in the C III ] line, as commonly observed in broad absorption-line quasars ( e.g., Gibson et al. 2009 ) . \nWe additionally see a signi /uniFB01 cant broad component in the best/uniFB01 t pro /uniFB01 le for the N IV ] λ 1486 line. This feature is not typically observed in luminous quasars at z /uniF088 4, but it is detected as a feature in the NIRSpec prism spectrum of the z = 10.6 galaxy GN-z11 ( Bunker et al. 2023 ) . Its presence in GN-z11 has been interpreted as evidence for high-density and nitrogen-enhanced gas ( Cameron et al. 2023; Senchyna et al. 2023 ) , both properties consistent with expectations for broadline regions around rapidly accreting and low-mass AGNs ( Hamann & Ferland 1999; Matsuoka et al. 2017 ) . \nOn the other hand, we do not observe broad components in the C IV and Mg II emission lines. The absence of C IV can be explained by the low S / N in its region of the NIRSpec spectrum, and the large /uniFB02 ux uncertainty at that wavelength range allows for an undetected broad C IV line that is of comparable brightness to the broad H β feature. The Mg II feature is more puzzling: the observed Mg II pro /uniFB01 le is best /uniFB01 t by narrow Gaussians for each of the doublet lines, consistent with galaxy ( or narrowline region ) emission, rather than the expected broad feature of an AGN. It is possible that this line is intrinsically broad but is affected by intervening absorption by the Mg II doublet: this requires a particular blueshift velocity of the absorption component, but is not uncommon in quasars. Alternatively, the AGN may be attenuated by dust, such that its UV emission lines are weaker than the Balmer lines, as implied by the best/uniFB01 t SED model, although this con /uniFB02 icts with the detection of broad Ly α ,NIV ] , and C III ] . Deeper spectra of this source would more effectively test for the presence of the broad UV lines expected for AGNs, especially in investigating the hypothesis for blueshifted absorption affecting the broad C III ] line and the Mg II line.", '4.5. Emission-line Flux Ratio Measurements': 'Despite potential lingering issues in the absolute /uniFB02 ux calibration of the NIRSpec instrument, we /uniFB01 nd that the relative /uniFB02 ux calibration is consistent, at least for pairs of emission lines near one another in wavelength. We measure [ OIII ] 5008 / [ OIII ] 4960 = 3.03 ± 0.13, which is consistent with the atomic physics calculation within errors ( Storey & Zeippen 2000 ) . We are con /uniFB01 dent in using the ratios of emission lines close in wavelength. Still, we acknowledge that additional /uniFB02 ux calibration and improvements of the NIRSpec instrument are needed to trust widely separated line ratios fully. We also note that our wavelength-dependent aperture correction using the differing spatial pro /uniFB01 le of CEERS\\_1019 across the three NIRSpec /uniFB01 lters may add additional systematic errors in distant line ratios. We thus only report ratios for lines that fall within the same NIRSpec /uniFB01 lter: G395M. Since we also use the NIRCam /uniFB01 lters to perform a wavelength-dependent /uniFB02 ux correction ( see Section 2.1.1 ) , the errors on the line ratios that spanacrossthetworedderNIRCam /uniFB01 lters ( F356W and F444W ) may be underrepresented. We report the following line ratios in this paper given the above caveats. \nWe list the measured line ratios for this source in Table 4. All reported line ratios are measured without using the broadline /uniFB01 t to the H β line, and only the narrow component from the dual-component /uniFB01 t ( See Figure 6 ) . For any of the line ratios reported where one line is not detected at a signi /uniFB01 cant level, we report the 1 σ upper limit for the line and mark it with an asterisk in the tables of line /uniFB02 uxes and ratios ( Tables 3 and 4 ) .', '5. Constraints from Continuum Emission': 'We explore /uniFB01 tting the photometric SED of this object with stellar and AGN emissions simultaneously to explore which dominates the observed continuum emission. We /uniFB01 rst use the FAST v1.1 ( Kriek et al. 2009; Aird et al. 2018 ) SED /uniFB01 tting code, including a star-forming galaxy and an AGN component, as described in Kocevski et al. ( 2023 ) . With this code, we try /uniFB01 ts using two sets of AGN templates. The /uniFB01 rst uses eight empirically determined AGN templates. These include /uniFB01 ve AGN-dominated templates from the Polletta et al. ( 2007 ) SWIRE template library ( namely, the Torus, TQSO1, BQSO1, QSO1, and QSO2 templates ) and three composite SEDs of X-ray-selected AGNs with absorption column densities of N H = 10 22 -23 , 10 23 -24 , and 10 24 -25 cm -2 from Silva et al. ( 2004 ) . See Appendix A of Aird et al. ( 2018 ) for additional details. We also try a second /uniFB01 t with a low-metallicity ( Z /uni0084 0.4 Z e ) AGN component from CLOUDY modeling, chosen to mimic the SED of a radio-quiet AGN ( see Section 6.5 for more details ) . We measure the ratio of light from the best/uniFB01 tting AGN model to the total stellar + AGN model in two wavelength windows: a rest-UV window at 0.15 -0.25 /uni03BC m and a restoptical window at 0.51 -0.6 /uni03BC m ( both designed to avoid strong emission lines ) . In both of these /uniFB01 ts, the photometry constrains the AGN to be subdominant to the stellar emission. With the /uniFB01 ducial AGN template, the stellar emission comprises 99.5% of the UV /uniFB02 ux and 85% of the optical /uniFB02 ux. With the lowmetallicity AGN model, stellar emission comprises 82% -83% of both the UV and optical /uniFB02 ux. \nWe try an alternative /uniFB01 t using the Cigale code ( Boquien et al. 2019; Yang et al. 2020, 2022 ) . In the /uniFB01 t, we adopt a modi /uniFB01 ed Schartmann et al. ( 2005 ) AGN accretion disk model. We allow the deviation from the default optical spectral slope ( the δ AGN \nTable 4 Emission-line Ratios for CEERS\\_1019 \nNote. Common emission-line /uniFB02 ux ratios as measured from JWST / NIRSpec for CEERS\\_1019. All of the ratios that include H β use the narrow component /uniFB02 ux from the dual-component Gaussian /uniFB01 t ( Section 3.4.3 ) , as reported in column 6 of Table 4, unless otherwise noted as using the Total H β , which is the value in column 4. \nparameter ) to vary from -1 to 1 and the polar dust extinction ( the E ( B -V ) PD parameter ) to vary from 0 to 1 ( see Section 4 of Yang et al. 2022 for details ) . Measuring the results in the same two wavelength windows as above shows that the stellar emission comprises 70% and 59% of the continuum emission in the rest-UV and rest-optical, respectively. \nThese results imply that the observed continuum emission is dominated by stellar light. We thus proceed to perform Bayesian SED modeling with stellar-only models to explore what constraints can be placed on the stellar population of the host galaxy. Our /uniFB01 ducial results come from the Prospector SED /uniFB01 tting code ( Johnson et al. 2021 ) , following Tacchella et al. ( 2022 ) by using both a continuity and bursty prior on the star formation history ( SFH ) . Both SFH priors lead to similar posterior distributions, implying that the data are informative and the impact of the prior is minimal. We /uniFB01 nd that this galaxy has a stellar mass of M å ≈ 10 9.5 ± 0.3 M e and is actively forming stars ( speci /uniFB01 c star formation rate, or sSFR ≈ 10 -7.9 ± 0.3 yr -1 ) , doubling its mass every ∼ 100 Myr. We /uniFB01 nd a mass-weighted age of t 34 Myr 50 29 119 = -+ . We also note that using the conversion from the star formation rate ( SFR ) to Balmer line /uniFB02 ux from Kennicutt & Evans ( 2012 ) , the H β /uniFB02 ux estimated from this SFR ( 1.3 -5.3 × 10 -18 erg s -1 cm -2 ) is fully consistent with our measured narrowline H β /uniFB02 ux ( 3.8 ± 0.7 × 10 -18 erg s -1 cm -2 ) within errors. We list the physical properties of this galaxy from our SED /uniFB01 tting in Table 5 and show the model /uniFB01 ts in Figure 7. \nWe also /uniFB01 t these data with BAGPIPES ( Carnall et al. 2018 ) . We generally followed the procedures in Papovich et al. ( 2022 ) , but we have now used Binary Population and Spectral \nTable 5 Physical Properties of CEERS\\_1019 \nNote. Physical properties of CEERS\\_1019 as derived through SED /uniFB01 tting to the HST and JWST photometry at the spectroscopic redshift. SED /uniFB01 tting with Prospector ( Johnson et al. 2021 ) using both a continuity and bursty SFH prior, as described in Tacchella et al. ( 2022 ) . Column ( 1 ) ; SFH prior used in the /uniFB01 t. Column ( 2 ) : absolute UV magnitude at rest-frame 1500 Å . Column ( 3 ) : UV spectral slope, b , as measured from the model spectra. Column ( 4 ) : stellar mass. Column ( 5 ) : SFR averaged over the past 50 Myr. Column ( 6 ) : speci /uniFB01 c SFR averaged over the past 50 Myr. Column ( 7 ) : dust attenuation at 5500 Å , from SED models. Column ( 8 ) : stellar metallicity from SED models to the HST photometry. The values in columns ( 4 ) to ( 8 ) are calculated from SED /uniFB01 tting as in Tacchella et al. ( 2022 ) . \nSynthesis ( BPASS ) v2.2.1 stellar population models ( Eldridge et al. 2017 ) , an SFH represented as a Gaussian mixture model ( Iyer et al. 2019 ) , and /uniFB01 t over a range of ionization parameters log U = [ -4, -1 ] to model the nebular emission. From these /uniFB01 ts, the galaxy has a stellar mass of M M log 9.3 0.1 = * , consistent with the /uniFB01 ts from Prospector, though a slightly higher log ( sSFR ) of -7.5 ± 0.2 yr -1 . The systematic uncertainties on stellar mass and SFR here are ≈ 0.2 dex. The inferred properties, such as stellar mass, SFR, and dust attenuation, are generally consistent with and within the systematic uncertainties of the Prospector /uniFB01 ts, but require a slightly higher SFR to account for differences in modeling assumptions. We show the best/uniFB01 tting Prospector and BAGPIPES models in Figure 7, alongside images of this galaxy in all observed /uniFB01 lters and photometric redshift results from EAZY.', '5.1. UV Slope ( β ) from Photometry': 'To measure the UV spectral slope, β , for this source, we /uniFB01 ta power-law function ( f λ ∝ λ β ; Calzetti et al. 1994 ) to the observed photometry. We only include /uniFB01 lters within the restframe 1500 -3000 Å range to avoid contamination from the Ly α break and strong emission lines. For this source, these /uniFB01 lters are F160W, F150W, F200W, and F277W. Using the EMCEE software ( Foreman-Mackey et al. 2013 ) , we measure the posterior distribution on β . CEERS\\_1019 has 1.76 0.13 0.12 b =--+ , where the uncertainty is taken as the 68% central width from the posterior. This is consistent at the ∼ 1 σ level with the value inferred from HST photometry from Tacchella et al. ( 2022 ) of 1.61 0.12 0.18 b =--+ . It is also consistent with the value of β measured from the Prospector model spectra, listed in Table 5. This value of β is comparable to the UV colors from other similarly bright sources ( Tacchella et al. 2022 ) , consistent with the low but non-negligible dust attenuation we /uniFB01 nd from our SED /uniFB01 t ( AV = 0.4 ± 0.2 ) .', '5.2. Morphology': "We investigate the morphology of this source using the /uniFB01 tting codes Galfit ( Peng et al. 2002, 2010 ) and statmorph ( Rodriguez-Gomez et al. 2019 ) . We use the Galfit least-squares /uniFB01 tting algorithm to /uniFB01 t the galaxy ' s surface brightness pro /uniFB01 le. We use a 100 × 100 pixel cutout of the F200W science image as input, the corresponding error array ( the ' ERR ' extension ) as the input sigma image, and the empirically derived PSFs. As an initial guess, we use the source location, magnitude, size, position angle, and axis ratios from the Source Extractor ( SE ) catalog. The Sérsic index is allowed \nto vary between 0.01 and 8, the magnitude of the galaxy between 0 and 45, the effective radius ( Re ) between 0.3 and 100 pixels, and the axis ratio between 0.0001 and 1. We also allow Galfit to oversample the PSF by a factor 9. We then visually inspect the best/uniFB01 t model and image residual for each source to ensure that the /uniFB01 ts are reasonable and that minimal /uniFB02 ux remained in the residual. \nWe /uniFB01 nd that a single Sérsic component poorly /uniFB01 ts the source. Visually, the source is extended asymmetrically, with three distinct components ( see Figure 8 ) . The optimum /uniFB01 tis obtained when the central region is /uniFB01 t with both a point source and a Sérsic component, and the two other regions to the west and the northeast are each /uniFB01 tted with their own Sérsic components. The requirement of a point-source component for the /uniFB01 t is consistent with the source having an AGN. We similarly performed /uniFB01 ts on the F356W and F444W images and again found that single-Sérsic /uniFB01 ts do not work because of the multiple components, even with the lower resolution at these longer wavelengths. The asymmetric nature of the source and the presence of the three separate components are consistent with the galaxy being involved in a major merger. We use statmorph to measure the concentration parameter and size as a function of wavelength on PSF-matched cutouts and /uniFB01 nd that C = 2.82 and Re = 6.49 pixels ( 0.91 kpc ) for F356W and C = 2.93 and Re = 5.64 pixels ( 0.79 kpc ) for F444W, suggesting that the F444W light is more concentrated than the F356W or F200W emission, supporting the visual impression in Figure 2. At the redshift of this galaxy, the F444W emission is dominated by the [ OIII ] emission line ( see Figure 7 ) .", '6.1. Black Hole /uniF0A0 Mass': 'In this section, we estimate the virial mass of the central BH, assuming that the observed broad H β emission traces the kinematics of gas in the broadline region. The measurement of BH masses with single-epoch spectra can be made using the width of the broad H β emission line and the rest-frame 5100 Å continuum luminosity, L 5100, which has been shown to correlate with the distance to the broadline region using reverberation mapping ( e.g., Kaspi et al. 2000; Cackett et al. 2021 ) . However, this assumes the rest-frame 5100 Å continuum luminosity is dominated by light from the AGN, which is likely not the case here ( see Section 6.1 ) . As a result, we instead use Equation ( 10 ) from Greene & Ho ( 2005 ) , which \nFigure 7. The images show 5 ¢ ¢ × 5 ¢ ¢ cutouts around this source in the CANDELS HST / ACS optical bands, CEERS NIRCam near-infrared bands, and CEERS MIRI bands. The source is a clear dropout in the optical and is well detected at 1 -8 /uni03BC m. The large inset plot shows the photometric SED from HST and JWST. We also show Spitzer / IRAC measurements from Finkelstein et al. ( 2022a ) with both TPHOT ( light gray ) and Gal /uniFB01 t ( dark gray ) ; this source was highly blended in IRAC, and the Gal /uniFB01 t measurements appear closer to the NIRCam measurements. The lines are best/uniFB01 t models from Prospector ( gray ) and BAGPIPES ( purple ) . The SED is dominated by stellar emission ( Section 5 ) , and these stellar models infer a massive ( log M / M e ∼ 9.5 ) and highly star-forming ( log sSFR ∼-7.9 Gyr -1 ) stellar population. The small inset panel shows constraints on the photometric redshift before ( green ) and after ( red and yellow ) the inclusion of JWST data. All three are consistent with the spectroscopic redshift, with those including JWST data placing much tighter constraints. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 8. JWST NIRCam 1.5 " × 1.5 " cutouts of CEERS\\_1019 in four different /uniFB01 lters ( F200W, F277W, F356W, and F444W ) and an RGB color composite ( with blue = F115W + F150W + F200W, green = F277W + F356W + F410M, and red = F444W ) with each /uniFB01 lter at its native resolution, highlighting the substructure visible at shorter wavelengths. The positions of the NIRSpec MOS shutters are overlaid. This source has a bright central component centered in the shutter and two extended components as discussed in Section 5.2. \n<!-- image --> \nemploys only the H β line luminosity, L H β , and width: \n( ) /uni239C /uni239F /uni239B /uni239D /uni239E /uni23A0 /uni239B /uni239D /uni239E /uni23A0 M L M 2.4 10 10 erg s FWHM 10 km s . 5 BH 6 H 42 1 0.59 H 3 1 2 = \' b b -- \nThis equation is based on the formula of Kaspi et al. ( 2000 ) , but with L H β substituted for L 5100 using the empirical correlation between Balmer emission-line luminosities and L 5100 reported in Greene & Ho ( 2005 ) . \nUsing the luminosity and width of the broad H β component, we derive a BH mass of ( ) M log M 6.95 0.37 BH = . This \nmass is 1 -2 dex lower than existing samples of luminous quasars with BH mass estimates at z > 5 and more comparable to the low-luminosity AGN reported in Kocevski et al. ( 2023 ) , found in these same CEERS NIRSpec data. Our measured mass implies that the AGN in CEERS\\_1019 is powered by the least massive BH known in the Universe during the EoR. \nTo determine the accretion rate onto the BH relative to the Eddington limit, we derive a bolometric luminosity, L Bol, from the broad H β line luminosity and compare it to the Eddington luminosity, L Edd, for our measured BH mass. Assuming an intrinsic broadline H α / H β ratio of 3.06 ( Dong et al. 2008 ) and \nTable 6 CEERS\\_1019 BH Properties \nNote. Column ( 1 ) : BH mass ( Section 6.1 ) . Column ( 2 ) : FWHM of the broad H β line ( Section 4.2 ) . Column ( 3 ) : luminosity of the broad H β line. Column ( 4 ) : bolometric luminosity. Column ( 5 ) : Eddington ratio ( L Bol / L Edd ) . \na bolometric correction of L Bol = 130 × L H α ( Richards et al. 2006; Stern & Laor 2012 ) , we obtain L Bol = 1.4 ± 0.6 × 10 45 erg s -1 . This results in an Eddington ratio, λ Edd = L Bol / L Edd,of 1.3 ± 0.5. This suggests that the BH is undergoing a rapid growth phase and is accreting at approximately its Eddington limit. \nWe note that while we observe a broad H β line, we do not detect AGN continuum emission at the level that might be expected given the Greene & Ho ( 2005 ) relationship between L H β and L 5100. This may indicate the AGN is moderately reddened due to a patchy obscuring medium, similar to what is found in red quasars, which show broad Balmer lines, yet have properties intermediate between type I and II quasars ( Greene et al. 2014; Glikman et al. 2023 ) . If this is the case and the H β emission is suppressed by dust extinction, then our estimated BH mass should be considered a lower limit to the true virial mass for this AGN. We report our measurements fo the BH in CEERS\\_1019 in Table 6.', '6.2. Electron Temperature and Te -based Metallicity': "The [ OIII ] 4364 / [ OIII ] 5008 + 4960 ratio can be used to measure the electron temperature ( Te ) of the galaxy ' s interstellar medium ( ISM ) . As these lines are all collisionally excited, a higher [ OIII ] 4364 emission relative to [ OIII ] 5008 + 4960 indicates higher-energy electrons are responsible for the excitation. This Te has been found to correlate with the ISM metallicity, providing a way to measure the metallicity of the source with these lines ( i.e., Kewley et al. 2019b ) . \nOur source has [ OIII ] 4364 / [ OIII ] 4960 + 5008 = 0.022 ± 0.007. We use the measured [ OIII ] 4364 / [ OIII ] 4960 + 5008 ratio to estimate T e using Equation ( 4 ) of Nicholls et al. ( 2020 ) : \n( ) ( ) T x x x x log 3.5363 7.2939 1.0000 1.6298 0.1221 0.0074 , 6 e 10 2 3 = + + -- \nwhere \n[ ] [ ] /uni239C /uni239F /uni239B /uni239D /uni239E /uni23A0 x log O O 1.657 0.003. III III 10 4364 4960 5008 = = - + \nThis yields ( ) T log 4.270 0.566 e 10 = and T e = 18630.755 ± 3.682 K for CEERS\\_1019. \nBy measuring Te in this way, we are only sensitive to the portion of the ISM emitting the [ OIII ] lines, which may not represent the entire galaxy. Any signi /uniFB01 cant gradients in density and / or ionization may lead to a mixture of different regions in the galaxy, and the Te from [ OIII ] is only sensitive to the highionization regions. \nWe then use Equation ( 1 ) from Pérez-Montero et al. ( 2021 ) to estimate the metallicity from Te as shown below: \n( ) ( ) t t 12 log O H 9.72 1.70 0.32 , 7 e e 2 + = -+ \nwhere te = Te in units of 10 4 K and thus our ' direct ' metallicity measurement for this source is: 12 + log ( O / H ) = 7.664 ± 0.508 or 0.095 0.065 0.209 -+ Z e ( Z e = 8.69; Asplund et al. 2021 ) . \nThis method is used for extreme emission-line galaxies ( EELGs ) , galaxies with extreme emission lines dominated by stellar photoionization, so it may not be strictly accurate for objects whose photoionization is dominated by an AGN. The above equation from Pérez-Montero et al. ( 2021 ) is adopted from Amorín et al. ( 2015 ) , who derived this relation based upon calibrations using local EELGs, after discarding AGN candidates based upon ' Baldwin -Phillips -Telervich ' ( BPT; Baldwin et al. 1981 ) diagram measurements.", '6.3. Electron Density from [ OII ] Doublet': 'The ratio between the two lines in the [ OII ] 3727 + 3729 doublet is often used to determine the electron density ( ne ) in starforming regions at temperatures T ∼ 10,000 = 20,000 K. This is because the excitation energy between the lines is on the order of the thermal electron energy. Thus the relative excitation rates depend only upon collision strengths ( Osterbrock 1989 ) . While the [ OII ] 3727 + 3729 doublet is blended at the resolution of the NIRSpec data ( R ∼ 1000 in the M gratings ) , the separation between the two line peaks at this redshift is large enough to distinguish. The doublet /uniFB01 t to this line ( Section 3.4.2 ) gives /uniFB02 ux values for both emission lines such that we can estimate a ratio between the two of [ OII ] 3729 / [ OII ] 3727 = 0.639 ± 0.255 with which to infer an electron density, n e = 1.9 ± 0.2 × 10 3 cm -3 ( Osterbrock 1989 ) . This is consistent with the feedback-free star formation model from Dekel et al. ( 2023 ) . Alternatively, measurements using the CIII ] doublet are sensitive to larger n e . We only detect the CIII ] 1908 line from this doublet, which could imply that CEERS\\_1019 has very high density, ∼ ne > 10 4 cm -3 ( Keenan et al. 1992 ) .', '6.4. Av from Balmer Decrement': "The ratios of Balmer lines can be used to measure nebular attenuation. In particular, gas with Case B recombination and a temperature of Te = 20,000 K ( as calculated for our object in Section 6.2 ) will have intrinsic Balmer ratios of H β / H γ = 2.110 and H β / H δ = 3.817 ( Osterbrock 1989 ) . We can estimate the dust attenuation E ( B -V ) with an assumed attenuation curve ( here, we use Calzetti 2001 ) , by comparing our observed line ratios to these intrinsic values. For the observed line ratios, we use the total line /uniFB02 ux rather than the narrowline-only /uniFB02 ux, because the H γ and H δ lines do not have suf /uniFB01 cient S / Nto separate broad and narrow components. This means that our estimated nebular attenuation may be a mix of attenuation affecting the AGN and the narrow lines. \nFrom the observed line ratios of H β / H γ = 2.82 ± 0.49 and H β / H δ = 6.42 ± 2.18, we /uniFB01 nd nebular attenuation of: \n1. E ( B -V ) H β / H γ = 0.64 ± 0.41; 2. E ( B -V ) H β / H δ = 0.83 ± 0.54. \nThe observed line ratios are statistically consistent with the intrinsic values, such that the implied E ( B -V ) values are \nFigure 9. The ' OHNO ' diagram, the line ratio diagnostic using [ O III ] 5008 / H β vs. [ Ne III ] 3870 / [ O II ] 3727 + 3729, with the black curve showing the boundary between star-forming and AGN regions at z ∼ 1 ( as de /uniFB01 ned in Backhaus et al. 2022, with the shaded region showing the star-forming area ) . The orange stars are from this work, representing the narrow component /uniFB01 t ( big star ) and the single-Gaussian /uniFB01 t ( to the full pro /uniFB01 le; small star ) of the H β emission line in our galaxy. The black polygons represent other high-redshift ( z > 5 ) galaxies with JWST / NIRSpec observations ( Trump et al. 2023; Kocevski et al. 2023; Tang et al. 2023 ) . Underplotted on both panels are photoionization models, with the left panel showing star-forming ( blue ) and AGN models ( red ) using the CLOUDY v17 code ( Ferland et al. 2017 ) and the right panel showing star-forming models ( green ) from Kewley et al. ( 2019a ) using the MAPPINGS V code ( Sutherland et al. 2018 ) . Each line of points represents a speci /uniFB01 c set of ionization parameters of log10 U = [ -2.5, -2.1, -1.5 ] for the CLOUDY models and log10 U = [ -2.5, -1.5 ] for the MAPPINGS models, with increasing ionization toward darker colors ( or toward the top right of each panel ) . For the MAPPINGS models, the lines are further separated by isobaric pressure ( denoted by line type ) . The sizes of the points represent the metallicities of the models. \n<!-- image --> \nsimilarly consistent with little to no dust. But the large error bars also allow for a broad range of nebular attenuation. In Section 5.1, the Prospector SED /uniFB01 t implied AV = 0.4 ± 0.2, which equates to E ( B -V ) = 0.1 ± 0.05. This is consistent within the large error bars of the Balmer decrement estimate for dust attenuation.", '6.5. Ionization Parameter': "To 48 place this source in the context of other star-forming galaxies and AGNs identi /uniFB01 ed in this early epoch, we investigate the ' OHNO ' line ratio diagnostic diagram. This diagram, comparing [ OIII ] 5008 / H β versus [ Ne III ] 3870 / [ OII ] 3727 + 3729, has been used at lower redshifts to identify ionization powered purely by star formation from that of an AGN ( e.g., Backhaus et al. 2022; Cleri et al. 2023, 2023 ) . Recent studies of high-redshift galaxies ( z > 5 ) using JWST spectroscopy have found most if not all of their sources occupying the AGN region of the diagram ( i.e., above the theoretical line dividing star formation from AGN, calibrated via low-redshift observations ) -results that seem to suggest strong ionization from AGNs and / or very-metal-poor starforming H II regions at these high redshifts ( e.g., Trump et al. 2023; Kocevski et al. 2023; Tang et al. 2023; Übler et al. 2023 ) . \nSuch is the case for CEERS\\_1019, with high measurements of the ionization indicators [ OIII ] 5008 / H β ≡ R3 = 10.44 ± 1.91 and [ OIII ] 4960 + 5008 / [ OII ] 3727 + 3729 ≡ O32 = 13.46 ± 1.18 ( see Table 4 ) . Figure 9 shows our galaxy on the ' OHNO ' diagram ( orange stars ) , using the narrow component of the H β /uniFB02 ux ( the bigger star -also the light green line in Figure 6; the smaller star shows the ratio with the full H β /uniFB02 ux ) . Additional sources from recent JWST studies focused on the EoR ( Trump \net al. 2023; Kocevski et al. 2023; Tang et al. 2023 ) are included as small gray polygons. The black curve in both panels denotes the boundary between star-forming and AGN regions as de /uniFB01 ned by Backhaus et al. ( 2022 ) , with the associated gray shaded region showing the z ∼ 1 ' star-forming ' region. Underplotted are photoionization models from both the CLOUDY v17 ( Ferland et al. 2017; left panel ) and MAPPINGS V ( Sutherland et al. 2018; right panel ) codes, with parameters chosen to showcase a range of possible galaxy properties at this redshift, including elemental abundances, stellar populations, and stellar and gas-phase metallicities. In this /uniFB01 gure, each line in the left panel represents CLOUDY models with a speci /uniFB01 c ionization parameter ( increasing log10 U, shown by the darker colors ) , where the sizes of the points in each line represent the speci /uniFB01 c gas-phase metallicity of said model. Similarly, each line in the right panel represents the MAPPINGS models for a speci /uniFB01 c ionization parameter ( similar to the left panel ) and isobaric pressure ( denoted by the line types ) . \nThe CLOUDY models shown in Figure 9 assume a planeparallel geometry with a nebular electron density of 10 3 cm -3 ( inferred from the /uniFB02 ux ratio of the [ OII ] 3727 + 3729 doublet; see Section 6.3 ) , scaled Solar elemental abundances, and cover ionization parameters from log10 U = [ -2.5, -2.1, -1.5 ] .The star-forming models use BPASS v2.0 ( Eldridge et al. 2017 ) /uniFB01 ducial binary stellar population models with an IMF that extends to 300 M e ( described by a slope of α = -1.30 from 0.1 to 0.5 M e and α = -2.35 from 0.5 to 300 M e ) and continuous star formation of 1 M e yr -1 . For these models, we /uniFB01 x the stellar and gas-phase metallicities, covering Z = 0.1 -0.5 Z e . The AGN models use the table agn model in CLOUDY, which approximates a ' typical ' radio-quiet AGN, covering the same range of ionization parameters as the star-forming models and spanning nebular metallicities of 0.05 -0.5 Z e . \nThe MAPPINGS models shown in the same /uniFB01 gure are the ' Pressure Models ' from Kewley et al. ( 2019a ) , assuming a \nplane-parallel geometry with a range of pressure ( log 10 ( P / k ) = 7 -9cm -3 ) , ionization parameter ( log10 U = [ -2.5, -1.5 ] ; also referred to as log10 Q = 8 -9 ) , and metallicity ( Z = 0.05 -1 Z e ) . These star-forming models use the Starburst99 ( Leitherer et al. 2014 ) stellar population models with a Salpeter IMF ( Salpeter 1955 ) extending to 100 M e . The stellar and gas-phase metallicities follow a prescription such that at lower metallicities, the models have increasingly enhanced α abundances ( see Nicholls et al. 2017 for more details ) . We have included these models in addition to the CLOUDY models ( which have Solar abundances scaled to chosen metallicity ) to showcase the range of potential galaxy properties for our highredshift galaxy. \nSimilar to that found in other spectroscopic studies in this epoch ( e.g., Trump et al. 2023; Kocevski et al. 2023; Tang et al. 2023 ) , the line ratio when using the single-component /uniFB01 t to the H β line for CEERS\\_1019 ( the smaller star ) lies in a region on this diagnostic that is dif /uniFB01 cult to differentiate between AGN and metal-poor high-ionization star-forming HII regions. This is not unexpected, as galaxies at higher redshifts have been shown to generally have higher ionization and lower metallicities in comparison to those at lower redshifts ( e.g., Shapley et al. 2003; Erb et al. 2010; Kewley et al. 2019a; Backhaus et al. 2022; Papovich et al. 2022; Sanders et al. 2023 ) . This suggests that at these high redshifts, successfully separating star-forming and AGN sources using these strong line diagnostics can be challenging -evidenced by the lower-metallicity AGN models overlapping with the starforming models in the left panel of Figure 9. Further discussion about the general utility of such emission-line diagnostics at high redshift is discussed in Section 7.3. \nHowever, when focusing on the /uniFB01 t to only the narrow component of the H β line for CEERS\\_1019 ( the bigger star ) , there is a clear distinction from the rest of the high-redshift sources shown in this /uniFB01 gure. Of the sources shown, our galaxy is the only one with enough S / NinH β to ( a ) see a broadening of the line pro /uniFB01 le, and ( b ) clearly measure both a broad and narrow component for the line. 49 Follow-up spectroscopy of these and other sources that fall in this strong line regime at high redshift may shed light on more ' hidden ' AGNs in the early Universe ( Kocevski et al. 2023 ) . \nFrom Figure 9, CEERS\\_1019 covers a similar location to the CLOUDY star-forming models with log 10 U ∼-1.9 to -1.5 ( for the single-component /uniFB01 ttoH β ; smaller star ) and the AGN models with log10 U ∼-2.1 ( for both the narrow and singlecomponent /uniFB01 ts to H β ; both stars ) . Similarly, the MAPPINGS models suggest agreement with log10 U ∼-2.5 to -1.5; however, this is also dependent upon the pressure assumed. These results add to the expectation that this galaxy has high ionization powering these strong line ratios. Indeed, this agrees well with the strong O32 line ratio measured for this galaxy, which is often indicative of highly ionized, metal-poor gas ( e.g., Schaerer et al. 2022; Williams et al. 2023 ) and a regime that could suggest Lyman continuum leakage ( e.g., Izotov et al. 2018 ) . As a comparison to the ionization parameters gleaned from Figure 9, we estimate this value using two relations from the literature. First, using the theoretical relation between O32, 12 + log ( O / H ) , and the ionization parameter described in Kobulnicky & Kewley ( 2004 ) , we derive log10 ∼-2.3. \nFinally, using the measured relation between O32 and the ionization parameter explored in Papovich et al. ( 2022 ) ,we derive /uniF0A0 log10 ∼-2.0. These results are all relatively consistent with one another and further highlight the strong ionizing nature of this source.", '7. Discussion': "The ' chicken or egg ' origin of BH seeds in the /uniFB01 rst galaxies remains unsolved. Theoretical predictions suggest a mix of ' light ' ( ∼ 10 2 M e ) Population III stellar remnants and / or ' heavy ' ( ∼ 10 5 M e ) seeds formed via the direct collapse of primordial gas, as summarized in recent reviews by Smith & Bromm ( 2019 ) , Inayoshi et al. ( 2020 ) , and Fan et al. ( 2022 ) , but the relative mix of each seed type remains unconstrained by observations. Regardless of the seeding mechanisms, our Universe is capable of forming extremely massive SMBHs very early, with the highest-redshift massive quasar at the time of this writing being z = 7.642 ( Wang et al. 2021 ) , existing /uniF088 700 Myr after the Big Bang. Observations with JWST, especially MIRI observations of obscured growth ( e.g., G. Yang et al. 2023, in preparation ) and NIRSpec observations of broad ( /uniF089 1000 km s -1 ) and / or high-ionization ( e.g., N V, He II, [ Ne V ] , and C IV ) emission lines can now allow the /uniFB01 rst real census of AGNs in low-mass ( M * < 10 10 M e ) hosts at high redshift, providing strong constraints on the initial BH seed distribution. Direct observations of early SMBHs will not only constrain mechanisms for early BH growth ( e.g., Ricarte &Natarajan 2018a, 2018b ) , but can also provide further insight into the role ionizing photons from AGNs played in the reionization of the IGM ( e.g., Finkelstein et al. 2019; Giallongo et al. 2019; Dayal et al. 2020; Grazian et al. 2020; Yung et al. 2021; Grazian et al. 2022 ) .", '7.1. Constraints on the Formation of This z = 8.68 AGN': "Discovering an actively accreting SMBH at such a high redshift further constrains the formation time for such objects. Here we consider plausible formation mechanisms for this source ( e.g., small versus large seeds ) , as well as discuss what its plausible descendants are. While the inferred mass of this BH is not much larger than the proposed range for DCBH seeds, it is unlikely that we are witnessing a DCBH soon after formation. The constraints we place on both the stellar mass and gas-phase metallicity of the host galaxy are signi /uniFB01 cantly higher than expected ( DCBH formation requires near-primordial conditions ) , implying that this object is a ' standard ' ( albeit very distant ) AGN accreting at ∼ the Eddington rate, with a continuum SED dominated by stellar emission ( in line with the stellar emission dominating the continuum SED; Section 5 ) , similar to the scenarios recently explored by Volonteri et al. ( 2023 ) . \nIt is therefore interesting to explore how this AGN was seeded: from a low-mass ( ∼ 10 -100 M e ) stellar seed or a highmass ( ∼ 10 4 -6 M e ) DCBH seed, summarized in Figure 10.In between these two scenarios could be a seed from a massive starburst cluster or young ultracompact dwarf galaxy ( Kroupa et al. 2020 ) labeled as ' Dense Star Cluster Seed ' in Figure 10. The purple curves show idealized BH mass tracks assuming a 100 M e stellar seed. Such a seed could plausibly form at z ∼ 30, but would be unable to accrete for ∼ 100 Myr due to radiative heating of the gas from the stellar progenitor ( e.g., Johnson & Bromm 2007; Jeon et al. 2014 ) . Assuming such a \nFigure 10. Mass of BHs vs. redshift. The gold star represents the result of this work. The green points denote published z = 6 -7.64 quasars, taken from the compilation of Inayoshi et al. ( 2020 ) and augmented by more recent z > 7 quasars from Fan et al. ( 2022 ) . The different lines show potential tracks of BH growth. The purple lines show a 100 M e Population III stellar remnant seed forming at z ∼ 30, with growth beginning after a 100 Myr delay due to progenitor gas heating. The dashed purple line shows constant Eddington growth, which is both likely unphysical and cannot reproduce our observed object mass. The solid line shows one example of how periods of superEddington growth ( 10 times Eddington for 10 Myr ) separated by periods of sub-Eddington growth ( 0.1 times Eddington for 50 Myr ) could plausibly create the observed source. The red lines show potential formation mechanisms from a small ( 3 × 10 4 M e ) DCBH forming at z = 15 ( dotted ) and a large ( 10 6 M e ) DCBH forming at z = 10.5 ( dashed ) , each growing at the Eddington rate. Either of the mechanisms that could explain our observed BH source, DCBH + Eddington or stellar + super-Eddington, are somewhat exotic scenarios, pushing standard assumptions. Probing SMBHs out to higher redshifts and lower masses will clarify the formation mechanisms of these objects. \n<!-- image --> \n100 Myr delay, the onset of BH growth would begin by z ∼ 18.5. As shown by the dashed purple line, Eddingtonlimited accretion in this scenario would be unable to reach the inferred BH mass by the redshift measured for this object. The enhanced stellar feedback environment the host galaxy must have experienced in the previous ∼ 100 Myr further complicates this. Building up to the observed Z /uniF089 0.01 Z e requires a large number of supernova explosions in a short period of time. This would lead to violent, turbulent mixing and heating of the gas, making accretion likely very inef /uniFB01 cient ( signi /uniFB01 cantly subEddington ) . \nHowever, there may also be periods of super-Eddington ( ' catch-up ' ) accretion. These would occur in short episodes, with minimal sub-Eddington accretion between them; such short bursts of BH growth could plausibly build up the ∼ 10 7 M e SMBH we observe in an otherwise stellar-emissiondominated galaxy ( e.g., Volonteri & Rees 2005; Madau et al. 2014; Inayoshi et al. 2016 ) . For example, Pezzulli et al. ( 2016 ) , when examining the evolution of merger tree simulations, /uniFB01 nd that super-Eddington accretion modes ( e.g., Haiman 2004; Silk 2005; Polletta 2008 ) can be a signi /uniFB01 cant component of SMBH growth in gas-rich environments, where up to 75% of the SMBH growth can be accounted for by periods of superEddington accretion, with intermittent phases of disruption resulting from the rapid depletion / replenishment of the bulge gas reservoir out of which the BHs accrete. \nIn Figure 10, we highlight one potential version of this scenario, with the purple solid line showing an object that starts with a seed mass of 100 M e , which begins growing at z = 18.5 with episodic 10 Myr periods of super-Eddington ( 10 times ) growth, followed by 50 Myr ' breathing ' periods with subEddington ( 0.1 times ) growth. Periodic super-/ sub-Eddington growth allows this hypothetical object to grow to ∼ 10 7 M e by \nz ∼ 8.7 from a small seed, matching our observations. While plausible ( in particular, the episodic growth with suppressed, sub-Eddington, accretion periods was predicted by the simulations of Jeon et al. 2012 and Massonneau et al. 2023 ) , this scenario is somewhat contrived, making a stellar seed somewhat unlikely. \nA DCBH origin may be more plausible. While the inferred metallicity implies the DCBH event occurred signi /uniFB01 cantly prior to the observed epoch, this could provide the needed time for a massive seed ∼ 10 6 M e to grow by the needed factor of ∼ 10 times, without also having to invoke super-Eddington conditions. Given the likely strong stellar feedback environment in the assembling galaxy, this DCBH scenario may be favored. We show two plausible DCBH tracks as the red lines in Figure 10. Both options, of a lower-mass 3 × 10 4 M e DCBH forming at z = 15 or a higher-mass 10 6 M e DCBH forming at z = 10.5, should they grow at the Eddington limit, could reproduce the observed mass of this object by z ∼ 8.7. \nLast, extrapolating the inferred BH growth tracks to lower redshifts provides insight into the ultimate descendants of early BHs, such as the one we observe here. As shown in Figure 10, this object is both too low-mass and observed too late to be the plausible progenitor of the z > 7 quasar population. However, should near-Eddington accretion be sustained for signi /uniFB01 cant periods of time, objects like this one could plausibly evolve into the massive z ∼ 6 -7 SDSS quasar population. Further constraining both the progenitors and descendants of highredshift SMBHs is possible, as the population of z > 8 SMBHs is coming into view. Given the discovery of this source in a relatively small data set, it is likely that further JWST spectroscopy -in particular, following up bright sources discovered over wide areas with the upcoming Nancy Grace Roman Space Telescope -will yield such a population ( e.g., Yung et al. 2023 ) .", '7.2. Evolution in the BH -Galaxy Mass Relationship': 'Figure 11 compares CEERS\\_1019 with other high-redshift ( z ∼ 6 ) AGNs ( Izumi et al. 2021; Kocevski et al. 2023 ) as well as the z = 0 M BH -M * relationship. Because they are easier to detect with pre-JWST instruments, most high-redshift AGNs are high luminosity. However, AGNs have a well-known luminosity-dependent bias ( i.e., the Lauer bias; Lauer et al. 2007 ) , such that higher-luminosity AGNs tend to have overmassive BHs. \nJWST has improved our ability to select low-luminosity accreting BHs ( e.g., Kocevski et al. 2023 ) , which are expected to lie closer to the intrinsic ( overall ) M BH -M * relationship. Figure 11 shows an estimate from the empirical TRINITY model ( Zhang et al. 2023a ) for how different bolometric luminosity thresholds bias the median M BH -M * relationship. With an estimated bolometric luminosity of L bol ∼ 10 45 erg / s and a host stellar mass of M * ∼ 10 9.5 M e , TRINITY would suggest that the present source would be expected to have a /uni0394 M BH ∼ 0.7 dex offset from the median relation. The present source is located around the upper envelope of the M BH -M * relation spanned by the AGN sample from Reines & Volonteri ( 2015 ) at z ∼ 0. The present source also has a lower M BH compared to the z = 5.55 AGN from Übler et al. ( 2023 ) , which is ∼ 10 times brighter. Qualitatively, this is consistent with the Lauer bias, that brighter AGNs tend to be overmassive BHs compared to their host galaxies. \nFigure 11. The predicted z = 8.7 median M GLYPH<129> -M * relation for quasars with different bolometric luminosity thresholds from the TRINITY model ( solid curves; Zhang et al. 2023a, 2023b ) . The pink shaded region is the 1 -σ spread around the median scaling relation for quasars ( ∼ 0.55 dex ) , which includes the random scatter in observed M GLYPH<129> when using virial estimates. This ( log-) normal scatter is nearly luminosity-independent, so we only show it for the brightest quasars for clarity. The black solid line is the predicted M GLYPH<129> -M * relation for all SMBHs at z = 8.7, and the black shaded region is the intrinsic + observed scatter around the intrinsic M GLYPH<129> -M * relation. The green solid line is the M GLYPH<129> -M * relation for AGNs brighter than [ ] L log erg s 45.1 bol , corresponding to the bolometric luminosity of the AGN in this work ( gold star ) . For comparison, we also show the z = 0 relation from Greene et al. ( 2016 ) with the black dashed line and the z = 0 AGN sample from Reines & Volonteri ( 2015 ) with the black crosses. The following data are shown in symbols color-coded by bolometric AGN luminosities: ( 1 ) z ∼ 6 quasars compiled by Izumi et al. ( 2021 ) and Übler et al. ( 2023; /uniFB01 lled circles ) ; and ( 2 ) the two z > 5 AGNs from Kocevski et al. ( 2023; pentagons ) . \n<!-- image --> \nIf the present source lies on the median M BH -M * for its luminosity, the Lauer bias estimate from TRINITY would suggest that the overall z = 8.7 median M BH -M * would have M BH = 10 6.3 M e at M * = 10 9.5 M e , i.e., there would be no overall evolution from the z = 0 M BH -M * relationship. If other sources are con /uniFB01 rmed to have similarly consistent masses compared to the z = 0 M BH -M * relationship, it would cement a tight relationship between galaxies and BHs that extends into the EoR, and would also provide a strong challenge to theoretical models that have predicted substantial evolution in the M BH -M * relation for high-redshift and low-mass galaxies ( see Habouzit et al. 2021 for a survey ) .', '7.3. The Ef /uniFB01 cacy of Line Ratio Diagrams in Identifying AGNs at Early Times': "Many of the canonical emission-line ratio diagnostics of AGN versus star formation using rest-frame optical emission lines are calibrated for low to moderate redshifts ( z /uniF088 1; e.g., Baldwin et al. 1981; Veilleux & Osterbrock 1987; Juneau et al. 2011, 2014; Trump et al. 2015; Backhaus et al. 2022 ) . As such, the divisions between the star-forming and AGN regions de /uniFB01 ned by these optical line ratio diagnostics are not necessarily valid at higher redshifts. \nSeveral works offer remedies to these low-redshift diagnostics by quantifying a redshift evolution to the division ( e.g., Coil et al. 2015; Cleri et al. 2023 ) , but these new divisions have not been extended to the EoR; in fact, these relations for these diagrams are shown to break down at high z in simulations ( Hirschmann et al. 2019, 2022 ) . The redshift-evolving diagnostics of these works employ a simple shift from the low-redshift divisions; however, there is insuf /uniFB01 cient knowledge \nof high-redshift AGNs to validate this practice for galaxies in the EoR. \nFor a different avenue to remediate these issues, other works have offered completely new emission-line ratio diagnostics with higher-ionization emission-line ratios, e.g., in the optical with He II / H β and [ Ne V ] / [ Ne III ] ( Katz et al. 2023; Cleri et al. 2023 ) and in the UV with, e.g., C III ] / He II, O III ] / He II, and C IV / He II ( e.g., Feltre et al. 2016; Hirschmann et al. 2019, 2022 ) . Unfortunately, many of these very high ionization lines are often weak and thus may not always have wellconstrained ratios, as is the case for the object studied in this work. Photoionization modeling ( e.g., from Cleri et al. 2023 ) , along with comparisons to data across a broad redshift range ( 0 /uniF088 z /uniF088 8.5 ) , suggest that sources of very highly ionizing photons ( > 54.42 eV; Berg et al. 2021 ) may easily be confounded with AGNs by traditional BPT-style diagnostics. This result has been shown with well-studied z ∼ 0 extreme metal-poor dwarf star-forming galaxies, which have been used as analogs to galaxies in the EoR ( e.g., Berg et al. 2019, 2021; Olivier et al. 2022 ) . \nMany recent studies from early JWST data have shown exactly this: high-ionization, low-metallicity star formation may produce line ratios consistent with AGNs in the lowerredshift diagnostics ( e.g., Brinchmann 2023; Katz et al. 2023; Trump et al. 2023; Trussler et al. 2022 ) . From the other side, other work has shown known AGNs that produce line ratios consistent with star-forming galaxies ( Übler et al. 2023 ) . This growing body of work suggests that the dichotomous classi /uniFB01 cation of a galaxy as either AGN-dominated or starformation-dominated is inadequate to accurately describe a galaxy ' s ionizing spectrum in the early Universe. \nThese works suggest a larger ' composite ' region of BPTstyle diagrams as a function of redshift. This allows for contributions to the ionizing spectrum from multiple sources ( e.g., an accreting BH and extreme metal-poor star formation ) . As more exotic systems in the early Universe are found, evolved versions of these diagnostics and their use with other information about the galaxy ( e.g., the analysis of broad Balmer lines and SED /uniFB01 ts as shown in this work ) will become increasingly necessary to discriminate between sources of ionization.", '7.4. Implications for the Reionization of This Overdense Region at z = 8.7': 'The presence of an AGN in this object adds another layer of intrigue to this region. This object is one of two spectroscopically con /uniFB01 rmed galaxies discussed by Larson et al. ( 2022 ) , which reside within a larger photometric overdensity of /uniFB01 ve bright HST-selected galaxies discussed by Finkelstein et al. ( 2022a ) . The CEERS NIRCam data also show further evidence of an overabundance of z ∼ 8.5 -9 galaxies in this /uniFB01 eld, as shown in Finkelstein et al. ( 2022b ) and studied in more detail by Larson et al. ( 2023, in preparation ) and Whitler et al. ( 2023 ) . Larson et al. ( 2022 ) discussed the ability of the galaxies in this region to ionize their surroundings, allowing Ly α to be visible. They found that an overdense region could provide the needed emissivity to reionize a large ( /uniF089 1 pMpc ) bubble, necessary for Ly α to redshift out of resonance. While at the observed epoch, the AGN does not appear to dominate the restUV emission, potential past periods of super-Eddington growth could have emitted large amounts of ionizing photons, potentially ionizing a large volume around this region.', '8. Conclusions': 'We present the discovery of an accreting SMBH at z = 8.679, using spectroscopy from NIRSpec and NIRCam / WFSS and imaging from NIRCam and MIRI from the JWST CEERS Survey ( Bagley et al. 2023; Finkelstein et al. 2022b; S. L. Finkelstein et al. 2023, /uniF0A0 in preparation ) . This source, denoted here as CEERS\\_1019, was initially identi /uniFB01 ed as a z ∼ 8 photometric Ly α -break dropout candidate by RobertsBorsani et al. ( 2016b ) , with spectroscopic con /uniFB01 rmation via Ly α emission using Keck / MOSFIRE by Zitrin et al. ( 2015 ) .We detect several strong rest-UV and rest-optical emission lines using medium-resolution JWST / NIRSpec spectroscopy ( R ∼ 1000 ) covering 1 -5 /uni03BC m. From this work, we measure a signi /uniFB01 cant broad component of the H β emission line with FWHM ∼ 1200 km s -1 , which we conclude originates from AGN activity. \nOur measurements are based on observations from several JWST instruments, but our key results are derived from the 1 -5 /uni03BC m medium-resolution grating spectra from NIRSpec. We use an automated line/uniFB01 nding code ( based on Larson et al. 2018 ) to identify signi /uniFB01 cant emission features in an unbiased and systematic way, running line-injection simulations to estimate accurate line /uniFB02 ux uncertainties. Due to the very high S / N of the [ OIII ] 5008 Å emission line, most of our emissionline /uniFB01 ts are tied to the redshift and FWHM of this line. We perform more customized /uniFB01 ts when required, including for known doublets, as well as observed broad lines. We also note that a few NIRSpec-detected lines are also observable in the NIRCam WFSS spectra, albeit at lower S / N. \nOur key result is that the observed H β emission line has a clear broad component, comprising ∼ half of the emission-line /uniFB02 ux, with an FWHM ∼ 1200 km s -1 . As this broad component is not seen in the stronger [ OIII ] lines ( as would be the case in large-scale out /uniFB02 ows ) , we conclude that its origin is from a broadline region around an AGN. This is supported by weak NV, N IV, and Mg II emission, as well as weak broad C III ] emission, and the morphology, which shows a compact point source among three Sérsic-like clumps. \nWe explore the properties of both this accreting SMBH as well as the host galaxy. Constraints from the continuum SED, including photometry from HST, as well as NIRCam and MIRI, show that the continuum emission is dominated by stellar light, particularly in the rest-UV, and that the stellar population is modestly massive ( log M / M e ∼ 9.5 ) and heavily star-forming ( log sSFR ∼-7.9 ) . \nFrom the width and /uniFB02 ux of the broad H β emission feature, we estimate the mass of the SMBH to be log ( M / M e ) = 6.95 ± 0.37 and that it is accreting at 1.2 ( ± 0.5 ) times the Eddington limit. From the ratios of narrow emission lines, we /uniFB01 nd that the gas in this galaxy is modestly metal-poor ( ∼ 0.1 Z e ) with little dust attenuation, dense, and highly ionized. Similar to other recent JWST results, CEERS\\_1019 sits elevated over most star-forming models in diagnostic line ratio diagrams. While this could be interpreted as evidence that the AGN signi /uniFB01 cantly contributes to the narrow lines, the presence of other presumably star-forming-dominated galaxies in a similar line ratio regime means that we cannot rule out stellardominated ionization. \nWe discuss the implications of the presence of this BH early in cosmic history. We /uniFB01 nd that it is dif /uniFB01 cult to explain an SMBH of this mass at z ∼ 8.7 with a stellar seed unless periodic episodes of super-Eddington accretion are possible. \nAlternatively, Eddington-limited accretion from a massive ( ∼ 10 4 -6 M e ) DCBH seed could reach the target mass by the observed epoch. Either scenario is somewhat exotic -the uncovering of a larger population of early SMBHs will place further constraints on their seeding and growth mechanisms. \nWe conclude by noting that while the broad H β component is statistically signi /uniFB01 cant ( 2.5 σ ) and is signi /uniFB01 cantly required by the /uniFB01 t ( /uni0394 BIC ∼ 3 ) , we can only detect this component in one Balmer line. However, the next few months will see a MIRI spectrum obtained by the MIRI Guaranteed Time Observation ( GTO ) team ( PID 1262 ) , covering H α . If, as we have concluded, a broadline AGN is present in this source, then the upcoming MIRI H α spectra should show a well-detected broad line.', 'Acknowledgments': 'We sincerely thank all of the engineers, scientists, technicians, staff, other humans, and their families who spent decades of their lives making JWST possible, with a special thanks to those who have spent countless hours this past year commissioning and operating the telescope ( Rigby et al. 2023 ) and providing calibration and pipeline updates. We also thank our other colleagues in the CEERS collaboration for their hard work and valuable contributions to this project. \nWe thank Xiaohui Fan, Dan Stark, and Rafaella Schneider for their helpful conversations. This work acknowledges support from the NASA / ESA / CSA JWST through the Space Telescope Science Institute ( STScI ) , operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-03127. Support for program No. JWST-ERS01345 was provided through a grant from the STScI under NASA contract NAS5-03127. \nR.L.L., S.L.F., and M.B. acknowledge that they work at an institution, the University of Texas at Austin, that sits on indigenous land. The Tonkawa lived in central Texas, and the Comanche and Apache moved through this area. We pay our respects to all the American Indian and Indigenous Peoples and communities who have been or have become a part of these lands and territories in Texas. We are grateful to be able to live, work, collaborate, and learn on this piece of Turtle Island. \nThe authors acknowledge the Texas Advanced Computing Center ( TACC; tacc.utexas.edu ) at the University of Texas at Austin for providing database and grid resources that have contributed to the research results reported within this paper. This work has used the Rainbow Cosmological Surveys Database, operated by the Centro de Astrobiologa ( CAB ) , CSIC-INTA, partnered with the University of California Observatories at Santa Cruz ( UCO / Lick, UCSC ) . \nT.A.H. and A.Y. are supported by appointment to the NASA Postdoctoral Program ( NPP ) at NASA Goddard Space Flight Center, administered by Oak Ridge Associated Universities under contract with NASA. C.P. thanks Marsha and Ralph Schilling for the generous support of this research. This work bene /uniFB01 ted from support from the George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University. D.K. acknowledges support from NASA grants JWST-ERS-01345 and JWST-AR02446. J.R.T. acknowledges support from NSF grant CAREER-1945546. D.B. and M.H.-C. thank the Programme National de Cosmologie et Galaxies and CNES for their support. R.A. acknowledges support from Fondecyt Regular 1202007. S.F. acknowledges funding from NASA through the \nNASA Hubble Fellowship grant HST-HF2-51505.001-A awarded by STScI. A.Z. acknowledges support from grant No. 2020750 from the United States -Israel Binational Science Foundation ( BSF ) and grant No. 2109066 from the United States National Science Foundation ( NSF ) and from the Ministry of Science & Technology, Israel. \nSome of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes ( MAST ) at the Space Telescope Science Institute. The speci /uniFB01 c observations analyzed can be accessed via doi:10.17909 / z7p0-8481 and at the CEERS Mast Archive Page. \nFacilities: HST ( ACS and WFC3 ) , JWST ( NIRCam, MIRI, NIRSpec, and NIRCam / WFSS ) , Spitzer ( IRAC and MIPS ) , Keck ( MOSFIRE ) , SCUBA-2, VLA, Chandra X-Ray Observatory, Herschel ( PACS and SPIRE ) , and the Texas Advanced Computing Center ( TACC ) . \nSoftware: BPASS v2.0 and v2.2.1 ( Eldridge et al. 2017; Stanway & Eldridge 2018 ) , CLOUDY v17.0 ( Ferland et al. 2017 ) , EAZY ( Brammer et al. 2008 ) , Prospector ( Johnson et al. 2021 ) , Cigale ( Boquien et al. 2019; Yang et al. 2020, 2022 ) , astropy ( The Astropy Collaboration et al. 2018 ) , topcat ( Taylor 2011 ) , Gal /uniFB01 t ( Peng et al. 2002, 2010 ) , statmorph ( Rodriguez-Gomez et al. 2019 ) , BAGPIPES ( Carnall et al. 2018 ) , MAPPINGS V ( Sutherland et al. 2018; Kewley et al. 2019a ) , FAST v1.1 ( Kriek et al. 2009; Aird et al. 2018 ) , IDL Astronomy Library: idlastro.gsfc.nasa.gov ( Landsman 1993 ) , matplotlib ( Hunter 2007 ) , NumPy ( Harris et al. 2020 ) , photutils ( Bradley et al. 2020 ) , SourceExtractor ( Bertin & Arnouts 1996 ) , SciPy ( Virtanen et al. 2020 ) , STScI JWST Calibration Pipeline ( jwst-pipeline.readthedocs.io; Rigby et al. 2023 ) .', 'Appendix A Far-infrared and Submillimeter Constraints': 'Recent studies have shown that a signi /uniFB01 cant fraction of highredshift AGNs exhibit large rest-frame infrared luminosities, with values ranging from 10 12 to 10 13 L e ( e.g., Decarli et al. 2018 ) . Here we report on the far-infrared and submillimeter photometric constraints on this source, exploiting the deep \n<!-- image --> \n<!-- image --> \nancillary data in the /uniFB01 eld. As shown in Figure 12, a ∼ 6 σ SCUBA-2 detection ( yellow contours ) with a /uniFB02 ux of S 1.9 0.3 850 m = m mJy ( Zavala et al. 2017 ) is found 2 " away from the position of CEERS\\_1019 ( red circle ) . Zavala et al. ( 2018 ) associated this emission as originating from a nearby z phot ≈ 3 galaxy ( named 850.44 ) marked by a blue cross in Figure 12. This nearby source is also detected with Spitzer at 24 /uni03BC m and at 100 /uni03BC m with Herschel, as shown in panels 2 and 3 of Figure 12. A recent survey was conducted with the JVLA at 3 GHz ( PI: M. Dickinson; see also Jimenez-Andrade et al. 2023, in preparation ) , which also found detectable emission at this location, likely associated with the nearby source ( the right panel in Figure 12 ) . \nIf this submillimeter detection was instead emitting from CEERS\\_1019 at z = 8.679, it would imply an infrared luminosity of L IR ∼ 3 × 10 12 L e . Although this is in line with the luminosity of z ∼ 7 quasars detected with the Atacama Large Millimeter / submillimeter Array ( e.g., Decarli et al. 2018 ) , it would exceed the Eddington limit of our target by a factor of ∼ 10. This supports the assumption that the submillimeter emission arises ( mainly ) from a different neighboring source.', 'Appendix B X-Ray Constraints': 'The Chandra X-ray Observatory took an 800 ks exposure over the EGS /uniFB01 eld ( Nandra et al. 2015 ) , but there is no emission detected at the location of CEERS\\_1019 in these data ( see Figure 13 ) . Adopting a 0.5 -10 keV sensitivity of 8.22 × 10 -16 erg cm -1 s -1 ( Nandra et al. 2015 ) and assuming a photon index ( Γ ) of 1.4, we estimate an upper limit of LX < 10 44.2 erg s -1 . This constraint places CEERS\\_1019 around or below the knee luminosity ( L X * ) of the AGN X-ray luminosity function ( i.e., the Seyfert regime ) at lower redshifts ( z ≈ 0 -5; e.g., Aird et al. 2015 ) . Thanks to its unprecedented sensitivity, JWST begins to uncover the Seyfert-like AGN population in the early universe. \n<!-- image --> \nFigure 12. From left to right: JWST / NIRCam F277W; Spitzer / MIPS 24 /uni03BC m; Herschel / PACS 100 /uni03BC m; and VLA 3 GHz ( north is up, east is to the left ) . All the 20 " × 20 " cutouts are centered on the coordinates of the 6 σ SCUBA-2 / 850 /uni03BC m detection ( yellow contours ) found around the location of CEERS\\_1019 ( red circle ) . The blue cross marks the position of a z ∼ 3 galaxy detected at 24 /uni03BC m and 3 GHz, which is likely the correct counterpart and the main contributor to the SCUBA-2 detection. \n<!-- image --> \nFigure 13. Full-band ( 0.5 -7 keV ) X-ray images from Nandra et al. ( 2015 ) . The format is similar to Figure 12. The left panel shows the original image, where each white pixel indicates one ( or multiple ) X-ray photon ( s ) . The right panel is a smoothed version of the left. Noise dominates in the region around the source. \n<!-- image -->', 'Appendix C NIRSpec Prism Spectrum': 'The NIRSpec / PRISM observation of CEERS\\_1019 ( see Figure 14 ) was not included in the analysis of this source, as it was observed in 2022 December and was contaminated by a short. When these CEERS observations were rescheduled for 2023 February, the telescope had rotated such that the MSA \nhad to be redesigned, and this source was no longer included. We include the spectrum here for completeness and to illustrate that several lines are still visible in the spectrum, including NIV ] ,CIII ] ,H β , and [ OIII ] , despite the contamination. Recent publications, including Heintz et al. ( 2022; where they identify the source as CEERS-z8684 ) and Harikane et al. ( 2023 ) , have shown this PRISM spectrum and also detected these lines. \nFigure 14. 2D and 1D spectra of the NIRSpec / PRISM spectrum of CEERS\\_1019, taken in 2022 December, which encountered a short contaminating a majority of the data. We show the spectra here for completeness and also to illustrate that many of the emission lines are visible, including N IV ] ,CIII ] ,H β , and [ O III ] , even though the data are contaminated. \n<!-- image -->', 'ORCID iDs': 'Rebecca L. Larson https: // orcid.org / 0000-0003-2366-8858 Steven L. Finkelstein https: // orcid.org / 0000-00018519-1130 \nDale D. Kocevski https: // orcid.org / 0000-0002-8360-3880 Taylor A. Hutchison https: // orcid.org / 0000-00016251-4988 \nJonathan R. Trump https: // orcid.org / 0000-0002-1410-0470 Pablo Arrabal Haro https: // orcid.org / 0000-0002-7959-8783 Volker Bromm https: // orcid.org / 0000-0003-0212-2979 Nikko J. Cleri https: // orcid.org / 0000-0001-7151-009X Mark Dickinson https: // orcid.org / 0000-0001-5414-5131 Seiji Fujimoto https: // orcid.org / 0000-0001-7201-5066 Jeyhan S. Kartaltepe https: // orcid.org / 0000-00019187-3605 \nAnton M. Koekemoer https: // orcid.org / 0000-00026610-2048 \nCasey Papovich https: // orcid.org / 0000-0001-7503-8482 Nor Pirzkal https: // orcid.org / 0000-0003-3382-5941 Sandro Tacchella https: // orcid.org / 0000-0002-8224-4505 Jorge A. Zavala https: // orcid.org / 0000-0002-7051-1100 Micaela Bagley https: // orcid.org / 0000-0002-9921-9218 Peter Behroozi https: // orcid.org / 0000-0002-2517-6446 Jaclyn B. Champagne https: // orcid.org / 0000-00026184-9097 \nJustin W. Cole \nhttps: \nIntae Jung \nhttps: \nAlexa M. Morales \nhttps: \nGuang Yang \nhttps: \nHaowen Zhang \nhttps: \nAdi Zitrin \nhttps: \n// \n// \norcid.org \norcid.org \n// \n/ \n/ \n0000-0002-6348-1900 \n0000-0003-1187-4240 \norcid.org \norcid.org \n/ \n/ \n0000-0003-4965-0402 \n0000-0001-8835-7722 \norcid.org \n// \norcid.org \n/ \n/ \n0000-0002-4321-3538 \n0000-0002-0350-4488 \nRicardo O. Amorín https: // orcid.org / 0000-0001-5758-1000 Denis Burgarella https: // orcid.org / 0000-0002-4193-2539 Caitlin M. Casey https: // orcid.org / 0000-0002-0930-6466 Óscar A. Chávez Ortiz https: // orcid.org / 0000-00032332-5505 \nIsabella G. Cox https: // orcid.org / 0000-0002-1803-794X Katherine Chworowsky https: // orcid.org / 0000-00034922-0613 \nAdriano Fontana https: // orcid.org / 0000-0003-3820-2823 Eric Gawiser https: // orcid.org / 0000-0003-1530-8713 Andrea Grazian https: // orcid.org / 0000-0002-5688-0663 Norman A. Grogin https: // orcid.org / 0000-0001-9440-8872 Santosh Harish https: // orcid.org / 0000-0003-0129-2079 Nimish P. Hathi https: // orcid.org / 0000-0001-6145-5090 Michaela Hirschmann https: // orcid.org / 0000-00023301-3321 \nBenne W. Holwerda https: // orcid.org / 0000-00024884-6756 \nStéphanie Juneau https: // orcid.org / 0000-0002-0000-2394 Gene C. K. Leung https: // orcid.org / 0000-0002-9393-6507 Ray A. Lucas https: // orcid.org / 0000-0003-1581-7825 Elizabeth J. McGrath https: // orcid.org / 0000-00018688-2443 \nPablo G. Pérez-González https: // orcid.org / 0000-00034528-5639 \nJane R. Rigby https: // orcid.org / 0000-0002-7627-6551 Lise-Marie Seillé https: // orcid.org / 0000-0001-7755-4755 Raymond C. Simons https: // orcid.org / 0000-00026386-7299 \nAlexander de la Vega https: // orcid.org / 0000-00026219-5558 \n// \n// \nBenjamin J. Weiner https: // orcid.org / 0000-0001-6065-7483 Stephen M. Wilkins https: // orcid.org / 0000-0003-3903-6935 L. Y. Aaron Yung https: // orcid.org / 0000-0003-3466-035X', 'References': "Aird, J., Coil, A. 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2024A&A...691A..88P	Context. IRAS 16293E is a rare case of a prestellar core being subjected to the effects of at least one outflow. Aims. We want to disentangle the actual structure of the core from the outflow impact and evaluate the evolutionary stage of the core. Methods. Prestellar cores being cold and depleted the best tracers of their central regions are the two isotopologues of the trihydrogen cation that are observable from the ground orthoHSUB2SUBDSUPSUP and paraDSUB2SUBHSUPSUP. We used the Atacama Pathfinder EXperiment APEX telescope to map the paraDSUB2SUBHSUPSUP emission in IRAS 16293E and collected James Clerk Maxwell Telescope JCMT archival data of orthoHSUB2SUBDSUPSUP. We compared their emission to that of other tracers including dust emission and analysed their abundance with the help of a 1D radiative transfer tool. The ratio of the abundances of orthoHSUB2SUBDSUPSUP to paraDSUB2SUBHSUPSUP can be used to estimate the stage of the chemical evolution of the core. Results. We have obtained the first complete map of paraDSUB2SUBHSUPSUP emission in a prestellar core. We compare it to a map of orthoHSUB2SUBDSUPSUP and show their partial anticorrelation. This reveals a strongly evolved core with a paraDSUB2SUBHSUPSUPorthoHSUB2SUBDSUPSUP abundance ratio towards the centre for which we obtain a conservative lower limit from 3.9 at 12 K to 8.3 at 8 K while the high extinction of the core is indicative of a central temperature below 10 K. This ratio is higher than predicted by the known chemical models found in the literature. ParaDSUB2SUBHSUPSUP and indirectly orthoHSUB2SUBDSUPSUP is the only species that reveals the true centre of this core while the emission of other molecular tracers and dust are biased by the temperature structure that results from the impact of the outflow. Conclusions. This study is an invitation to reconsider the analysis of previous observations of this source and possibly questions the validity of the deuteration chemical models or of the reaction and inelastic collisional rate coefficients of the HSUPSUPSUB3SUB ltinlineformula idFI1gtltalternativesgttexmath idtexeq1ltCDATAmathrmH3gttexmathmmlmath xmlnsmmlhttpwww.w3.org1998MathMathML displayinline idmmleq1mmlmsubsupmmlmrow classMJXTeXAtomORDmmlmi mathvariantnormalHmmlmimmlmrowmmlmn3mmlmnmmlmrow classMJXTeXAtomORDmmlmommlmommlmrowmmlmsubsupmmlmathltinlinegraphic xmlnsxlinkhttpwww.w3.org1999xlink idimgeq1 mimesubtypepng mimetypeimage xlinkhrefaa4735123eq1.pnggtltalternativesgtinlineformula isotopologue family. This could impact the deuteration clock predictions for all sources.	2024-11-01T00:00:00Z	['2024arXiv240910093P', 'arXiv:2409.10093', '10.1051/0004-6361/202347351', '2024A&A...691A..88P', '10.48550/arXiv.2409.10093']	['astrochemistry', 'radiative transfer', 'ISM: abundances', 'ISM: molecules', 'ISM: structure', 'ISM: individual objects: IRAS 16293E', 'Astrophysics - Astrophysics of Galaxies']	First map of DSUB2SUBHSUPSUP emission revealing the true centre of a prestellar core Further insights into deuterium chemistry	2,024	189	0.51	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.10093.pdf	{'First map of D 2 H + emission revealing the true centre of a prestellar core: further insights into deuterium chemistry': 'L. Pagani 1 , A. Belloche 2 , and B. Parise 2 * \n- 1 LERMA & UMR8112 du CNRS, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, F- 75014 Paris, France\n- e-mail: [email protected]\n- 2 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany \nReceived 04 / 07 / 2023; accepted ...', 'ABSTRACT': 'Context. IRAS 16293E is a rare case of a prestellar core being subjected to the e ff ects of at least one outflow. \nAims. We want to disentangle the actual structure of the core from the outflow impact and evaluate the evolutionary stage of the core. Methods. Prestellar cores being cold and depleted, the best tracers of their central regions are the two isotopologues of trihydrogren cation which are observable from the ground, ortho-H2D + and para-D2H + . We used the Atacama Pathfinder EXperiment (APEX) telescope to map the para-D2H + emission in IRAS 16293E and collected James Clerk Maxwell Telescope (JCMT) archival data of ortho-H2D + . We compare their emission to that of other tracers, including dust emission, and analyse their abundance with the help of a 1D radiative transfer tool. The ratio of the abundances of ortho-H2D + and para-D2H + can be used to estimate the stage of the chemical evolution of the core. \nResults. We have obtained the first complete map of para-D2H + emission in a prestellar core. We compare it to a map of orthoH2D + and show their partial anti-correlation. This reveals a strongly evolved core with a para-D2H + / ortho-H2D + abundance ratio towards the centre for which we obtain a conservative lower limit from 3.9 (at 12 K) up to 8.3 (at 8 K) while the high extinction of the core is indicative of a central temperature below 10 K. This ratio is higher than predicted by the known chemical models found in the literature. Para-D2H + (and indirectly ortho-H2D + ) is the only species that reveals the true centre of this core, while the emission of \nother molecular tracers and dust are biased by the temperature structure that results from the impact of the outflow. This study invites to reconsider the analysis of previous observations of this source and possibly questions the validity isotopologue family. This \nConclusions. of the deuteration chemical models or of the reaction and inelastic collisional rate coe ffi cients of the H + 3 could impact the deuteration clock predictions for all sources. \nKey words. ISM: abundances - ISM: clouds - ISM: structure - Astrochemistry - Molecular processes - ISM: individual objects : IRAS 16293E', '1. Introduction': "IRAS 16293E is a well-known prestellar core (PSC) at a distance of 141 pc (Dzib et al. 2018) which is under the influence of at least one outflow coming from the multiple protostar system IRAS 16293-2422 (e.g. Wootten & Loren 1987, Castets et al. 2001, Lis et al. 2002, Stark et al. 2004). A second outflow, visible in the Spitzer / IRAC data and coming from the Young Stellar Object (YSO) WLY 2-69 was also reported by Pagani et al. (2016). This outflow points towards the south of the PSC, close to the HE2 position (Castets et al. 2001) but whether this represents a fortuitous alignment or reveals an actual impact is not yet established. IRAS 16293-2422 has attracted considerable attention as well, with the SMA (Jørgensen et al. 2011) and ALMA (program PILS, Jørgensen et al. 2016, and follow-up papers), and the first (and still unique) detections of para-H2D + and orthoD2H + in absorption in front of the Class 0 objects (Bruenken et al. 2014, Harju et al. 2017). IRAS 16293E has also gained new attention with the advent of ALMA observations combined with CSO maps (Lis et al. 2016) after the detection of ND3 (Roue ff et al. 2005), which confirm that most deuterated species such as \nN2D + or DCO + peak at the outflow-PSC interface, with the exception of ND2H and ND3, the emission of which peak about 10 '' east of the DCO + emission peak. In a preliminary study we showed that Herschel continuum observations of the source with PACS and SPIRE reveal the temperature gradient induced by the outflow while ortho-H2D + emission traces the cold part of the core, away from the working surface, and approximately coincidental with the core seen in absorption in Spitzer IRAC midinfrared images (Pagani et al. 2016). The recent study by Kahle et al. (2023) of a large set of observations performed with the Atacama Pathfinder EXperiment (APEX) telescope towards both the YSOs and the PSC improves the census of molecular abundances and better di ff erentiates the various parts of this complex region, which confirms our understanding of its structure. \nIt has been shown that the deuterium chemistry evolves in a rather monotonic way during the PSC formation phase and measuring the abundance of deuterated species could help to measure the time the cloud took to contract to the PSC core under scrutiny (Flower et al. 2006, Pagani et al. 2009, Pagani et al. 2013). Though it is easier to measure deuteration of species like NH3, HCO + , or N2H + , it has been advocated that measuring directly the abundance of the deuteration vector, namely the H + 3 isotopologues, would be more precise to characterize the evolu- \ntion of the deuteration (Parise et al. 2011b, Harju et al. 2017). However single point measurements are often ambiguous, provide only line of sight averaged values, often decreasing the contrast between species. Only maps of the species allow us to fully understand the physics and chemistry of the PSCs. \nIn this paper, we present the first map of para-D2H + , towards the prestellar core IRAS 16293E, and compare it with archival data of ortho-H2D + , dust emission and N2H + . Section 2 presents the observations. In Sect. 3, we analyse the emission of both deuterated isotopologues of H + 3 , and compare them to dust absorption and emission, and N2H + emission. We present a 1D modified Monte-Carlo model in Sect. 4 to quantify the abundance of both isotopologues, and discuss the various results in Sect. 5. Our conclusions are given in Sect. 6.", '2. Observations': "We observed para-D2H + (JK aKc : 110-101) at 691.6604434 GHz (Jusko et al. 2017) with the Swedish-ESO PI Instrument for APEX (SEPIA660) receiver on the APEX 12-m telescope equipped with Fast Fourier Transform (FFT) backends, each of 65536 channels of 61 kHz width. The observations were performed in several runs on ESO time (3 / 5 / 2021, 28-29 / 6 / 2021, project E-0105.C-0251A) and MPI time (2426 / 3 / 2022, 1 / 4 / 2022, 4 / 4 / 2022, project M-0109.F-9501C-2022). We also included observations with the 7 pixel CHAMP + camera taken on 7 / 9 / 2010 (one pointing, project M-085.F-00162010). The weather was exceptionally good (precipitable water vapour, PWV = 0.2-0.3 mm) during most of the SEPIA660 observations, and very good (PWV ≈ 0.5 mm) for the CHAMP + observations. Apart from the seven positions observed with CHAMP + and two positions observed with SEPIA660 in May 2021, which were observed in position switching mode, all observations were done in the OTF mode to cover a 50 '' × 50 '' region (June 2021) extended to 50 '' × 72 '' in 2022 as the preliminary results indicated an unexpected emission to the south. The various reference positions are summarized in Table A.1. The total observing time is about 25 hours. The angular resolution is ∼ 9 '' . The pointing was checked every hour, mainly on the nearby object RAFGL1922. The stability of the pointing is better than 3 '' . We used CLASS 1 to subtract first-order baselines computed on the velocity range -24 to + 31 km s -1 , and smoothed the data (spatially, using XY\\_MAP) to 14 '' and (spectrally) to 244 kHz ( ≈ 0.1 km s -1 ) to improve the sensitivity. The peak signalto-noise ratio (SNR) is 7 in brightness temperature and 12 in integrated intensity (in the range 2.81-4.18 km s -1 ). The rms noise level in integrated intensity is 6.7 mK km s -1 (T ∗ a scale) towards the centre of the map and reaches ≈ 11 mKkms -1 at the edges. We assumed a main beam e ffi ciency of η MB ≈ 0.48 ± 0.048 for CHAMP + observations 2 and η MB ≈ 0.46 ± 0.046 for SEPIA660 observations (Perez-Beaupuits, priv. comm.) which were applied to each set before merging them. \nWe retrieved two sets of unpublished ortho-H2D + (JK aKc : 110-111) observations at 372.421340 GHz (Jusko et al. 2017) from the James Clerk Maxwell 15-m Telescope (JCMT) archive 3 , from projects M07AU29 (2007) and M09AU01 (2009). They were obtained with the 16 pixel Heterodyne Array Receiver Program (HARP) receiver with the Auto Correlation Spectral Imaging System (ACSIS) backend (61 kHz sampling). The atmospheric opacity at 225 GHz ranged from 0.052 to 0.034 \n(M07) and from 0.04 to 0.02 (M09). The M07 data consists of a 600-s long single pointing of 15 pixels (one corner detector was dead) towards the core centre. The M09 data (14 effective pixels) cover 232 '' × 113 '' , one square jiggle map centred on the core (1200 seconds integration) with an adjacent jiggle map to the East (300 seconds integration). The M09 central map was presented in our conference poster (Pagani et al. 2016). The original angular resolution of 13.2 '' was smoothed to 14 '' (after third-order baseline subtraction on a baseline extending from -21 to + 28 km s -1 , resampling and smoothing with CLASS / XY\\_MAP). The frequency sampling has been Hanningsmoothed to 122 kHz ( ≈ 0.1 km s -1 ). We assumed a main beam e ffi ciency of η MB ≈ 0.6 4 . The peak SNR is 11 in brightness temperature and 19 in integrated intensity (in the range 2.924.28 km s -1 ). The average integrated intensity rms noise is 23 mKkms -1 for the central map, twice as high for the eastern map of which we only use the edge to reach the emission limit of the core. The same observing run also provided us with N2H + ( J :4 3) observations at 372.6734633 GHz (Cazzoli et al. 2012), representative of many of the species observed towards the PSC (Lis et al. 2016, Kahle et al. 2023). Both D2H + and H2D + data sets are resampled on a 7 '' basis. \nWe also used the JCMT SCUBA-II 850 µ m continuum observations obtained and published by Pattle et al. (2015). We refer the reader to that paper for a detailed description of the observations. We retrieved Spitzer IRAC data from the archive 5 . As far as we know, the PSC image from these observations has never been published apart from our conference poster (Pagani et al. 2016).", '3. Analysis': "Figure 1 shows the map of para-D2H + (left panel). This is the first time that a map of para-D2H + is presented. The doubly deuterated ion follows precisely the IRAC-1 absorption image seen at 3.6 µ m (central panel). Since D2H + is expected to peak in the central coldest part of clouds, and since the total absence of stars at 3.6 µ m reveals high extinction (typically AV ≥ 100 mag), these two maps do trace the centre of the IRAS16293E PSC, while the dust emission map peak (red cross), which is very sensitive to the temperature increase due to the outflow collision with the PSC edge, is displaced towards the impact point. Therefore the dust emission map does not reveal the true cold centre of the PSC. In the right panel, the upper part of the IRAC4 image of the PSC also seen in absorption is partially hidden (reduced contrast) by PAH emission in relation with another outflow mostly seen in the IRAC images. In the same panel, N2H + ( J :4 - 3) contours (in orange) also point towards the impact point, known as the DCO + peak (e.g. Lis et al. 2016). \nThe left panel of Fig. 2 shows the ortho-H2D + emission superimposed on the IRAC-1 image. It is again clearly shifted to the East compared to the dust emission peak and N2H + ( J :4 - 3) line emission peak (marked by the red cross) as we already reported in Pagani et al. (2016) and has been confirmed by Kahle et al. (2023). It is remarkable that both dust and N2H + emissions peak at the same place, probably revealing the exact position of the outflow impact on the PSC. On the other hand, like for para-D2H + emission, dust absorption as seen by IRAC images, is correlated with ortho-H2D + emission. Dust in absorption is not sensitive to temperature e ff ects and this explains the \nFig. 1. Continuum and line emission images of IRAS 16293E. Left: Colour map of para-D2H + integrated intensity emission with white contours (colour scale and white contours from 0.01 to 0.07 by 0.01 K km s -1 plus a contour at 0.075 K km s -1 ; contour spacing of 0.01 K km s -1 = 1.5 σ ) Centre: IRAC-1 image at 3.6 µ m. White contours trace para-D2H + as in the left panel. Red contours trace the SCUBA-II 850 µ m emission (20 to 160 by 20 [ ≈ 10 σ ] MJy sr -1 ). Right: IRAC-4 image at 8.0 µ m. White contours trace para-D2H + as in the left panel. Orange contours trace the N2H + ( J :4 - 3) integrated intensity (0.3 to 2.3 by 0.4 [ = 12 σ ] K km s -1 ). The yellow cross marks the centre of the D2H + emission ( α J2000: 16 h 32 m 30.51 s , δ J2000: -24 · 28 ' 53.7 '' , see also Fig. 2). The red cross marks the peak position of the dust and N2H + ( J :4 - 3) emissions. The black cross (left panel) marks the position of the observation of D2H + and H2D + by Vastel et al. (2004). The yellow arrow traces the direction of the outflow from the IRAS16293-2422 protostellar system and the red arrow the outflow from WLY 2-69. The grey and white disks represent the JCMT and APEX beams after smoothing to 14 '' . \n<!-- image --> \nFig. 2. Left: IRAC-1 image at 3.6 µ m. White contours trace orthoH2D + . The filled grey contour marks the strongest emission (contours from 0.10 to 0.34 by 0.06 [ = 2.6 σ ] K km s -1 ). Right: para-D2H + and ortho-H2D + maps. Same map of para-D2H + (black contours) as in Fig. 1 superimposed on the ortho-H2D + map (colour scale, plus white contours as in left panel). The yellow, black, and red crosses are as in Fig. 1. D2H + , H2D + , and N2H + spectra along the cut traced by the dashed line are displayed in Fig. 3 \n<!-- image --> \ndecorrelation between dust absorption and emission. The dust absorption reveals the highest extinction part of the PSC and as expected this coincides with the emission peak of ortho-H2D + and para-D2H + . Another noticeable feature is the horse-shoe shape of the ortho-H2D + emission surrounding the PSC centre (the topmost grey-filled contour in Fig. 2, left side). This \northo-H2D + horse-shoe shape clearly surrounds the emission peak of para-D2H + as shown in the right panel of Fig. 2. This indicates that the ortho-H2D + emission is weaker towards the centre of the PSC. Since it is only the top contour of ortho-H2D + emission which shows this feature, it means that only the inner part of the PSC shows a diminished emission and therefore a diminished ortho-H2D + abundance, compared to the outer layers. The intensity di ff erence between the ortho-H2D + peak emission at a right ascension o ff set of + 7 '' from the centre and the central position is 66 mK km s -1 , with a statistical noise of 29 and 21 mKkms -1 , respectively. The di ff erence is 2.3 σ , but if the ortho-H2D + abundance had been constant, the emission would have increased towards the centre, like for para-D2H + , instead of decreasing. Therefore the di ff erence with respect to a situation where the abundance would be constant is much higher than 2.3 σ and statistically significant. In Sect. 4, we will model the case of a constant ortho-H2D + abundance to quantify the expected di ff erence. \nThe pointed observations of ortho-H2D + and para-D2H + that were previously reported by Vastel et al. (2004) towards the position marked by a black cross in Figs. 1 & 2 are found to be inconsistent with our more sensitive mapping observations towards the same position. For para-D2H + , they reported a peak temperature of 0.34 K ( ± 0.077 K ?) 6 and a linewidth of 0.29 ± 0.07 km s -1 , to be compared with our 83 ± 17 mK and a linewidth of 0.48 ± 0.07 km s -1 ; for ortho-H2D + , they report a peak temperature of 1.31 K ( ± 0.22 K ?) 6 and a linewidth of \n0.36 ± 0.04 km s -1 , to be compared with our 0.31 ± 0.03 K and a linewidth of 0.61 ± 0.04 km s -1 . For a gas temperature in the range 16-25 K from former studies (Loinard et al. 2001, Castets et al. 2001, Lis et al. 2002, Stark et al. 2004, Lis et al. 2016) that applies to the DCO + emission peak region, the thermal linewidth alone is expected to be between 0.38 and 0.48 km s -1 for paraD2H + and between 0.43 and 0.54 km s -1 for ortho-H2D + , which are both inconsistent with the small linewidths reported by Vastel et al. (2004).", '4. Radiative transfer modelling of para-D 2 H + and ortho-H 2 D +': "Modelling the abundance of both H + 3 isotopologues is needed to quantify our findings. The shape of the PSC is not symmetrical but still relatively simple. There is however no easy way to derive its density and temperature structures, due to the complexity of its environment. Star counts or reddening measurements are impossible towards the centre of the core due to the lack of stars even in the sensitive Spitzer data (Fig. 1). 8 µ m continuum absorption measurements (Bacmann et al. 2000, Lefèvre et al. 2016) are hampered by the superposition of nebulosity emission from the IRAS 16293-2422A / B outflows and possibly some Polycyclic Aromatic Hydrocarbon (PAH) emission. All emissions, whether dust or molecular, are strongly biased by the temperature gradient induced by the outflow impact on the PSC, and the density profile is also possibly perturbed on that side, away from a simple Plummer-like profile (Whitworth & Ward-Thompson 2001). To disentangle density, temperature, and abundances for molecules, or dust properties for continuum emission, a thorough analysis in 3D of the core must be performed. This is beyond the scope of the present paper and will be addressed in the future. Here, we present a model combining three partially di ff erent one-dimensional (1D) spherical models. \nThe first step is to build 1D gas density and temperature structures of the PSC. The only tracer from the edge to the centre of the PSC is dust, especially in emission around 300 GHz but its properties are not accurately known, and since the temperature and density profiles cannot be easily retrieved from the dust emission alone (Pagani et al. 2015), we have constructed the core structure from the following considerations: \n- -except for the south-western extension, the emission of both para-D2H + and ortho-H2D + is relatively round and symmetrical. We consider that the bulk of the original core, before the outflow impacts its western side, is still present and close enough to a spherical object.\n- -The eastern part of the core is unperturbed and its H2 density can be described with a Plummer-like profile (Tafalla et al. 2002): \nn(H2)(r) = 1 . 5 × 10 6 1 + ( r 3 × 10 16 ) 2 . 5 cm -3 . (1) \nThe peak column density is 1.3 × 10 23 cm -2 , i.e. a visual extinction of AV ≈ 140 mag, compatible with the total absence of stars seen through the core even in the Spitzer MIR maps. It is also consistent with the dust emission at 850 µ mtowards the same spot (which is a weak constraint given the large uncertainty on the dust emissivity, Demyk et al. 2017a,b). \nWe also apply this profile to the West side. Using a di ff erent density profile would locally change the abundance profile of the ions but would not impact the results for the central region of the core on first approximation. \n- -In the centre of cold cores, temperatures are usually in the 8 - 12 K range (Caselli et al. 2008). We thus considered central temperatures of 8, 10, and 12 K and imposed a temperature of 12 K in the external layers on the eastern side (the exact value has no impact in the 11-16 K range).\n- -The N2H + ( J :4 - 3) emission peaks about 20 '' west of the PSC centre in a place that is warmed by the outflow impact. Former studies proposed temperatures between 16 and 25K (Loinard et al. 2001, Castets et al. 2001, Lis et al. 2002, Stark et al. 2004, Lis et al. 2016), and we display results for the median value of 20K but also checked the 16 and 25 K cases.\n- -The para-D2H + spectra from -14 '' to -28 '' (and possibly -35 '' ) in Fig. 3 are slightly stronger than their symmetrical ones eastward, and since the position of the peak of N2H + emission is in the middle of this range, we set the temperature increase at -20 '' .\n- -We divided the sphere in three parts: each time we ran a 1D spherical model but we extracted the emission from a fraction of the sphere only. We stitched the three extracted portions together and convolved the output to 14 '' . This is graphically explained in Appendix B. The spherical model is discretized into 20 layers of 1.14 × 10 16 cm thickness. This corresponds to 5.4 '' thickness per layer for a distance of 141 pc. The total size is 216 '' , which completely covers the core. \nThough the warm part could be di ff erent in its density profile due to the possible compression from the outflow, if any (the outflow could be grazing the PSC and have a limited compression e ff ect only), it is not possible to assess the actual profile at this stage until we have built a 3D model to explore that possibility. However the warm part is far enough from the horse-shoe central region and has no influence on the modelling of the cloud centre, as our tests have shown. Similarly, the impact of the temperature between 16 and 25 K was found to be negligible. \nThe modelled density, temperature, and isotopologue abundance profiles are displayed in Fig. 4. We adjusted the abundance, non-thermal linewidth and radial velocity to fit the widths and strengths of the ortho-H2D + and para-D2H + lines (see synthetic spectra in Fig.3). Linewidths show some inconsistencies as revealed by Gaussian fits to the observed spectra (for more details, see Appendix C). This pseudo-1D modelling is performed with a 1D Monte-Carlo code adapted from Bernes (1979)'s radiative transfer program. For the non-LTE modelling of the two ions, we used the inelastic collisional rate coe ffi cients from Hugo et al. (2009). The abundances of both ions relative to H2 are displayed in the central panel of Fig. 4 while their ratio is displayed in the bottom panel. \nThe modelled lines are displayed in Fig. 3 and the same fit is traced in terms of integrated intensity in Fig. 5, where the centre of the ortho-H2D + horse-shoe shape is clearly visible. In Fig. 5, the observed intensities (full lines, with their individual error bars) are reproduced within 1 σ by the model (dashed line, same colour). Because the density keeps increasing towards the centre, the ortho-H2D + intensity drop needs a strong abundance decrease to explain the decorrelation between the two (unlike para-D2H + , the intensity of which keeps increasing). Since the para-D2H + abundance is found to be almost constant in the centre of the cloud (between 1.15 and 1.7 × 10 -10 , see below), we tested the case of a constant abundance for ortho-H2D + . The predicted intensity is shown in Fig. 5 with a dashed orange line. Towards the central position, the di ff erence reaches 7 σ and a uniform abundance of ortho-H2D + is clearly excluded. \nFig. 3. From top to bottom: N2H + ( J :4 - 3), ortho-H2D + (JK aKc : 110-111) and para-D2H + (JK aKc : 110-101) spectra in black, extracted along the cut displayed in Fig. 2, with best-fit model of ortho-H2D + and para-D2H + overlaid in red. The three weakest para-D2H + spectra have been smoothed to 0.21 km s -1 . The velocity axis in each box runs from 1 to 6 km s -1 . Green dashed lines mark the 3 σ individual noise levels and the systemic velocity (3.6 km s -1 ). \n<!-- image --> \nFig. 4. Pseudo-1D model profiles of the core. Upper box: n(H2) Plummer-like profile in black. In red, temperature profile treated separately for the East (left) and the West (right) parts of the core, with three di ff erent minimum temperatures (10 K in plain line, 8 and 12 K in dashed lines). Middle box: ortho-H2D + (black) and para-D2H + (red) abundance profiles relative to H2 and computed separately for each side for the 10K model, error bars are 1 σ deviation from best model. Arrows mark upper limits. In particular, for ortho-H2D + in the centre of the core, the dotted line represents the abundance to stay just below 1 σ di ff erence with the observations, and the upper error bar represents a di ff erence of 2 σ with the observations. Lower box: para-D2H + / orthoH2D + ratio for the 10K model. Error bars are computed by taking the opposite error bars of both species when available, otherwise lower or upper limits are indicated by arrows. The long dashed and dash-dotted lines mark a ratio of 1 and 2, respectively, 2 being the upper limit of the models of Bovino et al. (2021) at 10 K. \n<!-- image -->", '5. Discussion': "The H2D + / D2H + contrast reported in Sect. 3 is expected from chemical evolution of the deuteration of the H + 3 ion (Roberts et al. 2003, Walmsley et al. 2004) that indicates that H + 3 can be converted to H2D + , itself converted to D2H + and finally to D + 3 , the latter being potentially the most abundant ion in the end. The e ff ect is maximum at the peak density because the higher the density, the faster the chemical reactions, other parameters being equal. This is the first time that we directly witness the intermediate step of this transformation (H + 3 and D + 3 are not observable in dense clouds). This confirms that the original centre of the PSC is not coincident with the peak emission of species such as N2H + , N2D + , and DCO + , but 22 '' apart, which di ff ers from what is usually found in other objects (e.g., Tafalla et al. 2002, Pagani et al. 2007). This must result from the temperature enhancement produced by the molecular outflow hitting the PSC. This heating possibly explains the similar extent of H2D + and D2H + towards the working surface of the PSC, in the southwest direction. It may also explain the small secondary peak of para-D2H + as a result of the temperature increase compensating the drop of abundance, although further work is needed to reach a firm conclusion. \nFig. 5. Observed intensities along the horizontal cut (full lines with their error bar, corresponding to the spectra from Fig. 3) and corresponding model intensities (dashed lines). For ortho-H2D + , we present an additional model (orange dashed line) for which the abundance is kept constant across the core. Para-D2H + (in red) observations and model have their own intensity axis, also in red, on the right side of the plot. \n<!-- image --> \nThe column densities of both species derived from the modelling in Sect. 4 are nearly equal towards the centre, N(orthoH2D + ) peaks at 2.1- 4.8 × 10 13 cm -2 (at 12 K - 8 K), and N(para-D2H + ) at 0.94 - 4.2 × 10 13 cm -2 . The maximum paraD2H + / ortho-H2D + column density ratio is 1.9 + 2 . 1 -1 . 2 , (at 10 K, ranging from 1.1 at 12 K to 3.1 at 8 K) 7 . However, the derived abundance profiles yield a stronger contrast in the inner shells as demonstrated in Fig. 5. For the two most inner shells (total radius of 10.8 '' ), the abundance of ortho-H2D + can be set to zero while the line of sight emerging intensity computed by the model is still too high (but less than 1 σ away from the observations). Since we asymptotically reach a di ff erence of + 0.8 σ for the central position by decreasing the abundance by almost two orders of magnitude, we selected the abundance that provides exactly 1 σ di ff erence as the reference and computed the upper limit of this abundance by setting the error to 2 σ . There is no lower limit since we cannot make the model weaker than the observations for this central spectrum. This is because the large ortho-H2D + abundance needed to reproduce the intensity of the line towards the walls of the horse-shoe cavity is also su ffi cient to reproduce the intensity of the line for the two sightlines across the horse-shoe hole, the same walls probably standing in front and in the back of the cavity, the observer having no privileged point of view (a situation similar to the case of L1506C, Pagani et al. 2010). In the centre of the cavity we find a very high para-D2H + / ortho-H2D + abundance ratio of ∼ 20. To check whether this ratio could be compatible with known models when taking into account the uncertainties of the line modelling, we computed a conservative lowest value of the uncertainty range of this ratio. We estimated an upper limit to the abundance of ortho-H2D + at 2 σ above the observations and found abundances of 1.8 × 10 -11 at 12 K, 3.05 × 10 -11 at 10 K, and 4.0 × 10 -11 at 8 K. We also computed a lower abundance for para-D2H + by departing from the best fit abundance by -1 σ (X[para-D2H + ] = 6.95 × 10 -11 , 1.3 × 10 -10 , and 3.3 × 10 -10 at 12, 10, and 8K respectively). The para-D2H + / ortho-H2D + ratio lowest limit ranges from 3.9 to 8.3 (at 12 K - 8 K). \nThis para-D2H + / ortho-H2D + abundance ratio is high and beyond all predictions made by various models whether 0D (Pagani et al. 2009, Parise et al. 2011a), 1D (Pagani et al. 2013) or 3D (Bovino et al. 2019, 2021) which reach values around 1-2 in the densest parts of the cores, for a temperature of 10 K. Parise et al. (2011a) also found a too high ratio when analysing the HMM1 prestellar core in the LDN 1688 star forming region with a low temperature (7 K) but could reconcile the model and observations at 12 K. Here, the 12 K value is possibly marginally consistent with the Bovino models if we interpolate by eye between their 10 and 15 K cases (Bovino et al. 2021, their Fig. B.2). However, clouds with AV ≥ 100 mag. should be very cold in their centre, below 10 K following the models of Zucconi et al. (2001) and observations of Crapsi et al. (2007) and Pagani et al. (2007). These observations coupled to the present simple 1D modelling, if confirmed by a more accurate 3D modelling, either suggest that current chemical models are not yet close enough to reality to trace correctly the deuteration evolution of the tri-hydrogen ion in the cold medium or that the reaction rate coe ffi cients or inelastic collisional rate coe ffi cients of the H + 3 family with H2 and HD from Hugo et al. (2009), which were computed semiclassically, are not accurate enough and need to be improved. We found that increasing the collision rate coe ffi cients of paraD2H + with H2 by a factor 6 or increasing the ortho-H2D + + HD → para-D2H + + para-H2 reaction rate coe ffi cient by a factor 3 would change the para-D2H + / ortho-H2D + ratio by a factor 2. Other similar changes can also be considered.", '6. Conclusions': "We present in this article the first map of para-D2H + in the interstellar medium, obtained with the APEX telescope toward the prestellar core IRAS 16293E. The main conclusions of this work are: \n- 1. Maps of para-D2H + , and to a lesser degree, of ortho-H2D + reveal the position of the core centre before it was compressed and heated by at least one outflow coming from nearby YSOs. The new reference position for the core centre is α J2000: 16 h 32 m 30.51 s , δ J2000: -24 · 28 ' 53.7 '' .\n- 2. We derived peak column densities of ∼ 1-3 × 10 13 cm -2 for both ions (at 10 K).\n- 3. We derived a conservative lower limit of the paraD2H + / ortho-H2D + abundance ratio in the centre of the PSC in the range 3.9-8.3 at 12 - 8 K while the core extinction (AV ≥ 100 mag) is indicative of a central temperature below 10 K. This ratio is higher than what published chemical models predict for temperatures below or around 10 K.\n- 4. Temperature e ff ects induced by the outflow impact at the hotspot make line emission of most species bright. Therefore an analysis limited to the hotspot could miss the real peak abundance of species, in particular ND3 and other deuterated species, the emission of which peaks away from the orthoH2D + and para-D2H + abundance peaks. \nTo confirm the present model, a thorough modelling of the core to disentangle density and temperature variations induced by the outflow impact needs to take into account all available information (line emissions with several transitions per species, dust emission and absorption lower limit) in a 3D model, though the main di ffi culty of this model will be to constrain the line of sight distribution of all parameters. \nThe high para-D2H + / ortho-H2D + ratio measured in IRAS 16293E could suggest, if confirmed, that the inelastic collisions \nor reaction rate coe ffi cients of the H + 3 isotopologues need to be revised or that the chemical models need to be refined in order to improve our understanding of the deuteration process in star forming regions. This might have an impact on the deuteration chemical clock used to study many star forming regions. \nAcknowledgements. Based on observations with the Atacama Pathfinder EXperiment (APEX) telescope. At the time of the first observations presented in this paper, APEX was a collaboration between the Max Planck Institute for Radio Astronomy, the European Southern Observatory, and the Onsala Space Observatory. This research has made use of NASA's Astrophysics Data System Abstract Service. It has also made use of observations from Spitzer Space Telescope and from the NASA / IPAC Infrared Science Archive, which are operated by the Jet Propulsion Laboratory (JPL) and the California Institute of Technology under contract with NASA. This research makes a large use of the CDS (Strasbourg, France) services, especially Aladin (Bonnarel et al. 2000), Simbad and Vizier Ochsenbein et al. (2000). We are indebted to C. de Breuck for the observations in the ESO time and to all the observers at APEX. 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S., Sandell, G., et al. 2004, ApJ, 608, 341\n- Tafalla, M., Myers, P. C., Caselli, P., Walmsley, C. M., & Comito, C. 2002, ApJ, 569, 815\n- Vastel, C., Phillips, T. G., & Yoshida, H. 2004, ApJ, 606, L127\n- Walmsley, C. M., Flower, D. R., & Pineau des Forêts, G. 2004, A&A, 418, 1035 Whitworth, A. P. & Ward-Thompson, D. 2001, ApJ, 547, 317, aDS Bibcode: 2001ApJ...547..317W\n- Wootten, A. & Loren, R. B. 1987, ApJ, 317, 220 \nZucconi, A., Walmsley, C. M., & Galli, D. 2001, A&A, 376, 650', 'Appendix A: Reference positions for the APEX para-D 2 H + observations': 'Di ff erent reference positions were used for the various projects with APEX towards IRAS 16293E. We list them in Table A.1', 'Appendix B: Model building': "Fig. B.1. Steps to build an asymmetrical model from 1D spherical models. Based on a single density profile (eq. 1), we compute two models with di ff erent temperature profiles in the outer parts (either 12 or 20 K, top row, see Fig. 4). We apply two di ff erent abundance profiles (central row) for each ion for the East and West parts, the West part is computed for both temperature profiles. We extract the parts corresponding to each region of interest (sampled with a 1 '' resolution) and we assemble them in a cube (RA, Dec, Frequency, bottom row left). Finally, we convolve the cube with a 14 '' half-power beamwidth Gaussian beam to compare to the observations (bottom row right). We extract the spectra along the dash-dot line and compare them to the observations (see Fig. 3). \n<!-- image --> \nThe model is based on a single density profile (eq. 1) and we computed the emission on a 1 '' step basis (model diameter of 216 '' ) for various temperature and abundance profiles. \n- 1. for positive o ff sets (East side) we used the pristine model and extracted the emission from the half sphere towards the East.\n- 2. for o ff sets between 0 and -15 '' , we used the same model but on the West side with a di ff erent abundance profile except in the central shell which is forcibly common to both sides, being unresolved, (the centre being common to the East and West sides, we technically use only the West side and therefore the East stops at 5 '' , and the West goes from -15 to + 5 '' )\n- 3. for the external heated part, selected beyond -15 '' from the centre, we used a 1D model with warm outer layers beyond a radius of 20 '' in the Western part. \nFor the ortho-H2D + case shown in Fig. B.1, the strong dissymmetry is due to the fact that the emission is extended to the East \nand not as much to the West. The ortho-H2D + abundance profiles (Fig. 4, central panel) reflect this di ff erence by decreasing beyond -30 '' on the Western side, while it keeps increasing beyond + 30 '' on the Eastern side. One should note that the warm layers beyond -20 '' contribute to positions closer to the cloud centre, because being a 1D onion model, they are present on the line of sight of these closer positions in front and behind. The line of sight crosses a cold layer inserted between two hot layers. This could be justified by the outflow interacting with the surface of the PSC rather than ploughing its way to the centre, since the ortho-H2D + and para-D2H + lines do not seem to be kinematically perturbed in this warmer region. This problem will be examined in detail in a future work. The model is not representative of the emission towards the North or the South of the PSC.", 'Appendix C: Linewidth and Gaussian fits of the D 2 H + APEX and H 2 D + JCMT spectra': "The lines of both ions are optically thin (peak opacity of 0.6 and 0.7 for para-D2H + and ortho-H2D + , resp.), and in first approximation consistent with a Gaussian shape. We have therefore adjusted a Gaussian to each of the spectra. These Gaussian fits are displayed in Fig. C.1. To adjust the Monte-Carlo model, we modified the isotopologue abundance profile across the core to minimize the integrated intensity di ff erence between the observations and the modelled line. Since the lines are optically thin, the exact shape of the line has little impact on its integrated intensity but of course, we take care that the linewidth be compatible with the observations. However, from the Gaussian fit, we found that the linewidth of the two isotopologues are not equal. At the edges, this is probably a noise limitation (e.g. the 21 '' para-D2H + spectrum), but in the centre, the para-D2H + spectra at o ff sets 0, -7 '' , and -14 '' are clearly larger than the ortho-H2D + ones, e.g., for the 0 position: \n- -δ V(para-D2H + ) = 0.658 ± 0.074 km s -1\n- -δ V(ortho-H2D + ) = 0.524 ± 0.037 km s -1 \nH2D + being lighter than D2H + , one would expect the former to display lines slightly wider than the latter for similar thermal turbulence or macroscopic movements. The reverse could indicate that the core inside the horse-shoe, depleted in H2D + (see Sect. 3), has a signature of collapse of its own traced by D2H + alone, but we did not manage to reproduce this di ff erence with our model. The volume is probably too small to dominate the linewidth formation without perturbing the H2D + linewidth too. The other possibility is that this di ff erence be mostly due to noise (1.8 σ di ff erence) and we settled the model linewidth to be intermediate between these two values, making the D2H + modelled line slightly stronger than the Gaussian fit but narrower, and conversely for H2D + . \nTable A.1. Reference positions for the APEX observationsNotes. ( a ) O ff sets are with respect to equatorial coordinates except if marked with (H) for horizontal wobbler switching \nFig. C.1. JCMT spectra of ortho-H2D + J KK ' :110-111 (top row, velocity axis from 1 to 6 km s -1 ), and APEX spectra of para-D2H + J KK ' :110-101 (middle row, velocity axis from 1 to 6 km s -1 ) across the cloud (the position of the cut is indicated in Fig. 2). Gaussian fits are overlaid in red and green, respectively. The same Gaussian fits with same colour code are reproduced in the bottom row, after normalization to the peak temperature for the + 7 '' spectrum, for a better comparison of the ratio variation and linewidth di ff erences between the two isotopologues. \n<!-- image -->"}
2024A&A...691A.192Z	GRS 1716249 is a stellarmass black hole in a lowmass Xray binary that underwent a giant outburst in 201617. In this paper we use simultaneous observations from the Hard Xray Modulation Telescope InsightHXMT and the Nuclear Spectroscopic Telescope Array NuSTAR to determine its basic parameters. The observations were performed during the softest part of the outburst and the spectra show clear thermal disk emission and reflection features. We fit the Xray energy spectra using the joint fitting method of the continuum and reflection components with the kerrbb2 relxill model. Since there is a possibility that the distance to this source was previously underestimated we used the latest distance parameter of 6.9 kpc in our study in contrast to previous works where the distance was set at 2.4 kpc. Through a spectral fitting of the black hole mass at 6.4 MSUBSUB we observe a strong dependence of the derived spin on the distance aSUBSUB 0.972SUB0.005SUBSUP0.004SUP at an assumed distance of 2.4 kpc and aSUBSUB 0.464SUB0.007SUBSUP0.016SUP at an assumed distance of 6.9 kpc at a confidence level of 90. When considering the uncertainties in the distance and black hole mass there will be a wider range of spin with aSUBSUBlt 0.78. The fitting results with the new distance indicate that GRS 1716249 harbors a moderate spin black hole with an inclined i 40 50 accretion disk around it. Additionally we have also found that solely using the method of reflection component fitting while ignoring the constraints on the spin from the accretion disk component will result in an extremely high spin.	2024-11-01T00:00:00Z	['10.1051/0004-6361/202449646', '2024A&A...691A.192Z', '2024arXiv240911927Z', '10.48550/arXiv.2409.11927', 'arXiv:2409.11927']	['accretion', 'accretion disks', 'black hole physics', 'stars: individual: GRS 1716-249', 'X-rays: binaries', 'Astrophysics - High Energy Astrophysical Phenomena']	Revised spin for the black hole in GRS 1716249 given a new distance determination	2,024	189	0.48	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.11927.pdf	{'A Revised Spin of the Black Hole in GRS 1716-249 with a New Distance': 'S. J. Zhao 1 , 2 , L. Tao 1 ⋆ , Q. Q. Yin 1 ⋆⋆ , S. N. Zhang 1 , 2 , R. C. Ma 1 , 2 , P. P. Li 1 , 2 , Q. C. Zhao 1 , 2 , M. Y. Ge 1 , L. Zhang 1 , J. L. Qu 1 , S. Zhang 1 , X. Ma 1 , Y. Huang 1 , J. Q. Peng 1 , 2 , and Y. X. Xiao 1 , 2 \n- 1\n- Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China\n- 2 University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China \nReceived ; accepted', 'ABSTRACT': 'GRS 1716-249 is a stellar-mass black hole in a low-mass X-ray binary that underwent a gaint outburst in 2016 / 17. In this paper we use simultaneous observations of Insight-HXMT and NuSTAR to determine its basic parameters. The observations were performed during the softest part of the outburst, and the spectra show clear thermal disk emission and reflection features. We have fitted the X-ray energy spectra using the joint fitting method of the continuum and reflection components with the kerrbb2 + relxill model. Since there is a possibility that the distance to this source was previously underestimated, we use the latest distance parameter of 6.9 kpc in our study, in contrast to previous work in which the distance was set at 2.4 kpc. Through spectral fitting of fixing black hole mass at 6.4 M ⊙ , we observe a strong dependence of the derived spin on the distance: a ∗ = 0 . 972 + 0 . 004 -0 . 005 at an assumed distance of 2.4 kpc and a ∗ = 0 . 464 + 0 . 016 -0 . 007 at an assumed distance of 6.9 kpc, at a confidence level of 90%. If considering the uncertainties in the distance and black hole mass, there will be a wider range of spin with a ∗ < 0.78. The fitting results with the new distance indicate that GRS 1716-249 harbors a moderate spin black hole with an inclined ( i ∼ 40 -50 · ) accretion disk around it. Additionally, we have also found that solely using the method of the reflection component fitting but ignoring the constraints on the spin from the accretion disk component will result in an extremely high spin. \nKey words. X-ray binaries - black hole physics - accretion, accretion disk - stars: individual: (GRS 1716-249)', '1. Introduction': 'Spin is an important parameter of a black hole (BH), which, together with its mass, determines the spacetime around it and influences the accretion of matter and the formation of jets (for a review, see Reynolds 2021). Generally, the BH spin a ∗ is defined as a ∗ = Jc / GM 2 , where J represents the angular momentum of the black hole, G stands for the gravitational constant and c denotes the speed of light. The magnitude of the spin determines the radius of the innermost stable circular orbit (ISCO), R ISCO. A larger value of a ∗ corresponds to a smaller R ISCO. For example, when a ∗ = 0 (Schwarzschild black hole), R ISCO is equal to 6 R g (where R g = GM / c 2 is called the gravitational radius); when a ∗ = 1 (extreme Kerr black hole), R ISCO is equal to 1 R g. Currently, there are two main methods for measuring black hole spin through spectral fitting: continuum fitting (Zhang et al. 1997; McClintock et al. 2014) and reflection component fitting (Fabian et al. 1989; Reynolds & Fabian 2008; García et al. 2014, 2018b). Both methods require the common prerequisite that the inner edge of the accretion disk is located at the position of the ISCO ( R in = R ISCO). \nAccording to the standard accretion disk theory (Shakura & Sunyaev 1973; Novikov & Thorne 1973), the spin of a black hole a ff ects the temperature of the inner disk, which in turn affects the thermal emission of the accretion disk. The method of continuum fitting is precisely used to constrain the spin of the black hole by fitting the spectrum of the disk. However, the use \nof this method requires that the distance to the source ( D ), the inclination of the accretion disk ( i ), and the mass of the black hole ( M BH) are known (the reasons can be found in McClintock et al. 2011). Two commonly used models for fitting the disk component are kerrbb (Li et al. 2005) and kerrbb2 (McClintock et al. 2006), which consider the full relativistic e ff ects. The key fitting parameters of these models include the black hole spin and the mass accretion rate ( ˙ M ). \nThe typical reflection features are the iron line (at ∼ 6-7 keV) and the Compton hump (at ∼ 20-30 keV). The iron line emitted from the inner disk close to the black hole is influenced by the Doppler e ff ect, relativistic beaming e ff ect, and gravitational redshift, resulting in a broadened and distorted profile (Reynolds & Nowak 2003; Miller 2007). The gravitational redshift mainly affects the red wing of the iron line. The larger the black hole spin, the further the red wing extends to lower energies. This is because a larger spin allows the inner disk to be closer to the black hole and experience a stronger gravitational potential well. The core of the reflection component fitting is to fit the profile of the iron line. This method does not depend on M BH and D and can also be used to constrain i . \nIn theory, there are billions of stellar-mass black holes in the Milky Way galaxy (Brown & Bethe 1994). However, there are only around twenty X-ray binary systems containing a dynamically confirmed black hole (Remillard & McClintock 2006; Özel et al. 2010; Corral-Santana et al. 2016). GRS 1716-249 is one of them. This source was discovered in 1993 by the CGRO / BATSE and Granat / SIGMA telescopes (Harmon et al. 1993; Ballet et al. 1993). After about 23 years of quiescence, GRS 1716-249 had \nanother outburst detected by MAXI on December 18, 2016 (Negoro et al. 2016). During this outburst, when the source was in the hard intermediate state, it is believed that the accretion disk was a standard disk with the inner edge located at the ISCO (Bassi et al. 2019). Additionally, the spectrum showed significant power-law (PL) components from the corona, as well as prominent reflection features (broad iron line and Compton hump) (Bassi et al. 2019; Tao et al. 2019). Therefore, this source presents an ideal case for measuring the black hole spin because both continuum fitting and reflection component fitting methods can be combined to simultaneously constrain the parameters of the black hole. \nThe system parameters of this source, especially the distance, have not been well constrained. della Valle et al. (1994) estimated the distance to be 2.2-2.8 kpc. Masetti et al. (1996) provided a lower limit on the black hole mass of 4.9 M ⊙ , a companion star mass of ∼ 1.6 M ⊙ , an orbital period of ∼ 14.7 hr, and a distance of 2 . 4 ± 0 . 4 kpc. By fixing the distance at 2.4 kpc, Tao et al. (2019) found the black hole spin of a ∗ > 0 . 92, the accretion disk inclination of i ∼ 40 -50 · , and the black hole mass of M BH < 8 M ⊙ , and Chatterjee et al. (2021) used a two-component advective flow (TCAF) model to constrain the black hole mass to be between 4.5-5.9 M ⊙ . However, Saikia et al. (2022) argue that the distance of 2.4 kpc is an underestimate. They conservatively estimated the distance of GRS 1716-249 to be 4-17 kpc based on the global optical / X-ray correlation and suggested a most likely value of D ∼ 4-8 kpc based on the dynamics of the binary system. Casares et al. (2023) presented evidence for a 6.7 hr orbital period and used the empirical relationship between the quiescent r -band magnitude and the orbital period to constrain the distance to 6 . 9 ± 1 . 1 kpc, consistent with the results given by Saikia et al. (2022). Furthermore, Casares et al. (2023) also provided an orbital inclination of 61 ± 15 · and a BH mass of 6 . 4 + 3 . 2 -2 . 0 , M ⊙ at 68% confidence. \nDue to the distance-dependent nature of continuum fitting methods, di ff erent distances may yield di ff erent spin measurement results. Therefore, in this paper, we used two di ff erent distance parameters, the previously used 2.4 kpc (Masetti et al. 1996) and the most recent 6.9 kpc provided by Casares et al. (2023), to highlight the impact of distance changes on spin measurements. We employed a joint fitting method using continuum and reflection spectra to constrain the spin, utilizing data from Insight-HXMT and NuSTAR simultaneous observations during the hard intermediate state in the 2016-2017 outbrst. In the following section (Sect. 2), we will describe the observations and data reduction. Sect. 3 will present the spectral fitting and results. The obtained results will be discussed in Sect. 4 and summarized in Sect. 5.', '2. Observations and data reduction': "In contrast to Tao et al. (2019), which utilized simultaneous Swift and NuSTAR data, in this study we use simultaneous Insight-HXMT and NuSTAR data. Insight-HXMT can provide high-statistics spectra up to 150 keV for this source, enabling effective constraint of the PL and reflection components, and the lower energy limit of 1 keV for Insight-HXMT ensures accurate modeling of the disk component. Moreover, unlike the Swift data used by Tao et al. (2019), the spectra are not prone to distortion due to pile-up e ff ects as Insight-HXMT does not su ff er from this issue. \nInsight-HXMT observed GRS 1716-249 twice during the 2016-2017 outburst. According to the spectral classification of Bassi et al. (2019), both observations are in the hard intermedi- \nFig. 1. MAXI / GSClight-curve (2-20 keV) of GRS 1716-249. The solid red vertical line represents the simultaneous Insight-HXMT and NuSTAR observations used in this work. \n<!-- image --> \nFig. 2. The light curves of Insight-HXMT / LE (top pannel) and NuSTAR / FPMA (bottom panel). The black dashed vertical line represents the division point for the two data sets. \n<!-- image --> \nate state. NuSTAR observed the source three times in the hard intermediate state. The second observation by Insight-HXMT (obsID P0114335002) is strictly simultaneous with the third NuSTAR observation (obsID 90301007002), which started on July 28, 2017, and ended on July 30, 2017. The simultaneous observations by Insight-HXMT and NuSTAR are indicated on the outburst light curve obtained from MAXI / GSC (see Fig. 1). The e ff ective exposure times for Insight-HXMT / LE and NuSTAR / FPMAare 26ks and 89 ks, respectively (see Table 1). Since the observation span exceeds two days and the source's luminosity, that is, accretion rate, changes drastically during this period, we split the observations of Insight-HXMT and NuSTAR into two data sets (see Fig. 2) to more accurately measure the spin. The information on the divided data is provided in Table 2.", '2.1. Insight-HXMT': 'Weperform data reduction using the Insight-HXMT Data Analysis Software ( HXMTDAS v2.05 1 ) and the calibration database files ( CALDB v2.06 ). To determine the good time intervals, we established the following criteria: (a) pointing o ff set angle should \nTable 1. Observation information of Insight-HXMT and NuSTAR \nNotes. For Insight-HXMT and NuSTAR, the observation logs of LE and FPMA are listed as a representation, respectively. \nTable 2. Information of Data Set 1 and 2. \nNotes. For Insight-HXMT and NuSTAR, the observation logs of LE and FPMA are listed as a representation, respectively. \nbe less than 0.04 · ; (b) Earth elevation angle should be greater than 10 · ; (c) there should be a minimum time interval of 300 s from the crossing of the South Atlantic Anomaly region; (d) the geomagnetic cuto ff rigidity should exceed 8 GV. The background for the Low Energy (LE), Medium Energy (ME), and High Energy (HE) telescopes were generated using the scripts lebkgmap , mebkgmap , and hebkgmap , respectively, based on the Insight-HXMT background models (Liao et al. 2020b; Guo et al. 2020; Liao et al. 2020a). The response files for LE, ME, and HE were generated using lerspgen , merspgen , and herspgen , respectively. Following the recommendation of the Insight-HXMT calibration group, the combined spectra are rebinned as follows: The spectra of LE, ME and HE are respectively rebinned with 1000, 800 and 600 counts per bin at least. Additionally, systematic errors of 1%, 1%, and 2% are applied to the LE, ME, and HE spectra, respectively.', '2.2. NuSTAR': "The nupipeline routine of NuSTARDAS v1.9.7 in HEASoft v6.29 with CALDB v20211115 , is employed to process the cleaned event files. The nuproducts tool is then used to extract the source events, by adopting a circular region surrounding the source with a radius of 80 '' to optimize the signal-to-noise ratio of the spectra 2 . The corresponding background extraction region is a nearby source-free circle with a radius of 80 '' . The spectra are rebinned with 50 counts per bin at least.", '3. Analysis and results': "To obtain more reliable results, we performed a joint fit of the spectra from Data Sets 1 and 2. The energy bands used for Insight-HXMT data are 1-7 keV for LE, 10-35 keV for ME, and 35-150 keV for HE. For NuSTAR, we select the energy band of 3-79 keV. The spectral fitting is conducted using XSPEC v12.12.0 (Arnaud 1996). The abundances are set to WILM (Wilms et al. 2000), and the cross-sections are set to VERN (Verner et al. 1996). All errors are estimated via the Markov chain Monte Carlo algorithm (MCMC) with a length of 600000. \nDue to the source being in the hard intermediate state (Bassi et al. 2019), where the spectra exhibit prominent disk emission and reflection features (Tao et al. 2019), we perform a joint fitting of the spectra using both a continuum and reflection spectral model. The model selected for the fitting is tbabs(kerrbb+relxill) . Additionally, a multiplicative constant model ( constant ) is included to account for the normalization discrepancies between di ff erent telescopes. Therefore, our fitting model is M1 = constanttbabs(kerrbb+relxill) , as shown in Table A.1. The kerrbb model is a multi-temperature blackbody model that describes the spectrum emitted by a geometrically thin, steadystate accretion disk around a Kerr black hole, taking into account full relativistic e ff ects (Li et al. 2005). The relxill 3 model is a relativistic disk coronal reflection model that describes the reflection produced by the corona (with a cuto ff PL spectrum) illuminating the inner regions of the accretion disk (Dauser et al. 2014; García et al. 2014). In the kerrbb model, we fix the BH mass at 6.4 M ⊙ (Casares et al. 2023) and the normalization at 1, \nTable 3. Joint fitting parameters of Insight-HXMT and NuSTAR with M1 ( constanttbabs(kerrbb+relxill)) . \nNotes. All errors are quoted at the 90% confidence level. The probability distributions of parameters for M1B obtained from Data Set 1 through MCMC are presented in Fig. B.1; ⋆ indicates that the parameters between di ff erent data sets are linked; N H is the X-ray absorption column density in units of 10 22 atoms cm -2 ; ˙ M is the e ff ective mass accretion rate of the disk in units of 10 15 g s -1 ; f h is the spectral hardening factor; q in is the inner emissivity index; R b represents the transition radius between the inner and outer emissivity indices, measured in R ISCO units, and is capped at 100 R ISCO; a ∗ is the BH spin; i is the disk inclination angle in units of deg; Γ is the photon index of the incident cuto ff PL spectrum; ξ is the ionization parameter of the accretion disk in units of erg cm s -1 ; A Fe is the iron abundance of the accretion disk in units of solar abundance; E cut is the cuto ff energy in units of keV; R f is the reflection fraction; norm is the normalization of relxill in units of 10 -3 ; Flux is the 0.1-100 keV unabsorbed fluxes in units of 10 -8 erg cm -2 s -1 . \nand link the BH spin ( a ∗ ) and the disk inclination ( i ) to that of relxill . In the relxill model, we fix the inner radius of the accretion disk ( R in) to -1 since the disk inner edge is at the ISCO (Bassi et al. 2019), and presume that within the range from R in to a specific break radius ( R b), the emissivity is defined by a single inner emissivity index ( q in), and beyond R b, it generally adopts the standard r -3 ( q out = 3) profile. For di ff erent data sets, some parameters are linked and allowed to vary freely, including the X-ray absorption column density ( N H), a ∗ , i , and iron abundance ( A Fe). Other parameters are independent between di ff erent data sets, such as the mass accretion rate ( ˙ M ) and the spectral hardening factor f h in kerrbb , and q in, R b, the photon index ( Γ ), ionization parameter (log ξ ), cuto ff energy of the incident spectrum ( E cut), reflection factor ( R f ), and normalization in relxill . To determine the BH spin under the most recent measured distance and investigate the influence of distance on spin measurement, we perform two sets of fits: fixing the distance D in the kerrbb at 2.4 kpc ( M1A ; See Table A.1) and 6.9 kpc ( M1B ), respectively. \nThe two fittings at di ff erent distances yield similar goodness of fit, and their detailed fitting parameters are listed in Table 3. In both models M1A and M1B , the accretion rate shows a \ngradual decrease from Data Set 1 to Data Set 2, which aligns with the light curve's variability; both models yield N H ∼ 0.7, i ∼ 44 · , Γ ∼ 1 . 9, log ξ ∼ 3 . 8, and E cut ≳ 900 keV, indicating that these parameters do not exhibit strong distance dependence. The spin fitting results exhibit noteworthy changes, where an increase in distance transitions high spin values to moderate ones. The relationship between a ∗ and f h shown in Fig. B.1 demonstrates an inverse correlation, in agreement with the findings of Salvesen & Miller (2021). This suggests that f h has a significant impact on spin fitting. To reduce this impact and achieve reliable spin measurements, we have substituted kerrbb in M1 with kerrbb2 (McClintock et al. 2006). Unlike Kerrbb , Kerrbb2 incorporates the spectral hardening e ff ect using two search tables for f h, each corresponding to di ff erent viscosity parameters: α = 0.01 and 0.1. These look-up tables are generated via bhspec (Davis et al. 2005), relying on non-LTE atmosphere models within an α -viscosity framework. Other features of Kerrbb2 , such as Doppler boosting, gravitational redshift, and returning radiation, remain consistent with kerrbb . The updated model is now M2 = constanttbabs(kerrbb2+relxill) , as shown in Table A.1. We chose α = 0.1 (Steiner et al. 2011) in M2 and \nTable 4. Joint fitting parameters of Insight-HXMT and NuSTAR with M2 ( constanttbabs(kerrbb2+relxill)) . \nNotes. All errors are quoted at the 90% confidence level. The probability distributions of parameters for M2A and M2B obtained from Data Set 1 through MCMC are presented in Figs. B.2 and B.3, respectively. \nTable 5. Fitting results of partial parameters with M2 by assuming di ff erent distances. \nNotes. All errors are quoted at the 90% confidence level. The joint fitting results of BH spin, disk inclination, accretion rate of Data Set 1, iron abundance, and goodness of fitting at di ff erent distances with M2 are listed in the table (values without parentheses). For distances of 2.4 and 3 kpc, besides the results with the iron abundance freely fitted, we also include the fitting parameters set to a fixed iron abundance of 5 (values in parentheses). \nkept the other parameter settings the same as in M1 , such as keeping the black hole mass fixed at 6.4 M ⊙ . Additionally, similar to M1 , we fixed the distance parameter at two specific values: D = 2.4 kpc ( M2A ; See Table A.1) and D = 6.9 kpc ( M2B ), respectively. \nThe fitting results with M2 are showed in Table 4 and Fig. 3. Compared to M1 , the fitting results of M2 did not exhibit notable di ff erences, with M2A leading to a high spin value of a ∗ ∼ 0.97, and M2B resulting in a lower spin value of a ∗ ∼ 0.46. These dis- \nes in spin are attributed to the di ff erent distance assumptions. To investigate the influence of distance variations on spin measurements, by maintaining a constant BH mass, we conduct additional spectral fits using M2 with di ff erent distance parameters. The resulting spin values, inclination angles, accretion rate of Data Set 1, iron abundance, and goodness of fit are detailed in Table 5. The analysis revealed that the best-fit iron abundance exhibits two distinct sets of values: for distances of 2.4 and 3 kpc, \nFig. 3. Spectra (black for Insight-HXMT / LE, red for InsightHXMT / ME, green for Insight-HXMT / HE, gray for NuSTAR / FPMA and blue for NuSTAR / FPMB), model components of M2B , and spectral residuals of M2A and M2B . The black solid line is the total model fitted to the data, and the yellow and purple dotted lines show the kerrbb2 and relxill spectral components, respectively. The models are plotted based on the best-fit parameters obtained from Insight-HXMT. \n<!-- image --> \nwe observe very high values of A Fe > 9 . 3; in contrast, for distances between 4 and 8 kpc, the iron abundance stabilizes around A Fe ∼ 5. The latter A Fe is similar to the values measured for other BH binaries, such as GX 339-4 with A Fe = 5 ± 1 (García et al. 2015), V404 Cyg with A Fe ∼ 5 (Walton et al. 2017), and Cyg X1 with A Fe = 4 . 7 ± 0 . 1 (Parker et al. 2015). To examine the e ff ect of di ff erent A Fe, we fixed A Fe at 5 for distances of 2.4 and 3 kpc during the fitting process. The corresponding results are shown in Table 5 in parentheses, where a ∗ shows a slight decrease and i becomes more consistent with the cases of larger distances. Our analysis demonstrates a significant decrease in the spin with increasing distance. This emphasizes the importance of accurate distance determination when using continuum spectrum fitting for BH spin measurements. Any inaccuracies in the distance estimation can lead to deviations in the measured spin values, even if the reflection component modeling is also employed.", '4. Discussion': 'In this paper, we perform a joint fitting of the simultaneous spectra of GRS 1716-249 observed by Insight-HXMT and NuSTAR. The data used are obtained during the softest phase of the out- \nFig. 4. Comparison diagram of di ff erent models. Top panel: Di ff erent models and their disk components and reflection components. Bottom panel: The ratio of each model relative to M2B . \n<!-- image --> \nburst, where the disk component is most prominent and the inner edge of the disk is located at the ISCO (Bassi et al. 2019). Additionally, the presence of significant reflection features in the spectra allows us to measure the spin and inclination of the black hole using a combined fitting method of the continuum and reflection components.', '4.1. System Parameters with Updated Distance': 'The fitting of the continuum is intrinsically linked to the distance, as altering the distance will impact the determination of the inner disk radius, consequently influencing the constraint on the spin. By fitting the spectra using M2 with varying distance parameters, we obtain distinct spin results (refer to Table 5). Notably, assuming a distance of 2.4 kpc, we obtain a near-extreme spin, aligning with the findings in Tao et al. (2019). However, assuming a distance of 6.9 kpc based on the recent works of Saikia et al. (2022) and Casares et al. (2023), resulted in a moderate spin of a ∗ ∼ 0 . 46, implying that this source is not a rapidly rotating BH. This suggests that the spin was previously overestimated under the prior distance assumption. The disk inclination, even with varying distance assumptions, remains nearly constant \nTable 6. Joint fitting parameters of Insight-HXMT and NuSTAR with Tbabs(diskbb+relxill) . \nNotes. All errors are quoted at the 90% confidence level. The definition of each parameter is the same as that in Table 3. \naround 43 -47 · , which is within the error margin of the orbital inclination of 61 ± 15 · (with 68% confidence) as reported by Casares et al. (2023). \nIn the joint fitting of the continuum and reflection spectra, the distance measurement primarily a ff ects the fitting of the continuum spectrum model ( kerrbb2 ). Therefore, to investigate the impact of distance changes on the model, we plotted the best-fit M2 with its disk component Kerrbb2 and reflection component relxill at di ff erent distances (shown as solid lines in Fig. 4). The variation in distance did not result in significant changes in the total models, with di ff erences of less than 2% in each energy band (see Fig. 4 bottom panel). Importantly, we found that the energy corresponding to the peak flux of the disk component ( E disk peak ) did not show significant changes. Since the observed flux is fixed, an increase in distance leads to an increase in the accretion rate, as indicated by the parameter ˙ M in Table 5. According to accretion disk theory (Shakura & Sunyaev 1973), when other parameters remain constant, an increase in the accretion rate causes E disk peak to shift towards higher energies, while a decrease in spin leads to an increase in R in ( = R ISCO), resulting in a decrease in E disk peak . Therefore, to maintain a constant E disk peak , the model compensates for the increase in accretion rate due to larger distances by reducing the spin. This corresponds to the inverse correlation between a ∗ and ˙ M shown in Figs. B.1-B.3. \nFurthermore, when other parameters remain constant, an increase in BH mass will cause E disk peak to shift towards lower energies. In this case, the model compensates for the mass in- \ncrease by increasing the spin. Therefore, within the constraints of distance and mass provided by Casares et al. (2023), setting the lower limit (upper limit) of distance and the upper limit (lower limit) of mass yields the upper (lower) limit of the spin. When fixing the distance at 5.8 kpc and the BH mass at 9 . 6 M ⊙ ( M3 ; See Table A.1), we obtain a moderately high spin value of a ∗ = 0 . 757 + 0 . 023 -0 . 011 , with a goodness of fit of χ 2 / dof = 6416.4 / 6358 (see Fig. B.4 for the parameter probability distribution of Data Set 1). When fixing the distance at 8 kpc and the BH mass at 4 . 4 M ⊙ ( M4 ; See Table A.1), we obtain a near non-rotating black hole with a ∗ = -0 . 008 + 0 . 011 -0 . 013 , with a goodness of fit of χ 2 / dof = 6499.9 / 6358. The red and green dashed lines in Fig. 4 represent M3 and M4 . Therefore, considering the upper and lower limits of distance and mass provided by Casares et al. (2023), there will be a wider range of spin with a ∗ < 0.78. \nThe above discussion indicates that the variation in the measured spin values with changes in distance and mass is a result of the degeneracy in the model. A similar phenomenon was observed when measuring the spins of LMC X-1 and Cyg X-1 using continuum spectra (Zdziarski et al. 2024). Their analysis shows that the disk components dominate the spectra, so E disk peak derived from fitting matches the energy of the peak flux seen in the spectra ( E obs peak ). Increasing the hardening factor or convolving with Comptonization components outside the disk components will shift the E disk peak towards higher energies. To maintain E disk peak corresponding as closely as possible to E obs peak , the remaining free parameters in the continuum spectrum, spin, and accretion rate, will decrease accordingly to counteract the e ff ects of the hardening factor or Comptonization component. \nFurthermore, the inconsistency in iron abundance at di ff erent distances as shown in Table 5 ( A Fe > 9.3 for D = 2.4 and 3 kpc; A Fe ∼ 5 for D = 4-8 kpc) primarily reflects not only the degeneracies among the parameters of the continuum model but also a certain level of competition between the continuum and reflection components, particularly in the low energy band of ≲ 2 keV (García et al. 2018a; Tomsick et al. 2018). For instance, M2 with D = 2.4 and 3 kpc, as depicted in Fig. 4, exhibits a relatively higher proportion of disk component and a diminished reflection component compared to other models. The shape of the reflection spectrum in the low energy is significantly influenced by the configuration of the incident spectrum and the parameters of the disk as the reflector, leading to intertwined dependencies among various parameters in the overall model and rendering the parameter space exceptionally complicated. Zdziarski et al. (2024) reported local minimum iron abundance of A Fe ≲ 1 and global minimum values of A Fe ∼ 8 at ∆ χ 2 ≈ -0 . 6 while fitting the spectra of LMC X-1 using continuum and reflection components ( kerrbb2+reflkerr ), thereby corroborating this point. Therefore, for obtaining more rational and precise parameters during spectral fitting, a comprehensive analysis of the entire parameter space is imperative to assess potential biases introduced.', '4.2. Further Insights into Spin Measurements': 'Given this, we attempted to replace kerrbb2 with diskbb , which only has two parameters, disk temperature T in in units of keV and Norm, without considering the impact of distance, and re-conduct the spectral fitting ( M5 = constanttbabs(diskbb+relxill) ; See Table A.1). The fitting results are shown in Table 6, which are very close to the fitting results of M1A . We obtain an extreme spin of a ∗ ≳ 0 . 99, which is similar to the results obtained by Draghis et al. (2023a) using the reflection component fitting method, that is, using the \nmodel diskbb combined with a reflection model. Furthermore, utilizing diskbb Norm ≈ 5000-6000 and i ≈ 48 · of M5 , and assuming a distance of D = 5.8-8 kpc, a hardening factor of f h = 1 . 3 (from M1A ), and a boundary condition correction factor ζ = 0 . 412, we obtain the inner disk radius R in ≈ 35-53 km using the formula R in = √ Norm / cos i × ζ × f 2 h × D / 10kpc (Kubota et al. 1998). Assuming the source mass of 4.4-9.6 M ⊙ (Casares et al. 2023), the gravitational radius R g is about 6.5-14.3 km, thus M5 yields R in ≈ 2 . 4 -8 . 2 R g. By setting R in = R ISCO, the spin derived from the diskbb component is a ∗ ∼ -0 . 72 -0 . 88, which is inconsistent with the result of extreme spin obtained from the reflection model. \nSimilar to GRS 1716-249, when constraining the spin of some other sources, such as LMC X-3 (Draghis et al. 2023a; Steiner et al. 2014), H 1743-322 (Draghis et al. 2023b; Steiner et al. 2012), and MAXI J1820 + 070 (Draghis et al. 2023b; Guan et al. 2021; Zhao et al. 2021), results obtained solely from the reflection component fitting method yield close to extreme spin values, while the continuum spectrum fitting method yields low spins of a ∗ ≲ 0 . 3. The examples above demonstrate that when fitting the spectrum, using either the continuum or the reflection component fitting method separately resulted in di ff erent inner radii of the accretion disk, leading to di ff erent spin values and hence physically inconsistent results. The joint fitting method of the continuum and reflection components may help to avoid this issue by considering the jointly optimal parameter space during the fitting process.', '5. Conclusion': 'In conclusion, we re-evaluated several key parameters of the black hole GRS 1716-249 utilizing simultaneous data from Insight-HXMT and NuSTAR. This paper, through the application of a combined fitting method for the continuum and reflection components and the integration of updated distance and black hole mass, found that the black hole has a moderate spin and a moderately inclined accretion disk. Given the source distance of 6.9 kpc and the black hole mass of 6.4 M ⊙ , a ∗ is 0 . 464 + 0 . 016 -0 . 007 and i is 43 . 8 + 0 . 5 -0 . 8 · with 90% confidence level. Taking into account the uncertainties in distance and black hole mass, the spin range extends with a ∗ < 0.78. \nAcknowledgements. We thank the anonymous referee for useful comments that have allowed us to improved this manuscript. This work made use of data from the Insight-HXMT mission, a project funded by China National Space Administration (CNSA) and the Chinese Academy of Sciences (CAS). This work is supported by the National Key R&D Program of China (2021YFA0718500). We acknowledge funding support from the National Natural Science Foundation of China (NSFC) under grant Nos. 12122306, 12333007 and 12027803, the CAS Pioneer Hundred Talent Program Y8291130K2, the Strategic Priority Research Program of the Chinese Academy of Sciences under grant No. XDB0550300, the Scientific and technological innovation project of IHEP Y7515570U1 and the International Partnership Program of Chinese Academy of Sciences (Grant No.113111KYSB20190020) .', 'References': "- Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. \nJacoby & J. Barnes, 17 Ballet, J., Denis, M., Gilfanov, M., et al. 1993, IAU Circ., 5874, 1 Bassi, T., Del Santo, M., D'Aı, A., et al. 2019, MNRAS, 482, 1587 Brown, G. E. & Bethe, H. A. 1994, ApJ, 423, 659 \n- Casares, J., Yanes-Rizo, I. V., Torres, M. A. 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Various models and the setting of the black hole mass ( M ) and distance ( D ) parameters in the disk components. \n```\nM1 = constanttbabs(kerrbb+relxill) with M = 6 . 4 M ⊙ M1A = M1 with D = 2.4 kpc M1B = M1 with D = 6.9 kpc M2 = constanttbabs(kerrbb2+relxill) with M = 6 . 4 M ⊙ M2A = M2 with D = 2.4 kpc M2B = M2 with D = 6.9 kpc M3 = constanttbabs(kerrbb2+relxill) with M = 9 . 6 M ⊙ and D = 5.8 kpc M4 = constanttbabs(kerrbb2+relxill) with M = 4 . 4 M ⊙ and D = 8 kpc M5 = constanttbabs(diskbb+relxill)\n``` \nFig. B.1. Probability distributions of the parameters for M1B obtained from Data Set 1 through MCMC. \n<!-- image --> \nFig. B.2. Probability distributions of the parameters for M2A obtained from Data Set 1 through MCMC. \n<!-- image --> \nFig. B.3. Probability distributions of the parameters for M2B obtained from Data Set 1 through MCMC. \n<!-- image --> \nFig. B.4. Probability distributions of the parameters for M3 obtained from Data Set 1 through MCMC. \n<!-- image -->'}
2022PhRvD.105b3520A	We present the first cosmology results from largescale structure using the full 5000 degSUP2SUP of imaging data from the Dark Energy Survey DES Data Release 1. We perform an analysis of largescale structure combining three twopoint correlation functions 3 2 pt i cosmic shear using 100 million source galaxies ii galaxy clustering and iii the crosscorrelation of source galaxy shear with lens galaxy positions galaxygalaxy lensing. To achieve the cosmological precision enabled by these measurements has required updates to nearly every part of the analysis from DES Year 1 including the use of two independent galaxy clustering samples modeling advances and several novel improvements in the calibration of gravitational shear and photometric redshift inference. The analysis was performed under strict conditions to mitigate confirmation or observer bias we describe specific changes made to the lens galaxy sample following unblinding of the results and tests of the robustness of our results to this decision. We model the data within the flat CDM and w CDM cosmological models marginalizing over 25 nuisance parameters. We find consistent cosmological results between the three twopoint correlation functions their combination yields clustering amplitude SSUB8SUB0.77 6SUB0.017SUBSUP0.017SUP and matter density SUBmSUB0.33 9SUB0.031SUBSUP0.032SUP in CDM mean with 68 confidence limits SSUB8SUB0.77 5SUB0.024SUBSUP0.026SUP SUBmSUB0.35 2SUB0.041SUBSUP0.035SUP and dark energy equationofstate parameter w 0.9 8SUB0.20SUBSUP0.32SUP in w CDM . These constraints correspond to an improvement in signaltonoise of the DES Year 3 3 2 pt data relative to DES Year 1 by a factor of 2.1 about 20 more than expected from the increase in observing area alone. This combination of DES data is consistent with the prediction of the model favored by the Planck 2018 cosmic microwave background CMB primary anisotropy data which is quantified with a probabilitytoexceed p 0.13 0.48. We find better agreement between DES 3 2 pt and Planck than in DES Y1 despite the significantly improved precision of both. When combining DES 3 2 pt data with available baryon acoustic oscillation redshiftspace distortion and type Ia supernovae data we find p 0.34 . Combining all of these datasets with Planck CMB lensing yields joint parameter constraints of SSUB8SUB0.81 2SUB0.008SUBSUP0.008SUP SUBmSUB0.30 6SUB0.005SUBSUP0.004SUP h 0.68 0SUB0.003SUBSUP0.004SUP and mSUBSUBlt0.13 eV 95 C.L. in CDM SSUB8SUB0.81 2SUB0.008SUBSUP0.008SUP SUBmSUB0.30 2SUB0.006SUBSUP0.006SUP h 0.68 7SUB0.007SUBSUP0.006SUP and w 1.03 1SUB0.027SUBSUP0.030SUP in w CDM .	2022-01-01T00:00:00Z	['arXiv:2105.13549', '2021arXiv210513549D', '2022PhRvD.105b3520A', '10.1103/PhysRevD.105.023520', '10.48550/arXiv.2105.13549']	['Astrophysics - Cosmology and Nongalactic Astrophysics']	Dark Energy Survey Year 3 results Cosmological constraints from galaxy clustering and weak lensing	2,022	189	0.67	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	871	https://arxiv.org/pdf/2105.13549.pdf	{'No Header': '25', 'Cosmological Constraints from Galaxy Clustering and Weak Lensing': 'T. M. C. Abbott, 1 M. Aguena, 2 A. Alarcon, 3 S. Allam, 4 O. Alves, 5, 6, 2 A. Amon, 7 F. Andrade-Oliveira, 6, 2 J. Annis, 4 S. Avila, 8 D. Bacon, 9 E. Baxter, 10 K. Bechtol, 11 M. R. Becker, 3 G. M. Bernstein, 12 S. Bhargava, 13 S. Birrer, 14 J. Blazek, 15, 16 A. Brandao-Souza, 17, 2 S. L. Bridle, 18 D. Brooks, 19 E. Buckley-Geer, 20, 4 D. L. Burke, 7, 21 H. Camacho, 6, 2 A. Campos, 22 A. Carnero Rosell, 23, 2, 24 M. Carrasco Kind, 25, 26 J. Carretero, 27 F. J. Castander, 28, 29 R. Cawthon, 11 C. Chang, 20, 30 A. Chen, 5 R. Chen, 31 A. Choi, 32 C. Conselice, 18, 33 J. Cordero, 18 M. Costanzi, 34, 35, 36 M. Crocce, 28, 29 L. N. da Costa, 2, 37 M. E. da Silva Pereira, 5 C. Davis, 7 T. M. Davis, 38 J. De Vicente, 39 J. DeRose, 40 S. Desai, 41 E. Di Valentino, 18 H. T. Diehl, 4 J. P. Dietrich, 42 S. Dodelson, 22, 43 P. Doel, 19 C. Doux, 12 A. Drlica-Wagner, 20, 4, 30 K. Eckert, 12 T. F. Eifler, 44, 45 F. Elsner, 19 J. Elvin-Poole, 32, 46 S. Everett, 47 A. E. Evrard, 48, 5 X. Fang, 44 A. Farahi, 5, 49 E. Fernandez, 27 I. Ferrero, 50 A. Ferté, 45 P. Fosalba, 28, 29 O. Friedrich, 51 J. Frieman, 4, 30 J. García-Bellido, 8 M. Gatti, 12 E. Gaztanaga, 28, 29 D. W. Gerdes, 48, 5 T. Giannantonio, 52, 51 G. Giannini, 27 D. Gruen, 53, 7, 21 R. A. Gruendl, 25, 26 J. Gschwend, 2, 37 G. Gutierrez, 4 I. Harrison, 54, 18 W. G. Hartley, 55 K. Herner, 4 S. R. Hinton, 38 D. L. Hollowood, 47 K. Honscheid, 32, 46 B. Hoyle, 42 E. M. Huff, 45 D. Huterer, 5 B. Jain, 12 D. J. James, 56 M. Jarvis, 12 N. Jeffrey, 19, 57 T. Jeltema, 47 A. Kovacs, 23, 24 E. Krause, 44 R. Kron, 4, 30 K. Kuehn, 58, 59 N. Kuropatkin, 4 O. Lahav, 19 P.-F. Leget, 7 P. Lemos, 19, 13 A. R. Liddle, 60, 61, 62 C. Lidman, 63, 64 M. Lima, 65, 2 H. Lin, 4 N. MacCrann, 66 M. A. G. Maia, 2, 37 J. L. Marshall, 67 P. Martini, 32, 68, 69 J. McCullough, 7 P. Melchior, 70 J. Mena-Fernández, 39 F. Menanteau, 25, 26 R. Miquel, 71, 27 J. J. Mohr, 42, 72 R. Morgan, 11 J. Muir, 7 J. Myles, 53, 7, 21 S. Nadathur, 19 A. Navarro-Alsina, 17 R. C. Nichol, 9 R. L. C. Ogando, 2, 37 Y. Omori, 20, 30, 7 A. Palmese, 4, 30 S. Pandey, 12 Y. Park, 73 F. Paz-Chinchón, 25, 52 D. Petravick, 25 A. Pieres, 2, 37 A. A. Plazas Malagón, 70 A. Porredon, 32, 46 J. Prat, 20, 30 M. Raveri, 12 M. Rodriguez-Monroy, 39 R. P. Rollins, 18 A. K. Romer, 13 A. Roodman, 7, 21 R. Rosenfeld, 74, 2 A. J. Ross, 32 E. S. Rykoff, 7, 21 S. Samuroff, 22 C. Sánchez, 12 E. Sanchez, 39 J. Sanchez, 4 D. Sanchez Cid, 39 V. Scarpine, 4 M. Schubnell, 5 D. Scolnic, 31 L. F. Secco, 12, 30 S. Serrano, 28, 29 I. Sevilla-Noarbe, 39 E. Sheldon, 75 T. Shin, 12 M. Smith, 76 M. Soares-Santos, 5 E. Suchyta, 77 M. E. C. Swanson, 25 M. Tabbutt, 11 G. Tarle, 5 D. Thomas, 9 C. To, 53, 7, 21 A. Troja, 74, 2 M. A. Troxel, 31 D. L. Tucker, 4 I. Tutusaus, 28, 29 T. N. Varga, 72, 78 A. R. Walker, 1 N. Weaverdyck, 5 R. Wechsler, 53, 7, 21 J. Weller, 72, 78 B. Yanny, 4 B. Yin, 22 Y. Zhang, 4 and J. Zuntz 60', '(DES Collaboration)': "1 Cerro Tololo Inter-American Observatory, NSF's National Optical-Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile 2 Laboratório Interinstitucional de e-Astronomia - LIneA, \nRua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil \n3 Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA \n4 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA \n5 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA \n6 Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil \n7 Kavli Institute for Particle Astrophysics & Cosmology, \nP. O. Box 2450, Stanford University, Stanford, CA 94305, USA \n8 Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain \n9 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK \n10 Institute for Astronomy, University of Hawai'i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA \n11 Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison, 1150 University Avenue Madison, WI 53706-1390 \n12 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA \n13 Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK \n14 Graduate School of Education, Stanford University, 160, 450 Serra Mall, Stanford, CA 94305, USA \n15 Department of Physics, Northeastern University, Boston, MA 02115, USA \n16 Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland 17 \nInstituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil \n18 Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, \nUniversity of Manchester, Oxford Road, Manchester, M13 9PL, UK \n19 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK \n20 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA \n21 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA \n22 Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15312, USA \n23 Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain \n24 Universidad de La Laguna, Dpto. AstrofÃsica, E-38206 La Laguna, Tenerife, Spain \nCenter for Astrophysical Surveys, National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA \n26 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA \n27 Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of \nScience and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain \n28 Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain \n61 \n66 \n45 \n50 \n54 \n32 \n38 \n37 \n29 Institute of Space Sciences (ICE, CSIC), Campus UAB, \nCarrer de Can Magrans, s/n, 08193 Barcelona, Spain \n30 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA \n31 Department of Physics, Duke University Durham, NC 27708, USA \nCenter for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA \n33 University of Nottingham, School of Physics and Astronomy, Nottingham NG7 2RD, UK \n34 \n39 \nAstronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy \n35 \nINAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy \n36 \nInstitute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy \nObservatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil \nSchool of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia \nCentro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain \n40 \nLawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA \n41 Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India \nFaculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany \n43 NSF AI Planning Institute for Physics of the Future, \nCarnegie Mellon University, Pittsburgh, PA 15213, USA \n44 Department of Astronomy/Steward Observatory, University of Arizona, \n933 North Cherry Avenue, Tucson, AZ 85721-0065, USA \nJet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA \n46 \nDepartment of Physics, The Ohio State University, Columbus, OH 43210, USA \n47 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA \n48 Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA \nDepartments of Statistics and Data Science, University of Texas at Austin, Austin, TX 78757, USA \n49 \nInstitute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029 Blindern, NO-0315 Oslo, Norway \n51 \nKavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK \n52 \nInstitute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK \n53 \nDepartment of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA \nDepartment of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK \n55 \n56 \nDepartment of Astronomy, University of Geneva, ch. d'Écogia 16, CH-1290 Versoix, Switzerland \nCenter for Astrophysics \n\| \nHarvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA \n57 \nLaboratoire de Physique de l'Ecole Normale Supérieure, ENS, \nUniversité PSL, CNRS, Sorbonne Université, Université de Paris, Paris, France \nAustralian Astronomical Optics, Macquarie University, North Ryde, NSW 2113, Australia \n59 Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA \n60 Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK \nInstituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, 1769-016 Lisboa, Portugal \n62 \nPerimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada \n63 Centre for Gravitational Astrophysics, College of Science, \nThe Australian National University, ACT 2601, Australia \n64 The Research School of Astronomy and Astrophysics, Australian National University, ACT 2601, Australia 65 Departamento de Física Matemática, Instituto de Física, \nUniversidade de São Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil \nDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK \n67 \nGeorge P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, \nand Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA \n68 Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA \n69 Radcliffe Institute for Advanced Study, Harvard University, Cambridge, MA 02138 \n70 Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA \n71 Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain \n72 \nMax Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany \n73 Kavli Institute for the Physics and Mathematics of the Universe (WPI), \nUTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan \n74 ICTP South American Institute for Fundamental Research \nInstituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil \n75 Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA \n76 School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK \n77 Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 \n78 Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, Scheinerstr. 1, 81679 München, Germany (Dated: March 22, 2022) \n42 \n58 \nWe present the first cosmology results from large-scale structure using the full 5000 deg 2 of imaging data from the Dark Energy Survey (DES) Data Release 1. We perform an analysis of large-scale structure combining three two-point correlation functions (3 × 2pt): (i) cosmic shear using 100 million source galaxies, (ii) galaxy clustering, and (iii) the cross-correlation of source galaxy shear with lens galaxy positions, galaxy-galaxy lensing. To achieve the cosmological precision enabled by these measurements has required updates to nearly every part of the analysis from DES Year 1, including the use of two independent galaxy clustering samples, modeling advances, and several novel improvements in the calibration of gravitational shear and photometric redshift inference. The analysis was performed under strict conditions to mitigate confirmation or observer bias; we describe specific changes made to the lens galaxy sample following unblinding of the results and tests of the robustness of our results to this decision. We model the data within the flat Λ CDM and w CDM cosmological models, marginalizing over 25 nuisance parameters. We find consistent cosmological results between the three two-point correlation functions; their combination yields clustering amplitude S 8 = 0 . 776 +0 . 017 -0 . 017 and matter density Ω m = 0 . 339 +0 . 032 -0 . 031 in Λ CDM, mean with 68% confidence limits; S 8 = 0 . 775 +0 . 026 -0 . 024 , Ω m = 0 . 352 +0 . 035 -0 . 041 , and dark energy equation-of-state parameter w = -0 . 98 +0 . 32 -0 . 20 in w CDM. These constraints correspond to an improvement in signal-to-noise of the DES Year 3 3 × 2pt data relative to DES Year 1 by a factor of 2.1, about 20% more than expected from the increase in observing area alone. This combination of DES data is consistent with the prediction of the model favored by the Planck 2018 cosmic microwave background (CMB) primary anisotropy data, which is quantified with a probability-to-exceed p = 0 . 13 to 0 . 48 . We find better agreement between DES 3 × 2pt and Planck than in DES Y1, despite the significantly improved precision of both. When combining DES 3 × 2pt data with available baryon acoustic oscillation, redshift-space distortion, and type Ia supernovae data, we find p = 0 . 34 . Combining all of these data sets with Planck CMB lensing yields joint parameter constraints of S 8 = 0 . 812 +0 . 008 -0 . 008 , Ω m = 0 . 306 +0 . 004 -0 . 005 , h = 0 . 680 +0 . 004 -0 . 003 , and ∑ m ν < 0 . 13 eV (95% CL) in Λ CDM; S 8 = 0 . 812 +0 . 008 -0 . 008 , Ω m = 0 . 302 +0 . 006 -0 . 006 , h = 0 . 687 +0 . 006 -0 . 007 , and w = -1 . 031 +0 . 030 -0 . 027 in w CDM.", 'I. INTRODUCTION': "The discovery of the accelerated expansion of the universe [1, 2] led to a new standard model of cosmology, which is dominated by a spatially smooth component with negative pressure called dark energy. Over the intervening two decades, the evidence for the presence of dark energy has become much stronger thanks to data from an impressive variety of cosmological probes. Modern cosmological measurements using type Ia supernovae [3-11], cosmic microwave background (CMB) fluctuations [12-14], galaxy clustering [1521], and weak gravitational lensing [22-28] are in agreement with a spatially flat universe with about 30% matter (visible and dark) and 70% dark energy. \nHowever, the physical nature of the dark energy that causes accelerated expansion remains unknown. The simplest and best-known phenomenological model for dark energy is the energy density of the vacuum, incorporated in the field equations of General Relativity by the cosmological-constant term Λ [29]. The resulting Λ Cold Dark Matter ( Λ CDM) model serves as a benchmark for tests with current and future data. Beyond Λ CDM, there exists a rich set of other potential models to explain cosmic acceleration, including evolving scalar fields, modifications to general relativity, and other physically-motivated possibilities. This has spawned an active research area focused on describing and modeling dark energy and its effects on the expansion rate and the growth of density fluctuations [30, 31]. \nThe quest to understand dark energy has spawned a worldwide effort to better measure the growth and evolution of cosmic structure in the universe. The current generation of observations is spearheaded by the so-called Stage-III dark energy experiments, which include the Dark Energy Sur- \nvey (DES) 1 [32-34], the Hyper Suprime-Cam Subaru Strategic Program (HSC) 2 [27, 35, 36], the Kilo-Degree Survey (KiDS) 3 [28, 37], and the Extended Baryon Oscillation Spectroscopic Survey (eBOSS) 4 [38]. These surveys have demonstrated the feasibility of ambitious large-scale structure analyses, and featured extensive tests of theory, development of state-of-the-art systematics calibration, and new rigor in protecting analyses against observer bias before the results are revealed. These surveys have, thus far, provided constraints consistent with the Λ CDM model, and contributed to tightening the constraints on several of the key cosmological parameters related to dark matter and dark energy. \nLarge-scale structure (LSS) in the universe provides a powerful set of tools to probe dark energy. The statistics and temporal growth of cosmic structure complement the largely geometrical sensitivity to dark energy of type Ia supernovae and the CMB. For nearly half a century, measurements of the galaxy two-point correlation function, a statistic describing the spatial clustering of galaxies, have provided pioneering cosmological constraints and early evidence for the Λ CDM model [15, 16, 39-52], as well as recent, highprecision constraints on the cosmological parameters [20, 5365]. Another aspect of LSS that is sensitive to both dark matter and dark energy is cosmic shear, slight distortions of the shapes of distant background galaxies due to weak gravitational lensing of light passing through the structures between these sources and us. While the interpretation of galaxy clustering is complicated by galaxy bias [66, 67], cosmic shear \nmeasurements are more directly related to the distribution of mass. First detections of cosmic shear [68-71] have been followed by an impressive maturing of this probe, with increasingly more competitive constraints on cosmological parameters [23-25, 72-78]. Finally, galaxy-galaxy lensing, the cross-correlation of lens galaxy positions and source galaxy shapes, provides a link between galaxy clustering and cosmic shear. Galaxy-galaxy lensing measurements have also matured to the point where their combination with galaxy clustering breaks degeneracies between the cosmological parameters and bias, thereby helping to constrain dark energy [7992]. The combination of galaxy clustering, cosmic shear, and galaxy-galaxy lensing measurements powerfully constrains structure formation in the late universe, while strongly selfcalibrating many astrophysical or systematic parameters in the model. \nThe stakes have become higher with recent evidence for possible tensions between parameters as measured by different cosmological probes. These tensions may indicate new physics beyond Λ CDM - or else could be due to unaccounted-for systematics or an underestimation of uncertainty in some probes. Potentially most significant among these is the 'Hubble tension,' indicated by a ∼ 4-6 σ discrepancy between measurements of the Hubble constant inferred from the primary CMB anisotropies [13] and higher values measured from a local distance anchor, such as, most prominently the astronomical distance ladder (e.g., [93, 94]) or masers [95], though some measurements also indicate a lower value [96, 97] in better agreement with the CMB. The Hubble tension may indicate new physics, and it is crucial to improve measurements, revisit assumptions and systematics [e.g., 97, 98] and invest in novel, independent methods and probes [99, 100]. \nAdditionally, several experiments that are sensitive to the growth of structure have historically preferred, on average, lower values of the parameter S 8 ≡ σ 8 (Ω m / 0 . 3) 0 . 5 relative to that predicted by the CMB anisotropy, where the amplitude of mass fluctuations σ 8 is scaled by the square root of matter density Ω m . This parameter is predicted to be higher with the CMB [13] than is measured in lensing (e.g., [27, 33, 101, 102]). The difference has been claimed by other experiments to be as large as 2 -3 σ . Other probes of the late universe, in particular spectroscopic galaxy clustering [103], redshift-space distortions (RSD) [21], and the abundance of galaxy clusters [104, 105], also tend to favor a lower S 8 than that measured by the CMB on average (assuming the Λ CDM model). \nPreviously, the DES Collaboration analyzed data from its first year of observations, which covered 1514 deg 2 , and constrained cosmological parameters using galaxy clustering and gravitational lensing in Λ CDM and w CDM [33], constrained beyondw CDM models [106], and carried out numerous other tests of the standard cosmological framework [64, 101, 105, 107-114]. Along with the aforementioned KiDS and HSC observations and analyses, the DES Y1 analysis emphasized redundancy using two shape measurement methods that are independently calibrated, several photometric redshift estimation and validation techniques, and two in- \nependent codes for predicting the measurements and performing a likelihood analysis. \nThis paper presents key cosmological constraints from the first three years of observations (henceforth Y3) of DES. The DES Y3 data set analyzed here uses images covering nearly 5000 sq. deg., or more than three times the area of Y1. It also dramatically increases the number of source and lens galaxies, and introduces new techniques for the analysis and treatment of statistical and systematic errors. As in Y1, we rely on a key cosmological probe of photometric LSS surveys, the socalled '3 × 2pt' analysis, consisting of three two-point correlation functions: (i) w ( θ ) , the angular correlation function of the lens galaxies; (ii) γ t ( θ ) , the correlation of the tangential shear of sources with lens galaxy positions; and (iii) ξ ± ( θ ) , the correlation functions of different components of the ellipticities of the source galaxies. We use these measurements only on large angular scales, for which we have verified that a relatively simple model describes the data, although even with this restriction we must introduce 25 free parameters to capture astrophysical and measurement-related systematic uncertainties. The paper is built upon and uses tools and results from 29 accompanying papers [115-143] that are summarized in App. A. We summarize in App. B the major updates to the analysis that are different from the DES Y1 3 × 2pt analysis. \nThe cosmological quantity that is best constrained by the 3 × 2pt analysis is the overall amplitude of matter clustering in the low redshift universe, parameterized by S 8 . The precise measurement of S 8 in this paper allows a powerful test for consistency between the growth of structure and the expansion history in the broad class of cosmic acceleration models based on General Relativity (GR) and dark energy. Implementing this test requires a CMB anchor for the matter clustering amplitude at high redshift, and the test becomes sharper and more general when supernova and baryon acoustic oscillation (BAO) data are used to constrain the expansion history. DES probes matter clustering out to z ≈ 1 , so it also constrains dark energy models on its own through the history of structure growth over this redshift range. The degeneracy between Ω m and σ 8 in S 8 is broken partly by this redshift evolution and partly by the shape of the correlation functions, and it can be broken more strongly using external data that are sensitive to Ω m . Lensing measurements depend on the expansion history through the distance-redshift relation. This dependence affects our analysis, but the geometric constraints from DES weak lensing are not as strong as those from current supernova and BAO data. \nIn subsequent sections of the paper, we focus first on the DES data sets in Sec. II and measurements of the three twopoint correlation functions in Sec. III. We describe the modeling and analysis in Sec. IV, then turn to the primary results, tests, and parameter constraints from combining these measurements with additional measurements from DES, the CMB, and other external supernova, BAO, and RSD data in Sec. V. We conclude in Sec. VIII.", 'II. DARK ENERGY SURVEY DATA': "The Dark Energy Survey (DES) was a six-year observing program using the Dark Energy Camera (DECam [32]) at the Blanco 4m telescope at the Cerro Tololo Inter-American Observatory (CTIO) in Chile. The survey covered 5000 deg 2 in grizY bandpasses with approximately 10 overlapping dithered exposures in each filter (90 sec in griz , 45 sec in Y ) covering the survey footprint. In this paper, we utilize data taken during the first three years of DES operations (DES Y3), which made up DES Data Release 1 (DR1 [144]). This analysis uses imaging data covering the full 5000 deg 2 survey footprint for the first time, at approximately half the full-survey integrated exposure time. Preparing the imaging data for cosmological analysis is an exacting, multi-year process, and analysis of the final six-year data set is now in its early stages. The data is processed, calibrated, and coadded to produce a photometric data set of 399 million objects that is further refined to a 'Gold' sample for cosmological use [115, 145, 146]. The Gold sample includes selection requirements (cuts) on minimal image depth and quality, additional calibration and deblending, and quality flags to identify problematic photometry and regions of the sky with substantial photometric degradation (e.g., around bright stars). The Gold galaxy sample extends to a signal-to-noise > 10 (extended) limiting magnitude of 23 in i -band. The final Gold sample used in this work after all cuts contains 319 million objects. \nIn addition to the wide-field Gold sample, we rely on data from the DES deep fields [116] covering a subset of the 27 deg 2 DES transient search regions and the separate COSMOS field [147]. These images are taken in ugrizY bandpasses with DECam, and also have overlapping VIDEO [148] or UltraVISTA [149] imaging for near-IR photometry in Y JHK over most of the area. Coadd images are constructed from the best images (i.e., with smallest point-spread function (PSF) full-width half-maximum (FWHM)) with a goal of attaining a depth approximately 10 × the typical wide-field coadd image depth. From these coadd images, we produce a deep catalog of 2.8 million objects that has 10 σ limiting magnitude of 25 in i -band, and photometric variance 0.1 × the typical widefield variance. This catalog helps to validate and calibrate our wide-field data in several ways. First, it is used to create an input model space for representative objects to draw onto wide-field-quality images in the Balrog [121] or weak lensing image simulations [120]. Balrog is used to test the survey selection function, which describes the probability that an object type drawn from a complete galaxy population will be detected in our wide-field survey, and the weak lensing simulations are used to test our shear calibration. The deep catalog also serves as a stepping-stone in our redshift inference methodology [123]. It allows us to map available spectroscopic or many-band deep photometric observations into the ugrizJHK bandpass space of our deep catalog, for which we have 1.68 million sources with matched near-infrared photometry covering an area of 5.88 deg 2 . This is then mapped through Balrog onto wide-field galaxy information.", '1. Shapes': 'The DES Y3 shear catalog [117] is derived using the METACALIBRATION pipeline [150, 151], which infers the ellipticity and similar photometric properties of objects using information from the r,i,z -bands. The pipeline is similar to that used in the DES Y1 analysis [152], but with a number of updates, including improved PSF solutions [118], improved astrometric solutions [115], and the inclusion of an inverse-variance weighting for the galaxies. METACALIBRATION is able to self-calibrate the initial estimate of the shear field from the measured galaxy shapes, including sample selection biases. The current METACALIBRATION implementation, however, does not correct for a shear-dependent detection bias [153] that is coupled with object blending effects, which we find to cause a multiplicative bias in the shear at the level of 2-3%. This residual bias is calibrated using image simulations [120]. Objects are included in the catalog if they pass a number of selection cuts designed to reduce potential systematic biases [117]. After additional footprint masking to match the lens catalogs, the final DES Y3 shear catalog yields 100 million galaxies covering an area of 4143 deg 2 , with a weighted effective number density n eff = 5 . 9 per arcmin 2 and corresponding shape noise σ e = 0 . 26 . \nThe catalog has passed a variety of empirical tests [117], mostly aimed at identifying residual additive biases in the shear estimates. Systematic errors related to PSF modeling were shown to be negligible for the DES Y3 analysis, due to improved PSF modeling [118]. The B-mode signal was also shown to be consistent with zero. Other tests included the dependence of the shear estimates on galaxy and survey properties. \nWhile shear calibration is typically viewed as separable from redshift inference, which is described in the following section, we also account for the first time for how blending correlates the ensemble shear calibration in each redshift bin with corrections to the effective shape of the n ( z ) of each of four redshift bins [120]. These corrections stem from a blending-detection bias, which biases both the ensemble average shear (some blends will only be detected as separate objects depending on the shear) and redshift distribution in a potentially correlated way. One way to treat these effects coherently is to model the multiplicative shear calibration as a scaling of the total number density in each redshift bin, and fit these effects fully in redshift space. We would model then the shear calibration bias as e i j = ∫ ∞ 0 n γ ( z ) γ i j ( z ) + c , for additive shear bias c , observed ellipticity e , true ellipticity γ , shear component j , and redshift bin i . In practice we continue to separate a scalar multiplicative ( m ) shear bias component to be compatible with existing codes, where n γ ( z ) ∝ (1 + m ) n ( z ) . \nFIG. 1. The source (top), MagLim lens (middle), and redMaGiC lens (bottom) redshift distributions. The histograms are normalized to integrate to the total weighted galaxy density (arcmin -2 ) in each tomographic bin. The equivalent 1 σ uncertainties on the redshift distributions are indicated by the shaded regions. The distributions have been corrected by non-zero mean and width offsets derived in the relevant photoz uncertainty models. We adopt MagLim as our fiducial lens sample in this work, and use only redshift bins 1-4. \n<!-- image -->', '2. Photometric redshifts': "The full redshift inference process for the source galaxies [123] relies on connecting information about deep-field galaxies to those in the wide field that are used for the cosmological analysis [154, 155]. All galaxies in the deep fields with similar properties are clustered together into different 'phenotypes' via a self-organizing map (SOM), while the same is also separately done for all galaxies in the wide field. The deep-field galaxies have much lower photometric noise and additional wavelength information (i.e., overlapping infrared photometry), so more phenotypes can be uniquely identified than for the wide-field galaxies. A redshift distribution is inferred for each of the deep-field galaxy phenotypes using overlapping spectroscopic [156-160], and photometric COSMOS [147] and PAUS [161, 162] redshift measurements. We then create a probabilistic mapping between the deep- and wide-field phenotypes using the Balrog simulation [121]. For example, if a given wide-field galaxy phenotype was mapped uniquely onto \na single deep-field galaxy phenotype, its redshift distribution would be determined by the available redshift measurements of the deep-field galaxies that share that particular phenotype. In practice, the mapping is much more complicated: each wide-field phenotype has a non-zero probability of coming from many deep-field phenotypes, but the algorithm for generating an n ( z ) for that galaxy phenotype is simply a weighted average. A given redshift bin is defined by a unique subset of many wide-field galaxy phenotypes, and its n ( z ) then follows by averaging over these phenotypes. The four source redshift bins have edges z ∈ [0 . 0 , 0 . 36 , 0 . 63 , 0 . 87 , 2 . 0] . \nThe process we use to account for uncertainties accumulated in each step of this process is summarized in Ref. [123]. These are due in part to shot-noise and cosmic variance in the redshift samples and deep fields [163], and photometric calibration uncertainty. At low redshift, this uncertainty is primarily due to uncertainties in the photometric calibration, while at high redshift it is due to a combination of cosmic variance and uncertainties in the redshift samples. Uncertainty in the n ( z ) due to these effects are modeled or measured, and we generate many realizations of the redshift distribution, n i ( z ) , that appropriately sample the joint space of this uncertainty without relying on a simple parameterization like mean and width. \nThe emerging set of redshift realizations suffer from one further source of uncertainty that has not been explicitly included before: blending. Galaxies that are nearby one another when projected on the sky can actually be very far apart. Detection and measurement algorithms can misinterpret these blends and report not only incorrect shapes but also incorrect number densities or redshifts. To account for this effect, we created realistic simulations [120] and apply the same detection and measurement pipeline used for the DES data to obtain the 'observed' number density, shape, and photometric redshift of a known simulated object population that matches our deep field data, from which the impact of blending can be inferred. The result is a likelihood model describing the impact of blending on the joint shear calibration and n ( z ) shape that we add to the n i ( z ) [120]. \nWe empirically constrain the likelihood of each n i ( z ) using information from galaxy clustering on small scales that is not used in the primary 3 × 2pt observables [125]. We know that galaxies are likely to be found near other galaxies due to gravitational clustering. Therefore, if there is a galaxy in a given direction whose redshift is known, it is likely that nearby galaxies on the sky are at a similar redshift. We make use of multiple galaxy samples with well-determined redshifts and cross-correlate them with the wide-field source sample, thereby obtaining a likelihood for each of the n i ( z ) , which is jointly sampled with the models that produce the n i ( z ) before we account for blending effects. We produce several thousand n i ( z ) samples. We show the final redshift distributions and their uncertainties in the top panel of Fig. 1. \nTo sample over n i ( z ) in a likelihood analysis, we introduce a set of hyper parameters to our model that rank the n i ( z ) in multiple dimensions [127]. Ref. [127] has demonstrated that our constraints on the variation in n i ( z ) (i.e., shown in Fig. 1) are sufficiently precise that uncertainty in higher-order modes in the n i ( z ) , besides the mean redshift, were not ex- \npected to impact our cosmological constraints at a significant level in Y3. Thus in practice we simply sample the mean of the redshift distribution in the four redshift bins within a Gaussian prior based on the measured variance in the mean of each n i ( z ) . \nFinally, the measured two-point functions themselves further constrain the possible values of the redshift distributions via self-calibration in 3 × 2pt. We further augment this by explicitly using a set of the ratios of the galaxy-galaxy lensing signal on small scales between source redshift bins sharing the same lens bin [128], which contains information not used in the standard 3 × 2pt analysis. These scales are too difficult to model robustly in full, but the ratios are to first order independent of cosmological model and depend primarily on the redshift distribution and intrinsic alignment parameters, and to a lesser degree on any redshift dependent bias in shear calibration. This small-scale shear ratio likelihood is jointly sampled in the cosmological analyses.", 'B. Lens Galaxies': "We have selected two galaxy populations (MagLim and redMaGiC) that serve as 'lenses' in galaxy-galaxy lensing measurements and for galaxy clustering measurements. The fiducial results presented in this work use the MagLim sample. We now describe the two lens samples.", '1. MagLim sample': "We have selected a magnitude-limited lens sample [135, 141], which results in 10.7 million galaxies. This 'MagLim' sample is defined with a magnitude cut in the i -band that depends linearly on redshift, i < 4 z + 18 , where z is the photometric redshift estimate from the Directional Neighborhood Fitting (DNF) algorithm [115, 164]. This selection was optimized for w CDMconstraints [135]. \nThe MagLim sample is divided into six tomographic bins from z = 0 . 2 to z = 1 . 05 , with bin edges z = [0 . 20 , 0 . 40 , 0 . 55 , 0 . 70 , 0 . 85 , 0 . 95 , 1 . 05] . The redshift distributions from DNF are shown in the middle panel of Fig. 1 and have been validated using galaxy clustering cross-correlations [122]. Weights are derived to account for correlations in the number density with survey properties [136]. Further validation and characterization of the sample is described in Refs. [136, 141]. After unblinding, we discovered issues with the sample above z = 0 . 85 , which lead to disagreement between the galaxy clustering and galaxy-galaxy lensing signal, and contribute to a substantially poor model fit to any cosmological models considered in this work (i.e., the two right-most panels of Fig. 2). This led us to remove these redshift bins in the fiducial analysis, which is discussed further in Secs. V A &VC.", '2. redMaGiC sample': "This sample is selected with the redMaGiC algorithm [165], which results in 2.6 million galaxies. redMaGiC selects Luminous Red Galaxies (LRGs) according to the magnitudecolor-redshift relation of red sequence galaxies, calibrated using spectroscopic redshifts. The sample has a luminosity threshold L min and approximately constant comoving density. The redMaGiC sample has approximately 30% narrower redshift distributions than MagLim, but approximately onefourth the number of objects. \nWe split the redMaGiC sample into five tomographic bins, selected on the redMaGiC redshift point estimate quantity. The bin edges used are z = [0 . 15 , 0 . 35 , 0 . 50 , 0 . 65 , 0 . 80 , 0 . 90] . The first three bins use a luminosity threshold of L min > 0 . 5 L ∗ (the 'high density' sample). The last two redshift bins use a luminosity threshold of L min > 1 . 0 L ∗ (the 'high luminosity' sample). The redshift distributions are computed by stacking samples from a non-Gaussian redshift PDF of each individual redMaGiC galaxy. Each distribution is built from several draws of the redshift PDF and are shown in the bottom panel of Fig. 1. The mean and RMS width of the redshift distributions are validated using galaxy clustering cross-correlations in Ref. [122]. \nWeights are derived to account for correlations in the number density with survey properties [136]. Further validation and characterization of the sample is also described in Refs. [136, 140]. We find a potential residual systematic in the redMaGiC sample at all redshifts, which does not impact Λ CDMinference and is also discussed in Sec. V C.", 'III. TWO-POINT MEASUREMENTS': 'To extract cosmological information from the lens and source catalogs, we compute three sets of two-point correlation functions , which each measure information about how mass in the Universe is clustered. There are two fields representing the matter distribution that we can access with a galaxy survey: 1) the galaxy density field and 2) the weak lensing shear field. These two fields lead to these three sets of measured two-point functions. \nGalaxy Clustering : The two-point function between lens galaxy positions in redshift bins i and j , w ij ( θ ) , describes the excess (over random) number of galaxy pairs separated by an angular distance θ . The estimator for w ij ( θ ) and its measurement and validation process are described in detail in Ref. [136]. We only use the auto-correlations of the measured w ii ( θ ) in our analysis; these are shown with their uncertainties in Fig. 2 for MagLim and in App. C for redMaGiC. \nGalaxy-Galaxy Lensing : The two-point function between lens galaxy positions and source galaxy tangential shear in redshift bins i and j , γ ij t ( θ ) , describes the over-density of mass around galaxy positions. The matter correlated with the lens galaxy alters the path of the light emitted by the source galaxy, thereby distorting its shape. The estimator for γ ij t ( θ ) and its measurement and validation process are described in \nFIG. 2. The measured w ( θ ) correlation functions for each tomographic bin i of the MagLim lens galaxies (indicated by the i, i label in each panel). The best-fit Λ CDM model from the fiducial 3 × 2pt analysis is plotted as the solid line in the top part of each panel, while the bottom part of each panel shows the fractional difference between the measurements and the model prediction, ( w obs. -w th. ) /σ w (with y -axis range ± 5 σ ). In both the top and bottom part of each panel, 1 σ error bars are shown. Small angular scales where the linear galaxy bias assumption breaks down are not used in the cosmological analysis; these scales are indicated by grey shading. Bins 5 & 6 are not used in the final analysis. \n<!-- image --> \nFIG. 3. The measured γ t ( θ ) correlation functions for each tomographic bin combination using the MagLim sample. In each panel, the label i, j refers to MagLim lens tomographic bin i and the source bin j The best-fit Λ CDMmodel from the fiducial 3 × 2pt analysis is plotted as the solid line in the top part of each panel, with dotted curves indicating a negative model fit. The bottom part of each panel shows the fractional difference between the measurements and the model prediction, ( γ obs. t -γ th. t ) /σ γ t (with y -axis range ± 5 σ ). In both the top and bottom part of each panel, 1 σ error bars are included. Small angular scales where the linear galaxy bias assumption breaks down are not used in the cosmological analysis; these scales are indicated by grey shading. Bins 5 & 6 are not used in the final analysis. \n<!-- image --> \n⋃[˜∐√(∫∐√]} \nFIG. 4. The measured small-scale shear ratio values for each tomographic bin combination using the MagLim sample, with 1 σ error bars indicated. The x-axis identifies the two source bins that make up the measured ratio. The best-fit cosmological model from the fiducial 3 × 2pt analysis is over-plotted as the solid line for each set of lens-bin shear ratios. \n<!-- image --> \n∖∖∖∖∖∖∖∖∖ \n⋃}√√̂˜∖̂]{(∫∐√]} \ndetail in Ref. [137]. The measured γ ij t ( θ ) and their uncertainties are shown in Fig. 3 for MagLim and in App. C for redMaGiC. In addition, we include small-scale shear ratio information below the scale cuts used for γ t . These ratios are constructed from γ t measurements using different source galaxy bins, while keeping the lens bin fixed. This effectively erases their dependence on the galaxy power spectrum, but keeps information about redshift calibration, shear calibration, and galaxy intrinsic alignment. A detailed description of shear ratios and their validation can be found in Ref. [128]. Fig. 4 shows the shear-ratio measurement and uncertainties. This shear ratio data is included when analyzing all combinations of the three primary two-point functions in our analyses, unless otherwise noted. \nCosmic Shear : The correlation between source galaxy shears in redshift bins i and j is described by the two functions ξ ij ± ( θ ) , which are the sum and difference of the products of the tangential- and cross-components of the projected shear. The estimator for ξ ij ± ( θ ) and its measurement and validation process are described in detail in Refs. [142, 143]. The measured ξ ij ± ( θ ) and their uncertainties are shown in Fig. 5. \nThe total data vector includes measurements from five or six lens redshift bins and four source redshift bins, shown in Fig. 1, split into 20 logarithmic angular bins between 2.5 and 250 arcmin, for a total of 1300 elements (not including shear-ratio). After bin pair removal for w ( θ ) , imposing a post-unblinding maximum lens redshift cut, and other scale cuts, 462 elements remain in the final 3 × 2pt data vector. The scale cut choices and their validation are described in Refs. [119, 129, 140, 143], but are generally set to control the impact of unmodeled non-linear effects (e.g., baryonic effects on the matter power spectrum or higher-order galaxy bias) to better than 0 . 3 σ in the Ω m -S 8 plane in Λ -and w CDM. A choice of scale cuts that require biases meet our fiducial requirements in Λ CDM-only ( Λ CDM-optimized) leaves 508 data points. All measurements are made using TreeCorr [166]. We find a total signal-to-noise S/N = 87 for the 3 × 2pt data vector after fiducial scale cuts, where S/N ≡ ξ data C -1 ξ model / √ ξ model C -1 ξ model , with covariance matrix C and best-fit model ξ model . This is a factor of 2.1 improvement over the DES Year 1 3 × 2pt S/N . \nAll of these measurements are related to the underlying clustering of matter in the Universe, but in different ways. The relationship between the galaxy density and the underlying matter density is complex [140] and needs to be modeled with care. Alternately, the shape distortions depend more directly on the intervening matter, but the measurements themselves - especially of shapes and the redshift distribution require greater care. Our work on this calibration is summarized in Appendix A. The advantage of using all of these measurements is that the systematic difficulties differ from one to another, but they all measure the same underlying matter field. Hence, by comparing the results from each set, we obtain a measure of consistency and additional ability to self-calibrate systematics, thereby giving confidence that we are correctly inferring information about the clustering of matter and the cosmological model.', 'IV. ANALYSIS': "To infer parameters p from the measured two-point functions, we compare data organized in a 'data vector' ˆ D , \nˆ D ≡ { ˆ w i ( θ ) , ˆ γ ij t ( θ ) , ˆ ξ ij ± ( θ ) } , (1) \nto a theoretical model prediction organized in a vector T M of two-point correlation functions that are computed using the parameters p of a given model M , \nT M ( p ) ≡ { w i ( θ, p ) , γ ij t ( θ, p ) , ξ ij ± ( θ, p ) } , (2) \nassuming a Gaussian likelihood, \nL ( ˆ D \| p , M ) ∝ e -1 2 [ ( ˆ D -T M ( p ) ) T C -1 ( ˆ D -T M ( p ) ) ] . (3) \nHere C is the data covariance, which is obtained through analytic modeling as described and validated in Ref. [130]. \nWe construct a posterior probability distribution for the parameters p of the theoretical model given the data ˆ D as \nP ( p \| ˆ D , M ) ∝ L ( ˆ D \| p , M ) P ( p \| M ) , (4) \nwhere P ( p \| M ) is a prior probability distribution on the parameters. The proportionality constant is given by the inverse of the Bayesian Evidence \nP ( ˆ D \| M ) = ∫ d p L ( ˆ D \| p , M ) P ( p \| M ) , (5) \nwhich corresponds to the marginalized probability of a dataset being produced under a given theoretical model. \nThis section summarizes the theoretical model and parameterization we use for T M ( p ) , which is described in more detail in Ref. [129] and validated in Refs. [119, 129, 139, 140]. For clarity, we drop the parameter argument of the theoretical model predictions, such that, e.g., the predicted clustering signal is simply denoted as w i ( θ ) . \nWe report the mean in each parameter, along with the 68% confidence limit (CL) of posterior volume around the mean. \n- \n- \n- \n- \n- \n- \n- \n- \n- \n- \n- \n- \nFIG. 5. The measured ξ ± ( θ ) correlation functions for each tomographic bin combination, with labels as described in Fig. 3. The best-fit Λ CDM model from the fiducial 3 × 2pt analysis is plotted as the solid line in the top part of each panel, while the bottom part of each panel shows the fractional difference between the measurements and the model prediction, ( ξ obs. ± -ξ th. ± ) /σ ξ ± (with y -axis range ± 5 σ ). In both the top and bottom part of each panel, 1 σ error bars are included. The shaded regions (both light and dark) indicate scales not used in the fiducial analysis, primarily due to uncertainties in the impact of baryonic effects. The lighter shaded regions indicate scales that are used in an Λ CDM-optimized analysis, which meets our criterion for scale cuts described in Sec. IV in Λ CDMonly. \n<!-- image --> \nFor completeness, we also report the best-fit maximum posterior values. We have used both a parameter-level and χ 2 criterion for limiting the contribution of any systematic error to bias in the cosmological parameters. The threshold for this criterion is intended to limit the expected total bias in the 2D marginalized Ω m -S 8 plane from several independent potential sources of model bias to be contained within the 68% C.L. region [129] ( < 0 . 3 σ for any single contribution). The difference between the mean and best-fit values can give an indication of the magnitude of projection or non-Gaussian effects in the marginalized parameter posteriors. The estimated impact of projection or volume effects in the DES Year 3 3 × 2pt posteriors are tested and summarized in Ref. [129]. We also provide a 2D figure of merit (FoM) defined for two parameters as FoM p 1 ,p 2 = (det Cov( p 1 , p 2 )) -1 / 2 [167, 168]. The FoM is proportional to the inverse area of the confidence region in the space of the two parameters, and can be considered a summary statistic that enables a straightforward comparison of constraining power of experiments or analysis scenarios. \nThe analysis was designed and validated without access to the true cosmological results to protect against confirmation or \nobserver bias. This process is described in detail in App. D.", 'A. Model': "Wemodel the observed projected (lens) galaxy density contrast δ i obs (ˆ n ) as a combination of projected galaxy density contrast and modulation by magnification, δ µ , \nδ i obs (ˆ n ) = δ i g (ˆ n ) + δ i µ (ˆ n ) (6) \nfor position vector ˆ n , where i and j represent the redshift bin. The observed shear signal γ is modeled as the sum of gravitational shear, γ G , and intrinsic alignments, glyph[epsilon1] I , \nγ j α (ˆ n ) = γ j G ,α (ˆ n ) + glyph[epsilon1] j I ,α (ˆ n ) , (7) \nwith α the shear components. While B-modes produced by higher-order weak lensing effects are negligible for our analysis, it is important to account for B-modes generated by intrinsic alignments in the computation of cosmic shear two-point \nTABLE I. The model parameters and their priors used in the fiducial flat Λ CDM and w CDM analyses. The parameter w is fixed to -1 in Λ CDM. The parameters are defined in Sec. IV B. \ncorrelation functions. In Fourier space, this decomposition can be written as \nγ j E ( glyph[lscript] ) = κ j ( glyph[lscript] ) + glyph[epsilon1] j I , E ( glyph[lscript] ) , γ j B ( glyph[lscript] ) = glyph[epsilon1] j I , B ( glyph[lscript] ) , (8) \nwith the convergence field \nκ j (ˆ n ) = ∫ dχW j κ ( χ ) δ m (ˆ n χ, χ ) , (9) \nwhere δ m is the 3D matter density contrast. The galaxy density contrast δ g is related to δ m via a linear galaxy bias b i . The tomographic lens efficiency is \nW j κ ( χ ) = 3Ω m H 2 0 2 ∫ χ H χ dχ ' n j s ( χ ' ) χ a ( χ ) χ ' -χ χ ' . (10) \nχ is the comoving distance, χ H the comoving distance to the horizon, n s ( χ ) the source galaxy number density distribution, and a ( χ ) the scale factor.", '1. Two-point statistics': 'The angular power spectra C ( glyph[lscript] ) of these observed fields can be written as \nC ij EE ( glyph[lscript] ) = C ij κκ ( glyph[lscript] ) + C ij κ I E ( glyph[lscript] ) + C ji κ I E ( glyph[lscript] ) + C ij I E I E ( glyph[lscript] ) C ij BB ( glyph[lscript] ) = C ij I B I B C ij δ obs E ( glyph[lscript] ) = C ij δ g κ ( glyph[lscript] ) + C ij δ g I E ( glyph[lscript] ) + C ij δ µ κ ( glyph[lscript] ) + C ij δ µ I E ( glyph[lscript] ) C ii δ obs δ obs ( glyph[lscript] ) = C ii δ g δ g ( glyph[lscript] ) + C ii δ µ δ µ ( glyph[lscript] ) + C ii δ RSD δ RSD ( glyph[lscript] ) +2 C ii δ g δ µ ( glyph[lscript] ) + 2 C ii δ g δ RSD ( glyph[lscript] ) + 2 C ii δ RSD δ µ ( glyph[lscript] ) . (11) \nWith the exception of the galaxy clustering power spectra C δ obs δ obs , which are evaluated using the method described in Ref. [169], we calculate the angular cross-power spectrum between two fields A,B using the Limber approximation \nC ij AB ( glyph[lscript] ) = ∫ dχ W i A ( χ ) W j B ( χ ) χ 2 P AB ( k = glyph[lscript] + 1 2 χ , z ( χ ) ) , (12) \nwith P AB the corresponding three-dimensional power spectrum, which is specified by the parameterization choices summarized in IV B. The kernels W ij A,B correspond to W j κ for shear and the lens galaxy density n i l for position. The twopoint correlation functions within an angular bin [ θ min , θ max ] are related to the projected power spectra as \nw i ( θ ) = ∑ glyph[lscript] G 0 ( glyph[lscript], θ min , θ max ) C ii δ obs δ obs ( glyph[lscript] ) γ ij t ( θ ) = ∑ glyph[lscript] G 2 ( glyph[lscript], θ min , θ max ) C ij δ obs E ( glyph[lscript] ) (13) ξ ij ± ( θ ) = ∑ glyph[lscript] G 4 , ± ( glyph[lscript], θ min , θ max ) [ C ij EE ( glyph[lscript] ) ± C ij BB ( glyph[lscript] ) ] , \nwith G n analytic functions detailed in Refs. [129, 130] .', 'B. Parameterization and Priors': "We sample the posterior of these measurements in two cosmological models: flat Λ CDM and w CDM, with the sum of the three neutrino masses as a free parameter, where the impact of neutrino mass on the power spectrum is modeled via a fitting function [ ? ]. Λ CDM contains three energy densities in units of the critical density: the total matter density Ω m , the baryonic density Ω b , and the massive neutrino density Ω ν . We vary Ω ν h 2 , where h is the Hubble parameter, as a free parameter, while noting that it is often fixed in other cosmological analyses to be zero or to the minimum mass allowed by oscillation experiments m ν = 0 . 06 eV [170]. \nThe other cosmological parameters we vary within Λ CDM are the Hubble parameter h , the amplitude of primordial scalar density perturbations A s , and the spectral index n s of the power spectrum. We assume a flat model, with Ω Λ = 1 -Ω m . In w CDM, we allow for a free dark energy equation-of-state parameter w that is constant in time (in Λ CDM, this is fixed to w = -1 , corresponding to a cosmological constant). Thus Λ CDMincludes six free cosmological parameters and w CDM contains seven. The prior ranges for cosmological parameters in Table I are either motivated by physical constraints (e.g., an accelerating universe requires w < -1 / 3 ), or for parameters that are not strongly constrained by the DES data, typically given a range that encompasses five times the 68% C.L. from relevant external constraints. In analyses that sample external CMB likelihoods, we include the optical depth τ as a free parameter. \nWe will typically refer to the amplitude of density perturbations at z = 0 in terms of the RMS amplitude of mass on scales of 8 h -1 Mpc in linear theory, σ 8 . The constraints on the amplitude and density of matter fluctuations are degenerate in our analysis, and we will also refer to the parameter S 8 , which describes the width of the posterior in the direction roughly orthogonal to the primary degeneracy direction for cosmic shear in the σ 8 -Ω m plane [171], though this does not hold exactly for 3 × 2pt and changes with effective redshift. \nIn addition to these cosmological parameters, our fiducial analysis includes an additional 25 free parameters, for a total of 31 (32) parameters in Λ CDM ( w CDM). These additional parameters describe astrophysical and systematic contributions to the measured signal. The effective linear galaxy bias of lens galaxies in each redshift bin is parameterized by a scalar b i . We also test and apply a nonlinear galaxy bias model (with one extra free parameter per redshift bin) [119, 140, 141], which is described in App. E 2. The intrinsic alignment of galaxies [172, 173] is modeled with the Tidal Alignment and Tidal Torquing (TATT) model [174], which is parameterized by an amplitude a i and redshift power-law η i parameter (with redshift pivot z 0 = 0 . 62 ), for each of the (1) tidal alignment- and (2) tidal torquing-sourced terms in the model, as well as an effective source galaxy bias parameter b TA , which is described in further detail in Refs. [129, 174]. The TATT model contains the commonly employed nonlinear linear alignment (NLA) model in the a 2 = b TA = 0 subspace. The amplitude of the lens magnification term in Eq. 6 depends on the slope of the lens sample's luminosity and size distribution at the sample detection limit. The corresponding parameter C i l is calibrated from the data, as described in Ref. [139], and held fixed to that value. Nonlocal effects in γ t can significantly contaminate larger angular scales with nonlinear information due to integration of the projected mass within a given angular separation from the center of the halo. This is mitigated by analytically marginalizing over a free point-mass contribution to γ t in all analyses [175]. \nPhotometric redshift systematics are parameterized by an additive shift to the mean redshift of each bin, ∆ z i l for lenses and ∆ z i s for sources, where the true redshift distribution is related to the photometric redshift distribution n pz such that \nn i ( z ) = n i pz ( z -∆ z i ) . (14) \nIn addition, differences in the width of the lens redshift distribution are important at DES Y3 precision, which we parameterize by a stretch σ i z , such that \nFinally, uncertainty in the shear calibration bias is parameterized by m i , where the measured ellipticity e j is related to the true shear γ j in each bin by \nn i ( z ) = σ i z n i pz ( σ i z [ z -〈 z 〉 ] + 〈 z 〉 ) . (15) \ne i j = (1 + m i ) γ i j . (16) \nThe full set of parameters (cosmological, astrophysical, and systematic) and their priors are summarized in Table I. \nDifferences in the redMaGiC analysis are described in App. C.", 'C. Likelihood Analysis': 'Our likelihood analysis uses two independently developed analysis and inference pipelines, COSMOSIS [176] and COSMOLIKE [177], which have been validated against one another to ensure they produce consistent predictions of the observables and final cosmological constraints. A comparison of the theory predictions from COSMOSIS and COSMOLIKE is presented in Ref. [129]. The residual offset of χ 2 < 0 . 2 between 3 × 2pt model data vectors obtained from both codes in this analysis at a reference cosmology is found to have negligible impact on parameter constraints and we conclude that both pipelines can be used interchangeably.', '1. CosmoSIS': 'This pipeline uses the CAMB Boltzmann code [178, 179] to compute underlying background quantities and the linear matter power spectrum, and the HALOFIT [180] version presented in Ref. [181] for the non-linear power spectrum. It then generates theory predictions following the model described in section IV A, and using the Fast-PT method [182] for non-linear galaxy bias and the TATT model for intrinsic alignments. Non-Limber integrals are computed following the method of [169]. Accuracy parameters throughout the pipeline are chosen by requiring the log-likelihood to differ by less than 0.05 from a high precision calculation. For chains including Planck CMBmeasurements [13], we use the Planck 2018 public likelihood code [183]. 5 \nThe version of COSMOSIS 6 used for the analysis may be found in the des-y3 branch of the repositories. The COSMOSIS runs presented here use the PolyChord sampling \nmethod [184, 185], for both posterior samples and Bayesian evidence. The shear calibration values m i are used as fast parameters. The PolyChord parameters we use for our fiducial runs are: fast\\_fraction = 0 . 1 , live\\_points = 500 , num\\_repeats = 60 , tolerance = 0 . 01 , and boost\\_posteriors = 10 . 0 . \nAnalyses with the COSMOSIS pipeline also use the nonGaussian covariance matrix from COSMOLIKE described below.', '2. CosmoLike': 'This pipeline uses the CLASS Boltzmann code [186] to compute underlying background quantities and linear and non-linear matter power spectra, using the HALOFIT version presented in Ref. [181] for the latter. The theory predictions are calculated using the model described in section IV A, relying on the FAST-PT method [182, 187] to evaluate integrals over perturbation theory kernels for non-linear galaxy bias and the TATT intrinsic alignment model. The computation of non-Limber integrals for galaxy clustering further employs the FFTLog implementation of Ref. [169]. \nThe evaluation time of angular two-point statistics in COSMOLIKE is optimized through a series of interpolation schemes, for which runtime-optimized accuracy settings are validated through comparison to high-accuracy evaluations with slow runtime. Most DES Y3 likelihood analyses with COSMOLIKE employ the EMCEE [188] sampler, c.f. [189] for a detailed comparison of sampler configurations for COSMOLIKE likelihood analyses. \nThe COSMOCOV module [190] of COSMOLIKE is used to generate covariances for DES Y3 analyses, which include Gaussian and non-Gaussian terms [177] and account for the effect of the survey geometry on shape and shot-noise terms [101].', 'D. Tests on Simulations': "Our model and many other components of our analysis have been validated end-to-end on a suite of 18 cosmological simulations 7 [119, 191-193]. Validation is performed on the mean of measurements from all 18 of these simulations without shape noise, including photoz s and marginalizing over all cosmological and nuisance parameters. We have verified that we can recover the correct cosmology with our fiducial analysis to within approximately 0 . 3 σ in the 2D σ 8 -Ω m plane ( Λ CDM) and w -Ω m plane ( w CDM). We have shown that for a more stringent test in the absence of photometric redshift and shear calibration uncertainties (using true redshifts) our model is able to reproduce the mean ξ ± and w + γ t measurements from our 18 simulations with a χ 2 of 1.4 for cosmic \nshear (207 data points), 4.5 for w ( θ ) (53 data points), and 9.1 for γ t ( θ ) (232 data points). These χ 2 numbers are relative to the fiducial covariance for a single DES Y3 realizations, but with a measurement that is the average of 18 realizations without shape noise. Thus, they represent the potential systematic χ 2 contribution due to model inaccuracies, and shouldn't be interpreted as a goodness-of-fit metric. \nIn addition to these model tests, we have also investigated the systematic uncertainty inherent to our redshift inference process using these simulations. We have shown that the three independent source redshift n ( z ) estimates - SOMPZ, source-lens clustering, and shear ratios - produce consistent constraints on the redshift distribution. We have also performed our fiducial analysis using a redMaGiC-like lens sample, assuming source redshift distributions that are calibrated using the same three methods and lens redshift distributions estimated from redMaGiC. We found that the constraints from this analysis are consistent with those that use the true redshift distributions from the simulation. The final constraining power is similar between redMaGiC and MagLim, so we do not repeat the simulated analysis twice.", 'E. Quantifying internal and external consistency': 'To quantify consistency of internal and external data sets, we define a priori a process to guide decisions and conclusions before seeing the cosmological constraints. For internal consistency, we calculate the Posterior Predictive Distribution (PPD) [132] and derive a (calibrated) probability-to-exceed p . In short, the idea is to draw realizations of a particular subset of the data vector for model parameters drawn from the posterior of the same subset (goodness-of-fit tests) or a disjoint subset (consistency tests). These realizations are then compared to actual observations and a distance metric is computed in data space, which is then used to compute the p -value. We test the goodness-of-fit for the two combinations of two-point functions, ξ ± and w + γ t , and the combination of all three after confirming they are mutually consistent. In all cases, we require as part of the unblinding criteria defined in App. D that p > 0 . 01 . The validation of the use of PPD for these tests is described in Ref. [132]. \nTo quantify consistency with external experiments, we have explored a variety of metrics in order to calibrate expectations, which are described in Ref. [133]. We studied the particular case of quantifying consistency between DES 3 × 2pt and Planck CMB using both simulated DES Y3 data and real Y1 data. The consistency metrics can be divided into two categories: parameter-based, which measure relative deviations in the multi-dimensional parameter space, and Evidence-based, which also account for how well the individual and combined data sets fit the model. We discuss results in terms of at least one metric from each category: the parameter difference and Suspiciousness [194], along with the Evidence ratio to compare to DES Y1. These metrics can produce a probability-toexceed, and we require the same criterion of p < 0 . 01 to conclude there exists evidence for inconsistency between probes. \nDetailed results from these consistency studies are shown \nFIG. 6. Marginalized constraints on the three parameters σ 8 , S 8 = σ 8 √ Ω m / 0 . 3 , and Ω m in the Λ CDM model from cosmic shear ( ξ ± , blue), galaxy clustering and galaxy-galaxy lensing ( γ t + w ( θ ) , orange) and their combination (3 × 2pt, solid black). We also show a Λ CDM-optimized 3 × 2pt analysis that is valid for Λ CDM using smaller angular scales in cosmic shear (dashed black). The marginalized contours in this and further figures below show the 68% and 95% confidence levels. The top and side panels show 1D marginalized constraints with the 68% confidence region indicated. \n<!-- image --> \nin App. F.', 'A. Λ CDM': "The principal cosmological test of the DES Y3 data is to compare our data to the currently favored Λ CDMmodel. The model has six cosmological parameters, but also 25 nuisance parameters for a total of 31 free parameters (listed in Table I) in our fiducial analysis with the MagLim lens sample. Recall also that, in the fiducial analysis, we use all four source-galaxy bins, but only the first four (out of six total) MagLim lensgalaxy bins. \nWe first concentrate on comparing two subsets of 3 × 2pt \nmeasurements: those from cosmic shear ( ξ ± ) and those from the combination of galaxy clustering and galaxy-galaxy lensing ( w + γ t ). It is logical to compare these two subsets because each can constrain the Λ CDM parameters to a similar precision, yet the constraints from the two subsets contain independent information, since they are sensitive to the underlying matter density fluctuations in different ways. Moreover, comparing the two subsets of the 3 × 2pt measurements provides an internal consistency test. \nWe must first check that the ξ ± and w + γ t measurements are a good fit to the data, mutually consistent, and that their combination (3 × 2) is a good fit to the model. For each of the two measurements individually, we find a PPD result for model goodness-of-fit p ( ξ ± ) = 0 . 21 and p ( w + γ t ) = 0 . 02 . The PPD result for consistency between the two model constraints is p ( ξ ± \| w + γ t ) = 0 . 30 , meaning that it is appropriate to combine ξ ± and w + γ t . The joint 3 × 2pt goodness-of-fit is p ( ξ ± + γ t + w ) = 0 . 04 . Finally, the shear-ratio data has goodness-of-fit p = 0 . 03 in this joint best-fit model. All of these p values meet our original criterion of p > 0 . 01 defined in App. D. \nThe marginalized constraints from each probe and the 3 × 2pt combination in the parameter space spanned by σ 8 , S 8 , and Ω m are shown in Fig. 6. This is also summarized in Table II and Fig. 7, which show the numerical constraints on these three parameters. The DES Y3 3 × 2pt constraints on the key parameters are \nS 8 = 0 . 776 +0 . 017 -0 . 017 (0 . 776) Ω m = 0 . 339 +0 . 032 -0 . 031 (0 . 372) σ 8 = 0 . 733 +0 . 039 -0 . 049 (0 . 696) . (17) \nThe 3 × 2pt contours in these parameters are not centered on the overlap of ξ ± and γ t + w due to degeneracies in the higher dimensional parameter space. \nWe also perform two alternative 3 × 2pt analyses that use smaller scales. First, we perform a Λ CDM-optimized analysis that includes smaller-scale information in cosmic shear. This analysis meets our parameter bias requirements in Λ CDM (i.e., Sec. IV), but not w CDM. The 3 × 2pt results from the optimized analysis are shown in the row labeled ' Λ CDM-Opt.' in Table II and Fig. 7. The optimized results are consistent with the fiducial analysis, but are about 30% more constraining in the 2D marginalized Ω m -σ 8 plane. The second alternative analysis utilizes a more complicated nonlinear bias model in order to model smaller scale information in γ t + w ( θ ) , and is described in App. E. The 3 × 2pt results from the nonlinear analysis, shown in the row labeled 'NL bias' in Table II and Fig. 7, are consistent with the fiducial analysis and lead to an increase of 15% in constraining power in the Ω m -σ 8 plane. \nWhile we found no significant evidence of internal inconsistency with the Λ CDM model using the final MagLim lens selection, we have identified potential systematic modes in the data at high redshift for the MagLim sample and at all redshifts for the redMaGiC sample. We had agreed before seeing any cosmological results that we would pursue potential systematics in the case where the results failed to suffi- \nFIG. 7. Summary of marginalized constraints (mean and 68% CL) and maximum posterior values (crosses) on S 8 , Ω m , and σ 8 in Λ CDM. 'Ext. Lowz ' data consists of external SNe Ia, BAO, and RSD, while 'All Ext.' data consists of external SNe Ia, BAO, RSD, and Planck CMB with lensing. The top section shows constraints using only DES data, the middle section only external data, and the bottom section combinations of DES and external data. \n<!-- image --> \nFIG. 8. Constraints on the galaxy bias ( b g ) and effective intrinsic alignment (IA) amplitude from tidal alignment ( a 1 ) and tidal torquing ( a 2 ) are shown per redshift bin. Constraints using both lens samples (MagLim and redMaGiC) are shown. The galaxy bias is expected to be different for both lens samples, but the IA amplitude constraints, which are a property of the source galaxy sample, are consistent. We do not necessarily expect a 1 and a 2 to be consistent with one another. We sample over a power-law evolution of the IA amplitude, so the redshift evolution is forced to be smooth in a i . \n<!-- image --> \niently fit any of the models considered in this work ( Λ CDM and w CDM) at p < 0 . 01 . Including MagLim lens bins 5 and 6 caused a very poor model fit to both models, with p ≈ 5 × 10 -4 . Based on this criterion, we applied a highz cut to limit the MagLim sample to approximately the same redshift range of redMaGiC post-unblinding. This change is discussed further in App. D. The two lens samples are compared and further details of this are discussed in Sec. V C, but all issues that have been uncovered appear to be mostly orthogonal to the 3 × 2pt Λ CDM parameter dimensions - that is, they do not significantly impact the inferred cosmological parameters, and the cosmological parameters inferred from the two lens samples are consistent. This resilience of the 3 × 2pt combination of data and its ability to self-calibrate potential systematics in a subset of the two-point functions is one of the main motivations for pursuing this cosmological probe for large-scale structure. \nWe find that the DES Y3 3 × 2pt analysis is able to add information beyond the prior for 15 parameter dimensions in the model, three of which are cosmological. The cosmological modes that DES 3 × 2pt most improves with respect to the prior are obtained with the Karhunen-Loève decomposition of the posterior and prior covariance, and are: \np 1 = σ 8 Ω 0 . 77 m = 0 . 317 +0 . 015 -0 . 014 , p 2 = Ω m σ -1 . 16 8 = 0 . 49 +0 . 16 -0 . 15 , p 3 = hn 1 . 24 s Ω -0 . 39 b = 2 . 11 +0 . 45 -0 . 42 . (18) \nThe combined 3 × 2pt data is also able to simultaneously constrain a variety of 'astrophysical' parameters that encode how galaxies are connected to the underlying dark matter perturbation field, namely the linear and nonlinear bias parameters and intrinsic alignment of galaxies. Constraints for these model \nFIG. 9. Marginalized constraints on the two parameters Ω m and w in the w CDM model from DES Y3 3 × 2pt. A dotted line indicates w = -1 as given by the cosmological constant. \n<!-- image --> \nparameters are shown in Fig. 8. We find slightly higher galaxy bias constraints for redMaGiC galaxies than in the DES Y1 analysis using a similar redMaGiC sample. We find a preference for a slightly smaller intrinsic alignment amplitude than DES Y1. This value is consistent with the DES Y1 analysis, but is also consistent with zero intrinsic alignment.", 'B. w CDM': 'We also fit our data to the w CDMmodel, in order to test for evidence that the dark energy equation of state departs from its cosmological-constant value of w = -1 . In w CDM, the dark energy density evolves with time with a constant w , such that ρ DE ∝ (1 + z ) 3(1+ w ) . We show marginalized parameter posteriors for this model in Fig. 9 and parameter values in Table III and Fig. 10. \nWefindsimilar levels of agreement between ξ ± or γ t + w ( θ ) as in Λ CDM, and a similarly good fit to the data within the w CDM model, but do not show constraints from these subsets of the data due to increased prior influence and parameter volume effects. The DES Y3 3 × 2pt constraint on the matter density and dark energy equation of state parameter are \nΩ m = 0 . 352 +0 . 035 -0 . 041 (0 . 339) , w = -0 . 98 +0 . 32 -0 . 20 ( -1 . 03) . (19) \nTo determine if there is a preference for the w CDM model over the Λ CDMmodel, we compute the Bayes factor \nR = P ( ˆ D \| ΛCDM) P ( ˆ D \| w CDM) . (20) \nAvalue of R greater than unity implies that the w CDMmodel is not favored. We find R = 4 . 3 . This indicates that the lateuniverse large-scale structure probed by DES does not show evidence of needing the more complex dark energy density scenario of the w CDM model. We discuss further in Sec. VII more stringent tests of the Λ CDM model that leverage data across the age of the Universe.', 'C. Lens sample comparison': "As optical surveys cover larger fractions of the sky and probe higher redshifts, photometric galaxy clustering becomes both more powerful and more difficult to calibrate. Previous DES analyses used a luminous red sample of galaxies with constant comoving density, redMaGiC. To ensure robustness, we pursued two lens samples for DES Y3: a magnitudelimited lens sample, MagLim, and the redMaGiC sample. The redMaGiC sample was optimized for better understood and smaller photometric redshift errors. The MagLim sample was optimized for w CDM constraints, balancing increased number density vs. less well-constrained photoz s, while allowing selection to higher redshifts than possible with redMaGiC. Comparing the inferred cosmological parameters of our models from these two very different samples, which have fewer than 20% overlapping objects, allows us to infer potential uncorrected systematics from lens sample selection or photoz calibration of the lenses. \nMeasurements based on the second redshift sample, redMaGiC, also have an acceptable overall model fit to the Λ CDMand w CDMmodels. The cosmic shear in this model is also consistent with the combination of galaxy-galaxy lensing and galaxy clustering. These measurements and model fits are shown in App. C. We find a PPD result for model goodnessof-fit p ( ξ ± ) = 0 . 25 and p ( w + γ t ) = 0 . 04 , while the PPD result for consistency between the two model constraints is p ( ξ ± \| w + γ t ) = 0 . 02 . The joint 3 × 2pt goodness-of-fit is p ( ξ ± + γ t + w ) = 0 . 02 . \nThe marginalized constraints of each individual probe and the 3 × 2pt combination on S 8 and Ω m in Λ CDMare shown in Fig. 11, while they are shown for S 8 , w , and Ω m in w CDMin Fig. 12. Both figures compare redMaGiC results to the fiducial 3 × 2pt using the MagLim sample. As described above, the cosmic shear or γ t + w ( θ ) data alone are consistent with the 3 × 2pt model fit, though the γ t + w ( θ ) data on their own prefer a smaller S 8 value. This arises from the strong degeneracy between σ 8 and galaxy bias in γ t + w ( θ ) . Alone, it prefers a lower value of σ 8 and higher value of galaxy bias. Adding cosmic shear information effectively fixes the value of S 8 along that degeneracy, which brings the galaxy bias in 3 × 2pt back down to a value more consistent with the DES Year 1 redMaGiC galaxy bias constraints. \nThese results are consistent with the MagLim results and passed our unblinding requirements, including having a sufficiently good model fit to Λ CDM. However, after unblinding the results with redMaGiC we found evidence of internal tension in the data. Because w ( θ ) is not able to constrain cosmology on its own, this has limited impact on the combination \nTABLE II. Summary of marginalized parameter constraints in Λ CDM. The mean and 68% CL are provided for each cosmological parameter, followed by the maximum posterior value in parentheses, except for neutrino mass, for which the 95% upper bound is given. Parameters that are not significantly constrained are indicated by a dash. All data have been re-analyzed with model and prior choices matching the DES Y3 3 × 2pt analysis. \nFIG. 10. Summary of marginalized constraints (mean and 68% CL) and maximum posterior values (crosses) on S 8 , Ω m , and w in w CDM. 'Ext. Lowz ' data consists of external SNe Ia, BAO, and RSD, while 'All Ext.' data consists of external SNe Ia, BAO, RSD, and Planck CMB with lensing. The top section shows constraints using only DES data, the middle section only external data, and the bottom section combinations of DES and external data. \n<!-- image --> \nTABLE III. Summary of marginalized parameter constraints in w CDM. The mean and 68% CL are provided for each cosmological parameter, followed by the maximum posterior value in parentheses, except for neutrino mass, for which the 95% upper bound is given. Parameters that are not significantly constrained are indicated by a dash. All data have been re-analyzed with model and prior choices matching the DES Y3 3 × 2pt analysis. \nFIG. 12. A comparison of the marginalized w CDM constraints of the two lens samples. Dashed black contours show the 3 × 2pt constraints based on the redMaGiC lens sample. The 3 × 2pt redMaGiC constraints marginalizing over a free X lens parameter (dotted black) and the 3 × 2pt MagLim constraints (solid black) are also shown. The inferred cosmological parameters from 3 × 2pt are generally consistent, but in particular the redMaGiC results are sensitive to the impact of X lens in w CDM, showing substantial shifts in the inferred parameter values. \n<!-- image --> \nm \nFIG. 11. A comparison of the marginalized Λ CDMconstraints of the two lens samples. Dashed contours show the cosmic shear (blue), galaxy-galaxy lensing and clustering (orange), and 3 × 2pt (black) constraints based on the redMaGiC lens sample. The 3 × 2pt redMaGiC constraints marginalizing over a free X lens parameter are also shown (dotted black), and the 3 × 2pt MagLim constraints (solid black). The inferred cosmological parameters from 3 × 2pt are consistent in all three cases. \nof galaxy clustering and galaxy-galaxy lensing and no discernible impact on the 3 × 2pt combination in Λ CDM. Nevertheless, it is important to understand the source of this internal tension in redMaGiC results and judge its impact on cosmological inference. To do so, we modeled this inconsistency of the redMaGiC clustering and galaxy-galaxy lensing amplitudes with a systematic parameter X lens , which is related to the connection of the galaxy-galaxy lensing and galaxy clustering two-point functions to the matter two-point function: \nw ii ( θ ) = b 2 i ξ ii mm ( θ ) γ ij t ( θ ) = X lens b i ξ ij mm ( θ ) (21) \nwhere b i is the galaxy bias connecting the observable γ t or w ( θ ) to the matter correlation function ( ξ mm ) or spectrum and X lens is the same for all redshift bins i . We expect X lens = 1 in Λ CDM, if there are no systematic contributions to the signals. The fiducial model described in earlier sections is thus identical to the model including in Eq. 21 with an additional constraint X lens = 1 . \nWe show the result of marginalizing over a free X lens in the redMaGiC 3 × 2pt analyses in Figs. 11 and 12. We find a negligible impact on the primary cosmological parameters in Λ CDM, particularly S 8 . We find X lens = 0 . 877 +0 . 026 -0 . 019 , strongly inconsistent with X lens = 1 in Λ CDM. If we fix X lens to this value in the redMaGiC γ t + w ( θ ) analysis, the \n<!-- image --> \nm \ncontour in Fig. 11 shifts upward to agree with cosmic shear in S 8 . The value of X lens is correlated with the equation-of-state parameter w , so the redMaGiC w CDM constraint is strongly affected by this potential systematic. Adding the single free parameter X lens in Λ CDM leads to an improvement in χ 2 of 25, while adding a free w leads to an improvement in χ 2 of 7. Thus, X lens clearly leads to a better model fit. \nAfter unblinding the results with redMaGiC, but before unblinding those with MagLim, we decided to use the MagLim sample for our fiducial cosmological analysis if it showed no indication of this scale- and redshift-independent effect that is present in redMaGiC. This potential systematic was studied at length between the unblinding of the redMaGiC sample and the MagLim sample. Studies of this effect are discussed in much more detail in [136, 139, 140]. We have demonstrated that the effect (and its relative impact vs. the clustering am- \nplitude of the MagLim sample) is roughly independent of redshift, angular scale, or position in the survey footprint. \nAfter initial submission of this paper, we found that relaxing the goodness-of-fit requirement for the red galaxy model selection in redMaGiC leads to a cosmological model fit consistent with X lens = 1 and no significant change to the cosmological parameter results. This test suggests that a colordependent photometric issue is the source of X lens ! = 1 , and is plausibly connected to background subtraction. A specific fix for this systematic at the image level has not been identified, but these results pinning down the likely source of X lens are described further in Ref. [140]. Further study of this effect and pipeline modifications will continue for the final DES Year 6 analyses. \nglyph[negationslash]", '1. Summary of possible non-systematic causes of X lens = 1': "There are several classes of non-systematic explanations for X lens , all of which we believe are implausible given our data. These possible explanations are: \nStochastic Bias : While the effect of X lens on clustering and galaxy-galaxy lensing looks very similar to stochasticity, a decorrelation between the galaxy and matter distributions, predictions from galaxy bias models make this interpretation unlikely. In configuration space, perturbative stochastic terms are expected to contribute only at small separations r ∼ R glyph[star] (the Lagrangian size of halos), and to statistics that involve zero-lag correlators [67]. \nLensing-is-low : Ref. [90] reported that the γ t signal around luminous red galaxies is lower than expected from a model conditioned on their autocorrelation, which resembles the pattern seen in our redMaGiC sample. But with the possible exception of the large scale results in Ref. [195, 196], the lensing-is-low result [90] applies to models that fit to scales sensitive to complexities of the small scale dark matter-galaxy connection. There is still debate within the lensing-is-low literature as to whether the effect can be accounted for by additional complexity in these small scale models [197]. \nThe DES Y1 results (which also used a redMaGiC sample) do not support the lensing-is-low scenario, nor do the DES Y3 results for MagLim in the redshift range of the redMaGiC sample. The DES Y3 results for the redMaGiC sample show what could be interpreted as galaxy-galaxy lensing being 1015% percent lower than galaxy clustering at fixed cosmology (Planck 2015 in the case of Ref. [90]; DES 3 × 2pt cosmology in the DES Y3 results for X lens ). However, the more plausible cause is that the clustering of the Y3 redMaGiC sample is anomalously high, as indicated by internal consistency tests of the individual data vectors. While we are still studying the X lens < 1 anomaly, we currently do not believe that it supports a conclusion that galaxy-galaxy lensing is 'low.' \nFundamental physics : Any dynamical modifications to either the Poisson equation or the shear equation generally changes the galaxy and matter distributions but their correlation is maintained, i.e. X lens = 1 is maintained at linear scales. Beyond this possibility, any separation of the impact of relative 'bias' between the two types of matter (apparent in \nlensing vs. clustering) at the level of 15% would require significant fluctuations in the dark matter field, which would have substantial ramifications in other observables that we have not seen. Therefore, we conclude that a fundamental-physics explanation for X lens < 1 would probably have to be very finetuned. \nglyph[negationslash]", '2. Potential systematics in w ( θ ) and γ t vs. X lens = 1': 'glyph[negationslash] \nWe now continue discussing the X lens = 1 anomaly by comparing redMaGiC to MagLim, and commenting on potential systematics in galaxy clustering and galaxy-galaxy lensing as the cause of the anomaly. \nWe find the redMaGiC sample shows X lens < 1 at high significance at all scales and redshifts. The highest two redshift bins of the MagLim sample, which have been removed from the analysis, also indicate X lens < 1 at high significance, which is clearly visible in the model fit in those two bins of Fig. 2. In the redshift range overlapping the redMaGiC sample, we find no evidence of a non-unity X lens for MagLim. We discard the two high redshift bins for the MagLim sample as a conservative choice. Based on our investigations so far and current understanding of theoretical extensions beyond w CDM, we do not believe these anomalies are indications of new physics. We have found plausible but unverified indications that the origin may lie in potential systematics, e.g., associated with the photometric uncertainty or background subtraction for large or faint objects, or in the de-reddening process. \nThese issues are the subject of ongoing investigations, which will be crucial for understanding photometric clustering and its combination with galaxy-galaxy lensing in DES Y6 and beyond. However, while these measurements are potentially impacted at a level we can measure by some as yet unidentified systematic, this does not have a significant impact on Λ CDM cosmology when the three two-point functions are combined within 3 × 2pt. For the MagLim sample our tests indicate that both the Λ CDMand w CDMconstraints are robust. This self-calibration effect is one of the primary motivations for combining these different probes of the same underlying matter density field into the 3 × 2pt observable.', 'VI. COMPARISON WITH OTHER DES DATA': "DES has produced competitive cosmological constraints using its four primary probes: galaxy clustering and weak gravitational lensing (3 × 2pt), type Ia supernovae (SNe Ia), galaxy cluster counts and masses, and BAO. Together, these probes have been demonstrated to provide dark energy constraints that can be competitive with the best combined external constraints [107]. We describe each of them briefly below. \nType Ia supernovae : The DES SNe Ia sample has 207 spectroscopically confirmed SNe in the redshift range 0 . 07 < z < 0 . 85 . The sample-building and analysis pipelines are described in a series of papers that detail the SN search and discovery [146, 198, 199]; simulations [200]; photometry [201]; \nFIG. 13. A comparison of the marginalized constraints on parameters in the Λ CDM model from a variety of DES probes: large-scale structure and weak lensing (3 × 2pt; Y3 - black solid, Y1 reanalyzed - black dashed), type Ia supernovae (purple), galaxy cluster number counts and masses (orange and green), and BAO. The combination of DES Y3 3 × 2pt, SNe Ia, and BAO is shown in blue. Going from Y1 to Y3, we find approximately a factor of two improvement in the 3 × 2pt constraint in Ω m -S 8 plane. \n<!-- image --> \ncalibration [145, 202]; spectroscopic follow-up [203]; and selection bias [204-206]. The methodology and systematic uncertainties are found in Ref. [207]. These were used to constrain cosmology [208] and the Hubble constant [209]. These analyses included additional external low-redshift SNe that we do not use in this analysis. We compute the SNe likelihood using a module [210] implemented in COSMOSIS, which reproduces the results in Ref. [208]. The constraint from only DES SNe on Ω m is shown in Fig. 13 (purple). \nGalaxy clusters : The DES Y1 redMaPPer catalog consists of ∼ 6500 clusters with richness larger than λ = 20 in the redshift range z ∈ [0 . 2 , 0 . 65] . The first cosmological analysis of DES clusters [105] ('Clusters 1': orange contours in Fig. 13), which combines cluster counts data and mass estimates from the stacked weak lensing analysis of [211], found a larger than 2 σ tension with the other DES probes. This \nis driven by low-richness systems, and has been interpreted as unmodeled systematics that affect the stacked weak lensing signal of the optically selected sample. This interpretation is supported by the analysis of [111] ('Clusters 2': green contours in Fig. 13), which recovers results consistent with the other DES probes by combining cluster abundances with the large-scale auto-correlations of galaxy and cluster position and cross-correlations of cluster position with galaxy position and shear from DES Y1 data (4 × 2pt+N). The conclusions of [105] are further corroborated by the analysis of [212] which derive cosmological posteriors consistent with [33] by analyzing the DES Y1 redMaPPer cluster abundances, but replacing the stacked weak lensing mass estimates of [211] with multiwavelength follow-up data from the SPT-SZ 2500 deg 2 survey [213]. \nBaryon acoustic oscillations : A sample of 7 million galaxies from the DES Y3 'Gold' catalog is selected in the redshift range 0 . 6 < z < 1 . 1 [214] and used to measure the scale of the BAO feature in the distribution of galaxies at an effective redshift z eff = 0 . 835 [215]. We use a likelihood from Ref. [215] for the ratio of the angular diameter distance D A at z eff and the sound horizon distance at the drag epoch, r d , which is implemented in COSMOSIS. The simulated galaxy catalogs used in the analysis to derive the uncertainty of the measurement are described in Ref. [216]. While the BAO and 3 × 2pt analyses probe common sky area and redshift range, and the measurements of this work include scales impacted by the BAO feature, the overlap in galaxy sample is small and the method for inferring the BAO distance ratio likelihood is insensitive to cosmology, so we neglect this non-zero correlation when combining the measurements. This will be further validated in future work that combines and studies all final DES Y3 probes. \nDES Year 1 3 × 2pt : We reanalyze the DES Y1 3 × 2pt data in the Y3 model and prior space, but do not update the scale cuts or marginalize over a free point mass for galaxy-galaxy lensing. We also make no changes in priors on systematic parameters (e.g., photoz or shear calibration parameters). \nThe comparison of these cosmological constraints in Λ CDM using the DES probes is shown in Fig. 13. The combination of DES Y3 3 × 2pt, SNe Ia, and BAO data is also shown in (blue). While the constraint in S 8 is driven primarily by 3 × 2pt, there is substantial gain in other parameter dimensions due to the additional data. The marginalized parameter values are summarized in Tables II & III and Figs. 7 & 10.", 'VII. COMPARISON WITH EXTERNAL DATA': 'It has been demonstrated that various combinations of lowredshift data and high-redshift data from the CMB can independently fit the Λ CDM model. However, the most stringent tests of the model will come from combining these data and testing whether the model can simultaneously fit the diverse set of cosmological probes available to us at all redshifts simultaneously. These data sets are sensitive to the growth of density perturbations, the expansion and geometry of the Universe, or both, and are sourced from a variety of very dif- \nerent physical processes. The combination of the independent external low-redshift probes with DES Y3 data further reduces the potential impact of any residual systematic effects in the low-redshift anchor of the test, while the combined DES probes have been carefully calibrated from the same data and consistently protected against confirmation bias. Both of these considerations give us further confidence, for complementary reasons, in testing the Λ CDMmodelacross the age of the Universe.', 'A. External data sets': 'The likelihoods from data sets external to DES include: \nType Ia supernovae : The Pantheon sample [217] combines the distance measurements from 1048 SNe ranging from 0 . 01 < z < 2 . 3 , supplementing Pan-STARRS1 measurements with other available samples. \nBaryonic acoustic oscillations and redshift-space distortions : We use the constraints from SDSS measurements of BAO and RSD in eBOSS DR16. 8 When using constraints on f ( z ) σ 8 ( z ) , where f ( z ) is the growth rate, from RSD, we use the released covariance matrices between the constraints from the BAO and RSD measurements. These measurements are expressed in terms of the Hubble distance D H , sound horizon distance r d , comoving distance D M , and volume-average distance D V = ( D 2 M D H z ) 1 / 3 . The measurements include (from low to high redshift measurements): \n- · The measurement of D V at an effective redshift of z eff = 0 . 15 using the Main Galaxy Sample (MGS) [218] and adding to fσ 8 measurement from Ref. [219].\n- · A re-analysed version of BOSS DR12 measurements of D M and D H from BAO, and fσ 8 from RSD, at z eff = 0 . 38 and 0.51 [220].\n- · The eBOSS DR16 measurements of D M /r d , D H /r d from BAO, and adding fσ 8 from the full-shape information, at z eff = 0 . 698 using Luminous Red Galaxies (LRG) [221, 222].\n- · The eBOSS DR16 measurements of D V /r d when using BAO alone and D M /r d , D H /r d , fσ 8 when using BAO and the full-shape information, at z eff = 0 . 845 using Emission Line Galaxies (ELG) [223],\n- · The eBOSS DR16 measurements of D M /r d , D H /r d from BAO, and adding fσ 8 from the full-shape information, at z eff = 1 . 48 using the Quasar Sample (QSO) [224, 225],\n- · The eBOSS DR16 measurements of D M , D H at z eff = 2 . 33 using the Lymanα forests [226]. This data set only has information from BAO. \nCMB : We use the likelihoods from the Planck 2018 data release [13, 183]. Our fiducial combination of Planck likelihoods includes: \n- · The Plik likelihood of the temperature power spectrum C TT glyph[lscript] in 30 ≤ glyph[lscript] ≤ 2508 and the E -mode power spectrum C EE glyph[lscript] and the cross power-spectrum between temperature and E -mode C TE glyph[lscript] in the range 30 ≤ glyph[lscript] ≤ 1996 .\n- · The Commander likelihood of the temperature power spectrum C TT glyph[lscript] in 2 ≤ glyph[lscript] ≤ 29 .\n- · The SimAll likelihood of the E -mode power spectrum C EE glyph[lscript] in 2 ≤ glyph[lscript] ≤ 29 . \nWe also use the likelihood of the lensing potential φ power spectrum C φφ glyph[lscript] measured by Planck in the range 8 ≤ glyph[lscript] ≤ 400 , either in combination with our fiducial combination of Planck likelihoods described above or alone. In the latter case we use the likelihood marginalized over the CMB power spectrum. Planck CMB will refer to the primary CMB anisotropy data (without lensing) unless otherwise stated. \nBig Bang nucleosynthesis : We construct an Ω b h 2 constraint based on observations of damped Lymanα systems [227]. The primordial deuterium-to-hydrogen ratios measured from these systems can be translated to constraints on Ω b h 2 via Big Bang nucleosynthesis (BBN) calculations, but different assumptions on the BBN physics, in particular on the rate of the d ( p, γ ) 3 He nuclear reaction, yield different final constraints on Ω b h 2 . Our constraint conservatively incorporates the two major categories of such assumptions, namely the theoretical approach presented in Ref. [227] and the experimental measurement-based approach from Ref. [228]. Specifically, we adopt the mean and the statistical uncertainty on Ω b h 2 from Ref. [228], and in addition introduce a systematic uncertainty defined by the difference between 1) the mean from Ref. [228] and 2) an inverse-variance weighted average of the two respective means [227, 228]. This results in our adopted constraint of 100Ω b h 2 = 2 . 195 ± 0 . 028 . \nLocal Hubble parameter : We use a local h prior from SH0ES [93], which constrains h = 0 . 732 ± 0 . 013 using a local distance ladder that depends on measurements of Cepheids and type Ia supernovae. \nWe use versions of these likelihoods implemented as modules in COSMOSIS, which are used to obtain the constraints presented in the following.', 'B. High redshift vs. low redshift in Λ CDM': 'One of the most stringent tests of Λ CDM is to compare the prediction of the state of the Universe and amplitude of perturbations from the epoch of recombination, which we can observe from the CMB, to the current day, which we observe with low-redshift surveys like DES. At the time of the CMB, the Universe was very hot and dense, and its physics was dominated by radiation. DES is most sensitive to a period in the Universe approximately eight billion years later, where perturbations have grown by several orders of magnitude and nonlinear growth is important. DES observes a volume of the Universe spanning nearly nine billion years of its evolution. The volumes probed by large low-redshift surveys provide significant additional information on potential changes to the evo- \nFIG. 15. A comparison of marginalized parameter constraints from three similarly constraining sets of cosmological probes in w CDM. Combined external BAO, RSD, and SNe Ia data (Ext. Lowz ) are shown in orange, the combination of DES galaxy clustering and weak lensing data (3 × 2pt) is shown in black, and Planck CMB (no lensing) data is shown in green. The combination of Ext. Lowz data with DES 3 × 2pt is shown in purple and this combined additionally with Planck CMB (w/ lensing) is shown in blue. \n<!-- image --> \nFIG. 14. A comparison of marginalized constraints from three similarly constraining sets of cosmological probes in Λ CDM. Combined external BAO, RSD, and SNe Ia data (Ext. Lowz ) are shown in orange, the combination of DES galaxy clustering and weak lensing data (3 × 2pt) is shown in black, and Planck CMB (no lensing) data is shown in green. The three share a common parameter space in the Ω m -S 8 plane at their 68% CL bounds. The combination of Ext. Lowz data with DES 3 × 2pt is shown in purple and this combined additionally with Planck CMB (w/ lensing) is shown in blue. \nlution of perturbations or growth of the Universe over time, allowing them to strongly test the nature of dark energy. \nBy taking precise measurements of the Λ CDMmodel from CMB observations and predicting what we should observe in terms of the amplitude of perturbations or matter density in the late Universe, we can test whether our observations from surveys like DES agree with those predictions. If they do not agree at high significance, we have demonstrated that Λ CDM cannot describe the full evolution of the Universe. There has been considerable debate about the tendency of late Universe measurements to prefer slightly lower matter density or amplitude of clustering relative to measurements from the CMB (e.g., [13, 27, 33, 101, 102]). As more powerful data becomes available, like the current DES Y3 analysis, we can determine whether these measurements converge towards or away from the Planck CMB prediction. \n<!-- image --> \nm \nWe compare three similarly constraining and complementary subsets of available cosmological probes in Λ CDM and w CDM in Figs. 14 & 15. The external low-redshift SNe Ia, BAO, and RSD data primarily constrain Ω m and w , while the DES 3 × 2pt data adds substantial information on A s or σ 8 , which helps to further constrain Ω m and w through degeneracy-breaking of correlated parameters. These external low-redshift data sets complement the DES weak lensing and large-scale structure information by probing the growth and geometry of the cosmological model in fundamentally different ways. The CMB is able to tightly constrain both Ω m and A s or σ 8 in Λ CDM, but is comparable in constraining power to DES 3 × 2pt in w CDM, since it primarily has access to information limited to the surface of last scattering at z ≈ 1100 . The combination of DES 3 × 2pt with the other low-redshift data provides substantial gain in A s , σ 8 , Ω m , and w . We list marginalized parameter constraints for these probes in Tables \nFIG. 17. A comparison of the marginalized parameter constraints in the w CDM model from the Dark Energy Survey with predictions from Planck CMB data (no lensing; green). We show the fiducial 3 × 2pt (solid black) and the combined Y3 3 × 2pt and Planck (orange) results. \n<!-- image --> \nm \nFIG. 16. A comparison of the marginalized parameter constraints in the Λ CDM model from the Dark Energy Survey with predictions from Planck CMB data (no lensing; green). We show the fiducial 3 × 2pt (solid black) and the combined Y3 3 × 2pt and Planck (orange) results. \nII & III.', '1. Consistency results': 'We show the comparison of DES 3 × 2pt and the Planck CMB data for the Λ CDM and w CDM models in Figs. 16 & 17. Visually, we find better agreement in the overlap of the marginalized Ω m -S 8 parameters with the DES Y3 3 × 2pt data than found in the DES Y1 analysis [33], despite substantial improvements to the precision of both DES and Planck predictions. This is qualitatively unchanged when using the more precise, optimized Λ CDM version of the analysis that uses more small scale information - the DES contour shrinks, but asymmetrically in the direction of the CMB prediction. \nWe evaluate the consistency of the DES and Planck data in several ways, including shifts in parameter space and the Bayesian evidence. These are described further in Sec. IV E \n<!-- image --> \nm \nand full results are provided in App. F. We find a parameter difference of 1.5 σ ( p = 0 . 13 ) in the cosmological model space and a Suspiciousness of 0 . 7 ± 0 . 1 σ , corresponding to p = 0 . 48 ± 0 . 08 . This generally leads to the conclusion that despite substantially increased precision from both experiments, we find no significant evidence against the Λ CDM model from comparing these data sets. Agreement between DES and Planck in these metrics has improved relative to the comparison of DES Y1 3 × 2pt and earlier Planck results, which gave a parameter difference of 2 . 2 σ and Suspiciousness of 2 . 4 ± 0 . 2 σ [133]. The combined DES and Planck CMB contour is shown in orange in Figs. 16 & 17. \nWe repeat this exercise for the full combined low-redshift data, including DES 3 × 2pt, all BAO, and external SNe Ia and RSD data. This comparison is shown in Figs. 14 & 15, and is highly complementary, as the external probes are sensitive to both growth and geometry in the model in ways the DES 3 × 2pt data is not, and come from a variety of different exper- \nents. We find better agreement between all of these lowredshift probes and Planck CMB predictions than in the comparison with DES 3 × 2pt data alone, with a parameter difference of 0.9 σ or p = 0 . 34 . These results indicate that we can combine all these available cosmic probes into a single joint result in the following subsection. \nThere are several reasonable motivations for caution in the interpretation of any strong evidence for or against cosmological consistency in tests like this. It is worth noting that while we have multiple redundant low-redshift sources of information for each main cosmological probe used, it would be useful to have a second, blinded large-scale CMB polarization experiment to increase confidence in the test at the highz limit. While polarization data is required to break degeneracies in the cosmological parameters with the optical depth τ , we also repeat the caution from Ref. [13] against over-interpreting the Planck polarization results and the sensitivity of the final parameter constraints to assumptions made in the construction of the likelihood, which can lead to a <1 σ shift toward the DES posterior relative to the fiducial Planck likelihood. Similar shifts are seen based on certain analysis choices in the DES results as well, which are shown in App. E. Neither the shift in Planck posteriors or those from other analysis choices in DES contribute to a significant change in the final interpretation of the comparisons. Finally, the DES Y3 analysis has uncovered potential systematics connected to photometry (e.g., Sec. V C). While there is evidence that these do not impact the cosmological results, and thus would not impact this comparison of data sets, they have not been connected to a specific source. However, these are examples of unresolved uncertainties that call for additional care in interpreting any statements about the consistency of early- and late-universe probes in Λ CDM, which should be addressed for future more precise analyses.', 'C. Joint cosmological constraints in Λ CDMand w CDM': 'We find that external low-redshift (BAO+RSD+SNe Ia), Planck CMB, and DES 3 × 2pt data sets are able to provide three independent, highly complementary, and similarly powerful constraints on parameters related to dark matter and dark energy in the Λ CDM and w CDM models, as seen in Figs. 14 & 15. Given the results of the above consistency tests, detailed in App. F, these data sets are each consistent with one another, and thus can be combined into a joint constraint on the models. We present these joint results in Figs. 14 & 15 and a summary in Figs. 7 & 10 and Tables II & III. In the Λ CDM model, we find \nS 8 = 0 . 812 +0 . 008 -0 . 008 (0 . 815) Ω m = 0 . 306 +0 . 004 -0 . 005 (0 . 306) σ 8 = 0 . 804 +0 . 008 -0 . 008 (0 . 807) . (22) \nIn the w CDMmodel, \nσ 8 = 0 . 810 +0 . 010 -0 . 009 (0 . 804) , Ω m = 0 . 302 +0 . 006 -0 . 006 (0 . 298) , w = -1 . 031 +0 . 030 -0 . 027 ( -1 . 001) . (23) \nWe find R = 7 . 8 , indicating that there is also no preference for w CDMover Λ CDMin the full joint data analysis. \nThese data sets together are able to provide unprecedented precision on the cosmological parameters of the models. In Λ CDM, we are able to constrain σ 8 , S 8 , h , Ω b , and n s to less than 1%; Ω m and A s to about 1%; τ to about 10%; and place an upper limit on the sum of neutrino masses of ∑ m ν < 0 . 13 eV (95% CL). In w CDM, we are able to constrain n s to less than 1%; Ω m , Ω b , h , and A s to about 1-2%; w to about 3%; τ to about 10%; and place an upper limit on the sum of neutrino masses of ∑ m ν < 0 . 17 eV (95% CL). Individually, the three subsets of data constrain σ 8 and Ω m in Λ CDM with FoM between 2000 and 4000, while combined, they reach a FoM of 34,000. This clearly demonstrates the highly complementary nature of these three independent data sets.', 'D. Comparison of Lensing Probes': 'Weare able to probe the distribution of large-scale structure via weak gravitational lensing (cosmic shear) in two very different ways, using either the shapes of galaxies or the CMB photons as tracers for the reconstruction of deflections in the path of the light. These probe the same physical phenomenon via independent sources and measurement methods, which are sensitive to different types of systematics. While the effective kernel of CMB lensing [229] is sensitive to higher redshift structure than galaxy lensing, their comparison provides a significant validation of the robustness of modern weak lensing results. We show this comparison in Fig. 18, where we find very good agreement between the two cosmic shear measurements and the full 3 × 2pt measurement from DES. \nIn addition to DES, other concurrent photometric surveys HSC [27, 36] and KiDS [28, 102] are also pursuing precision weak lensing measurements using galaxy shapes. These three surveys span a range of depth and survey area tradeoffs, with HSC being deepest, DES widest, and KiDS using the widest wavelength coverage. We over-plot recent results from each of the surveys with our DES cosmic shear and 3 × 2pt results in Fig. 19. Unlike other comparisons in this work, these external survey data have not been re-analyzed within a consistent model and prior space. Thus, no direct or rigorous comparison can be made about data consistency. We defer a detailed discussion of the consistency of concurrent photometric weak lensing surveys and their combination (e.g., Ref. [230]) to a future work. The apparent orthogonal direction of the KiDS+BOSS+2dFLenS 3 × 2pt contours to the DES 3 × 2pt contours is driven by the very strong constraint coming from spectroscopic clustering, similar to the orientation of the DES γ t + w ( θ ) constraint. \nFIG. 18. A comparison of weak lensing constraints on the Λ CDM model. Weak lensing of the CMB is shown in green, weak lensing of galaxies in DES is shown in blue, and the combined DES 3 × 2pt data is shown in black. \n<!-- image --> \nFIG. 19. The DES Λ CDM-optimized 3 × 2pt and cosmic shear, HSC and KiDS cosmic shear, and KiDS lensing + BOSS+2dFLenS spectroscopic 3 × 2pt data results are over-plotted for the Λ CDM model. Unlike other comparisons in this work, these external survey data have not been re-analyzed within a consistent model and prior space. Thus, no direct or rigorous comparison can be made about data consistency. \n<!-- image --> \nFIG. 20. Marginalized constraints on h and Ω m in the Λ CDMmodel are compared to the SH0ES local determination of h . Planck CMB data and the combination of BAO and BBN data provide comparable uncertainties on h compared to the local constraint. Adding DES 3 × 2pt to BAO and BBN improves the constraint on h slightly due to 3 × 2pt providing additional information on Ω m , while the combination of DES 3 × 2pt and all non-local external data provide a constraint on h that is a factor of 3-4 more powerful than the local determination. \n<!-- image -->', 'E. Constraints on the Hubble parameter': 'There is an interesting disagreement in local measurement of the Hubble parameter h and marginalized constraints on h from cosmological experiments. Multiple local measurements prefer a higher value of the expansion velocity, such as, most prominently, the astronomical distance ladder (e.g., h = 0 . 732 ± 0 . 013 [93] with Cepheid variable stars; h = 0 . 733 ± 0 . 040 [94] with Mira variable stars), or masers (e.g., h = 0 . 739 ± 0 . 030 km / s / Mpc [95]). These local measurements stand in contrast to constraints from the CMB by Planck , which prefer h = 0 . 665 +0 . 013 -0 . 006 (when the neutrino mass density is varied) [13]. However, there are also local measurements with lower values reported ( h = 0 . 696 ± 0 . 019 [96] with tip of the red giant branch distance ladder; h = 0 . 674 +0 . 041 -0 . 032 with strong lensing when combining the TDCOSMO+SLACS data set [97]). The Hubble tension may indicate new physics and it is crucial to improve measurements, revisit assumptions [e.g., 97, 98], check for consistencies among different measurements, and invest in novel, independent methods and probes [99, 100]. \nWecan also constrain the value of h independently of CMB data using a combination of BAO, BBN constraints on Ω b h 2 , and DES 3 × 2pt measurements. Constraints on h and Ω m in Λ CDM are summarized in Fig. 20. The determination of h \nFigure 21 shows marginalized constraints on the sum of neutrino masses, where neutrino mass and density Ω ν are related via ∑ m ν = 93 . 14Ω ν h 2 eV. We model massive neutrinos as three degenerate species of equal mass. As expected, DES does not constrain neutrino mass: whether alone or in \n<!-- image --> \n∑ \nFIG. 21. Marginalized constraints on the sum of neutrino masses in the Λ CDM model. We show the DES fiducial 3 × 2pt constraints (black), DES 3 × 2pt combined with external BAO, RSD, and SNe Ia (orange), Planck CMB constraints (green), and DES 3 × 2pt combined with all of these external data sets. The upper panel shows the one-dimensional marginalized posteriors for ∑ m ν , with shaded 95% confidence regions. The lower panel shows 68 and 95% CL for Ω m and ∑ m ν . \nusing BAO and BBN is of similar constraining power to that of the CMB and agrees very well with the CMB constraint on h . Adding DES 3 × 2pt data slightly improves the constraint on h and shifts it to higher values by about 1 σ . Combining DES 3 × 2pt data with BAO, RSD, SNe Ia, and Planck CMB (w/ lensing) leads to a marginalized constraint on h \nh = 0 . 680 +0 . 004 -0 . 003 (0 . 681) (24) \nthat is 3-4 times more powerful than any current local measurement of h . Constraints on other cosmological parameters are summarized in Tables II & III. We find no significant impact on the other cosmological parameters by adopting this high-redshift anchor for the expansion rate vs a local prior on the expansion rate from Ref. [93]. The final joint constraint on h is consistent with the Planck - or BAO+BBN-only constraints and slightly less than 4 σ offset relative to the local h by SH0ES.', 'F. Neutrino Mass': 'combination with external BAO, RSD, and supernova data, the marginalized posterior of the sum of neutrino masses is bounded by its prior. Given this, DES data is expected to add very little direct information on neutrinos. \nBeyond constraints on neutrino mass themselves, a motivation for looking at the neutrino mass constraint is to highlight a feature of the relationship between Planck and lowz constraints on Ω m , as shown in e.g. Fig. 14. As has been previously discussed [183], a geometric degeneracy means that CMB-only constraints are unable to distinguish between Ω m and ∑ m ν , but combining CMB data with low redshift expansion history constraints from BAO and/or SNe can break that degeneracy. This also illustrates that the parts of the Planck CMBposterior at higher Ω m also have relatively high neutrino mass. This is further supported by the behavior of constraints when neutrino mass is fixed, as shown in App. E. \nLowering the clustering amplitude has a similar impact on Ω m , which we see as an increase in the upper limit on ∑ m ν of 23% when combining DES 3 × 2pt with Planck . Combining the DES 3 × 2pt data, other low-redshift data, and Planck CMB, we find an upper limit \n∑ m ν < 0 . 13 eV (95% CL) , (25) \nnearly a factor of three reduction from the CMB-only constraint.', 'VIII. CONCLUSIONS': "We have described the 3 × 2pt measurements, calibration, modeling, and analysis from the first three years of DES data. The substantial improvement in statistical power of the DES Y3 data, which cover an area of sky about three times that of DES Y1, has required substantial improvements in almost every part of the data processing, analysis and inference. The specific improvements relative to the DES Y1 analysis are detailed in App. B, but we also briefly summarize here. At the catalog level, some of the most important updates for this analysis include improved PSF modeling and two complementary lens sample selections. We have substantially revised our shear and redshift inference and calibration processes. This includes more realistic image simulations to derive corrections on shear and redshift bias due to blending and detection, and a redshift inference process that combines spectroscopic and deep, multi-band photometric redshifts from DES deep-field data, cross-clustering between source and highquality photoz and spectroscopic samples, and small-scale galaxy-galaxy lensing shear ratio information. Finally, we have updated several components of the analysis, including blinding, modeling of non-Limber and RSD contributions to the clustering signal, mitigation of nonlocal effects in γ t , improved validation of the covariance matrix, and improved, calibrated metrics for evaluating the internal and external consistency of data sets. \nThe statistical power of the DES Year 3 data set has posed unique challenges for precision cosmological inference, some of which were unforeseen. We have identified some puzzling \nresults from our photometric lens samples and galaxy clustering, which have not been identified with a clear source. This can be seen at very large significance as an apparent disagreement in the clustering and lensing amplitudes at all redshifts and angular scales for redMaGiC, and in the highest redshifts of MagLim not used in the fiducial analysis. After unblinding our analysis, it was necessary to make two important revisions to the fiducial analysis plan: 1) we made the MagLim galaxy sample our fiducial lens sample owing to the decorrelation of the lensing and clustering amplitudes inferred from the redMaGiC sample, and 2) we dropped the two highest redshift bins of the MagLim sample as they contributed to a very poor fit to all models considered in this paper. Sections V A & V C provide the detailed rationale. These decisions were made after extensive, careful investigations of possible systematics, which did not reveal problems that we could address. Further investigations are already underway and may reveal the source of these issues, but our robustness tests so far indicate that any potential changes to the results will lie well within our quoted uncertainties. \nWe have achieved a factor of two improvement in statistical power relative to the DES Y1 3 × 2pt analysis in the σ 8 -Ω m marginalized parameter plane, providing competitive cosmological constraints relative to both the combination of all other external non-lensing low-redshift data and the predictions from the Planck CMB data. We find consistent cosmological constraints from cosmic shear and the combination of galaxy clustering and galaxy-galaxy lensing, as well as consistent cosmological results from 3 × 2pt utilizing two lens samples, MagLim and redMaGiC. For the fiducial 3 × 2pt analysis in Λ CDM, we find constraints on the clustering amplitude S 8 = 0 . 776 +0 . 017 -0 . 017 (0.776) and matter density Ω m = 0 . 339 +0 . 032 -0 . 031 (0.372). In the w CDM model, we find Ω m = 0 . 352 +0 . 035 -0 . 041 (0.339), and dark energy equation of state parameter w = -0 . 98 +0 . 32 -0 . 20 ( -1 . 03 ). \nThe low-redshift measurements of the matter clustering amplitude by some galaxy surveys have tended to find lower variance relative to the prediction from Planck CMBanisotropies, which may indicate some inconsistency between the low- and high-redshift Universe within the Λ CDM model, with claims of up to 2-3 σ significance. The DES Y3 3 × 2pt analysis is an ideal experiment to test whether this is a real problem with the Λ CDMmodel. There have been substantial improvements in constraining power for both DES and Planck since the DES Y1 3 × 2pt analysis, yet we continue to find that DES 3 × 2pt data and the combination of 3 × 2pt with BAO and external SNe Ia and RSD data that the low-redshift Universe are consistent with predictions in the Λ CDM model from measurements at the time of the CMB from Planck . We find all three independent data set combinations (DES 3 × 2pt; BAO, RSD, and SNe Ia; and Planck CMB) to be mutually consistent within Λ CDM. Despite caveats on the precision with which we can make this statement discussed in Sec. VII B, this is the most powerful test of the standard cosmological model to date, comparing predictions from measurements of acoustic peaks in the early plasma of the Universe when it was 380,000 years old to measurements of large-scale structure from low-redshift surveys like DES spanning nearly nine billion years of cosmic \nevolution to the current day. \nCombining DES 3 × 2pt, CMB, BAO, RSD, and SNe Ia data allows us to place the most precise constraints on the Λ CDM and w CDM models to date. We find S 8 = 0 . 812 +0 . 008 -0 . 008 (0.815) and Ω m = 0 . 306 +0 . 004 -0 . 005 (0.306) in Λ CDM; σ 8 = 0 . 812 +0 . 008 -0 . 008 (0.804), Ω m = 0 . 302 +0 . 006 -0 . 006 (0.298), and w = -1 . 031 +0 . 030 -0 . 027 ( -1 . 00 ) in w CDM. Additionally, we find an independent constraint on the Hubble parameter combining DES 3 × 2pt, BAO, and BBN data of h = 0 . 676 +0 . 009 -0 . 009 (0 . 673) , which is consistent with the Planck prediction for h . From the combination of DES 3 × 2pt, CMB, BAO, RSD, and SNe Ia data, we find h = 0 . 680 +0 . 004 -0 . 003 (0 . 681) . This is slightly closer to the local h measurement by SH0ES than Planck , but a factor of three to four more constraining than either the local or Planck measurement of h . We are also able to constrain the sum of neutrino masses to be m ν < 0 . 13 eV (95% CL) in Λ CDM. \n∑ While we have shown that the inferred cosmological constraints from the fiducial analysis are robust, there remains significant work to fully characterize the underlying causes of these potential systematics and examine other potential theoretical causes in extended model spaces beyond the dark energy models considered in this work. Further understanding the potential systematic issues related to differences between photometric clustering and galaxy-galaxy lensing; improvements in how we deal with shear calibration and redshift inference in the presence of blending; and finding ways to improve systematic floors in our redshift inference are all important next steps. This continued followup work will be critical to the final DES Year 6 analyses and future 'Stage IV' photometric surveys like the Euclid Space Telescope, 9 the Nancy G. Roman Space Telescope, 10 and the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST). 11 \nThe novel advances required for the DES Y3 analyses, summarized in the accompanying 29 papers [115-143], set the stage for these future precision low-redshift large-scale structure and weak lensing studies. DES has utilized only half its final data set in the DES Y3 3 × 2pt and BAO analyses, and future SNe Ia and galaxy cluster analyses promise even larger improvements in statistical power. The legacy analyses of the full DES data will be a focus of the next several years leading up to the start of Stage IV dark energy experiments.", 'ACKNOWLEDGMENTS': "Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of \nIllinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. \nThe Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, NFS's NOIRLab, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&MUniversity, and the OzDES Membership Consortium. \nBased in part on observations at Cerro Tololo InterAmerican Observatory at NSF's NOIRLab (NOIRLab Prop. 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We acknowledge support from the Brazilian Instituto Nacional de Ciência e Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-2). \nThis manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. \nThis research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC0205CH11231. This work also used resources on Duke Compute Cluster (DCC), the CCAPP condo of the Ruby Cluster at the Ohio Supercomputing Center [231], and computing resources at SLAC National Accelerator Laboratory. We also thank the staff of the Fermilab Computing Sector for their support. 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Dailey, arXiv e-prints , arXiv:1112.3108 (2011), arXiv:1112.3108 [astro-ph.CO].\n- [236] S. Pandey et al. (DES), Phys. Rev. D 102 , 123522 (2020), arXiv:2008.05991 [astro-ph.CO].", 'Appendix A: Summary of Associated Papers': 'This paper is built on the results presented in 29 accompanying papers. A useful way to navigate these is to divide them up into five categories: \nCatalog Papers : The link between the raw images and the two-point functions from which cosmology is extracted is a set of catalogs. The Gold catalog (Sevilla-Noarbe et al. [115]) uses coadd images to identify galaxies and their properties. This is a first step to almost all ensuing work. Estimating the redshifts of those galaxies hinges in large part on the much deeper catalog of galaxies from the DES deep fields with overlapping near-infrared photometry presented in Hartley, Choi et al. [116]. Finally, the shear catalog is presented in Gatti, Sheldon et al. [117], which uses a new PSF measurement described in Jarvis et al. [118]. \nSimulations : To test our models and calibration of the data, we rely on large suites of cosmological and image-level simulations. The Buzzard simulations (DeRose et al. [119]) are generated from cosmological N-Body simulations, populated with realistic galaxy samples, and then used in end-to-end analyses to stress test our modeling and methods. To calibrate blending and detection biases, we generated multiple image simulations, described in MacCrann et al. [120], where shear and redshift biases are evaluated. Balrog (Everett et al. [121]) is a tool developed to inject realistic images of galaxies into real DES images to evaluate the survey selection function. This is critical for multiple purposes, including our photometric redshift inference. The full DES detection and measurement pipelines are run on the simulations described in MacCrann et al. [120] and Everett et al. [121]. \nPhotometric Redshifts : The redshift distributions for the redMaGiC and MagLim lens galaxy samples are validated using cross-correlations with spectroscopic galaxies in Cawthon et al. [122], which informs our priors on the uncertainty of the redshift distribution. The redshift inference process for the source galaxies is a much more involved process. The overview of this work is presented in Myles, Alarcon et al. [123], which builds on the self-organizing map formalism developed in Buchs, Davis et al. [124], and incorporates constraints from the cross-correlation with both redMaGiC and spectroscopic galaxies described in Gatti, Giannini et al. [125]. A followup analysis applying this methodology to the MagLim lens redshift distributions is described in Giannini et al. [126]. The sampling of the resulting realizations of the redshift distribution are described in Cordero, Harrison et al. [127]. Finally, we also add additional information from the posterior from the shear ratio likelihood, which is most sensitive to photoz parameters and is described in Sánchez, Prat et al. [128]. \nAnalysis : The modeling outlined in the text is described and tested in detail in Krause et al. [129]. One of the most important pieces of the analysis is the generation of the covariance matrix of the two-point functions. The way in which this is generated and tested is described in Friedrich et al. [130]. As mentioned in the text, we remain blinded to the data throughout the analysis to avoid unconscious bias; this blinding has multiple levels, but one of the key tiers is described in Muir \net al. [131]. Methods to evaluate the internal consistency of our data and to evaluate consistency with external data sets were calibrated and described in Doux et al. [132] & Lemos et al. [133]. Efficient sampling of the likelihood can save months of analysis time, and ways to optimally, and more importantly robustly, sample our likelihood are summarized in Lemos et al. [134]. Finally, the optimization of the MagLim lens sample is described in Porredon et al. [135]. \nResults : The galaxy clustering weights and measurement are presented in Rodríguez-Monroy et al. [136]; galaxygalaxy lensing in Prat et al. [137] and in Sánchez, Prat et al. [128] for the smaller-scale shear ratio measurements; and the weak lensing convergence mass map in Jeffrey, Gatti et al. [138]. The results of combining clustering with galaxygalaxy lensing are presented in three papers, one of which focuses on magnification (Elvin-Poole, MacCrann et al. [139]), another on redMaGiC and the bias model (Pandey et al. [140]), and the last on the MagLim sample (Porredon et al. [141]). Finally, the cosmic shear results are presented in two papers, one of which focuses on observational systematics (Amon et al. [142]) and the other on biases in theoretical modeling (Secco, Samuroff et al. [143]).', 'Appendix B: Differences relative to DES Year 1 analysis': "In this Appendix, we summarize the major differences in the DES Y3 analysis relative to DES Y1. \nData Processing : DES Y3 contains significantly more data than DES Y1 [144], which are processed with an improved version of the DESDM system [146]. Photometric calibration is performed with the forward global calibration method [145], which significantly improves the relative photometric calibration compared to the calibration techniques applied in DES Y1 [233]. The Y3 catalogs also introduce more robust morphological classification based on multi-epoch fitting and improved flagging to enhance the quality of the galaxy sample. Further details on improvements to the DES data set and object catalogs can be found in Sevilla-Noarbe et al. [115]. \nCatalog-level : We have produced a calibrated deep-field data reduction for use in the analysis. We have constructed only one shape catalog (METACALIBRATION; [117]), but produced two very different lens samples to compare cosmological results [136, 140, 141]. We use a new PSF model (PIFF) [118]. \nSimulation & Calibration : We have produced the Balrog simulation to characterize the wide-field survey selection function of objects derived from the deep-field coadd images [121]. We have produced a new suite of image simulations for shear calibration that are more realistic and better match the data [120]. We have produced a new methodology for shear calibration that explicitly accounts for the impact of blending and detection biases as a function of redshift, including modifications to the effective n ( z ) [120]. We have developed a new redshift inference and calibration framework, mixing spectroscopic and photometric redshift information, to produce Bayesian posterior n ( z ) samples [122-128]. \nModeling : We have improved the rigor of constraining po- \nTABLE IV. The parameter differences for the redMaGiC analysis relative to Table I. \ntential bias from model approximations, with explicit accuracy goals at the χ 2 and parameter levels [129]. We have included non-Limber and redshift-space distortion (RSD) contributions to the clustering theory [129]. We have included the impact of lens magnification [129, 139]. We have made updates to the covariance modeling and improved the rigor of validation tests in both χ 2 and parameter space [130]. We utilize a new intrinsic alignment model that allows for 'red', 'blue', and mixed alignment modes to second order in perturbation theory [174]. We have developed a nonlinear galaxy bias model for the analysis [140]. We have updated our methodology for determining scale cuts using explicit accuracy goals in both χ 2 and parameter space [129]. \nAnalysis : We have utilized a new summary-statistic-level blinding scheme [131]. We have introduced a new shearratio likelihood [128]. We have introduced a new crossclustering redshift likelihood [123, 125]. We have implemented a method for sampling over the full-shape n ( z ) samples [127]. We analytically marginalize over uncertainty in the lens sample clustering weights [136]. We analytically marginalize over a point-mass contribution to γ t to mitigate non-local effects [129]. We marginalize over the widths of the lens n ( z ) [122, 136]. We have updated and calibrated a new set of internal and external consistency tests [132, 133]. We have updated requirements on sampling and evidence precision and now use the PolyChord sampler [134].", 'Appendix C: Differences in redMaGiC analysis': 'The redMaGiC lens analysis differs in several ways due to the different redshift range, binning, and sample selection relative to the MagLim lens analysis. The parameterization and prior changes relative to the fiducial analysis are listed in Table IV. In particular, we do not vary the width of the redshift distributions in the first four redMaGiC lens bins, since constraints from clustering on the n ( z ) agree well with the predic- \nFIG. 22. The measured w ( θ ) correlation functions for each tomographic bin combination used in the redMaGiC analysis, which is indicated by the i, j label in each set of panels. The best-fit Λ CDM model from the analysis using redMaGiC is plotted as the solid line in the top part of each panel, while the bottom parts of each panel shows the fractional difference between the measurements and the model prediction, ( w obs. -w th. ) /σ w (with y -axis range ± 5 σ ). The best-fit model with fixed X lens is shown in black, while the best-fit model marginalizing over X lens is shown in blue. Both the top and bottom part of each panel includes 1 σ error bars. Small angular scales where the linear galaxy bias assumption breaking down are not used in the cosmological analysis; these scales are indicated by grey shading. \n<!-- image --> \n2 \nFIG. 23. The measured γ t ( θ ) correlation functions for each tomographic bin combination using the redMaGiC sample, with labels as described in Fig. 3. The best-fit Λ CDM model from the analysis with fixed X lens is plotted as the solid line in the top part of each panel, with dotted curves indicating a negative model fit. The best-fit model marginalizing over X lens is shown in blue. The bottom part of each panel shows the fractional differences between the measurements and the model prediction, ( γ obs. t -γ th. t ) /σ γ t (with y -axis range ± 5 σ ). In both panels, 1 σ error bars are included. Angular scales not used in the cosmological analysis are indicated by grey shading, which are excluded on small scales where the linear galaxy bias assumption breaks down. \n<!-- image --> \nFIG. 24. A test of the convergence of the theoretical 3 × 2pt covariance, showing marginalized parameter constraints using two iterations of the covariance that use the best-fit cosmological parameters from the previous analysis. \n<!-- image --> \nm \ntions from the redMaGiC photoz algorithm [122]. The magnification parameters are also fixed to different values based on measurements from Balrog [139]. We also show the redMaGiC galaxy clustering and galaxy-galaxy lensing data vectors and best-fit cosmological model in Figs. 22 & 23.', 'Appendix D: Observer Bias & Validation Process': 'The role of the observer in determining how to proceed in an analysis or measurement and when to accept a result as final has been demonstrated to contribute to uncharacterized bias in results (e.g., [234, 235]). To protect against this, all cosmologically relevant measurements, calibrations, model validation, and fiducial analysis plans were performed blinded. This was done in three stages, to provide redundant protection: \n- 1. The ellipticity in the shape catalog was blinded by a random factor in b ∈ [0 . 9 , 1 . 1] , transforming η i → bη i , where η i = 2 e i arctanh( e ) /e and e 2 = e 2 1 + e 2 2 . The factor b is generated in a random way by hashing a short string phrase, and this phrase was known only to the people making the catalog.\n- 2. The two-point correlation functions were coherently shifted by an unknown vector in cosmological parameter space ( Ω m , w ), as described in Ref. [131]. This produces a new data vector corresponding to an unknown w CDM cosmology, and has the benefit of leaving all \n- three two-point correlation functions internally consistent.\n- 3. All parameter values and the axes of posterior plots inferred from chains were randomly shifted. \nThese protections were removed one by one as the analysis matured. \nFirst, the catalog-level blinding was removed when the validation and planned calibration of the catalog were finalized, to allow null tests and non-cosmological measurements to be remade for the papers while the data vectors were still blinded. In practice, this also only occurred after model validation tests had been completed and other analysis and calibration plans were also finalized. \nOnce the data vector, shear and redshift calibration, all model validation and choices, and plans for testing internal and external consistency were finalized, we proceeded with a pre-determined set of tests on the blinded results. The enumerated tests below were those that would have led us to reconsider whether to proceed with unblinding and pursue potential systematic causes. In addition to this, other validation checks were also performed at this point on the blinded data vector and posteriors, but were not required for unblinding. \n- 1. Verify the final independent redshift posteriors are consistent (i.e., from SOMPZ, the shear ratio, and clustering cross-correlations).\n- 2. Verify no posteriors of systematic parameters concentrate at the edge of their priors in ways that are not understood.\n- 3. Verify the goodness-of-fit for each subset of the data (i.e., cosmic shear and galaxy clustering+galaxygalaxy lensing). We used the posterior predictive distribution (PPD) with a quantitative requirement p > 0 . 01 .\n- 4. Verify that cosmic shear and galaxy clustering+galaxygalaxy lensing are consistent with each other, with the same quantitative requirement.\n- 5. If any of the previous tests failed in Λ CDM, a passing condition in w CDMwould also be sufficient. \nThe theoretical covariance matrix associated with the 3 × 2pt data is calculated with an assumption about the true cosmological and other parameters in our model that may be different from what we find after unblinding our data. Using a covariance matrix that assumes the wrong cosmology can bias the cosmological inference process. To mitigate this, we recalculate the covariance matrix at the best-fit cosmology of the initial 3 × 2pt analysis and run all final chains with this covariance. We confirm that this process has converged by comparing the result of this analysis with an analysis based on yet a third covariance calculated at the second best-fit cosmology. This is shown in Fig. 24. \nWe unblinded the redMaGiC sample before the MagLim sample, finding it to pass all unblinding checks. After unblinding it became apparent from additional internal consistency (PPD) tests that we had statistically significant evidence \nfor a potential systematic error in the clustering part of the data vector. This was explored at length in parallel to the final validation of the MagLim sample leading up to its unblinding. We verified, though did not require, that there was no evidence that MagLim suffered from the same effect before unblinding it. The MagLim sample, however, did fail the χ 2 criterion above for both models after updating the parameter values for the covariance matrix after the initial unblinding chains, with an excess χ 2 ≈ 100 . This was not seen before unblinding due partly to a poor choice of galaxy bias values for the initial covariance, leading to the updated covariance matrix being 40-50% smaller in the w ( θ ) block. \nWe had agreed that it was implausible that our data should be able to distinguish a nonw CDM model at such a large χ 2 before unblinding, and so we had planned to explore potential causes for the poor model fit before proceeding in such a case, correcting any problems that were then discovered before unblinding. Unfortunately, this effect was hidden due to the initial covariance before parameter updates being weaker in the w ( θ ) blocks, and so this was a post-unblinding correction. However, the solution (imposing an upper redshift cut for the MagLim sample) is something we would almost surely have pursued before unblinding had this problem been apparent then. These lens sample issues and related tests are summarized further in Sec. V C and in Refs. [136, 140, 141]. \nThe list of changes made to the analyses post-unblinding are as follows. For redMaGiC, we investigated how the lens weights were calculated due to indications that an alternative method that used a principal component (PC) basis of the observing condition maps produced a non-trivial shift in the clustering signal in the direction to correct the observed excess clustering parameterized by X lens . However, it was determined that this basis was contaminated by true largescale structure modes. A simplified basis limited to the first 50 PCs to remove any contaminated modes was used in the final analysis that gives consistent results with the original weights used at unblinding. An additive component was added to the w ( θ ) covariance block of both lens samples that accounts for potential over-correction and differences between the two weights methods. The MagLim sample used this final weighting and covariance when unblinded. As discussed previously, the highest two redshift bins of the MagLim sample were also removed to resolve a very poor χ 2 fit to any dark energy model considered and a strong indication of inconsistency between w ( θ ) and γ t approaching X lens = 0 . 55 in the highest redshift bin. There was no indication of the impact of X lens for MagLim generally within the redshift range overlapping redMaGiC. This change in redshift limit resulted in a shift of 0 . 78 σ in the Ω m -S 8 plane, almost fully in the direction of increasing Ω m , relative to the parameter values found at unblinding. This is consistent with shifts observed when removing parts of the redshift range of the data, so not clearly evidence of systematic impact on the parameter values. \n<!-- image --> \nm \nFIG. 25. A test of the impact of alternative redshift analysis choices on the inferred cosmology from 3 × 2pt. We compare the fiducial 3 × 2pt analysis (black) to an analysis where we marginalize over the ensemble of n ( z ) realizations directly via Hyperrank instead of their effective mean redshifts (blue) and to an analysis where we remove the shear-ratio data (orange).FIG. 26. A test of the impact of alternative analysis choices on the inferred cosmology from 3 × 2pt. We compare the fiducial 3 × 2pt analysis (black) to an analysis where we marginalize over a nonlinear bias model using smaller scales in γ t and w ( θ ) in blue, an analysis that marginalizes over free lens magnification bias parameters in orange, and an analysis that uses the NLA IA model in green. \n<!-- image --> \nTABLE V. The MagLim galaxy bias constraints (mean with 68% CL) from the fiducial linear bias analysis and the nonlinear bias analysis. Due to a nontrivial second peak in the posterior of the b 2 4 , we also show in parentheses the 1D marginalized peak value. \nFIG. 27. The validation of the model and inference pipeline on the Buzzard simulation suite, where the true cosmology is indicated by the cross. An analysis of a synthetic data vector (black) at the true Buzzard cosmology based on the true redshift distributions with fixed shear and photoz parameters is compared to a full analysis of the mean data vector of 18 simulation realizations (blue) including all nuisance parameters and n ( z ) distributions inferred in the same way we do using the real survey data. \n<!-- image -->', 'Appendix E: Alternative analysis choices and robustness': 'In addition to the validation described in the main text of this work and the associated papers described in App. A, we discuss several other analysis modifications in this Appendix that test the robustness of our result.', '1. Photometric redshifts': 'In Fig. 25 we show the fiducial 3 × 2pt analysis compared to two variations in how we use information related to the photoz s of our source sample. First, we simply remove the shear-ratio part of the data vector, which helps to constrain the photoz parameters, but does not contribute directly to constraining cosmological parameters. Second, we marginalize over the full ensemble of n ( z ) realizations. Before unblind- \nFIG. 28. A comparison of the fiducial 3 × 2pt analysis (black solid) with one that fixes the neutrino mass density to fixed is minimum value (black dashed). We make a similar comparison for the Planck CMB data (green solid and dashed). \n<!-- image --> \nng, we decided to simplify the way we use the redshift information for the sources and marginalize over the more typical set of parameters that encode shifts in the mean redshift of the average n ( z ) from the ensembles. This choice was driven by the additional computational expense of marginalizing over the n ( z ) realizations directly, but was demonstrated in simulated analyses to produce consistent cosmological results at DES Y3 precision. To test this, we run a single chain to show the potential impact of this choice on the real data. While all three results are consistent, with a shift of 0.53 σ between Hyperrank and marginalizing over ∆ z , it is likely that future analyses will need to marginalize over the full ensemble of n ( z ) realizations directly using something like the Hyperrank process. Finally, we utilize the SOMPZ framework to rederive the redshift distributions of the magnitude limited sample [126], instead of relying on the fiducial DNF redshifts. The 3 × 2pt cosmology inferred from this alternate redshift distribution is fully consistent with the fiducial result. \nTABLE VI. Summary of internal goodness-of-fit tests with the posterior predictive distribution for analyses with the redMaGiC and MagLim samples. The first and second columns indicate the subset of data d considered. Realizations of d are generated for model parameters drawn from its own posterior and compared to actual observations of d , following the method developed in Ref. [132]. The third column shows the (calibrated) probability-to-exceed for the full data test used for comparison. The other columns indicate the probability-to-exceed for different subsets used for comparison.TABLEVII. Summary of internal consistency tests with the posterior predictive distribution for analyses with the redMaGiC and MagLim samples. We consider the consistency between different two-point functions, as listed in the first column. The probability-to-exceed p ( a \| b ) is obtained by generating realizations of a for model parameters drawn from the posterior of b , and comparing these realizations to the actual observations of a . The third and fourth columns indicate the probability-to-exceed for the redMaGiC and MagLim samples respectively.', '2. Nonlinear bias modelling': 'The fiducial analysis of galaxy clustering and galaxygalaxy lensing assumes a linear galaxy bias model, which requires removing significant small-scale information in our data vector that exhibits substantial nonlinear behavior. This keeps the analysis simpler with fewer free parameters, but potentially wastes this additional information to further constrain cosmology on small scales. In our fiducial model, we only utilize large scales above 8 h -1 Mpc for w ( θ ) and 6 h -1 Mpc for γ t measurements. To remedy this, we have developed a nonlinear galaxy bias model and analysis that can utilize this smaller scale information from the lens sample at the expense of marginalizing over additional bias parameters. This is similar to going to smaller scales in cosmic shear and having to marginalize over additional baryonic effect freedom. In both cases, the improvements are limited by needing to simultaneously constrain these additional parameters. \nOur model is a hybrid 1-Loop effective field theory model, having five free parameters, as detailed in Refs. [119, 129, 140, 141]. In Ref. [236] we validated this model using 3D correlation function measurements in redMaGiC and MagLim mock catalogs and find that this nonlinear bias model agrees with significantly higher signal-to-noise measurements at bet- \nter than 2% above scales of 4 h -1 Mpc. For cosmological inferences we fix three of these parameters based on theoretical considerations and validation on these mock catalogs in order to reduce parameter degeneracies and potential parameter volume effects. We have validated the analysis using multiple buzzard simulation realizations[119, 140, 141] (see Sec. IV D for details), finding that this model gives a cosmological bias of less than 0.3 σ in the Ω m -S 8 plane for both the w ( θ ) + γ t and 3 × 2pt probes. \nIn Fig. 26 we show the cosmological constraints when applying this model to our data. The galaxy bias constraints are summarized in Table V. We find cosmological constraints in w CDM to be consistent with the fiducial analysis. We find a similar improvement in constraining power for Λ CDM with this nonlinear bias analysis as with the Λ CDM-optimized analysis that instead incorporates additional small scale information in cosmic shear. With DES Year 6 data, it is possible we could better optimize these choices of how to utilize smallscale information in Λ CDManalyses to substantially improve constraining power.', '3. Lens magnification bias parameters': 'In the fiducial analysis, we fix the lens magnification coefficients C i l to values derived in Ref. [139]. In Fig. 26 we show the impact of freeing these coefficients with a wide flat prior between -6 and 10 in each lens bin. We find the cosmological constraints in Λ CDMto be consistent with the fiducial analysis. Further investigations into the lens magnification coefficients and their impact on cosmological constraints can be found in Ref. [139].', '4. Intrinsic alignment models': 'In the fiducial analysis, we use the full TATT intrinsic alignment model. In Fig. 26 we show the impact of limiting the intrinsic alignment model to the more commonly used NLA model with free redshift power-law evolution. This is the same as fixing the parameters A 2 = b TA = 0 in the TATT model. We find consistent cosmological parameters, with a \nTABLE VIII. A comparison of metrics testing the consistency of independent data sets within the Λ CDM model. We show results in units of σ , with a derived probability-to-exceed p shown in parentheses. Overall, we find no significant (defined as p < 0 . 01 ) evidence of disagreement between the DES 3 × 2pt, external low-redshift, or Planck CMB data sets. The details of the metrics and derivation of p are described fully in Ref. [133]. \ngain of only 13% relative to TATT in the σ 8 -Ω m parameter plane. The use of NLA was not motivated a priori as the fiducial IA model, but due to the IA amplitude parameters being constrained to be small in TATT, A 2 = b TA = 0 is not a poor approximation.', '5. Simulation validation tests': 'We reproduce the fiducial analysis on a suite of 18 Buzzard simulations described in Sec. IV D. This is shown in Fig. 27, where the true cosmology is indicated by the cross. We compare two simulated analysis. The first analysis uses a synthetic, noiseless data vector based on the true n ( z ) from Buzzard and without marginalizing over shear or photoz parameters. The second reproduces the full analysis on the mean data vector of all 18 simulation realizations, which marginalizes over all nuisance parameters and uses an photoz n ( z ) that is inferred from the same process we apply to the real survey data. We find that the two simulated analyses agree very well with each other and with the true cosmology for each simulated data vector.', '6. Neutrino mass': 'Finally, we also show a version of the analysis that fixes the neutrino mass density at the minimum allowed mass in Fig. 28, comparing to a similar fixed neutrino mass density of the Planck CMB data.', 'Appendix F: Details of Internal and External Consistency': "Results of the final internal consistency tests are reported in Tables VI and VII. Table VI reports results of the goodnessof-fit tests where realizations of a subset of the data vector are generated for model parameters drawn from its own posterior. \nWe show results for analyses using the MagLim lens sample and for redMaGiC. Table VII reports extra consistency tests between the two-point functions that were not included as unblinding criteria. In these consistency tests, the data vector is split in two, and one part is used to predict the other. For instance, the first row of Table VII shows the result of a test where realizations of cosmic shear are generated for model parameters drawn from the 2 × 2 pt posterior, and compared to observations of cosmic shear. All pre-defined unblinding requirements were met for these internal data combinations, and both 3 × 2pt and the cosmologically-constraining subsets of the data for both lens samples show consistency with each other and with Λ CDM. However, there persists evidence of potential issues with the consistency of some parts of the combination of galaxy clustering and galaxy-galaxy lensing relative to the best-fit model of the full 3 × 2pt or cosmic shear data. In particular, for redMaGiC this seems slightly more likely to be sourced primarily from the clustering data. \nWe report detailed consistency metrics between external data set pairs in Table VIII. The details of the metrics and derivation of probability-to-exceed p are described fully in Ref. [133]. Overall, we find no significant (defined as p < 0 . 01 ) evidence of disagreement between the DES 3 × 2pt, external low-redshift, or Planck CMB data sets shown in Figs. 14 & 15. We show both a method measuring parameter differences and a method based on the evidence ('Suspiciousness') for the two primary comparisons of DES 3 × 2pt with external data sets. The evidence ratio is also shown, and all three metrics give qualitatively consistent results. \nAll data set combinations are largely in agreement, according to all tension metrics. Some differences between methods are likely due to non-Gaussianity of both posteriors involved, as well as the different approaches that the methods employ in treating the impact of priors on tension quantifications, but in all cases are below a fraction of a sigma. We also note that the Eigentension result for the case of DES 3 × 2pt vs complementary external low-redshift probes may not be capturing the full tension; the two data sets have comparable constraining powers, for which case Eigentension is not optimized."}
2023MmSAI..94d..88M	The popularity of pulsating stars resides in their capacity of determining several crucial and relevant parameters such as heliocentric distances ages metallicity gradients and reddening. RR Lyrae stars are old stellar tracers and have been detected in nearly all nearby galaxies that have been searched for these stars with just a few exceptions of very low mass dwarfs. Less common but also of great importance are Anomalous Cepheids indicators of either old or intermediateage population depending on their stellar origin. Classical Cepheids are only found within young stellar populations and because of their brighter absolute magnitudes they can be detected in galaxies farther than the Local Group. This paper presents a concise review built upon the aforementioned pulsating stars in Local Group dwarf galaxies and some of their applications to infer important properties of their host galaxies.	2023-12-01T00:00:00Z	['arXiv:2409.04993', '10.36116/MEMSAIT_94N4.2023.88', '10.48550/arXiv.2409.04993', '2024arXiv240904993M', '2023MmSAI..94d..88M']	['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies']	Pulsating Stars in Local Group Dwarf Galaxies	2,023	189	0.37	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	0	https://arxiv.org/pdf/2409.04993.pdf	{'No Header': 'Mem. S.A.It. Vol. 94, 88 © SAIt 2023 \n<!-- image -->', 'Pulsating stars in Local Group dwarf galaxies': "C. E. Mart'ınez-V'azquez 1 \nGemini Observatory / NSF's NOIRLab, 670 N. A'ohoku Pl., 96720 Hilo, HI, USA e-mail: [email protected] \nReceived: 04 October 2023; Accepted: 21 November 2023", 'Abstract.': 'The popularity of pulsating stars resides in their capacity of determining several crucial and relevant parameters such as heliocentric distances, ages, metallicity gradients and reddening. RR Lyrae stars are old stellar tracers and have been detected in nearly all nearby galaxies that have been searched for these stars, with just a few exceptions of very low mass dwarfs. Less common but also of great importance are Anomalous Cepheids, indicators of either old or intermediate-age population, depending on their stellar origin. Classical Cepheids are only found within young stellar populations, and because of their brighter absolute magnitudes, they can be detected in galaxies farther than the Local Group. This paper presents a concise review built upon the aforementioned pulsating stars in Local Group dwarf galaxies and some of their applications to infer important properties of their host galaxies. \nKey words. Stars: Variables: RR Lyrae stars, Anomalous Cepheids, Classical Cepheids Dwarf galaxies: Resolved stellar populations', '1. Introduction': "Pulsating stars are considered standard candles because they obey well established periodluminosity relations in optical and infrared bands (Catelan & Smith 2015). Therefore, they are used very often to derive accurate and precise distances. Coming from very di ff erent population, Cepheids (Population I) and RR Lyrae stars (RRLs; Population II) can be used as one of the first rungs in the cosmic distance ladder thanks to their recurrent and periodic brightness variation. \nBy far, the most frequent and common pulsating stars found among Local Group (LG) dwarf galaxies are RRLs (Monelli & Fiorentino 2022, and references therein). The discovery of the first RRL star was made \nby Williamina Fleming in the late 19th century through her analysis of plates from Solon Bailey's cluster survey in 1893 (Pickering et al. 1901). Thanks to their well-know periodluminosity relation (also known as Henrietta Leavitt's Law), the popularity of this type of stellar pulsator has increased exponentially, especially in recent decades. The extensive detection of RRL stars in surveys like ASAS, Catalina, DES, Gaia, OGLE, PanSTARRS or ZTF, coupled with their role as stellar tracers of ancient stellar population (Walker 1989; Savino et al. 2020), positions them as powerful tools to study old stellar structures. In addition, RRLs have been frequently used to detect / confirm new low mass (ultra-faint) and low surface brightness (ultra-di ff use) dwarf \ngalaxies and to obtain accurate distances to those systems, where the large contamination by field stars makes challenging the determination of distances from isochrone fitting (see e.g., Bechtol et al. 2015; Drlica-Wagner et al. 2015; Torrealba et al. 2016, 2019). In addition, in those systems where the rate of RRLs is statistically significant, RRLs can be utilized as metallicity tracers, giving insight into the chemical evolution of the old population they belong to (see e.g., Sculptor - Mart'ınezV'azquez et al. 2015, 2016a; Eridanus II Mart'ınez-V'azquez et al. 2021b). Thus helping us not only to reveal the formation history and chemical evolution of their host galaxy but also to provide clues about the contribution of dwarf galaxies in the formation of the halo of the larger galaxies they are bound to. \nAmong the brightest and most easily recognized pulsating stars in LG galaxies are Classical Cepheids (CCs) and Anomalous Cepheids (ACs). However, these two types of pulsators belong to di ff erent stellar populations. While CCs are young ( ∼ 100 -300 Myr, De Somma et al. 2020), ACs can belong to either old ( ≳ 10 Gyr) or intermediate-age ( ∼ 1 -6 Gyr) population since ACs can form through two di ff erent channels: originating either from the evolution of blue straggler stars (binary channel), or else from the evolution of single, metal-poor (Z < 0.0004, Fiorentino et al. 2006), relatively young ( ≲ 6 Gyr) stars, respectively. Interestingly, (Mateo et al. 1995) reported a correlation between the frequency of ACs (known as log S ) and the absolute V magnitude of the host galaxy (M V ), while Fiorentino & Monelli (2012) found that the frequency of ACs (at a fixed M V ) is higher in systems that harbor intermediate age population. This is attributed to the contributions of the two channels in generating ACs. However, in purely old galaxies such as Sculptor or Eridanus II, only the progeny of binary stars is expected. \nThe detection of CCs, ACs and RRLs in LG dwarf galaxies is of great importance not only to assess precise and accurate distances to those systems but also to be able to study the radial profiles of di ff erent populations -and therefore trace di ff erent ages-, measure their \nmetallicity gradients and help reconstruct the star formation history in galaxies where using population synthesis methods (which requires reaching the main-sequence) is di ffi cult or impossible. \nIn the following sections, I will highlight some of the results obtained from pulsating variable stars in classical, ultra-faint and ultradi ff use dwarf galaxies. I further discuss how the identification of these pulsating stars provides valuable insights into the star formation history, distance determination and possible tidal disruption of the host galaxy. Finally, I will conclude this manuscript by providing some RRL period-luminosity relations in the SDSS pass-band system to show the potential of having an homogeneous distance calibrator in the near future with the advent of the Vera C. Rubin LSST survey.", '2. Deciphering the star-formation with pulsating stars in classical dwarf galaxies': "Dwarf galaxies are the most abundant type of galaxy in the Universe. The term dwarf galaxies is used to refer to low luminosity galaxies (typically fainter that M V ∼ -17) dominated by dark matter (e.g., Tolstoy et al. 2009). The LG hosts a large number of dwarf galaxies of di ff erent morphological types: dwarf spheroidals (dSph; devoid of gas and with no recent star formation history, also called early types ), dwarf irregulars (dIrr; gas rich with an active star formation histories) and dwarf transition types (dT; with intermediate properties between the two groups). A classification based on the full star formation histories (SFHs) of dwarf galaxies defines two types of dwarf galaxies: slow and fast dwarf galaxies. Slow dwarfs formed a small fraction of their stellar mass at an early epoch, and continued forming stars until the present (Gallart et al. 2015). All dIrr with available full SFHs can be classified as slow dwarfs. Also, some dSphs like Leo I, Fornax or Carina have important intermediate-age and even young populations, and thus SFHs that resemble those of dIrrs. These can also be classified as slow dwarfs. Fast dwarfs are those that started their evolu- \nFig. 1. Color-magnitude diagrams of three classic dwarf galaxies (Leo A, Leo I and Sculptor) with di ff erent morphology and star formation histories. Highlighted are their RRLs (red dots) and Cepheids (blue stars), either CCs or ACs. From left to right, the dominant stellar component transitions from a notable young population to prominent old population. The ratio of RRLs over Cepheids increase also from left to right. It is evident how these pulsating stars prove invaluable in distinguishing among various systems that have experienced distinct star formation histories. \n<!-- image --> \non with a dominant star formation event and their period of star formation activity was short (less than a few Gyr). Most dSph galaxies are fast dwarfs, but not all. \nNumerous works have searched for variable stars within dwarf galaxies (see compilation of references for variable star studies in LG dwarf galaxies made by Mart'ınez-V'azquez et al. 2019 and Monelli & Fiorentino 2022). In the context of this review, I will just mention three galaxies that combine di ff erent morphology classification and star-formation histories to show how di ff erent star-forming rates impact the population of CCs, ACs and RRLs. \nLeo A is a predominantly young dIrr galaxy with a considerable delay in its star formation history. Cole et al. (2007) refer to this galaxy as an extreme case of a 'late bloomer' in the LG since they find that over 90% of all the star formation that ever occurred in Leo A happened more recently than 8 Gyr ago (with the peak of the star formation rate happening at 1.5 Gyr) and ongoing star formation to this day (Leˇsˇcinskait˙e et al. 2022). Bernard et al. (2013) identified 156 Cepheids and only 10 RRLs. About the 90% of the detected Cepheids have periods shorter than 1.5 d. In this case \nalso, a comparison with theoretical models using evolutionary tracks for stars that ignite helium in the core in degenerate (ACs) and nondegenerate conditions (CCs) indicate that some of the fainter stars classified as CCs could be ACs. In comparison with Leo I, the density of RRLs in Leo A is very low. This agrees with the star-formation history obtained by Cole et al. (2007) where only a very small amount of star formation occurred in the first few Gyr after the Big Bang. \nLeo I is a dSph galaxy with a slow starformation history. Fiorentino et al. (2012) study its pulsating stars and conclude that Leo I contains the largest sample of Cepheids and the largest Cepheids to RRL ratio found in a dSph (106 RRL stars and 51 Cepheids). In addition, because of its extended and recent star formation history (star formation enhancements 13, 5.5, 2, and 1 Gyr ago, after which it was substantially quenched; Ruiz-Lara et al. 2021), its Cepheids traces a unique mix of ACs (blue extent of the red-clump, partially electrondegenerate central helium-burning stars) and short-period CCs (blue-loop, quiescent central helium-burning stars). \nOn the other hand, Sculptor is a dSph galaxy with a fast star-formation history. de Boer et al. (2012) showed that the star formation history of Sculptor is dominated by old ( > 10 Gyr) stars, and the majority of its total mass in stars was formed between 14 and 7 Gyr ago, with a peak at 13-14 Gyr ago. In terms of pulsating stars, the galaxy does not contain any CC (because of the absence of any recent star-formation), but according to the search for variable stars made by Mart'ınez-V'azquez et al. (2016b), it contains a plethora of RRL stars (536) and a few (4) ACs. \nThese three illustrative cases comprise two slow galaxies with di ff erent stars formation histories (an extended and a very recent one) and one fast dwarf galaxy. These cases help us realize how powerful detecting di ff erent types of pulsating stars -such as RRLs, ACs and CCs that belong to di ff erent stellar population- are in order to determine the star formation history of the galaxies. Figure 1 shows the colormagnitude diagrams for the three aforementioned galaxies, highlighting their RRLs and Cepheid stars.", "3. The role of pulsating stars in 'Newly discovered' dwarf galaxies": 'In the past two decades, we have witnessed the discovery of about 60 new ultra-faint, and two new ultra-di ff use, dwarf galaxies. These have been detected thanks to large-area, deep, multi-color imaging sky surveys carried out with the Dark Energy Camera (e.g., DES, SMASH, MagLites, DELVE), Hyper Suprime Cam, Pan-STARRS and Gaia. The low masses and stellar densities, coupled with the high contamination by field stars, make the determination of morphological parameters and distances for these galaxies a challenging task. A compelling approach to improve the distance determination to these ultra-faint systems -and thus clarify their nature- is to detect RRL members. Moreover, RRLs can also provide valuable insight into the properties of the old stellar population of the host.', '3.1. Ultra-faint dwarf galaxies': "The Sloan Digital Sky Survey (SDSS) discovered a new class of objects, the ultra-faint dwarf (UFD) galaxies, the first examples being Willman 1 and Ursa Major I (Willman et al. 2005a,b). These UFDs extend the spectrum of properties of classical LG dwarf galaxies to a lower mass regime (M V > -7 . 7 mag, Simon 2019). Since these first discoveries, more than 50 UFDs have been found in the Milky Way (MW) neighbourhood (see e.g. Drlica-Wagner et al. 2020; Cerny et al. 2023b, and references therein). UFDs are possibly the oldest and most primitive of galaxies (Bose, Deason & Frenk 2018; Simon 2019). According to the hierarchical galaxy formation model (White & Frenk 1991), large galaxies are built up by the accretion of smaller galaxies; thus, UFDs may be representative of the basic building blocks of the galaxy formation process. \nThere are several studies in the literature searching for variable stars in UFDs. UFDs are prone to harbor at least one RRL if the absolute V magnitude of the host system is M V ≲ -3 . 5 (Mart'ınez-V'azquez et al. 2019). Table 1 shows the updated number of RRL stars (up to the year 2023) associated to ultrafaint systems sorted by their heliocentric distances. The motivation to search for RRL stars in these ultra-faint systems is to get precise distance measurements to help improve estimates of parameters such as the absolute visual magnitude and the physical size of the host. As an example, the first detection of RRL stars in Grus I (2), Phoenix II (1), and in Grus II (1) allowed Mart'ınez-V'azquez et al. (2019) to measure distances of these galaxies for the first time using stellar candles, which placed them farther away than predicted in their discovery papers. These refined distance implied larger sizes for these systems with a 30% change for Phoenix II and 5% for Grus I and Grus II. RRL stars can also be instrumental in studying the extent of the systems and identifying potential tidal stripping. Garling et al. (2018) identified three extra-tidal RRL stars in Hercules in addition to its nine already known RRL stars within its tidal radius (Musella et al. 2012), which suggest that Hercules has been tidal stripped. \nTable 1. Compilation of the number of RRL stars found in ultra-faint dwarf (UFD) systems up to the year 2023. \nNotes .- The table is sorted by heliocentric distances, ranging from 21.5 kpc (Dra2) to 763 kpc (And11). Systems marked with an asterisk ( ∗ ) are now confirmed clusters. \nIn particular, the location of one of the outer RRLs is aligned with Hercules' orbit and consistent with the debris being in that direction. \nThe most exhaustive investigation concerning the association of RRL stars with UFDs was carried out by Vivas, Mart'ınezV'azquez & Walker (2020) using the Gaia DR2 RRL catalog (Clementini et al. 2019). Vivas, Mart'ınez-V'azquez & Walker (2020) found 47 RRL stars associated with 14 dif- \nMW ultra-faint satellites within 100 kpc. They identified RRLs for the first time in Tucana II, finding additional members in Ursa Major II, Coma Berenices, Hydrus I, Bootes I and Bootes III, and distinguishing possible candidate extra-tidal RRLs in Bootes I, Bootes III, Sagittarius II (cluster), Tucana III, Eridanus III (unclassified system), and Reticulum III. Recently, several independent studies kept searching and detecting RRL \nFig. 2. Number of RRL stars in dwarf galaxies as a function of the absolute V magnitude of the host galaxy, MV. The di ff erent colors represent the area (in rh units) of the variable star search for a given galaxy. The black and green dashed lines show the linear fit between log NRRL and MV for those galaxies where the variability search areas was greater or equal than 2 rh obtained by Mart'ınez-V'azquez et al. (2019) and this work after including updates (up to the year 2023). The left panel is in semi-logarithmic scale while the right panel is a zoom-in of the left panel on the faint dwarf regime (MV > -7 . 7 mag) in a linear scale. The black and green lines represent the same respective fits in the left and right panels. \n<!-- image --> \nstars as member of newly discovered galaxies (DELVE 2: Cerny et al. 2021a, Centaurus I: Mart'ınez-V'azquez et al. 2021c, Eridanus IV: Cerny et al. 2021b, Pegasus IV: Cerny et al. 2023a) or already know UFDs (Pisces II, Pegasus III: Garofalo et al. 2021, Eridanus II: Mart'ınez-V'azquez et al. 2021b), Cetus III, Tucana IV: Stringer et al. 2021). In these studies, distant RRL stars have been identified at ∼ 6 rh from Tucana IV and Centaurus I and ∼ 10 rh from Pegasus IV. The detection of the aforementioned distant RRLs may suggests that a past tidal disruption could have happened in these galaxies. Nonetheless, to confirm this scenario, radial velocities of these stars are needed to determine whether those RRLs are (or are not) members of those UFDs. \nFigure 2 shows a comprehensive analysis up to the year 2023 of the number of RRLs in dwarf galaxies. This figure is an updated version of Figure 10 in Mart'ınez-V'azquez et al. (2019) and it includes as a color-map the information about the variable star search area (in rh units) for each galaxy. An updated fit of the number of RRL stars as a function of MV for those galaxies where the variability search \ncovers an area greater or equal than 2rh (green dashed line in Figure 2) is given by: \nlog N RRL = -0 . 296 ± 0 . 013 M V -0 . 797 ± 0 . 107(1) \nwhich is nearly identical to the relation provided by Mart'ınez-V'azquez et al. (2019) (black dashed line in Figure 2) but with slightly improved precision in the coe ffi cients. \nEither by utilizing this correlation or by examining Figure 2, we can deduce that the discovery of new UFDs through the identification of cluster of RRLs (Baker & Willman 2015) will be e ff ective only for those UFDs brighter than MV ∼-4 mag. Stringer et al. (2021) quantified the sensitivity of this search using a suite of simulated satellite galaxies generated by Drlica-Wagner et al. (2020) where the expected number of RRLs was predicted by using the previous version of Eq. (1). They found that an RRL-based search is more sensitive than those using resolved stellar populations in the regime of large (r > 500 pc) and low surface-brightness dwarf galaxies. However, the isochrone matched-filter searches remain more sensitive for satellites with MV > -5 mag, due to the small number of RRLs expected in these galaxies. \nFig. 3. Census of RR Lyrae stars (upper left), anomalous Cepheids (upper right), classical Cepheids (bottom left) and type II Cepheids (bottom right) in LG dwarf galaxies. The size of the symbols is a representation of the number of stars of each type that a galaxy contains. The upper panels depict the LMC and SMC as unfilled symbols to enhance visualization. As both galaxies are known for harboring a substantial population of RR Lyrae stars and anomalous Cepheids, their symbols overlap with others, and if filled, they would obscure them. \n<!-- image -->", '3.2. Ultra-diffuse dwarf galaxies': 'Torrealba et al. (2016, 2019) discovered two new MW dwarf galaxies (Crater II and Antlia II) of a type unprecedented in the MW vicinity, the so-called ultra-di ff use galaxies (UDG). These galaxies have stellar masses comparable to classical dwarfs like Sculptor or Fornax but they are much more extended systems. Their surface brightnesses are fainter than 30 mag arcsec -2 . Recent publications show that these galaxies host hundreds of RRL stars (99 Crater II, Joo et al. 2018; Vivas et al. 2020; 318 Antlia II; Vivas et al. 2022) plus several ACs (7 Crater II, 8 Antlia II) located within 2 r h . \nFrom the spatial distribution of the RRLs in Crater II, Vivas et al. (2020) detect a more elongated shape ( ϵ = 0 . 24) than found when selecting giant branch stars ( ϵ = 0 . 12). From \ntheir high quality light curves, they refine the distance of Crater II at 117 ± 4 kpc and measure a metallicity dispersion of 0.2 dex, consistent with that found for the RGB members stars by spectroscopy (Caldwell et al. 2017; Fu et al. 2019), and with its relatively narrow red giant branch (Walker et al. 2019). \nAntlia II also shows an elongated shape ( ϵ = 0 . 28) from its population of RRLs (Vivas et al. 2022), although a more extended shape has been measure by their spectroscopic members ( ϵ = 0 . 60, Ji et al. 2021). It is slightly farther (average distance to Antlia II based on the RRLs is 124.1 kpc), more massive and extended than Crater II (rh is ∼ 3 times larger), and it exhibits a distance gradient of 2.72 kpc deg -1 along the semi-major axis of the galaxy with the southeast side being farther away than the northwest side. The elongation along the \nFig. 4. Distribution of dwarf galaxies in the sky up to ∼ 2 Mpc. The red dot symbols are those galaxies where variable star searches have been conducted beyond 2 rh. Blue circles are those that do not meet the above criteria but where variable searches have been carried out. Black crosses are galaxies with no variability studies to the date. To allow the readability of the figure, only galaxies with variable star searches covering an area larger than 2 rh have been labeled. \n<!-- image --> \nline of sight is likely due to the ongoing tidal disruption of Antlia II. In particular, Vivas et al. (2022) show that a model in which Antlia II is tidally disrupting explains the observed distance gradient from the RRLs.', '4. Updated census of pulsating stars': "Using the compilation from Mart'ınez-V'azquez et al. (2019); Vivas, Mart'ınez-V'azquez & Walker (2020) and recent updates (see Table 1) for RRL stars and the compilation of Monelli & Fiorentino (2022) for CCs, ACs and type II Cepheids, Figure 3 shows the amount of each of these pulsating stars (weighted by the symbol size) in LG dwarf galaxies. Looking at these maps, one can clearly notice how RRLs are the most abundant pulsating stars detected in the LG. \nFigure 4 displays the distribution of all the known dwarf galaxies up to ∼ 2 Mpc. \nThe chart highlights whether variability studies have been carried out in these galaxies or not. To date, there are 129 dwarf galaxies within 2 Mpc and to the best of my knowledge variability studies have been carried out in 88 of them. However, there are only 57 dwarf galaxies where their variable star searches span an area up to or beyond 2 half-light radii. From the 41 without variable studies yet, only 5 of them have distance moduli below 20.5 mag and thus can be searched with Gaia for RRL stars.", '5. Towards an homogeneous RR Lyrae distance scale': "In the past few decades, the use of the SDSS pass-bands has become significant in big survey cannons such as PanSTARRS and DECam. The upcoming Vera Rubin LSST survey will also observe in the ugrizy bands. \nStudies based on star formation histories of LG galaxies show that all systems contain an old population (see e.g., Weisz et al. 2014). Since old populations are ubiquitous, RRL stars can be found almost everywhere in our Local neighborhood. \nIn order to get precise and homogeneous distances using RRL stars, it is crucial to build period-luminosity relations in the SDSS bands. Recently, Marconi et al. (2022) provided new theoretical period-luminosity-metallicity relations for the Vera C. Rubin LSST filters. \nHowever, only a few empirical periodluminosity relations in the SDSS bands have been derived (Sesar et al. 2017; Vivas et al. 2019; Mart'ınez-V'azquez et al. 2021b), utilizing systems that exhibit distinct metallicity distributions. \nDeriving a well-calibrated set of periodluminosity relations in the SDDS bands that spans various metallicity and period ranges is essential as the Vera C. Rubin LSST survey begins detecting RRL stars, aiming for accurate distance measurements. This e ff ort will enhance our ability to gain a more detailed understanding of the Galactic structure.", '6. Conclusion and Final Remarks': "About 130 known dwarf galaxies within a heliocentric distance of 2 Mpc reside in our LG. Out of them 44% have a catalog of variable \nFigure 5 presents the period-luminosity relations of the only three galaxies with enough RRL stars and well-sampled light curves observed with DECam in the g , r , i bands (Sextans: Vivas et al. 2019, Crater II: Vivas et al. 2020, and Eridanus II: Mart'ınezV'azquez et al. 2021b). The zero-point differences are mainly driven by the metallicity dependence. Since Sextans and Crater II share the same mean metallicity of [Fe / H] ∼ -2 . 0 dex (Battaglia et al. 2011; Ji et al. 2021) both data set overlap and share a common period-luminosity relation in the g-band (upper panel). Eridanus II is a more metal-poor system, [Fe / H] ∼ -2 . 4 dex (Li et al. 2017). This is reflected in its fainter absolute magnitudes, i.e., larger zero-point. \n<!-- image --> \n<!-- image --> \nFig. 5. Period versus absolute magnitude plots in the g , r , i bands for di ff erent dwarf galaxies observed with DECam (dots: Sextans, squares: Crater II, stars: Eridanus II). The di ff erent colors represent the di ff erent types of RRLs (blue: RRab, orange: RRc, green: RRd). The periods of the RRcd stars have been fundamentalized. The solid lines are the empirical period-luminosity relations derived by fitting the data (black: Sextan, Crater II or both ( ⟨ [Fe / H] ⟩ ≈ -2 . 0); red: Eridanus II ( ⟨ [Fe / H] ⟩ ≈ -2 . 4). \n<!-- image --> \nstars that at least reach 2 r h and 32% have not yet been searched for variables (see Figure 4). \nWe are continuing our observations on these interesting and important systems, and \nanticipate a harvest of new galaxies to study following the commencement of the Vera C. Rubin LSST survey. \nAdditionally, future imaging surveys will lead to the detection of new UFDs and UDGs (see, e.g., Mutlu-Pakdil et al. 2021). The characterization of these systems is of great value to study the dark matter (Nadler et al. 2021) and trace the mass assembly history of the MW. The determination of accurate and precise distances is crucial in determining the nature (mass and size), orbits, and pericentric passages of these Galactic building blocks. \nThe advent of extremely large telescopes allow the detection of RRL stars up to 6 Mpc and, therefore, will enable the study of old populations in systems where the main-sequence cannot be resolved. As shown in this review, the identification and analysis of pulsating stars that belong to di ff erent stellar populations such as Cepheids and RRLs, will be crucial to decipher the star-formation history and to investigate age gradients in their host galaxies. \nAcknowledgements. C.E.M.-V. thanks all her collaborators and many other researchers for their contribution and hard work searching for pulsating variable stars in LG galaxies. 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2024EPJD...78..130Y	The Sun is a standard reference object for astrophysics and also a fascinating subject of study in its own right. Xray and extreme ultraviolet movies of the Suns atmosphere show an extraordinary diversity of plasma phenomena from barely visible bursts and jets to coronal mass ejections that impact a large portion of the solar surface. The processes that produce these phenomena heat the corona and power the solar wind remain actively studied and accurate atomic data are essential for interpreting observations and making model predictions. For the Suns interior intense effort is focused on resolving the solar problem a discrepancy between solar interior models and helioseismology measurements and atomic data are central to both element abundance measurements and interior physics such as opacity and nuclear reaction rates. In this article topics within solar interior and solar atmosphere physics are discussed and the role of atomic data described. Areas of active research are highlighted and specific atomic data needs are identified. An image of a solar active region obtained with the 193 A channel of SDOAIA showing plasma at around 1.5 million degrees.	2024-10-01T00:00:00Z	['10.48550/arXiv.2409.09166', '2024arXiv240909166Y', '2024EPJD...78..130Y', 'arXiv:2409.09166', '10.1140/epjd/s10053-024-00915-6']	['Astrophysics - Solar and Stellar Astrophysics', 'Physics - Space Physics']	Applications of atomic data to studies of the Sun	2,024	189	0.47	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	0	https://arxiv.org/pdf/2409.09166.pdf	{'Peter R. Young 1,2': '1 Heliophysics Division, NASA Goddard Space Flight Center, Greenbelt, MD, 20771, USA. \n2 Department of Mathematics, Physics & Electrical Engineering, Northumbria University, Newcastle Upon Tyne, NE1 8ST, UK. \nContributing authors: [email protected];', 'Abstract': "The Sun is a standard reference object for Astrophysics and also a fascinating subject of study in its own right. X-ray and extreme ultraviolet movies of the Sun's atmosphere show an extraordinary diversity of plasma phenomena, from barely visible bursts and jets to coronal mass ejections that impact a large portion of the solar surface. The processes that produce these phenomena, heat the corona and power the solar wind remain actively studied and accurate atomic data are essential for interpreting observations and making model predictions. For the Sun's interior intense effort is focused on resolving the 'solar problem,' (a discrepancy between solar interior models and helioseismology measurements) and atomic data are central to both element abundance measurements and interior physics such as opacity and nuclear reaction rates. In this article, topics within solar interior and solar atmosphere physics are discussed and the role of atomic data described. Areas of active research are highlighted and specific atomic data needs are identified. \nKeywords: The Sun, Solar atmosphere, Solar interior, Laboratory astrophysics", '1 Introduction': 'Solar Physics encompasses studies of the solar interior and the solar atmosphere. The latter extends far from the Sun and transitions to the solar wind, thus there is significant overlap with Heliospheric Physics particularly with regard the origin of the solar wind and how it is accelerated. Atomic data are needed in order to model radiative emissions from the solar plasma and compare with remote sensing observations. They are also required for modeling the solar interior, in particular opacities and nuclear cross-sections. This article is based on an invited talk [1] given at the \n14th Atomic Spectra and Oscillator Strengths conference 1 held in Paris in July 2023. It describes recent advances in applying atomic data to Solar Physics and highlights areas where new data are needed. \nSolar Physics is a field that is strongly motivated by new observational data, particularly from space. The US space agency, NASA, dominates solar space observations as shown in Table 1. Fourteen current and future major space missions are listed, with six led by NASA and four having NASA as a major partner. The growing strength of China and India is reflected in the three recent missions from these countries. \nTable 1 Major Solar Physics space missions. Acronyms are given in Table 3. \n1 NASA a major partner. \n2 Expected launch dates. \nGiven the dominant impact of NASA on the field, it is appropriate to expand on the role of Solar Physics within NASA. The Science Mission Directorate (SMD) within NASA funds basic science, including space missions and research. There are four major science divisions within SMD: Earth Science, Planetary Science, Astrophysics and Heliophysics. Solar Physics falls within the latter, which also includes Heliospheric Physics (study of the solar wind), and physics of the near-Earth plasma environment (magnetosphere, ionosphere, thermosphere and mesosphere). One focus of Heliophysics is Space Weather, the effect of solar activity on the Earth, particularly its effect on astronauts and space- and ground-based infrastructure. \nA dedicated Division for Heliophysics ensures there is regular cadence of NASA Solar Physics missions. However, support for atomic data calculations and measurements within NASA Heliophysics is more limited than for NASA Astrophysics. For Astrophysics and Solar Physics, atomic data are needed to interpret the radiative emissions measured by telescopes. Heliophysics includes studies of solar wind and Earth plasmas that can be directly sampled by instrumentation, hence atomic data are not required for interpreting these data. Thus demand for new atomic data within Heliophysics is mostly confined to the remote-sensing Solar Physics community. Of course there is a wide overlap between atomic data needs for Astrophysics and Solar Physics, so the \nlatter community benefits from advances made for Astrophysics. \nSection 2 summarizes some properties of the solar interior and how it is studied, and Section 3 gives examples of how atomic data is applied to studies of the interior and photosphere. Properties of the solar atmosphere are summarized in Section 4, and applications of atomic data are described in Section 5. A summary is given in Section 6. Acronyms for Solar Physics missions and instruments mentioned in this article are given in Table 3.', '2 The Solar Interior and Photosphere': "The Sun's interior comprises the core where nuclear fusion occurs, the radiative zone where energy is transferred outwards by photons, and the convection zone where energy is transferred by convective motions. The boundaries between the regions are at 0.25 R ⊙ and 0.70 R ⊙ . The solar rotation changes at the latter boundary (termed the tachocline) from uniform in the inner region to differential in the outer region, which is believed to play an important role in the solar dynamo. The solar photosphere-usually considered the surface of the Sun-is where the Sun transitions from being optically thick to radiation to optically thin. The thickness of this layer is only around 500 km and the temperature ranges from around 6200 K at Rosseland mean optical depth unity to close to 4000 K at the uppermost parts. The photosphere is the boundary between the solar interior and the atmosphere, and for this article I assign it to the interior. This is principally because the measurement of element abundances in the photosphere is particularly relevant to solar interior studies (Section 3.3). \nInterest in the interior has a strong overlap with Astrophysics, where the Sun is a reference object for topics such as stellar structure, stellar dynamos and element composition in the universe. In contrast to the highly dynamic solar atmosphere, the solar interior is mostly viewed as a constant except for the rhythm of the 11-year solar cycle. The physics of the solar interior are well-established to the extent that Standard Solar \nModels (SSMs) exist that yield excellent agreement with observed quantities, with the notable exception of the 'solar problem' described later. \nObservationally, the interior is mostly studied from ground-based observatories. When viewed at high spatial and temporal resolution, the photosphere shows continuous convective motions organized into granules and supergranules of sizes around 1 and 30 Mm, respectively. Helioseismology is the measurement and interpretation of oscillations of the solar surface, usually obtained from Doppler shifts of visible absorption lines. Wave modes of different frequencies penetrate to different depths in the interior hence helioseismology can yield quantities such as pressure, density and sound speed in the interior-crucial for testing SSMs. \nThere are numerous ground-based solar telescopes around the world. Large telescopes such as the Goode Solar Telescope (California), the Daniel K. Inouye Solar Telescope (Hawaii), the Swedish Solar Telescope and the GREGOR telescope (both in the Canary Islands) yield images of the photosphere down to a resolution of around 0.1 '' (70 km). The Global Oscillation Network Group (GONG) is a network of six smaller-scale telescopes distributed so as to give almost 24-hour coverage of the Sun. Each telescope observes the entire solar disk and the data are used for helioseismology. Space-based visible telescopes can yield 24-hour coverage of the Sun and are free from distortion effects due to atmospheric seeing, however the sizes of the telescopes are limited. The most important space instrument is the SDO/HMI, which obtains full disk Dopplergrams for helioseismology and magnetograms for studies of the solar magnetic field. \nIn the 21st Century a great deal of attention has been paid to what has been referred to as the 'solar abundance problem' or the 'solar modeling problem' (here simply 'solar problem' is preferred). As discussed in Section 3.3, element abundances are derived from modeling of solar photospheric spectra and used as input to SSMs. Advanced 3D models have replaced earlier 1D models and yielded abundances that can be significantly different, depending on the analysis method and lines studied. SSMs that previously gave results that agreed well with parameters derived from helioseismology no longer do so with \nsome of the newer abundance results [2]. For example, predictions of sound speed profiles, the depth of the convection zone, and the helium abundance in the solar envelope are all modified when the photospheric abundances change. Atomic data are highly relevant to this problem, as discussed in the following section.", '3 Applications of Atomic Data to the Solar Interior and Photosphere': 'There are three main areas to which atomic data are important for the solar interior. Nuclear cross-section rates and atomic opacities are critical for SSMs, while measurements of element abundances in the photosphere depend on atomic cross-sections and oscillator strengths.', '3.1 The solar core: nuclear reactions': "Nuclear reactions in the Sun's core provide the energy source for the Sun's radiative emissions, and the dominant process is the proton-proton (pp) chain that produces 4 He. A combination of experimental and theoretical cross-sections are required for various reactions that comprise the pp chain and its sub-branches. For example, the initial reaction that fuses two protons to yield deuteron has a cross-section that is too small to be measured in the laboratory. For reaction crosssections that can be measured in the laboratory, the energy range is often higher than the energies relevant in the Sun and hence theory is used to extrapolate to lower energies. Measurements and theory yield the S -factor, which is derived from the cross-section by removing some of the rapidlyvarying energy terms. Figure 1 shows the S -factor for the 2 H + p → 3 He + γ reaction. Agreement between the theoretical calculation (solid line) and the measured values is excellent. Comprehensive reviews of the nuclear cross-sections that recommended best values and uncertainties were published in 1998 and 2011 [3, 4], and a further update has recently been prepared [5]. \nModels for the nuclear processes can be checked indirectly through their input into Standard Solar Models followed by comparisons with helioseismology, and directly through their prediction of neutrino fluxes. The 'solar neutrino \nFig. 1 A comparison of four experimental measurements of S for the 2 H + p → 3 He + γ reaction with a theoretical calculation (black line). The broad red line shows the fit to the experimental data. The inset planel shows a close-up of the low energy points. From [4]. \n<!-- image --> \nproblem,' whereby the flux of electron neutrinos at the Earth was a factor three lower than expected was resolved in the early 2000s following measurements by Super-Kamiokande [6] and the Sudbury Neutrino Observatory [7] that demonstrated that electron neutrinos change their flavor during their passage to the Earth, explaining the lower fluxes. \nIn the push for improved accuracy for neutrino measurements, underground experiments have become a priority in order to reduce contamination by cosmic rays. An example is the Laboratory for Underground Nuclear Astrophysics (LUNA) at Gran Sasso, Italy, which specializes in low-energy reactions for Astrophysics (Figure 1).", '3.2 Opacity': "M. Schwarzschild [8] stated in 1958: 'For the astronomer who tries to reconstruct the stellar interior the opacity is by far the most bothersome factor in the entire theory.' and it continues to be a major focus for solar scientists \nand atomic physicists. Opacity is the degree to which radiation is absorbed by a plasma and it comprises the individual processes photoexcitation (bound-bound), photoionization (boundfree), inverse bremsstrahlung, and photon (Thomson) scattering. In the past, stellar interior codes used as input the Rosseland mean opacity, which is the total opacity integrated over frequency. Modern codes such as Modules for Experiments in Stellar Astrophysics [9], CESAM2K [10] and the Toulouse-Geneva Evolution Code [11] directly use opacity tables from, e.g., the Opacity Project [12] or OPAL [13]. \nThe opacity has a dependence on both element abundances and temperature, such that an element's contribution changes with location within the interior. The region close to the convective/radiative zone boundary is of particular interest as the opacity here effectively determines where the boundary occurs. The boundary has been accurately measured from helioseismology [14] giving a good point of comparison with SSMs. The largest contributors to opacity near the boundary are oxygen, iron and neon [15]. The oxygen opacity remains uncertain due to controversy over the photospheric oxygen abundance. Recent values are generally smaller than the standard oxygen abundance from the 1990s [16, 17], with one value up to 34% lower [18], while another is 17% lower [19] but consistent within the uncertainties. Neon can not be directly measured in the photosphere [18], but can be measured relative to oxygen in the solar atmosphere [20]. Hence a lower oxygen abundance indirectly lowers the neon abundance, too, further reducing the opacity. The result is a significant reduction in the convection zone depth in SSMs, giving a large discrepancy with the empirical value [21]. \nThe opacity data used in SSMs are largely from theoretical calculations, and the discrepancy could be resolved if the theoretical data underestimate actual opacities by 15-20%. The main sources of opacities are the Opacity Project and OPAL, but good agreement is found between the two with no evidence of errors that could explain the solar problem [22]. \nOpacities have been measured in the laboratory since the 1980s [24], and recent work has focused on iron opacities in the energy region \nFig. 2 Comparisons of model (red line) and measured (black line) opacities for Cr, Fe and Ni at a temperature of 180 eV and a density of 3 × 10 22 cm -3 . Significant differences are found for Fe, but not Cr and Ni. From [23]. \n<!-- image --> \ncorresponding to the solar tachocline. In a surprising result [25], the measured opacities for iron (charge states 16+, 17+ and 18+) were found to be higher than the theoretical values by amounts ranging from 30% to 400%, depending on wavelength. A follow-up work [23] on Cr and Ni opacities found significantly smaller discrepancies (Figure 2), which suggests physics problems specifically for Fe at the temperatures and densities sampled by the experiment. New theoretical work has been performed for Fe xvii [26] and Fe xvii-xix [27, 28] to investigate the source of the experimental discrepancy, but the opacities remain relatively unchanged.", '3.3 Photospheric abundances': "Element abundances derived from the Sun's photospheric spectrum are important inputs to SSMs. They are derived in a two-step process whereby a 3D hydrodynamic model of a small area of the solar surface that includes the sub-surface convection zone and photosphere is first created. Temporal snapshots are then extracted from the simulation and subject to radiative transfer models to predict the absorption profile of the line of interest. The emission in a line depends on the level populations of the atom or molecule, which can be assumed to be in local thermodynamic equilibrium (LTE). More commonly in modern calculations, non-LTE is assumed whereby the balance between multiple atomic processes is solved to yield the populations. Unlike earlier 1D models, the 3D models do not have any free parameters that can \nFig. 3 An example of modeling a photospheric absorption line to infer the element abundance. The line is Ca i 6439 ˚ A, and three modeled profiles are shown. The lower panel shows the differences between the observed profile and modeled profiles. From [32]. \n<!-- image --> \nbe adjusted and are found to accurately reproduce parameters such as the photospheric temperature [29], giving confidence in their accuracy. The abundance itself is usually obtained by fitting the synthetic line profile to one or more solar spectra [30], but fits to the equivalent widths can also be performed. \nThe initial results from the 3D models generated controversy in the early 2000s, as the derived C, N and O abundances were significantly lower than the values previously used in Astrophysics. Due to the importance of these elements to SSMs (see above) the results have been revisited several times and generally been confirmed [18], although there remain differences between different authors [31]. \nAtomic data play a critical role in the abundance measurements in several ways. Most directly, accurate oscillator strengths are needed for the strengths of the absorption lines. To fit the profile shapes, Stark and van der Waals broadening parameters are needed. For the non-LTE level population calculations, cross-sections for electron and hydrogen collisions, and photoionization are also needed, together with accurate energy levels [33]. \nThe process of validating the abundances obtained by the 3D NLTE modeling is complex. Taking oxygen as an example, both atomic and molecular (OH, CO) transitions can be used and care has to be taken to account for blending both in the cores of the lines and the wings. Centerto-limb variation of the lines' observed profiles needs to be modeled depending on the observed spectra being fit (e.g., disk-center, near-limb, or disk-averaged), but can also be used to constrain \natomic parameters [34]. Due to the large differences with earlier 1D models, it is typical to compare the 3D model results with various flavors of 1D models. For example, averaging the 3D hydrodynamic model over space and time to create a 1D model, or using theoretical and semi-empirical models 1D models with the same equation of state, radiative transfer methods and opacities [e.g., 18, 34]. \nFigure 3 [from 32] illustrates the change in an absorption line profile from LTE to non-LTE, and also the impact of using accurate atomic data (the NLTE curve) versus simple approximations to the data (the vR+A curve, where 'vR' and 'A' refer to standard formulae for electron excitation and ionization rates due to van Regemorter and Allen). The example is the Ca i 6439 ˚ A line, and the same calcium abundance and macroscopic broadening parameters are used for the three curves. In this case the accurate atomic data do not have a significant impact on the derived abundance, but do lead to an improved fit to the line profile.", '4 The Solar Atmosphere': "The Sun's atmosphere is strikingly revealed during total solar eclipses, where lobes and plumes of emission can be seen extending 2-3 R ⊙ from the solar limb. The emission is principally photospheric light scattered by free electrons in the Sun's corona. The high temperature of the corona (1-2 MK) was established in the 1940s and implied that the bulk of the corona's emission should be at extreme ultraviolet (EUV) and X-ray wavelengths. Multiple spacecraft now routinely image the corona at EUV wavelengths, including NASA's flagship SDO, which obtains full-disk images in seven EUV wavelengths at 12 s cadence and 1.2 '' (900 km) spatial resolution. Figure 4(c) shows a section of an SDO/AIA image obtained at a wavelength of 193 ˚ A. \nThe complex structure of the corona seen in this image is driven by the convective motions in the subsurface layers, which stretch, squeeze, twist and move the magnetic field that passes through the photosphere. The low density of the corona means the plasma is constrained to move along magnetic field lines, giving loops and plumes of emission according to if the magnetic field is closed or open. The 3D coronal magnetic field can not be routinely measured, but estimates are \npossible through coronal seismology [35], radio gyroresonance emission [36] and polarization of coronal forbidden lines [37]. Instead, the coronal field is usually extrapolated from the photospheric field, which is routinely measured by SDO/HMI. Figure 4(a) shows the line-of-sight (LOS) magnetic field strength in the photosphere, as measured by SDO, with white and black denoting strong fields of opposite polarity and gray denoting areas of weak magnetic field. The solar surface is uniformly covered with a network of small-scale, weak magnetic field structures that are organized by the sub-surface convective motions. Large-scale strong field is associated with active regions, such as the one in the center of Figure 4(c). These generally have a bipolar structure. \nThe temperature of the photosphere falls with height to a minimum of 4000 K, marking where the chromosphere begins. This is a region where temperature increases to 10,000 K at heights of around 10Mm and it can be studied through lines in the visible (particularly H α and Ca ii H & K) and UV (Mg ii h & k). The UV continuum is also important for the chromosphere, and Figure 4(b) shows an image from the SDO/AIA 1700 ˚ A channel. Comparing with the upper panel, the dark sunspots can be identified with the most intense magnetic field. Intermediate magnetic field strengths correlate with areas of brightness in the chromosphere that are termed plage. The temperature rises sharply from the chromosphere to the corona through a thin layer called the transition region: the large temperature gradient is due to the large conductive flux from the hot corona. \nThe wavelength region 150-1600 ˚ A is particularly important for observing the solar atmosphere as it features emission lines from the chromosphere to the corona. Transition region lines are mostly found above an approximate dividing line at 400 ˚ A, and coronal lines below this wavelength. A group of strong resonance lines of Fe ix-xiv between 170-212 ˚ A are particularly important. Interest in the far-UV and EUV has motivated a wide range of space instrument designs and current and future concepts are discussed in [38]. \nFig. 4 Images of the Sun from 23 October 2014 at 12:00 UT. Panel (a) shows the photospheric LOS magnetic field from SDO/HMI, scaled within ± 2000 G; Panel (b) shows a chromospheric continuum image at 1700 ˚ A from SDO/AIA; and Panel (c) shows a coronal emission line image at 193 ˚ A, also from SDO/AIA. A logarithmic intensity scaling is used for the latter two Panels. The blue lines on Panel (c) indicate the sub-regions shown in Panels (a) and (b). \n<!-- image -->", '5 Solar Atmosphere Atomic Data Requirements': 'The low density of the corona (around 10 9 cm -3 ) means the plasma is not in thermodynamic equilibrium and so it is necessary to model the atomic processes that excite and de-excite atomic levels. The most important processes are electron excitation and spontaneous radiative decay. Depending on the ion, transition data for up to 1000 levels may be required to yield accurate emissivities for transitions observed in the solar spectrum. Over 200 ions give rise to measurable lines in observed spectra and thus a huge amount of atomic data are required to accurately model coronal spectra. The CHIANTI atomic database has provided these data to the community for almost 30 years and is described in Section 5.1. Modeling the balance between ionization states is also crucial for modeling the coronal spectrum, and electron collisional ionization and recombination (both radiative and dielectronic) are the key processes. Timescales are \nsignificantly longer than for electron excitation and so level balance can be treated separately from ion balance, greatly simplifying the problem. In addition, total rates out of the ground state are usually sufficient. These rates are also provided in CHIANTI. \nIn the transition region and chromosphere, additional atomic processes can be important for low charge species (typically 3+ or less). For example, metastable levels can have populations comparable to the ground state and so levelresolved ionization and recombination rates may be needed. Charge transfer, i.e., the exchange of an electron between the ion and H or He, and photoionization can compete with electron ionization and recombination. Many strong chromospheric lines are optically thick and so radiative transfer calculations are necessary for modeling. Modeling of solar atmosphere structures that fully takes into account matter-radiation interactions is currently only possible in 1D [39], an example being the RADYN code [40] that is widely used for \nsolar flare modeling. 3D models of solar atmosphere structures need to include the magnetic field, and so magnetohydrodynamic simulations are performed. Examples include the Bifrost [41] and MURaM [42] codes. Fully including the effects of radiative transfer on the plasma is beyond current computational capabilities, and so simple recipes are employed such as lookup tables [43]. \nExamples of recent work in applying atomic physics to solar atmosphere data are discussed below, and the CHIANTI atomic database is described.', '5.1 The CHIANTI atomic database': 'CHIANTI was named for the wine-growing region in Tuscany near Florence where a number of meetings were held in the formative years of the project. The first version was released in 1996 [44] and, unusually for the time, it was completely open source. CHIANTI consists of a database of atomic parameters for ions, plus software packages written in IDL and Python that take these data and allow the user to compute the radiative emission from a plasma. \nThe core data holdings are electron excitation rates, spontaneous radiative decay rates and level energies. Only fine structure ( J -resolved) levels are included in order to obtain accurate transition wavelengths. Figure 5 shows ions for which CHIANTI has data, and the number of levels for each ion is indicated. \nA crucial part of CHIANTI has been the assessment of electron collision strengths through a graphical procedure that follows that recommended by [45]. This has been invaluable for identifying problematic data, and the team has frequently communicated with atomic physicists to understand and resolve discrepancies. The team members are also practising spectroscopists who use CHIANTI for their research, and they have performed many benchmark analyzes to assess the accuracy of the database, e.g., [46-48]. \nAs can be seen from Figure 5 there remain many ions for which CHIANTI does not have data. These are either because the ion does not yield significant lines in the solar spectrum, or because fine-structure electron excitation data do not exist for the ion. \nAt UV and EUV wavelengths emission lines are mostly resolved and so high accuracy data are \nneeded for specific emission lines. The lines generally come from n = 2 , 3 states, and so model atoms do not need to be very large to model the UV lines. At X-ray wavelengths, however, lines typically can come from n = 4 , 5 , 6 states, so much larger atomic models are needed. Since line density at X-ray wavelengths is high and solar X-ray spectrometers generally have low spectral resolution then there is less requirement for high accuracy data, however.', '5.2 Line identifications: Fe VII and Fe IX': 'Fe vii and Fe ix yield hundreds of weak emission lines in the 150-300 ˚ A region that result from 3 p 5 3 d 3 and 3 p 4 3 d 2 configurations, respectively. These configurations have 110 and 111 fine structure levels that exhibit very significant level mixing making line identification difficult. [49-51] have utilized solar EUV spectra from Hinode/EIS and high-resolution laboratory spectra to perform new line identifications for these ions. Figure 6 shows some of the new identifications in the 170185 ˚ A wavelength region. A CHIANTI synthetic spectrum is displayed that contains only lines from the two ions. The red lines are the new identifications that were added to CHIANTI 10.1 [52]. The displayed region contains the very strong Fe ix 171 ˚ Aline, and the new data will be relevant to the upcoming MUSE and Solar-C missions (Table 1) which will observe this wavelength range. \nDespite the progress for these two ions, 51% and 82% of the 3 p 5 3 d 3 and 3 p 4 3 d 2 levels do not have experimental energies. To make further progress more accurate electron scattering calculations need to be performed for both ions in order to yield more accurate predicted line emissivities. In addition, new high resolution laboratory spectra over a wide wavelength band (e.g., 100 ˚ A) are required that can discriminate between neighboring charge states. A wide wavelength region is crucial for identifying Ritz combinations, as done for Fe ix [51].', '5.3 Experimental energies: Mg VII and Si VII': "The National Institute for Standards and Technology (NIST) maintains the Atomic Spectra Database (ASD) [54] that is the standard reference \nFig. 5 A table showing the ions that are modeled in the CHIANTI atomic database. The number in each box indicates the number of fine-structure levels of the ion model. \n<!-- image --> \nFig. 6 CHIANTI synthetic spectrum in the 170-185 ˚ A region showing only Fe vii and Fe ix lines. The blue line shows the spectrum from CHIANTI 10.0 and the red lines shows new lines that were added in CHIANTI 10.1. The spectra were generated with the differential emission measure curve of [53] and a density of 10 9 cm -3 . \n<!-- image --> \nfor ion wavelengths. It is compiled from laboratory and astrophysical wavelength measurements that are combined to yield an optimized set of energy levels from which Ritz wavelengths are calculated. An alternative set of reference wavelengths is provided by B. Edl'en in a series of papers between 1983 and 1985 for isoelectronic sequences from Be through F. Unlike the NIST compilation, experimental wavelength measurements are supplemented by theoretical calculations and extrapolations along the sequences. The Edl'en work is also limited to n = 2 configurations. \nTable 2 Reference wavelength comparison for Si vii and Mg vii . \n1 Configuations: 2 s 2 2 p 4 -2 s 2 p 5 . \nThe CHIANTI team has made use of both NIST and Edl'en data, and have found significant differences in some cases. [55] highlighted problems for C-like Mg vii and O-like Si vii and examples are given in Table 2. These lines are all observed by Hinode/EIS, which can measure wavelengths to a precision of 1 km s -1 and an accuracy of 5 km s -1 . The wavelength differences between the ASD and Edl'en wavelengths are thus very significant. \nBy combining wavelengths measured from Hinode/EIS with forbidden transition wavelengths from Astrophysical nebula observations (which were unavailable at the time of the NIST and Edl'en compilations), a new set of reference wavelengths could be generated for the two ions [55]. Better agreement was found with the Edl'en values than the NIST values. \nThis example demonstrates the importance of keeping the NIST database up to date with the latest developments in space-based wavelength measurements. This requires continued funding for NIST as well as active collaboration of astrophysicists with the NIST team. \nFig. 7 Ion fractions for Si iii and Si iv computed from the Dufresne advanced models (red) and CHIANTI 10.1 (blue). \n<!-- image -->", '5.4 Improved ionization balance modeling': "CHIANTI models the ionization balance for an element through a balance between electron collisional ionization and recombination, as first introduced in CHIANTI 6 [56]. An update was performed in CHIANTI 10 [57] to model the suppression of dielectronic recombination (DR) at high densities. In a series of papers [58-61], R. Dufresne and colleagues have developed advanced atomic models for several isonuclear sequences that incorporate additional effects that impact the ionization balance, particularly for low charge states in the chromosphere and low temperature transition region. \nFigure 7 compares ion fractions for Si iii and Si iv from the advanced models [61] with those from CHIANTI 10.1 [52]. The advanced models include density effects (level-resolved ionization/recombination and DR suppression), photoionization and charge transfer, and a constant pressure of 3 . 0 × 10 14 Kcm -3 was assumed. \nCHIANTI 11 is scheduled to be released in 2024 and will include modified versions of the Dufresne advanced models for the elements C, N, O, Ne, Mg, Si and S. The models will incorporate the density effects of level-resolved ionization/recombination, DR suppression and charge transfer on the ionization balance for these elements. While the new models represent a significant advance over the previous CHIANTI modeling, three areas can be identified where new atomic calculations would be valuable. \n1. DR suppression . The method for calculating this dates back to 1974 [62], where suppression \nfactors as a function of density were computed. The more recent work of [63] applied these factors to recent DR rates. Given the importance of DR suppression [58, 64] in shifting transition region ions to lower temperatures, a modern calculation of the suppression factors is highly desirable. \n- 2. Level-resolved ionization rates . Ionization rates from the ground states of ions can be calculated and 'calibrated' against laboratory-measured rates in many cases [65]. For the advanced models discussed above, rates from metastables are required that are generally not available in the literature. An empirical formula [66] can be used to scale the metastable rates to the ground rates [61]. Systematic ab initio calculations along isoelectronic sequences of metastable rates are desirable, together with laboratory measurements for comparison.\n- 3. Recombination rates . N. Badnell and colleagues [67-69] have systematically produced radiative and dielectronic recombination rates for isoelectronic sequences that are used in CHIANTI and other plasma codes. The data are available for all isoelectronic sequences from H to Si, plus the Ar sequence. Data for the P, S and Cl sequences are needed, and for sequences of the fourth row, particularly for iron ions.", '6 Summary': 'A number of topics in Solar Physics have been highlighted where atomic physics data are important. A division has been made between the solar interior (Section 2) and the solar atmosphere (Section 4). Although Solar Physics is a mature field, the Sun remains an important plasma laboratory that can be studied in great detail and provides a testing ground for Astrophysics and plasma theories. High quality atomic data will therefore continue to be required in the future, both theoretical calculations and laboratory measurements. \nFor further details and background on some of the topics covered in this article the following articles and books are recommended. The book of Pradhan & Nahar [70] details the atomic processes important to Astrophysics and how they are used to interpret spectra. The review of Barklem [33] discusses the role of atomic physics in measuring \nthe abundances of late-type stars, which is relevant to the Sun. The review of Adelberger et al. [4] gives a detailed summary of nuclear rates, including those relevant to the Sun. A short review of how various physical processess, including opacity and abundances, impact SSMs is given by Serenelli [71], and Asplund et al. [18] give an overview of how 3D NLTE models are used to obtain photospheric abundances. Atomic data and plasma diagnostics relevant to the solar atmosphere are extensively covered in the book of Phillips et al. [72] and the review article of Del Zanna & Mason [73].', '7.2 Data availability': 'There are no associated data with this article.', '7.3 Author contribution': 'Acknowledgements. The author thanks the referees for valuable comments that have improved the manuscript. Funding for this work was provided by the NASA Heliophysics Data Resource Library, the GSFC Internal Scientist Funding Model competitive work package program, and the Hinode project.', 'References': "- [1] Young, P.R.: Applications of Atomic Data to Studies of the Sun. Zenodo (2024). https: //doi.org/10.5281/zenodo.10908729 . https: //doi.org/10.5281/zenodo.10908729\n- [2] Basu, S., Antia, H.M.: Helioseismology and solar abundances. Phys. 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2024Natur.626..737L	The mergers of binary compact objects such as neutron stars and black holes are of central interest to several areas of astrophysics including as the progenitors of gammaray bursts GRBsSUP1SUP sources of highfrequency gravitational waves GWsSUP2SUP and likely production sites for heavyelement nucleosynthesis by means of rapid neutron capture the rprocessSUP3SUP. Here we present observations of the exceptionally bright GRB 230307A. We show that GRB 230307A belongs to the class of longduration GRBs associated with compact object mergersSUP46SUP and contains a kilonova similar to AT2017gfo associated with the GW merger GW170817 refs. SUP712SUP. We obtained James Webb Space Telescope JWST midinfrared imaging and spectroscopy 29 and 61 days after the burst. The spectroscopy shows an emission line at 2.15 microns which we interpret as tellurium atomic mass A 130 and a very red source emitting most of its light in the midinfrared owing to the production of lanthanides. These observations demonstrate that nucleosynthesis in GRBs can create rprocess elements across a broad atomic mass range and play a central role in heavyelement nucleosynthesis across the Universe.	2024-02-01T00:00:00Z	['2024Natur.626..737L', 'arXiv:2307.02098', '10.48550/arXiv.2307.02098', '10.1038/s41586-023-06759-1', '2023arXiv230702098L']	['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	Heavyelement production in a compact object merger observed by JWST	2,024	189	0.7	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	131	https://arxiv.org/pdf/2307.02098.pdf	{'JWST detection of heavy neutron capture elements in a compact object merger': "Andrew Levan 1,2* , Benjamin P. Gompertz 3 , Om Sharan Salafia 4,5 , Mattia Bulla 6,7,8 , Eric Burns 9 , Kenta Hotokezaka 10,11 , Luca Izzo 12,13 , Gavin P. Lamb 14,15 , Daniele B. Malesani 1,16,17 , Samantha R. Oates 3 , Maria Edvige Ravasio 1,4 , Alicia Rouco Escorial 18 , Benjamin Schneider 19 , Nikhil Sarin 20,21 , Steve Schulze 21 , Nial R. Tanvir 15 , Kendall Ackley 2 , Gemma Anderson 22 , Gabriel B. Brammer 16,17 , Lise Christensen 16,17 , Vikram S. Dhillon 23,24 , Phil A. Evans 15 , Michael Fausnaugh 19,25 , Wen-fai Fong 26,27 , Andrew S. Fruchter 28 , Chris Fryer 29,30,31,32 , Johan P. U. Fynbo 16,17 , Nicola Gaspari 1 , Kasper E. Heintz 16,17 , Jens Hjorth 12 , Jamie A. Kennea 33 , Mark R. Kennedy 34,35 , Tanmoy Laskar 1,36 , Giorgos Leloudas 37 , Ilya Mandel 38,39 , Antonio Martin-Carrillo 40 , Brian D. Metzger 41,42 , Matt Nicholl 43 , Anya Nugent 26,27 , Jesse T. Palmerio 44 , Giovanna Pugliese 45 , Jillian Rastinejad 26,27 , Lauren Rhodes 46 , Andrea Rossi 47 , Stephen J. Smartt 43,46 , Heloise F. Stevance 46,48 , Aaron Tohuvavohu 49 , Alexander van der Horst 32 , Susanna D. Vergani 44 , Darach Watson 16,17 , Thomas Barclay 50 , Kornpob Bhirombhakdi 28 , Elm'e Breedt 51 , Alice A. Breeveld 52 , Alexander J. Brown 23 , Sergio Campana 4 , Ashley A. Chrimes 1 , Paolo D'Avanzo 4 , Valerio D'Elia 53,54 , Massimiliano De Pasquale 55 , Martin J. Dyer 23 , Duncan K. Galloway 38,39 , James A. Garbutt 23 , Matthew J. Green 56 , Dieter H. Hartmann 57 , P'all Jakobsson 58 , Paul Kerry 23 , Danial Langeroodi 12 , James K. Leung 59,60,39 , Stuart P. Littlefair 23 , James Munday 2,61 , Paul O'Brien 15 , Steven G. Parsons 23 , Ingrid Pelisoli 2 , Andrea Saccardi 44 , David I. Sahman 23 , Ruben Salvaterra 62 , Boris Sbarufatti 4 , Danny Steeghs 2,39 , Gianpiero Tagliaferri 4 , Christina C. Thone 63 , Antonio de Ugarte Postigo 64 , David Alexander Kann 65 \n- 1 Department of Astrophysics/IMAPP, Radboud University, 6525 AJ Nijmegen, The Netherlands.\n- 2 Department of Physics, University of Warwick, Coventry, CV4 7AL, UK. 3 Institute for Gravitational Wave Astronomy and School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK.\n- INAF - Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807,\n- 4 Merate (LC), Italy.\n- 5 INFN - Sezione di Milano-Bicocca, Piazza della Scienza 2, I-20146, Milano (MI), Italy.\n- 6 Department of Physics and Earth Science, University of Ferrara, via Saragat 1, I-44122 Ferrara, Italy. \n7 \n- INFN, Sezione di Ferrara, via Saragat 1, I-44122 Ferrara, Italy. 8 INAF, Osservatorio Astronomico d'Abruzzo, via Mentore Maggini snc,\n- 64100 Teramo, Italy.\n- 9 Department of Physics & Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA.\n- 10 Research Center for the Early Universe, Graduate School of Science, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan.\n- 11 Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan.\n- 12 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen N, Denmark.\n- 13 INAF-Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131, Napoli, Italy.\n- 14\n- Astrophysics Research Institute, Liverpool John Moores University, IC2 Liverpool Science Park, Liverpool, L3 5RF, Liverpool, UK. 15 School of Physics & Astronomy, University of Leicester, University Road, Leicester, LE1 7RH, UK. \n16 \n- Cosmic Dawn Center (DAWN), Denmark.\n- 17 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen N, Denmark.\n- 18 European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692 Villanueva de la Ca˜nada, Madrid, Spain. \n19 Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA. \n20 Nordita, Stockholm University and KTH Royal Institute of Technology Hannes Alfv'ens vag 12, SE-106 91 Stockholm, Sweden. \n- 21 The Oskar Klein Centre, Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden.\n- 22 International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia.\n- 23 Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom.\n- 24 Instituto de Astrof'ısica de Canarias, E-38205 La Laguna, Tenerife, Spain.\n- 25 Department of Physics & Astronomy, Texas Tech University, Lubbock TX, 79410-1051, USA.\n- 26 Center for Interdisciplinary Exploration and Research in Astrophysics, Northwestern University, 1800 Sherman Ave., Evanston, 60208, IL, USA. 27 Department of Physics and Astronomy, Northwestern University, 2145\n- Sheridan Road, Evanston, 60208-3112, IL, USA.\n- 28 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218.\n- 29 Center for Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM 87545.\n- 30 Department of Astronomy, The University of Arizona, Tucson, AZ 85721.\n- 31 Department of Physics and Astronomy, The University of New Mexico, Albuquerque, NM 87131.\n- 32 Department of Physics, The George Washington University, Washington, DC 20052.\n- 33 Department of Astronomy and Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA. 34 School of Physics, Kane Building, University College Cork, Cork, Ireland.\n- 35 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, The University of Manchester, M13 9PL, UK.\n- 36\n- Department of Physics & Astronomy, University of Utah, Salt Lake City, UT 84112, USA.\n- 37 DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark. School of Physics and Astronomy, Monash University, Clayton,\n- 38 Victoria 3800, Australia. \n- 61 Isaac Newton Group of Telescopes, Apartado de Correos 368, E-38700 Santa Cruz de La Palma, Spain. 62 INAF/IASF-MI, Via Alfonso Corti 12, I-20133, Milano, Italy. 63 Astronomical Institute of the Czech Academy of Sciences, Friˇcova 298, 251 65 Ondˇrejov, Czech Republic. 64 Artemis, Observatoire de la Cˆote d'Azur, Universit'e Cˆote d'Azur, Boulevard de l'Observatoire, F-06304 Nice, France. 65 Hessian Research Cluster ELEMENTS, Giersch Science Center,\n- Max-von-Laue-Straße 12, Goethe University Frankfurt, Campus Riedberg, D-60438 Frankfurt am Main, Germany. \nCorresponding author(s). E-mail(s): [email protected];", 'Abstract': "The mergers of binary compact objects such as neutron stars and black holes are of central interest to several areas of astrophysics, including as the progenitors of gamma-ray bursts (GRBs), sources of high-frequency gravitational waves and likely production sites for heavy element nucleosynthesis via rapid neutron capture (the r -process). These heavy elements include some of great geophysical, biological and cultural importance, such as thorium, iodine and gold. Here we present observations of the exceptionally bright gamma-ray burst GRB 230307A. We show that GRB 230307A belongs to the class of long-duration gamma-ray bursts associated with compact object mergers, and contains a kilonova similar to AT2017gfo, associated with the gravitational-wave merger GW170817. We obtained James Webb Space Telescope mid-infrared (mid-IR) imaging and spectroscopy 29 and 61 days after the burst. The spectroscopy shows an emission line at 2.15 microns which we interpret as tellurium (atomic mass A=130), and a very red source, emitting most of its light in the mid-IR due to the production of lanthanides. These observations demonstrate that nucleosynthesis in GRBs can create r -process elements across a broad atomic mass range and play a central role in heavy element nucleosynthesis across the Universe. \nKeywords: Gamma-ray burst, Nucleosynthesis, Neutron star merger \nGRB 230307A was first detected by the Fermi Gamma-ray Burst Monitor (GBM) at 15:44:06 UT on 7 Mar 2023 [1]. It was an exceptionally bright gamma-ray burst with a duration of T 90 ∼ 35 s and a prompt fluence of (2.951 ± 0.004) × 10 -3 erg cm -2 in the 10-1000 keV band [2] (Figure 1). These properties place this event at the peak of the distribution of the class of 'long-soft' GRBs. The measured fluence makes it the second-brightest GRB ever detected [3]. \nIn addition to Fermi , the burst was also detected by several other high-energy instruments (see Methods). The multiple detections enabled source triangulation by \nthe InterPlanetary Network (IPN). The Neil Gehrels Swift Observatory (hereafter Swift ) tiled the initial ∼ 2 sq. deg IPN region [4] which revealed one candidate X-ray afterglow [5] consistent with the final 8 sq. arcmin IPN localization [6]. \nWe obtained optical observations of the field of GRB 230307A with the ULTRACAM instrument, mounted on the 3.5m New Technology Telescope (NTT) at La Silla, Chile. Visual inspection of the images compared to those obtained with the Legacy Survey [7] revealed a new source coincident with the Swift X-ray source, and we identified it as the optical afterglow of GRB 230307A [8]. This identification was subsequently confirmed via imaging by several additional observatories [9-11]. The location was also serendipitously imaged by the Transiting Exoplanet Survey Satellite (TESS) from 3 days before to 3 days after the GRB [12]. \nFollowing the precise localisation, we obtained extensive follow-up observations, in the optical and near infrared with the Gemini South Telescope and the Very Large Telescope (VLT); in the X-ray with the Swift /XRT and the Chandra X-ray observatory; and in the radio with the Australia Telescope Compact Array (ATCA) and MeerKAT in South Africa. This campaign included spectroscopy with the VLT Xshooter instrument, as well as the Multi Unit Spectroscopic Explorer (MUSE) integral field spectrograph. The latter provides redshift information for many galaxies in the field, including, in particular, a bright spiral galaxy at z = 0 . 0646 ± 0 . 0001 offset 30.2 arcseconds (38.9 kiloparsec in projection) from the burst position (Figure 2). Of the optically detected galaxies in the field, this is the one with the lowest probability of being an unrelated chance alignment, and hence is most likely to be the host (see also [13]). \nOur ground-based campaign included imaging from 1.4 to 41 days after the burst (see Supplementary Information Table 1). At 11 days, infrared observations demonstrated a transition from an early blue spectral slope, to a much redder one with i -K > 2 . 9(AB). This extremely red colour appeared similar to the expectations for a kilonova, powered by the decay of unstable isotopes of heavy elements synthesised by rapid neutron capture within the ejecta produced during the merger of a neutron star and another compact object [14-16]. Based on this detection, we requested James Webb Space Telescope (JWST) observations (GO 4434, 4445, PI Levan), which were initiated on 5 April 2023. At the first epoch (+28.9 days after GRB), we took 6colour observations with the Near Infrared Camera (NIRCam) (Figure 2), as well as a spectrum with the Near Infrared Spectrograph (NIRSpec) covering 0 . 5 -5 . 5 microns (Figure 3). \nThe NIRCam observations reveal an extremely red source that is only weakly detected in the bluer bands, where F150W(AB) = 28 . 11 ± 0 . 12 mag, but rises sharply through the mid-IR to F444W(AB) = 24 . 4 ± 0 . 01 mag. The NIRSpec observations also exhibit this steep rise. A faint galaxy is detected in these data at z = 3 . 87, offset approximately 0.3 arcseconds from the burst position. However, the burst's properties are inconsistent with an origin at this redshift, in particular because the implied isotropic equivalent energy release would exceed all known GRBs by an order of magnitude or more (see Supplementary Information). A second epoch of JWST observations was obtained approximately 61 days after the burst. These observations showed that the source had faded by 2.4 magnitudes in F444W, demonstrating a rapid \ndecay expected in the kilonova scenario and effectively ruling out alternatives (see Supplementary Information). \nSome of the burst properties are remarkably similar to those of the bright GRB 211211A, which was also shown to be accompanied by a kilonova [17-19]. In particular, the prompt emission consists of a hard pulse lasting for ∼ 19 seconds, followed by much softer emission (Figure 1). The prompt emission spectrum is well modelled by a double broken power-law with two spectral breaks moving rapidly through the gammaray band (see Methods), suggesting a synchrotron origin of the emission [20]. The X-ray afterglow is exceptionally faint, much fainter than most bursts when scaled by the prompt GRB fluence (see Figure 1 and Supplementary Information). The development of the optical and IR counterpart is also similar to GRB 211211A, with an early blue colour and a subsequent transition to red on a timescale of a few days. In Figure 4, we plot the evolution of the counterpart compared with the kilonova AT2017gfo [2128], identified in association with the gravitational-wave detected binary neutron star merger, GW170817 [29]. AT2017gfo is the most rapidly evolving thermal transient ever observed; much more rapid than supernovae or even, for example, fast blue optical transients [30]. The counterpart of GRB 230307A appears to show near identical decline rates to AT2017gfo both at early times in the optical and IR, and later in the mid-IR [31]. These similarities are confirmed by a joint fit of afterglow and kilonova models to our multi-wavelength data (see Supplementary Information). Our JWST observations rule out supernovae: for any redshift z < 1, a supernova would need to be more than 100 times fainter than the canonical GRB-supernova, SN 1998bw, to be compatible with our observations. Therefore, we conclude that GRB 230307A is a long-duration GRB formed from a compact object merger. This falls into a class that includes GRB 211211A [17-19, 32], GRB 060614 [33], GRB 111005A [34] and GRB 191019A [35], among others. \nThe JWST observations provide a view of a kilonova in the mid-IR with high spatial resolution and sensitivity. On timescales of ∼ 30 days, it is apparent that the kilonova emits almost all of its light in the mid-infrared, beyond the limits of sensitive groundbased observations (effectively limited to below 2.5 microns). This is consistent with previous model predictions [36], but has not previously been observationally confirmed. Late-time studies of such emission in the nebular phase must therefore be conducted in the mid-IR. Strikingly, despite its powerful and long-lived prompt emission that stands in stark contrast to GRB 170817A, the GRB 230307A kilonova is remarkably similar to AT 2017gfo. This was also the case for the long-lived merger GRB 211211A [1719, 32], and suggests the kilonova signal, particularly the red component, is relatively insensitive to the GRB. \nOur NIRSpec spectrum shows a broad emission feature with a central wavelength of 2.15 microns, visible in both epochs of JWST spectroscopy. At longer wavelengths, the spectrum displays a slowly rising continuum up to 4.5 microns followed by either an additional feature or change of spectral slope. The colours of the counterpart at this time can be readily explained by kilonova models (see Supplementary Information Section 6). \nA similar emission-like feature is also visible in the later epochs of X-shooter observations of AT2017gfo [23], measured at 2.1 microns by [37]. This further strengthens both the kilonova interpretation and the redshift measurement of GRB 230307A (Figure 3). We interpret this feature as arising from the forbidden [Te III] transition between the ground level and the first fine structure level of tellurium, with an experimentally-determined wavelength of 2.1044 microns [38]. The presence of tellurium is plausible, as it lies at the second peak in the r -process abundance pattern, which occurs at atomic masses around A ≈ 130 [39]. It should therefore be abundantly produced in kilonovae, as is seen in hydrodynamical simulations of binary neutron star mergers with nucleosynthetic compositions similar to those favoured for AT2017gfo [40]. Furthermore, the typical ionization state of Te in kilonova ejecta is expected to be Te III at this epoch because of the efficient radioactive ionization [41]. Tellurium has recently been suggested as the origin of the same feature in the spectrum of AT2017gfo [42]. A previous study [37] also identified this tellurium transition and noted that the observed feature is most likely two blended emission lines. However, alternative transitions from heavy r -process elements have been considered for this feature [e.g. Ce III; 37]. Tellurium can also be produced via the slower capture of neutrons in the s -process. Indeed, this line is also seen in planetary nebulae [43]. The detection of [Te III] 2 . 1 µ m extends on the earlier detection of strontium, a first r -process peak element, in the early time photospheric emission of AT2017gfo [44]. The mass of Te III estimated from the observed line flux is ∼ 10 -3 M ⊙ (see Supplementary Information 6.2). Although weaker, we also note that the spectral feature visible at 4.5 microns is approximately consistent with the expected location of the first peak element selenium and the near-third peak element tungsten [45]. \nDetailed spectral fitting at late epochs is challenging because of the breakdown of the assumptions regarding local thermodynamic equilibrium (LTE) which are used to predict kilonova spectra at earlier ages, as well as fundamental uncertainties in the atomic physics and electron transitions in the highly complex electron shells of r -process elements. However, these observations provide a calibration sample for informing future models. The red continuum emission indicates a large opacity in the mid-IR at low temperatures, e.g., ∼ 10 cm 2 g -1 at ∼ 700 K. Since the Planck mean opacity under LTE is expected to be less than 1 cm 2 g -1 at ≲ 1000 K [46], this large opacity may suggest that lanthanides are abundant in the ejecta. Indeed, it has been shown that non-LTE effects can increase the lanthanide opacity in mid-IR at late times [47]. Therefore, systematic studies of non-LTE opacity are necessary to answer the question whether lanthanides are the origin of the red emission at this epoch. \nA fit to our combined MUSE + JWST data for the host galaxy (Supplementary Information) suggests a relatively low mass ( ∼ 2 . 5 × 10 9 M ⊙ ) dominated by an older ∼ 10 Gyr population, but with a second more recent burst of star formation. These properties are entirely consistent with the population of short GRB host galaxies [48, 49]. The host normalised offset of the burst from the host galaxy places it in the top 10% of those seen for short GRBs [48, 50]. The offset could readily be achieved by a binary with a velocity of a few hundred km s -1 and a merger time of hundreds of millions of years. Alternatively, in the second epoch of JWST observations, a faint source is detected in the F150W images at the location of the transient. This may \nrepresent continued emission from the transient. However, its absolute magnitude of M F 150 W ∼ -8 . 5 is comparable to the absolute magnitude of globular clusters in which dynamical interactions could be at play to create merging systems at enhanced rates [51]. Future observations should readily distinguish between a fading afterglow or underlying cluster. \nIt is striking that GRB 230307A is an extremely bright GRB, with only the exceptional GRB 221009A being brighter [52]. Of the ten most fluent Fermi /GBM GRBs, two are now associated with kilonovae (230307A and 211211A), three are associated with supernovae, and the nature of the remaining five appears unclear (see Table 4). For bright GRBs, there may be a significant contribution from mergers. Indeed, such a conclusion can also be reached by considering the energetics. Both GRB 230307A and GRB 211211A have isotropic equivalent energies of E iso > 10 51 erg. The majority of local GRBs for which the connection between GRBs and core-collapse supernovae has been established are much less energetic (typically E iso > 10 49 -50 erg) and it has been suggested that they represent a separate population powered via different processes [53]. For more energetic bursts in the local Universe (where supernovae can still readily be discovered) the fraction of long GRBs with and without supernovae appear similar (see Supplementary Information). If a substantial number of long GRBs are associated with compact object mergers, they provide an essential complement to gravitationalwave (GW) detections. Firstly, joint GW-GRB detections, including long GRBs, can push the effective horizons of GW detectors to greater distances and provide much smaller localisations. In the case of GRB 230307A, the distance of 300 Mpc could have provided a robust detection in gravitational waves for the relevant O4 sensitivity [17, 54]. Secondly, long GRBs can be detected without GW detectors, providing a valuable route to enhancing the number of kilonova detections. Thirdly, JWST can detect kilonova emission at redshifts substantially beyond the horizons of the current generation of GW detectors, enabling the study of kilonovae across a greater volume of the Universe. \nThe duration of the prompt γ -ray emission in these mergers remains a challenge to explain. In particular, the natural timescales for emission in compact object mergers are much shorter than the measured duration of GRB 230307A. Previously suggested models that may also explain GRB 230307A include magnetars [55, 56], black hole - neutron star mergers [57, 58], or even neutron star - white dwarf systems [19, 59]. Recent results have also shown that the jet timescale does not directly track the accretion timescale in compact object mergers, and that long GRBs may be created from very short lived engines [60], and hence from binary neutron star mergers without magnetars. \nThere is evidence that the kilonova in GRB 230307A produced elements across a wide range of atomic mass. The detection of second peak elements in the spectrum of a kilonova demonstrates that nuclei with atomic masses around A ∼ 130 are being created in the mergers of compact objects. Many second peak elements have important biological roles. For example iodine is essential for mammals and may have been used by the single cell Last Universal Common Ancestor [61]. The creation of these elements in compact object mergers, which can have long delay times, may have important consequences for the time at which certain evolutionary channels become plausible. \nFig. 1 The high energy properties of GRB 230307A. The left panel shows the light curve of the GRB at 64 ms time resolution with the Fermi /GBM. The shaded region indicates the region where saturation may be an issue. The burst begins very hard, with the count rate dominated by photons in the hardest (100-900 keV) band, but rapidly softens, with the count rate in the hard band being progressively overtaken by softer bands (e.g. 8-25 keV and 25-100 keV) beyond ∼ 20s. This strong hard-to-soft evolution is reminiscent of GRB 211211A [20] and is caused by the motion of two spectral breaks through the γ -ray regime (see Methods). The right panel shows the X-ray light curves of GRBs from the Swift X-ray telescope. These have been divided by the prompt fluence of the burst, which broadly scales with the X-ray light curve luminosity, resulting in a modest spread of afterglows. The greyscale background represents the ensemble of long GRBs. GRB 230307A is an extreme outlier of the > 1000 Swift -GRBs, with an extremely faint afterglow for the brightness of its prompt emission. Other merger GRBs from long bursts occupy a similar region of parameter space. This suggests the prompt to afterglow fluence could be a valuable tool in distinguishing long GRBs from mergers and those from supernovae. \n<!-- image --> \nFig. 2 JWST images of GRB 230307A at 28.5 days post burst. The upper panel shows the wide field image combining the F115W, F150W and F444W images. The putative host is the bright face-on spiral galaxy, while the afterglow appears at a 30-arcsecond offset, within the white box. The lower panel shows cut-outs of the NIRCAM data around the GRB afterglow location. The source is faint and barely detected in the bluer bands but very bright and well detected in the red. In the red bands, a faint galaxy is present northeast of the transient position. This galaxy has a redshift of z = 3 . 87, but we consider it to be a background object unrelated to the GRB (see Supplementary Information). \n<!-- image --> \nFig. 3 JWST/NIRSpec spectroscopy of the counterpart of GRB 230307A taken on 5 April 2023. The top panel shows the 2-D spectrum rectified to a common wavelength scale. The transient is well detected beyond 2 microns but not short ward, indicative of an extremely red source. Emission lines from the nearby galaxy at z = 3 . 87 can also be seen offset from the afterglow trace. The lower panel shows the 1D extraction of the spectrum in comparison with the latest (10-day) AT2017gfo epoch and different kilonova models. A clear emission feature can be seen at ∼ 2.15 microns at both 29 and 61 days. This feature is consistent with the expected location of [Te III], while redder features are compatible with lines from [Se III] and [W III]. This line is also clearly visible in the late time spectrum of AT2017gfo [37, 42] \n<!-- image --> \nFig. 4 A comparison of the counterpart of GRB 230307A with AT2017gfo (the kilonova associated with GW170817) at z = 0 . 065. Beyond ∼ 2 days, the kilonova dominates the counterpart (see Supplementary Information). The decay rates in both the optical and IR are very similar to those in AT2017gfo. These are too rapid for any plausible afterglow model (e.g. as a power-law, they decay faster than t -3 . 5 over a prolonged period). There is also good agreement in the late time slope between the measurements made at 4.4 microns with JWST and 4.5 microns for AT2017gfo with Spitzer [31]. \n<!-- image -->", '2 Observations': 'Below we outline the observational data that were used in this paper. Magnitudes are given in the AB system unless stated otherwise. We utilize cosmology resulting from the Planck observations [62]. All uncertainties are given at the 1 σ level unless explicitly stated.', '2.1 γ -ray observations': 'GRB 230307A was first detected by Fermi /GBM and GECAM at 15:44:06 UT on 7 Mar 2023 [1]. It had a duration of T 90 ∼ 35s and an exceptionally bright prompt fluence of (2.951 ± 0.004) × 10 -3 erg cm -2 [2]. The burst fell outside of the coded field-of-view of the Swift Burst Alert Telescope (BAT), and so did not receive a subdegree localisation despite a strong detection. However, detections by Swift , GECAM [63], STIX on the Solar Orbiter [64], AGILE [65], ASTROSAT [66], GRBalpha [67], VZLUSAT [68], Konus-WIND [69] and ASO-HXI [70] enabled an enhanced position via the InterPlanetary Network to increasingly precise localisations of 1.948 deg 2 [71], 30 arcmin 2 [72], and ultimately to 8 arcmin 2 [6]. This was sufficiently small to enable tiling with Swift and ground-based telescopes.', '2.1.1 Fermi /GBM data analysis': 'In Figure 1, we plot the light curve of GRB 230307A as seen by the Fermi /GBM in several bands, built by selecting Time Tagged Event (TTE) data, binned with a time resolution of 64 ms. The highlighted time interval of 3-7 s after trigger are affected by data loss due to the bandwidth limit for TTE data [73]. \nFor the spectral analysis, we made use of the CSPEC data, which have 1024 ms time resolution. Data files were obtained from the online archive 1 . Following the suggestion reported by the Fermi Collaboration [73], we analysed the data detected by NaI 10 and BGO 1, which had a source viewing angle less than 60 · , and excluded the time intervals affected by pulse pile-up issues (from 2.5 s to 7.5 s). The data extraction was performed with the public software gtburst , while data were analysed with Xspec . The background, whose time intervals have been selected before and after the source, was modelled with a polynomial function whose order is automatically found by gtburst and manually checked. In the fitting procedure, we used intercalibration factors among the detectors, scaled to the only NaI analysed and free to vary within 30%. We used the PG-Statistic, valid for Poisson data with a Gaussian background. The best-fit parameters and their uncertainties were estimated through a Markov Chain Monte Carlo (MCMC) approach. We selected the time intervals before and after the excluded period of 2.5-7.5 s due to instrumental effects. In particular, we extracted 2 time intervals from 0 to 2.5 s (1.25 s each) and 14 time intervals from 7.5 s to 40.5 (bin width of 2 s, except the last two with integration of 5 s to increase the signal-to-noise ratio), for a total of 16 time intervals. We fitted the corresponding \nspectra with the two smoothly broken power-law (2SBPL) function [74, 75], which has been shown to successfully model the synchrotron-like spectral shape of bright long GRBs, including the merger-driven GRB 211211A [20]. \nFrom our spectral analysis we found that all spectra up to ∼ 20 s are well modelled by the 2SBPL function, namely they are described by the presence of two spectral breaks inside the GBM band (8 keV-40 MeV). In particular, in the time intervals between 7.5 and 19.5 s, the low-energy break E break is coherently decreasing from 304 . 3 +5 . 2 -2 . 6 keV down to 52 . 1 +4 . 3 -5 . 1 keV, and the typical νF ν peak energy E peak is also becoming softer, moving from ∼ 1 MeV to 450 keV. The spectral indices of the two power-laws below and above the low-energy break are distributed around the values of -0.82 and -1.72, which are similar to the predictions for synchrotron emission in marginally fast-cooling regime (i.e. -2 / 3 and -3 / 2). This is consistent with what has been found in GRB 211211A [20]. We notice, however, that in all spectra the highenergy power-law above E peak is characterised by a much softer index (with a mean value of -4 . 10 ± 0 . 24) with respect to the value of ∼ -2.5 typically found in Fermi GRBs. This suggests that the spectral data might require a cut-off at high energy, although further investigations are needed to support this. From 19.5 s until 40.5 s (the last time interval analysed), all the break energies are found to be below 20 keV, close to the GBM low energy threshold. In the same time intervals, the peak energy E peak decreases from 682 . 4 +3 . 2 -6 . 1 to 123 . 1 +5 . 4 -4 . 9 keV, and the index of the power-law below the peak energy is fully consistent (mean value of -1 . 45 ± 0 . 06) with the synchrotron predicted value of -1.5.', '2.2.1 NTT - Afterglow discovery': 'Following the refinement of the IPN error box to an area of 30 arcmin 2 [72], we obtained observations of the field of GRB 230307A with the ULTRACAM instrument [76], mounted on the 3.5m New Technology Telescope (NTT) at La Silla, Chile. The instrument obtains images in 3 simultaneous bands, and is optimised for short exposure, low dead-time observations [76]. We obtained 10 × 20 s exposures in two pointings in each of the Super SDSS u , g and r , bands (where the Super SDSS bands match the wavelength range of the traditional SDSS filters but with a higher throughput; Dhillon et al. 77). The observations began at 01:53:21 UT on 2023-03-09, approximately 34 hr after the GRB. The images were reduced via the HIPERCAM pipeline [77] using bias and flat frames taken on the same night. Visual inspection of the images compared to those obtained with the Legacy Survey [7] revealed a new source coincident with an X-ray source identified via Swift /XRT observations [5], and we identified it as the likely optical afterglow of GRB 230307A [8]. The best available optical position of this source (ultimately measured from our JWST observations, see below) is RA(J2000) = 04:03:26.02, Dec(J2000) = -75:22:42.76, with an uncertainty of 0.05 arcseconds in each axis. The IPN error box and the footprint of the ULTRACAM observations are shown in Figure 5. \nThis identification was subsequently confirmed via observations from a number of additional observatories, including [9-11, 78-80]. We acquired two further epochs of \nobservations with ULTRACAM on the following nights with 10 × 20s exposures in the Super SDSS u , g and i , bands. Aperture photometry of the source is reported in Table 1, and is reported relative to the Legacy survey for the gri bands, and to SkyMapper for the u -band.', '2.2.2 TESS': 'The prompt and afterglow emission of GRB 230307A was detected by TESS, which observed the field continuously from 3 days before the Fermi trigger to 3 days after at a cadence of 200 s [12]. A reference image was subtracted from the observations to obtain GRB-only flux over this period. The measured flux in the broad TESS filter (600nm - 1000nm) is corrected for Galactic extinction and converted to the I c band assuming a power-law spectrum with F ∝ ν -0 . 8 . We then bin the light curve logarithmically, taking the mean flux of the observations in each bin and converting to AB magnitudes. A systematic error of 0.1 magnitudes was added in quadrature to the measured statistical errors to account for the uncertainties in the data processing. These data are presented in Table 1.', '2.2.3 Swift /UVOT': 'The Swift Ultra-violet/Optical Telescope [UVOT; 81] began observing the field of GRB 230307A ∼ 84 . 6 ks after the Fermi /GBM trigger [1]. The source counts were extracted using a source region of 5 arcsec radius. Background counts were extracted using a circular region of 20 arcsec radius located in a source-free part of the sky. The count rates were obtained from the image lists using the Swift tool uvotsource . A faint catalogued unrelated source also falls within the 5 arcsec radius, this will affect the photometry, particularly at late times. We, therefore, requested a deep template image in white in order to estimate the level of contamination. We extracted the count rate in the template image using the same 5 arcsec radii aperture. This was subtracted from the source count rates to obtain the afterglow count rates. The afterglow count rates were converted to magnitudes using the UVOT photometric zero points [82, 83].', '2.2.4 Gemini': 'We obtained three epochs of K-band observations using the FLAMINGOS-2 instrument on the Gemini-South telescope. These observations were reduced through the DRAGONS pipeline to produce dark and sky-subtracted and flat-fielded images [84]. At the location of the optical counterpart to GRB 230307A, we identify a relatively bright K-band source in the first and second epochs, with only a upper limit in epoch 3. We report our photometry, performed relative to secondary standards in the VISTA hemisphere survey [85], in Table 1.', '2.2.5 VLT imaging': "We carried out observations of the GRB 230307A field with the 8.2-m VLT telescopes located in Cerro Paranal, Chile. The observations were obtained with the FORS2 camera (mounted on the Unit Telescope 1, UT1, ANTU) in B , R , I and z bands at \nmultiple epochs, and with the HAWK-I instrument (mounted on the Unit Telescope 4, UT4, Yepun) in the K band at one epoch. All images were reduced using the standard ESO (European Southern Observatory) Reflex pipeline [86]. The source was detected in the FORS2 z -band image at ∼ 6.4 days after the Fermi /GBM detection. A single r ' -band observation of the GRB 230307A was also executed with the 2.6m VLT Survey Telescope (VST) after 2.37 days from the GRB discovery. In later observations the source was not detected (see bottom right panels of Fig. 5) and the upper limit values at 3 σ level are reported in Table 1.", '2.2.6 VLT spectroscopy': 'To attempt to measure the redshift of GRB 230307A and the nearby candidate host galaxies, we obtained spectroscopy with the VLT utilising both the X-shooter and MUSE instruments, mounted respectively on the Unit Telescope 2 (UT2, Kueyen) and on UT4 (Yepun). \nX-shooter spectroscopy, covering the wavelength range 3000-22000 ˚ A was undertaken on 2023-03-15. Observations were taken at a fixed position angle, with the slit centred on a nearby bright star. X-shooter data have been reduced with standard esorex recipes. Given that only two of the four nod exposures were covering the GRB position, resulting in a total exposure time of 2400s on-source, we have reduced each single exposure using the stare mode data reduction. Then, we have stacked the two 2D frames covering the GRB position using dedicated post-processing tools developed in a python framework [87]. We further obtained observations with the MUSE integral field unit on 2023-03-23. The MUSE observations cover multiple galaxies within the field, as well as the GRB position, and cover the wavelength range 4750-9350 ˚ A. MUSE data have been reduced using standard esorex recipes embedded within a single python script that performs the entire data reduction procedure. Later, the resulting datacube has been corrected for sky emission residuals using ZAP [88]. The MUSE observations reveal the redshifts for a large number of galaxies in the field, including a prominent spiral G1 at z = 0 . 0646 [see also 80] and a group of galaxies, G2, G3 and G4 at z = 0 . 263.', '2.3 X-ray afterglow': "Swift began tiled observations of the IPN localisation region with its X-ray Telescope [XRT; 89] at 12:56:42 on 8 Mar 2023 2 [90]. XRT made the first reported detection of the afterglow (initially identified as 'Source 2') with a count rate of 0 . 019 ± 0 . 004 cts -1 [91] and later confirmed it to be fading with a temporal power-law index of 1 . 1 +0 . 6 -0 . 5 [92]. XRT data were downloaded from the UK Swift Science Data Centre [UKSSDC; 93, 94]. \nWe further obtained observations with the Chandra X-ray observatory (programme ID 402458: PI Fong/Gompertz). A total of 50.26 ks (49.67 ks of effective exposure) of data were obtained in three visits between 31 March 2023 and 2 April 2023. The source was placed at the default aim point on the S3 chip of the ACIS detector. At the location of the optical and X-ray afterglow of GRB 230307A, we detect a total of \n12 counts, with an expected background of ∼ 1, corresponding to a detection of the afterglow at > 5 σ based on the photon statistics of [95]. To obtain fluxes, we performed a joint spectral fit of the Chandra and Swift /XRT data. The best fitting spectrum, adopting uniform priors on all parameters, is a power law with a photon index of Γ = 2 . 50 +0 . 30 -0 . 29 when fitting with a Galactic N H = 1 . 26 × 10 21 cm -2 [96] and zero intrinsic absorption (neither XRT nor Chandra spectra have sufficient signal to noise to constrain any intrinsic absorption component). The resultant flux in the 0.3 - 10 keV band is F X (1 . 7 d) = 4 . 91 +0 . 89 -0 . 79 × 10 -13 erg cm -2 s -1 during the XRT observation and F X (24 . 8 d) = 1 . 19 +0 . 87 -0 . 62 × 10 -14 erg cm -2 s -1 during the Chandra observation. Due to the low count number, the Chandra flux posterior support extends to considerably below the reported median, with the 5th percentile being as low as F X , 5th = 3 × 10 -15 erg cm -2 s -1 . If a uniform-in-the-logarithm prior on the flux were adopted, this would extend to even lower values. Chandra and XRT fluxes are converted to 1 keV flux densities using the best fit spectrum (Table 2).", '2.4 ATCA': 'Following the identification of the optical afterglow [97], we requested target of opportunity observations of GRB 230307A (proposal identification CX529) with the Australia Telescope Compact Array (ATCA) to search for a radio counterpart. These data were processed using Miriad [98], which is the native reduction software package for ATCA data using standard techniques. Flux and bandpass calibration were performed using PKS 1934-638, with phase calibration using interleaved observations of 0454-810. \nThe first observation took place on 2023-03-12 at 4.46 d post-burst, which was conducted using the 4 cm dual receiver with frequencies centered at 5.5 and 9 GHz, each with a 2 GHz bandwidth. The array was in the 750C configuration 3 with a maximum baseline of 6 km. A radio source was detected at the position of the optical afterglow at 9 GHz with a flux density of 92 ± 22 µ Jy but went undetected at 5.5 GHz (3 σ upper limit of 84 µ Jy). A further two follow-up observations were also obtained swapping between the 4 cm and 15 mm dual receivers (the latter with central frequencies of 16.7 and 21.2 GHz, each with a 2 GHz bandwidth). During our second epoch at 10.66 d we detected the radio counterpart again, having become detectable at 5.5 GHz with marginal fading at 9 GHz. By the third epoch, the radio afterglow had faded below detectability. We did not detect the radio transient at 16.7 or 21.2 GHz in either epoch. All ATCA flux densities are listed in Table 2.', '2.5 MeerKAT': "We were awarded time to observe the position of GRB 230307A with the MeerKAT radio telescope via a successful Director's Discretionary Time proposal (PI: Rhodes, DDT-20230313-LR-01). The MeerKAT radio telescope is a 64-dish interferometer based in the Karoo Desert, Northern Cape, South Africa [99]. Each dish is 12 m in diameter and the longest baseline is ∼ 8 km allowing for an angular resolution of \n∼ 7 arccsec and a field of view of 1 deg 2 . The observations we were awarded were made at both L and S-band. \nGRB 230307A was observed over three separate epochs between seven and 41 days post-burst. The first two observations were made at both L and S4-band (the highest frequency of the five S-band sub-bands), centred at 1.28 and 3.06 GHz with a bandwidth of 0.856 and 0.875 GHz, respectively. Each observation spent two hours at L-band and 20 minutes at S4-band. The final observation was made only at S4-band with one hour on target. Please see the paper by MPIfR for further details on the new MeerKAT S-band receiver. \nEach observation was processed using oxkat , a series of semi-automated Python scripts designed specifically to deal with MeerKAT imaging data [100]. The scripts average the data and perform flagging on the calibrators from which delay, bandpass and gain corrections are calculated and then applied to the target. The sources J04086545 and J0252-7104 were used at the flux and complex gain calibrator, respectively. Flagging and imaging of the target field are performed. We also perform a single round of phase-only self-calibration. We do not detect a radio counterpart in any epoch in either band. The rms noise in the field was measured using an empty region of the sky and used to calculate 3 σ upper limits which are given in Table 2.", '2.6 JWST observations': 'We obtained two epochs of observations of the location of GRB 230307A with JWST. The first on 5 April 2023, with observations beginning at 00:16 UT (MJD=60039.01), 28.4 days after the burst (under programme GO 4434, PI Levan), and the second on 8 May 2023, 61.5 days after the burst (programme 4445, PI Levan). The observations were at a post-peak epoch because the source was not in the JWST field of regard at the time of the burst and only entered it on 2 April 2023. \nAt the first epoch, we obtained observations in the F070W, F115W, F150W, F277W, F356W and F444W filters of NIRCam [101], as well as a prism spectrum with NIRSpec [102]. In the second epoch we obtained NIRCam observations in F115W, F150W, F277W and F444W and a further NIRSpec prism observation. However, in the second epoch the prism observation is contaminated by light from the diffraction spike of a nearby star and is of limited use, in particular at the blue end of the spectrum. We therefore use only light redwad of 1.8 microns. However, even here we should be cautious in interpreting the overall spectral shape. However, the feature at 2.15 microns is visible in both the 29 and 61 day spectra. \nWe reprocessed and re-drizzled the NIRCam data products to remove 1 /f striping and aid point spread function recovery, with the final images having plate scales of 0.02 arcsec/pixel (blue channel) and 0.04 arcsec/pixel (red channel). \nIn the NIRCam imaging we detect a source at the location of the optical counterpart of GRB 230307A. This source is weakly detected in all three bluer filters (F070W, F115W and F150W), but is at high signal-to-noise ratio in the redder channels (see Figure 2). The source is compact. We also identify a second source offset (H1) approximately 0.3 arcseconds from the burst location. This source is also weakly, or non-detected in the bluer bands, and is brightest in the F277W filter. \nBecause of the proximity of the nearby star and a contribution from diffraction spikes close to the afterglow position we model point spread functions for the appropriate bands using WebbPSF [103], and then scale and subtract these from star position. Photometry is measured in small (0.05 arcsec (blue) and 0.1 arcsec (red)) apertures and then corrected using tabulated encircled energy corrections. In addition to the direct photometry of the NIRCam images we also report a a K-band point based on folding the NIRSpec spectrum (see below), through a 2MASS, Ks filter. This both provides a better broadband SED and a direct comparison with ground based K-band observations. Details of photometric measurements are shown in Table 1. \nFor NIRSpec, we utilise the available archive-processed level 3 two-dimensional spectrum (Figure 3). In this spectrum we clearly identify the trace of the optical counterpart, which appears effectively undetected until 2 microns and then rises rapidly. We also identify two likely emission lines which are offset from the burst position. These are consistent with the identification with H α and [O iii ] (4959/5007) at a redshift of z = 3 . 87. Both of these lines lie within the F277W filter in NIRCam and support the identification of the nearby source as the origin of these lines. \nWe extract the spectrum in two small (2 pixel) apertures. One of these is centred on the transient position, while the second is centred on the location of the emission lines. Since the offset between these two locations is only ∼ 0 . 2 arcseconds there is naturally some contamination of each spectrum with light from both sources, but this is minimised by the use of small extraction apertures. The resulting 1D spectra are shown in Figure 3. The counterpart is very red, with a sharp break at 2 microns and an apparent emission feature at 2.15 microns. The spectrum then continues to rise to a possible second feature (or a change in the associated spectral slope) at around 4.5 microns.', '4.1 Prompt emission': "GRB 230307A is an exceptionally bright GRB. It has the second highest fluence of any GRB observed in more than 50 years of GRB observations [52]. While it remains a factor of 50 less fluent than GRB 221009A, it is still a factor ∼ 2 brighter than GRB 130427A, the third brightest burst. Bursts with these extreme fluences are rare. In Figure 6, we plot the distribution of observed fluence for Fermi /GBM detected bursts. At the brighter end, the slope of the distribution is consistent with the expected -3 / 2 slope for a uniform distribution of sources. The extrapolation of this relation suggests that bursts like GRB 230307A should occur once every several decades. Notably, three bursts well above the extrapolation (GRB 130427A, GRB 230307A, GRB 221009A) may indicate that bright bursts arise more frequently than expected. However, observationally it is clear that GRB 230307A is, at least, a once-per-decade event. \nThe prompt light curve of GRB 230307A (Figure 1) shows two distinct emission features: an initial episode of hard emission from the trigger until ≈ 18 s, then a softer episode from ≈ 19 s onwards. These distinct episodes of hard and soft emission are strongly reminiscent of the long-duration merger GRB 211211A, but the initial pulse complex is ∼ 50 per cent longer in GRB 230307A when compared to the ∼ 12 s duration seen in GRB 211211A [17]. The relative durations of the initial pulse complex in the two GRBs bear a striking resemblance to their relative time-averaged peak energies [936 ± 3 keV vs 647 ± 8 keV; 2, 104]. In GRB 211211A, substantial spectral evolution was seen to drive the light curve, and the underlying radiation mechanism was identified as fast-cooling synchrotron emission [20]. The coherent development of the hardness ratio (Figure 1, lower) indicated similar spectral evolution in GRB 230307A, which the spectral analysis confirmed. Indeed, as described in Section 2.1.1, the time-resolved spectral analysis of the prompt emission revealed the presence of two spectral breaks in the GBM band, E break and E peak , coherently becoming softer from 7.5 s up to 19.5 s. Also, in this case, the spectral indices indicate synchrotron emission in the marginally fast-cooling regime. From 19.5 s onwards (approximately when the softer and dimmer emission episode starts), the low-energy break E break is continuously approaching the lower limit of the GBM band (8 keV), presumably crossing it to enter the X-ray regime. Unfortunately, the lack of simultaneous observations in X-rays with another telescope, e.g. Swift /XRT, prevents us from fully tracing the evolution of the spectral break down to X-rays at later times, as was done for GRB 211211A. \nThe time-averaged Fermi /GBM spectrum of GRB 230307A across the T 90 interval is best fit with a cutoff power-law with α = 1 . 07 ± 0 . 01 and cutoff energy 936 ± 3 keV [2]. From this, we calculate a hardness ratio (the ratio of the 50 - 300 keV photon flux to the 10 - 50 keV photon flux) of 0 . 88 +0 . 01 -0 . 02 . This is higher than the value for 211211A (0 . 57) but comfortably within the 1 σ distribution of hardness ratios for canonical long GRBs (i.e. with T 90 > 2 s) in the Fermi catalogue, which we calculate to be 0 . 66 +0 . 51 -0 . 29 from the data in von Kienlin et al. [105]. Like GRB 211211A before it, GRB \n230307A appears to have 'typical' long GRB properties in terms of its time-averaged hardness ratio and its T 90 . This strengthens the case for a significant number [17, 18] of long-duration GRBs having been mistakenly identified as stellar collapse events. \nHowever, in some ways, GRB 230307A differs significantly from several of the other brightest GRBs. For example, the afterglow was relatively faint, while the burst was very bright. In Figure 7, we plot the prompt fluence in the 15-150 keV band against the X-ray afterglow brightness at 11 hours [updated from 106, 107]. The general trend between the afterglow brightness and fluence is seen; the best-fit slope to this relation is approximately one. So, while there is substantial scatter, there is a direct proportionality between the fluence and the afterglow brightness. Notably, while the afterglow and prompt emission of GRB 221009A were exceptionally bright (after correcting for the heavy foreground extinction), they were in keeping with this relatively broad relationship. GRB 230307A is different. Here we extrapolate the X-ray flux to 11 hours based on the measured X-ray flux at ∼ 1 day and the decay slope. We also re-calculate the GRB 230307A fluence in the relevant 15-150 keV energy band for comparison to Swift /BAT. This burst is a notable outlier in the relation, with a faint X-ray flux for its extraordinary prompt brightness. The afterglow brightness depends both on the energy of the burst and the density of the interstellar medium; it is, therefore, possible that the location in this fluence - afterglow brightness plane is indicative of a lowdensity medium, which would be consistent with expectations for such a large GRB host offset. \nIt is also of interest that another burst in a similar location is GRB 211211A. This long burst has a clear signature of kilonova emission within its light curve. If GRB 230307A is a similar event, faint afterglows (relative to the prompt emission) may be an effective route for disentangling mergers from collapsars. \nTo further compare the ratio between the X-ray brightness and the γ -ray fluence, we retrieve the X-ray light curve of all Swift -detected GRBs from the Swift Burst Analyser [108] and limit the sample to 985 long GRBs and 55 short GRBs with at least two XRT detections and measured BAT fluence. The fluences are taken from the Swift /BAT Gamma-Ray Burst Catalog 4 [109] and represent the measurements from 15 to 150 keV integrated over the total burst duration. We add to this sample the GRBs 170817A [off-axis GRB; 110] and 221009A [brightest GRB detected to date; 52]. For the former, we retrieve the X-ray light curve from Hajela et al. [111] and use a γ -ray fluence of 2 . 4 × 10 -7 erg cm -2 [110]. For the latter, we take the X-ray light curve from the Swift Burst Analyser and assume a fluence of 0 . 007 erg cm -2 (corrected from the 1-10000 keV fluence in [52] to the 15-150 keV band). Following [112], we resample the X-ray light curves and normalise them by the γ -ray fluence on a grid defined by the observed F X / Fluence ratios and the time-span probed by the data. If no data are available at a specific time of the grid, we linearly interpolate between adjacent observations but do not extrapolate any data. Hence the paucity of observations at later times reflects the last time at which sources were detected by the Swift /XRT. \nShort and long GRBs occupy the same part of the F X / Fluence vs time parameter space (Figure 1). In contrast, GRB 230307A has an unprecedentedly low F X / Fluence ratio that is almost 10-fold lower than the faintest GRBs at the same time. To \nemphasise the uniqueness of GRB 230307A, we also show in the same figure the Swift /BAT-detected GRBs 050925, 051105A, 070209, 070810B, 100628A, 130313A, 170112A that evaded detection with Swift /XRT. The limits on their F X / Fluence ratio (shown by downward pointing triangles in that figure) are consistent with the observed range of F X / Fluence ratios, ruling out a selection bias against GRBs with lower than usual F X / Fluence. Intriguingly, GRBs 080503, 191019A and 211211A had markedly low F X / Fluence ratios during the shallow decline phase of their X-ray light curves. Furthermore, GRB 211211A reached a value of 1 . 2 × 10 -9 s -1 at 120 ks, comparable to GRB 230307A.", '4.2 Counterpart Evolution': 'Although the afterglow of GRB 230307A was promptly detected thanks to TESS, this data was not available to the community for several days. Further follow-up was, therefore, much slower, and the counterpart was not discovered until the localisation was narrowed down to several sq. arcminutes, approximately 24 hours after the burst. The result is that the counterpart is poorly sampled (particularly in colour) during the early phases, while later observations suffer from typically modest signal-to-noise. \nThe TESS observations detected a relatively bright (though not exceptional given the fluence of the burst) outburst, coincident with the prompt emission, likely peaking at I < 15 [12]. The afterglow was much fainter, apparently no brighter than I = 18 in the minutes to hours after the burst was detected. It was relatively flat during this period, with a power-law through the first to last TESS observations decaying as F ( t ) ∝ t -0 . 2 . The TESS and ground-based observations can be consistently modelled with a forward shock afterglow + kilonova (see Section 6.1). \nThere are no simultaneous colours at the time of the first ground-based afterglow detections (1.4 days), although extrapolation of the r -band detection with ULTRACAM to the WHITE detection with the Swift /UVOT suggests a relatively red colour (WHITE-r = 1.6 ± 0.4) . However, such an interpretation is difficult due both to the large photometric errors and the width of the WHITE filter on the Swift /UVOT. \nOptical observations obtained multiple colours at an epoch ∼ 2 . 4 days post-burst. These show the afterglow to have a blue colour with g = 22 . 35 ± 0 . 26, i = 21 . 68 ± 0 . 09 and z = 21 . 8 [80]. This is consistent with GRB afterglows in general(i.e F ν ∝ ν -β gives β ≈ 1). Observations in the near infrared (NIR) were not undertaken until ∼ 10.4 days post-burst. However, these reveal a relatively bright K-band source. The inferred i-K(AB) > 2.9 at this epoch is very red. Interpreted as a change in the spectral slope, it is β ≈ 2 . 5. The K-band light hence appears to be in significant excess with respect to the afterglow expectations based both on optical data and on the X-ray light curve. \nIt is relevant to consider if such an excess could arise via extinction. However, this is not straightforward to explain. For a generic β = 1 slope we expect i -K ( AB ) ≈ 1 . 1. At z = 0, to obtain i -K ( AB ) = 2 . 9 would correspond to a foreground extinction of A V ≈ 4. However, this would also predict g -i ≈ 3, which is entirely inconsistent with the earlier observations. This problem becomes more acute for higher redshifts, where the bluer bands probe increasingly into the UV. \nThe IR excess becomes extremely prominent by the time of the JWST observations. At 28.5 days, the source is detected in all bands but is very faint in the NIRCam blue \nchannel (F070W, F115W, F150W) and rises rapidly (in F ν ) through the redder bands (F277W, F356W, F444W). Expressed as a power-law, this is β ≈ 3 . 1 in the 2-5 micron region, and β ≈ 1 between 0.7-1.5 microns. This does not match the expectations for any plausible spectral break in a GRB afterglow or any plausible extinction (where one would expect the slope to steepen towards the blue). This strongly implies that the red excess seen in the K-band at ten days and with JWST at 28.5 days is some additional component. Indeed, in the JWST observations, the other component, beginning at around 2 microns, is very clearly visible in both photometry and spectroscopy. \nThis component evolves exceptionally rapidly. In the K-band, the inferred decay rate from 11.5 to 28.5 days is ∼ t -3 . 5 expressed as a power-law or ∼ 0 . 25 mag per day, if exponential. This is much faster than observed in GRB afterglows or supernovae. It is, however, consistent with the expectations for kilonovae. As shown in Figure 4, the overall evolution shows substantial similarity with AT2017gfo. To constrain the temporal and spectral evolution within a plausible physical model more accurately, we fit the multi-band photometry with afterglow and kilonova models. The outputs of these models are described in detail in section 6.', '4.3 Identification of the host galaxy': "Deep optical imaging of the field identifies several relatively bright galaxies in the vicinity of the sky position of GRB 230307A. Our preferred host galaxy is the brightest of these, which we denote as G1. It lies at z = 0 . 065 and is offset 30 arcseconds (40 kpc in projection) from the location of the afterglow. Following the method of [113] this galaxy has a probability of chance alignment of P chance ∼ 0 . 09 (see also [80]). Although this is not extremely low, and so is only suggestive of a connection to the transient, we note that i) the luminosity of the late time counterpart at this redshift is very similar to AT2017gfo and ii) the spectral feature seen at 2.1 microns in AT2017gfo matches with the emission feature seen in the JWST spectroscopy of GRB 230307A. This is a broad line, but assuming they have the same physical origin, they fix the redshift to the range 0 . 04 < z < 0 . 08. G1 is the only galaxy within this range in the field. The physical properties of this galaxy are outlined in section 4.4. \nOur MUSE observations provide redshifts for this galaxy and several others, also identifying a small group of galaxies (G2, G3, G4) at a common redshift of z = 0 . 263. All of these galaxies have P chance values substantially greater than our preferred host. Furthermore, because of the larger redshifts, the implied offsets from GRB 230307A are ≫ 100kpc. This is larger than seen for any short GRB with a firmly identified host. We, therefore, disfavour these as plausible host galaxies for GRB 230307A. \nDeep JWST observations reveal no evidence of a directly underlying host galaxy for GRB 230307A, as would be expected if it had a collapsar origin. In particular, at late times, the faint source at the counterpart's location is consistent with a pointsource (i.e. a subtraction of the PSF constructed by WebbPSF yields no significant residuals). However, we identify a faint galaxy, undetected in the blue and with F277W =27 . 9 ± 0 . 1, offset only 0.3 arcseconds from the burst position. We designate this galaxy H1. \nOur NIRSpec observations provide a redshift of z = 3 . 87 for H1 based on the detection of [O III] (5007) and H α . At this redshift, the offset is only ∼ 1 . 3 kpc. \nAlthough many z ∼ 4 galaxies are extremely compact [114], it seems likely that some stellar population from this galaxy does extend under the burst position, and there may be marginal evidence for extension in this direction in the F444W image. However, this region is neither UV-bright nor an emission line region where one may expect to observe massive stars. \nThe galaxy photometry, performed in 0.1-arcsecond apertures and subsequently corrected for encircled energy assuming point-source curves is F070W > 29.0, F115W=28.4 ± 0.3, F150W=28.6 ± 0.4, F277W=27.9 ± 0.1, F444W=28.3 ± 0.1, and the galaxy is only robustly detected in the redder bands (see Figure 2). We note that because of the proximity of the afterglow, we use a smaller aperture than may be optimal, although the galaxy is also compact. \nWe can estimate the probability of chance alignment of this source with the GRB position via various routes. In principle, one can use number counts of galaxies on the sky in the multiple bands. These have recently been updated based on the first observations with JWST to provide number counts in appropriate bands [115]. We find that P chance , following the approach of [113] to be in the range ∼ 3-6 % for F277W and F444W (with no bound in the filters where the galaxy is undetected). Alternatively, we also estimate the probability directly from the data. We extract sources within the field via Source Extractor to create a mask of objects within the field. In the brightest detection (F277W) approximately 5% of the image is covered with objects of equal or brighter magnitude to H1, and we note that the burst position is not contained within this mask. This suggests that in this particular field, P chance > 5%. \nThe absolute magnitude of H1 is M i ∼ -17 . 7, and the H α star formation rate is approximately 1 M ⊙ yr -1 . The half-light radius of the galaxy is approximately 0.1 arcseconds (700 pc). Although limited information is available, these values are generally consistent with those of the long GRB population. The burst offset from its host galaxy is ∼ 2 . 5 half-light radii. This is large but within the range seen for long-GRBs [116]. \nIn our X-shooter and MUSE observations there is no trace visible in 1D or 2D extractions at the source position, although a weak continuum is seen in the X-shooter spectrum when heavily binned. This is consistent with its faint magnitude at the time of the observations. At the location of Ly α at z = 3 . 87 we place limits of F < 2 . 5 × 10 -17 erg s -1 cm -2 assuming an unresolved line. \nWe also examined both spectra for any emission lines at other redshifts. This is worthwhile given the strong emission lines often seen in long GRB hosts [117], which may make emission line redshifts possible, even if the host itself is undetected. However, despite deep observations, there are no visible emission lines consistent with no directly underlying host galaxy, consistent with a compact object merger, but not a collapsar. \nUnsurprisingly, there are also numerous faint galaxies in the JWST images. However, all of these have large P chance values, and we do not consider them plausible host galaxies. \nTaken a face value, the probability of chance alignment for G1 (our preferred host) and H1 ( z = 3 . 87) is similar. However, the luminosity, lightcurve evolution and spectroscopic feature at the redshift of G1 offer strong support for it as the host galaxy of GRB 230307A. Furthermore, there is no straightforward, reasonably viable \nphysical model that could explain the burst's extreme properties at z = 3 . 87. This scenario would require extreme energetics, exceptionally rapid evolution and yields unphysical outcomes in standard GRB or supernovae scenarios. We outline this in detail in section 7.1.", '4.4 Host galaxy properties': 'To better understand the properties of G1, the likely GRB host galaxy, we performed a fit to both the MUSE spectrum and photometric measurements from the far-UV to the mid-IR. For the photometric measurements, we retrieved science-ready coadded images from the Galaxy Evolution Explorer (GALEX) general release 6/7 [118], DESI Legacy Imaging Surveys (LS; [119]) data release 9, and re-processed WISE images [120] from the unWISE archive [121] 5 . The unWISE images are based on the public WISE data and include images from the ongoing NEOWISE-Reactivation mission R7 [122, 123]. We measured the brightness of the galaxy G1 using the Lambda Adaptive Multi-Band Deblending Algorithm in R ( LAMBDAR [124]) and the methods described in [125]. We augment the SED with Swift /UVOT photometry in the u band and our 6-band JWST/NIRCAM photometry. The photometry on the UVOT images was done with uvotsource in HEASoft and an aperture encircling the entire galaxy. For JWST photometry, we used a 6-arcsec circular aperture, which allows us to gather all the observed light observed in JWST filters from the host galaxy. All measurements are summarised in Table 7.4. \nTo derive the main physical properties of the host galaxy, such as its stellar mass, we employ two separate methodologies based on the photometric and spectroscopic data available for the host, and finally compare the results to assess the robustness of our conclusions. We first fit the multi-wavelength (0.1-4.4 µ m) dataset using the prospector python package [126], which allows us to model the host galaxy spectrum starting from its main constituents, namely a set of stellar population base spectra, built from the Flexible Stellar Population Synthesis (FSPS) package [127], and combined with a specific star-formation history (SFH) model. Moreover, we have also considered a fixed attenuation model based on the Calzetti [128] attenuation curve, and an additional nebular model originating from the gas component, which is built using the Cloudy photo-ionization code [129], and considering the FSPS stellar population as ionising sources. We have adopted a parametric SFH model, which is described by a delayed-exponential model where the star-formation rate varies as a function of time t = t age -t lt , with t lt being the lookback time [126], as SFR ∝ ( t/τ ) exp( -t/τ ), with τ being the e -folding time. We finally have used the dynesty Speagle [130] ensemble sampler to reconstruct the posterior distribution. \nThe results of the prospector analysis are shown in Fig. 8. We obtain a mass value of the living stars of M ∗ = 2 . 37(+0 . 24 , -0 . 35) × 10 9 M ⊙ yr -1 . The mass of all stars ever formed is 0 . 20(+0 . 02 , -0 . 04) dex larger. The light-weighted stellar age resulting from the fit is 1 . 13(+1 . 49 , -0 . 36) Gyr. \nAn alternative to parametric SED fits is to use synthetic stellar population SEDs as templates and combine them to fit the galactic spectra (the underlying assumption being instantaneous star formations rather than continuous functions of time). We \ncan use the spectral synthesis from the BPASSv2.2.2[131, 132] binary populations and create templates with hoki [133] that are compatible with the ppxf fitting package[134], as described in [135]. Because SED fitting has a high level of degeneracies (see [134]), at first we do not fit all 13 BPASS metallicities at once with ppxf , as this can result in unphysical results (see discussion in Stevance et al. 135); instead we fit the metallicities individually to find which ones result in the best fits on their own. We find that a low Z (0.001) population and solar metallicity population (Z=0.014) result in decent fits, but the low metallicity population fails to predict a young stellar component that is seen in the images, whilst the solar metallicity fit fails to accurately match the H β and neighbouring absorption features in the blue part of the spectrum. So we then fit the galaxy simultaneously with Z=0.001 and Z=0.014 templates, and retrieve a good fit shown in Figure 9 alongside the recovered SFH. \nWe find evidence of three main stellar populations: > 95% of the mass is found in lower metallicity (Z=0.001) stars with ages ranging from a few Gyr to 10 Gyr, with a peak of star formation around 5 Gyr; > 4.7% of the mass originates from a solar metallicity population (Z=0.014) that formed around 400 Myr ago; finally a small fraction ( < 0.05%) of the stellar mass in the host originates from the star-forming regions with ages a few Myrs. \nThe details of the age distributions and exact metallicity values can be model dependent so we also fit the integrated galaxy spectrum with the single stellar population synthesis code STARLIGHT [136], which uses stellar populations based on 25 different ages and six metallicity values [137], and a Chabrier IMF [138]. The SFH retrieved by this method is more complex and would require odd configurations (including some high metallicties at old ages and low metallicities around 100 Myr, which is counter-intuitive, unless inflow from pristine gas will trigger a burst of SFR), but it also finds that overall the galaxy is dominated by an old population with lower metallicity and has a younger component at higher metallicity. In Figure 9 we show a comparison of the STARLIGHT and BPASS fits in the bottom left panel and see that they are very similar, despite STARLIGHT containing 6 different metallicities and assuming solely single star populations. This highlights the level of degeneracy we face when performing galaxy SED fits. We leave further comparisons to a follow-up study dedicated to the host and the progenitor populations of GRB 230307A, where we will also present detailed, specially resolved, fits to the datacube including its kinematics. \nFor now we use the BPASS integrated fits to infer the stellar mass and the star formation rate of the host of GRB 230307A, as the fit and SFH is more convincing that the one obtained with STARLIGHT. We find that there are currently M ∗ = 1 . 65 × 10 9 M ⊙ of living stars (corresponding to 3.1 × 10 9 M ⊙ at ZAMS) in G1. Using the nebular component retrieved from subtracting the fit of the stellar component to the observed data, we can also estimate the star formation rate and metallicity. From the H α feature we estimate that the SFR is 5.47 ± 0 . 30 × 10 -1 M ⊙ yr -1 using the Kennicutt formulation [139], and using the N2 index, in the CALIFA formulation [140], we infer an oxygen abundance of 8.20 ± 0.16 (12 + log(O/H)). \nThere are qualitative similarities between the host of GRB 230307A and NGC 4993[135], the host of the first confirmed kilonova (they are both dominated by an older stellar populations and include a younger more metal rich component), but there \nare some key differences: NGC 4993 was a lenticular galaxy without a clear young component, whereas the host of GRB 230307A shows clear spiral arms and star forming regions. Another major difference is that the metallicity of the old population in this galaxy is 10 times lower than that of NGC 4993 (Z=0.001 compared to Z=0.010), which will influence the stellar evolution of potential progenitors. Finally, NGC 4993 had a large stellar (and presumably dark halo) mass M ∗ ≈ 10 11 M ⊙ [135], a factor of > 50 larger than the host of GRB 230307A. \nThe location of GRB 230307A relative to its host galaxy is consistent with these properties. In particular, the low mass of the galaxy suggests a modest gravitational potential such that binaries with velocities of a few hundred km s -1 can readily escape. The large offset also suggests that the binary is formed from the older stellar population.', '4.5 Properties of the brightest GRBs': 'GRB 230307A is the second brightest 6 burst observed in over 50 years of observations [52]. If it arises from a compact object merger, this implies that such bright bursts can be created in mergers. Indeed, such a picture appears likely based on GRB 211211A [17-19], the sixth brightest burst. Of the ten brightest bursts observed by the Fermi /GBM, and subsequently localised at the arcsecond level, three have apparently secure associations with supernovae (GRB 130427A, GRB 171010A, GRB 190114C), and two (GRB 211211A, GRB 230307A) are associated with kilonovae, and hence mergers. Of the remaining five, one lies at z = 1 . 4 and has energetics which suggest a collapsar; three have no redshift information, although one of these (GRB 160821A) lies in proximity to several galaxies at z = 0 . 19; and one is GRB 221009A whose associated with a supernova remains unclear [97, 141, 142], although recent observations suggest a collapsar with an associated supernova is most likely [143]. Within this very bright population, collapsars are likely as common as mergers.', '5 Event rates': "One key question of interest is the likely event rate for such merger GRBs. A simple estimate of the event rate associated to a single event is given by \nR = 1 Ω tV max . (1) \nHere Ω reflects the fraction of the sky covered by the detection mission, t the effective mission duration (accounting for the duty cycle) and V max the maximum co-moving volume within which a burst with the same properties could be identified. \nFor GRB 230307A, Ω = 0 . 65 (average for the Fermi /GBM) and t ≈ 15 years. V max is more complicated: as shown in Figure 6, the fluence distribution for GBM bursts extends to ∼ 10 -8 erg cm -2 and is likely complete to around 10 -6 erg cm -2 . Given the extreme brightness of GRB 230307A, it would likely have been recovered to a distance ∼ 50 times greater than its observed distance. If at z = 0 . 065 the inferred z max = 2 . 03 \nor V max = 630 Gpc -3 . In this case, the inferred rate of such bursts becomes extremely small, R ≈ 1 . 6 × 10 -4 Gpc -3 yr -1 . However, in practice, such bursts would not readily be identified at such redshifts since neither supernova nor kilonova signatures could be observed. A more realistic estimate would correspond to the distance at which associated supernovae can be either identified or ruled out with moderate confidence. In this case z max = 0 . 5 (also adopted by [144]), V max = 29 Gpc 3 , and R ≈ 3 . 5 × 10 -3 Gpc -3 yr -1 . \nThese rate estimates also assume that GRB 230307A is the only merger-GRB to have occurred within the 15-year lifetime of the Fermi /GBM. This is almost certainly not the case. Indeed, GRB 211211A was also identified by Fermi /GBM and has rather similar estimates of the intrinisc rate [5 . 7 × 10 -3 Gpc -3 yr -1 , 144]. \nHowever, even the interpretation of ∼ 2 events is problematic. In particular, the V/V max for GRB 230307A is 0 . 004, and for GRB 211211A = 0 . 005 (again assuming z max = 0 . 5). For a sample average of uniformly distributed sources of comparable energy or luminosity, we expect V/V max ∼ 0 . 5). That the initial identification of such a population should arise from bursts with such extreme V/V max values is surprising, but may reflect that these bursts are the brightest, which likely encouraged a detailed follow-up. However, it is improbable that they represent the only such bursts observed, and we should expect a much larger population. \nTo better quantify this, we extend our analysis to the Swift bursts and utilise the fluence of GRB 230307A converted to a 15-150 keV equivalent fluence using the observed spectral parameters. At z = 0 . 065, E iso (15-150) keV ∼ 7 × 10 51 erg, and for GRB 211211A E iso (15-150) keV = 2 × 10 51 erg. As expected, low energy events dominate the low redshift GRB population. However, at z < 0 . 5, there are 12 (out of 42) bursts with E iso ≳ 10 51 erg. This includes some further supernova-less GRBs, in particular GRB 060614 ( E iso = 9 × 10 50 erg), GRB 191019A ( E iso = 2 . 0 × 10 51 erg), and some bursts for which supernova searches have not been reported (e.g. GRB 150727A, GRB 061021, and the 'ultra-long' GRB 130925A). This sets an upper limit on the number of bursts at low redshift, which may be associated with mergers. In practice, selection effects would support a scenario where mergers generate a larger fraction of these bursts. In particular, the afterglows of GRB 230307A and GRB 211211A appear to be faint, despite the bright prompt emission. Such afterglows are difficult to find and may evade detection. In these cases redshifts may only be obtained from host galaxies. The associations may not be obvious if the bursts are offset from host galaxies at moderate redshifts. Such follow-up may occur late after the burst, or optical afterglow non-detections may lead to a lack of optical/IR follow-up because of uncertainty regarding the optical brightness of the event or suggestions it may be optically dark because of host galaxy extinction. Finally, given the afterglow brightness issues, it is possible that the small fraction of bright GRBs without redshift measurements may arise from a similar channel. These observations would imply that between 30-70% bursts at z < 0 . 5 and E iso ≳ 10 51 erg could arise from mergers, although it is likely less. A modest number of events at higher redshift is consistent with the observations, and would alleviate concerns regarding V/V max for GRB 211211A and GRB 230307A. \nThis fraction is surprisingly high given the strong evidence that long GRBs arise from broad-lined type Ic supernovae and short GRBs from compact object mergers. \nHowever, the dominant contributors to the long-GRB supernova connection occur at low energy, and belong to a population of low luminosity GRBs (LLGRBs) [145]. In a significant number of these, we may observe a energy source in the prompt emission separate from the highly relativistic jet seen in on-axis, energetic bursts. For example, the long-lived, soft nature of some bursts suggests a contribution from shock breakout or cocoon emission. If, for this reason, the luminosity function of collapsar GRBs is steeper at low luminosity than that of merger-GRBs, it is possible that at low luminosity the long GRB population is dominated by collapsars, while at high luminosity the contribution of mergers is significant. Such an interpretation is not without problems, given the star-forming nature of long-GRB hosts and their typically small offsets from their host galaxies. However, it is a logical investigation for future work.", '6.1 Light curve modelling': "In order to shed light onto the properties of the jet and, even more importantly, to separate the contribution of the kilonova from that of the jet afterglow in the UVOIR bands, we modelled the multi-wavelength light curves from radio to X-rays as a superposition of synchrotron emission from the forward shock driven by the jet into the interstellar medium (ISM), following [32, 146], and blackbody emission from the photophere of a kilonova, using the simple single-component model of [147]. \nThe forward shock synchrotron emission model has eight parameters, namely the isotropic-equivalent kinetic energy in the jet E K , its initial bulk Lorentz factor Γ 0 , its half-opening angle θ j , the ISM number density n , the fraction ξ N of ISM electrons that undergo diffusive shock acceleration in the forward shock, the fraction ϵ e of the shock downstream internal energy that is shared by such electrons, the slope p of the power law d N e / d γ ∝ γ -p that describes the Lorentz factor (as measured in the shock downstream comoving frame) distribution of the accelerated electrons as they leave the acceleration region, and the fraction ϵ B of the shock downstream internal energy that is shared by a small-scale, turbulence-driven, random magnetic field. The shock hydrodynamics is computed from energy conservation and accounts for the lateral expansion of the shock [146]. The effective electron energy distribution is computed accounting for the cooling induced by synchrotron and synchrotron-self-Compton emission, including an approximate treatment of the Klein-Nishina suppression of the Thomson cross section [146]. In computing flux densities, the synchrotron surface brightness of the shock is integrated over equal-arrival-time surfaces to account for the effects of relativistic aberration and latitude-dependent retarded times on the spectral shape [148]. \nThe kilonova model [147] assumes spherical ejecta expanding homologously, v = r/t , and featuring a power law density profile ρ ( r, t ) ∝ t -3 v -δ between a minimum and a maximum velocity, v ej ≤ v ≤ v ej , max . The density normalization is set by the total ejecta mass M ej . In general, the model allows for the ejecta opacity (assumed grey) κ to be piecewise-constant within the profile, but here we assume a uniform opacity across the ejecta for simplicity. The model divides the ejecta into 100 small \nshells and computes the heating rate and thermalization efficiency within each. This allows for the derivation of the internal energy evolution in each shell and eventually the computation of the photospheric luminosity L KN in the diffusion approximation. The fixed ejecta opacity also allows for the computation of the optical depth and hence for the identification of a photospheric radius, which then sets the effective temperature T KN by the Stefan-Boltzmann law. In our modelling of GRB 230307A, we computed the flux density by simply assuming pure blackbody emission with the given luminosity and effective temperature at each given time. We fixed v max = 0 . 6 c and left M ej , v ej , κ and δ as free parameters. \nTo carry out the model fitting, we defined an asymmetric Gaussian log-likelihood term for the i -th datapoint, which corresponds to an observation at time t i and in a band whose central frequency is ν i , as \nln L i = -1 2 ( F ν, m ( ν i , t i ) -F ν, obs , i ) 2 σ 2 i + f 2 sys F 2 ν, m -ln [ √ 2 π ( σ 2 l ,i + f 2 sys F 2 ν, m ) + √ 2 π ( σ 2 h ,i + f 2 sys F 2 ν, m ) ] , (2) \nwhere F ν, m ( ν, t ) is the flux density predicted by the model, F ν, obs ,i is the measured flux density, the one-sigma error reflects the potentially asymmetric error bars \nσ i = { σ l ,i if F ν, m ( ν i , t i ) ≤ F ν, obs ,i σ h ,i if F ν, m ( ν i , t i ) > F ν, obs ,i , (3) \nand we introduced a fractional systematic error contribution f sys , which we take as an additional nuisance parameter, to account for potential inter-calibration uncertainties between different instruments and for the fact that error bars typically only account for statistical uncertainties. For X-ray detections, we fit the integrated flux and the spectral index independently, with an analogous term for each (but with no systematic error contribution for the spectral index). Upper limits were treated simply by setting F ν, obs ,i equal to the reported upper limit, σ h ,i = F ν, obs ,i / 10 and σ l ,i = 10 F ν, obs ,i . The final log-likelihood was taken as the sum of these terms. \nIn order to derive a posterior probability density on our 13-dimensional parameter space, we assumed the priors reported in Table 6 and we sampled the posterior with a Markov Chain Monte Carlo approach using the emcee python module [149], which implements the Goodman and Weare [150] affine-invariant ensemple sampler. The medians and 90% credible intervals of the marginalised posteriors on each parameter obtained in this way are reported in Table 6. The posterior is visualised by means of corner plots in Figures 10 (jet afterglow parameters), 11 (kilonova parameters) and 12 (all parameters). \nThe left-hand panel in Figure 13 shows the observed light curve data (markers) along with the best-fitting model (solid lines). Dashed lines single out the contribution of the kilonova. The right-hand panel in the same figure shows some selected spectra, showing in particular the good agreement of the first JWST epoch with the blackbody plus power law spectrum implied by our model at those times. \nWhile the best-fit model demonstrates a relatively good agreement with most of the measurements, some discrepancies stand out, most prominently with the 61.5 d \nJWST data and with the 28.5 d Chandra detection. The former is not too surprising, as the assumptions in the kilonova model (in particular that of blackbody photopsheric emission, which is particularly rough in such a nebular phase, and that of constant and uniform grey opacity, due to recombination of at least some species) are expected to break down at such late epochs. The latter is linked to the steepening ('jet break') apparent at around 2 days in the model X-ray light curve, which in turn is mainly driven by the need to not exceed the optical and near-infrared fluxes implied by observations at around one week and beyond. In absence of these constraints, the fit would have accommodated a larger jet half-opening angle, postponing the jet break and hence allowing for a better match with the best-fit Chandra flux. On the other hand, as noted in Methods, this flux is rather uncertain, with the low-end uncertainty possibly extending to fluxes lower by one order of magnitude or more, depending on the adopted prior in the spectral analysis (see Methods). Still, such a discrepancy might indicate the presence of additional X-ray emission that is not accounted for by the model, as has been seen previously in e.g. [151, 152].", '6.2 Spectral analysis modeling': "The JWST/NIRSpec spectrum taken on 5 April 2023 exhibits a red continuum component with emission line features. The most distinctive feature is a broad emission line at 2 . 15 microns (in the rest frame, assuming z = 0 . 065). This may be a blend (visibly split in Figure 3) and a simultaneous fit of two Gaussians provides measured centroids of 20285 ± 10 ˚ A and 22062 ± 10 ˚ A. The line widths are both consistent at v FWHM = 19100kms -1 (0.064c). This 2 . 1 micron feature is quite similar in strength and width to the 2 . 07 micron feature in AT2017gfo at 10.5 days after merger [discussed in 37]. The AT2017gfo line also appears to be better fit as a blend of two features rather than a single transition, with line velocities of v FWHM = 38900kms -1 . While the average line centre is reasonably consistent between the two, the components inferred for AT2017gfo and the kilonova of GRB 230307A are each quite different. Reference [37] finds them at 20590 ˚ A and 21350 ˚ A and there is no consistent velocity shift that could be applied to match AT2017gfo with our JWST spectrum. Nevertheless, the similarity in their average line centroids, velocities and equivalent widths is striking, as demonstrated in Figure 3. \nWith a Doppler broadening parameter of ≲ 0 . 1 c , it is unlikely that the continuum component is formed as a result of the superposition of emission lines. Because kilonova radiation transfer at such late times is not yet fully understood, here we attempt to model the spectrum with the assumption that the emission consists of blackbody radiation from the photosphere and forbidden emission lines of heavy elements formed outside the photosphere. \nIf the continuum is described with blackbody radiation, the temperature and photospheric velocity are ≈ 670 K and ≈ 0 . 08 c , respectively. The continuum luminosity is estimated as ∼ 2 × 10 39 erg / s in the NIRSpec band and ∼ 5 × 10 39 erg / s if the blackbody emission extends to much longer wavelengths. Assuming this emission is entirely powered by radioactivity of r -process nuclei, these correspond to an ejecta mass of ∼ 0 . 03-0 . 07 M ⊙ [147]. With the ejecta mass and velocity, the opacity is required to be ≳ 5 cm 2 / g in order to keep the ejecta optically thick at 30 day. It is worth noting \nthat such a high opacity in the mid-IR indicates that the inner part of the ejecta is lanthanide rich [153-155]. \nForbidden emission lines in the infrared are expected to arise from fine structure transitions of low-lying energy levels of heavy elements. Most abundant ions are expected to produce the strongest lines. We attribute the strongest observed line at 2.15 microns to tellurium (Te) III from an M1 line list of heavy elements presented in [45], where the line wavelengths are experimentally calibrated according to the NIST database [156]. Te belongs to the second r -process peak. With the M1 line list, we model kilonova emission line spectra under the assumption that photons from forbidden lines produced outside the photosphere freely escape from the ejecta. The collision strengths of Te III are taken from an R-matrix calculation [43] and those of other ions are obtained by using an atomic structure code HULLAC[157]. The abundance pattern is chosen to be the solar r -process but we separate 'light' and 'heavy' elements at an atomic mass of 85 and introduce a parameter, the abundance ratio of the two (see figure 14). The ionization fractions are fixed to be ( Y +1 , Y +2 , Y +3 ) = (0 . 2 , 0 . 5 , 0 . 3) motivated by the Te ionization evolution in kilonova ejecta [41]. The line shape is approximated by a Gaussian with a line broadening velocity of 0 . 08 c , which is the same as the photospheric velocity. The mass in the line forming region is estimated by assuming that the observed line luminosity, 5 × 10 38 erg s -1 , is locally generated by radioactivity of r -process nuclei, corresponding to ∼ 0 . 02 M ⊙ . Given the abundance pattern and ionization state, the mass of Te III in the line forming region is ≈ 8 · 10 -4 M ⊙ . The electron temperature of the line forming region is then determined such that the total line luminosity agrees with the observed one. The estimated electron temperature is ∼ 3000 K, which is slightly higher than that derived from the pure neodymium nebular modeling [158]. This is because the cooling by tellurium ions is more efficient than neodymium. \nWe find that [Te III] 2 . 10 µ m line is indeed the most outstanding emission line around 2 microns. Several weaker lines also contribute to the flux around 3-4 microns. There is another potential line feature around 4 . 5 microns in the NIRSpec spectrum. The location of this feature is consistent with [Se III] 4 . 55 µ m and [W III] 4 . 43 µ m as pointed out by [45] for the kilonova AT2017gfo. From the spectral modeling, we obtain the total ejecta mass of ∼ 0 . 05-0 . 1 M ⊙ , which agrees with the one obtained from the light curve modeling ∼ 0 . 1 M ⊙ . \nHere we show a brief estimate of the Te III mass from the observed line at 2 . 15 microns ( 3 P 0 -3 P 1 ). The collisional excitation rate per Te III ion from the ground level ( 3 P 0 ) to the first exited level ( 3 P 1 ) is given by \nk 01 = 8 . 63 · 10 -6 n e √ T e Ω 01 g 0 e -E 01 /kT e s -1 , (4) \nwhere n e and T e are the thermal electron density and temperature, Ω 01 ≈ 5 . 8 is the collision strength [43], E 01 ≈ 0 . 6 eV is the excitation energy, g 0 is the statistical weight of the ground level. Assuming that the ejecta mass in the line forming region is 0 . 02 M ⊙ expanding with 0 . 08 c and the ions are typically doubly ionised, we estimate \nn e ∼ 3 · 10 5 cm -3 , and thus, the line emissivity per Te III ion is \nϵ 10 ≈ 2 . 5 · 10 -14 ( n e 3 · 10 5 cm -3 ) erg / s , (5) \nwhere T e = 3000K is used. Combining the line emissivity with the observed line luminosity in 2 . 25 ± 0 . 23 µ m, L line ≈ 3 · 10 38 erg / s, we obtain \nM (Te III) ≈ 10 -3 M ⊙ ( n e 3 · 10 5 cm -3 ) -1 ( L line 3 · 10 38 erg / s ) . (6) \nThe mass estimated from the line is somewhat dependent on T e and n e . However, we emphasise that, with T e ≈ 3000 K and n e ≈ 3 · 10 5 cm -3 , the line luminosity is consistent with the radioactive power in the line forming region. It is also interesting to note that the Te III mass of 10 -3 M ⊙ is in good agreement with the one obtained based on the same line seen in AT2017gfo at 10.5 day [42]. \nWhile we conclude that the observed line feature at 2.1 microns is most likely attributed to Te III, it is important to note that there are caveats associated with our modeling. One obvious caveat is that the model does not include E1 lines. Lanthanides and actinides have E1 transitions between low-lying levels in the mid-IR [discussed in 37]. Due to their lower abundances, these lines are expected to be weaker compared to the Te line if collisional excitation dominates the excitation processes. However, as we make an implicit assumption that their E1 lines contribute to the opacity in the mid-IR, they may produce P-Cygni like features, see, e.g., [37, 159]. For example, Ce III has a strong line at 2 . 07 µ m with log gf = -1 . 67 [159]. We estimate that its line optical depth is ≲ 0 . 1 at 700 K with ∼ 0 . 05 M ⊙ and ∼ 0 . 1 c even if Ce is purely in Ce III. However, more careful analyses including non-LTE effects are needed to quantify it. Another caveat is that the opacity of lanthanides is expected to have some wavelength dependence. Including this effect may also affect the spectral modelings.", '7 Alternative progenitor possibilities': 'Our interpretation of GRB 230307A provides a self-consistent model for the source in which the temporal and spectral evolution, as well as the source location, can be readily explained. The kilonova has marked similarities with AT2017gfo providing a robust indication of its origin, and we do not need to postulate new and unseen phenomena to explain it. However, it is also relevant to consider alternative possibilities. In particular, given the location of the galaxy at z = 3 . 87, it is important to consider if the burst could originate at that redshift.', '7.1 GRB 230307A as a high redshift, highly energetic GRB': "The nearby galaxy H1 (F277W(AB)=27.5 ± 0.1 , r proj = 0 . 3 arcsec) with a spectroscopic redshift of z=3.87 has a relatively low probability of occurring by chance ( ∼ 5 -10%, see section 4.3). This galaxy has a comparable P chance to G1 (the z = 0 . 065 galaxy). The host-normalised offset for H1 is ∼ 2 . 5, which is large but not unprecedented for long GRBs [116]. However, assuming the late time light at the GRB position \nis all from the transient, it does not lie on the stellar field of this galaxy, which is unusual, for example, in the samples of [116, 160, 161], there is only one (of > 100) sub-arcsecond localised GRB not on the stellar field of its host. \nAt z = 3 . 87, the inferred isotropic energy release and luminosity of GRB 230307A would be E iso = 1 . 2 × 10 56 erg and L iso = 1 . 7 × 10 56 erg s -1 (using a 64 ms peak flux). This is approximately an order of magnitude more energetic and two orders of magnitude more luminous than any other previously identified GRB [52]. \nIf at z = 3 . 87, we can have some confidence that GRB 230307A would be the most energetic burst ever detected by Swift or Fermi , including those without redshift or even afterglow identifications. In Table 4, we tabulate the most fluent GRBs observed by Fermi . Most of these have either redshifts or optical detections, which constrain z < 6 via the detection of the source in the optical band. This leads to a set of measured or maximum E iso values. For events without any redshift information, we can place a conservative upper redshift limit of z = 16. No GRBs detected by Fermi without a redshift can have energy over 10 56 erg unless they lie beyond z ∼ 20. Hence, GRB 230307A is sufficiently rare, if at z = 3 . 87, that events like it occur less frequently than once per decade across the Universe (i.e. no more than one in the combined lifetimes of Swift and Fermi ). \nThe energetics of the burst at this redshift lie would lie far beyond those of the general GRB population and beyond those suggested as the upper limit for GRB energetics [162]. The only population of core-collapse GRBs whose energetics have been suggested to approach this value are those from first-generation population III stars [163-165]. It is not expected that such stars should exist at z ∼ 4. However, while a pop-III origin may alleviate energy concerns, the properties of the GRB and its optical/IR counterpart do not resemble the predictions for pop-III stars. In particular, pop III GRBs are suggested to have particularly long durations given the mass and radii of their progenitors [165], and so require extremely long durations of engine activity to enable jet breakout. However, the ∼ 35 s duration of GRB 230307A and its rapid variability do not readily fit this expectation. \nIf one ascribes the GRB to a stellar collapse event, considering the afterglow's properties and associated supernovae is also relevant. Firstly, at 28.5 days, the JWST spectral observations are inconsistent synchrotron emission, suggesting that the counterpart must be dominated by another source in the mid-IR and the earlier K-band points. This excess, which in our preferred model is explained by a kilonova, would have to be due to the supernova or shock breakout if at z = 3 . 87. The K-band (rest frame B-band) light would reach a peak of M B ( AB ) < -23 . 5 on a timeframe of < 2-days (rest-frame) before decaying at a rate of > 1 mag day -1 for the next four days, or as a power-law decay, a rate of approximately t -4 . This appears too rapid for radioactively powered transients, at least based on the sample to date (note that at z = 3 . 87, the timescales are a factor of ∼ 5 faster than in the z = 0 . 065 scenario due to cosmological time dilation). The most likely option for such emission would be shock-breakout, which may begin blue but rapidly cool. There are simulations for the shock breakout associated with pop-III supernovae, which show an early peak [166]. However, this emission peaks in the UV to soft X-ray regime and at luminosities below that seen in GRB 230307A. Indeed, taking 28.5 days as a baseline; for a plausible \nmaximum Pop III radius (e.g. 2000 R ⊙ , [167]) and a luminosity of L bol ∼ 10 43 erg s -1 the inferred temperature is T ∼ 300 , 000 K. This is incompatible with the spectral shape seen in the counterpart to GRB 230307A which peaks at > 4 . 5 microns ( T < 3000K at z = 3 . 87). Dust or metal line blanketing could alleviate this discrepancy to some degree, but it would be extreme to explain the observations. It would also come at the cost of an even higher intrinsic luminosity. Conversely, the radius at which the luminosity and temperature would be consistent is extremely large ( ∼ 0 . 3 pc) and indeed would require super-luminal expansion to reach from a single explosion within the time since burst. These constraints become even more extreme for the Kband observations at 11.5 days, where the luminosity is > 50 times higher. However, we lack detailed information regarding the spectral shape at this time. We can conclude that a thermal transient launched at the time of GRB 230307A cannot explain the observed source at z = 3 . 87. \nWe should consider if GRB 230307A could be related to an explosion which bears little to no similarity to long-GRB progenitors. Given the inferred energetics, this is not an unreasonable proposition. However, the emission is too bright and too fast for, for example, the fast blue optical transients (e.g. [168]), the fastest of which have halftimes of ∼ 4 days [169, c.f. < 1 day for GRB 230307A at z = 3 . 87]. A further alternative may be a relativistic tidal disruption event. This would face significant challenges with the rapid variability timescales seen in the prompt emission and the non-nuclearity of the source within the galaxy at z = 3 . 87. Putting aside these concerns, the peak optical/IR luminosity is comparable to AT2022cmc [170, 171], but the evolution is too rapid and the dynamic range too large. \nFinally, it is possible that the red excess seen at later times is not directly related to the progenitor or the transient but is a result of the re-processing of the GRB radiation by material within the host galaxy. In particular, for GRB 211211A [172] suggest that an alternative explanation for the emission could be the heating of dust. However, this model also encounters significant issues at z = 3 . 87. In particular, the observed K -band excess is a rest-frame B-band excess, much bluer than expected for dust heating. If this represented the peak of the thermal spectrum, it would be above the sublimation temperature of the dust. Alternatively, if it were the blue tail of a much cooler black body, the luminosity would be extremely high. \nA final challenge to the high -z scenario is that the afterglow is detected in the UVOT-white and ground-based g -bands. These observations all have substantial sensitivity blue-ward of Ly α at z=3.87. A typical column from the intergalactic medium should attenuate ∼ 50% of the light in these bands, inconsistent with observations. Indeed, for a typical β = 1 spectrum, we expect to observe white -i = 2.8 and g -i = 1 . 9, approximately 3 and 4 σ away from the observed colours. There is significant variation in the absorption strength as a function of the line of sight, so a low (or near zero) absorption column would alleviate this tension. The sample of [173] implies that at z ∼ 4, perhaps 10% of galaxies have such low absorption sight lines. \nHence, while the proximate galaxy could indicate a high redshift, there are few other indications in the transient properties that would support this interpretation. In particular, neither standard thermal nor non-thermal emission can explain the observed counterpart properties. If the burst does arise from z = 3 . 87, it requires a \nnew kind of explosion, unlike any seen until now. In practice, such explosions could be extremely rare: the volumetric rate of GRBs with E iso > 10 56 erg is minimal, but postulating them is unnecessary when a robust, physically motivated explanation can be obtained for a lower redshift solution.", '7.2 Other cosmological scenarios': "We should also consider a further option which is that GRB 230307A does not reside at either z = 0 . 065 or z = 3 . 87 but is a chance super-position with both galaxies. In this scenario, the actual host is undetected or is one of the other galaxies within the field. The absence of direct redshift measurements makes placing constraints on this scenario challenging. However, we can use the non-detection of the late-time JWST magnitudes to limit the brightness of a supernova component at any redshift. \nTo quantify the exclusion of 'normal' long GRBs at intermediate redshift we utilize model light curves for SN 1998bw from MOSFiT [174] calculated at a range of redshift from 0 . 05 < z < 4 . 0 (we take z = 4 as an upper limit for the redshift the GRB based on the observed g-band detections together with the detection of continuum emission to 5300 ˚ A [13] and similar faint trace seen to ∼ 5100 ˚ Ain our X-shooter spectrum). At each redshift, we compare our observed JWST photometry in each band to the model predictions at that time and report the most constraining limit (e.g. the lowest ratio of F obs /F 98 bw ). This is shown in Figure 15. At all plausible redshifts, any supernova must be at least a factor ∼ 3 fainter than SN 1998bw at similar times. For any redshift where the burst energetics fall within the range seen in the bulk GRB population ( z < 1 . 2), any supernova must be a factor > 100 fainter than SN 1998bw. Hence, there is no route in which GRB 230307A can arise from a classical long GRB ( E iso < 10 55 erg with an associated broad-lined Ic supernova. The strength of this rejection is predominantly based on the faintness of the source in the bluer bands, whereas at the first epoch, the redder bands are substantially brighter than SN 1998bw at z = 3 . 87. \nIf the burst lies at an intermediate redshift, dust models may become more appealing. In particular, for a low to moderate redshift, the luminosty and timescales may be a suitable match (e.g. for GRB 211211A [172] find plausible explanations at z ∼ 0 . 5). However, in this case, the lack of a supernova to extremely deep limits would be surprising, as would the non-detection of the host galaxy.", '7.3 Galactic objects': 'In the absence of a robust absorption redshift, it is also necessary to consider if GRB 230307A could arise from a Galactic system. \nThe very faint magnitudes and extreme red colours observed at late times can effectively rule out X-ray binary outbursts. For example, with an M-dwarf companion (absolute magnitude 9), the distance to the source would be ∼ 100 kpc, and larger for any more massive star, while the late-time colours of the source are not stellar. In practice, given that the source is transient, and we may also expect some contribution from an accretion disc, the overall properties cannot be remedied within the accreting binary framework. \nThe inferred energetics for Galactic systems are E iso ∼ 5 × 10 43 ( d/ 10 kpc) 2 . This energetic output is within the bounds of giant outbursts from magnetars. For example, the giant flare of SGR 1806-20 had an inferred isotropic energy of E iso ∼ 2 × 10 46 erg (see [175], although subsequent downward revisions in its distance lower this somewhat [176]). However, GRB 230307A does not appear to meet the requirements of a magnetar. In particular, it is at high Galactic latitude (-36 degrees) and far from any plausible star formation that could give rise to a young neutron star. Furthermore, the emission in all magnetar outbursts is dominated by a very short pulse followed by decaying emission in which the pulse period of the neutron star is visible. This is not the case for GRB 230307A. \nGRBs have previously been suggested to arise from the tidal disruption and accretion of rocky material onto a neutron star [177], and such events are seen in the case of white dwarfs [178]. However, for accretion onto a neutron star, we would usually expect to observe a relatively soft outburst (e.g. in the model of [177] the temperature is ∼ 10 keV). The spectrum we observe for GRB 230307A consists of evolving synchrotron emission which does not contain a thermal component and is inconsistent with directly observing heating accreting material. Indeed, even for near direct accretion, it is not clear how such a spectrum would be formed in an accreting neutron star scenario. Indeed, in this case, the evolution to very low temperatures on a timescale of ∼ 60 days would also not be natural. \nHence, we conclude that no known Galactic systems could explain the observed properties of GRB 230307A.', '7.4 GRB 230307A as a white-dwarf - neutron star merger': "A final alternative is that GRB 230307A is related to the merger of a white dwarf and a neutron star. Although this is still a 'compact object merger', such mergers are very different from those of neutron stars with another neutron star or a black hole. In particular, simulations show that no r -process material is produced [179], and so we should not expect the very red emission. Although there are suggestions that GRB 211211A could have been produced by a WD-NS merger [19, 59], it is unclear if these could readily explain the detailed spectrophotometric evolution of GRB 230307A. \nWhite dwarf neutron star mergers are appealing, because the long-duration of the gamma-ray emission could suggest a less compact remnant merger event. The wider separation of the binary at the disruption of the white dwarf produces disk accretions times from 100 to 1000s, matching the long duration of these bursts [180]. However, we expect the accretion rate in the mergers to be low, producing less-powerful, and hence, less luminous GRBs. Current WD/NS merger simulations predict a range of light curves that span the emission from GRB 230307A. A Ca feature does exist that is close to the observed line feature, but the models are too blue to explain the shape of the spectra [179]. This is because WD/NS mergers do not produce elements much heavier than the iron peak elements. As we have noticed in matching kilonova models, these elements do not have strong lines beyond ∼ 20 , 000 ˚ A. The subsequent emission above these wavelengths is very weak in these models. In addition, WD/NS mergers are expected to have fairly week kicks to ensure that the binary remains bound and these mergers are expected to have much lower offsets than neutron star mergers. However, \nthe mass of the best-candidate host galaxy is sufficiently low that the observed offset for this burst can be attained [181]. \nTable 1 Optical and IR observations of the optical counterpart of GRB 230307A. Limits are given at the 3 σ level. \nTable 2 X-ray and radio observations of the afterglow of GRB 230307A. \nTable 3 Properties of possible host galaxies for GRB 230307A. Formally, because the galaxy is undetected in the r -band, P chance is unbounded. This probability is based on the magnitudes measured at other wavelengths. \nTable 4 The properties of the brightest 10 bursts observed by Fermi /GBM. Eight of these have afterglow identifications which place limits on the redshift, while it is likely that all originate from z < 16. As expected for the (observationally) brightest bursts there is a preference for low redshift, with the bursts all arising at much less than the mean for Swift GRBs [182]. In 3-4 cases a supernova has been identified, meaning the bursts arise from core collapse (with GRB 221009A still ambiguous at the time of writing). In 2 cases a kilonova origin appears more likely, while for the remaining 4 the absence of afterglows or the redshifts render such diagnostics impossible. However, it would appear that samples of bright, long-GRBs may contain significant number of compact object mergers. \nTable 5 Photometry (in the AB photometric system) of the host galaxy of GRB 230307A. No reddening correction was applied. \nTable 6 Light curve model parameters, priors and posterior medians and 90% credible intervals. \nFig. 5 Ground based optical and IR imaging of the afterglow of GRB 230307A. Panel (a) shows a legacy survey image of the field, the green trapezoid shows the IPN error box at the time of afterlgow discovery, with the blue dotted lines showing its ultimate refinement. The shaded regions represent observations taken with ULTRACAM 1.4 days post burst. Panel (b) shows the same image, zoomed in to a region around the afterglow. The red source in the centre is an unrelated foreground star (confirmed both by its measured proper motion and spectrum), and several other galaxies can also be seen. Panel (c) shows an ULTRACAM image, where the afterglow can be seen to the north-east of the star. The remaining panels (d,e,f) show the afterglow imaged on ∼ 10 -20 day timescales. At 10 days the source is undetected in deep FORS2 i-band imaging, but well detected in the K-band with Gemini-South. This very red colour was suggestive of an additional component over and above any afterglow, and motivated further follow-up. \n<!-- image -->", 'Author contribution statements': 'AJL led the project, including the location of the afterglow and kilonova and the JWST observations. BPG first identified the source as a likely compact object merger, was co-PI of the Chandra observations, and contributed to analysis and writing. OS contributed to afterglow and kilonova modelling and led the writing of these sections. MB was involved in kilonova modelling, EB contributed to interpretation, placing the burst in context and high energy properties. KH was involved in kilonova spectral modelling and identified the 2.15 micron feature. LI reduced VLT/MUSE and X-shooter observations and led the host analysis. GPL contributed to afterglow and kilonova modelling. ARE analysed the Chandra observations, BS reduced and analysed VLT observations. NS contributed to afterglow and kilonova modelling. SS was responsible for placing the burst afterglow in context and demonstrating its faintness. NRT \ncontributed to analysis, interpretation and writing. KA was involved in the ULTRACAM observations and interpretation. GA led the ATCA observations. GB reduced the JWST NIRCAM data. LC processed and analysed the MUSE observations. VSD is the ULTRACAM PI. JPUF contributed to the interpretation. WF was the PI on the Chandra observations and contributed to discussion. CF contributed to the theoretical interpretation. NG was involved in host analysis. KEH, GP, AR, SDV, SC, PDA, DH, MDP, CCT, AdUP and DA contributed to ESO observations and discussion. DW contributed to spectral and progenitor modelling and discussion. MJD, PK, SP, JM, SGP, IP, DIS contributed to the ULTRACAM observations. GL investigated potential similarities with other transients. AT, PAE, BS and JAK contributed to the Swift observations. MF extracted and flux-calibrated the TESS light curve. SJS analysed the JWST spectral lines, and contributed to interpretation and writing. HFS performed the BPASS-hoki-ppxf fits to the integrated MUSE flux and contributed the associate figure and text. All authors contributed to manuscript preparation through contributions to concept development, discussion and text.', 'Acknowledgements': "We dedicate this paper to David Alexander Kann, who passed on March 10. The final messages he sent were regarding follow-up of GRB 230307A, and we hope it would satisfy his curiosity to know the final conclusions. \nThis work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program #4434 and 4445. Support for Program numbers 4434 and 4445 was provided through grants from the STScI under NASA contract NAS5- 03127. \nThis paper is partly based on observations collected at the European Southern Observatory under ESO programme 110.24CF (PI Tanvir), and on observations obtained at the international Gemini Observatory (program IDs GS-2023A-DD-106), a program of NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci'on y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog'ıa e Innovaci'on (Argentina), Minist'erio da Ciˆencia, Tecnologia, Inova¸c˜oes e Comunica¸c˜oes (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). Processed using the Gemini IRAF package and DRAGONS (Data Reduction for Astronomy from Gemini Observatory North and South). \nAJL, DBM and NRT were supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 725246). \nN. Sarin is supported by a Nordita fellowship. Nordita is supported in part by NordForsk. B. Metzger is supported in part by the NSF (grant AST-2002577). \nJ.H. and D.L. were supported by a VILLUM FONDEN Investigator grant (project number 16599). G.P.L. is supported by a Royal Society Dorothy Hodgkin Fellowship (grant Nos. DHF-R1-221175 and DHF-ERE-221005). G.L. was supported by a research grant (19054) from VILLUM FONDEN. K. H. is supported by JST FOREST Program (JPMJFR2136) and the JSPS Grant-in-Aid for Scientific Research (20H05639, 20H00158, 23H01169, 20K14513). KEH acknowledges support from the Carlsberg Foundation Reintegration Fellowship Grant CF21-0103. S. Schulze acknowledges support from the G.R.E.A.T. research environment, funded by Vetenskapsr˚adet , the Swedish Research Council, project number 2016-06012. The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant No. 140. JPUF is supported by the Independent Research Fund Denmark (DFF-409000079) and thanks the Carlsberg Foundation for support. SJS acknowledges funding from STFC Grant ST/X006506/1 and ST/T000198/1. VSD and ULTRACAM are funded by STFC grant ST/V000853/1 AAB acknowledges funding from the UK Space Agency. MN is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 948381) and by UK Space Agency Grant No. ST/Y000692/1. H.F.S is supported by the Eric and Wendy Schmidt AI in Science Postdoctoral Fellowship, a Schmidt Futures program. DS acknowledges funding from STFC grants ST/T000406/1, ST/T003103/1, ST/X001121/1. MER acknowledges support from the research programme Athena with project number 184.034.002, which is financed by the Dutch Research Council (NWO). POB acknowledges funding from STFC grant ST/W000857/1. D.K.G acknowledges support from the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), through project number CE170100004.", 'Data availability': 'JWST data are directly available from the MAST archive. Chandra and Swift data are also in the public domain. ESO and Gemini data are stored in their respective archives and will be available to all once the proprietary period expires. Data can be obtained from the corresponding author between the date of publication and the end of the proprietary period. This research has made use of Fermi data which are publicly available and can be obtained through the High Energy Astrophysics Science Archive Research Center (HEASARC) website at https://heasarc.gsfc.nasa.gov/W3Browse/ fermi/fermigbrst.html', 'Code availability': "Much analysis for this paper has been undertaken with publically available codes and the details required to reproduce the analysis are contained within the manuscript. \nFig. 6 The distribution of measure fluence from the Fermi /GBM catalogue [105]. The solid line shows an expected slope of -3 / 2 for a uniform distribution. The faint end deviates from this line because of incompleteness. At the brighter end, there are three bursts which appear to be extremely rare, GRB 130427A, GRB 221009A [52] and GRB 230307A. To indicate the apparent rarity we also plot lines representing the expected frequency of events under the assumption of a -3 / 2 slope. We would expect to observe bursts akin to GRB 230307A only once per several decades. \n<!-- image --> \nFig. 7 A comparison between the prompt fluence (in the 15-150 keV band) and the X-ray flux at 11 hours for Swift GRBs with GRB 221009A and GRB 230307A added, updated from [106, 107]. The general 1:1 trend between the prompt fluence and X-ray brightness can clearly be seen, although it has a significant scatter, although a very rare event, GRB 221009A apparently lies on the same relation. However, GRB 211211A and GRB 230307A are clearly outliers to this relation with GRB 230307A occupying a region devoid of other GRBs. This very faint afterglow compared to the prompt emission may be related to a location at large projected offset from its host galaxy in which the density of the ambient medium is very low. \n<!-- image --> \nFig. 8 Spectral energy distribution (SED) of the galaxy G1 from 1000 to 60,000 ˚ A (black data points) and its best fit with the prospector SED fitting code (grey shaded curve). In the top right, we also report the values of the model parameters and their 1 σ uncertainties. The red squares represent the model-predicted magnitudes. The fitting parameters are shown in the upper-left corner. The abbreviation 'n.o.f.' stands for the number of filters. \n<!-- image --> \nFig. 9 SED fit (top left panel) and Star Formation History (right hand panel of the host galaxy obtained with BPASSv.2.2.2 -hoki templates fit with ppxf . We also include a comparison of the final fits obtained with BPASS (2 metallicities) and STALRIGHT (6 metallicites) in the bottom left panel to highlight how similar both fits are. \n<!-- image --> \nFig. 10 Corner plot of posterior probability density from multi-wavelength light curve fitting, limited to the relativistic jet afterglow parameter space. Histograms on the diagonal show the marginalized posterior probability densities on the parameters constructed from our MCMC posterior samples. Dashed black lines show the 90% credible interval, while red lines show the medians. The remaining plots show the one, two and three sigma equivalent contours of the joint posterior probability densities of parameter pairs. Red lines and squares mark the medians. \n<!-- image --> \nFig. 11 Corner plot of posterior probability density from multi-wavelength light curve fitting, limited to the kilonova parameter space. Similar to figure 10, but for the kilonova model parameters. \n<!-- image --> \nFig. 12 Corner plot of posterior probability density from multi-wavelength light curve fitting. Similar to figures 10 and 11, but showing all model parameters, including the nuisance parameter f sys . \n<!-- image --> \n1 2 \n3 \n4 \n5 \nFig. 13 Multi-wavelength light curves and model predictions. Markers in the figure show the observed flux density at the position of GRB 230307A in various bands (see legend in the left-hand panel) and at various times. Downward-facing triangles represent upper limits. The optical and near infrared flux densities are multiplied by the numbers reported in the legend for presentation purposes. The butterfly-shaped filled regions in the right-hand panel encompass flux densities consistent at one, two and three sigma (progressively lighter shades) with the Swift /XRT and Chandra detections in the 0.3-10 keV band, according to our analysis and adopting a uniform prior on the flux. Solid lines of the corresponding colours show the predicted light curves (left-hand panel) and spectra (right-hand panel) of our afterglow (forward shock only) plus kilonova model at the central frequencies of the bands. Dashed lines single out the contribution of the kilonova. \n<!-- image --> \nFig. 14 Abundance used in the spectral modeling. The abundance is chosen based on the solar rprocess residuals. The abundance of the 'light' elements ( A < 85) is reduced relative to the solar pattern. The locations of selenium (Se), tellurium (Te) and tungsten (W) are marked. \n<!-- image --> \nFig. 15 Limits on supernova similar to SN 1998bw as a function of redshift, based on the most constraining detection with JWST . At any redshift for which GRB 230307A would not be the most energetic GRB ever observed, any supernova is at least a factor of ∼ 100 fainter than SN 1998bw. \n<!-- image -->", 'References': "- [1] Fermi GBM Team: GRB 230307A: Fermi GBM Final Real-time Localization. GRB Coordinates Network 33405 , 1 (2023)\n- [2] Dalessi, S., Roberts, O.J., Meegan, C., Fermi GBM Team: GRB 230307A: Fermi GBM Observation of a very bright burst. GRB Coordinates Network 33411 , 1 (2023)\n- [3] Burns, E., Goldstein, A., Lesage, S., Dalessi, S.: Grb 230307a: possibly the second highest grb energy fluence ever identified. 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2024arXiv240905220P	The interiors of many planets consist mostly of fluid layers. When these layers are subject to superadiabatic temperature or compositional gradients turbulent convection transports heat and momentum. In addition planets are fast rotators. Thus the key process that underpins planetary evolution the dynamo action flow patterns and more is rotating convection. Because planetary interiors are inaccessible to direct observation experiments offer physically consistent models that are crucial to guide our understanding. If we can fully understand the laboratory model we may eventually fully understand the original. Experimentally reproducing rotating thermal convection relevant to planetary interiors comes with specific challenges e.g. modelling the central gravity field of a planet that is parallel to the temperature gradient. Three classes of experiments tackle this challenge. One approach consists of using an alternative central force field such as the electric force. These are however weaker than gravity and require going to space. Another method entails rotating the device fast enough so that the centrifugal force supersedes Earths gravity. This mimics the equatorial regions of a planet. Lastly by using the actual lab gravity aligned with the rotation axis insight into the polar regions is gained. These experiments have been continuously refined during the past seven decades. We review their evolution from the early days of visualising the onset patterns of convection over central force field experiments in spacecrafts liquid metal experiments to the latest optical velocity mapping of rotating magnetoconvection in sulfuric acid inside highfield magnets. We show how innovative experimental design and emerging experimental techniques advanced our understanding and painted a more realistic picture of planetary interiors including Earths liquid metal outer core.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.05220', 'arXiv:2409.05220', '2024arXiv240905220P']	['Physics - Geophysics', 'Astrophysics - Earth and Planetary Astrophysics']	Seven decades of exploring planetary interiors with rotating convection experiments	2,024	189	0.22	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.05220.pdf	{'No Header': '<!-- image -->', 'Seven decades of exploring planetary interiors with rotating convection experiments': "Sept décennies d'exploration des intérieurs de planètes par les expériences de convection en rotation", 'Alban Pothérat , ∗ , a and Susanne Horn , a': "a Centre for Fluid and Complex Systems, Coventry University, Mile Lane, CV1 2NL Coventry, United Kingdom \nE-mails: [email protected] (A. Pothérat), [email protected] (S. Horn) \nAbstract. The interiors of all the planets in the solar system consist of layers, most of which are made out of fluids. When these layers are subject to superadiabatic temperature or compositional gradients, turbulent convection takes place that transports heat and momentum. In addition, planets are fast rotators. Thus, the key process that underpins planetary evolution, the existence of dynamo action or lack thereof, the observable flow patterns and much more, is rotating convection. Because planetary interiors are remote and inaccessible to direct observation, experiments offer crucial, physically consistent models capable of guiding our understanding and complementing numerical simulations. If we can fully understand the fluid dynamics of the laboratory model, we may eventually fully understand the original. Experimentally reproducing rotating thermal convection relevant to planetary interiors comes with very specific challenges, in particular, modelling the central gravity field of a planet that is parallel to the temperature gradient. Three distinct classes of experiments have been developed to tackle this challenge. One approach consists of using an alternative central force field such as the electric one. This comes with the caveat that these forces are typically weaker than gravity and require going to space. Another method entails rotating the device fast enough so that the centrifugal force exceeds and effectively supersedes Earth's gravity. This mimics the equatorial and lower latitude regions of a planet. Lastly, by using the actual lab gravity aligned with the rotation axis, insight into the polar and higher latitude regions is gained. These experiments have been continuously refined during the past seven decades. Here, we review their evolution, from the early days of visualising the onset patterns of convection, over central force field experiments in spacecraft, ultrasound velocity measurements in liquid metals, to the latest optical velocity mapping of rotating magnetoconvection in sulfuric acid inside high-field magnets. We show how innovative experimental design coupled with emerging experimental techniques has advanced our understanding of planetary interiors and helped us paint a more realistic, detailed picture of planetary interiors, including Earth's liquid metal outer core. \nRésumé. Les intérieurs de toutes les planètes du système solaire sont constituées de couches internes, dont la pluspart sont fluides. Quand ces couches sont sujettes à un gradient de température ou de composition chimique super-adiabatiques, un écoulement convectif s'installe qui transporte chaleur et moment angulaire. De plus, les planètes tournent extrêment vite. Ainsi, le processus-clé qui sous-tend l'évolution des planètes est la convection en rotation. Alors que les intérieurs de planètes sont eloignés et impropres à l'observation directe, les expériences de laboratoire présentent des modèles cohérents à même de guider notre compréhension et de complémenter les simulations numériques. Si nous parvenons à comprendre la mécanique \ndes fluides des modèles de laboratoire, nous finirons peut-être par comprendre l'original complètement. Cependant, reproduire expérimentalement une convection en rotation pertinente pour les intérieurs de planètes présente une gageure en soi, ne serait-ce qu'au niveau de la modélisation de la gravité centrale d'une planète et de son gradient de température parallèle. Trois classes d'expériences distinctes sont apparues pour relever ce défi. Une première approche consiste à utiliser une autre force centrale, telle que la force de Coulomb. L'inconvénient est que ces forces sont typiquement plus faibles que la gravité ambiente et nécessitent donc d'aller dans l'espace. Une autre méthode implique de faire tourner le dispositif suffisament vite pour que la force centrifuge dépasse et même remplace la gravité terrestre. Cette technique imite les conditions des regions équatoriales et à faible latitude d'une planète. Finalement, utiliser la véritable gravité terrestre alignée avec l'axe de rotation nous renseigne sur les régions polaires et à haute latitude. Ces expériences ont été perfectionnées au cours de plus de sept décenies. Nous parcourons ici leur évolution, depuis les premières visualisations des motifs au départ de la convection, en passant par les expériences à gravité centrale conduites à bord d'engins spatiaux, l'avènement de la vélocimétrie à ultrasons dans le métaux liquides, jusqu'aux dernières cartographies de champs de vitesse par méthodes optiques dans des écoulements de magnoconvection en rotation dans de l'acide sulphurique à l'intérieur d'aimants à champs fort. Nous montrons comment des concepts expérimentaux innovants couplés aux techniques expérimentales émergentes ont poussé notre compréhension des intérieurs de planètes et nous ont aidé à en dépeindre une image toujours plus détaillée et plue réaliste, en particulier du cœur liquide de la Terre. \nKeywords. rotating convection, experimental fluid mechanics, measurement techniques, planetary interiors, turbulent convection. \nFunding. A.P. is supported by EPSRC (grant EP/X010937/1) and the Leverhulme Trust (grant RPG-2017-366). S.H. is suported by the UKRI Horizon Europe guarantee, which was selected by the European Research Council (ERC), grant no. EP/X034402/1 (MAGNADO). SH further received funding by EPSRC, grant no. EP/V047388/1. \nElectronic supplementary material. Supplementary material for this article is supplied as a separate archive available from the journal's website under article's URL or from the author. \nThis article is a draft (not yet accepted!)", '1. Introduction': 'In 1906, Richard D. Oldham [1], wrote \'Of all regions of the earth none invites speculation more than that which lies beneath our feet, and in none is speculation more dangerous: yet, apart from speculation, it is little that we can say regarding the constitution of the interior of the earth. [...] the central substance of the earth has been supposed to be fiery, fluid, solid, and gaseous in turn, till geologists have turned in despair from the subject, and become inclined to confine their attention to the outermost crust of the earth, leaving its centre as a playground for mathematicians\' . Then the seismograph changed geophysics forever. Oldham analysed earthquake data and was indeed able to see inside the Earth. He concluded that the Earth had a mantle and core based on different travelling behaviours of seismic waves. In 1936, Inge Lehmann further proved seismologically that there was an inner solid core [2]. \nSeismology provides us with an accurate measure of the core mantle and inner core boundaries (CMB, ICB) and unequivocal evidence that the outer core is liquid. Recent advanced seismological techniques have been revealing the detailed structure of Earth\'s mantle and inner core. They also complement other approaches, e.g. numerical simulations, high-pressure experiments or mineral physics modelling [3-8]. The outer core is a very different story. Its liquid nature makes it less amenable to seismology-based constraints. The main reason lies in the outer core\'s comparatively low viscosity and subsequent lack of shear strength. The outer core has a viscosity of about ν oc ≈ 10 -6 m 2 /s which is about 10 24 and 10 16 times lower than those of the mantle [9] and the inner core [10], respectively. \nThe low viscosity of Earth\'s outer core has another crucial consequence: Rotation matters. The relative importance of rotation is expressed through the Ekman number, Ek = ν /(2 Ω H 2 ), where \nν is the viscosity, H is the layer thickness, and Ω is the rotation rate. Earth\'s rotation rate is about Ω E = 7.27 × 10 -5 rad/s, and the mantle, outer core and inner core have respective thicknesses of Hm = 2885km, Hoc = 2270km, and H i c = 1216km. This gives Ekm ≈ 10 9 , Ekoc ≈ 10 -15 and Ek i c ≈ 47, respectively. Throughout the entire interior of the Earth, convection drives motion and transports heat and momentum. But the high Ek in the mantle and inner core, means convection in those two regions is practically unaffected by rotation, whereas the low Ek in the outer core implies that convection there is strongly constrained by rotation [11-14]. Another distinct feature of the outer core is that it is both made of electrically conducting metal and liquid. Because of it, convection is vigorous enough to power dynamo action and so generates and sustains Earth\'s magnetic field. \nIn short, the outer core is dynamically very different compared to the mantle or the inner core. It is the layer of the Earth that is least accessible through conventional means, in particular, seismology. Hence, the outer core is still largely a \'playground\' for mathematicians, numericists, and, importantly, experimentalists. This by no means implies that it is all mere speculation or that we should not study the problem. Instead, to account for the large uncertainties on the actual conditions in Earth\'s interior, we need to create models that are rigorously built upon the known principles of mathematics, physics, and most crucially fluid dynamics. But as Willis puts it [15], "[. . . ] it is less difficult to imitate one of nature\'s processes than to understand either the imitation or, through it, the original" . In the study of outer core dynamics, it is experiments that most reliably ensure that our understanding is built on firm ground, and shall be the focus of this review. \nWe will first discuss briefly the theoretical framework that is used to understand both the laboratory experiments and the planetary core processes these experiments aim to \'imitate\' . Then, we give an overview of the major design principles and their historical development. In the main part, we review individual experiments, the challenges involved in experimental endeavours and the engineering strategy to tackle them. We shall also see how boldly extrapolating these models to the Earth, and also, other planets encourages further exploration along this path while flagging up its limitations at the same time.', '1.1. Governing equations and control parameters': 'To understand the challenges in bringing planetary physics to life in a small contraption at laboratory scale, we shall start by summarising the main theoretical ingredients that both are usually assumed to share, i.e. rotating thermal convection. The most common and simplest framework describing rotating convection experiments exploring planetary interiors, are the Navier-Stokes equations complemented by the transport equation for temperature T . The basic configuration is the Rayleigh-Bénard convection (RBC) one [16,17], made of a fluid layer, heated from below and cooled from above (in the sense of gravity) with a superadiabatic temperature gradient. The fluid in planetary cores modelled by lab experiments, is usually assumed to be Newtonian and to be described within the Oberbeck-Boussinesq approximation [18-20]. Thus, its material properties, kinematic viscosity ν and thermal diffusivity κ are homogeneous, temperature- and pressure-independent. The density is approximated as a Taylor expansion around T = Tm up to first order, i.e. it is assumed to be linearly dependent on the temperature, \nρ = ρ m (1 -α ( T -Tm )), (1) \nwith α being the constant isobaric expansion coefficient, α ≡-1 ρ ∂ T ρ fl fl p . The subscript m denotes the reference value, typically, the arithmetic mean temperature between the top and bottom boundary temperatures, Tm = ( Tt + T b )/2, and ρ m = ρ ( Tm ). The second term in the density is small except in the gravitational and centrifugal buoyancy terms [21-25]. The temperature \nis the actual temperature in experiments and the superadiabatic part of the temperature when considering planetary interiors [26]. Planetary rotation is assumed constant, Ω ˆ e z , and the origin placed at the centre of the planet. The governing equations in dimensional form under these assumptions are: \n∇· u = 0, (2) \nDt u = ν ∇ 2 u -∇ p + 2 Ω u × ˆ e z -Ω 2 r α ( T -Tm )ˆ e r + g α ( T -Tm )ˆ e g , (3) \nDtT = κ ∇ 2 T . (4) \nThe unit vector ˆ e g points in the direction of gravity, e.g. in the spherical radial direction in spherical geometries or in the vertical direction in plane geometries, such as cylinders or annuli; ˆ e r is the cylindrical radial unit vector perpendicular to the direction of rotation ˆ e z . \nThese equations are more meaningful in nondimensional form. Choosing as reference quantities, the fluid layer thickness H , the temperature difference between the hot and cold boundary, in the case of Earth, the inner core boundary (ICB) and core-mantle boundary (CMB), ∆ , the freefall velocity U f f = p g α H ∆ , and the corresponding time and pressure scales, H / U f f and ρ mU 2 f f , we have: \n∇· u = 0, (5) \nDt u = s Pr Ra ∇ 2 u -∇ p + s Pr RaEk 2 u × ˆ e z -Fr γ T r ˆ e r + T ˆ e z , (6) \nDtT = r 1 RaPr ∇ 2 T . (7) \nIn this form, the problem is governed by following nondimensional control parameters: the Ekman number, which expresses the ratio of viscous to Coriolis forces, the Rayleigh number, which is the ratio of buoyancy to viscous forces, the Prandtl number, which is the ratio of viscous to thermal diffusivities, the Froude number, which is the ratio of centrifugal to gravitational forces, and the radius-to-height aspect ratio (or alternatively, the diameter-to-height aspect ratio) \nEk = ν 2 Ω H 2 , Ra = α g ∆ TH 3 νκ , Fr = Ω 2 R g , γ = R H , Γ = 2 R H . (8) \nThe form of eqs. (5-7) brings out three further classical parameters, the free-fall based Reynolds, Rossby, and Péclet numbers, \nRe = s Ra Pr , Ro = s RaEk 2 Pr , Pe = π RaPr . (9) \nThis Rossby number is also often referred to as convective Rossby number to distinguish it from the output parameter. Similarly, also the Reynolds and Péclet numbers can be output parameters when defined with measured velocities [14].', '1.2. Main physical processes': "The eqs. (3) and (6) include both the centrifugal and the standard gravitational buoyancy terms. Both act in a very similar way, that is, denser (colder) fluid moves in the direction of ˆ e r and -ˆ e g , and lighter (warmer) fluid moves in the direction of -ˆ e r and ˆ e g , respectively. Specifically, in a plane geometry, cold fluid moves radially outwards and downwards and warm fluid moves radially inwards and upwards. The Froude number expresses which of the buoyancy terms is dominant [22-25,27]. In experiments, Fr is always non-zero and in some experiments, it is the design principle that Fr ≫ 1 such that the centrifugal acceleration supersedes the gravitational one and so becomes the effective gravity (see section 3). \nIn most planetary settings, however, centrifugal buoyancy is negligible at the largest scale, e.g. for Earth's outer Fr ≃ 1.72 × 10 -3 . Thus, in most theoretical and numerical studies, Fr is set to zero. In this case, the main effect of the Coriolis force, on convective and other flows, can be seen by considering the limit of fast rotation Ek → 0, Ro → 0 in the governing equations: away from boundary layers, the main balance is geostrophic , i.e. between the Coriolis force and the pressure gradient, so the curl of eq. (6) imposes that \n∂ z u = O ( Ek , Ro ) → 0, (10) \n∇ r φ · u = O ( Ek , Ro ) → 0, (11) \nwhere φ is the azimuthal coordinate and ∇ r φ is the two-dimensional divergence in the r -φ plane. Hence, at the leading order, the Coriolis force makes the flow quasi-two-dimensional and quasi-horizontally solenoidal, a result known as the Taylor-Proudman Constraint (TPC) [28-30]. It implies that a cylindrical fluid parcel must conserve its height through motion, or equivalently, that the flow must follow surfaces of constant height along the rotation direction, called geostrophic surfaces . Since the TPC restricts possible fluid motions, the onset of stationary convection takes place at higher critical Rayleigh numbers Rac than without rotation. In a plane Rayleigh-Bénard configuration, Rac ∼ Ek -4/3 , and the non-rotating convection cells at onset give way to much thinner spiral cells, of length scale ℓ c ∼ Ek 1/3 . Importantly, in low-Prandtl fluids ( Pr < 0.68) such as liquid metals, the onset modes are oscillatory [31-36]. The critical Rayleigh number and length scales are then Pr -dependent, namely, Rac ∼ ( Ek / Pr ) -4/3 and ℓ c ∼ ( Ek / Pr ) 1/3 . At higher criticality, the cellular structures turn into convective Taylor columns, and then plumes and geostrophic turbulence (see e.g. the recent reviews by Kunnen [11] and Ecke & Shishkina [37]). \nIn planetary and experimental contexts ( Ek → 0, Ro → 0, Fr may be small or large), unlike in classical RBC, the time-averaged temperature gradient can be misaligned with the gravity and therefore the main pressure gradient, at least locally. This may be due to the Coriolis, the centrifugal or other forces. The misalignment drives a baroclinic flow (see the review by Harlander within this Special Issue). For example, when Fr > 0, there is a flow even at arbitrarily low imposed temperature gradients, or in Earth's core outside the polar region, baroclinic flows naturally arise as the TPC aligns the flow with rotation rather than gravity [38,39]. From the curl of (3), the Coriolis force constrains the baroclinic motion into an azimuthal wind u φ which is controlled by the thermal wind balance . In cylindrical coordinates, and using the previous nondimensionalisation, it reads \n∂ zu φ = Ro GLYPH<181> ∂ r T + Fr γ r ∂ zT ¶ . (12) \nHere, the effects of the first and second terms in the rhs of eq. (12) are illustrated by the two examples mentioned above. In the geostrophic limit, in which Fr → 0, the gradient part of the Coriolis force is O (1) and balanced by the pressure gradient, but its curl part may be balanced by buoyancy, which is O ( Ro ). In this case, the thermal wind does not break the Proudman-Taylor Constraint, even at O ( Ro ), and so follows the geostrophic contours.", '1.3. Planetary and Experimental peculiarities': 'In planetary context, the spherical shell geometry constrains the expression of these processes. Since planets are extremely fast rotators, the flow tends to follow geostrophic contours because of the TPC. Spherical shells being axisymmetric, these are cylinders aligned with the rotation axis, so radial motions are impeded whereas azimuthal flows are favoured by rotation. One of these contours plays a particularly important role: the so-called tangent cylinder (TC) extruded from the equatorial boundary of the inner solid core along the rotation separates polar regions located \nFigure 1. Sketches of the fundamental geometries used for the study of rotating convection in the study of planetary core convection. The gravitational acceleration (or any proxy) g and the temperature gradient ∇ T and their respective direction are indicated by dark green and lavender arrows. The angular rotation is assumed to be vertical, Ω = Ω ˆ ez , and the axis of rotation is marked by the vertical black dash-dotted line. The boundaries are demarcated by blue for the cold boundary Tc and pink for the hot boundary T h . (a) Spherical shell; R i is the inner radius, Ro is the outer radius, and H = Ro -R i is the spherical shell gap width. The grey cylinder of radius R i is the tangent cylinder (TC). The grey annulus is detailed in (b), and the small grey cylinder in the polar region is detailed in (c). (b) Busse annulus ; R i is the inner radius, Ro is the outer radius, H = Ro -R i is the annulus gap width, ¯ R = R i + H /2 = Ro -H /2 is the mean radius, L is the mean height, and χ is the sloping angle of the endwalls. The endwalls are typically thermally insulating. (c) Cylinder; Ro is the radius, H is the height of the cylinder. The endwalls are typically thermally insulating. \n<!-- image --> \nbelow the poles from equatorial regions (see figure 1a). If a fluid parcel was to cross the TC, it would see either a doubling or a halving of the domain\'s height along ˆ e z , which the TPC strongly opposes. Hence the TC acts as mechanical boundary between the polar and equatorial regions. For the same reason, radial motion near the equatorial plane is very strongly suppressed by the jump in domain height there. Thus, because of the Earth\'s solid inner core, the liquid outer core is split into two regions with different types of rotating convection. \nIn the equatorial region outside the TC, the gravity is rather perpendicular to the rotation. Near the equatorial plane the gravity and temperature gradients drive motion in the plane, and therefore perpendicular to Ω ˆ e z . This motion is therefore not as strongly opposed by the TPC. Instead the rotation tends to elongate these cells along ˆ e z and so forms quasi-geostrophic columns extending across the entire northern and southern hemispheres, named Busse columns , after Busse\'s pioneering experimental work with the Busse annulus represented in figure 1(b) [40]. The variation in domain height in the equatorial region and in the Busse annulus incurs a linear damping on the geostrophic structures called β -effect [30, 42]. The reduced domain height at low latitudes prevents Busse columns from entering that region so they tend to remain close to the inner core. Nevertheless the β -effect incurred by the sloping outer boundary forces Busse columns to oscillate radially and form thermal Rossby waves , as sketched in figure 2. Hindman & Jain [43] offer an elegant explanation of their basic mechanism: Considering a column of height L , "If a spinning column near the equator is pushed toward the rotation axis, the column \nequatorial plane, top view \nFigure 2. Sketch of the general propagating behaviour of (incompressible) Rossby waves, adapted from ref. [41]. Visualised is part of the equatorial plane of a rotating spherical shell. Initially, a series of fluid columns is at rest and aligned along the mid radius. If these columns are displaced in a sinusoidal manner, then the vortices that move outwards (indicated by the green arrows) must reduce in length to conserve mass, thus, they get squashed and acquire retrograde vorticity to conserve potential vorticity. Similarly, the vortices that move inwards (indicated by the purple arrows) must extend in length, they get stretched and acquire prograde vorticity. The effect on the neighbouring columns is that the columns to the west are pushed away from the rotation axis and the columns to the east are pushed towards the rotation axis (red arrows). This results in the phase of the initial sinusoidal pattern propagating westwards (prograde) which is known as Rossby wave. \n<!-- image --> \ngrows in height as the chord length of the column\'s axis increases. In an incompressible fluid, this vortex stretching is accompanied by a commensurate narrowing of the column to conserve mass. Subsequently, as the column compresses laterally, the column must spin faster to conserve angular momentumabout its own axis [25,44]. [..] This conservation principle is enforced by the constancy of the potential vorticity. As L increases, [the vorticity component aligned with the rotation axis] must also increase. The resulting induced vorticity causes the neighbouring column to the west to be pushed outward, away from the rotation axis, and the column to the east to be pushed inward, toward the rotation axis. These newly pushed columns conserve their own potential vorticity (i.e., angular momentum) and induce spinning columns further down the belt to also move inward and outward. The result is a prograde-propagating Rossby wave where the spinning columns dance back and forth, toward and away from the rotation axis." One of the main design challenges in imitating this phenomenology arises out of the vertical direction of gravity in the lab, which makes it difficult to reproduce equatorial convection in experiments using Earth\'s gravity. Here the centrifugal forces comes to the experimentalist\'s rescue. Since the region outside the TC is annular, it can be simulated by rotating an annulus with sloping end walls sufficiently fast for the centrifugal force to exert an apparent radial gravity. This was the essence of Busse\'s revolutionary idea [45]. \nIn the polar regions inside the TC, the temperature gradient, the gravity and the rotation are mostly aligned, and topographic effects are minor: the convection there is expected to develop somewhatsimilarly to rotating RBC (RRBC). Experimentally, one would expect to capture its main features by rotating a vertical cylindrical vessel around its axis at sufficiently large speed for the Coriolis force to dominate, as in figure 1(c). Unfortunately, at the scale of the laboratory, the centrifugal force becomes important at rapid rotation, and even more so for cylinders of low aspect ratios Γ = 2 Ro / H [22]. Such a cylinder may represent the entire TC or a smaller, concentric \ncylinder inside it (see figure 1(a,c). A further issue arises from the nature of the TC: while it may be legitimate to model it with a rigid boundary, the thermal boundary condition there are both unknown and crucial: if adiabatic, isothermal surfaces remain horizontal all the way across the cylinder, whereas if isothermal, very strong baroclinicity would drive a flow near the cylinder side wall [38, 39, 46]. Since there is no reason to think that either of these ideal cases applies to planetary cores, actual TCs must be reproduced in experiments. \nFinally, a key question that arises when experimentally modelling either region is the choice of working fluid, a choice that is closely linked to the envisaged measurement techniques and the specific aims pursued in building such devices. The Earth\'s outer core is mostly made of liquid iron, whose physical properties can be estimated from Iron\'s melting point [47, 48]: ρ ≃ 10 4 kg/m 3 , ν ≃ 7 × 10 -6 m 2 /s, κ ≃ 4 × 10 -6 m 2 /s, α ≃ 10 -5 K -1 , so 0.1 ≤ Pr ≤ 1 [49]. Liquid metals would therefore seem the obvious choice but have only been used in a handful of experiments. The main issue with liquid metals besides their direct or indirect costs, toxicity, and chemical reactivity is that they are opaque and so preclude any direct optical visualisation or any of the convenient LASER-based techniques used in transparent fluids to map velocity and temperature fields. For these reasons, high Pr fluids such as water, but also silicon oils and a few more exotic fluids, appear as cheap and attractive alternatives for the purpose of understanding the interplay between convection and rotation in planetary geometries. For the purpose of studying the effect of magnetic fields however, liquid metals are usually the fluid of choice. However, neither liquid metals nor the high-Prandtl fluids mentioned above afford the combination of electromagnetic effects with the advantage of optical visualisation. Sulfuric acid recently emerged as a solution to do just that [46,50-52].', '1.4. Historical evolution through four classes of experiments': "The constraints imposed by terrestrial gravity, led to four fundamental classes of rotating convection experiments that explore planetary interiors. However, with an outer core thickness of H ≃ 2.26 × 10 6 m, a rotation of one revolution per (Earth) day, a mid-thickness gravity of g ≃ 7m/s 2 [49], the regimes of the outer core of Earth are too extreme for any laboratory experiment: Ek ≃ 10 -15 , Ra ≃ 10 22 to 10 30 , Ro ∈ [10 -6 , 10 -3 ], Re ∈ [10 11 , 3 × 10 15 ] and Pe ∈ [10 21 , 10 31 ]. Other planets offer no solace either as they operate in just as extreme regimes [13,49]. Hence, experiments first focused on understanding the fundamentals of convection (onset, regimes, heat flux) to later extrapolate them to these extreme regimes, and incorporate some of the more complex specifics of the Earth (inhomogeneities, magnetic field). A timeline summarising the history of about seven decades of experiments (1953-2024) is given in figure 3, and a large A3 version of it is provided as supplementary material. \nCentral force field. It is extremely hard to produce self-gravitating spheres in miniature form within a terrestrial laboratory environment, thus, proxy-gravitational forces have so far been the only way to generate central force fields similar to Earth's gravity and suitable for studying rotating convection. Examples are the dielectrophoretic force which was utilised in two space experiments (GFFC, 1985-1999 [53] and GeoFlow, 2008-2020 [54]) and very recently the pycnoclinic acoustic radiation force which also works on Earth (Los Angeles, 2023 [55]). These experiments are detailed in section 2. \nCylindrical radial gravity. This class of experiments simulates the regions outside the TC. Since gravity is mostly cylindrical-radial there, these experiments use annular vessels of various shapes: spherical, hemispherical, cylindrical and variations thereupon, detailed in section 3. Guided by these constraints, the first experiments where pioneered by Busse in the 1970's in a rotating \nannulus, with slanted end-walls reproducing the β -effect and focussed mostly on the onset of convection [45]. Simpler cylindrical annuli with flat end-walls were later used to study inhomogeneous heat flux [56] and ultimate convection [57]. The first and only experiment in a full sphere in 1986 is due to Chamberlain [58]. Aside of this, spherical annuli became the geometry of choice for groups in Los Angeles [40], Cambridge [58], Bayreuth [59], Baltimore [60, 61], Grenoble [47] and Maryland [62]. The mid 2000's mark the appearance of experiments on rotating magneto-convection in spherical annuli in Grenoble [63] and Maryland [62], using gallium and sodium, respectively. \nAxial gravity. Experiments relevant to the polar region started with in the convection community, rather than with geophysicists. The combination of axial gravity and rotation, with the geometry of the geostrophic contour makes cylindrical vessels the obvious choice for this type of experiment. Indeed the very first experiment on rotating convection was conducted on the 20 th of November 1953 in Chicago by Fultz, Nakagawa, & Frenzen [64]. Since then, cylinders have remained the experimentalists' favourite to study rotating convection with axial gravity. A very brief summary of the numerous experiments is given in section 4. \nTangent Cylinder Dynamics. The radial and axial strands met in 2003 when Aurnou [65] built the first experiment dedicated to study the TC, with a raised heater inside an hemispherical dome filled with water. It wasn't until 2014 that this idea was pushed when Aujogue et al. used electrically conducting sulfuric acid and the availability of high magnetic fields up to 10 T to produce the study of magneto-rotating convection with PIV visualisations in a configuration directly relevant to planets [38,46,52,66]. This was followed by a much improved rebuild using a cylinder instead of a hemisphere [39] (see section 4).", '2.1. Outer space: The Geophysical Fluid Flow Cell (GFFC) and the GeoFlow experiments': "Spherical geometries with a central force field may at first glance be essential for creating a laboratory model of Earth's outer core, and other geophysical and astrophysical systems. Unfortunately, on the ground, the terrestrial gravitational potential prevents such a set-up. Thus, realistic experiments where the angle between the Coriolis and buoyancy force varies, are not straightforwardly realisable. We first focus on the two experiments that went to space to escape the terrestrial limitations: the Geophysical Fluid Flow Cell (GFFC) [53, 67-69] and the GeoFlow experiment [54, 70-75], shown in figure 4. Neither experiment specifically aimed at understanding Earth's core, but both constitute unique laboratory experiments of (hemi)spherical rotating convection. \nThe main underlying principle of both the space experiments is that if an electric potential is applied to a dielectric fluid, the neutral non-charged molecules are polarised, and a force acts in the direction of the strongest electric field region [54, 68]. The idea was originally suggested by Smylie [77, 78] but Hart et al. [53] later independently realised them for the first time in a spherical experiment. These polarisation forces are also known as dielectrophoretic forces, and are density- and permittivity-dependent. They act in a very similar manner to traditional buoyancy forces resulting from either the gravitational or centrifugal potential, but here the potential is E · E , with E being the electric field. The key advantage is that in spherical geometries these forces act effectively like a central gravity. Hence, in a rotating experiment, the angle electohydrodynamic 'gravity' vector and the rotation vector vary latitudinally in a similar manner as in planetary and stellar interiors. \n<!-- image -->", 'axial gravity': 'cylindrical radial gravity \ncentral force field \ntangent cylinder \n<!-- image --> \nwith magnetic fields \nrcublueHTML0066CC Figure 3. Time line of the laboratory experiments relevant for planetary core convection from 1953 till 2024. As starting point the year of the date of submission of the first publication, or, where available, the date of the first experimental run is used, the end point is the year of the publication of the last publication of the experimental data. The experiments are either referred to by the name of the device or by the place they were conducted if the experimentalists were less creative with their naming convention than with the experimental design. The main experimentalist(s) are also given together with a selection of the relevant publications. The background colours indicate the class of experiment: pink - axial gravity, i.e. either cylinders or cuboid; blue - cylindrical radial gravity, i.e. cylindrical or spherical annulus, hemispherical or spherical shell; green central force fields, i.e. spheres and hemispheres (the dotted areas indicate when the space missions took place); purple - tangent cylinder geometries. The working fluids in the experiments are indicated by the different droplets. If also rotating magnetoconvection experiments were performed, a magnet symbol is added. An A3 scale version of the figure is also provided as a supplementary file. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nsodium ( Pr … \n- 0.010)\n- mercury ( Pr … 0.025) \ngallium ( Pr … 0.025) \nSulfur ( Pr … 0.15) \nhelium ( Pr … 0.49) \nnitrogen ( Pr … 0.7) \nsulfur hexafluorid ( Pr … 0.8) \nwater (2 glyph[lessorsimilar] Pr glyph[lessorsimilar] 10) \n<!-- image --> \n<!-- image --> \nsucrose solution ( Pr … 10) \nnovec ( Pr … 10) \nsulfuric acid ( Pr … 12) \nsilicone oil (40 glyph[lessorsimilar] Pr glyph[lessorsimilar] 1000) \n1-nonanol (125 glyph[lessorsimilar] Pr glyph[lessorsimilar] 180) \nethylene glycol ( Pr … 145) \nglycerol ( Pr … 2500)', '2008-2020 GeoFlow': "Figure 4. Left panels (a-f): the Geophysical Fluid Flow Cell (GFFC) experiment; (a) Space Shuttle Challenger with Spacelab 3 that flew GFFC to space; (b) sketch of the cross section of the GFFC test cell [68]; (c) rotating turntable with power supply, circulation pumps, test cell, and associated circuitry [68]; (d) the full GFFC experiment including optical imaging system [67]; (e,f) unwrapped convection planforms showing banana cells and pronounced polar disturbances, the tangent cylinder is at 43 · , (e) shows north-south fringes (sensitive to latitudinal temperature gradients) and (f) shows east-west fringes (sensitive to longitudinal temperature gradients) [76]. Right panels (g-j): the GeoFlow experiment; (g) photo of the experimental setup with the International Space Station (ISS), image by ESA; (h,i) Wollaston Shearing Interferometry which visualises the thermal gradients based on the changes of the temperature-dependent refractive index, the columnar cells are coloured using automatic pattern recognition [75], (h) spiralling columnar cells and (i) almost straight cells; (j) the full GeoFlow experiment with adaption optics and rotary tray [70]. \n<!-- image --> \nMore specifically, one can approximate the functional dependence of the permittivity to the first order, as for the density in the Oberbeck-Boussinesq approximation (1): \nϵ = ϵ m (1 -γ ( T -Tm )), (13) \nwhere ϵ m is the ambient permittivity and γ is the dielectric variability. The values for the different fluids used in the experiment are given in table 1. Since in all fluids γ ≈ α , the resulting system can be treated with an equivalent version of the Oberbeck-Boussinesq approximation [18, 19]. Further, if the electrical field E behaves like a spherical capacitor [73], we have \nE ( r ) = R i Ro Ro -R i 1 r 2 V rms ˆ e r , (14) \nwhere V rms is the applied AC voltagthe shells. The resulting electro-hydrodynamic 'gravity' is then \nF e = ge γ T ˆ e r with ge ( r ) = 2 ϵ mV 2 rms ρ m GLYPH<181> R i Ro Ro -R i ¶ 2 1 r 5 . (15) \nThus, gravity varies radially but falls off much steeper with 1/ r 5 compared to planetary gravities, closer to 1/ r 2 . This, however, is argued to be only of secondary importance [68] even though the inner boundary gravity is much stronger than at the outer boundary, i.e. g i ≫ go . These values \nare given in table 1. Ra is the usual Rayleigh number, but is defined using the outer value go . rotation rates and vessel size to small values. \nThe Geophysical Fluid Flow Cell (GFFC) flew to space the first time on the 29 th of April 1985 together with two squirrel monkeys, 24 rats and seven astronauts [53, 67, 68]. The principal investigator was John Hart of the University of Colorado at Boulder, it was managed by NASA's Marshall Space Flight Center. The experiment was conducted in the Spacelab, a microgravity laboratory developed by ESA and flown by the NASA space shuttle Challenger. The specific mission was Spacelab 3 (STS 51-B; 29 April-6 May 6, 1985), the first in which a strict low-gravity environment was maintained in orbit, here in a low Earth orbit of 400 km altitude. GFFC 's second space flight in 1995 was with the US Microgravity Laboratory-2 (USML-2) on board of the space shuttle Columbia (STS-73; 20 October- 5 November 1995) [69, 76]. The first mission provided about 110 hours of experimental data mainly in the form of about 50,000 film images. During the second mission 29 separate 6-hour runs (174 hours in total) were carried out. \nThe set-up itself is a spherical shell rotating at up to 3 rad/s, but only its northern hemisphere was used in the experiment, as depicted in figure 4, The inner shell was made of polished nickel of radius R i = 2.402cm This ensured the absence of conductive electric currents that may lead to other and undesired fluid instabilities, as the associated period was short compared to the charges' relaxation time of the working fluid. The southern hemisphere was filled with Teflon to avoid the effect of non-radial electric fields occurring in vicinity of the inner sphere's mechanical support. The working fluid was a dielectric silicone oil Dow Corning with Pr = 8.4. The inner sphere was heated and the outer sphere was cooled with a computer-regulated temperature gradient of up to 20K ± 0.1K. Individual heater elements also allowed to study differential, latitudinally varying, heating, and several experiments were run with a hotter north pole (a cooler pole would have obscured visualisations). As optical techniques shadowgraph and back-focus Schlieren visualisations were used, which give information about the temperature structures. Successive images from the fixed camera only allowed flow reconstruction in weakly timedependent flows but not for more chaotic and turbulent flows. During the second flight, an additional video camera was installed, but it failed at the 19th run. Originally, it was also planned to use spiropyrane, a photochromic dye that is activated by UV light, as a flow tracer for velocity measurements during the Spacelab 3 mission. However, the dye degraded over the time of 1.5 years when it was not accessible during integration into the space module, thus, not permitting the extraction of accurate velocity data. \nThe GeoFlow experiment was very similar in conception, and was integrated in the Fluid Science Laboratory of the European Columbus module on the International Space Station (ISS). This crucial difference allowed for much longer, extensive campaigns with more than 2,500 hours of scientific run time [54,70-75]. It allowed daily data transfer from the orbit and thereby almost real-time data analysis. The GeoFlow experiment went to space the first time on 7 August 2008 and stayed there until January 2009 ( GeoFlow I ), with three follow-up missions (March 2011May 2012; December 2012-May 2013; November 2016-February 2017; ( GeoFlow II, IIb, IIc ). The head of the GeoFlow TeamwasChristoph Egbers from the Brandenburg University of Technology Cottbus-Senftenberg. Unlike GFFC , GeoFlow was a full sphere with inherent limitations due to the heating supply shaft at the south pole [54,70]. The device had an inner radius of R i = 1.35cm and an outer radius of Ro = 2.70cm with the possible rotation rates of Ω = 63rad/s. The working fluid of GeoFlow I was the silicone oil M5 with Pr = 64. For GeoFlow II , the team improved the olfactory aspect and used the alcohol 1-nonanol, known in the cosmetics industry for its lemon fragrance, with Pr = 125 and Pr = 178. Most of the results were for very slow rotation rates that were mainly there to ensure the visualisation of the full latitudinal direction and more relevant to mantle convection [71-74]. Only the GeoFlow IIc experiment focused on rotating convection with parameters comparable to the GFFC experiment, with a minimum Ek = 2.64 × 10 -3 , and \nmaximum Ra = 1.59 × 10 5 . The fluid 1-nonanol also allowed to study both convection induced by the temperature difference between the spherical shells and due to the internal dielectric heating [75]. The flow visualisation technique used was Wollaston prism shearing interferometry which showed only projections of the thermal structure, since safety and weight limitations did not permit the use of tracer particles or larger optical systems [75]. Exemplar pictures are shown in figure 4. \nBoth experiments, GFFC and GeoFlow studied the stability properties, the formation of patterns and the transition between states, including chaos and turbulence. Many of the results had rather high values of Ek and Ro , as well as low values of Ra , thus, many of the results are practically in a rotation-unaffected regime [11,12,14]. These flows show a tessellate pattern, also found in more recent numerical simulations of non-rotating convection [79, 80]. The GeoFlow experiments also show mantle-like plumes influenced by non-Oberbeck-Boussinesq effects [71-73,81]. \nIn the rotating regime, the main results was the confirmation of the existence of columnar convection in radial gravity ('banana cells' [25]) in the equatorial region. Experiments also captured their drifting in prograde direction, illustrated in figure 4 [53,68,75]. In both experiments, columnar convection was very robust, likely because of the relatively high Prandtl number [11,82-86]. The experiments, especially GFFC , also showed that at higher supercriticalities, retrograde midlatitude convection modes develop, the TC becomes convectively active, and later on these midlatitude and polar modes become unstable. Likely, there were two effects at play here, a loosening of the rotational constraint, i.e. moving closer to the non-rotating regime expected at higher supercriticality [39]. The unique specificity of GFFC was that they could also observe the interactions of the columnar equatorial cells with the retrograde-propagating mid-latitude convection modes, which can only be captured with spherical radial gravity. These interaction initially caused a wavering, and finally the erosion of the columnar equatorial cells. They also found that the virtual boundary of the tangent cylinder broke down, with columnar convection only within 20 · of the equator whereas the tangent cylinder extended up to 43 · . The no-slip boundary conditions (in addition to the not very extreme control parameters) were likely the reason for not observing any zonal flows. However, these boundary conditions are arguably also most relevant for Earth's core than for gas giants. Many of the results are indeed reminiscent of more \nTable 1. Material properties used in the space experiments, ρ m is the mean density, ϵ m the mean permittivity, α the expansion coefficient, γ the dielectric variability, V rms the (maximum) applied a.c. voltage, and go and g i the values of the electro-hydrodynamic 'gravity' at the outer and inner radius of the (hemi)spherical shell, respectively. The silicone oil Dow Corning 0.65 cSt 200 Fluid was used in the GFFC experiment, silicone oil M5inthe GeoFlowI experiments, and 1-nonanol at different mean temperatures during the GeoFlow II experiments. Note that ϵ 0 = 8.854 × 10 -12 CV -1 m -1 is the vacuum permittivity. Taken from references [68] and [73,74]. \nmodern numerical spherical simulations of rotating thermal convection [80,87,88]. They further also showed that spherical shell thermal convection may be hysteretic, finding sometimes the same and sometimes different flows patterns depending on whether they increased or decreased Ra [69, 76]. The GFFC experiment also explored latitudinal inhomogeneous heating. They observed spiralling convection when the north pole was heated, which were interpreted as evidence for baroclinic waves. They showed that with increased differential heating, the interaction of the equatorial columnar cells with mid-latitude waves lead to triangular waves and ultimately, with turbulent structures moving downward from the pole, to full turbulence.", '2.2. ... and elsewhere: Kelvin forces and pycnoclinic acoustic gravity': "The main limitation of the space experiments' applicability to planetary core convection are the relatively low Ra and high Ek required to minimise centrifugal effects. is that the poor electric conductivity of dielectric materials renders the study of MHD effects practically infeasible. \nHaving said that, it may be worthwhile to embrace Coriolis-centrifugal convection, and so thanks to the combination of Ω 2 r and g , spot tornado-like vortices as they occur in planetary and stellar atmospheres [22,23,89-93]. \nFerrofluids offer an alternative route to create a central force field, with the benefit that some of them are also electrically conducting. Experiments of this kind have been conducted in non-rotating thermo-magnetic convection where spherical gravity is mimicked using permanent magnets as an inner core, and a ferrofluid is filling the outer core [94, 95]. However, neither rotating convection nor magnetoconvection have so far been realised. The idea is here that a the magnetised fluid is attracted towards higher magnetic fields. This attracting body force is referred to as Kelvin force and mimics buoyancy. The magnetisation depends on both the magnetic field and the temperature, a colder fluid is attracted more strongly than a warmer fluid. This is expressed through the pyromagnetic coefficient which plays the role of the the expansion coefficient. \nAcoustic gravity is another clever way of creating thermal rotating convection with a central force field in an Earth-bound laboratory setting [55]: Sound, when averaged over many cycles, exerts a force on density gradients in a gas. The so-called pycnoclinic acoustic radiation force -〈 v 2 ac 〉∇ ¯ ρ /2 yields an acoustic gravity of g ac = ∇〈 v 2 ac 〉 /2 [96, 97]. Here, ¯ ρ is the time-averaged density and vac is the acoustic velocity. Koulakis et al. use a rotating spherical plasma bulb with a radius of 1.5cm and filled with a weakly ionized sulfur gas. The gas is heated volumetrically by microwaves. The rotation is essential to guarantee quiescent conditions of the plasma. Then, the amplitude modulation of the microwave power generates a high amplitude, spherically symmetric acoustic standing wave. This creates an acoustic gravity that can briefly reach values of up to 1000 times Earth's gravity, so centrifugal effects become negligible. Gravity changes sign roughly at half the bulb radius, so confining the convective zone to the outer region, as in stars and Earth's core. While the parameters are only slightly more extreme than the space experiments, density differences of a factor of about two make it possible to study compressible rotating spherical convection too.", '3. Cylindrical radial gravity': 'Given the difficulties in modelling spherical rotating thermal convection with central force fields, geophysicists, planetary scientists and fluid dynamicists had to resort to alternative means to model planetary core convection. A very successful branch of laboratory research in this regard relied on centrifugal buoyancy to mimic equatorial and low latitude convection.', '3.1. The Busse Annulus': "The cylindrical annulus geometry was first conceptualised as a laboratory device with direct application to Earth's core by Busse in 1970 [25]. The underlying idea here is to use the cylindrically radially outward directed centrifugal force and then cool the inner boundary and heat the outer one. This ensures that buoyancy acts in the right direction, as sketched in figure 1. Only the product of gravity and temperature gradient matters physically. Busse let the centrifugal acceleration exceed Earth gravity by a factor of two or three [41] and envisioned slanted boundaries, where the height varies with cylindrical radial distance from the vertical rotation axis. The resulting β -effect, creates slowly drifting columnar thermal Rossby waves. These are nowadays also known as Busse columns and have been argued to exist in planetary cores [25]. Busse first derived a simple analytical model for the onset Rayleigh number, drift frequency and wave number in the small gap approximation with asymptotically small endwall slopes. He then materialised his theoretical ideas into an innovative laboratory set-up [45]. The concept of the Busse annulus turns the arduous three-dimensional spherical problem into a tractable two-dimensional one and offers a simple access to the spherical rotating convection. \nBusse (and collaborators) subsequently fleshed out the original idea [45, 98-107], including the application to magnetoconvection [108,109] and the geodynamo [110] (see also the reviews [41,111-113] and subsection 3.2 on the spherical geometry). The Busse Annulus has been subject to many theoretical and numerical studies up to recent days [114-119]. In particular, Calkins et al. [118] lifted the original restriction to small slopes and near criticality. He introduced an asymptotically reduced three-dimensional set of equation for quasi-geostrophic convection, thus, establishing a closer connection to spheres and spherical shells. \nHere, we focus on the experimental aspects of the Busse Annulus [45, 98, 99, 105, 112]. A sketch of the first laboratory realisation [45] is shown in figure 5 (a). Two concentric cylinders were mounted on two circular end plates. The outer cylinder was heated through a circulating thermostatically controlled reservoir enclosed in an outer nearly cubical plexiglas box. The inner cylinder was cooled through circulating water through two centred hollow shafts mounted on the endplates, and ball-and-socket couplings connected the stainless steal shaft to the cooling tubes. Baffles at the inside of the end plates ensured an efficient distribution towards the inner annulus wall. The shafts also served as rotation axis, and rotation rates of up to 400 rpm (41.9 rad/s) were achieved. The temperature difference was measured using a thermocouple in each of the water baths. Since the experiment concerned the onset of convection, typical temperature differences were less than 1 K. The inner cylinder was made of aluminium and three different sizes were used, R i = {13.34cm,3.81cm,3.18cm}. The outer cylinder was made of acrylic plastic (plexiglas) to allow for optical access, and the outer plexiglas cube helped to avoid optical distortions. The used radii were Ro = {13.94cm,4.775cm,4.74cm. The endwalls were made of Teflon and three different sloping angles were used χ = 0 (corresponding to horizontal plates) and χ = 22.5 · and 45 · . The mean heights were L = {28.6cm,6.97cm,6.69cm,1.96cm,0.95cm,0.45cm}. The working fluids were silicon-oil and water. Small amounts of Kalliroscope were used for flow visualisation. These are nearly neutrally buoyant particles that align with the shear, stroboscopic light was used to make the shear patterns visible and to measure the rotation rate. The onset of convection was determined visually. \nQuantitative measurements involved measuring a buoyancy parameter B versus an inverse Ekman number Ek = ν /( Ω L 2 ) (bar the factor 2 in both cases). In modern terminology, their buoyancy parameter is in fact the square of the (convective) Rossby number. This can be seen by recalling that Ro = τ r ot / τ buoy , i.e. the ratio of a rotational time scale, τ r ot = 1/ Ω andthebuoyancy time scale, τ buoy = H / p α ∆Ω 2 ¯ RH , defined in analogy to the usual free-fall time scale, with the temperature difference ∆ = T h -Tc , Ω 2 ¯ R being used as centrifugal gravity, and the characteristic \nlength scale being the gap width H . Thus, we have \nRo 2 = B = τ 2 r ot τ 2 buoy = α ∆Ω 2 ¯ RH H 2 Ω 2 = α ∆ ¯ R H . (16) \nAt the onset of convection, Busse & Carrigan saw the first experimental evidence of the retrograde drifting columns that where slightly tilted and helical. Their results showed decent agreement with the small gap linear theory [25]. Qualitatively, it was also found that the wavelength of the instability decreased with increasing rotation rate in the sloping endwall cases. The instabilities in this case were typically superposition of several waves and the columns showed beating phenomena. Dedicated experiments sought hysteresis and subcritical convection but found none of them. The β -effect arising out of the interplay of the Coriolis force with the top and bottom conical end walls was found to inhibit the convective instability compared to the flat case that develops columns, see figure 5 (b). At high rotation rates the experiment strikingly exhibits geostrophy, in agreement with the TPC, which imposes that the flow be z -independent in these regimes. \nZonal flows. For the observations of zonal flows, Busse & Hood modified the Busse annulus ( L = 12.6cm, H = 2.46cm, ¯ R = 3.51cm) to include radial curvature of top and bottom boundaries, either convex or concave [99]. The mean flow was measured by releasing electrolytic dye into the working fluid (water) by frequent pulsed currents, following a method devised by Baker [120], and taking photographs at intervals of 10 s. \nA similar experiment by Azouni et al. [98] used mercury and water as as working fluids. The outer cylinder was replaced by anodized aluminium to avoid the thermal loss through the acrylic sidewall ( L = 14.48cm, ¯ R = 3.917cm, H = 3.917cm and χ = 47.4 · ). They measured drift rates and amplitudes of the convection columns by combining and correlating five thermistor probes in the equatorial plane and spaced in azimuth. Three were embedded in the inner cylinder flush with the outer surface, two others were glued to the outer cylinder. \nEarth's gravity creates a thermal wind and a meridional circulation scaling as Ω -1 and Ω -1/2 , respectively. The thermal wind is antisymmetric relative to the equatorial plane, being retrograde in the lower part and prograde in the upper part, thus, it does not affect the dye injected in the equatorial plane too much. The meridional circulation is noticeable, on the other hand. It leads to an upflow near the hot outer wall and downflow near the cold inner wall, thus, there is an additional prograde zonal flow at the outer wall and a retrograde flow at the inner one. \nBoth experiments confirmed the existence of a mean zonal flow. The main differential rotation was argued to stem from a mean flow instability due to Reynolds stresses generated through strong mean zonal shear, shown schematically in figure 6. The convective columns in figure 6(a) get slightly tilted, e.g. by small fluctuations. This creates Reynolds stresses 〈 u ' r u ' φ 〉 , where u ' r and u ' φ are the fluctuating part of the radial and azimuthal velocity component, respectively. Depending on the initial tilt, either direction of the evolving mean flow is possible. If tilted in the prograde direction, figure 6(b), then prograde momentum (purple arrows) is carried outwards and retrograde momentum (green arrows) is transported inwards. Thus, the flow is retrograde at the inner boundary and prograde at the outer boundary. The resulting flow increases the initial tilt leading to a feedback process sustaining a mean flow, figure 6(c). Thus, there is a differential rotation in which the outer fluid rotates faster than the inner one. If the columns are tilted in the opposite direction, the mean flow direction is reversed. For similar reasons, for convex endwalls, thermal Rossby waves propagate faster on the outside than on the inside, thus, the columns spiral outwards and the mean flow is retrograde at the inner boundary and prograde on the outer boundary. The opposite happens for concave endwalls [41, 99]. Viscous stresses oppose the differential rotation and lead to an equilibrium. The differential rotation was argued \n(a) \n<!-- image --> \nFigure 5. (a) Schematic vertical cross-section of Busse and Carrigan's original annulus apparatus, adapted from ref. [45]. The north east lines pattern indicates the rotating parts and the crosshatch dotted pattern indicates the non-rotating parts, in particular, the plexiglas cube. The radius and height of the inner and outer cylinders were adaptable and varied during the experimental campaigns. (b) Photograph of two annuli stacked on top of each other, scale is in cm, adopted from ref. [45]. The upper part shows an annulus with constant height, showing strong convection columns. The lower part shows an annulus with conical endwalls, that results in an axisymmetric flow at the same parameters. The working fluid was water at 30 °C and Kalliroscope was used for visualisation. \n<!-- image --> \nto be rather insensitive to the cylindrical annulus sidewall, and, hence should also be found in a spherical shells [41]. Azouni et al. [98] found drift rates in agreement with his simplified theory, whereas the amplitudes decreased with increasing Rayleigh number. \nMore recent developments. Busse continued investigating flows in the Busse annulus . He and his collaborators found quasi-geostrophic chaotic states in air [112]. They also devised an elegant method to visualise convective patterns using thermochromatic liquid crystals in highPr fluids [105]. At such high Pr , the effects of rotation were much lower. They found different types of patterns, such as knot and hexaroll, which is predicted theoretically [107], but also patterns that still lack a theoretical description such as oblique rolls.", 'equatorial plane, top view (a)': "Figure 6. Creation of mean zonal flows in spherical shell and annuli geometries. (a) Convective columns, initially at rest; (b) Columns get tilted, here in prograde direction, e.g. by small fluctuations which create Reynolds stresses. Prograde momentum (purple arrows) is carried outwards and retrograde momentum (green arrows) is transported inwards. (c) Resulting prograde flow at the outer boundary and retrograde flows at the inner boundary increases the initial tilt leading to a feedback process sustaining a mean flow. \n<!-- image --> \nThe Busse annulus without sloping endwalls was later replicated by Jiang et al. [57, 121]; they rediscovered the centrifugal force as a proxy for gravity and called it supergravity. Their achievable parameters were comparable to Busse's original design, with rotation rates between 211rpm to 705rpm, focussing on the strongly supercritical regime. While they also rediscovered the mean flow instability, the constant radial height does not permit e.g. Rossby waves and thus, makes the experiment less geophysically relevant.", '3.2. Spheres, spherical annuli and hemispherical shells': 'Spherical geometries are a natural improvement of the Busse annuli , as they capture a more realistic β -effect and, in some setups, the Stewartson layers wrapped around the TC [30, 122]. Here too, centrifugal gravity is used to mimic the gravitational field in the equatorial region, still at the expense of the polar regions. As for the Busse annulus , since the centrifugal gravity points outwards, the temperature difference has to be reversed with the outer spherical wall held at constant hot temperature by either water [123] or air cooling [124]. \nIn spherical shell geometries, the radial temperature gradient is controlled by cold fluid (water, or even Kerosene [62]) circulating through the inner sphere. A solid shaft must be fitted to connect the inner sphere to the cooling circuit and to hold the solid sphere at the centre of the outer sphere. This, however, finalises the sacrifice of the polar regions. Because of this shaft, we refer to setups of this category as the spherical annulus configuration. Another non-planetary feature of the spherical shells is the misalignment of the temperature gradient and gravity that increases with latitude. This source of baroclinicity combined with the Coriolis force incurs an artificial azimuthal wind, and results must therefore be interpreted keeping its influence in mind [123]. \nSeveral experiments were built on this principle with variants trying to address some of its shortcomings. These experiments are classics of geophysical rotating convection and have been extensively reviewed [125-127]. We shall only recall their main outcomes and highlight some specificities of the experimental approach itself. \nExperiments at moderate Pr : water and oil. The first spherical annulus experiment was designed by Busse & Carrigan [40, 123], whose original sketch is reproduced in figure 7(a). The working fluids were water and ethylene glycol and the inner core radius was varied by using different inner spheres. High rotations up to 1000 rpm ensures the gravity is effectively cylindrical radial. The outer sphere (inner radius 10 cm) is made of transparent plastic and cooled by a temperature-controlled water bath. The flow was diagnosed by seeding the fluid with reflective platelets that align with the shear and taking snapshots through the outer sphere. From these visualisations, the authors were able to detect the onset of convection, and characterise the size of the onset structures. They mostly recovered Busse\'s linear theory for the onset of convection in a rotating shell with central gravity in the small-gap limit, and identified the onset structures as being Rossby waves originating near the inner sphere, at Rac ∼ Ek -4/3 , see original visualisations in figure 8 (a). The critical wavenumber of the Rossby waves followed the theoretical scaling law, but with a lower prefactor. A discrepancy between the experiment and the small gap theory arises due to presence of Stewartson layers developing along the TC. The Stewartson layers tend to stabilise the Rossby waves because they are precisely located where the waves originate. This effect becomes negligible in the limit Ek → 0. \nChamberlain & Carrigan [58] later modified the setup, by removing the inner sphere and thereby the Stewartson layers [58], see figure 7(b). The outer bath temperature was controlled to increase linearly in time to maintain a constant temperature gradient in the absence of inner cooling. After a short transient, the lag in the temperature response of the fluid inside the sphere leads to a constant temperature gradient there. The radial temperature profile is then the same as if the heat source was distributed in volume, up to a background temperature linearly increasing in time. This radial temperature profile was monitored by radially aligned temperature probes. The critical Rayleigh numbers were found in better agreement with Robert\'s linear theory for an internally heated sphere [132] than Busse\'s small-gap linear theory [25]. As in the previous experiment, however, the centrifugal wind incurred by the high-latitude baroclinicity prevented the authors from observing the retrograde drift of the Rossby waves. This led them to suggest \nFigure 7. Evolution of the spherical shell experiments since 1976: (a) 1972-1983: Busse & Carrigan\'s original spherical shell convection device with spherical inner cold boundary [40, 123], (b) 1984-1986: Chamberlain & Carrigan \'s modification of Carrigan & Busse original setup, with no inner sphere and linearly increasing outer wall temperature [58] (c) 1990-1993: Cordero & Busse\'s [59] (d) 1001-1994: Cardin & Olson (1994)\'s air-cooled setup [124] (e) 1999-2003: Sumita & Olson\'s modified version of Cardin & Olson\'s setup with a hemispherical annulus [61,128-130] (f) 1999-2007: Aubert and collaborators\' water and liquid metal experiment [47,131] \n<!-- image --> \nexperiments using a cylindrical annulus to eliminate that effect as Busse and collaborators did [98,99]. \nThe first measurement of the drift of Rossby waves with a spherical geometry came from Cordero & Busse [59] using a hemispherical shell. Baroclinic effects were minimised by rotating the hemisphere at a speed where the sum of gravity and centrifugal forces produced an apparent gravity with parabolic iso-values nearly parallel to the curvature of the sphere. The outer sphere was made of glass and heated by an external thermostatically-controlled bath of water. The inner sphere was made of brass with an internal water-cooling circuit. Using the same visualisation method as in spherical experiments [40] and thermistors, the authors captured the variations of the decay of the drift velocity with the Rayleigh number as well as the appearance of secondary rolls of larger wavelength near the outer sphere. \nFigure 8. Patterns of rotating convection in a spherical gap by order of criticality: (a) Rossby waves at the onset of convection originally observed by Busse & Carrigan (side view) [40], (b) spiralling columns at e Ra = 5.9 (top view through the transparent top lid of the hemispherical vessel) [128] (c) dual convection at e Ra = 19.3 (top view as for (b)) [128] (d) chaotic convection at e Ra = 50 (side view) [124], (e) same as (d), but top view. \n<!-- image --> \nThe next series of spherical annulus experiments was initiated by Cardin & Olson [60, 124], see figure 7(c). They also uses water but with a slightly larger outer sphere of radius 15 cm. Unlike previous spherical annuli, the outer temperature was thermostatically controlled at room temperature by the air current induced by the sphere\'s rotation. The rotation up to 400 rpm was somewhat lower than in Carrigan & Busse\'s experiment, so the proxy-gravity was not entirely cylindrical-radial but also had a significant vertical component. A new visualisation system relied on snapshots synchronised with a strobe light illuminating vertical planes and a fluid seeded with rheoscopic flakes to identify the structures. In the horizontal planes, the flow was visualised by releasing dye at the inner sphere. The new visualisation system made it possible to investigate the regimes beyond the onset of convection. They found that for moderate and higher supercriticality, the classical Rossby waves give way to a chaotic columnar regime. The flowis quasi-geostrophic, made of irregularly distributed columns extending over the entire shell height. Its basic pattern is made of retrograde vortices produced at the inner core boundary extending outward into prograde spiralling streets of retrograde vortices (see figure 8 (d)). This regime is consistent with equatorial mirror-symmetry [133] and non-periodic magnetic flux [134] inferred from geomagnetic data. Scaling up experimental velocities in this regime to Earth parameters yields estimates of 0.1 cm/s for the convective velocity, consistent with estimates from geomagnetic variations [135]. However, the extrapolated column size of 20km-radius columnsis unobservable and inconsistent with the large-scale geomagnetic data. Cardin & Olson attribute this discrepancy to the action of the Lorentz force in the core. \nThe transition to the chaotic regime and the fully developed stages of convection were further investigated by Sumita & Olson [128]. They modified Cardin & Olson\'s setup into a lower hemisphere, with copper inner and outer boundaries, closed at the equatorial plane with a transparent plexiglas lid. The copper parts enabled a better control of the temperature with a more homogenous distribution at the boundary. The lid offered a better visualisation window. and its near-thermally insulating properties left the mostly radial conducting thermal gradient unaffected. The temperature was monitored by thermistors inserted at adjustable depths through the lid. They rotated the experiment at a fixed speed of 206 rpm to generate a parabolic gravity iso-potential, with radial gravity seven times greater at the equator than at the pole. Under these conditions, Sumita & Olson observed the progressive filling of the gap by Rossby waves drifting in the retrograde direction for f Ra = Ra / Rac ≤ 8, see figure 8(b), as well as a radial increase in their azimuthal wave number: the effect is driven by the increasing slope enforcing an ever stronger TPC suppressing radial motion. At higher forcing f Ra > 8, this effect splits the flow into a turbulent inner region and strongly suppressed motion in the outer region where Rossby waves are expelled, a regime they coin dual convection , illustrated in figure 8 (c). \nFurther experiments in the same setup using silicon oil and water as working fluid enabled Sumita & Olson to explore much higher levels of supercriticality, up to f Ra = 612 [130]. In these regimes, the turbulent convection homogenises the temperatures to a nearly isothermal state with a thin boundary layer near the inner boundary and heat transfer follows a scaling law of the form Nu ∼ Ra 0.4 . This law, and a comparison of experimental thermal fluctuations to theory [124] led the authors to identify this regime as geostrophically turbulent. \nExperiments at low Pr : liquid metals. The advent of Ultrasound Doppler Velocimetry (UDV) [36, 136, 137], which unlike the name suggest does not rely on the Doppler effect [138], made it possible to measure velocities in the bulk flow of core fluid-like liquid metals. \nAubert and collaborators [47] took advantage of this technology to perform experiments in a spherical annulus geometry with a thick central shaft, and no inner sphere, see figure 7(f). The shaft provided cooling through a central cylindrical copper wall, so in planetary terms, Aubert et al.\'s experiment had a cylindrical core and a solid, impermeable TC. They performed experiments both in water and gallium. With water, direct visualisations with Kalliroscope were obtained through the outer sphere made of transparent plastic and maintained at thermostated room temperature by air currents as in Cardin & Olson\'s experiment [124]. With gallium, optical visualisation is not possible and the outer sphere was replaced by a copper sphere heated by a resisting wire wrapped around it. They measured velocities in both fluids across the gap with two probes along two directions within the equatorial plane. From these two measurements, they are able to reconstruct radial (convective) and azimuthal (zonal) velocities. For both fluids, the maximum convective velocity in the turbulent regime scales as ur ∼ ( α gQ / ρ Cp Ω 3 ( Ro -R i ) 4 ) 2/5 , a law that can be recovered by assuming a triple balance between Coriolis, inertia and the Archimedian (buoyancy) force, called CIA balance. Here, Q is the heat flux, Cp the heat capacity at constant pressure of the fluid. Velocities rescaled by this law are higher in gallium than water for the same criticality. The reason is the two-order-of-magnitude lower Pr of gallium. These high velocities drive a strong zonal flow, which forms as the condensate of an inverse energy cascade fed by the convective scales [131]. The condensate materialises at Rhines\' scale, which results from the balance between the β -effect and Reynolds stresses [139]. Energy dissipation at this scale and for the larger scales of turbulence occurs by Ekman friction. However, bold extrapolation of this phenomenology to Earth leads to overestimated velocities by an order of magnitude and too small scales, compared to geomagnetic data. The authors attribute the mismatch, again, to the absence of magnetic field in their experiments. \nGillet and collaborators conducted further experiments with the same setup up to e Ra = 80 \nfocusing on the zonal flow [131]. In the viscous regime of water experiments, the zonal flow scales with the onset convective scales. In the inertial regime reached with gallium, by contrast, they recovered the phenomenology of Aubert et al. [47] with a zonal flow scaling as Rhines\' scale. Based on this phenomenology, they showed that at these larger criticalities, the average zonal flow U φ scales with the rms radial convective flow e Ur as U φ ∼ e U 4/3 r . Unlike early experiments at low criticality in water, where the zonal flow expected from the drift of Rossby waves is weak, the zonal flow observed in this regime is at least an order of magnitude faster than the parasitic thermal wind incurred by the misalignment of temperature gradients and gravity. Hence, in these turbulent regimes at low Pr , the spherical annulus produces a more geophysically-relevant zonal flow than most of the water experiments. \nLastly, Shew & Lathrop [62] built a large spherical annulus (60 cm external diameter), with magnetic runs and dynamo applications in mind. A much larger setup was later built more specifically to create dynamo, which is still under development [140]. However, the Shew & Lathrop experiment remains to date the only rotating convection experiment running with liquid sodium. Measurements rely on thermocouples immersed within the fluid, and velocities are obtained by correlating neighbouring thermocouples and using Taylor\'s hypothesis. They focussed on the turbulent regime and observed small-scale convective motions with a strong, large-scale retrograde azimuthal flow. The local intensity of the zonal flow intensity fits a scaling of u θ ≃ 3.5 Ω α D ∆ T obtained from a simple balance between Buoyancy and the Coriolis force. As one would expect, they observe developed turbulence with weak convective heat transport and a near-diffusive temperature profile. They measured Reynolds numbers in the range Re ∼ 10 3 -10 4 , and a total radial heat flux following a Nu ∼ Ek -1/3 Ra , indicating a rotation-dominated heat transfer [11, 37]. Extrapolating these measurements to Earth leads to an estimated Rayleigh number of Ra ∼ 10 23 and convective velocities comparable to the azimuthal flow of ≃ 2 × 10 -4 m/s, from which they estimated the magnetic Reynolds number in the equatorial regions of the Earth to be Rm ≃ 2 × 10 2 . They identified a "knee" in the energy spectra where energy was injected by convection into the flow, which, extrapolated to Earth corresponds to a turnover time of 30 days, and a size of 1 km, again well below any observable scale. They also derived an estimate for the Joule dissipation in the range of 10 GW to 10 TW.', '3.3. Rotating magnetoconvection in spherical annulus experiments': 'Extrapolating rotating convection experiments to Earth yields estimates that are inconsistent with geomagnetic data, for which the Lorentz force usually takes the blame. It was therefore natural for rotating magnetoconvection experiments to emerge. There are essentially three strategies to go about these experiments: letting the fluid motion itself generate is own magnetic field, putting a magnet around the experiment, or bringing the experiment to a magnet. \nThe first strategy, dynamo experiments, however poses such a challenge in its own that no working concept of a convective dynamo experiment has been put forward yet. There are even claims that such experiments are impossible at laboratory scale [141]. \nThe second strategy has been pursued to date by only two magnetoconvection experiments with a spherical geometry by Shew & Lathrop and Gillet et al. [62,63]. These experiments rely on the rotating convection devices previously built by the Maryland and Grenoble groups [47,62,63]. The main drawback of this strategy is that only relatively weak magnets can be transported and fitted around existing setups, and thus, deliver limited magnetic fields. In the geophysical context, the Lorentz force is typically assessed relative to the Coriolis force by the Elsasser number Λ = σ B 2 0 /(2 ρ Ω ), which for the Earth is deemed to lie between 0.1 and 100 [13, 49, 124, 142, 143]. Shew & Lathrop [62] placed a 3mT Helmholtz coil around their spherical annulus providing Λ ≤ 1.9 × 10 -4 and saw a global decrease in heat flux with increasing magnetic field. Gillet and \ncollaborators [63], created a "hairy magnet", see figure 9, by wrapping 444 turns of a wire around meridional planes and through the central shaft of their spherical annulus to generate azimuthal field of up to 0.03 T, and Λ ≤ 9.94 × 10 -2 . They found a zonal flow following the non-magnetic phenomenology [63], but with larger azimuthal lengthscales and magnetically controlled zonal velocity scaling as ¯ U ∼ ( e Ur e U θ ) 2/3 ∼ e Ur ( l θ / l β ) 2/3 . Hence, both experiments showed the stabilising effect predicted at low Λ [144, 145]. Their relevance to the Earth is, however, questionable considering the Lorentz force may become destabilising at greater values of Λ around unity [27,143,144,146-148]. \nFinally, the third strategy, and the power of large magnets it affords, has only been exploited in set-ups targeting the polar region, namely LEE1 and LEE2, which took advantage of fields up to 10T. These are discussed in section 5. \n<!-- image --> \nFigure 9. Liquid metal experiments: (a) 2003-2005: Shew & Lathrop\'s sodium experiment with a Helmholtz coil generating an axial magnetic field [62] (b) 2001-2007: Gillet et al\'s Gallium experiment with their "hairy magnet" made of a long toroidal wire wrapped around the shell, generating an azimuthal magnetic field of up to 0.03 T [63]. \n<!-- image -->', '3.4. Experiments with inhomogeneous heating': 'In most experiments, great efforts are deployed to keep homogeneous temperature boundary conditions, so as to preserve the ideal character and to make the results universal. Yet, there is evidence that large-scale magnetic flux anomalies are driven by large-scale inhomogeneities at the CMB [149-151] and that they may drive magnetic field reversals [152-156]. Heat flux heterogeneity at the CMB may thus play a role in the interior dynamics. Hart [69, 76] made the first attempt at studying them but suffered from limitations inherent to space experiments, when the heat source failed, as discussed in section 2. Since then, two experiments studied their effect on rotating convection systematically. \nSumita & Olson [61, 129] adapted their experiment [128, 130] and created a local thermal inhomogeneity with a small rectangular heater attached to the outer boundary. Its size and and latitudinal position were varied. This extra heating models a local cold anomaly of the \nEarth\'s CMB where the outward heat flux locally is higher, since the temperature gradient in the experiment is reversed compared to the Earth\'s outer core. The total heat flux was monitored by comparing inlet and outlet temperatures of the inner sphere\'s cooling shaft. The intensity of the anomaly was quantified by the ratio of excess heat flux at the rectangular heater q h to the flux at the inner boundary Q , Q ∗ = q h / Q . The heterogeneity drives a local prograde (eastwards) flow. Two different regimes exist: For Q ∗ < 0.7, in the so-called local locking regime, this flow remains localised near the outer boundary. For stronger inhomogeneities, in the global locking regime, a large-scale spiral with a sharp front develops across the whole gap. The front separates warm and cold regions and is accompanied by a thin jet linking the outer and inner boundaries, that is responsible for an increase of global heat flux and large inhomogeneities at the inner boundary. The regimes are practically independent of the size of the heater but depend on its latitudinal position: at low latitudes, the prograde flow forms but does not develop into a front, whereas at the higher latitudes, the front enters a regime of periodic formation and destruction. Sumita & Olson estimated the heat flux beneath east Asia to be Q ∗ ≃ 2, based on the CMB temperature and its variations [157-159] and seismic models [160]. For this value, global locking is possible, so they proposed that "the existence of this front in the core may explain the Pacific quiet zone in the secular variation of the geomagnetic field and the longitudinally heterogeneous structure of the solid inner core." \nSahoo and Sreenivasan [56, 161] built a cylindrical annulus filled with water (height 37 cm, inner/outer radii 5 cm/14.2 cm), with flat, thermally insulating top and bottom boundaries, mostly radial centrifugal force and PIV in horizontal planes [56, 161]. At the inner boundary the temperature is imposed, while at the outer boundary, the heat flux is imposed by the heater fitted around the annulus. Unlike Sumita & Olson\'s experiment, the inhomogeneity is not local but takes the form of an azimuthal variation of boundary heat flux controlled by dividing the external heater into four independently controlled π /2 sectors. The system spanned inhomogeneities in the range 0 ≤ Q ∗ ≤ 2. Here q h is the amplitude of the azimuthal flux variation. \nExperiments are conducted with alternating high and low fluxes every π (one-fold symmetry) and every π /2 (two-fold symmetry). Both configurations lead to a slight reduction of the critical flux-based Rayleigh number for the onset of convection. The flow consists of counter-rotating vortex pairs separated by thin downwellings Similar to Sumita & Olson, the large scales are accompanied by small-scale motion. With a two-fold variation, the flow becomes homogeneous above about 30 times the critical Rayleigh number, most likely as a result of the strong azimuthal flow that exists in this regime. Homogenisation was not observed with a one-fold variation as convection in the low-flux sector is not able to transport motion across it. Sahoo & Sreenivasan interpreted the magnetic field inhomogeneities as signatures of CMB inhomogeneities to estimate that Q ∗> 2 for the Earth, implying regions of subadiabatic heat flux. Achieving these in the experiments, would, however, require adding active cooling to the outer wall. Nevertheless, such inhomogeneities in the Earth\'s magnetic field suggest that homogenisation does not take place in the equatorial regions of the core. This offers a potential method to constrain the Rayleigh number in the Earth\'s outer core.', '4. The default setup for axial gravity: cylinders': "Experiments in geometries where the natural terrestrial gravity, the temperature gradient and the imposed rotation are aligned (fig. 1 (c)) capture the part of the planet that the experiments with cylindrical radial gravity neglected - the polar region. The default set-up is a straight cylinder, the shape that naturally arises in rapidly rotating systems. Only three experiments were conducted in box geometries [162-164] which creates the undesired issue of developing corner flows that have noplanetary equivalent. These Cartesian geometries, however, ease visual access in experiments. \n<!-- image --> \nFigure 10. Flow patterns produced by localised heating of the outer boundary of a hemispherical shell [129]. (a) Inner core spiral developing in the global locking mechanism as a result of localised heating at the outer hemisphere, visualised by injection of dye [129]. (b) the front forming the spiral results from the collision of the eastward and westward jets originating at the heating anomaly, a mechanism suspected to take place in the Earth core (right). \n<!-- image --> \nConvection in rotating spherical convection onsets in the equatorial region, and thus was initially thought to be the most relevant for planetary interiors. Nowadays, it is, however, clear that the polar region is at least equally important in the more supercritical regimes where planets reside [165,166]. Thus, the importance of some of the classical rotating convection experiments with axial gravity to geophysical processes is sometimes only emphasised more strongly in hindsight [142, 143, 167-169]. Nonetheless, as visualised in fig. 3 , this configuration has been the most common and enduring one since the beginning of rotating convection experiments and shows no signs of losing its popularity. There are several recent thorough reviews detailing the experiments, their results and more [11,13,37,170] so that here we keep the discussion very brief. \nArguably the first systematic and quantitative experiments of rotating convection was a series of mercury experiments conducted by Fultz, Nakagawa, Frenzen and Goroff in the years 1953 to 1960 at the University of Chicago [64, 171-175], the first one conducted on the 20 November 1953 as shown in fig. 12(a). (A few earlier rotating experiments were also conducted by Nakagawa and Fultz at the University of Tokyo and Chicago, respectively, in water and air, however, the upper surface was cooled by evaporation which is a slightly different fluid dynamics problem from the Rayleigh-Bénard-like set-up we consider throughout this review [172].) These seminal experiments were very comprehensive. Since they were conducted in a liquid metal with Pr ≈ 0.025, oscillatory convection was naturally included. Moreover, Nakagawa had access to an electromagnet of a discarded 36.5 in cyclotron which had been reconditioned at the Enrico Fermi Institute for Nuclear Studies at the University of Chicago with a field strength of up to 1.3 T, shown in fig. 12(b) [176]. This allowed him to also study rotating magnetoconvection. These experiments were mainly concerned with the flow behaviour close to the onset of convection and the validation of linear stability results derived by Chandrasekhar and Elbert [27,143]. \nThe focus on relatively low supercriticalities remained until the end of the last century. The \nFigure 11. Horizontal velocity vectors (arrows) and shaded contours of axial vorticity (s -1 ) in horizontal planes of [56]'s experiment in an annulus with heated outer boundary heater on two or four sectors of independently adjustable heat flux (yellow and green colours indicate regions or high and low heat flux respectively). Plots are averaged in time, and shown for different levels of supercriticality f Ra . (a)-(c): one-fold variations with Q ∗ = 0.7, (d)-(f): two-fold variation at Q ∗ = 1 showing regionalisation of convection, (g)-(i): two-fold variation with Q ∗ = 2, homogenisation of convection takes place at sufficiently high forcing under the effect of the azimuthal flow. \n<!-- image --> \nmost common working fluid was water [162-164, 177-183] which allows direct optical access and visualisation, but prohibits oscillatory convection. The same applies to silicone oil as tested by Koschmieder [184] and [177]. These earlier experiments established the formation of regular cell patterns and convective Taylor columns as the typical structures of steady bulk convection, see fig. 12(c), but also found irregular vortex patterns; an example is shown fig. 12(d). Also, the \nheat transport and its scaling with the control parameters was investigated, especially in the few visually opaque experiments (either due to the fluid or by design) by Rossby in mercury [177], Donnelly and collaborators in cryogenic helium-I [185-187], and Aurnou and Olson in gallium [188]. None of the experiments were likely extreme enough to capture the geostrophic turbulent heat transport scaling one may expect in Earth's outer core or planetary interior, though. \nLikely the most crucial finding in that period, however, was the discovery of wall modes by Ecke and collaborators, shown in fig. 12(e) [180, 181, 189]. Wallmodes explained why the onset of convection as determined e.g. by heat transport measurements did not agree with Chandrasekhar's predictions but was indeed lower. The destabilising effect of the wall and the breaking of the symmetry results in structures that travel along the periphery of the rotating convection vessel and exponentially decay towards its centre. Initially thought to be mainly an onset phenomenon and an experimental restriction, wallmodes are now known to be much more persistent and transform non-linearly into a boundary-zonal flow affecting also strongly supercritical settings [36, 190-196]. Moreover, they may also exist along the virtual boundary of the tangent cylinder in Earth's outer core, as we will discuss in the next section 5. \nIn our current 21st century, the goal has changed, and is now rather to push turbulent rotating convection to the extreme, such as in the TROCONVEX experiment in fig.12(f). [11, 197]. The most popular working fluid is still water [197, 197-213], but also cryogenic helium [214, 215], silicone oils [86], sucrose solution [198], gallium [35, 36, 168, 216] , and pressurised SF6 [190, 217]. The connection to geophysics and planetary physics is sought much more explicitly, in particular, in the Calimero (Californian Model of Earth's Rotation), NoMag, RoMag (all three in Los Angeles, Aurnou and collaborators [86,198,218]) and TROCONVEX (Eindhoven, Kunnen and collaborators [209]) experiments. Notably, RoMag, is the only operative liquid metal rotating magnetoconvection experiment with thus, the closest resemblance to planetary core fluid. Thus, this meant establishing the possible regimes of geostrophic convection (see fig.12(e), convective Taylor columns, plumes, geostrophic turbulence, rotationally influenced turbulence), testing various scaling for the heat and momentum transport, length scales, the importance of centrifugal effects and much more. Several of these experiments in China, USA and The Netherlands, are still active and running and producing results (see fig. 3), thus, we may expect more to come that will enrich our understanding of the polar physics in planetary interiors in the future.", '5. Tangent Cylinder Dynamics': 'Rotating convection in cylinders, reviewed in the previous section 4, may represent the dynamics in the polar regions. However, those experiments do likely not fully capture the entirety of the polar region that extends up to the virtual tangent cylinder (TC) boundary, see fig. 1 (a). The sidewall boundary conditions in cylindrical experiments are usually impermeable and adiabatic. The boundary conditions of planetary tangent cylinder are, however, likely neither. The most recent data form the SWARM mission clearly shows a strong jet wrapped around the TC and, importantly, meandering in and out of the TC [219]. Similarly, dynamo simulations at extreme enough parameters show flows across the TC [166]. \nThus, several major questions arise: First, under which conditions can the Taylor-Proudman constraint (TPC, see section 1.2) be violated such that the TC boundary becomes permeable? Second, how much heat is transferred between the polar and equatorial region across the TC? Finally, given such drastic differences in boundary conditions, how well do the experiments in cylinders truly represent polar convection? \nFigure 12. (a) The first experiment of rotating convection in Chicago on the 20 November 1953 [64]. (b) The hydromagnetic laboratory at the University of Chicago where Nakagawa did his seminal rotating convection experiments. [176] (c) Visualisation of columns in rotating convection in water using thermotropic liquid-crystal capsules by Sakai, top and side view (top/bottom) [162]. (d) Visualisation of chaotic vortices in rotating convection in water using aluminium powder on the top surface by Boubnov and Golitsyn [163]. (e) Visualisation of wall modes in water using shadowgraph by Zhong et al. [180]. (f) Visualisation of the flow regimes in turbulent rotating convection in water using rheoscopic particles and illumination with a vertical light sheet; left to right: convective Taylor columns, plumes, geostrophic turbulence, rotationally influenced turbulence by Cheng et al. [197] \n<!-- image -->', "5.1. Aurnou et al.'s TC: What's new in Baltimore?": 'Aurnou and collaborators [65] undertook the first attempt at addressing these questions in their Baltimore lab. They built a setup with a hemispherical vessel (15.2 cm diameter) filled with water, sitting in a cylindrical bath also filled with water ensuring a cold, approximately homogeneous temperature at its outer boundary (figure 13(a)). A heater shaped like a hockey puck (10 cmdiameter, 3.6 cm high) was rested at the flat bottom of the hemisphere. The heater achieved two \nfunctions: First, it delivered a constant heat flux q into the fluid. Second, the outer edge of the puck formed a discontinuity in the height of the fluid domain in the same way as a planetary TC does. The choice of a puck shape for the heater, instead of a spherical one was motivated by the difference in gravitational fields between the Earth and the experiment: in the experiment, the axial gravity would misalign with a spherical boundary of near-constant temperature and drive unwanted baroclinic flows in its vicinity. By contrast, the flat upper surface of the heater is everywhere perpendicular to the gravity, and so eliminates that problem. Hence, in the lab, pucks are better representation of the Earth\'s inner core than spheres. \nAurnou and collaborators used dye visualisation and thermistors embedded near the heater surface to measure heat fluxes and velocities. They identified five successive regimes based on the flow structure observed as the thermal forcing ramps up during a transient experiment. These regimes map to those in cylindrical vessels, but for specificities inherent to the TC geometry. At low thermal forcing, no convection is detected inside or outside the TC but non-vertical residual thermal gradients drive a very weak thermal wind that suppresses convection and thus increases the critical Rayleigh number for the onset of convection inside the TC. Increasing the thermal forcing leads to a ring of vortices attached to the outside of the TC. These vortices result from a baroclinic instability and remain confined outside the TC due to the TPC. Inside the TC, convection is still absent. At higher thermal forcing, convection sets in within the TC at slightly higher critical Rayleigh number than in cylindrical vessels with adiabatic sidewalls because of the weak thermal wind. The onset structures are helical, prograde near the heater and retrograde near the top. They ignite near the TC boundary and their motion is confined within the TC by the TPC as visualised during a transient experiment in figure 14(b). Further increasing the thermal forcing, but keeping strong rotation, the helical structures fill the entire TC, in a regime that would nowadays be referred to as "rotation-dominated columnar regime" [11,82]. When rotation is less dominant over buoyancy, the flow structures do not extend over the full height of the volume \nThepresenceofbaroclinicity in background rotation drives a strong azimuthal wind with maximum velocity U max φ near the inner TC boundary. The scaling of its intensity with the convective buoyancy flux qB = α gq /( ρ Cp ), as U max φ ∼ 2.05( qB / Ω ) 0.52 ± 0.03 confirms this mechanism. The authors suggest that this mechanism and its confinement to polar regions within the TC may explain the polar vortices inferred there from the 1980 Magsat and 2000 Oersted satellite missions [220]. Extrapolation of their experimental observation to Earth leads to estimates of a few km for the diameter of columnar vortices within the TC and a flux Rayleigh number of the order of 10 31 . They do stress however that testing this thermal wind scaling for the Earth would require incorporating such effects as the Lorentz force, compressibility and other possible effects.', '5.2. Little Earth Experiment I (LEE1)': "Pothérat and collaborators used design ideas similar to Aurnou et al. [65] in the Little Earth Experiment (LEE1), a hemispherical dome of diameter 28.5 cm, and a puck-shaped heater 2.25 cm in height and 10 cm in diameter. They introduced two novelties with drastic consequences for the experimental realisation (figure 13(b)). First, they incorporated PIV measurements to capture the detail of the time-dependent velocity field in horizontal planes and in one meridional plane. Second, LEE1 targeted magnetostrophic regimes with Λ ∼ 1 [38,46,52]. \nSince sulfuric acid is typically 5 to 10 times less dense than a liquid metal, it is easy to see that to reach Λ ∼ 1 at Ek ∼ 10 -6 , magnetic fields of the order of 10 T are needed. To complicate matters, the magnetic field must pervade a sufficiently large volume to host a fluid dynamics experiment with its instrumentation i.e. a cylindrical volume of about 30 cm diameter over about 1m. At the time of LEE1's operation, only one magnet in the world was capable of delivering such a magnetic field: the 12 MW M10 magnet of the Grenoble High Magnetic Field laboratory \nFigure 13. Sketches of the three experiments reproducing the TC Geometry: (a) 2001-2003: Rotating vessel of Aurnou et al. showing the puck-shaped heater inside the hemispherical dome filled with water [65] (b) 2013-2019: Rotating part and static support of LEE1 set against a section of the Earth illustrating how the inner core, ICB and CMB are modelled in the experiment [38, 46, 52] (c) 2019-: Rotating part of LEE2, with cameras, mirrors and optical slip ring, where the fluid vessel is now cylindrical instead of hemispherical. The two large black cameras near the top and the two mirrors fitted at the bottom are part of the PIV system for vertical planes. The vertically travelling carriage near the bottom holds the lens generating the horizontal LASER plane for PIV in horizontal planes of adjustable height [39]. \n<!-- image --> \n(LNCMI-G) with up to 10T in a 376mm-diameter/2m-long warm bore (Nowadays, LNCMI-G offers up to 18 T in their new hybrid magnet [221]). Using such a a large magnet meant that only one strategy was reasonably possible for the design of LEE1 out of the three outlined in section 3.3: LEE1 had to be built to fit into the existing magnet. This imposed further severe constraints on the design. Materials rotating within the bore had to be non-conducting to avoid Foucault currents, and those in contact with the working fluid had to possess long-term resistance to highly concentrated acids. Electronic equipment and electric motors cannot operate in high magnetic fields and had to be deported either far above or far below the magnet. The space between the vessel and the inner wall of the magnet bore is only a few cm. Thus, visualisation from the side had to rely on small concave mirrors providing sufficient angular access to capture the entire height of the TC. Finally, LEE1's motor was located about 0.5 m underneath the bottom of magnet. The acquisition system (camera, laptop controlling PIV LASERS and recording thermocouple signals) was placed on a rotating platform about 2 m above the top of the magnet. In total, LEE1 is approximately 4.5 m in height and linked by a structure mostly made of different plastics. \nThe heater was made of a plastic heat exchanger circulated by ethylene-glycol heated in the static frame. The top of the heater delivering heat to the acid was made of SHAPPAL, a heat- \nFigure 14. Left: dye visualisation showing the regimes of rotating convection inside a TC during the instationary phase following the ignition of thermal forcing from a state of solid body rotation, by Aurnou et al. [65]. Top view of (a) the onset of convection and (b) helical plumes at Ra = 9.01 × 10 9 ; Ek = 9.65 × 10 -5 , (c) side view of baroclinic instability developing across the TC boundary, (d) side view of fully developed convection inside the TC before the onset of baroclinic instability at ( Ra = 1.10 × 10 10 , Ek = 4.11 × 10 -5 ). Arrows indicate the direction of rotation. (e) Plumes arising near the TC boundary at the onset of convection ( Ra = 4.44 × 10 9 , Ek = 4.26 × 10 -5 ). \n<!-- image --> \nconducting, electrically insulating and acid-resistant ceramic. This design avoided electric parts but its thermal inertia prevented any feedback control of the temperature difference. \nNon-magnetic rotating convection. LEE1 was first operated in water to focus on non-magnetic rotating convection in the TC. In the entire range of parameters investigated, the flow was mostly confined within the TC, i.e. the TPC was sufficiently well satisfied for for the TC boundary to be practically impermeable, with only some weak motion detected outside the TC, see figure 15 (a,d,g). The onset of convection follows similar laws to the onset of planar layer convection, but with slightly different prefactors: close for the critical Rayleigh number Rac = (32.3 ± 4) Ek -1.29 ± 0.05 , but lower for the critical wavenumber kc = (0.58 ± 0.08) Ek -0.32 ± 0.05 . Past the onset, both wall modes and bulk modes are observed, which implies that the TC boundary acts as a solid impermeable wall in conditions relatively close to onset [32]. Unlike in cylindrical tanks, however, wall modes were not detected below the onset of bulk convection. A possible explanation is that the TPC does not erect a boundary when the flow is still. In other words, below the onset of convection, the TC does not exist as a wall, so neither do wall modes. With increasing supercriticality, the flow follows a complex sequence of patterns to arrive at a state dominated by a single retrograde vortex for e Ra ≃ 10. At high latitude, this vortex is associated with a retrograde thermal wind located inside the TC, whose intensity follows a scaling nearly identical to that found by Aurnou et al. [65] Ro = (5.33 ± 0.3)( NuRaEk 3 Pr -2 ) 0.51 ± 0.04 . The heat flux showed a clear \ntransition between regimes of rotation-dominated convection scaling as Nu ≃ 0.38 Ek 2 Ra 1.58 and buoyancy dominated convection with Nu ≃ 0.2 Ek 0.33 which resembles convection in cylinders and plane layers [11,37]. \nMagneto-rotating convection. Unlike liquid planetary cores, LEE1 operates in the quasi-static MHDregimewherethefeedback of the flow onto the magnetic field is negligible, so the magnetic field is imposed, and constant, and dynamo processes are excluded [222]. The acid's low conductivity also precludes the occurrence of Alfvén waves or magneto-Coriolis waves, even though these can exist a low magnetic Reynolds numbers in liquid metals [223, 224] and bear relevance to those in planetary cores [225,226]. Hence, LEE1 focuses on capturing the feedback of the magnetic field on convection rather than the reverse. \nThe magnetic regimes differ from the nonmagnetic ones in a number of ways, see figure 15. At the most fundamental level, the Coriolis force now competes with the Lorentz force, so the TPC no longer applies. Instead, a Magnetic Taylor-Proudman constraint (MTPC) imposes a kinematic relation between the flow along the TC boundary ( u φ ) and the flow through it ( ur ): \n∆ r z 〈 ur 〉 t , φ + Λ ∂ 2 zz 〈 u φ 〉 t , φ = 0. (17) \nThis theory is confirmed using PIV in two horizontal planes at low and high latitude showing that a meridional flow through the TC boundary follows the MTPC [52]. As a result of this reorganisation of the flow, the azimuthal flow does not follow the thermal wind scaling and increased significantly with Λ . Applying the MTPC to the Earth would require a more general form to incorporate the complex geometry of the Earth's magnetic field. Nonetheless, the MTPC provides at least a plausible mechanism to account for the meandering of the zonal flow in and out the TC inferred from the SWARM data [219], and previous simulations suggesting that the TPC was broken at the TC boundary [166,227]. \nThe main effect of the Lorentz force inside the TC was a suppression of heat transfer. At Λ > 0.2, Nusselt numbers are systematically much lower than their counterparts at Λ = 0, but scale with a greater power of Ra (up to 1). Unlike in the non-magnetic case, the highest thermal forcing available, Ra ∼ 10 9 , is insufficient to reach a transition to a buoyancy-dominated regime [66]. Detecting the onset of rotating magneto-convection inside the TC would require far more magnet time than possible in M10 but several states of convection are detected with increasing the thermal forcing. Near the onset of convection within the TC, both bulk and wall modes display similar structures to the non-magnetic case. As the thermal forcing is increased these are progressively replaced by a large retrograde vortex occupying the middle of the TC, consistent with the very strong azimuthal flows observed there, see figure 15. This structure however appeared at comparably lower thermal forcing, and much higher intensity than in the non-magnetic case and displayed off-centre maximum vorticity [66]. These results suggest that the axial magnetic field favours the emergence of a large polar vortex, again a plausible, if not directly applicable mechanism for the emergence of polar vortices in the Earth's outer core [220].", '5.3. Little Earth Experiment II': 'LEE2 is a complete rebuild of LEE1 with a much greater control of both the rotation and heating, reduced vibrations and a 10-fold increase in PIV resolution. Its new vessel focuses on the TC itself so the heater diameter was increased and the outer dome replaced by a cylinder of diameter 22cmwithapuck-shapedheaterofdiameter15cmandyieldingaTCheightof H = 14.3 cm. Thus, the region outside the TC is now a rather thin rectangular annulus. The outer wall is still cooled by a bath of water at constant temperature, which imposes a cold temperature along the top and its side walls, i.e. only 3.5 cm away from the TC boundary. This creates a much greater source of \nFigure 15. PIV visualisation of the non-magnetic and MHD flow patterns inside the TC in LEE1 [38, 66] showing vorticity (color) and velocity (arrows) averaged in time in a plane located at 51 · latitude ( i.e. just below the point where the TC meets the hemispherical vessel wall). Values of f Ra are calculated with respect to the critical Rayleigh number for Λ = 0, even for cases where Λ > 0. The red circle indicates the position of the TC boundary whereas the blue circle corresponds to the vessel wall. The Non-magnetic case (a,d,g) exhibits large, merged columns in weakly supercritical regime. For higher criticality, wall modesare visible and for the highest criticality a large central retrograde vortex exists at the centre. At Λ = 0.15 (b,e,h), the effect of the magnetic field is not pronounced and the flow patterns resemble those at Λ = 0. For Λ = 0.33 (c,f,i), a large retrograde structure forms a much lower criticality, with zero-vorticity "eye" in its centre. \n<!-- image --> \nf \nf \nf \nbaroclinicity there than in LEE1 to potentially break away from the TPC, and so attain regimes of higher inertia that would otherwise require higher thermal forcing. \nInside the new heater, the fluid heat exchanger was replaced by two resistive wires joined together in a spiral shape, and fed with opposite currents to cancel out the total Lorentz force due to the heating current inside the high magnetic field. This makes it possible to implement a feedback control loop on the heater to control the temperature difference between any pair of thermistors placed inside the device. Four are located at the top of the heater surface. Two pairs are placed inside and outside of the side wall and the top wall. Measurement inside the vessel provides a more accurate input for the control of the temperature drop within the fluid. With this system, the axial temperature difference between the top of the heater and the underside of the top wall can be kept constant for longer periods of time and controlled to ± 0.1 K. The two thermistor pairs also measure the local heat flux across the outer cylinder top and side walls. \nThe PIV system now benefits from a movable vertical LASER plane whose axial position can be remotely controlled during operation, without having to restart the experiment: this makes it possible to scan the flow using several PIV planes during the same run. All signals are passed to the static frame via an optical slip ring capable of absorbing the bandwidth the PIV cameras. These are now located further away from the bore to be able to operate at fields up to 12 T. These changes make LEE2 considerably heavier, so the new, more powerful motor too had to be placed further way from the magnet and the total height of the device now reaches 6.5 m. LEE2 can also be operated outside M10 in a shorter non-magnetic configuration (2.5 m tall). \nLEE2\'s high PIV resolution made it possible to map the regimes of convection against the classification established for classical rotating RBC during the 2010\'s [11, 82]. By order of criticality, the flow starts with a regime without convection within the TC, but with a weak toroidal flow in the outer annulus driven by baroclinicity. At slightly higher forcing, the outer flow becomes unstable to baroclinic instabilities and exhibits a ring of co-rotating vortices similar to the "rim instabilities" identified by Aurnou et al. [65], and visible in figure 16 (a). Since this region has a constant height, these vortices are not Rossby waves, which they nevertheless resemble. Convection in the TC sets in at approximately the critical Rayleigh number for plane layer rotating RBC, and displays the same cellular regime at onset with the same lengthscale (figure 16 (b)). For slightly higher thermal forcing, the cellular network breaks into quasi-geostrophic columns with a symmetric spiral structure of prograde vorticity near the bottom (hot) wall and retrograde vorticity at the top (cold) wall. The structure is surrounded by a sheath of opposite vorticity (figure 16 (c)). Up to this point the TC boundary acts essentially as an impermeable wall separating the flowinside and outside the TC with the exception of thin jet induced by baroclinic vortices locally traversing the TC boundary near the top, where baroclinicity is strongest. Further increasing the thermal forcing, convection within the TC enters a plume regime, where columns become threedimensional, and lose their outer sheath of opposite vorticity. More jets cross the TC near the top and the TC boundary becomes locally permeable over practically the whole height but a small region near the bottom (16 (d)). Even higher forcing disrupts the plumes and the flow displays turbulent features with a wide range of scales from small, 3D scales to a small number (two to four) of large vortices. The global balance of force suggests these may still be quasi-geostrophic (QG). However they appear to be driven by strong baroclinic jets originating outside the TC. In this regime, the TPC is broken over the entire cylinder height but a strong zonal flow persists at its location (16 (e)). \nThese regimes broadly follow those found in classical rotating RBC [11, 37]. A key feature of the TC geometry is that these regimes become more and more distorted by baroclinicallyinduced inertia, which progressively loosens the TPC. The breakup of the TC boundary also enables the heat flux to escape laterally, and thus to bypass the suppression of axial heat transfer by rotation within the TC. As as result, the transition between them takes place at significantly \nFigure 16. Snapshots of axial vorticity obtained from non simultaneous PIV measurements in LEE2 at z = H /6, H /3, H /2,2 H /3,5 H /6 (from bottom to top) [39]. The pink circle marks the location of the TC boundary. Regimes encountered by order of criticality f Ra are similar to those encountered in rotating convection in a cylinder with solid side wall. However, the transitions between them takes place at lower criticality because of inertia incurred by the baroclinic flow outside the TC. In particular, quasi-geostrophic turbulence appears at f Ra = 37 vs. f Ra ≃ 80 in Aguirre-Guzman et al.\'s work [228]. The baroclinic flow incurs a local breakup of the TPC visible through structures straddling the TC boundary as low as at f Ra = 2.9 (in the plane at z = 5/6 H ). \n<!-- image --> \nlower thermal forcing. A further effect of the strong baroclinicity at the TC boundary is the suppression of wallmodes which, unlike in cylindrical vessels appear in none of these regimes. Theabsenceofwallmodesandthefactthattransitions take place over a narrower range of critical Rayleigh numbers implies that QG turbulence and large-scale vortices [82, 228-231] are more easily reachable in TC geometries with strong baroclinicity. \nThese results suggest that inertia induced by baroclinicity, or otherwise, near the TC boundary in Earth or other planets may favour the emergence of large structures at lower levels of criticality. This could potentially explain the large scales observed in the polar regions. Hence, inertia such as induced by baroclinicity offers a second possible route for the breakup of the TC besides magnetic effects. This mechanism too could favour the meandering of zonal flows in and out of the TC. LEE2 is poised to examine the competition between these mechanisms in detail with future campaigns in high magnetic fields at LNCMI-G.', '6. Discussion': 'In the end, what does the evolution of experiments modelling rotating convection in planetary interiors over the last 70 years tell us? For sure, experiments are hard, cumbersome, and inherently limited by the abysmal gap that separates the relative comfort of the laboratory from realistic planetary conditions: the gravity points in the wrong direction, the fluids are either too viscous or too opaque, the experiments are of the wrong shape, cannot rotate fast enough, reliable measurements are extremely difficult to obtain and the list goes on... Yet, none of these hurdles have deterred the ingenuity of experimentalists. From Nakagawa\'s early experiments in cylinders [172-174] to modelling equatorial regions with an annulus [45] and the solid core with a hockey puck [65] and the more recent mapping velocity fields in vessels with a tangent cylinder [38, 46, 52], they have kept focussing on targeted aspects of planetary interiors: they isolated equatorial and polar regions to obtain Earth-like gravity at least locally, they worked with transparent fluids to reveal the first flow patterns of the outer core [40], be it at expense of some of the lowPr convection patterns [35], they simulated radial gravity in space at the cost of limited ranges of parameters [53, 68, 69], etc. None of these experiments is a realistic representation of the Earth\'s outer core, but each of them has revealed key bits of knowledge, which accumulated, and confronted to geophysical data, simulations and mathematical models, have, little by little, shaped our current understanding of planetary cores. The key to their success is that however incomplete, experiments are the best proxy for actual physical processes taking place in planetary interiors, provided the relevant processes can be identified and characterised. \nYet, at a time when numerical simulations have become so accessible and so powerful, and when "realistic" or "Earth-like" simulations are emerging [232], it is worth asking whether there is still a place for cumbersome laboratory experiments. The answer lies in the respective merits of both approaches: simulations in extreme regimes will remain time-intensive, as least as costly as experiments and not easily repeatable for the foreseeable future. Unlike experiments, they do offer access to the entire time-dependent field of variables. But unlimited accessibility to an ocean of data is only useful when knowing where to look, and experiments on more targeted processes covering wide parameter ranges are ideally placed to point in the right direction. So are the analysis of real geophysical data and simple mathematical theories. Furthermore, even when simulations are perfectly resolved at all scales and all time scales, a very hard condition to meet, the results are still only as good as the mathematical model they are solving. Experiments, on the other hand, are no facsimile: they return the real physics of all the processes they incorporate and so provide a real-life test of the mathematical and numerical models. So the answer is yes, experiments are crucial and will continue to improve, just like numerical simulations, and just like the oldest methods in the field: geophysical observation and simple mathematical theories. None of these approaches work anywhere near as well in isolation as when they can draw on the progress and feedback of the other three. \nThe main issue faced by experimentalists and theoreticians alike is that of the extrapolation of their results to the Earth or other planets. Given the limitations of numerics and experiments and the large uncertainties on the actual conditions in planetary interiors, attempting a precise model combining all potentially relevant processes would certainly be in vain. Stevenson finds solace in a elegant way out of this conundrum [233], citing a similar point made by Urey about the chemical composition of planets [234]: "What Urey is doing is giving a license to study illposed problems. By ill-posed, I mean that the issue can be stated with precision but the knowledge needed to settle the issue is not yet entirely adequate. [...] The important lesson that we learn is this: It is not so important that you get the right answer: it is most important that you ask the right question." In this spirit, the question we may be able to ask may not concern so much the exact nature of the convection in planetary interiors but rather: "What processes arising from \nthe interplay of convection and rotation may we expect to play a role there?". Seven decades of exploring planetary interiors with experiments on rotating convection have certainly provided a wealth of answers to that question, but by no means all of them. \nSo what lies ahead for experiments? The colossal progress of experiments certainly owes to the audacious ideas of their creators, but not only. They also constantly drew on technological and scientific advances from other fields: from the dawn of spacecrafts in the 1960\'s it became possible to escape lab gravity twenty year later [53]. LASERs and digital signal processing led to the advent of optical velocity mapping, ultrasound velocimetry made it possible to measure velocity profiles in opaque metals [63, 136, 137]. At the same time, experimental design has incorporated more and more complex aspects of outer core dynamics: cylinders became slanted annuli, then spheres then spheres with a puck. Heating became inhomogeneous, internal [58,75,235] and now magnetic fields are becoming an increasingly important part of the mix. \nWhat is next is certainly found at the crossing of what is currently missing or insufficiently explored in terms of physical process and what is becoming technologically possible. It is the biased view of the authors that the greatest scope for progress may lie in incorporating MHD effects in the rotating convection experiments. Certainly the dynamo problem has been the focus of significant experimental endeavours, whose main focus was to find under which conditions a flow could spontaneously generate a magnetic field [236-240]. However, the generic character of this question and the magnitude of the challenge involved in building experimental dynamos have pushed aside the question of how actual planetary dynamos work. So understandably, none of the dynamo experiments directly model actual planetary dynamos and the playground of understanding planetary dynamos has been occupied by numericists [152, 166]. For planets, the question is to find which flow within the outer core produces a magnetic field consistent with observations. Crucially, the associated Lorentz force due to the magnetic field may drastically alter the answers that to question. In fact, the effects of the Earth\'s field is often invoked when extrapolating rotating convection experiments to Earth [47,65,124]. Yet, the Lorentz force is still subject to debate: proponents of the geostrophic scenario argue that it remains much smaller than the Coriolis force [241], while others argue that a magnetostrophic scenario where both forces are comparable is required to explain the flow patterns inferred from geomagnetic data [143,242]. Here the availability of large electromagnets offers scope for experiments in magnetorotating convection to advance these questions. These possibilities are only starting to be explored: fields up to 18 T in sufficiently large volumes to host laboratory experiments [221] even make it possible to reach extreme regimes where Alfvén waves may interact with convection [224, 243]. Such experiments may also benefit from decades of accumulated knowledge on liquid metal MHD turbulence [244-250] and MHD convection tracing back to Nakagawa\'s first experiments [172,188,251,252]. Such large magnets combined with measurement techniques developed for transparent electrolytes [50,51,253,254] now even offer the possibility of mapping velocity and pressure fields in magnetorotating convection at similar Ek to non-MHD experiments but with a Elsasser numbers within the range expected for the Earth (between 0.1 and 10) [46,52]. So perhaps now is the time for Hide\'s 1958 wisdom to finally come true [255]: "Now it is clear that extensions in several directions, especially into hydromagnetics, will have to be made if results of any geomagnetic interest are to be obtained." \nOther, non-MHD processes that have been well investigated outside the context of convection in the outer core can be integrated into rotating convection experiments to capture essential elements of the outer core dynamics: solidification and re-melting of the inner core releases or absorbs lighter elements into/from the liquid outer core, and so incurs significant compositional buoyancy [256]. Only one experiment focused on compositional convection in the context of the inner core [60]. It is sometimes dismissed on the ground that the large Schmidt numbers associated to it justify mimicking convection there with high Prandtl number thermal convection. \nThis analogy may not capture well the crucial conditions the core boundaries, and even less the interplay between thermal and compositional convection. \nVariations of physical properties too may play a several roles. In gas giants, they can lead to compressibility effects or to a radial regionalisation of convection forming layers with strongly differentiated convective patterns [257]. While less significant in the liquid core of Earth-like planets, density variations and the variation of other physical properties may still produce local stratification capable of locally suppressing convection: the presence of such a stratified layer has indeed been theorised near the core-mantle boundary [258]. \nThe role of the centrifugal force is somewhat less obvious: it remains very small at the scale of planetary cores, but may act at the scale of crucial convection patterns forming there [23]: tornadoes forming in the Sun offer a reminder not to discard the centrifugal force too quickly in large systems [89,92]. \nAnother seven decades will surely tell which of these, or other avenues convection experiments will travel along, but there is little doubt that even then, they will still be advancing in the journey to the outer core of the Earth.', 'Acknowledgements': 'We would like to thank John Brothold for his inspiring talk at the 18 th SEDI conference at Great Barrington, MA. 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2024arXiv240902158R	We present uniform modeling of eight kilonovae five following short gammaray bursts GRBs including GRB170817A and three following long GRBs. We model their broadband afterglows to determine the relative contributions of afterglow and kilonova emission. We fit the kilonovae using a threecomponent model in MOSFiT that accounts for ejecta geometry and find population median ejecta masses for the total blue kappaB 0.5 cm2 g purple kappaP 3 cm2 g and red kappaR 10 cm2 g components of Mej tot 0.0850.0400.110 Modot Mej B 0.0060.0040.015 Modot Mej P 0.0200.0100.034 Modot and Mej R 0.0510.0450.100 Modot 68 confidence. The kilonova of GW170817 is near the median of the sample in most derived properties while the sample indicates great diversity. We investigate trends between the ejecta masses and the isotropicequivalent and beamingcorrected gammaray energies Egamma iso Egamma as well as restframe durations T90 rest. We find long GRB kilonovae have higher median red ejecta masses Mej R gt 0.05 Modot compared to onaxis short GRB kilonovae Mej R lt 0.02 Modot. We also observe a weak scaling between the total and red ejecta masses with Egamma iso and Egamma though a larger sample is needed to establish a significant correlation. These findings imply a connection between mergerdriven long GRBs and larger tidal dynamical ejecta masses which may indicate that their progenitors are asymmetric compact object binaries. We produce representative kilonova light curves and find that the planned depths and cadences of the Rubin and Roman Observatory surveys will be sufficient for orderofmagnitude constraints on Mej B and for Roman Mej P and Mej R of future kilonovae at z lt 0.1.	2024-09-01T00:00:00Z	['2024arXiv240902158R', 'arXiv:2409.02158', '10.48550/arXiv.2409.02158']	['Astrophysics - High Energy Astrophysical Phenomena']	Uniform Modeling of Observed Kilonovae Implications for Diversity and the Progenitors of MergerDriven Long GammaRay Bursts	2,024	189	0.61	['EPRINT_HTML', 'EPRINT_PDF']	2	https://arxiv.org/pdf/2409.02158.pdf	{'Uniform Modeling of Observed Kilonovae: Implications for Diversity and the Progenitors of Merger-Driven Long Gamma-Ray Bursts': "J. C. Rastinejad, 1 W. Fong, 1 C. D. Kilpatrick, 1 M. Nicholl, 2 and B. D. Metzger 3, 4 \n1 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA \n- 2 Astrophysics Research Centre, School of Mathematics and Physics, Queen's University Belfast, BT7 1NN, UK \n3 Department of Physics and Columbia Astrophysics Laboratory, Columbia University, Pupin Hall, New York, NY 10027, USA 4 Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA", 'ABSTRACT': 'We present uniform modeling of eight kilonovae, five following short gamma-ray bursts (GRBs; including GRB170817A) and three following long GRBs. We model their broadband afterglows to determine the relative contributions of afterglow and kilonova emission. We fit the kilonovae using a three-component model in MOSFiT , and report population median ejecta masses for the total, blue ( κ B = 0 . 5 cm 2 g -1 ), purple ( κ P = 3 cm 2 g -1 ), and red ( κ R = 10 cm 2 g -1 ) components. The kilonova of GW170817 is near the sample median in most derived properties. We investigate trends between the ejecta masses and the isotropic-equivalent and beaming-corrected γ -ray energies ( E γ, iso , E γ ), as well as rest-frame durations ( T 90 , rest ). We find long GRB kilonovae have higher median red ejecta masses ( M ej , R ≳ 0 . 05 M ⊙ ) compared to on-axis short GRB kilonovae ( M ej , R ≲ 0 . 02 M ⊙ ). We also observe a weak scaling between the total and red ejecta masses with E γ, iso and E γ , though a larger sample is needed to establish a significant correlation. These findings imply a connection between merger-driven long GRBs and larger tidal dynamical ejecta masses, which may indicate that their progenitors are asymmetric compact object binaries. We produce representative kilonova light curves and find that the planned depths and cadences of the Rubin and Roman Observatory surveys will be sufficient for order-of-magnitude constraints on M ej , B (and, for Roman, M ej , P and M ej , R ) of future kilonovae at z ≲ 0 . 1. \nKeywords: kilonovae, r -process, gamma-ray bursts', '1. INTRODUCTION': "On August 17, 2017, the nearly coincident detection of a binary neutron star (BNS) merger through gravitational waves (GWs, GW170817; Abbott et al. 2017a,b) and a short gamma-ray burst (GRB 170817A; Goldstein et al. 2017; Savchenko et al. 2017) confirmed the long-theorized connection between these 'multimessenger' signals. Adding to this long-awaited breakthrough, an optical-near-IR counterpart ('AT 2017gfo') to the GW and GRB signals was observed (Arcavi et al. 2017; Coulter et al. 2017; Lipunov et al. 2017; Tanvir et al. 2017; Soares-Santos et al. 2017; Valenti et al. 2017) and strongly resembled theoretical predic- \ntions for a kilonova, a thermal transient powered by the radioactive decay of elements beyond the iron-peak formed through rapid neutron capture nucleosynthesis (' r -process'; e.g., Burbidge et al. 1957; Cameron 1957; Lattimer & Schramm 1974; Li & Paczy'nski 1998; Metzger et al. 2010). Indeed, the counterpart's rapid temporal evolution pointed to a relatively low ejecta mass ( ≲ 0 . 1 M ⊙ ), expected for neutron star mergers, while its reddening on ∼ day timescales indicated a high (compared to that of iron-group elements) ejecta opacity produced by newly-created heavy elements (e.g., Barnes & Kasen 2013; Tanaka & Hotokezaka 2013; Kasen et al. 2017). Modeling of AT 2017gfo demonstrated that the light curve was well-explained with a two-component model, in which each component is parameterized by a different ejecta mass, velocity, and opacity (e.g., Cowperthwaite et al. 2017; Drout et al. 2017; Kasen et al. \n2017; Kilpatrick et al. 2017; Tanvir et al. 2017; Troja et al. 2017; Villar et al. 2017b), providing evidence for multiple emission sources from the merger. \nThe first definitive kilonova opened a host of new questions and predictions that may only be studied with a wider population of observed events. Rapid progress on the theoretical end of kilonova studies has resulted in models that incorporate additional physics, including updated neutrino schemes and viewing angle dependencies (e.g., Bulla 2019; Nicholl et al. 2021; Wollaeger et al. 2021; Chase et al. 2022; Curtis et al. 2023), and a mapping of progenitors, remnants, and novel emission mechanisms to predicted light curve properties (e.g., Arcavi 2018; Gottlieb et al. 2018; Metzger et al. 2018; Kawaguchi et al. 2020a; Gompertz et al. 2023). In parallel, GW observations of diverse BNS and NSBH progenitors (e.g., Abbott et al. 2020a,b; The LIGO Scientific Collaboration et al. 2024) and recent kilonovae discoveries following LGRBs (Rastinejad et al. 2022a; Troja et al. 2022; Yang et al. 2022; Gillanders et al. 2023; Levan et al. 2024; Yang et al. 2024) further motivate expectations for observed kilonova diversity. Looking forward, constraints on light curve diversity are critical for kilonova search strategies with next-generation facilities such as the Rubin Observatory and the Roman Space Telescope (Roman; Margutti et al. 2018; Andreoni et al. 2024). \nDespite progress on the theoretical end, constraints on kilonova ejecta parameters from observations other than AT2017gfo remain limited due to the low rates of detected neutron star mergers (e.g., Mandel & Broekgaarden 2022; The LIGO Scientific Collaboration et al. 2021). Kilonovae are also relatively faint and fleeting transients, rendering them difficult to observe at distances beyond z ≈ 0 . 1 without rapid response and highly-sensitive telescopes. Several kilonovae following GRBs have been discovered contemporaneously and in archival data (e.g., Berger et al. 2013; Tanvir et al. 2013; Yang et al. 2015; Gompertz et al. 2018) and, when combined with deep upper limits, reveal a span of ≈ 100 in optical luminosity (e.g., Gompertz et al. 2018; Rossi et al. 2020; Rastinejad et al. 2021). However, the majority of previous fits to observed light curves have been with by multiple codes with varying assumptions (e.g., Lamb et al. 2019; Troja et al. 2019; O'Connor et al. 2021). Thus, direct comparisons of derived parameters (e.g., mass, velocity, composition) between individual events are inadvisable. \nPrevious uniform modeling of GRB kilonovae used a single component model (Ascenzi et al. 2019), prohibiting a search for diversity among emission components. Further, a uniform analysis has not been performed \nsince 2019 (Ascenzi et al. 2019), thus excluding three recent events (GRBs 200522A, 211211A and 230307A; Fong et al. 2021; O'Connor et al. 2021; Rastinejad et al. 2022a; Troja et al. 2022; Levan et al. 2024; Yang et al. 2024). Surprisingly, two of these kilonovae followed long-duration GRBs. The progenitors and/or emission mechanisms driving their longer γ -ray signals remain an unknown, with several observationally untested theories (e.g., Yang et al. 2022; Gottlieb et al. 2023). Beyond significantly expanding the sample size, these new events motivate an updated, multi-component modeling endeavor to explore kilonova diversity and compare γ -ray and kilonova properties across the short and long GRB populations. \nHere, we perform uniform, multi-component modeling of a sample of eight kilonovae discovered over 2005-2023. We provide compiled multi-wavelength light curves for future fitting efforts with additional models. In Section 2 we describe our sample selection and provide details for each event. In Section 3 we detail our modeling of the synchrotron afterglow component to extricate the kilonova emission. In Section 4 we describe our kilonova modeling procedure and report our results. In Section 5 we discuss the implications of our results for future work. We assume a cosmology of H 0 = 69.6 km s -1 Mpc -1 , Ω M = 0.286, Ω vac = 0.714 (Bennett et al. 2014) and report magnitudes in the AB system throughout this work.", '2.1. Sample Selection': "We begin with the list of claimed kilonovae compiled in previous literature and covering the full sample of short GRBs discovered in the postSwift era (Gompertz et al. 2018; Rastinejad et al. 2021). These include GRBs050709 (Jin et al. 2016), 050724A (Gao et al. 2017), 060614 (Yang et al. 2015), 070714B (Gao et al. 2017), 070809 (Jin et al. 2020), 130603B (Berger et al. 2013; Tanvir et al. 2013), 150101B (Troja et al. 2018), 160821B (Lamb et al. 2019; Troja et al. 2019), 170817A (GW170817) and 200522A (Fong et al. 2021; O'Connor et al. 2021). Though GRBs080503 and 150424A have been claimed as a kilonova candidate (Perley et al. 2009; Bucciantini et al. 2012; Rossi et al. 2022), we do not include these events as their redshifts, and thus their kilonova luminosities, remain uncertain (Perley et al. 2009; Fong et al. 2022). In addition, we also consider the more recent kilonova candidates GRB 211211A (Rastinejad et al. 2022a; Troja et al. 2022), and GRB 230307A \n(Levan et al. 2024; Yang et al. 2024), both of which were identified following recent merger-driven long GRBs 1 . \nWe limit our sample to events with sufficient observations (typically, multiple X-ray detections, several optical-near-IR detections past 2 days and at least one radio observation) for robust afterglow and multicomponent kilonova modeling. To evaluate the availability of X-ray and radio observations, we use the catalog of Fong et al. (2015), supplemented with data from the Gamma-ray Circular Notices (GCNs) and the literature (Fox et al. 2005; Mangano et al. 2007; Londish et al. 2006; Berger et al. 2013; Tanvir et al. 2013; Fong et al. 2014; Lamb et al. 2019; Troja et al. 2019; Mei et al. 2022; Rastinejad et al. 2022a; Troja et al. 2022; Levan et al. 2024; Yang et al. 2024). We restrict our sample to events with two or more optical-near-IR observations past 2 days for two reasons. First, this is the timescale on which we roughly expect the kilonova to significantly contribute to the observed flux. Second, multiple observations allow us to assess fading, which in turn constrains the ejecta mass and velocity, and/or color, giving insight to the relative contribution from different components (e.g., Barnes & Kasen 2013). To evaluate available optical and near-IR data for each burst we consider data collected and published in short GRB kilonova compilations (Gompertz et al. 2018; Rastinejad et al. 2021) for bursts prior to 2021 and the literature for those from 2021-2023. We do not include GRBs 050724A, 070714B and 150101B in our sample as they each have only a single optical detection past 2 days. We further remove GRB070809 from our sample as it was not observed in the radio and no optical data is available past ≈ 1 . 5 days. \nWe are left with eight events that meet our criteria: GRBs 050709, 060614, 130603B, 160821B, 170817A (GW170817), 200522A, 211211A, and 230307A. The redshift range of these events is relatively low for short GRBs ( z = 0 . 008 -0 . 554; Fong et al. 2022), but greater than the expected horizon for GW-detected compact object mergers in O4 and O5 (Abbott et al. 2016). We describe each burst and the dataset used in our analysis in the following sections. We list the main properties of each burst in Table 1, including the γ -ray durations, GRB fluence ( f γ ), isotropic-equivalent γ -ray energy ( E γ, iso ; calculated using the method of Fong et al. 2015), and galaxy type (Nugent et al. 2022; Levan et al. 2024). For Swift GRBs we collect values for f γ (15-350 keV) and the durations over which 50% and 90% of the \ngamma-ray fluence was detected (T 50 and T 90 ) from the Swift -BAT catalog 2 (Lien et al. 2016). For GRBs detected with other satellites we collect values from the GCNs and literature (Fox et al. 2005; Stanbro & Meegan 2016; Goldstein et al. 2017; Veres et al. 2023; Dalessi et al. 2023). In the following sub-sections, we summarize the burst discovery for each event in our sample and the source of data used in our analysis.", '2.2. GRB050709': 'GRB050709 was detected by the High Energy Transient Explorer II (HETE-II; Ricker et al. 2003) at 22:36:37 UT on 2005 July 9, with a T 90 = 0 . 07 s (Hjorth et al. 2005). Follow-up observations with the Chandra X-ray Observatory ( Chandra ) revealed a new Xray source within the HETE-II localization (Fox et al. 2005). Follow-up observations by the Swift X-Ray Telescope (XRT; Burrows et al. 2005) and Chandra confirmed this counterpart as fading and thus, likely related to GRB050709 (Fox et al. 2005). At the position of the X-ray source, an optical counterpart was detected, embedded in a star-forming galaxy at z = 0 . 161 (Fox et al. 2005; Hjorth et al. 2005). The Very Large Array (VLA) observed the position of the X-ray counterpart over four epochs, but did not detect any significant emission (Fox et al. 2005). \nImaging with the Hubble Space Telescope ( HST ) revealed long-lived emission in the F814W band, which has been attributed to a kilonova by numerous groups in the literature (Jin et al. 2016; Gompertz et al. 2018; Ascenzi et al. 2019). We combine the X-ray, radio (Fox et al. 2005) and optical-near-IR (Fox et al. 2005; Hjorth et al. 2005; Covino et al. 2006) datasets for our analysis. We present all observations used in our analysis in Table 5.', '2.3. GRB060614': 'GRB060614 was discovered by the Swift Burst Alert Telescope (BAT; Barthelmy et al. 2005) on 2006 June 14 at 12:43:48.5 UT with a T 90 = 109 ± 3 s (Barthelmy et al. 2006). Prompt follow-up by Swift -XRT and the Swift Ultra-Violet Optical Telescope (UVOT; Roming et al. 2005) revealed bright counterparts to the burst (Mangano et al. 2007). The early ultraviolet through optical afterglow spectral energy distribution (SED) provides evidence for low line-of-sight extinction and a z < 1 . 3 origin (Gehrels et al. 2006). Subsequent followup by ground-based observatories revealed an optical \nTable 1. GRB Sample & Properties \nNote -1 Milky Way extinction values taken from Schlafly & Finkbeiner (2011). \ncounterpart with a spectroscopic redshift of z = 0 . 125 (Price et al. 2006; Fugazza et al. 2006) on the outskirts of a star-forming galaxy at the same distance. The optical counterpart was monitored to late times with HST (Yang et al. 2015). \nNumerous deep imaging observations and spectra, extending out to ≈ 65 days, failed to reveal the expected counterpart to a long-duration GRB, a SN Ic-BL (Della Valle et al. 2006; Fynbo et al. 2006; Gal-Yam et al. 2006). This fact, combined with its intriguing gamma-ray properties, motivated a later analysis showing that the optical light curve reddened at later times, suggesting instead the presence of a kilonova (Yang et al. 2015). \nFor our analysis, we use the XRT and UVOT light curves (though we do not use UVOT detections past 10 days as the exposures are on ≳ day timescales; Mangano et al. 2007), ATCA radio upper limits (Londish et al. 2006) and combine the optical datasets in the literature (Cobb 2006; Della Valle et al. 2006; Fynbo et al. 2006; Schmidt et al. 2006; Xu et al. 2009; Yang et al. 2015).', '2.4. GRB130603B': 'GRB130603B was detected by Swift -BAT and the KonusWind Observatory (Melandri et al. 2013; Golenetskii et al. 2013) on 2013 June 13 at 15:49:14 UT with T 90 = 0 . 18 ± 0 . 02 s (Berger et al. 2013). Prompt Swift -XRT observations revealed an X-ray counterpart (Melandri et al. 2013). Rapid optical follow-up revealed the optical afterglow within the X-ray localiza- \nion (Levan et al. 2013). Spectroscopy of the afterglow identified a GRB origin of z = 0 . 356 (Foley et al. 2013; Thone et al. 2013). Additional multi-wavelength followup detected a radio counterpart and a well-sampled optical and X-ray afterglow (e.g., Fong et al. 2014). Later observations with HST revealed that the optical counterpart had significantly reddened, resulting in a bright near-IR detection and deep limits in the optical bands. This represented the first bona-fide claim of an r -process-enriched kilonova (Berger et al. 2013; Tanvir et al. 2013). \nThe kilonova detection of GRB130603B has been modeled by numerous groups in the literature, though with only a single detection, precise ejecta parameters remain uncertain (e.g., Berger et al. 2013; Tanvir et al. 2013; Hotokezaka et al. 2013; Barnes et al. 2016). We employ the multi-wavelength dataset compiled in Fong et al. (2014) and combine the optical through near-IR datasets published in the literature (Berger et al. 2013; Cucchiara et al. 2013; de Ugarte Postigo et al. 2014; Tanvir et al. 2013; Fong et al. 2014).', '2.5. GRB160821B': 'GRB160821B was detected by the Fermi Space Telescope Gamma-ray Burst Monitor (GBM; Meegan et al. 2009) and the Swift -BAT on 2016 August 21 at 22:29:13 UT. It was classified as a short GRB with T 90 = 0 . 048 ± 0 . 07 s ( Swift ; Lamb et al. 2019) and T 90 ≈ 1 s ( Fermi ; Stanbro & Meegan 2016). Swift -XRT quickly localized \na counterpart (Evans et al. 2016). UVOT observed the location but did not detect a counterpart (Breeveld & Siegel 2016). Prompt follow-up identified optical and radio counterparts on the outskirts of a bright galaxy at z = 0 . 1616 (Fong et al. 2016; Levan et al. 2016; Xu et al. 2016), motivating further multi-color follow-up by HST and ground observatories (e.g., Kasliwal et al. 2017; Lamb et al. 2019; Troja et al. 2019). \nThe afterglow and kilonova of GRB160821B have been analyzed by multiple groups in the literature (e.g., Kasliwal et al. 2017; Lamb et al. 2019; Troja et al. 2019), and show evidence for an early reverse shock (e.g., Lamb et al. 2019) and a bluer kilonova relative to AT 2017gfo (e.g., Rastinejad et al. 2021). For our analysis, we collect Swift and XMM-Newton (Troja et al. 2019) and VLA radio observations (Lamb et al. 2019; Fong et al. 2021) from the literature. We combine the optical-NIR datasets (Kasliwal et al. 2017; Lamb et al. 2019; Troja et al. 2019) and report where we draw specific measurements from in Table 5.', '2.6. GRB170817A/GW170817': "GRB170817A was detected by the Fermi -GBM and INTernational Gamma-ray Astrophysics Laboratory (INTEGRAL; Winkler et al. 2003) on 2017 August 17 at 12:41:06 UT, 1.7 s after the LVK-detected BNS merger GW170817 (Abbott et al. 2017b,a; Goldstein et al. 2017; Savchenko et al. 2017). The gamma-ray duration was T 90 = 2 . 0 ± 0 . 5 s (50-300 keV; Fermi ; Goldstein et al. 2017). An optical counterpart (AT 2017gfo) was first found δt = 0 . 452 days following the event discovery (Arcavi et al. 2017; Coulter et al. 2017; Lipunov et al. 2017; Tanvir et al. 2017; Soares-Santos et al. 2017; Valenti et al. 2017). The community observed the transient in exquisite temporal and color detail, revealing a fast-fading, reddening source with spectroscopic evidence of the radioactive decay of lanthanide elements (e.g., Chornock et al. 2017; Gillanders et al. 2022; Kasen et al. 2017; Kilpatrick et al. 2017; Nicholl et al. 2017; Pian et al. 2017; Smartt et al. 2017; Watson et al. 2019; Hotokezaka et al. 2023). \nWe utilize the optical-NIR dataset compiled in Villar et al. 2017b with observations from the literature (Andreoni et al. 2017; Arcavi et al. 2017; Coulter et al. 2017; Cowperthwaite et al. 2017; D'ıaz et al. 2017; Drout et al. 2017; Evans et al. 2017; Hu et al. 2017; Kasliwal et al. 2017; Lipunov et al. 2017; Pian et al. 2017; Pozanenko et al. 2018; Shappee et al. 2017; Smartt et al. 2017; Tanvir et al. 2017; Troja et al. 2017; Utsumi et al. 2017; Valenti et al. 2017) extending out to ≈ 25 days. Specifically, we employ the set of observations used in their analysis, as there are known inconsistencies amongst \nthe full dataset (Villar et al. 2017b). We do not collect multi-wavelength data to model this component as it is well-established that the off-axis afterglow did not contaminate observations on the timescales of the kilonova ( δt ≲ 30 days; e.g., Lyman et al. 2018; Fong et al. 2019; Margutti & Chornock 2020; Kilpatrick et al. 2021).", '2.7. GRB200522A': "GRB200522A was discovered on 2020 May 22 at 11:41:34 UT by Swift -BAT with T 90 = 0 . 62 ± 0 . 08 s (Evans et al. 2020). A prompt XRT counterpart location was reported (Evans et al. 2020), within which a catalogued galaxy with photometric redshift z phot = 0 . 4 ± 0 . 1 was noted (Alam et al. 2015; Fong et al. 2020). Subsequent observations identified a radio counterpart (Schroeder et al. 2020; Fong et al. 2021) and secured the host redshift to z = 0 . 554 (Fong et al. 2021; O'Connor et al. 2021). Follow-up by HST and ground observatories uncovered a fading optical-near-IR counterpart embedded in the host galaxy (Strausbaugh & Cucchiara 2020; Fong et al. 2021; O'Connor et al. 2021). \nWe draw the Swift -XRT observations from the United Kingdom Swift Science Data Centre (UKSSDC; Evans et al. 2007, 2009) and incorporate Chandra and VLA observations (Fong et al. 2021). We combine the optical-near-IR data in the literature (Fong et al. 2021; O'Connor et al. 2021).", '2.8. GRB211211A': 'GRB211211A was detected on 11 December 2021 at 13:09:59 UT by Swift -BAT, Fermi -GBM, INTEGRAL, and the CALET Gamma-ray Burst Monitor (Mangan et al. 2021; Stamatikos et al. 2021; Tamura et al. 2021; Minaev et al. 2021). It was reported as a bright, longduration GRB with T 90 = 50 . 7 ± 0 . 9 s ( Swift ) and T 90 = 34 . 3 ± 0 . 6 s ( Fermi ; Veres et al. 2023). Swift -XRT and UVOT promptly identified counterparts to the burst (Osborne et al. 2021; Belles et al. 2021). Notably, the detection of the afterglow in the UVW 2 filter limits the event origin to z < 1 . 4 (Rastinejad et al. 2022a). An early optical counterpart was discovered proximate to galaxy SDSS J140910.47+275320.8 (Zheng et al. 2021). Later spectroscopy revealed a featureless afterglow (Rastinejad et al. 2022a) and a galaxy redshift of z = 0 . 0763, rendering it one of the most nearby GRBs observed across all durations (Rastinejad et al. 2022a; Troja et al. 2022). \nMotivated by the low redshift of the putative host galaxy, the counterpart was followed in the optical and near-IR. These observations revealed a fast-fading, red transient with similar luminosities and behavior to AT2017gfo (Rastinejad et al. 2022a). Later ( δt ≈ 2-3 \nweeks), deep optical upper limits revealed no sign of a supernova counterpart to a luminosity lower than that of any known GRB-SN (Rastinejad et al. 2022a; Troja et al. 2022), further motivating an interpretation of the red excess as a kilonova. The kilonova of GRB 211211A has been modeled in the literature by numerous groups (e.g., Rastinejad et al. 2022a; Troja et al. 2022; Mei et al. 2022; Yang et al. 2022; Kunert et al. 2024). For our analysis, we use Swift -XRT observations from UKSSDC, XMM-Newton upper limits (Mei et al. 2022) and a VLA radio observation (Rastinejad et al. 2022a). We use the Swift -UVOT and optical-near-IR datasets from Rastinejad et al. 2022a, and add early optical data from Troja et al. 2022.', '2.9. GRB230307A': "GRB230307A was detected on 7 March 2023 15:44:06.67 UT by Fermi -GBMwith a duration of ≈ 35 s (Fermi GBM Team 2023; Dalessi et al. 2023). The burst was also observed by GECAM, the InterPlanetary Network (IPN), AGILE, AstroSAT, and GRBAlpha (Xiong et al. 2023; Kozyrev et al. 2023; Casentini et al. 2023; Navaneeth et al. 2023; Dafcikova et al. 2023), and was quickly noted as the second-brightest GRB seen by Fermi to date (Burns et al. 2023). The ULTRACAM instrument mounted on the 3.5 m New Technology Telescope (NTT) and Swift -XRT undertook wide-field searches for an optical and X-ray counterpart, respectively, discovering coincident candidates at δt = 1 . 4 -1 . 7 days (Burrows et al. 2023; Levan et al. 2024). The counterpart was offset 30.2 '' (38.9 kpc) from a bright spiral galaxy confirmed at z = 0 . 0646 (Levan et al. 2024), and an extensive multi-wavelength followup campaign was initiated. \nDespite GRB230307A's nominal long duration, its high-energy properties, including its spectral lag and Xray flux decay, provided evidence for a compact object merger origin similar to GRB 211211A. Near-IR followup from ground-based observatories and JWST revealed a late-time red excess, light curve shape, and spectral features expected for a kilonova (e.g., Gillanders et al. 2023; Gillanders & Smartt 2024; Levan et al. 2024; Yang et al. 2024). We combine the multi-wavelength datasets presented in the literature, which includes Chandra , Swift , and XMM X-ray observations, broad coverage by ground observatories, late-time HST and JWST observations, and ATCA and AMI-LA radio observations (Levan et al. 2024; Yang et al. 2024) for our analysis.", '3. AFTERGLOW MODELING': "In addition to a radioactive decay-powered kilonova, BNS mergers are expected to launch a relativistic jet \nwhose interaction with the surrounding medium produces broadband synchroton emission, or the 'afterglow' (e.g., Sari et al. 1998). The afterglow flux is a potential contaminant for modeling kilonovae, especially for on-axis events in which the afterglow often dominates the total luminosity at early times, motivating us to model the afterglows of each event in our sample, with the exception of GW 170817 (see Section 2.6). Our afterglow model uses the formulae of Granot & Sari (2002) and methods described in Laskar et al. (2014) to describe synchrotron emission from a forward shock (FS), produced by the interaction of the GRB's collimated jet and the surrounding medium, incorporating the effects of Inverse Compton cooling (Sari & Esin 2001; Laskar et al. 2015). \nThe parameters fit in this afterglow model are the jet isotropic-equivalent energy ( E K , iso ), the circumburst density of the surrounding medium ( n 0 ), the input electron distribution power law index ( p ), the fraction of energy deposited into non-thermal relativistic electrons ( ϵ e ), the line-of-sight extinction in B -band ( A B ; fixed to 0 in our fits except for GRB 130603B, see below), and the time of the jet break ( t jet ), which is directly related to the jet opening angle (Rhoads 1999; Sari et al. 1999). For each event we assume a radially-homogeneous ISMlike environment ( k = 0), as this is expected in the local environments for most NS mergers. We fix the energy deposited in magnetic fields ( ϵ B ) to 0.01, the median for short GRBs (Fong et al. 2015; Schroeder et al. 2024), as the model fails to converge to a reasonable solution when leaving this parameter free. We fit the afterglow model using the Markov Chain Monte Carlo (MCMC) emcee package (Foreman-Mackey et al. 2013), enforcing a minimum 10% uncertainty on all detections to capture realistic measurement errors. We run each fit using 128 walkers for 5000 iterations and discard the first 10% of steps as burn-in. For each event, we employ the redshifts and extinction values listed in Table 1 and use values from the literature as starting parameters. \nAs the kilonova and afterglow both contribute flux in the optical-to near-IR wavelengths, disentangling emission between the two can be difficult, especially at early times. In particular, observations of AT 2017gfo demonstrated that kilonovae may have early ( δt ≲ 1 day) blue emission due to large quantities of fast-moving lanthanide-poor ejecta or, potentially, additional energy sources such as free neutron decay (e.g., Metzger et al. 2015) or shock-heating (e.g., Kasliwal et al. 2017; Villar et al. 2017b; Arcavi 2018; Nicholl et al. 2021). Thus, we attempt to remain agnostic to the precise afterglow contribution in the optical and mask data in the range 10 13 -10 16 Hz (effectively, fitting only the radio and X- \nFigure 1. The X-ray and radio detections (circles) and 3 σ upper limits (triangles) of the seven events in our sample with on-axis afterglows. In each panel, we show the model light curves' median and 68% confidence range (shaded region) along with the X-ray (grey) and radio (colored and labeled) observations. Open symbols denote data that was masked in the afterglow fit (Section 3). We show two models for GRB 230307A. The first ('Model 1') is fit to the X-ray and radio data only and provides a poor fit to the early TESS observations. The second ('Model 2') is fit with the TESS, radio and X-ray observations. It provides a better constraint on the optical contribution but cannot fully account for the late-time X-ray and radio observations. \n<!-- image --> \nobservations) in our fits to GRBs 160821B, 200522A, 211211A and 230307A. For GRB230307A, we find this method significantly underestimates the flux of TESS optical observations at 0 . 01 ≲ δt ≲ 0 . 2 days (Figure 1; 'Model 1') while a fit including the TESS observations provides a worse fit to the X-ray and radio data ('Model 2'). We perform our analysis on the results from both models due to the uncertainty in the emission source. For GRB160821B, we exclude the radio detection at δt = 0 . 17 days as it is likely the result of an early reverse shock (Lamb et al. 2019), and incompatible with the standard FS model. We do not expect the reverse shock to significantly contaminate the optical flux on the timescales of observations we use to model the kilonova ( δt > 0 . 95 days). \nFor the remaining three events we include some early optical data in our fit either due to sparse X-ray and radio detections (GRBs 050709, 060614) or high line-of- \nsight extinction that will significantly affect our estimation of the afterglow flux contribution (GRB 130603B; Cucchiara et al. 2013; Fong et al. 2014, 2015). While it is possible that some kilonova flux may be contributing in these optical detections, previous works have shown that these points can be explained with an afterglow model only (e.g., Fox et al. 2005; Fong et al. 2014; Xu et al. 2009). Specifically, for GRB 050709, we include two early ( δt < 2 . 4 days) R -band optical observations. For GRB 060614, we include early RIJK -band data ( δt = 0 . 7 -2 days) in our afterglow fit. We exclude X-ray data of GRB060614 prior to δt = 0 . 5 days as numerous analyses favor an energy injection scenario to explain the afterglow plateau observed at δt ≲ 0 . 5 day that is incompatible with the standard afterglow model (Figure 1; e.g., Xu et al. 2009; Mangano et al. 2007). We do not expect the exclusion of the energy injection episode in GRB 060614 to affect our kilonova modeling \nas the FS is expected to dominate on the timescales and in the filters we use to model the kilonovae. Finally, for GRB130603B we include early ( δt ≲ day) opticalnear-IR detections to measure the line-of-sight dust, A B , which we propagate to the output models we use for subtraction. We also exclude the final XMM-Newton observation of GRB 130603B as it is known to be contaminated by an unrelated X-ray source (Rouco Escorial et al. 2023). We denote which observations were used in the afterglow fitting in Table 5. \nFor all events, we visually inspect the optical-near-IR afterglow models to ensure they are not brighter than measured values beyond the uncertainties. In general, we find that our derived afterglow physical parameters are consistent with those in the literature, though in some cases we find inconsistent values (typically, in the degenerate parameters E K , iso and n 0 ). Variations are likely the result of discrepancies between modeling codes or the fact that we primarily use X-ray and radio data only; however the inferred afterglow parameters are not used in any subsequent analysis 3 , so the specific values are not important for this work. We create model light curves at the same rest-frame wavelengths as the opticalnear-IR observations for 1000 random draws from the full posterior of each event. From these 1000 draws we calculate the median and 68% credible flux range, using the 68% flux range as our uncertainty on the afterglow model. We show our fits to the multi-wavelength data, along with their uncertainties, in Figure 1. Following interpolation of the afterglow light curves to the δt of each observation, we subtract the median afterglow model flux from each observed flux, producing an 'afterglowsubtracted' light curve. We combine the data uncertainty and model uncertainty at the time of the observation in quadrature. We present our afterglow-subtracted values and errors in Table 4.", '4.1. Description of Kilonova Model': "We employ the Python-based Modular Open Source Fitter for Transients ( MOSFiT ) code (Guillochon et al. 2017) to fit kilonova models to the afterglow-subtracted data, for which the kilonova contribution has nominally been isolated. From this modeling, we derive physical parameters describing the observed kilonova emission, which we parameterize in terms of ejecta mass ( M ej ), velocity ( V ej ) and temperature cooling floor ( T floor ). The \nTable 2. Kilonova Model Parameters & Priors \nlatter is the temperature below which the photosphere recedes into the ejecta. Hence, T floor represents an effective emission temperature for the optically-thin nebular phase to follow. We elect to use MOSFiT as its modular design affords us the flexibility to build new modules, freeze or adjust parameters and their priors, add constraints, and test several samplers without a high computational cost. In addition, MOSFiT is a well-tested method that has been used to fit or determine upper ejecta mass limits of several past kilonovae (e.g., Villar et al. 2017b; Nicholl et al. 2021; Rastinejad et al. 2022a; Coulter et al. 2024). All of the results presented in this section were performed with nested sampling, as implemented in the dynesty fitting routine (Speagle 2020). \nWe begin with MOSFiT 's existing three-component (blue, purple and red; see below) kilonova model (Villar et al. 2017a; Metzger 2019), which assumes analytic forms for the radioactive heating rate (Korobkin et al. 2012; equation 1 of Villar et al. 2017b) and the thermalization efficiency (Barnes et al. 2016; equation 2 of Villar et al. 2017b). The model calculates the bolometric luminosity assuming a central energy source and following an updated Arnett (1982) formalism (Chatzopoulos et al. 2012; equation 3 of Villar et al. 2017b). Here, each of the three components is parameterized by a constant 'grey' ejecta opacity ( κ ), the value of which correlates with lanthanide or electron fractions in portions of the total ejecta. Kilonova models comprised of two or three components were found to provide a better fit to the well-sampled AT 2017gfo compared to single component models (Cowperthwaite et al. 2017; Drout et al. 2017; Kasen et al. 2017; Kilpatrick et al. 2017; \nTanaka et al. 2017; Villar et al. 2017b). They are also physically motivated by simulations that predict multiple ejecta mechanisms with distinct elemental compositions prior to and following the NS merger (e.g., Metzger & Fern'andez 2014; Kasen et al. 2017). \nFor our three-component model we employ κ R = 10 cm 2 g -1 , κ P = 3 cm 2 g -1 , and κ B = 0 . 5 cm 2 g -1 for the 'red', 'purple' and 'blue' components, respectively (Tanaka et al. 2017). We expect these components to roughly map to the red (lanthanide-rich or Y e ≲ 0 . 2) tidal dynamical ejecta (e.g., Rosswog et al. 1999), the purple (moderately lanthanide-rich or 0 . 2 ≲ Y e ≲ 0 . 3) disk wind ejecta (e.g., Metzger & Fern'andez 2014; Just et al. 2015; Fern'andez & Metzger 2016; Lippuner et al. 2017), and the blue (lanthanide-poor or Y e ≳ 0 . 3) dynamical ejecta shocked at the NS contact interface and ejected near the poles (e.g., Sekiguchi et al. 2015) or ejected in a magnetized wind from the neutron star remnant prior to black hole formation (e.g., Metzger et al. 2018; Fong et al. 2021; Combi & Siegel 2023; Curtis et al. 2024). For each component in our model, we measure M ej , V ej and T floor (Table 2). We pursue a three-component model (as opposed to a two-component model) as this provides a better mapping of ejecta mechanism to opacity, enabling a search for trends tied to physical properties. \nThe geometry of the mechanism producing each ejecta component and the viewing angle of the observer are known to significantly impact the observed light curve (e.g., Darbha & Kasen 2020; Chase et al. 2022), and thus any parameter inference. Of particular relevance to this work, the assumption of an isotropic kilonova will likely introduce a bias in estimating the mass of material ejected along the line-of-sight. Thus, under this assumption, GRB events observed along the jet axis likely have blue and red ejecta components that are overestimated and underestimated, respectively. Instead, here we account for the geometry of the ejecta by modifying the original spherical kilonova model (Villar et al. 2017b) to an aspherical model, wherein a half-opening angle ( θ open ) defines a conical boundary between red ejecta, confined to the equatorial region, and the blue and purple ejecta, modeled in the direction of the poles. We use the half-opening angle prescription of Darbha & Kasen (2020), as implemented in MOSFiT by Nicholl et al. (2021). The viewing angle ( θ obs ; defined relative to the axis of the GRB jet) of each event in our sample is wellestablished, either as relatively pole-on due to the detection of a cosmological GRB or measured through highprecision astrometry in the case of AT 2017gfo (Mooley et al. 2022). Thus, we fix θ obs = 22 · (the central value in the range given) for AT 2017gfo (Mooley et al. 2022) and \nθ obs = 0 · for all other events in our sample. We allow the kilonova ejecta half-opening angle to be a free parameter and include an additional parameter ( σ ) to account for white noise in the likelihood function. These, in addition to M ej , V ej and T floor for each of the three components, comprise the 11 parameters measured for each kilonova. \nWe acknowledge that the number of free parameters exceeds the number of data points for some events in our sample. We perform several test fits fixing each component's T floor and θ open , finding consistent masses compared to runs where these parameters are left free. In the end, we opt to keep these parameters free in order to marginalize over the uncertainties on other parameters when measuring masses. Uniform use of the three-component model is critical to comparisons between kilonovae. In these cases we find very broad posteriors on the masses (e.g., Figure 4 and Table 3), but are still able to constrain them compared to the broad uniform prior. \nWe list the parameters and priors used in our kilonova model in Table 2. For M ej and V ej we elect to use the widest priors possible that correspond to physical values based on simulations (e.g., Radice et al. 2018; Nedora et al. 2021). For T floor we use the range 1000-4000 K following the reasoning of Nicholl et al. (2021) based on previous fits to AT 2017gfo with MOSFiT (Cowperthwaite et al. 2017; Villar et al. 2017b). For GRB 230307A, JWST detections at δt ≈ 29 and 61 days likely occur at epochs when nebular emission is either significantly contributing or dominating the observed flux. As kilonova nebular emission remains a challenge to properly model, we do not include the observations at δt ≈ 61 days in our kilonova fit. To accommodate the observations at δt ≈ 29 days and the mid-IR coverage of the JWST detections, we allow the T floor , P and T floor , R prior range to extend down to 800 and 440 K, respectively. We base our choice of T floor , R = 440 K on the blackbody fit to synthetic nebular lanthanide-rich kilonova spectra and Spitzer detections of AT 2017gfo (Kasliwal et al. 2022; Hotokezaka et al. 2021; Barnes & Metzger 2022). We choose a median value of T floor , P = 800 K as we expect the blackbody temperature of moderately lanthaniderich ejecta to fall in between that of lanthanide-poor and lanthanide-rich ejecta. \nWe fit the eight afterglow-subtracted optical-near-IR light curves at δt > 0 . 5 days (Table 4) with the threecomponent kilonova model presented above. We do not include data prior to 0.5 days in our kilonova fits as these timescales may be affected by emission mechanisms be- \nFigure 2. Afterglow-subtracted observations of the eight kilonovae in our sample along with best-fit (median and 68% confidence) model light curves. 3 σ upper limits are marked with triangles, while detections are marked with circles. We show light curves in the filters where observations (including upper limits) are available for each event. Two models are shown for GRB230307A based on the two afterglow models described in Section 3 and shown in Figure 1. \n<!-- image --> \nond radioactive decay, including energy injection in the afterglow (GRB 060614; Mangano et al. 2007), central engine activity (e.g., GRB 211211A; Hamidani et al. 2024a), or shock cooling (e.g., Rastinejad et al. 2022a). For our fits, we include only data points whose combined statistical and systematic (from afterglow fitting; Section 4) errors are < 0 . 5 mag and treat all observations with combined errors > 0 . 5 mag as upper limits. \nIn general, this results in detections prior to ≈ 1 day being treated as upper limits (e.g., Figure 2 and Table 4). For GRB050709, we employ a threshold of < 0 . 6 mag, as we find the inclusion of the two additional points provides a significantly better fit. We present the median and 68% confidence range for each parameter measured in our kilonova modeling in Table 3. \nTable 3. Kilonova Model Posteriors \nNote -'B', 'P', and 'R' subscripts refer to the blue, purple and red ejecta components, as described in Section 4.", '4.2. Results and Observed M ej Diversity': "In Figure 2 we plot median and 68% confidence range model light curves, constructed from 900 random draws of the full posterior of each GRB (with the exception of GRB170817A for which we use 50 random draws due to computational constraints). Overall, we find that the models provide reasonable fits to the data and follow the predicted behavior of kilonovae: rapid decay, especially in bluer bands, and reddening over time. As expected, events with better observational coverage correspond to tighter constraints on the light curves. For AT 2017gfo, we find that our model provides a good fit to the optical data and the majority of the near-IR observations, but overpredicts the late-time ( δt ≳ 20 day) K -band observations, a feature also noted in previous MOSFiT fits to this event (Villar et al. 2017b). We discuss comparisons to previous fits further in Section 4.4. \nFor GRB230307A we find that our choice of afterglow model significantly affects the shape of the model optical light curve at later times, reflected in the disparate best-fit values found for M ej , R and T floor , R . The fit to data subtracted with afterglow Model 1 (Figures 1 and 2; ''KN Fit 1') follows a typical fading behavior at δt ≳ 10 days. In contrast, the fit to data subtracted with afterglow Model 2 ('KN Fit 2'; in particular the JWST F070W detection at ≈ 29 days which is an upper limit in KN Fit 1) produces a flattening in the optical decay past δt ≳ 10 days. To produce this shape, KN Fit 2 requires higher values of T floor , B and T floor , R than KN Fit 1 to explain the emission at later times (Table 3). We \nfavor KN Fit 1 in our later analysis (though we show both for completeness), for several reasons. First, we prefer the afterglow model that provides a better fit to the X-ray and radio data than the early TESS observations (afterglow Model 1; Figure 1) as there are a number of proposed emission mechanisms, including a reverse shock, shock cooling and free-neutron decay (Metzger et al. 2015; Gottlieb et al. 2018), that may explain the early TESS excess but could not account for a late-time radio and X-ray excess. Second, the true TESS bandpass is wider than the nominal I c -band reported in the literature (e.g., Yang et al. 2024), and may also explain excess emission relative to the model. Third, KN Fit 1 results in more physically realistic values for T floor , R , as they are similar to the blackbody temperature that approximates kilonova nebular emission at the wavelengths where the nebular observations occur (e.g., 440 K; Barnes & Metzger 2022). Finally, fits to both afterglow-subtracted datasets without JWST observations are more consistent with KN Model 1. \nIn Figure 3 we show the median and 68% confidence range for M ej and V ej for each component and the total ejecta. In keeping with previous works (e.g., Rastinejad et al. 2021), our fits place the tightest constraints on the blue and purple component parameters and the coarsest constraints on the red component. This is in large part due to the traditional use of more sensitive optical telescopes for afterglow searches, especially for the kilonova candidates detected prior to GW 170817. Notably, our fits to events with just two or three detec- \nFigure 3. Median M ej and V ej (68% confidence) for the blue (top left), purple (top right), and red (bottom left) components as well as the total ejecta (bottom right). We mark long GRBs with diamonds, GW 170817/AT 2017gfo with a star, and the remaining short GRBs with circles. Our analysis highlights that GW 170817/AT 2017gfo is a 'typical' kilonova. \n<!-- image --> \ntions and deep upper limits (GRBs 050709 and 200522A) place order-of-magnitude or tighter constraints on M ej , B and M ej , P . In Section 5.1 we analyze and compare the kilonova properties of long and short GRBs. \nFocusing on the blue component, our analysis finds that all except one event prefer M ej , B ≤ 0 . 01 M ⊙ (Table 3). We also observe a general trend between increasing M ej , B and V ej , B (Figure 3), though the large error bars on several events prohibit a firm conclusion. Taking 1000 draws from the posterior of each event (and discarding GRB230307A KN Fit 2), we calculate a median of M ej , B = 0 . 006 +0 . 015 -0 . 004 M ⊙ (68% confidence; Figure 4). \nThe blue kilonova emissions may be attributed to either dynamical ejecta heated at the contact surface between the NSs and ejected along the the axis of the jet (e.g., Oechslin et al. 2007; Sekiguchi et al. 2015) or post-merger disk ejecta experiencing neutrino irradiation from a NS remnant, which lowers the lanthaniderichness of any ejecta (e.g., Metzger & Fern'andez 2014; Miller et al. 2019). Notably, GRB200522A is a significant outlier in M ej , B (Figure 4) with M ej , B = 0 . 046 ± 0 . 009 M ⊙ , consistent with past findings (Fong et al. 2021). We slightly favor a disk-wind source (rather than a shock-heated dynamical source) to explain the majority of GRB 200522A's larger M ej , B as BNS merger sim- \nions that measure shock-heated dynamical ejecta do not produce M ej , B ≳ 0 . 01 M ⊙ , even when spanning a range in NS masses, mass ratios and two NS equations of state (Sekiguchi et al. 2015). In contrast, simulations measuring a disk-wind mass in the case of a long-lived NS remnant produce M ej , B ≈ 0 . 03 M ⊙ (e.g., Lippuner et al. 2017). The extreme luminosity of GRB 200522A's kilonova has previously been explained with the creation of a magnetar remnant, which may provide an additional blue emission source (Fong et al. 2021). \nTurning to the purple component, we find a population median of M ej , P = 0 . 020 +0 . 034 -0 . 010 M ⊙ (68% confidence; Figure 4). This range is broadly consistent with expectations of disk component masses (e.g., Lippuner et al. 2017). We find that several events (GRBs 050709, 130603B and 230307A) prefer lower ejecta velocities compared to those found for the blue or red components. This trend supports the purple component's source as a disk wind which is likely ejected with slower speeds compared to the dynamical red and blue components ( V ej ≈ 0 . 01 -0 . 1 c ; e.g., Metzger & Fern'andez 2014; Fern'andez et al. 2015). We find that GRB211211A is an outlier on the higher end in V ej , P . As our fit to this event finds high velocities for all three components, we posit this is likely a reflection of the fast-decaying UV- \nFigure 4. The median and 68% confidence range of the M ej found for each GRB in the blue (top left), purple (top right), and red (bottom left) components as well as the total ejecta mass (bottom left). Long GRBs, short GRBs and AT 2017gfo are marked with diamonds, circles, and a star, respectively. We show the median and 68% confidence range across all events in the grey vertical line and shaded region, and mark the log-uniform prior for each component in the grey horizontal line. We also mark upper limits on the component ejecta masses from observations of short GRB afterglows with left-facing light purple triangles (Section 4.3). We show our two fits for GRB 230307A, marking KN Fit 2 with an open symbol as we disfavor this fit in our analysis (Section 4). GW 170817/AT 2017gfo falls comfortable within the 68% confidence range for each ejecta mass. \n<!-- image --> \nl emission observed at δt ≈ 0 . 5 -2 days, which previous fits have explained with a shock cooling model (e.g., Rastinejad et al. 2022a). \nFinally, for the red component, we find a population median of M ej , R = 0 . 051 +0 . 100 -0 . 045 M ⊙ (68% confidence; Figure 4). Amongst the three components, this is highest median, and the widest range in ejecta masses, though we note there are poor constraints on this value for GRBs050709, 130603B and 200522A. The red kilonova component can be ascribed to the neutron-rich ejecta tidally stripped from the NS surfaces as the compact objects slowly inspiral (e.g., Lattimer & Schramm 1974; Li & Paczy'nski 1998; Metzger et al. 2010; Tanaka & Hotokezaka 2013). Generally, it is expected that red dynamical ejecta mass will increase with larger asymmetry \nin the progenitor masses (particularly for NSBH events; e.g., Foucart et al. 2014) or higher spins (e.g., Kyutoku et al. 2015; Shibata & Hotokezaka 2019). Notably, the three highest M ej , R median values are found for the three LGRB events, GRBs060614, 211211A and 230307A. We further discuss the implications of this finding and other trends with GRB properties in Section 5.1. \nAcross the sample, we derive a median total ejecta mass of M ej , tot = 0 . 085 +0 . 110 -0 . 040 M ⊙ (68% confidence). In every component and the total, the ejecta mass of GW170817/AT2017gfo falls comfortably within the 68% credible range found for all events. This indicates that AT2017gfo may be considered a 'representative' kilonova (Figure 4) compared to the seven kilonovae analyzed here. In contrast, the kilonova of GRB 160821B \nfalls below the median ejecta masses found for all GRBs, rendering it a critical point in probing kilonova diversity.", '4.3. Constraints on M ej from Additional Short GRB Observations': "Next, we briefly explore the possibility that our kilonova sample is observationally biased towards more luminous events. As luminosity roughly scales with a fractional power of M ej (e.g., Metzger et al. 2018), a missing population of low-luminosity kilonovae would translate to a population with lower M ej than those reported here.To evaluate this, we employ observations of seven short GRBs with upper limits and afterglow detections that are less luminous compared to AT 2017gfo when matched in rest-frame time and band at their known redshifts (see Rastinejad et al. 2021, for details). These bursts 4 are GRBs050509B (Bloom et al. 2006; Cenko et al. 2005; Hjorth et al. 2005; Castro-Tirado et al. 2005), 080905A (Rowlinson et al. 2010a; Nicuesa Guelbenzu et al. 2012), 090515 (Rowlinson et al. 2010b), 100206 (Nicuesa Guelbenzu et al. 2012; Perley et al. 2012), 130822A Cenko et al. (2013); Rastinejad et al. (2021), 150120A (Rastinejad et al. 2021) and 160624A (O'Connor et al. 2021; Rastinejad et al. 2021). For this analysis, we assume all detections are dominated by afterglow flux and treat them as upper limits. \nIn the observed frame for each event (determined with the redshift catalog of Fong et al. 2022), we generate three sets of one-component kilonova light curves, characterized by either the blue, purple, and red fixed opacity value mentioned in Section 4.1. We fix T floor = 1000 K and use the corresponding component's median velocity found in Section 4.2. For each component, we produce kilonova models log-spaced in M ej over the range M ej = 0 . 001 -0 . 5 M ⊙ . We compare our GRB observations to the set of kilonova models, and record the highest M ej in each component allowed by the upper limits for each event. In Figure 4 we plot constraining ( < 0 . 5 M ⊙ ) upper limits on the respective component ejecta masses. We note that our use of one-component models translates to a conservative upper limit as flux from other components is not taken into account in our procedure. \nSimilar to previous findings, short GRB observations are most constraining of M ej , B (Figure 4; purple triangles). Past, deep rest-frame near-IR coverage is sparse. Thus, historical short GRB upper limits do not place meaningful constraints on M ej , P and M ej , R (Figure 4). In the blue component, we find one event less massive \nthan AT2017gfo (GRB080905A; M ej , B < 0 . 002 M ⊙ ) and an additional two events with upper limits below the median value found in Section 4.2 (GRBs 050509B and 130822A; M ej , B < 0 . 005 M ⊙ ). Overall, the existing upper limits span the range of blue ejecta masses. Thus, it is difficult to conclude at present if the population that do not have detected kilonovae also have lower ejecta masses.", '4.4. Comparison to Previous Kilonova Fits': 'Comparing our results in Table 3 to those from previous fits, we find our results are generally consistent with those in the literature. We find variation in absolute differences, an expected outcome given the range of kilonova modeling codes used, which we further discuss here. \nFor AT2017gfo, our fit produces a larger M ej , R and a smaller M ej , B and M ej , P compared to a previous MOSFiT run (Villar et al. 2017b), though a similar value for the total ejecta mass is reached ( M tot ≈ 0 . 07 M ⊙ ). We ascribe this difference in the relative component masses to our addition of the geometry prescription (Section 4.1; Darbha & Kasen 2020), which is expected to increase the amount of red mass relative to the blue for viewing angles less than the θ open (and is observed in Villar et al. 2017b though a larger θ obs and different asymmetry prescription was used). Nicholl et al. (2021) also model AT2017gfo with MOSFiT , incorporating constraints from GWobservations, and find comparable M ej , B and M ej , P . Their analysis finds a smaller M ej , R ( ≈ 0 . 001 M ⊙ ), which we attribute to their constraints on the tidal dynamical ejecta based on the BNS mass ratio and chirp mass (Nicholl et al. 2021). Compared to two-component (red and blue components only) fits to AT 2017gfo, we obtain M ej , B , R values that are within the range but on the upper end of those measured in the literature (e.g., Arcavi et al. 2017; Chornock et al. 2017; Kasen et al. 2017; Kasliwal et al. 2017; McCully et al. 2017; Pian et al. 2017; Smartt et al. 2017; Troja et al. 2017; Anand et al. 2023). \nOf the remaining kilonovae in our sample, the majority have measured M ej and V ej . However, previous fits to these events use a variety of models and methods to measure the kilonova parameters and exact comparisons are not advisable. For GRB 060614, a previous estimate found M ej , tot ≈ 0 . 1 M ⊙ , on the lower end of our estimate (Yang et al. 2015). For GRB130603B, previous fits find a wide span in M ej , tot = 0 . 01 -0 . 1 M ⊙ (Berger et al. 2013; Tanvir et al. 2013; Barnes et al. 2016), broadly consistent with our results. For GRB 160821B, Troja et al. (2019) constrain a lanthanide-rich mass to < 0 . 006 M ⊙ and a lanthanide-poor mass to ≈ 0 . 01 M ⊙ . \nFor the same burst, Lamb et al. (2019) find a dynamical ejecta mass of (1 . 0 ± 0 . 6) × 10 -3 M ⊙ and a post-merger ejecta mass of (1 . 0 ± 0 . 6) × 10 -2 M ⊙ . Within errors, our results are consistent with both findings but are on the upper end of the given ranges. \nGRB211211A was fit with a three-component model in MOSFiT that included a shock heating prescription, finding M ej , B = 0 . 01 ± 0 . 001 M ⊙ , M ej , P = 0 . 01 ± 0 . 02 M ⊙ and M ej , R = 0 . 02 +0 . 02 -0 . 01 M ⊙ (Rastinejad et al. 2022a). They find consistent results when performing joint afterglow and kilonova modeling (Rastinejad et al. 2022a). Within errors, the blue and purple component masses are consistent with our results, but we find a larger median M ej , R by a factor of two. Additional M ej , tot estimates of this kilonova span 0 . 02 -0 . 1 M ⊙ (Troja et al. 2022; Yang et al. 2022; Kunert et al. 2024). For GRB230307A, M ej , tot of 0 . 05 +0 . 15 -0 . 05 M ⊙ (Levan et al. 2024) and ≈ 0 . 08 M ⊙ (Yang et al. 2024) are found in the literature. Taken together, our results are generally consistent with those found in the literature. Our median values are on the upper end of previous estimates, consistent with comparisons to AT 2017gfo modeling.', '4.5. Caveats to Kilonova Model': "We find that in several cases the error bars on M ej (particularly for the blue component) are unrealistically small. While a powerful tool to infer physical properties, MOSFiT makes a series of simplying assumptions that likely result in an underestimation of the true errors, similar to the conclusion made for MOSFiT modeling of tidal disruption events (Mockler et al. 2019). \nIn particular, the assumption of a constant grey opacity may significantly affect the model posteriors, as M ej , V ej and κ are degenerate parameters in predicting the shape of the light curve. Notably, κ may vary up to an order of magnitude on the timescales of our observations (0 . 1 ≲ δt ≲ 30 days) as the ejecta temperature and density directly impact the elements' ionization states (e.g., Tanaka et al. 2020; Banerjee et al. 2024). We quantify the minimum systematic error introduced by our assumed opacities by running fits of AT2017gfo in which each component's opacity is a free parameter whilst holding the other two components' κ values constant and keeping the same prior ranges listed in Table 2. We then determine the difference in derived component M ej relative to the masses inferred in Section 4.1 with fiducial, fixed opacities. Specifically, we explore the effects of the range κ R = 5 -30 cm 2 g -1 (compared to κ R = 10 cm 2 g -1 in our Section 4.1 fits), κ P = 1 -5 cm 2 g -1 ( κ P = 3 cm 2 g -1 ), and κ B = 0 . 2 -2 . 5 cm 2 g -1 ( κ B = 0 . 5 cm 2 g -1 ). We use a minimum of κ B = 0 . 2 cm 2 g -1 as it corresponds to \nthe minimum value dictated by Thompson scattering of ionized elements (Paczynski 1983). We base the remaining ranges on calculations for kilonova opacities at δt ≈ 1 day (Banerjee et al. 2024; Tanaka et al. 2020). \nBased on the three fits with free κ values, we find the median errors on the component masses are ∆ M ej , R = ( -0 . 04 , +0 . 0008) M ⊙ , ∆ M ej , P = ( -0 . 004 , +0 . 02) M ⊙ and ∆ M ej , B = ( -6 . 5 × 10 -5 , 0 . 0006) M ⊙ . For the total mass, fitting with κ as a free parameter results in only modestly lower total ejecta masses of which the median value is δM ej , tot = -0 . 001 M ⊙ . We find that our fits varying κ P result in the most significant source of uncertainty. This result can naturally be explained as values of κ P more similar to κ B or κ R will divert a portion of the luminosity typically explained by M ej , B or M ej , R to M ej , P . This results in a different ratio between the two component masses but an overall similar M ej , tot . We conclude that our choice of opacity (within the range of values explored here) does not significantly impact our ejecta mass results. We do not include these uncertainties where we compare between our uniform fits. \nWe caution that, in addition to the effect explored above, the kilonova models used in this analysis do not account for the effect of jet-ejecta interaction, shock cooling, central engine activity, or magnetic fields, all of which may play a large role in determining ejecta masses (e.g., Radice et al. 2018; Ciolfi & Kalinani 2020; Nativi et al. 2021; Shrestha et al. 2023; Curtis et al. 2024; Hamidani et al. 2024b). In addition, several pieces of kilonova physics are still not well understood even in state-of-the-art simulations, such as the uncertainty in wavelength-dependent opacities, nuclear heating rate, and thermalization efficiencies (Barnes et al. 2021; Bulla 2023; Brethauer et al. 2024; Sarin & Rosswog 2024). Further, our two-stage analysis of the afterglow and kilonova may result in additional uncertainty and bias in the derived kilonova parameters compared to a joint models (e.g., Wallace & Sarin 2024). To account for this point, we exclude data at < 0 . 5 days in our kilonova fits, the timescale on which the afterglow is most likely to dominate, and propagate the uncertainties in our afterglow model to the data passed to the kilonova model (Sections 3 and 4.1). \nIn light of these uncertainties, we emphasize that the aim of this work is to perform uniform modeling on a sample of kilonovae, allowing for an exploration of diversity and correlations with γ -ray and environment properties. For all objects in our sample besides AT 2017gfo, the datasets are relatively sparse (e.g., Section 2 and Figure 2), limiting our ability to investigate each uncertainty listed above. We provide all observations (Ta- \nle 5), including afterglow-subtracted photometry (Table 4), for the community to model with other existing, or future, codes.", '5.1. Comparison to γ -ray Properties and Implications for Long/Short Progenitors': 'Motivated by the unknown progenitor properties and/or mechanism driving merger-origin long GRBs, we next examine any trends between the kilonova ejecta and γ -ray properties. Historically, one factor in explaining the divide in γ -ray duration between BNS and stellar progenitors is their order-of-magnitude difference in the mass reservoir surrounding the central compact object. The significantly higher masses and larger physical size of long GRB progenitor stars lead to a longer timescale for accretion, translating to a longer-lived jet (e.g., Tchekhovskoy et al. 2011). To produce a long-lived GRB jet from a neutron star merger, previous works have posited the progenitors are a white dwarf-neutron star binary (Yang et al. 2022; Wang et al. 2024) or an NSBH (e.g., Rastinejad et al. 2022a; Gompertz et al. 2022), or that the merger remnant is a magnetar (e.g., Rastinejad et al. 2022a; Gompertz et al. 2022). \nHere, we explore the theory that a long-lived, massive ( ≈ 0 . 2 M ⊙ ) accretion disk, a product of an asymmetric binary merger, is capable of powering longer-lived and more energetic GRBs (Gottlieb et al. 2023). The merger of an asymmetric binary, whether it is a BNS or an NSBH with a favorable mass ratio ( Q ≈ 3 -5; e.g., Kawaguchi et al. 2016), will produce a greater amount of lanthanide-rich tidal dynamical ejecta compared to a symmetric binary (e.g., Hotokezaka et al. 2013; Kyutoku et al. 2018; Kawaguchi et al. 2020b). Within our modeling framework, this translates to an expected trend between M ej , R and the duration and/or energy of the GRB. \nTo investigate these possible trends, we compare the kilonova ejecta masses (Table 3) with the values of E γ, iso (described in Section 2.1), beaming-corrected E γ , and T 90 , rest (converted to the rest-frame using their respective redshifts; Table 1). To calculate E γ , we gather jet opening angle measurements ( θ j ) from the literature (Xu et al. 2009; Rastinejad et al. 2022a; Rouco Escorial et al. 2023). All events in our sample have measured θ j values from X-ray observations, with the exception of GRB050709 for which a lower limit is reported (Rouco Escorial et al. 2023). For GRB 230307A we use 5000 random draws from our afterglow model 1 (Section 3), as it is fit to the combined X-ray light curves from the literature, therein providing a tighter constraint on the jet break than previous analyses (Levan et al. 2024; Yang \net al. 2024). Our analysis finds θ j = 3 . 95 +1 . 72 -0 . 65 deg. We calculate the beaming-corrected energy, given by, \nE γ = [1 -cos( θ j )] × E γ, iso . (1) \nIn Figure 5, we show the results of this analysis. We observe that M ej , B (with the exception of the outlier GRB200522A; Section 4.2), M ej , R and M ej , tot generally increase with higher values of E γ, iso and E γ . We further observe that M ej , R increases with longer T 90 . The trends with E γ, iso , E γ , and T 90 are most apparent with M ej , R (Figure 5, third column), though the error bars for several short GRB masses and θ j preclude a firm conclusion. We do not observe any apparent trends between M ej , P (Figure 5, second column) and γ -ray properties. \nWe briefly test the statistical significance of any trends between the ejecta masses and the γ -ray properties. We apply the Pearson correlation coefficient (r-score) test to the datasets in each panel of Figure 5. We caution that this test is agnostic to a model, and thus does not probe all underlying physical motivations. We randomly draw values from the ejecta mass posteriors and calculate r- and p-scores with the E γ, iso , E γ , and T 90 values for 1000 iterations, producing distributions of 1000 rand p-values. We then determine the fraction of random draws that imply a significant correlation between the ejecta mass and γ -ray properties using a threshold of p < 0 . 05. We do not find that a significant fraction of the p-scores favor a correlation between the ejecta masses and any γ -ray properties. The strongest correlation is between M ej , R and T 90 , where 32% of p-scores indicate a significant correlation. \nThough we do not find any statistically significant correlations, we observe that GRBs with T 90 , rest ≳ 2 s have higher median red ejecta masses ( M ej , R ≳ 0 . 05 M ⊙ ) compared to typical short GRBs ( M ej , R ≲ 0 . 02 M ⊙ ) potentially hinting at a bimodality. We observe a similar pattern with M ej , B and E γ (with the exception of GRB200522A), though this component may also originate in the post-merger disk wind (Section 4.2). The implication that M ej , B seems to trend more strongly with γ -ray properties compared to M ej , P may also suggest that long GRB mergers produce relatively blue disk winds, perhaps due to energy injection from the GRB. At present, our results could indicate asymmetric binaries as the progenitors of merger-driven long GRBs. However, to establish any statistically significant correlations, a larger population of joint GRB-kilonova detections with well-constrained ejecta masses is necessary. \nFinally, we note that while asymmetric binaries are uncommon in the observed population of BNS systems (e.g., Tauris et al. 2017), LVK observations have revealed an asymmetric BNS (GW190425; Abbott et al. \nFigure 5. The blue (first column), purple (second), red (third) and total (fourth) ejecta masses for each on-axis GRB event plotted against the respective γ -ray properties: E γ, iso (top row), beaming-corrected E γ (middle row), and T 90 (bottom row). Long GRBs are marked with diamonds, while short GRBs are marked with circles. As there is only a lower limit on the jet half-opening angle of GRB 050709 (Section 5.1) we mark the resulting upper limit on E γ with a triangle. We observe potential trends between E γ, iso , E γ and the red and total ejecta mass. \n<!-- image --> \n2020a) and an NSBH (GW230519; The LIGO Scientific Collaboration et al. 2024) merger. These detections may point to these asymmetric binaries being common in the Universe, with potential rates able to explain the increasing (but highly uncertain) rates of long GRBs from mergers.', '5.2. Kilonova Contribution to Universal r -Process': "At present, kilonovae are the only observationallyconfirmed source of r -process production in the Uni- \nverse. Indirect observational evidence may favor the existence of a second, 'faster' heavy element nucleosynthesis channel. Specifically, to explain observations of r -process-enhanced metal poor stars in the Milky Way and dwarf galaxies (e.g., Ji et al. 2016; Hansen et al. 2017; Frebel 2018) a significant fraction of NS mergers are required to have short delay times from enrichment to star formation (e.g., Zevin et al. 2022). Simulations have demonstrated that collapsar and magnetohydrodynamical (jet-driven) supernovae could be this \nsecond channel (e.g., Siegel et al. 2019; Mosta et al. 2018; Halevi & Mosta 2018). At present, observations of these candidates are limited and those that exist do not support r -process enrichment (Blanchard et al. 2023; Anand et al. 2024; Rastinejad et al. 2024). Here, we explore if the average r -process yield from kilonovae, calculated using the median ejecta masses calculated in Section 4 along with the current NS merger rates, is capable of producing the estimated r -process abundance in the Milky Way. In this analysis, we assume the kilonovae in our sample are created by BNS mergers only, though we note in Section 5.1 the kilonovae following long GRBs are favored to be from asymmetric mergers that may be NSBH events. We make this assumption for simplicity, as only a small fraction of NSBH events are expected to produce kilonovae and the rate of NSBH events is subdominant compared to the rates of BNS mergers (e.g., Abbott et al. 2023; Mandel & Broekgaarden 2022). \nTo calculate the r -process enrichment in the Milky Way from NS mergers, we employ the equation from Rosswog et al. (2018): \nM r ∼ 17 000 M ⊙ [ R BNS 500Gpc -3 yr -1 ][ ¯ m ej 0 . 03 M ⊙ ][ τ gal 1 . 3 × 10 10 yr ] (2) \nwhere R BNS is the rate of NS mergers, ¯ m ej is the average kilonova ejecta mass, and τ gal is the age of the Milky Way, which we fix to 1 . 3 × 10 10 yr. We employ the range in BNS merger rate calculated from the Gravitational Wave Transient Catalog 3 (GWTC-3) of R BNS = 10 -1700 (Abbott et al. 2023). For ¯ m ej we employ our median for the eight kilonovae in our sample, M ej , tot = 0 . 085 +0 . 108 -0 . 044 M ⊙ , including the uncertainties due to the opacity (Section 4.5). Taking these values together, we find a wide range of M r ≈ 500 - 400,000 M ⊙ for BNS mergers, mostly driven by the large uncertainty in BNS merger rate. \nThis range for M r encompasses the estimate of total r -process mass in the Milky Way, M r, MW ≈ 23 , 000 M ⊙ (Venn et al. 2004; Battistini & Bensby 2016; Hotokezaka et al. 2018) and, on the lower end, leaves room for the existence of a second r -process source. As discussed in Sections 4.3-4.5, our ¯ m ej value is likely to be an overestimate of the true value for two reasons. First, less luminous, and thus less massive, kilonovae are likely to have been missed due to observational biases (Section 4.3). Second, in keeping with fits to AT 2017gfo, our MOSFiT model derives ejecta mass values on the upper end of ranges from previous fits (Section 4.4). However, we anticipate that the uncertainty in ¯ m ej in subdominant compared to the uncertainty in BNS merger rate, as our values are generally comparable with literature values, \nwhere they exist, (Section 4.4) and likely do not vary beyond an order of magnitude. \nThe specific star formation rate of the host galaxy is a dominant factor in governing what fraction of ejected r -process elements enrich later generations of stars (Nugent et al., in prep.). Notably, in comparison to the quiescent host galaxy of GW 170817/AT 2017gfo (Blanchard et al. 2017; Levan et al. 2017), the hosts of the kilonovae in our on-axis GRB sample are all star-forming (Nugent et al. 2022; Levan et al. 2024; Table 1). Though losses may still be significant for these latter events, it is more likely that they enriched their galaxies with heavy elements compared to AT 2017gfo.", '5.3. Future Kilonova Observations': "We next consider the implications of our sample and model light curves for future wide-field kilonova searches, either triggered with a GRB or GW event or untriggered (e.g, Smartt et al. 2017; Doctor et al. 2017; Kasliwal et al. 2017; Andreoni et al. 2020). For these searches, an understanding of the span in kilonova light curve diversity is critical in the vetting process, which has yet to be enabled with observations beyond AT2017gfo. \nFor each of the seven on-axis GRB events in our sample, we create 900 light curves in the observed-frame rizJHK -bands. Each light curve is based on a random draw from the posterior (we use GRB 230307A Model 1 in our fit given the reasoning in Section 4.2) and matched to the approximate rest-frame band in Figure 6. We create 50 light curves for GRB 170817A/AT 2017gfo due to computational constraints and iterate over these 18 times. Due to the tight posteriors on this event, we do not expect this process to affect our conclusions. Across the random draws, we calculate the median, 68% and 90% credible region in luminosity space for the sample of kilonovae. In Figure 6 we show the median and kilonova luminosity range in each filter. Taken together, our results indicate that future kilonovae will span ≳ one order of magnitude in luminosity. \nAs we are motivated by both targeted and untargeted kilonova searches in large surveys, we also plot the single image 5 σ depths for the Rubin Observatory (Bianco et al. 2022), the WINTER J -band limiting magnitude (Frostig et al. 2022) and the 'Wide Tier' ( RZ -band) and 'Deep Tier' ( JH -band) limiting magnitudes for Roman High Latitude Time Domain Survey (HLTDS; Rose et al. 2021), shifted to z = 0 . 1. We caution that our results are based mostly on on-axis events, and the bolometric luminosity of kilonovae may vary up to factor of ∼ 10 with viewing angle (Darbha & Kasen 2020). In the riz -bands, Rubin observatory will be sensitive \nFigure 6. The median (solid line), 68% credible range (dotted lines) and 90% credible range (shaded region) in luminosity space shown in the rizJHK -bands and calculated using random draws (light grey lines) from each event. We also show the expected depths of several current and upcoming wide-field observatories that plan to engage in kilonova searches (Margutti et al. 2018; Rose et al. 2021; Bianco et al. 2022; Frostig et al. 2022; Andreoni et al. 2024). \n<!-- image --> \nto the full and upper end of the 68% credible range of kilonovae at z = 0 . 1 out to δt ≈ 3 and ≈ 7 days, respectively. As shown with our analysis GRB 200522A, in Section 4.2, a single epoch of simultaneous rest-frame optical observations on these timescales is sufficient for order-of-magnitude constraints on M ej , B . We therefore find that order-of-magnitude constraints on M ej , B are possible for kilonovae observed by Rubin at one epoch. We acknowledge that large outstanding challenges exist to distinguish these rare events from other transients, especially in just one or two epochs of observations, and encourage further development of automatic vetting tools. \nRoman's HLTDS will be a powerful tool in measuring the properties of low-redshift kilonovae if events can be identified. Indeed, Roman is capable of detecting z = 0 . 1 kilonovae in the optical and near-IR out to δt ≈ 2 and 3 weeks, respectively. If the Roman HLTDS observations occur at a five-day cadence, Roman is poised to observe z = 0 . 1 kilonovae over ≈ 2-3 \nepochs in the optical and ≈ 3-5 epochs in the near-IR. Two to three epochs is sufficient for obtaining better than order-of-magnitude constraints on M ej -V ej , as we have demonstrated in Section 4. Finally, WINTER will be sensitive to the most luminous kilonovae, but not the 68% credible range, at z = 0 . 1. \nIn addition to wide-field searches in the nearby Universe, well-localized Swift GRBs continue to be a promising method to detect kilonovae. These events come with their own set of challenges, including higher redshifts, bright afterglows and low rates. Further, the more coarsely localized detections of GRBs by Fermi , Space Variable Objects Monitor (SVOM; Atteia et al. 2022), the InterPlanetary Network (IPN; e.g., Atteia et al. 1987; Svinkin et al. 2022) and other γ -ray telescopes offer a second route to finding kilonovae using 'targeted' wide-field surveys. However, for all GRB kilonovae, the higher average redshifts render follow-up with large-aperture ground-based telescopes and spacebased observatories critical. Indeed, seven of the eight \nkilonovae in our sample had key observations with HST that were critical to detections on ≳ week timescales. Looking to the future, JWST can obtain similar observations for events out to z ≈ 1 (e.g., Rastinejad et al. 2021). For lower-redshift events JWST has the power to capture the kilonova spectral energy distribution (SED) in the nebular phase, an important element in refining the mapping from observations to M ej .", '6. CONCLUSIONS': "We have compiled and collated the multi-wavelength light curves of eight kilonovae from the GCNs and literature. Five of these events follow short GRBs, while three events follow long GRBs, allowing us to explore trends between γ -ray and kilonova ejecta properties. We uniformly model the afterglows of seven events with on-axis GRBs, producing 'afterglow-subtracted' light curves. We fit the afterglow-subtracted light curves with a threecomponent kilonova model in MOSFiT that accounts for geometric viewing effects. Our fits provide reasonable fits to the data, and we compare our posteriors to those in the literature. Our major conclusions are as follows: \n- · Our fits unveil a wide span in derived kilonova properties, namely M ej and V ej , implying that the progenitors and/or remnants of these mergers are also diverse. We determine that the luminous kilonova of GRB 200522A has a significantly more massive M ej , B compared to the sample of events (or the luminosity may be boosted by nonradioactive heating like a magnetar; Fong et al. 2021), while the kilonova of GRB 160821B is the least massive of the total sample.\n- · While well-sampled events provide the tightest constraints, we also find value in kilonovae with a single color measurement, particularly if their colors are unique (e.g., GRBs 130603B and 200522A).\n- · We discuss the main uncertainties in our modeling (Section 4.5) and compare our results to previous fits with other modeling codes. We emphasize that all observations used in this work, including our 'afterglow-subtracted' light curves, are provided for future modeling endeavors in Tables 4 and 5.\n- · We demonstrate that GW170817/AT2017gfo is a 'representative' kilonova in each components' M ej and V ej (Table 3), when compared to the seven kilonovae in our sample and deep upper limits from the literature. Given this, our estimate of the total Milky Way r -process mass produced by kilonovae does not change significantly when using the median ejecta mass of our sample compared to previous estimates made for AT 2017gfo.\n- · We explore trends between our derived ejecta masses and E γ, iso , beaming-corrected E γ and T 90 . Overall, we do not find any statistically significant correlations but observe that long GRB kilonovae have larger median M ej , R compared to short GRB kilonovae. We hypothesize that this is indicative of an asymmetric binary merger origin for longerlived GRBs. A larger sample of well-studied kilonovae following short and long GRBs will be critical to confirming this hypothesis.\n- · We produce median, 68% and 90% confidence range light curves in a variety of bands based on the posteriors of the eight events in our sample. Comparing these light curves to the expected depths of upcoming surveys, we anticipate that Rubin and Roman will be sensitive to the majority of the kilonova luminosity range for z ≲ 0 . 1 and are capable of order-of-magnitude mass constraints. \nHere, we have shown that the existing sample of kilonovae, the majority of which are detected at a fixed viewing angle, demonstrates diversity and trends with γ -ray properties. Widening the sample of these events requires dedicated strategies for observational pointings (e.g., Margutti et al. 2018; Coulter et al. 2024) and kilonova candidate vetting (e.g., Rastinejad et al. 2022b) that take into account the full diversity of compact binary merger EM counterparts and environmental properties. The advent of next-generation GW detectors, deep wide-field surveys, and new γ -ray instruments, when combined with dedicated search strategies, opens the doors for unprecedented exploration into the physics of compact binary mergers, jets, and kilonovae.", '7. ACKNOWLEDGEMENTS': 'The authors thank Ore Gottlieb, Genevieve Schroeder, Anya Nugent, Tanmoy Laskar, Igor Andreoni, Sylvia Biscoveneau, Nick Kaaz, Ben Margalit and Michael Fausnaugh for helpful conversations regarding this manuscript. \nJ.C.R. acknowledges support from the Northwestern Presidential Fellowship. The Fong Group at Northwestern acknowledges support by the National Science Foundation under grant Nos. AST-2206494, AST-2308182, and CAREER grant No. AST-2047919. W.F. gratefully acknowledges support by the David and Lucile Packard Foundation, the Alfred P. Sloan Foundation, and the Research Corporation for Science Advancement through Cottrell Scholar Award 28284. \nTable 4 . Afterglow-subtracted Observations \nTable 4 continued \nTable 4 (continued) \nTable 4 continued \nTable 4 (continued) \nTable 4 continued \nTable 4 (continued) \nTable 4 continued \nTable 4 (continued) \nTable 4 continued \nTable 4 (continued) \nTable 4 continued \nTable 4 (continued) \nTable 4 continued', 'Rastinejad et al.': "Table 5 (continued) \nNote - Observations are not corrected for Galactic nor local extinction. \n† Starred observations were employed in our afterglow analysis. \nReferences : (1) Fox et al. 2005, (2) Hjorth et al. 2005, (3) Covino et al. 2006, (4) UKSSDC (Evans et al. 2007, 2009), (5) Mangano et al. 2007, (6) Schmidt et al. 2006, (7) Xu et al. 2009, (8) Cobb 2006, (9) Fynbo et al. 2006, (10) Della Valle et al. 2006, (11) Yang et al. 2015, (12) Londish et al. 2006, (13) Fong et al. 2014, (14) de Ugarte Postigo et al. 2014, (15) Cucchiara et al. 2013, (16) Berger et al. 2013, (17) Tanvir et al. 2013, (18) Troja et al. 2019, (19) Lamb et al. 2019, (20) Kasliwal et al. 2017, (21) Fong et al. 2021, (22) O'Connor et al. 2021, (23) Mei et al. 2022, (24) Ito et al. 2021, (25) Troja et al. 2022, (26) Xiao et al. 2024, (27) Kumar et al. 2021, (28) Rastinejad et al. 2022a, (29) Strausbaugh & Cucchiara 2021, (30) Mao et al. 2021, (31) Moskvitin et al. 2021, (32) Gupta et al. 2021, (33) Yang et al. 2024, (34) Levan et al. 2024.", 'REFERENCES': "Abbott, B. 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2024PhRvD.110j4044F	We develop a relativistic scenario of fast magnetic reconnection process for general magnetohydrodynamical plasmas around Kerr black holes. Generalizing the Petschek model we study various properties of the reconnection layer in distinct configurations. When current sheet forms in the zeroangularmomentum ZAMO frame which corotates with the black hole the reconnection rate for both radial and azimuthal configurations is decreased by spacetime curvature. However when the current sheet forms in a nonZAMO frame which rotates either faster or slower than the black hole detail analysis establishes that for any given slow rotations subrelativistic at most and mildly relativistic inflow the ZAMO observer will find asymmetric reconnection rates for radial configuration. It is decreased on one side of the current sheet and is increased on the other side in comparison to the unrotation limit. This is valid to both the SweetParker and the Petschek scenario. The results clarify the effects of rotation on the reconnection layer in the laboratory frame in the flat spacetime limit.	2024-11-01T00:00:00Z	['2024arXiv240905434F', 'arXiv:2409.05434', '10.48550/arXiv.2409.05434', '10.1103/PhysRevD.110.104044', '2024PhRvD.110j4044F']	['General relativity', 'alternative theories of gravity', 'Astrophysics - High Energy Astrophysical Phenomena']	Fast magnetic reconnection in Kerr spacetime	2,024	189	0.39	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	3	https://arxiv.org/pdf/2409.05434.pdf	{'Fast magnetic reconnection in Kerr spacetime': 'Zhong-Ying Fan 1 , Yuehang Li 2 , Fan Zhou 2 , Minyong Guo 2 , 3 ∗ \n- 1 Department of Astrophysics, School of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, P. R. China\n- 2 School of physics and astronomy, Beijing Normal University, Beijing 100875, P. R. China\n- 3 Key Laboratory of Multiscale Spin Physics, Ministry of Education, Beijing 100875, P. R. China \n100871, P.R. China', 'Abstract': 'We develop a relativistic scenario of fast magnetic reconnection process, for general magnetohydrodynamical plasmas around Kerr black holes. Generalizing the Petschek model, we study various properties of the reconnection layer in distinct configurations. When current sheet forms in the zeroangular-momentum (ZAMO) frame which corotates with the black hole, the reconnection rate for both radial and azimuthal configurations is decreased by spacetime curvature. However, when the current sheet forms in a non-ZAMO frame, which rotates either faster or slower than the black hole, detail analysis establishes that for any given slow rotations (subrelativistic at most) and mildly relativistic inflow, the ZAMO observer will find asymmetric reconnection rates for radial configuration: it is decreased on one side of the current sheet and is increased on the other side in comparison to the unrotation limit. This is valid to both the Sweet-Parker and the Petschek scenario. The results clarify the effects of rotation on the reconnection layer in the laboratory frame in the flat spacetime limit.', '1 Introduction': "Magnetic reconnection is a fundamental process for magnetohydrodynamical plasmas, which converts magnetic energy into plasmas particle energy rapidly. It is generally considered to be a competitive mechanism to explain many high energy phenomenons in astrophysics [1, 2], such as coronal mass ejections, magnetospheric substorms, stellar flares and gamma-ray bursts. \nDespite that magnetic reconnection was usually studied in the non-relativistic regime [1, 2], it was recently recognized that special relativistic effect is of paramount importance for strongly magnetized plasmas [3, 4]. This is the case when the magnetic energy density exceeds the plasmas rest mass density greatly so that the Alfv'en wave speed approaches to the speed of light. Generalization of magnetic reconnection to relativistic regimes has been conducted by a number of authors, both analytically [5-8] and numerically [9-12]. It was found that relativistic effect is extremely important for magnetic energy conversion and particle accelerations [13-16]. In particular, the highly efficient conversion of magnetic energy into nonthermal particle energy [17-20] suggests that relativistic reconnection is a primary candidates to explain the high energy emissions in magnetically dominated environments like pulsars and jets from gammar-ray bursts and from active galactic nuclei. \nIn contrast, the effects of spacetime curvature of black holes to magnetic reconnection are much less explored. In [21], generalization of Sweet-Parker model to curved spacetime was first developed by Asenjo and Comisso. It was established that for various current sheet configurations, the reconnection rate will be decreased by the spacetime curvature in comparison to the flat spacetime limit. Inclusion of collisionless effect and gravitational electromotive force for magnetic reconnection in curved spacetime are further explored in [22, 23]. However, for the Sweet-Parker scenario, an essential equetion is the reconnection rate is very small (without collisionless effect): it is of the order ∼ S -1 / 2 , where S = L/η is the relativistic Lindquist number. Since the Lindquist number is generally very large in astrophysical environments, the characteristic time scale of energesis in this model is usually much larger than the observation time scale. It is widely known that this issue can be resolved in a Petschek like model, in which the reconnection rate is significantly improved to ∼ (ln S ) -1 . \nIn this work, our first motivation is to develop the Petschek model for relativistic magnetohydrodynamic plasmas around rotating black holes. We will study the effects of spacetime curvature to the reconnection rate and other important properties of the reconnection layer for various current sheet configurations. Our second motivation is to study magnetic reconnection for a general rotating reconnection layer. It was assumed in [21] that plasmas corotates with the black hole and the reconnection was studied for the zero-angular-momentum (ZAMO) observers. Since generically the current sheet may form in a non-ZAMO frame which rotates either faster or slower than the ZAMO frame, we would like to study the general situation from the view of the ZAMO observer carefully. We explore a general rotating disk in the equatorial plane of the black hole. We will show that for slow rotations (subrelativistic at most) and mildly relativistic inflow, the ZAMO observer will always find asymmetric reconnection rates for radial configuration: it is decreased on one side of the current sheet and is increased on the other side in comparison to the unrotation limit. \nThe paper is organized as follows. In section 2, we introduce the governing equations for relativistic magnetohydrodynamical plasmas in curved spacetime. In section 3, we explore various properties of Petschek model for radial current sheet configuration in the ZAMO frame. We compute the reconnection rate and establish that the gravitational potential gives the leading order corrections to decrease the reconnection rate in the weak gravity regimes. In section 4, we study various properties of the reconnection layer in azimuthal direction for ZAMO observers. In section 5, we study both the Sweet-Parker and the Petschek model in the non-ZAMO frame. We focus on a rotating reconnection layer in radial configuration. We briefly conclude in section 6.", '2 Preliminaries': "Consider relativistic magnetohydrodynamic (MHD) fluids in curved spacetime. The fluids is governed by the continuity equation [24] \n∇ µ ( ρU µ ) = 0 , (2.1) \nthe energy-momentum conservation \nand the Ohm's law \nU ν F µ ν = η ( J µ -ρ e U µ ) , (2.3) \nwhere ρ ( ρ e ) is the mass (charge) density, h is the enthalpy density, p is the pressure, U µ is the fluid four velocity and J µ is the electric current. Besides, MHD fluids obeys the Maxwell's equation \n∇ ν F µν = J µ . (2.4) \nThese provide a complete set of equations for MHD in a fixed background. \nConsider rotating black holes, whose metric can be written in the form of \nds 2 = -α 2 dt 2 + 3 ∑ i =1 ( h i dx i -αβ i ) 2 , (2.5) \nwhere h 2 0 = -g 00 , h 2 i = g ii . Here β i is related to the black hole rotations β i = h i ω i /α,ω i = -g 0 i /h 2 i and α 2 = h 2 0 + ∑ i ( h i ω i ) 2 . An advantage of this coordinate is one can introduce the ZAMO frame { ˆ t , ˆ x i } [25], under which the spacetime is locally flat ds 2 = η µν d ˆ x µ d ˆ x ν . One has \nd ˆ t = αdt, d ˆ x i = h i dx i -αβ i dt , (2.6) \nor the inverse transformation \ndt = α -1 d ˆ t , dx i = h -1 i ( d ˆ x i + β i d ˆ t ) . (2.7) \nThroughout out this paper, we use hats to stand for all vectors or tensors in the ZAMO frame. \nTo study magnetic reconnection in the ZAMO frame, we rewrite the energy-momentum equation (2.2) in the form of \n∇ ν S µν = H µ , (2.8) \nwhere \n∇ ν ( hU µ U ν ) = -∇ µ p + J ν F µ ν , (2.2) \nS \nµν \n= \nhU \nµ \nU \nν \n, \nH µ = -∇ µ p + J ν F µ ν . (2.9) \nIt was established in [26] that any equation of this form in the ZAMO frame can be expressed into \n1 h 1 h 2 h 3 3 ∑ j =1 ∂ ∂x j [ αh 1 h 2 h 3 h j ( ˆ S ij + β j ˆ S i 0 ) ] + ˆ S 00 h i ∂α ∂x i -α 3 ∑ j =1 ( G ij ˆ S ij -G ji ˆ S jj + β j G ij ˆ S 0 i -β j G ji ˆ S 0 j ) + 3 ∑ j =1 σ ji ˆ S 0 i = α ˆ H i , (2.10) \nwhere G ij = -(1 /h i h j ) ∂h i /∂x j and σ ij = -(1 /h j ) ∂ ( αβ i ) /∂x j . \nSince we are interested in studying magnetic reconnection in Kerr black holes, we present the metric in the Boyer-Lindquist (BL) coordinates ( x 0 , x 1 , x 2 , x 3 ) = ( t , r , θ , ϕ ) \nh 0 = √ 1 -2 r g r ρ 2 , h 1 = Σ √ ∆ , h 2 = Σ , h 3 = Πsin θ ρ , (2.11) \nwhere r g = GM is the gravitational radius and a = J/J max ≤ 1 is the rotation parameter ( J is the angular momentum and J max = GM 2 ). The functions ρ, ∆ and Π are given by \nΣ 2 = r 2 + a 2 r 2 g cos 2 θ , ∆ = r 2 -2 r g r + a 2 r 2 g , Π 2 = ( r 2 + a 2 r 2 g ) 2 -∆ ( ar g sin θ ) 2 . (2.12) \nIn addition, the black hole rotates only around the ϕ -direction ω 1 = ω 2 = 0 , and ω 3 = 2 r 2 g ar/ Π 2 so that β i = β ϕ δ iϕ . \nTo study magnetic reconnection in a Petschek-like scenario, we consider a quasi-two-dimensional electric current sheet (with the thickness far less than the length δ ≪ L ) and adopt quasi-stationary conditions ( ∂ t ≈ 0). While the current sheet can form in different locations around the black holes, we will focus on two configurations in the equatorial plane of the black hole. We will assume the current sheet is comoving with the ZAMO frame at first. The situation of a general rotating reconnection layer will be studied later.", '3 Reconnection layer in radial direction': "Consider reconnection layer in the radial direction at first, see the illustration figure Fig. 1. In a Petschek model, there are three patches in the current sheet: i) The inflow region. The plasma is viewed as the ideal MHD fluid. In particular, the ingoing magnetic fields upstream of the sheet are not antiparallel any longer. The field lines enter the diffusion region with an inclination angle. This is extremely \nFigure 1: Raidal magnetic reconnection layer of Petschek model in Kerr spacetime. Here y ≡ r -r C . \n<!-- image --> \nimportant. The magnetic field line bends down and is reconnected at the neutral line. It turns out that even if the magnetic field has small perturbations around a uniform background, the reconnection rate will be improved significantly. ii) The diffusion region. In this region, magnetic reconnection develops via Sweet-Parker process. Previously this region was studied carefully in [21]. The reconnection rate is evaluated as \nM = ˆ v i ∼ X Y h 1 ∣ ∣ ∣ C ∼ ( Y h 1 η ) -1 / 2 ∣ ∣ ∣ C . (3.1) \nHowever, in a Petschek model, this region is extremely small, compared to the sheet length Y ≪ L . iii) The outflow region. In this region, the shape of the current sheet is generally complicated, with the thickness depending on the radial coordinates. Consider an arbitrary point O on the boundary ∆ = ∆( y ) , y = r -r C . The boundary approaches asymptotically to a straight line, as will be shown later. \nTo proceed, we work in the quasi-stationary limit ∂ t ≈ 0. Since the sheet is put in the radial direction, we assume ˆ v ϕ ≈ 0 ≈ ˆ B ϕ and ∂ ϕ ≈ 0. The electric current and the reconnected electric field are given by J = ˆ J ϕ e ϕ , E = ˆ E ϕ e ϕ , where e ϕ stands for unit vector in azimuthal direction 1 . Without loss of generality, we focus on the upper half region of the sheet. We set ˆ v in = -ˆ v i e θ and ˆ B in = -ˆ B 0 e r + ˆ B 1 , where B 0 is a constant magnetic field. Since we are interested in small perturbations around a uniform magnetic field, we assume \| ˆ B 1 \| ≪ ˆ B 0 . The outflow velocity ˆ v o in the ZAMO frame can be calculated using the r -component of the energy-momentum equation \n∂ ∂r ( h ˆ γ 2 ( ˆ v r ) 2 ) + ∂ ∂θ ( h 1 h -1 2 h ˆ γ 2 ˆ v r ˆ v θ ) + h ˆ γ 2 ∂ ln α ∂r = -∂p ∂r -h 1 ˆ J ϕ ˆ B θ . (3.2) \nNotice that the second term on the l.h.s cannot be dropped since O is located on the boundary. This term gives nontrivial contributions when integrating the equation along θ -direction. Notice that we consider \na short current sheet L ≪ r g and focus on weakly gravity regimes so that gravitational tidal force could be ignored when performing integrations (however when collisionless effect is included, gravitational electromotive force will be important and cannot be ignored even in the weak gravity regimes [23]). In principle, integrating the equation from C to O , we are able to derive the outflow velocity but the result depends on the sheet thickness ∆( y ) and the magnetic field ˆ B θ in the outflow region. So let's first study general behaviors of these quantities. \nThe first estimation of the thickness ∆( y ) can be achieved from flux conservation. The inflow flux ∂ θ ( αh 1 h 3 ρ ˆ γ i ˆ v i ) / ( αh 1 h 2 h 3 ) ≈ ρ ˆ γ i ˆ v i / ∆( y ) must balance the outflow flux \n∂ r ( αh 2 h 3 ρ ˆ γ o ˆ v o ) / ( αh 1 h 2 h 3 ) ≈ ρ ˆ γ o ˆ v o / ( yh 1 ( y ) ) . This gives \n∆( y ) ≈ ˆ γ i ˆ v i ˆ γ o ˆ v o yh 1 ( y ) . (3.3) \nNext, the Ohm's law on the current sheet leads to \nˆ γ ˆ E ϕ + ˆ γ ˆ v r ˆ B θ = η ˆ J ϕ . (3.4) \nSince beyond the sheet the plasmas is ideal MHD fluid η ≈ 0, the electric field at the center incident point is ˆ E ϕ \| i ≈ ˆ v i ˆ B 0 . Quasi-stationary approximation implies the electric field on the sheet is homogeneous so ˆ E ϕ o ≈ ˆ E ϕ \| i . In addition, the electric current ˆ J ϕ at O can be evaluated from the Maxwell's law \nˆ J ϕ ∣ ∣ ∣ o ≈ h -1 2 ∂ θ ˆ F ϕθ ≈ ˆ B 0 ∆( y ) . (3.5) \nIntegrating (3.4) along thickness of the sheet and using (3.3), we deduce \n∆( y ) -ˆ γ i ˆ B o ˆ γ o ˆ B 0 yh 1 ( y ) -η ˆ γ o ˆ v i = 0 , (3.6) \nwhere we have set ˆ B θ ∣ ∣ ∣ O ≡ -ˆ B o . This connects thickness of the sheet to the reconnected magnetic field on the boundary. Close to the diffusion region, the second term can be dropped so that ∆ ≈ η/ ˆ v i ≈ X , consistent with the Sweet-Parker process [21]. Otherwise, in the far outflow region y ≈ L , the third term can be dropped so that asymptotically \n∆( y ) yh 1 ( y ) ∣ ∣ ∣ y ≈ L ≈ ˆ γ i ˆ B o ˆ γ o ˆ B 0 . (3.7) \nCombing the result with (3.3), one finds the reconnection rate is related to the asymptotic thickness or the magnetic field \nˆ v i ≈ ˆ B o ˆ B 0 ≈ ∆( y ) yh 1 ( y ) ∣ ∣ ∣ y ≈ L , (3.8) \nwhere in the second equality we have assumed the outflow is mildly relativistic ˆ γ o ≈ 1, which will be verified later. This tells us the boundary approaches asymptotically to a straight line, with a rate ˆ v i . \nTo determine the full shape of the boundary, we integrate the momentum equation (3.2) along the sheet thickness and obtain \nˆ B o ˆ B 0 = h ˆ v 2 i ˆ B 2 0 [ 2 y∂ y ( yh 1 ∆ ) + ˆ γ o yh 1 ∆ ] + h ˆ γ 2 o ˆ B 2 0 ∆ yh 1 y∂ y ln α, (3.9) \nwhere we have adopted (3.3). For hot relativistic plasmas h = 4 p and at the neutral line, pressure balance implies p = ˆ B 2 0 / 2. By plugging this equation into (3.6), we deduce \n( 1 -2ˆ γ o y∂ y ln α ) ∆( y ) -2ˆ v 2 i yh 1 [ 2ˆ γ -1 o y∂ y ( yh 1 ∆ ) + yh 1 ∆ ] -η ˆ γ o ˆ v i = 0 . (3.10) \nThis solves the sheet thickness given the inflow and the outflow velocity at an arbitrary point on the boundary. \nIn the far outflow region y ≈ L ≫ Y , (3.9) implies \nˆ B o ˆ B 0 ∼ ˆ v 2 i yh 1 ∆ ∼ ˆ v i , (3.11) \nin agreement with previous estimations. On the other hand, near the diffusion region y ∼ Y , ∆ ∼ X , one has \nˆ B o ˆ B 0 ≈ 6ˆ v 3 i η yh 1 ( y ) . (3.12) \nThis implies that in the outflow region, the reconnected magnetic field grows and approaches to a constant asymptotically. \nTo solve the outflow velocity at O , we integrate the momentum equation (3.2) along the radial direction. We deduce \nh ( ˆ γ 2 o ˆ v 2 o + ˆ γ 2 o ˆ v o ˆ v i yh 1 ∆ + ˆ γ 2 o y∂ y ln α ) = p ( 1 + 2 yh 1 ∆ ˆ B o ˆ B 0 ) . (3.13) \nThis, together with (3.6), (3.9) (or (3.10)) provides a complete set of equations to solve quantities ∆ , ˆ B o , ˆ v o at any position on the outflow-inflow boundary, with a given reconnection rate. It turns out that the situation could be much simplified since only mildly relativistic outflow solution is permitted. To see this, consider the far region y ≈ L and using (3.7), (3.8), we deduce from (3.13) \n( 1 + ˆ γ o ) ˆ γ o 2 ˆ v 2 o + ˆ γ 2 o L∂ y ln α ∣ ∣ ∣ y ≈ L ≈ 1 4 ( 1 + 2ˆ γ o ) . (3.14) \nThis is a cubic equation for ˆ γ o , of which the general solution is complex. However, clearly an ultra \nrelativistic solution is not allowed. We may take ˆ γ o ≈ 1 and then 2 \nˆ v o = ( 1 -L∂ y ln α 2 ) 1 / 2 ∣ ∣ ∣ y ≈ L , ˆ γ o = ( 1 + L∂ y ln α 2 ) -1 / 2 ∣ ∣ ∣ y ≈ L . (3.16) \nThe results are essentially the same as the Sweet-Parker-like model [21]. This does not conflict with our setup since the Petschek model improves the reconnection rate rather than the kinetic energy of the outflow significantly. \nFinally, to complete our derivations, we compute the reconnection rate, according to the initial condition of the magnetic field in the inflow region. The more the magnetic field line bends down, the larger the reconnection rate becomes. Consider the upper half region ˆ B in = -ˆ B 0 e r + ˆ B 1 . We assume \| ˆ B 1 \| ≪ ˆ B 0 and the ingoing magnetic field can be well described by a static potential. On the boundary, continuity of magnetic field lines demands \nˆ B ⊥ = ˆ B in ⊥ . (3.17) \nThe slope of the boundary at O can be specified as \ntan ψ ≡ ∆( y ) yh 1 ( y ) . (3.18) \nIt follows that ˆ B ⊥ = -ˆ B o cos ψ and ˆ B in ⊥ = ˆ B 0 sin ψ + ˆ B 1 ⊥ . However, since \| ˆ B 1 \| ≪ ˆ B 0 , ψ should be very small ( this is a very good approximation for typical reconnection rate ˆ v i = 0 . 1 ∼ 0 . 001 ≪ 1). Hence in the far outflow region ψ ≈ ˆ v i ≈ ˆ B o / ˆ B 0 . One has ˆ B 1 ⊥ ≈ ˆ B 0 ψ -ˆ B o ≈ -2ˆ v i ˆ B 0 . In the lower half region, symmetry of the current sheet implies ˆ B 1 ⊥ ≈ 2ˆ v i ˆ B 0 . Therefore, the ˆ B 1 field in the inflow region can be viewed as the static field produced by two monopoles ˆ B 1 ⊥ ≈ ∓ 2ˆ v i ˆ B 0 . Using standard approach, the magnetic field at the center incidence point i can be evaluated as \nˆ B r 1 ∣ ∣ ∣ i ≈ 2ˆ v i ˆ B 0 π ln ( L 2 Y 2 ) ≈ 4ˆ v i ˆ B 0 π ln ( ˆ v 2 i S i h 1 ( r o ) ) , (3.19) \nwhere in the second equality L/Y ≈ ˆ v 2 i Sh 1 ( r o ) if L ≪ r g according to (3 . 1). Since ˆ B r in = -ˆ B 0 + ˆ B r 1 < 0, it implies that the reconnection rate can be significantly improved to ˆ v i ∼ (ln S ) -1 . We deduce \nˆ v i ≈ k ( ln ( ˆ v 2 i Sh 1 ( r o ) ) ) -1 , k = π ˆ B r 1 4 ˆ B 0 ∣ ∣ ∣ i . (3.20) \nSince ˆ B r 1 < ˆ B 0 , a typical choice is k = π/ 8 when ˆ B r 1 = ˆ B 0 / 2. This tells us that gravity effect decreases the reconnection rate with respect to the flat spacetime limit. This can be seen even more clearly by \nˆ γ o ≈ 1 . 176970 -0 . 276516 L∂ y ln α ∣ ∣ ∣ y ≈ L , ˆ v o ≈ 0 . 527315 -0 . 321662 L∂ y ln α ∣ ∣ ∣ y ≈ L . (3.15) \nFigure 2: Numerical solution for sheet thickness ∆, magnetic field ˆ B o / ˆ B 0 and outflow velocity on the inflow-outflow boundary. Here we take r C = 99 r g / 10 , L = r g / 10 , k = π/ 8 , S = 10 18 , which gives the reconnection rate ˆ v i ≈ 0 . 012. We have set a = 9 / 10 , r g = 10000. \n<!-- image --> \nexpanding the result in large r o \nˆ v i ≈ k [ ln ( ˆ v 2 i S ) + r g r o + (2 -a 2 ) r 2 g 2 r 2 o + · · · ] -1 . (3.21) \nIn Fig. 2, an example of numerical solutions for the sheet thickness ∆, the magnetic field ˆ B o and the outflow velocity ˆ v o in Kerr black holes is presented. Here we take r C = 99 r g / 10 , L = r g / 10 , k = π/ 8 , S = 10 18 , a = 9 / 10 , r g = 10000, which results to a reconnection rate ˆ v i ≈ 0 . 012. We consider mildly relativistic outflow ˆ γ o ≈ 1, which is justified by the numerical solution. The length scales of the diffusion region X ∼ 10 -13 , Y ∼ 10 -10 are highly suppressed compared to the length of sheet L = 1000. The shape of the boundary approaches to a straight line with the slope ∼ ˆ v i rapidly. The reconnected magnetic field increases monotonically and approaches to ˆ v i ˆ B 0 asymptotically.", '4 Reconnection layer in azimuthal direction': "Consider reconnection layer in the azimuthal direction, see Fig. 3. Again there are three patches in the current sheet. Without loss of generality, we focus on the upper right region. In the inflow region, the initial magnetic field upstream of the current sheet is specified as: ˆ B in = -ˆ B 0 e ϕ + ˆ B 1 with a small incidence angle \| ˆ B 1 \| ≪ ˆ B 0 . In the diffusion region. Sweet-Parker process dominates, with the reconnection rate [21] \nM∼ h 1 X Y ∣ ∣ ∣ C ∼ ( Y h 3 ηr ) -1 / 2 ∣ ∣ ∣ C . (4.1) \nTo study the outflow region, we construct a local orthogonal frame in the r -ϕ plane, with C located at the origin. We define x = r -r C , y = h 3 ϕ . The thickness of the current sheet depends on y , that is ∆ = ∆( y ) as well as the reconnected magnetic field ˆ B r and the outflow velocity ˆ v o . \nAgain we work in the quasi-stationary limit ∂ t ≈ 0. We assume ˆ v θ ≈ 0 ≈ ˆ B θ and ∂ θ ≈ 0. The current and the electric field are given by J = ˆ J θ e θ , E = ˆ E θ e θ and we take ˆ v in = -ˆ v i e r . Consider an arbitrary \nFigure 3: Reconnection layer in azimuthal direction. Here x ≡ r -r C , y ≡ h 3 ϕ . \n<!-- image --> \npoint O on the boundary. The ϕ -component of the energy-momentum equation is given by \n∂ ∂y ( h ˆ γ 2 ˆ v ϕ ( ˆ v ϕ + β ϕ ) ) + ∂ ∂r ( h -1 1 h ˆ γ 2 ˆ v r ˆ v ϕ ) = -∂p ∂y -ˆ J θ ˆ B r , (4.2) \nwhere the second term on the l.h.s cannot be dropped since O is located on the boundary. We consider r g /r o ≪ 1 so that ˆ v ϕ ≫ β ϕ ∼ r 2 g /r 2 o , valid to mildly relativistic outflow. Again we focus on a short current sheet in weak gravity regimes so that when integrating the momentum equation, we ignore the contributions from the gravitational tidal force. \nThe balance between the inflow flux ∼ ρ ˆ γ i ˆ v i / ( h 1 ∆) and the outflow flux ∼ ρ ˆ γ o ˆ v o /y gives rise to the first estimation of the sheet thickness \n∆( y ) ≈ ˆ γ i ˆ v i ˆ γ o ˆ v o y h 1 . (4.3) \nThe Ohm's law on the current sheet gives \nˆ γ ˆ E θ + ˆ γ ˆ v ϕ ˆ B r = η ˆ J θ . (4.4) \nAgain at the center incidence point, the plasmas is viewed as ideal MHD so that the electric field is given by ˆ E ϕ \| i ≈ ˆ v i ˆ B 0 . In addition, the electric field on the current sheet is homogeneous ˆ E ϕ o ≈ ˆ E ϕ \| i as long as the thickness of the sheet is very small. The electric current at O can be evaluated from the Maxwell's law \nˆ J θ ∣ ∣ ∣ o ≈ h -1 1 ∂ r ˆ F θr ≈ ˆ B 0 h 1 ∆( y ) . (4.5) \nCombining these results, we deduce \n∆( y ) -ˆ γ i ˆ B o ˆ γ o ˆ B 0 y h 1 -η ˆ γ o ˆ v i h 1 = 0 , (4.6) \nwhere we have set ˆ B r ∣ ∣ ∣ O = -ˆ B o . This connects the thickness of the current sheet to the reconnected magnetic field on the boundary. Close to the diffusion region, the second term can be dropped so that ∆ ≈ η/ ˆ v i h 1 ≈ X , consistent with the Sweet-Parker like model [21]. Otherwise, in the far outflow region y ≈ L , the third term can be dropped so that asymptotically \nh 1 ∆( y ) y ∣ ∣ ∣ y ≈ L ≈ ˆ γ i ˆ B o ˆ γ o ˆ B 0 . (4.7) \nCombing the result with (4.3), one finds the reconnection rate is related to the asymptotic thickness or the magnetic field \nˆ v i ∼ ˆ B o ˆ B 0 ≈ h 1 ∆( y ) y , (4.8) \nwhere we have assumed the outflow is mildly relativistic ˆ γ o ≈ 1, which will be verified later. It implies that asymptotically the shape of the boundary approaches to a straight line, with the slope ∼ ˆ v i . \nIntegrating the momentum equation (4.2) along the radial direction yields \nˆ B o ˆ B 0 = h ˆ v 2 i ˆ B 2 0 [ 2 y∂ y ( y h 1 ∆ ) + ˆ γ o y h 1 ∆ ] , (4.9) \nwhere we have adopted (4.3). Furthermore, by plugging this equation into (4.6), we deduce \nh 1 ∆( y ) -2ˆ v 2 i y [ 2ˆ γ -1 o y∂ y ( y h 1 ∆ ) + y h 1 ∆ ] -η ˆ γ o ˆ v i = 0 . (4.10) \nThis determines the shape of the current sheet given the plasmas velocities ˆ v i and ˆ v o . It implies in the far outflow region y ≈ L , ∆( y ) ∼ ˆ v i y/h 1 so that \nˆ B o ˆ B 0 ∼ ˆ v 2 i y h 1 ∆ ∼ ˆ v i , (4.11) \nin agreement with previous estimations. On the other hand, near the diffusion region y ∼ Y , ∆ ∼ X , one finds instead \nˆ B o ˆ B 0 ≈ 6ˆ v 3 i η y . (4.12) \nThese results imply that in the outflow region, the reconnected magnetic field grows monotonically and approaches to a constant asymptotically. \nTo solve the outflow velocity at O , we integrate the momentum equation (4.2) along the azimuthal direction. We deduce \nh ( ˆ γ 2 o ˆ v 2 o + ˆ γ 2 o ˆ v o ˆ v i y h 1 ∆ ) = p ( 1 + 2 y h 1 ∆ ˆ B o ˆ B 0 ) . (4.13) \nThis, together with (4.6), (4.9) or (4.10) provides a complete set of equations to solve various quantities ∆ , ˆ B o , ˆ v o at an arbirtary point on the inflow-outflow boundary. It turns out that only mildly relativistic outflow solution is permitted. To see this, consider the far outflow region, one has \n( ˆ γ o 2 ˆ v 2 o + ˆ γ 2 o ˆ v o )∣ ∣ ∣ y ≈ L ≈ 3 4 , (4.14) \nwhich gives ˆ γ o ˆ v o ∣ ∣ ∣ y ≈ L ≈ 1 / 2. \nFinally, to complete our derivations, we consider the inflow region and derive the reconnection rate, according to the initial condition of the magnetic field. Again we assume the magnetic field can be well described by a static potential. \nFigure 4: Azimuthal reconnection layer: numerical solutions for sheet thickness ∆, magnetic field ˆ B o / ˆ B 0 and outflow velocity on the inflow-outflow boundary. Here we take r C = 99 r g / 10 , L = r g / 10 , k = π/ 8 , S = 10 18 and h 1 is evaluated at r = 10 r g . This results to the reconnection rate ˆ v i ≈ 0 . 012. We have set a = 9 / 10 , r g = 10000. \n<!-- image --> \nThen standard analysis implies that the ˆ B 1 field in the inflow region can be viewed as the static field produced by two monopoles with ˆ B 1 ⊥ ≈ ∓ 2ˆ v i ˆ B 0 . Evaluating the magnetic field at the incidence point i yields \nˆ B ϕ 1 ∣ ∣ ∣ i ≈ 2ˆ v i ˆ B 0 π ln ( L 2 Y 2 ) ≈ 4ˆ v i ˆ B 0 π ln ( ˆ v 2 i Sh 3 ( r o ) /r o ) , (4.15) \nwhere in the second equality we have adopted L/Y ≈ ˆ v 2 i Sh 3 ( r o ) /r o if L ≪ r g . Again the reconnection rate can be significantly improved to ˆ v i ∼ (ln S ) -1 . One has \nˆ v i ≈ k ( ln ( ˆ v 2 i Sh 3 ( r o ) /r o ) ) -1 , k = π ˆ B ϕ 1 4 ˆ B 0 ∣ ∣ ∣ i . (4.16) \nAgain the gravity effect decreases the reconnection rate with respect to the flat spacetime limit. The leading corrections is given by \nˆ v i ≈ k [ ln ( ˆ v 2 i S ) + a 2 r 2 g 2 r 2 o + · · · ] -1 . (4.17) \nUnlike the radial case, rotation of the black hole dominates in the curvature effects. \nIn Fig. 4, we present numerical solutions for the sheet thickness ∆, the magnetic field ˆ B o / ˆ B 0 and the outflow velocity ˆ v o . To compare with the radial case, we choose the same parameters for the plasmas as well as the background. This gives the reconnection rate ˆ v i ≈ 0 . 012. Clearly all the results are consistent with previous analytical analysis.", '5 Reconnection layer in non-ZAMO frame': "In curved spacetime, the motion of plasmas is generally complicated and the electric current sheet may rotate either faster or slower than the black hole. Here we would like to explore the effect of rotation of the reconnection layer on the magnetic reconnection from the view of a ZAMO observer. The relevant topic in Minkowski spacetime is the magnetic reconnection for a rotating current sheet in the laboratory frame. \nAgain we work in quasi-stationarity and consider circular stable orbits. We consider a typical example at first: the Einstein's rotating disk, which orbits around the ZAMO frame with a constant angular velocity ω . General rotating disk will be discussed later. \nBefore introducing the rotating disk frame, we first rewrite the Kerr metric in cylindrical coordinates ( ˆ t , ˆ r , ˆ ψ, ˆ z ) for a ZAMO observer \nds 2 = -d ˆ t 2 + d ˆ r 2 + ˆ r 2 d ˆ ψ 2 + d ˆ z 2 , (5.1) \nwhere ˆ x 1 = ˆ r cos ˆ ψ, ˆ x 3 = ˆ r sin ˆ ψ, ˆ z = ˆ x 2 . Notice that these coordinates are not adapted to local measures. Vectors (or tensors) measured in the local rest frame (LRF) are related to those in the cylindrical coordinates via vielbeins 3 . It will be convenient for us to work directly with a new coordinate d ˆ y = ˆ rd ˆ ψ , which absorbs the warped factor in the polar direction. This is equivalent to building a orthonormal frame for local observers under cylindrical coordinates. \nTo proceed, we introduce the rotating disk frame { ¯ t , ¯ r , ¯ ψ, ¯ z } , defined as \nd ¯ t = d ˆ t , d ¯ r = d ˆ r , d ¯ ψ = d ˆ ψ -ωd ˆ t , d ¯ z = d ˆ z . (5.2) \nIn this frame, the local metric becomes hypersurface non-orthogonal \nds 2 = -(1 -ω 2 ¯ r 2 ) d ¯ t 2 + d ¯ r 2 + ¯ r 2 d ¯ ψ 2 +2 ω ¯ r 2 d ¯ td ¯ ψ + d ¯ z 2 . (5.3) \nHowever, it will be cumbersome to directly study a rotating reconnection layer for the ZAMO observer. The existence of the mixing metric component ¯ g tψ will lead to various issues. Our strategy is examining the properties of the reconnection layer for a comoving observer at first and then translating the results into the ZAMO frame. We define the corotating frame (CRF) { ˜ t , ˜ r , ˜ y , ˜ z } to be \nd ˜ t = d ¯ t -ω ¯ r 2 1 -ω 2 ¯ r 2 d ¯ ψ, d ˜ r = d ¯ r , d ˜ y = ¯ rd ¯ ψ, d ˜ z = d ¯ z , (5.4) \nin which the metric becomes hypersurface orthogonal \nds 2 = -q 2 d ˜ t 2 + d ˜ r 2 + q -2 d ˜ y 2 + d ˜ z 2 , (5.5) \n√ \nwhere q = 1 -ω 2 ˜ r 2 . Notice that the spatial metric becomes non-Euclidean for the comoving observers. This is a characteristic feature for a general rotating disk. It is extremely important since measure of (current sheet) length is essentially not local and will effect the reconnection rate significantly. Without loss of generality, we put the frame at ˆ ψ = 0 so that ˆ r (or ˜ r ) is parallel to radial direction of the black hole. Vectors or tensors measured by a ZAMO observer are related to those in the CRF via the coordinate \nFigure 5: Radial magnetic reconnection layer of Sweet-Parker configuration in corotating frame. \n<!-- image --> \ntransformations \nd ˆ t = d ˜ t + ω ˜ r 1 -ω 2 ˜ r 2 d ˜ y , d ˆ x 1 = d ˜ r , d ˆ x 2 = d ˜ z , d ˆ x 3 = ω ˜ rd ˜ t + 1 1 -ω 2 ˜ r 2 d ˜ y . (5.6)", '5.1 Sweet-Parker model': "Firstly, we consider a Sweet-Parker model, with the current sheet located in the ˜ r -direction. We assume ˜ v z = 0 = ˜ B z and ˜ ∂ z ≈ 0. The electric and magnetic fields are related to the filed strength tensor as 4 \n˜ E i = -q -1 ˜ F 0 i , ˜ B i = 1 2 q -1 ϵ ijk ˜ F jk . (5.7) \nHere the q -1 factor for the electric field comes from the timelike Killing vector whereas the same factor for the magnetic field comes from the three volume of the spatial slice. Without loss of generality, we focus on the first quadrant of Fig. 5, where we set ˜ B in = -˜ B 0 ˜ e r , ˜ v in = -˜ v i ˜ e y and ˜ B out = -˜ B o ˜ e y , ˜ v out = ˜ v o ˜ e r . From continuity equation, the inflow flux ˜ ∂ y (˜ ρ ˜ γ i ˜ v i ) ≈ ˜ ρ ˜ γ i ˜ v i ˜ δ must balance the outflow flux ˜ ∂ r (˜ ρ ˜ γ o ˜ v o ) ≈ ˜ ρ ˜ γ o ˜ v o ˜ L . This gives \n˜ δ = ˜ γ i ˜ v i ˜ γ o ˜ v o ˜ L. (5.8) \nHere ˜ γ is the Lorentz factor in the CRF. Normalization of four-velocity requires ˜ γ i = (1 -ω 2 ˜ r 2 -q -2 ˜ v 2 i ) -1 / 2 and ˜ γ o = (1 -ω 2 ˜ r 2 -˜ v 2 o ) -1 / 2 . Notice that reality of the Lorentz factor requires q ≥ q -1 ˜ v i and q ≥ ˜ v o . \nThis could be understood as the disk is slowly rotating compared to the plasmas velocities. In fact, this is a consequence of causality, as will be shown later. According to the Ohm's law, evaluation of the electric field at i and o yields \nE z \| i = q -2 ˜ v i ˜ B 0 , E z \| o = q -2 ˜ v o ˜ B o . (5.9) \nHere and below q (and ˜ r ) is understood to be evaluated at O . Again we assume the electric field on the current sheet is uniform so that \n˜ B o = ˜ v i ˜ B 0 ˜ v o ≈ ˜ δ ˜ B 0 ˜ L , (5.10) \nfor mildly relativistic outflow ˜ γ o ≈ 1. The result is in agreement with magnetic flux conservation. \nTo derive the outflow velocity, we consider the energy-momentum equation in ˜ r -direction. To leading order in large ˜ r , one has \n∂ ˜ r ( h ˜ γ 2 ˜ v 2 ) ≈ -( ∂ ˜ r p + q ˜ J z ˜ B y ) . (5.11) \nOn the other hand, according to Maxwell's equation \n˜ J z ∣ ∣ ∣ O ≈ ˜ ∂ y ˜ F zy ∣ ∣ ∣ O ≈ q ˜ B 0 ˜ δ . (5.12) \nIntegrating the equation (5.11) yields h ˜ γ 2 o ˜ v 2 o ≈ p + ˜ B 2 0 . This gives \n˜ γ o ˜ v o ≈ 1 . (5.13) \nIndeed the outflow is mildly relativistic. To study the reconnection rate, consider the Ohm's law on the current sheet \nq ˜ γ ( ˜ E z + ˜ v ˜ B y ) = η ˜ J z . (5.14) \nOn the neutral line ˜ v = 0 so that ˜ E z \| C ≈ η ˜ J z /q . This leads to \n˜ J z ∣ ∣ ∣ C ≈ ˜ v i ˜ B 0 qη . (5.15) \nSince ˜ J z \| C ≈ ˜ J z \| O in our approximation, one finds ˜ v i = q 2 η/ ˜ δ . It turns out that \n˜ v i = q u i , u i = h -1 / 2 1 ∣ ∣ ∣ o S -1 / 2 , (5.16) \nwhere u i is the reconnection rate locally measured by the comoving observer, which is irrelevant to motion of the plasmas. In contrast, the reconnection rate for the ZAMO observer does rely on rotation of the plasmas. Using (5.6) and ˆ U µ = ˜ U ν ∂ ˆ x µ /∂ ˜ x ν , we compute the four-velocity at i: ˆ U 0 = ˜ γ i (1 -q -2 ˜ v i ω ˜ r ) and ˆ U 3 = ˜ γ i ( ω ˜ r -q -2 ˜ v i ). This gives rise to \nˆ v i ≡ -ˆ U 3 / ˆ U 0 = q -2 ˜ v i -ω ˜ r 1 -q -2 ˜ v i ω ˜ r ∣ ∣ ∣ o . (5.17) \nFigure 6: Plots for the difference ˆ v i -u i as a function of u i . In the left panel ω > 0 and the solid/dashed/dotdashed line corresponds to ω ˜ r = 0 . 1 / 0 . 2 / 0 . 3, respectively. In the right panel ω < 0 and the solid/dashed/dotdashed line corresponds to ω ˜ r = -0 . 1 / -0 . 2 / -0 . 3, respectively. \n<!-- image --> \nClearly the result does not follow from the speed superposition principle in special relativity owing to the warped factor q -2 . This can be traced back to the fact that the spatial space in CRF is non-Euclidean. Nevertheless, causality is protected since ˆ v i ≤ 1 because of u i ≤ q , which guarantees reality of the Lorentz factor ˜ γ i . It turns out that in comparison to the rate u i for the comoving observer, the reconnection rate ˆ v i measured by the ZAMO observer could be either decreased or increased depending on rotation of the disk. There exists a critical value of u i \nu ± c = -(1 -q ) ± √ (1 -q ) 2 +4 qω 2 ˜ r 2 2 ω ˜ r , (5.18) \nwhere u + c ( u -c ) corresponds to ω > 0 ( ω < 0). Consider ω > 0 at first. In this case, the disk rotates slower than the ZAMO frame. One has u -c < 0 , u + c ≤ 1 and u + c decreases as ω ˜ r increases. Since ˆ v i -u i ∼ ω ( u i -u + c )( u i -u -c ), it follows that ˆ v i < u i if u i < u + c and ˆ v i > u i if u i > u + c . However, for subrelativistic inflow, the reconnection rate is generally decreased, see the left panel of Fig. 6. This can be seen even more clearly for a non-relativistic disk which has ω ˜ r ∼ 0 and u + c ≈ 1 -ω ˜ r/ 4 approaching to the speed of light. Hence for mildly relativistic inflow (for example for ω ˜ r ≤ 0 . 5, u i ≤ 0 . 8 ≤ u + c ), rotation of the reconnection layer will lead to a further decrease of the reconnection rate for the ZAMO observer. \nHowever, when the disk rotates faster than the ZAMO frame ω < 0, the situation will be quite different. The critical velocity is always superluminal u -c > 1 for a timelike disk. Since u + c < 0 in this case, it implies that ˆ v i is always larger than u i , see the right panel of Fig. 6. \n̸ \nBy symmetry of the configuration, the inflow velocity in the lower half plane of CRF is given by ˜ v in = ˜ v i ˜ e y , where ˜ v i is given by (5.16). However, in the ZAMO frame, the reconnection rate is obtained by (5.17), with ω → -ω . Consequently, for any given ω = 0, the ZAMO observer will always find the asymmetric rates: one side of the current sheet has a larger reconnection rate whereas the other side has a smaller one, in comparison to the unrotation limit. This is correct for mildly relativistic inflow no matter the reconnection layer rotates faster or slower than the ZAMO frame. \nHowever, when the current sheet forms in azimuthal direction, the reconnection rate for the ZAMO observer will not receive any contributions from the rotation. We refer the readers to the appendix for details about this case.", '5.2 Petschek model': "We proceed to study a general rotating reconnection layer in the Petschek model, located in the radial direction (the illustration figure is the same as Fig. 1 but now all quantities have a tilde). Again we assume ˜ v z = 0 = ˜ B z and ˜ ∂ z ≈ 0 ≈ ˜ ∂ t . We focus on the first quadrant. We set ˜ B in = -˜ B 0 ˜ e r + ˜ B 1 , ˜ v in = -˜ v i ˜ e y and ˜ B out = -˜ B o ˜ e y , ˜ v out = ˜ v o ˜ e r , where O is an arbitrary point on the inflow-outflow boundary. According to previous discussions, in the diffusion region, Sweet-Parker process dominates and \n˜ v i ∼ ˜ X ˜ Y ∼ q ( ˜ Y η ) -1 / 2 . (5.19) \nTo estimate thickness of the current sheet at O, we evaluate the inflow flux ˜ ∂ y (˜ ρ ˜ U y ) ∼ ˜ ρ ˜ γ i ˜ v i / ∆(˜ r ) and the outflow flux ˜ ∂ r (˜ ρ ˜ U r ) ∼ ˜ ρ ˜ γ o ˜ v o / ˜ r . The balance between the two implies that \n∆(˜ r ) = ˜ γ i ˜ v i ˜ γ o ˜ v o ˜ r ∣ ∣ ∣ O . (5.20) \nOn the current sheet, the Ohm's law gives \nq ˜ γ ( ˜ E z + ˜ v r ˜ B y ) = η ˜ J z . (5.21) \nWe assume the electric field on the current sheet is uniform. Evaluation of the electric field at the incidence point i yields ˜ E z \| O = ˜ E z \| i = q -2 ˜ v i ˜ B 0 . Besides, using the Maxwell's law, ˜ J z \| O ≈ ˜ ∂ y ˜ F zy ≈ q ˜ B 0 / ∆. Then integrating (5.21) along ˜ y -direction yields \n∆(˜ r ) -˜ γ i ˜ B o ˜ γ o ˜ B 0 ˜ r -q 2 η ˜ γ o ˜ v i = 0 , (5.22) \nwhere we have adopted (5.20). Close to the diffusion region, the second term is negligible, one has ˜ v 2 i ∼ q 2 η/ ˜ Y , consistent with previous analysis. On the other hand, in the far outflow region, the third term is negligible so that ˜ v i ∼ ∆ / ˜ r ∼ ˜ B o / ˜ B 0 . \nTo proceed, consider the energy-momentum tensor equation at ˜ r -direction \n∂ ∂ ˜ r ( h ˜ γ 2 ( ˜ v r ) 2 ) + ∂ ∂ ˜ y ( h ˜ γ 2 ˜ v r ˜ v y ) ≈ -∂p ∂ ˜ r -q ˜ J z ˜ B y , (5.23) \nwhere we have dropped metric derivative terms which are subdominant in large ˜ r . Integrating (5.23) along ˜ y -direction and using (5.20), we deduce \n˜ B o ˜ B 0 = h ˜ v 2 i ˜ B 2 0 ( 2˜ r∂ ˜ r ( ˜ r ∆ ) + ˜ γ o ˜ r ∆ ) . (5.24) \nSubstituting this equation into (5.22), we deduce \n∆(˜ r ) -2˜ v 2 i ˜ r ( 2ˆ γ -1 o ˜ r∂ ˜ r ( ˜ r ∆ ) + ˜ r ∆ ) -q 2 η ˜ γ o ˜ v i = 0 . (5.25) \nAgain h = 4 p for hot relativistic plasmas and p = ˜ B 2 0 / 2 according to pressure balance at the neutral line. For mildly relativistic outflow, this equation solves the thickness of the current sheet given the inflow velocity. \nTo derive the outflow velocity at O , we integrate the momentum equation (5.23) along ˜ r -direction. We deduce \nh ( ˜ γ 2 o ˜ v 2 o + ˜ γ 2 o ˜ v o ˜ v i ˜ r ∆ ) = p ( 1 + 2˜ r ∆ ˜ B o ˜ B 0 ) . (5.26) \nThe outflow is indeed mildly relativistic since ˜ γ o ˜ v o ≈ 1 in the far outflow region. \nThe equations (5.24), (5.25) and (5.26) give a complete set of equations to solve the various quantities in the outflow region once the reconnection rate is given. The later depends on the initial condition of the magnetic field in the inflow region ˜ B in = -˜ B 0 ˜ e r + ˜ B 1 . Again we assume \| ˜ B 1 \| ≪ ˜ B 0 and the ingoing magnetic field can be well described by a static potential. Derivation of ˜ B 1 can follow the standard approach. One has ˜ B 1 ⊥ ≈ -2 q -1 ˜ v i ˜ B 0 . This implies that the ˜ B 1 field in the inflow region can be viewed as the static field produced by two monopoles ˜ B 1 ⊥ ≈ ∓ 2 q -1 ˜ v i ˜ B 0 . The magnetic field at the center incidence point i can be evaluated as \n˜ B r 1 ∣ ∣ ∣ i ≈ 2˜ v i ˜ B 0 qπ ln ( ˜ L 2 ˜ Y 2 ) ≈ 4˜ v i ˜ B 0 qπ ln ( q -2 ˜ v 2 i Sh 1 ( r o ) ) , (5.27) \nwhere in the second equality ˜ L/ ˜ Y ≈ q -2 ˜ v 2 i Sh 1 ( r o ). We deduce \n˜ v i ≈ qk ( ln ( q -2 ˜ v 2 i Sh 1 ( r o ) ) ) -1 , k = π ˜ B r 1 4 ˜ B 0 . (5.28) \nAgain ˜ v i = qu i , where u i is the reconnection rate measured by the comoving observer locally. Similar to the Sweet-Parker case, the rate ˆ v i for the ZAMO observer is given by (5.17). It implies that despite the configuration of current sheet is quite different for the two cases, the reconnection rate is changed in the same manner by rotation of the disk.", '5.3 General rotating disk': "Having studied the Einstein's rotating disk, we would like to relax the condition and explore the reconnection layer in a general rotating frame. Perhaps the simplest way to do this is letting ω in (5.2) to be position dependent. While this is correct, here we would like to work in the other way around. Without introducing cylindrical coordinates, we directly define the rotating frame using the BL coordinates as \nd ¯ t = αdt, d ¯ x i = h i dx i -α ¯ β i dt , (5.29) \n̸ \nwhere ¯ β i = ¯ β ϕ δ iϕ . Generally ¯ β ϕ = β ϕ and hence this frame does not corotate with the black hole. It is easy to see that this frame is related to the ZAMO frame as \nd ¯ t = d ˆ t , d ¯ x 1 = d ˆ x 1 , d ¯ x 2 = d ˆ x 2 , d ¯ x 3 = d ˆ x 3 -ϖd ˆ t , (5.30) \nwhere ϖ ≡ ¯ β ϕ -β ϕ . The metric turns out to be \nds 2 = -d ¯ t 2 +( d ¯ x 1 ) 2 +( d ¯ x 2 ) 2 +( d ¯ x 3 + ϖd ¯ t ) 2 , (5.31) \nwhich is hypersurface non-orthogonal. To compare with the previous case, we would like to relabel the coordinates properly and define the comoving observer as \nd ˜ t = d ¯ t -ϖ 1 -ϖ 2 d ¯ x 3 , d ˜ r = d ¯ x 1 , d ˜ y = d ¯ x 3 , d ˜ z = d ¯ x 2 . (5.32) \nIn this frame, the metric becomes hypersurface orthogonal \nds 2 = -q 2 d ˜ t 2 + d ˜ r 2 + q -2 d ˜ y 2 + d ˜ z 2 , (5.33) \nwhere q = √ 1 -ϖ 2 . Notice that \| ϖ \| < 1 for a timelike disk. Clearly when ϖ = ω ˜ r , it reduces to the Einstein's rotating disk. This is consistent with our intuitive expectations that a general rotating disk can be obtained by letting ω in (5.2) to be position dependent. \nIt is immediately seen that for reconnection layer in radial direction, the inflow velocity ˜ v i will acquire a factor of q according to (5.16) or (5.28) for Sweet-Parker or Petschek scenario. That is ˜ v i = qu i , where the meaning of u i is the same as before. It turns out that the reconnection rate ˆ v i for the ZAMO observer reads \nˆ v i = q -2 ˜ v i ∓ ϖ 1 ∓ q -2 ˜ v i ϖ ∣ ∣ ∣ o , (5.34) \nwhere '+' ('-') corresponds to the lower/upper half plane of the current sheet. Without presenting more details, we claim that for a generally slowly rotating disk (subrelativistic at most), the reconnection rate ˆ v i will be decreased by the rotation on one side of the configuration and be increased on the other side, provided the inflow is mildly relativistic.", '6 Conclusion': "We have developed a scenario for fast magnetic reconnection process for general relativistic magnetohydrodynamical plasmas around Kerr black holes. Generally speaking, the current sheet may form in a non-zero-angular-momentum (non-ZAMO) frame in the black hole background. As a first step toward this, we study magnetic reconnection in the ZAMO frame at first, which corotates with the black hole. We compute the reconnection rate and analyze various important properties of the reconnection layer for two configurations in the equatorial plane of the black hole. We show that compared to the flat spacetime \nlimit, both mass and rotation of the black hole can decrease the reconnection rate. The former (later) dominates for the radial (azimuthal) configuration. \nWhen the current sheet forms in a non-ZAMO frame, we consider a typical example: the Einstein's rotating disk, which orbits around the ZAMO frame with a constant angular velocity. In this case, we studied both the Sweet-Parker and the Petscheck scenario. We established that for any given slow rotations (subrelativistic at most) and mildly relativistic inflow, the ZAMO observer will always find asymmetric reconnection rates for radial configuration: it is decreased on one side of the current sheet and is increased on the other side in comparison to the unrotation limit. This clarifies the effects of rotation on the reconnection layer in the laboratory frame in the flat spacetime limit. These results are valid to a generally rotating reconnection layer, no matter it rotates faster (clockwise) or slower (anti-clockwise) than the ZAMO frame (in the laboratory frame). \nIn this work, we focused on the reconnection layer moving in a circular stable orbit in the equatorial plane of the black hole. It will be very interesting to relax the condition and study more general situations, which are relevant to astrophysical events. Our results also have potential applications to energy extraction from black holes via magnetic reconnection, see recent developments in this direction [27]. It is also interesting to consider collisionless effects as well as the gravitational electromotive force in magnetic reconnection. We leave these to future research.", 'Acknowledgments': 'The work is partly supported by NSFC Grant No. 12275004 and 12205013. Z.Y. Fan was supported in part by the National Natural Science Foundations of China with Grant No. 11805041 and No. 11873025 and also supported in part by Guangzhou Science and Technology Project 2023A03J0016.', "A Sweet-Parker in azimuthal direction for Einstein's disk": "Consider the Sweet-Parker model in Einstein's disk, with the current sheet located in the ˜ y -direction. Again we assume ˜ v z = 0 = ˜ B z and ˜ ∂ z ≈ 0. We set ˜ B in = ˜ B 0 ˜ e y , ˜ v in = -˜ v i ˜ e r and the outflow velocity ˜ v \| O = ˜ v o ˜ e y , see Fig. 7. \nFrom continuity equation, the balance of inflow and outflow flux gives \n˜ δ = ˜ γ i ˜ v i ˜ γ o ˜ v o ˜ L. (A.1) \nFigure 7: Rotating reconnection layer in azimuthal direction in CRF. \n<!-- image --> \nUsing the Ohm's law, evaluation of the electric field at i and o yields \nE z \| i = q -2 ˜ v i ˜ B 0 , E z \| o = q -2 ˜ v o ˜ B o . (A.2) \nHere ˜ B r \| O ≡ -˜ B o and q is understood to be evaluated at O . Since the electric field on the current sheet is uniform, one finds \n˜ B o = ˜ v i ˜ B 0 ˜ v o ≈ ˜ δ ˜ B 0 ˜ L , (A.3) \nfor mildly relativistic outflow ˜ γ o ≈ 1. Again it is consistent with magnetic flux conservation. \nTo derive the outflow velocity, we consider the energy-momentum equation in ˜ y -direction. To leading order, one has \n˜ ∂ y ( h ˜ γ 2 ˜ v 2 ) ≈ -( q 2 ˜ ∂ y p + q ˜ J z ˜ B r ) . (A.4) \nThe current at O can be evaluated as \n˜ J z ∣ ∣ ∣ O ≈ ˜ ∂ r ˜ F zr ∣ ∣ ∣ O ≈ ˜ B 0 q ˜ δ . (A.5) \nIntegrating the equation (A.4) yields h ˜ γ 2 o ˜ v 2 o ≈ q 2 ( p + ¯ B 2 0 ), where ¯ B 0 = q -1 ˜ B 0 is the magnetic field measured by a local observer. Because of h = 4 p for hot relativistic plasmas and p = ¯ B 2 0 / 2, this gives \n˜ γ o ˜ v o ≈ q . (A.6) \nSince q ≤ 1, the outflow is still mildly relativistic. However, this effects the the reconnection rate and the scaling symmetry significantly. Consider the Ohm's law on the current sheet \n˜ γ ( q ˜ E z -q -1 ˜ v ˜ B r ) = η ˜ J z . (A.7) \nOn the neutral line ˜ v = 0 so that ˜ E z \| C ≈ η ˜ J z /q . This leads to \n˜ J z ∣ ∣ ∣ C ≈ ˜ v i ˜ B 0 qη . (A.8) \nUsing ˜ J z \| C ≈ ˜ J z \| O , we obtain \n˜ v i = η ˜ δ . (A.9) \nNotably, this together with (A.1) and (A.3) is invariant under the scaling of LRF, which implies ˆ L → q -1 ˜ L, ˆ B 0 → q -1 ˜ B 0 , ˆ v o → q -1 ˜ v o and ( ˆ δ , ˆ B o , ˆ v i ) → ( ˜ δ , ˜ B o , ˜ v i ). Combining (A.9) with (A.1), we deduce ˜ δ = q -1 / 2 √ η ˜ L . The reconnection rate for the comoving observer turns out to be \n˜ v i = ( r h 3 ∣ ∣ ∣ O ) 1 / 2 S -1 / 2 . (A.10) \nAccording to the transformation (5.6), this is exactly equal to the rate measured by the ZAMO observer, namely ˆ v i = ˜ v i . 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2024MNRAS.535L..13G	In a recent paper On the time dependency of inlineformulatexmath idTM0003 notationLaTeXa0texmathinlineformula the authors claim that they have tested one of the predictions of the Scale Invariant Vacuum SIV theory on MOND by studying the dependence of the Modified Newtonian Dynamics MOND acceleration at two data sets lowz inlineformulatexmath idTM0005 notationLaTeX3.2times 104le zle 3.2times 102texmathinlineformula and highz inlineformulatexmath idTM0007 notationLaTeX0.5le zle 2.5texmathinlineformula. They claim both samples show a dependency of inlineformulatexmath idTM0008 notationLaTeXa0texmathinlineformula from z. Here the work mentioned above is revisited. The explicit analytic expression for the zdependence of the inlineformulatexmath idTM0011 notationLaTeXa0texmathinlineformula within the SIV theory is given. Furthermore the first estimates of the inlineformulatexmath idTM0012 notationLaTeXOmega mtexmathinlineformula within SIV theory give inlineformulatexmath idTM0013 notationLaTeXOmega m0.28pm 0.04texmathinlineformula using the lowz data only while a value of inlineformulatexmath idTM0014 notationLaTeXOmega m0.055texmathinlineformula is obtained using both data sets. This much lower inlineformulatexmath idTM0015 notationLaTeXOmega mtexmathinlineformula leaves no room for nonbaryonic matter Unlike in the mentioned paper above the slope in the zdependence of inlineformulatexmath idTM0017 notationLaTeXA0log 10a0texmathinlineformula is estimated to be consistent with zero Zslope for the two data sets. Finally the statistics of the data are consistent with the SIV predictions in particular the possibility of change in the sign of the slopes for the two data sets is explainable within the SIV paradigm however the uncertainty in the data is too big for the clear demonstration of a zdependence yet.	2024-11-01T00:00:00Z	['2024MNRAS.535L..13G', 'arXiv:2409.11425', '2024MNRAS.tmpL..78G', '2024arXiv240911425G', '10.48550/arXiv.2409.11425', '10.1093/mnrasl/slae085']	['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	Elucidating the zdependence of the MOND acceleration aSUB0SUB within the scale invariant vacuum SIV paradigm	2,024	190	0.33	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.11425.pdf	{'Elucidating the z -dependence of the MOND acceleration ( a 0 ) within the Scale Invariant Vacuum (SIV) paradigm': 'Vesselin G. Gueorguiev 1 , 2 ⋆ \n- 1 Institute for Advanced Physical Studies, Sofia, Bulgaria\n- 2 Ronin Institute for Independent Scholarship, Montclair, NJ, USA \nAccepted 2024 September 6. Received 2024 July 16; in original form 2024 April 17', 'ABSTRACT': "In a recent paper: 'On the time dependency of a 0 ' the authors claim that they have tested 'one of the predictions of the Scale Invariant Vacuum (SIV) theory on MOND' by studying the dependence of the Modified Newtonian Dynamics (MOND) acceleration at two data sets, lowz (3 . 2 × 10 -4 ≤ z ≤ 3 . 2 × 10 -2 ) and highz (0 . 5 ≤ z ≤ 2 . 5). They claim 'both samples show a dependency of a 0 from z '. Here, the work mentioned above is revisited. The explicit analytic expression for the z -dependence of the a 0 within the SIV theory is given. Furthermore, the first estimates of the Ω m within SIV theory give Ω m = 0 . 28 ± 0 . 04 using the low-z data only, while a value of Ω m = 0 . 055 is obtained using both data sets. This much lower Ω m leaves no room for non-baryonic matter! Unlike in the mentioned paper above, the slope in the z -dependence of A 0 = log 10 ( a 0 ) is estimated to be consistent with zero Z-slope for the two data sets. Finally, the statistics of the data are consistent with the SIV predictions; in particular, the possibility of change in the sign of the slopes for the two data sets is explainable within the SIV paradigm; however, the uncertainty in the data is too big for the clear demonstration of a z -dependence yet. \nKey words: cosmology - dark matter - cosmological parameters - cosmology: theory", '1 INTRODUCTION': "Modern physics is well understood based on two main contemporary pillars: Einstein's General Relativity (EGR) and the Relativistic Quantum Field Theory. However, there are some perplexing observations about the motion of stars within galaxies and clusters. Within the popular current model of Cosmology and Astrophysics, the resolution of these perplexing phenomena is often associated with concepts such as Dark Matter and Dark Energy Bertone & Hooper (2018). However, for over 30 years, there has not been a definite detection of any new particles or fields. An alternative to the Dark Matter approach to resolving the observational discrepancies in galaxies and clusters of galaxies is the idea of the Modified Newtonian Dynamics (MOND, Milgrom (1983)) that has steadily gained support in the Astrophysics communities. While the concept of Dark Matter is a natural continuation of the matter paradigm into a non-luminous matter to explain the observational fact of the flat rotational curves, the MOND idea does not need extra matter 1 ; instead, it mod- \n- ⋆ E-mail: Vesselin at MailAPS.org; Orcid ID: 0000-0002-20226432;\n- 1 Some fashionable models of gravity are trying to explain MOND and its fundamental acceleration a 0 (Pazy 2013; Verlinde 2017), but these models have failed the observational tests by Lelli et al. (2017), while the relativistic implementation of MOND suggests imperceptible variation of a 0 'to redshift unity or even beyond it' (Bekenstein & Sagi 2008). \n© \nifies the dynamics once the observed acceleration g = v 2 /r falls below the certain cut-off value a 0 ≫ g . In this deep MOND regime, one expects scale invariance to be present in the system under study (Milgrom 2009). \nScale invariance is an old idea introduced by Weyl as early as 1918 Weyl (1922, 1970) as a gauge invariant gravity, where along with the metric tensor g µν there is a connexion vector κ µ controlling the length change dl = lκ µ dx µ , and a scalar field λ that describes the gauge freedom g µν → λ 2 g µν . The shortcomings of the original Weyl geometry pointed out by Einstein (1918) were addressed by the introduction of the Weyl Integrable Geometry (WIG) (Eddington 1923), where the connexion vector satisfies κ µ = -∂ µ ln λ . Consequently, Dirac (1973) and Canuto et al. (1977) have applied the idea to formulate scale invariant cosmology and tried to fix λ based on Dirac's Large Numbers Hypothesis (Dirac 1974). The recent reincarnation of the notion of scale invariance was introduced by Maeder (2017a,b), where the scalar field λ was fixed to be only time-dependent by the requirement of homogeneity and isotropy of space. In doing so, the specific functional form of λ ( t ) is determined by the requirement that the macroscopic vacuum must be scale invariant and thus introducing the Scale Invariant Vacuum (SIV) paradigm (Maeder & Gueorguiev 2023). This new approach has been explored only in the past few years by Gueorguiev & Maeder (2024) as a potential alternative to the standard cosmological model of dark energy plus cold dark matter paradigm (ΛCDM) Planck Collaboration: Ade et al. (2016). This new alterna- \nTable 1. Results from simple statistical analysis for the two sets of redshifts; the first set of lowz data with 0 . 00032 ≤ z ≤ 0 . 032 (Marra et al. 2020) and the second highz data with 0 . 5 ≤ z ≤ 2 . 5 (Del Popolo & Chan 2024c). \npossible connection to dark matter and dark energy (Maeder & Gueorguiev 2020a). It has been shown recently by Maeder (2023) that the MOND fundamental acceleration a 0 could be derived within the SIV-paradigm, and the result depends on the cosmological parameters such as Hubble constant H 0 and the total current mass fraction Ω m = ρ m /ρ c , where ρ c = 3 H 2 0 / (8 π G ) is the critical density. \nBy taking this result at face value along with the epochdependent scale factor λ ( t ), it is natural to expect that the SIV-derived MOND acceleration a 0 may have an epochdependent value, just as it is the case for the mass content of the Universe ρ m , and the Hubble parameter as well ( H = ˙ a/a , where a ( t ) is the usual FLRW expansion factor). \nIn this respect, the recent papers by Del Popolo & Chan (2024a,c) have initiated interesting research about testing the connection between MOND by Milgrom (1983) and its possible justification within the SIV paradigm by Maeder (2023). In doing so, they studied the z -dependence of a 0 using observational data but didn't derive the explicit z -dependence, nor did they discuss the relevant SIV model parameters for Ω m . As a new model different from ΛCDM, one should expect that some of the standard cosmological parameters may have different values within the SIV model. In this case, Ω m is a model parameter to be determined, while the Hubble constant H 0 is a model-constraining observational parameter. \nIn what follows, I will present my analyses of the z -dependency of the MOND acceleration a 0 along with the specific z -dependent expression of a 0 within SIV. Furthermore, the results of the statistical analyses will be utilized to perform one of the first determinations of the SIV parameter Ω m representing the fraction of the total matter-energy content of the Universe. The results will illustrate a puzzling situation that needs a better understanding of the data or the model utilized.", '2 RESULTS FROM STATISTICS': 'Before going into more detail about the SIV theory, it is important to note that a simple statistical analysis of the two main variables A 0 = log 10 ( a 0 ) with a 0 in km/s 2 and Z = log 10 ( z ) based on data reported by Del Popolo & Chan (2024c) gives averaged values ¯ A 0 = -13 . 07 ± 0 . 06 and ¯ Z = -2 . 49 ± 0 . 04 (see Table 1) for the lowz data set from Marra et al. (2020). \nFor notational and pragmatic reasons, I maintain the choice of the main variables to be the dimensionless A 0 and Z . Using, the log 10 on a 0 in km/s 2 keeps the corresponding range of A 0 between -15 and -10, while the log 10 on z is in the range -4 to 1. Since a 0 should be in km/s 2 for evaluating log 10 ( a 0 ) to compute A 0 , that is, A 0 = log 10 ( a 0 / (km/s 2 )) for \nTable 2. Values of the Z-slopes and intercepts for the current work and the values deduced in ref. (Del Popolo & Chan 2024c). \na 0 in arbitrary units, then the units of A 0 are dimensionless, and so are the corresponding average values ¯ A 0 . \nThe highz data shown in Del Popolo & Chan (2024c) is based on the work done by Nestor Shachar et al. (2023). It contains only 17 highz Galaxies for which the inferred a 0 is less than 1 . 2 × 10 -13 km/s 2 ; that is, A 0 < -12 . 92. Such selection criteria cut a large data segment from the highz data, while it has not been applied to the lowz data, thus introducing a bias. In my opinion, a fair, unbiased, and appropriate data selection procedure, if any, should be applied to both sets. Here, I disagree with such a selection criteria applied to highz data and not to the lowz data, so I use all 100 data points in Nestor Shachar et al. (2023) as seen in Fig. 1, both data sets have a compatible spread of A 0 -values. The relevant values follow the calculation of a 0 using Eq. (5) in Del Popolo & Chan (2024c) based on the data from Nestor Shachar et al. (2023). In doing the calculations, an error was noticed in the initial evaluations by Del Popolo & Chan (2024a) due to units conversion; upon communicating with the authors, this was later recognized by Del Popolo & Chan (2024b) and corrected in the subsequent version of their paper Del Popolo & Chan (2024c); the proper conversion is necessary to obtain a 0 in km/s 2 for the correct evaluation of the corresponding A 0 . It is worth noticing that the corresponding simple statistical analysis gives ¯ A 0 = -12 . 60 ± 0 . 03 and ¯ Z = 0 . 17 ± 0 . 02 for all of the 100 highz data points (see Table 1). Thus, the Z -slope based on these two aggregated data sets is ∆ ¯ A 0 / ∆ ¯ Z = ( -12 . 6+13 . 07) / (0 . 17+2 . 49) ≈ 0 . 18 ± 0 . 03 indicating a change in the MOND acceleration as seen from Table 1. This is an overly simplified estimate of the Z -slop based on both sets, marked in Table 2 with an asterisk. \nThe statistical analyses can be taken further, as done in the paper by Del Popolo & Chan (2024c), where they do a linear fit to the two sets and derive a Z -slope of 0 . 6 ± 0 . 2 with an intercept of A 0 = -11 . 6 ± 0 . 5 for the lowz data. Simple linear regression on the same data gives agreement with the zero slope since the result is a Z -slope of 0 . 12 ± 0 . 13 with a different intercept of A 0 = -12 . 8 ± 0 . 3. Such intercept should be regarded as related to Z = 0; thus, to the value of a 0 at z =1 and not at z = 0. Therefore, this corresponds to about 11% or only 2% change of A 0 from ¯ A 0 = -13 . 07 near z ≈ 0 to A 0 = -11 . 6 or A 0 = -12 . 8 near z ≈ 1. \nRegarding the highz data, the Z -slope of -0 . 2 ± 0 . 4 is consistent with zero while the intercept of A 0 = -9 . 61 ± 0 . 08 for the highz data is questionable due to the applied data selection criteria and the possible error of their analyses mentioned earlier. The simple linear regression on the full data set derived from Nestor Shachar et al. (2023) gives again agreement with the zero Z -slope since the result is Z -slope is 0 . 01 ± 0 . 2 with an intercept of A 0 = -12 . 603 ± 0 . 05. Notice', 'MOND acceleration at varius redshift values': 'Figure 1. Lowz and highz data sets are discussed in the text. For this figure only, I have dropped the outlier related to the NGC2976 Galaxy data in the lowz data set since its value A 0 = -18 . 32 is too low compared to the other values shown; however, the data point has been used in the various data analyses presented. Unlike in Del Popolo & Chan (2024c), where only 17 highz data points have been used, here I have shown all 100 data points related to the data in Nestor Shachar et al. (2023). For the relevant error bars, the reader is referred to the paper by Del Popolo & Chan (2024c). \n<!-- image --> \nthat for the current analysis, the lowz and highz data sets have intercepts that agree with each other unlike those in Del Popolo & Chan (2024c). There must be an agreement between these two intercepts since they should reflect the value of the MOND acceleration at z = 1 .', '3 SIV FRAMEWORK': "Within SIV the fundamental MOND acceleration a 0 can be related to the Hubble constant H 0 and the current matter content of the Universe Ω m Maeder (2023); Maeder & Gueorguiev (2020b). For the derivations and formulas to be used in this section, I will denote the MOND fundamental acceleration a 0 by a M whenever appropriate to avoid confusion with the expansion scale factor a but will use a 0 in the absence of such a problem. This is done to avoid confusion with the FLRW expansion scale factor a , which by convention should be denoted by a 0 at the current epoch; and to also avoid awkward notation a 00 for the current value of the MOND acceleration. \nWithin SIV, there is an extra velocity-dependent term, denoted as dynamical acceleration Maeder & Gueorguiev (2020a): \nd 2 -→ r dt 2 = -G t M ( t ) r 2 -→ r r + κ ( t ) d -→ r dt , (1) \nwhere κ = -˙ λ/λ is the time component of the SIV connexion vector, with a simple functional form κ = 1 /t within the SIV gauge, where the SIV cosmic time t is a dimensionless parameter such that t ∈ [ t in , t 0 = 1]. Here, t in is the moment of the Bing Bang when the FLRW scale factor satisfies \na ( t in ) = 0, happening near t in = Ω 1 / 3 , while at the current epoch a ( t 0 ) = 1 and time is set so that t 0 = 1. Within SIV the conformal-scale factor is λ = 1 /t and is used to perform Weyl transformation g ' µν = λ 2 g µν that relates EGR metric g ' µν to the metric g µν within the WIG framework (Maeder & Gueorguiev 2020a). \nTo arrive at an expression for the MOND acceleration a 0 , one considers the ratio of the magnitudes of the Newtonian acceleration g N = GM/r 2 to the additional acceleration κ ( t ) v in (1), where v denotes the magnitude of the velocity -→ v = d -→ r /dt : \nx = κvr 2 GM . (2) \nNow, one can use the relation given by the instantaneous radial acceleration v 2 /r = GM/r 2 to eliminate the speed v from the expression of x given by (2); then using g N = GM/r 2 to remove GM , one arrives at: \nx = κvr 2 GM = κ √ r 3 GM = κ √ r g N . (3) \nWhen the dynamic acceleration dominates over the Newtonian acceleration ( x ≫ 1), one has: \ng = g N + xg N ≈ xg N = κ √ rg N . (4) \nTherefore, one arrives at the MOND type relation g ∼ √ a 0 g N from which one can deduce an expression for a 0 : \na 0 ≈ κ 2 r. (5) \nThe above expression indicates a possible r dependence of a 0 , thus demoting a 0 of its fundamental parameter status within MOND, which may be testable with future high-precision \ndata. Such dependence may explain the variance of a 0 . To restore the fundamental character of the above expression (5) as in MOND, one could consider the limit r → r H , where the Hubble radius reflects the influence of the Universe causally connected to the object studied. Thus, the time-dependent MOND acceleration within SIV is the upper bound of (5) given by the expression: \nκ 2 r → κ 2 r H = κ 2 c/H = a M ( t ) , (6) \nDuring the matter-dominated epoch, SIV has an analytic form for expansion scale-factor a ( t ) (Jesus 2017; Maeder & Gueorguiev 2020a): \na ( t ) = ( t 3 -Ω m 1 -Ω m ) 2 / 3 ⇒ H = 2 t 2 t 3 -Ω m . (7) \nThus, from (6) with SIV time in units t ∈ [ t in , t 0 = 1], one obtains for the MOND fundamental acceleration: \na M ( t ) = c ( t 3 -Ω m ) 2 t 4 . (8) \nTo express a M in the usual time units, where τ in = 0 at the Big Bang when the scale factor a = 0 while the age of the Universe now is τ 0 = 13 . 8 billion years, one has to use the chain rule for differentiation, that is: \na M ( τ ) = c ( ˙ λ λ ) 2 a ˙ a = cκ ( τ ) κ ( t ) H ( t ) = ( dt dτ ) a M ( t ) . (9) \nThe value of dt/dτ is assessed based on the assumption that the following relation provides a connection between the two time-scales ( t -t in ) / ( t 0 -t in ) = ( τ -τ in ) / ( τ 0 -τ in ) where t in = Ω 1 / 3 m , t 0 = 1 , τ in = 0 , and τ 0 = 13 . 8 billion years (Maeder & Gueorguiev 2020a). Thus, one has: \ndt/dτ = (1 -t in ) /τ 0 , (10) \nand therefore: \na M ( τ ) = c ( 1 -Ω 1 / 3 m τ 0 ) ( t 3 -Ω m 2 t 4 ) = \n= c 2 b ( 1 -b τ 0 )( x 3 -1 x 4 ) , \n(11) (12) \nwhere x = t/b , Ω m = t 3 in = b 3 . Set ˜ a = a ( 1 -b 3 b 3 ) 2 / 3 and revisit (7) to get: \na = ( b 3 1 -b 3 ) 2 / 3 ( x 3 -1 ) 2 / 3 ⇒ ˜ a = ( x 3 -1 ) 2 / 3 . (13) \nThus, upon utilization of (12) and the above substitution, the MOND acceleration (9), as a function of the scale factor, becomes: \na M ( τ ) = c 2 b ( 1 -b τ 0 ) ˜ a 3 / 2 (˜ a 3 / 2 +1) 4 / 3 . (14) \nNext, look at the z -dependence, and use a = 1 / ( z +1); thus, when z is 0, or 1, and even 2, then a is 1, or 1 / 2, and correspondingly 1 / 3. Therefore, one has: \na M ( z ) = a M ( z = 0) × ( 1 z +1 ) 3 / 2 ( ˜ a 3 / 2 0 +1 ) 4 / 3 ( ( 1 z +1 ) 3 / 2 ˜ a 3 / 2 0 +1 ) 4 / 3 (15) \n= a M 0 × ( z +1) -3 / 2 ( (1 -Ω m ) ( z +1) 3 / 2 +Ω m ) -4 / 3 (16) \nwhere ˜ a 0 = ( 1 -Ω m Ω m ) 2 / 3 was utilized. The above expression (16) provides the formula for the explicit z -dependence of the MOND fundamental acceleration within the SIV framework. It can be used to test this SIV prediction against the observational data. When evaluated at z = 1 and 2, one finds that a M ( z ) is about 79% and 58% of the current value a M 0 = a M ( z = 0) for Ω m = 0 . 3. Thus, the z -dependence of the MOND acceleration is weak, and it is likely buried within the scatter of the current observational data and its uncertainty (see Del Popolo & Chan (2024c)); as one can see later, the value of a M 0 , at best, is accurate within 13% while the 1 σ error of the data on A 0 reported by Del Popolo & Chan (2024c) usually translates in more than 29% uncertainty for observationally deduced MOND acceleration data points. \nThe value of a M 0 could be used to assess the parameter Ω m within the SIV theory. That is, by using (12) one has: \na M 0 = c (1 -b )(1 -b 3 ) 2 τ 0 = c (1 -Ω 1 / 3 m )(1 -Ω m ) 2 τ 0 . (17) \nOne can solve for Ω m by taking log 10 of (17): \nA 0 -log 10 ( c 2 τ 0 ) = log 10 [( 1 -Ω 1 / 3 m ) (1 -Ω m ) ] , (18) \nfor lowz data with ¯ A 0 = -13 . 07 this gives Ω m = 0 . 28. Such a value of Ω m aligns with the comparative study of the SIV and ΛCDM cosmologies that demonstrated only light adjustments in Ω m if the expansion factors a ( t ) of these models are to be very similar Maeder (2017a). However, it contrasts with the MOND idea of dark matter redundancy. Notably, the range of values of A 0 based on (18) for Ω m given by 0 . 01 , 0 . 1 , 0 . 3 , 0 . 6 , 0 . 8 are -12 . 57 , -12 . 78 , -13 . 098 , -13 . 67 , -14 . 31. \nBased on (17), the corresponding fractional uncertainties are related in the following way: \n∆ a M 0 a M 0 = ∆Ω m Ω m Ω m (1 -Ω m ) + 1 3 Ω 1 / 3 m ( 1 -Ω 1 / 3 m ) . (19) \nThe slope of the z -dependence, m = d log 10 a M ( z ) dz when taking log 10 of (16), is then: \nm = 1 -(1 + 3(1 + z ) 3 / 2 )Ω m (1 + z )(1 + ((1 + z ) 3 / 2 -1)Ω m ) ln(100) . (20) \ngiving m 0 = (1 -4Ω m ) / ln(100) at z = 0, which is positive only for Ω m < 0 . 25. While for the Z -slope one has: \ndA 0 dZ = ( dA 0 dz )( dz dZ ) = m ln(10) z = m ln(10)10 Z , (21) \nwhere dz/dZ = (exp( Z ln(10))) ' is utilized. Based on the data provided: A 0 = log 10 a M 0 = -13 . 067 ± 0 . 056 for a M 0 in km/s 2 . The fractional uncertainty is ∆ A 0 / \| A 0 \| = 0 . 0043 = ∆( a M 0 ) / ( \| A 0 \| a M 0 ln(10)). That is, ∆( a M 0 ) /a M 0 = 0 . 13; therefore, using (19) one has Ω m = 0 . 28 ± 0 . 04. The z -slope is then m 0 = -0 . 01 ± 0 . 3, while the Z -slope is -0 . 0001 using Z = -2 . 49. This results practically in a horizontal line that changes very little from being -13 . 0667 at Z = -4 to -13 . 0671 at Z = 2. This change is well within the current error ( ± 0 . 06) for the lowz . For the highz data, one can now evaluate m 1 to be -0 . 11; therefore, the Z -slope is -0 . 25 in agreement with the Z -slope of -0 . 2 ± 0 . 4 reported by Del Popolo & Chan (2024c).", '4 DISCUSSION AND CONCLUSION': "It is still inconclusive about the z -dependence of the MOND fundamental acceleration, but such dependence is present within the SIV theory (16); furthermore, the SIV expression (20) does suggest that there is a change in the sign of the slope when going through z = (1 / 3 × (1 -Ω m ) / Ω m ) 2 / 3 -1 with m 0 positive for Ω m < 0 . 25 and negative otherwise; this could explain the change in the sign of the slope of the two data sets as noticed by Del Popolo & Chan (2024c). However, the corresponding value of A 0 at z = 1, which is the intercept, is about the same as ¯ A 0 based on low-z with ¯ A 0 = -13 . 07, but Del Popolo & Chan's highz intercept differs significantly from the corresponding ¯ A 0 values around A 0 = -12 . 6. However, if one is to embrace the matching values of A 0 at z = 1, that is, to use A 0 = -12 . 6 ± 0 . 05 along with (17) in (16), Then, the value of Ω m is significantly lower; that is, one gets Ω m = 0 . 055. In this case, the sign change of the slopes will be happening around z = 2 . 7. For Ω m = 0 . 055, equation (18) gives A 0 = -12 . 69, which is lower than A 0 = -12 . 6 but is bigger than A 0 = -13 . 07. Note that such a low value for Ω m does not leave much room for any dark matter. Such a result is better aligned with the MOND view about dark matter but seems to be a drastic departure from the need for dark matter and dark energy as required by ΛCDM. This approach may be favorable as a method of determining Ω m since it relies on the two data sets and their consistent Z -intercepts, which is in contrast to the first method presented that utilized only the lowz data and assumed that ¯ A 0 = -13 . 07 at ¯ Z = -2 . 49 is sufficiently close to z = 0 even though z → 0 would imply Z → -∞ . Therefore, further studies are needed to determine the correct Ω m values within the SIV framework. Thus, more precise data analyses are needed along with improved uncertainties of the observational data points (for the currently used data see Fig. 1) to confirm the z -dependence of the MOND fundamental acceleration and to potentially test the SIV theory via its model prediction for a 0 ( z ) as well as to deduce the relevant SIV model parameters. \nIn conclusion, the long-standing mystery of galactic rotation curves has fueled the development of Modified Newtonian Dynamics (MOND). This work presents a significant contribution by providing the first explicit analytic expressions for the z-dependence of the fundamental MOND acceleration ( a 0 ) within the framework of the Scale Invariant Vacuum (SIV) theory (16). This novel approach goes beyond previous studies Del Popolo & Chan (2024a,c). Furthermore, we leverage existing observational data to perform the firstever estimation of the cosmological matter density parameter (Ω m ) within the SIV framework. The current analysis yields a value of Ω m = 0 . 28 ± 0 . 04 based on low-z data and Ω m = 0 . 055 based on the consistency of both data sets at z = 1, potentially removing the need for dark matter entirely. The above is a puzzling result as to why the two methods presented to determine the value of Ω m within SIV result in relevant values within the ΛCDM model. \nOn the one hand, the SIV value 28% for Ω m deduced by using only the dataset with z ≈ 0 is close to the ΛCDM model of about 30% (Planck Collaboration: Ade et al. 2016), while on the other hand, the value 5 . 5% deduced by using both datasets (via the A 0 intercept at z = 1) is close to the baryon matter value within the ΛCDM model of about 5%. This could be just a numerical coincidence, or there may \nbe some deeper reason for why the values are like that. For example, it may be related to the transition from a matterdominated epoch to a cosmological constant (dark-energy) dominated epoch within the ΛCDM model. However, within the SIV paradigm, one does not expect dark-matter and darkenergy components. For example, the energy density Ω Λ E due to the Einstein Cosmological Constant Λ E , which within the ΛCDM model is estimated to be 70%, does not exist within SIV but is replaced by Ω λ = -2 H ˙ λ λ that also compliments Ω m to 1 (assuming flat Universe Ω k = 0) within the SIV paradigm (Maeder & Gueorguiev 2020a). \nInterestingly, the data suggests an almost flat z -dependence of A 0 = log 10 ( a 0 ), contrasting with previous claims by Del Popolo & Chan (2024a,c). While the current data limitations prevent the definitive confirmation of the z -dependence (16), the observed trends are consistent with SIV predictions. SIV offers a unique explanation for the potential sign change in the slopes previously indicated across different redshift ranges. Future higher precision data will be crucial for definitively resolving the presence or absence of z-dependence in a 0 .", 'ACKNOWLEDGMENTS': 'The author is grateful for the moral and financial support by particularly close private parties during the various stages of the research presented and to Prof. A. Maeder for the long and fruitful scientific collaboration over the years. This research does not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. However, the publishing charges were waived graciously by the Author Support, Oxford Journals, at Oxford University Press - for which the author is very grateful!', 'DATA AVAILABILITY': 'No new data was generated or analyzed in support of this research.', 'REFERENCES': "Bekenstein J. D., Sagi E., 2008, Phys. Rev. D, 77, 103512 \nBertone G., Hooper D., 2018, Reviews of Modern Physics, 90, \n- 045002\n- Canuto V., Adams P. J., Hsieh S. H., Tsiang E., 1977, Phys. Rev. D, 16, 1643\n- Del Popolo A., Chan M. H., 2024a, Physics of the Dark Universe, 43, 101393\n- Del Popolo A., Chan M. H., 2024b, Physics of the Dark Universe, 43, 101414\n- Del Popolo A., Chan M. H., 2024c, Physics of the Dark Universe, 43, 101415\n- Dirac P. A. M., 1973, Proceedings of the Royal Society of London Series A, 333, 403\n- Dirac P. A. M., 1974, Proceedings of the Royal Society of London Series A, 338, 439\n- Eddington A. S., 1923, The mathematical theory of relativity, 2nd edition edn. Cambridge University Press, https://www. gutenberg.org/files/59248/59248-pdf.pdf \nEinstein A., 1918, Sitzungsberichte der Koniglich, Preussischen Akademie der Wissenschaften zu Berlin, p. 478 \n```\nGueorguiev V. G., Maeder A., 2024, Symmetry, 16, 657 Jesus J. F., 2017, arXiv e-prints, p. arXiv:1712.00697 Lelli F., McGaugh S. S., Schombert J. M., 2017, MNRAS, 468, L68 Maeder A., 2017a, ApJ, 834, 194 Maeder A., 2017b, ApJ, 849, 158 Maeder A., 2023, MNRAS, 520, 1447 Maeder A., Gueorguiev V. G., 2020a, Universe, 6, 46 Maeder A., Gueorguiev V. G., 2020b, MNRAS, 492, 2698 Maeder A., Gueorguiev V. G., 2023, Symmetry, 15, 1966 Marra V., Rodrigues D. C., de Almeida ' A. O. F., 2020, MNRAS, 494, 2875 Milgrom M., 1983, ApJ, 270, 365 Milgrom M., 2009, ApJ, 698, 1630 Nestor Shachar A., et al., 2023, ApJ, 944, 78 Pazy E., 2013, Phys. Rev. D, 87, 084063 Planck Collaboration: Ade et al. 2016, A&A, 594, A13 Verlinde E. P., 2017, SciPost Physics, 2, 016 Weyl H., 1922, Space - Time - Matter. Dutton, https://www. gutenberg.org/files/43006/43006-pdf.pdf Weyl H., 1970, Raum, Zeit, Materie. Vorlesungen uber allgemeine Relativitatstheorie.. Berlin: Springer, https://archive.org/ details/raumzeitmateriev00weyl/page/n5/mode/2up\n``` \nThis paper has been typeset from a T E X/L A T E X file prepared by the author."}
2024ApJ...975..164H	While there has been an increase in interest in the possibility of quasiperiodic oscillations QPOs in blazars the search has hitherto been restricted to sources with wellsampled light curves. Objects with light curves that include gaps have been to our knowledge overlooked. Here we study two such curves which have the interesting feature of pertaining to relatively highredshift blazarsFSRQs PKS 215583 and PKS 2255282observed by the Fermi Large Area Telescope. Their redshifts border the cosmic noon era of galaxy formation and merging and their light curves exhibit a distinctive pattern of repetitive high and low gap dominant states for 15.6 yr. To accommodate for the gaps in the curves data are integrated over extended time intervals of 1 month and 2 months. The resulting curves were also examined using methods suitable for sparsely sampled data. This investigation of PKS 215583 and PKS 2255282 suggests QPOs with periods of 4.69 0.79 yr 3 and 6.82 2.25 yr 2.8 respectively. The probability density functions of the blazars fluxes along with the correlation between their flux and spectral index were also analyzed. Given the epochs in which the objects are observed the plausibility of a binary black hole scenario as an origin of the apparent periodicity was examined. We estimated the prospective parameters of such a system using a simple geometric model. The total masses were estimated and found to be consistent in principle with independent dynamical measurements of the central black hole masses in the two host galaxies.	2024-11-01T00:00:00Z	['10.48550/arXiv.2409.10622', '2024arXiv240910622H', '10.3847/1538-4357/ad7a6e', 'arXiv:2409.10622', '2024ApJ...975..164H']	['Active galactic nuclei', 'Gamma-ray astronomy', 'Time series analysis', 'Non-thermal radiation sources', 'Galaxy mergers', 'Blazars', '16', '628', '1916', '1119', '608', '164', 'Astrophysics - High Energy Astrophysical Phenomena']	Quasiperiodic Ray Modulations in the Blazars PKS 215583 and PKS 2255282	2,024	190	0.5	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.10622.pdf	{'No Header': 'Draft version September 18, 2024', 'Quasi-periodic γ -ray modulations in the blazars PKS 2155-83 and PKS 2255-282': 'M. A. Hashad, 1, 2 Amr A. EL-Zant, 2 Y. Abdou, 3 and H. M. Badran 3 \n1 Basic Sciences Department, Modern Academy for Engineering and Technology, Maadi, 11585, Cairo, Egypt 2 Centre for Theoretical Physics, The British University in Egypt, Sherouk City, 11837, Cairo, Egypt 3 Physics Department, Faculty of Science, Tanta University, Tanta, 31527, Gharbia, Egypt', 'ABSTRACT': "While there has been an increase in interest in the possibility of quasi-periodic oscillations (QPOs) in blazars, the search has hitherto been restricted to sources with well-sampled light curves. Objects with light curves that include gaps have been, to our knowledge, overlooked. Here, we study two such curves, which have the interesting feature of pertaining to relatively high redshift blazars - FSRQs, PKS 2155-83 and PKS 2255-282 - observed by Fermi -LAT. Their redshifts border the 'cosmic noon' era of galaxy formation and merging, and their light curves exhibit a distinctive pattern of repetitive high and low (gap dominant) states for 15 . 6 years. To accommodate for the gaps in the curves, data is integrated over extended time intervals of 1 month and 2 months. The resulting curves were also examined using methods suitable for sparsely sampled data. This investigation of PKS 2155-83 and PKS 2255-282 suggests QPOs with periods of 4 . 69 ± 0 . 79 yr (3 σ ) and 6 . 82 ± 2 . 25 yr (2 . 8 σ ), respectively. The PDFs of the blazars' fluxes, along with the correlation between their flux and spectral index, were also analyzed. Given the epochs the objects are observed, the plausibility of a binary black hole scenario as an origin of the apparent periodicity was examined. We estimated the prospective parameters of such a system using a simple geometric model. The total masses were estimated, and found to be consistent, in principle, with independent (dynamical) measurements of the central black hole masses in the two host galaxies. \nKeywords: Active galactic nuclei; Gamma-rays; Time series analysis; Period search", '1. INTRODUCTION': "Collimated plasma jets are launched from active galactic nuclei (AGN) born of spinning black holes and strongly magnetized accretion disks. Blazars are a class of AGN whose relativistic jets nearly point to the line of sight connecting it to the Earth, and are dominated by non-thermal emission (Blandford et al. 2019; Madejski & Sikora 2016; Urry & Padovani 1995). Owing to the jet's alignment and its narrow opening, Doppler enhancement is expected of the blazar emission and, subsequently, contract the timescales of its variability (Yan et al. 2018). \nErratic variability in blazars' emission has been observed over almost all the electromagnetic spectrum and over a wide range of timescales (Abhir et al. 2021; Bhatta & Dhital 2020; Liao et al. 2014; Bglyph[suppress]la˙zejowski et al. \nCorresponding author: M. A. Hashad \[email protected] \n2005). In particular, virtue of the continuous all-sky monitoring of the Fermi Large Area Telescope ( Fermi -LAT; Atwood et al. 2009), launched in 2008, one can seek out and check for quasi-periodicity in γ -ray blazars with periods up to a few years. Numerous recent results have indeed found evidence for the existence of such quasi-periodic variability in blazars (Ren et al. 2023; Hashad et al. 2023; Zhang et al. 2020; Pe˜nil et al. 2020; Benkhali et al. 2020; Tavani et al. 2018; Sandrinelli et al. 2018; Prokhorov & Moraghan 2017). \nAfter six years of Fermi -LAT data gathering, the first evidence of a significant QPO ( ∼ 3 σ ) in γ -ray LC has been reported for PG 1553+113 with a period of 2.18 yr (Ackermann et al. 2015). One remarkable case is PKS 2247-131, which manifests a short periodicity ( ∼ 34.5 days) in its γ -ray LC from November 2016 to June 2017 with six cycles at high significance (5 . 2 σ ) (Zhou et al. 2018). This QPO has been interpreted in terms of a helical structure in the jet. Using 12 years of Fermi -LAT data, Pe˜nil et al. (2022) have examined the γ -ray LCs of the most promising 24 periodicities reported in \nthe literature. Five blazars with γ -ray QPOs observed with significance ≳ 3 σ have been found, one FSRQ, PKS 0454-234, and four BL Lacs, OJ 014, PG 1553+113, S5 0716+714, and PKS 2155-304. Observing such periodic signals could provide insight into blazars' nature and black hole (BH)-jet systems. \nIndeed, the mechanism producing possible γ -ray QPO in blazars is not entirely understood. Scenarios like jet precession, pulsational accretion flow instabilities, and the existence of binary black hole systems have been proposed (e.g., Caproni et al. 2017; Sobacchi et al. 2016; Ackermann et al. 2015). The broadness of the possibilities is reflected in the fact that, within the variety of models of QPOs in blazars, the QPOs can originate from intrinsic as well as apparent origins; in the intrinsic scenario, QPO is assumed to exist in the relativistic jet's co-moving frame, while an apparent origin refers to a periodically changing viewing angle and associated Doppler factor, which in turn boosts the observed flux periodically. \nThe reported searches for quasi-periodicity usually focus on objects with LCs that have significant detections in most bins. Typically, the minimum detection significance of the bins is set to be ≥ 3 σ . As a result, important attributes in LCs with many upper limits or gaps are generally neglected. \nIn this work, we introduce the results of a QPO search in two moderate redshifts blazars: flat-spectrum radio quasars (FSRQs), PKS 2155-83 ( z = 1 . 865), and PKS 2255-282 ( z = 0 . 926). Both exhibit the distinguishing behavior of repetitive high states and extremely faint epochs. These faint epochs are considered low (quiescent) states where upper limits and gaps frequently exist, particularly for time-binning of narrow time intervals. Here, we examine longer binning intervals and use methods that are suitable for time series with gaps, in order to attempt to reveal possible QPO signals. The redshifts, bordering the 'cosmic noon' era of galaxy formation and merging (in some definitions, PKS 2155-83 being well within that epoch), may hint at the apparent QPO being suitable candidates to be associated with supermassive black hole (SMBH) merger events (Mezcua et al. 2024; Yang et al. 2020; Volonteri et al. 2015; Madau & Dickinson 2014; Komossa et al. 2015; Begelman et al. 1980). \nOur search is based on Fermi -LAT data over about 15.6 years. A detailed analysis of Fermi -LAT data is presented in Section 2. The resulting LCs are then examined for possible quasi-periodicities, using various methods, in Section 3. Section 4 summarizes the results and discusses the findings.", '2.1. The Fermi-LAT Light Curves': "The Fermi Gamma-ray Space Telescope is a space mission with two scientific instruments: the LAT and the Gamma-ray Burst Monitor (GBM). The Fermi -LAT high-energy γ -ray telescope covers the energy range from about 20 MeV up to 1 TeV and owns a large effective area ( ∼ 8000 cm 2 at 1 GeV), with ∼ 2 . 4 sr field of view and a point-spread function (PSF) of < 0 . 8 · above 1 GeV (Atwood et al. 2009). LAT scans the whole celestial sphere every 3 hours. The instrument took off on 11 June 2008, and is still in operation. In the course of its 15-year operation, it detected high-energy gamma rays from assorted classes of objects with the most severe conditions, including but not limited to gamma-ray blazars. \nWe consider here the LCs of two blazars detected by the Fermi observatory, searching within them for quasi-periodic signals. The distant FSRQ PKS 215583 is at RA = 330h36m00 . 00s and Dec = 83d35m60 . 0s, J2000 (Fabricius et al. 2021; Mauch et al. 2003). It was observed by the Fermi -LAT in a high state on 5 January 2010, with a γ -ray flux F ( E > 100 MeV) of (1 . 4 ± 0 . 3) × 10 -6 photons cm -2 s -1 , which is more than an order of magnitude larger than the average flux during the first 11 months of observations (Wallace 2010). According to the Fermi -LAT 4th source catalog (4FGL-DR3; Abdollahi et al. 2022), PKS 215583 (4FGL J2201.5-8339) has an average detection significance of 41 . 88 σ with a predicted photon number of 4247.39 and flux fractional variability of 67 . 86 ± 14 . 84%. During the 15.6 years of Fermi -LAT observations, the object has had eminent behavior, where it seems to release four high states in 2010, 2014, 2019, and 2023 throughout its low state, as shown in the LCs in Fig. 1. This behavior may underline a featured origin, as the repeated high states look periodic. \nSimilarly, another distant FSRQ, PKS 2255-282 with RA = 344h30m00 . 00s and Dec = -27d53m60 . 0s, J2000 (Jones et al. 2009) - shows three high states (2009-2013, 2017-2021, and 2023-up to the end of data) throughout its low state, as shown in Fig. 2. An average detection significance of 66 . 57 σ , a predicted photon number of 7383.77, and flux fractional variability of 84 . 67 ± 18 . 17%, were reported from the source in Fermi's fourth catalog (Abdollahi et al. 2022). Radio measurements at 15 GHz (Lister et al. 2009) and 22 and 43 GHz (Charlot et al. 2010) suggested a compact object with a core-dominated structure. The object was in a high state on 26 February 2012, with a γ -ray average daily flux above 100 MeV of (1 . 0 ± 0 . 3) × \n10 -6 photons cm -2 s -1 . This exemplifies a boost factor of ∼ 14 above its average flux in Fermi's second catalog (Dutka et al. 2012). It was the first significant Fermi -LAT source to be detected at such a high value of flux, although a γ -ray flare was detected earlier from the object by EGRET in December 1997 (Macomb et al. 1999). The EGRET outburst lasted from 30 December 1997 till 12 January 1998, with possible weak variability of a short timescale of several days (Tornikoski et al. 1999). The total flux above 100 MeV was (1 . 6 ± 0 . 3) × 10 -6 photons cm -2 s -1 with a peak flux of (4 . 8 ± 1 . 1) × 10 -6 photons cm -2 s -1 . This was higher than the quiescent upper limits to emission, based upon previous EGRET observations, by a factor of 20, placing PKS 2255-282 among EGRET's brightest blazars. Before this outburst, PKS 2255-282 had been in the field of view of EGRET several times but was not detected in γ -rays, see Fig. 2 in Macomb et al. (1999). In November 1997, EGRET recorded a flux above 100 MeV of (4 . 7 ± 2 . 3) × 10 -7 photons cm -2 s -1 (3 σ detection), which is about 10 times smaller than the flux from the outburst of January 1998. The count rate dropped by roughly a factor of 3 by the end of the outburst period (Tornikoski et al. 1999). As EGRET was in reduced field mode during the outburst period, the 9.2 σ detection only corresponds to 51 ± 9 source counts. With such sparse data, it was difficult to locate γ -ray variability. However, many photons were clustered around two separate times, from 2.2 to 3.1 January 1998 and from 9.1 to 10.5 January 1998. It is noteworthy that a prolonged quiescent state with reported upper limits preceded the outburst observed by EGRET and this was, to some extent, repeatedly observed later on by Fermi -LAT. \nFor the sake of performing the periodicity search for both objects PKS 2155-83 and PKS 2255-282, we proceed as follows: We generated LCs with one month (1m) and two months (2-m) of time binning, encompassing the approximately 15.6 years (MJD: 54683-60369) of LAT data in the energy range 100 MeV-500 GeV (Figures 1 and 2). A 1-m binned LC with detected bins contingent on test statistics > 4 ( TS = 2log( L/L 0 ), where L and L 0 are the maximum likelihood of the models with and without a source at the target position) was constructed to allow inspection of the flux variation in relatively moderate details. The choice of 2-m binning with TS > 9 is motivated by keeping the time intervals long enough to diminish the missing values, reduce fluctuations, and provide better statistics of the underlying variations within the data on the cost of losing the time resolution. The detection ratios (the number of detections in the LC to the total number of bins) in 1-m and 2-m are comparable and estimated to be about 64% and \n81% for PKS 2155-83 and PKS 2255-282, respectively. The LCs were then reduced with the maximum likelihood method using Fermitools version 2.2.0 1 by the implementation of the Python package fermipy 2 (Version: 1.2.2; Wood et al. 2017). The instrument response function P8R3 SOURCE V3 was used with 'SOURCE' class photons. \nIn this analysis, the photons within the region of interest (a 15 · × 15 · square centered on the position of the source of interest) were selected. Photons were then modeled by accounting for the point sources in the 4FGL catalog that positioned around the source of interest (up to 20 · ). Moreover, the background emission was also modeled with including a galactic component (the Milky Way's diffuse γ -ray emission; gll iem v 07.fits file) and an extragalactic one (the isotropic γ -ray emission from celestial and residual charged-particle backgrounds; iso P 8 R 3 SOURCE V 3 v 1.txt file). A cut on the zenith angle larger than 90 · was imposed to exclude γ -ray augmentation from the earth limb. Additionally, we used the recommended data quality cuts (DATA QUAL > 0)&&(LAT CONFIG== 1) and removed time periods coinciding with gamma-ray bursts and solar flares detected by the LAT. A 0 . 1 · spatial binning and eight logarithmic energy bins per decade were adopted. The normalizations of all sources within 3 · away from the ROI center and the galactic and isotropic diffuse backgrounds, as well as the normalization and spectral index of the target source, were left free to vary in the likelihood analysis over the full time range of the observation. All other parameters were set at their catalog values. The routines gta.optimize() and gta.fit() were iteratively run till a good fit quality is achieved (fit quality = 3). To construct the LCs, we split the data for each source into 1-m and 2-m bins and conducted a full likelihood fit in each bin. This is done while utilizing the parameters' values obtained from the full time range analysis. The spectral parameters of the target except the scale parameter were left free during the fit. The normalizations of sources within 3 · from the center of the ROI along with the normalizations of diffuse components, were left to vary. \nThe observed γ -ray fluxes of both objects display striking variation (Figures 1 and 2). The average flux of the 1-m binned LCs is; F av = (6 . 12 ± 4 . 64) × 10 -8 photons cm -2 s -1 and F av = (1 . 05 ± 1 . 02) × 10 -7 photons cm -2 s -1 for PKS 2155-83 and PKS 2255-282, respectively and for the 2-m binned \nLCs is F av = (6 . 16 ± 0 . 53) × 10 -8 photons cm -2 s -1 and F av = (1 . 04 ± 0 . 90) × 10 -7 photons cm -2 s -1 for PKS 2155-83 and PKS 2255-282, respectively. \nPhoton spectral indices (Γ av ) are estimated to be 2 . 39 ± 0 . 39 (1-m) and 2 . 40 ± 0 . 29 (2-m) for PKS 215583, and 2 . 41 ± 0 . 34 (1-m) and 2 . 45 ± 0 . 26 (2-m) for PKS 2255-282. Both sources show soft intrinsic γ -ray spectra like most FSRQs (Madejski & Sikora 2016) for both 1-m and 2-m binned LCs. This probably explains the absence of γ -ray detection from these sources at higher energies. PKS 2155-83 (Pearson correlation coefficient ρ 1 -m = 0 . 28) and PKS 2255-282 ( ρ 1 -m = 0 . 01) showed no flux-photon index correlation along the full time range of observation implying that both objects may favor the association with apparent geometrical effects.", '2.2. Flux Distribution': "An essential feature of an astronomical source's variability is the distribution of the photon flux. The distinction between the two most common Gaussian and log-normal distributions can help characterize the inherent physical process causing the observed variability (Shah et al. 2018; Rieger 2019; Shah et al. 2020; Morris et al. 2019). A Gaussian distribution reflects a linear random process, where the flux variation is to be indicated by the distribution width (Sinha et al. 2018). In this case, an additive statistical model is implied, with the linear summation of components taking part in building up the emission (e.g., shot-noise or a linear summation of many 'mini-jets'). Recent findings, on the other hand, preferentially advocate the lognormal distribution for blazars at various wavelengths and timescales (Romoli et al. 2018; Shah et al. 2018; Abramowski et al. 2010; Sinha et al. 2017; Wang et al. 2023). \nThe log-normality of blazars' fluxes can be linked to the existence of a multiplicative process (Rieger 2019). Accretion disk fluctuations could be a possible origin for such action, as in the case of X-ray binaries (Lyubarskii 1997; Arevalo & Uttley 2006). In this case, the accretion rate of mass varies as the result of an independent accretion disk's density fluctuations on a timescale that corresponds to the local viscous time scales. The fluctuations proliferate inward and provide a multiplicative process as they couple together in the innermost part of the disk (Lyubarskii 1997; King et al. 2004; Arevalo & Uttley 2006). If the instabilities in the accretion flow exhibit a quasi-periodic nature, the resulting QPOs will also propagate to the jet, and corresponding emission may be observed (Rieger & Volpe 2010). By analogy, such origins for the log-normality may favor a binary \nFigure 1. PKS 2155-83 LCs in the energy range from 100 MeV to 500 GeV between 4 August 2008, and 29 February 2024. Showing (a) the one month (1-m) binned LC, (b) the photon spectral index of (a), (c) two months (2-m) binned LC, and (d) the photon spectral index of (c). In panels (a) and (c), filled black points denote significant detections, with TS > 4 for the 1-m binned LC and TS > 9 for the 2-m binned LC. Downward gray arrows denote 95% confidence level upper limits. The gray vertical columns approximately delineate periods of high states, inferred from the generalized Lomb Scargle periodogram. The periodic signals' uncertainty is indicated by the width of the gray columns. The photon index is plotted only for the detected bins. In all panels, vertical error bars are 1 σ error. \n<!-- image --> \nFigure 2. The same as Figure 1, but for the object PKS 2255-282. \n<!-- image --> \nblack hole scenario for the possible quasi-periodic time signal. Log-normal flux distributions may also originate from cascade-related scenarios, such as magnetospheric inverse-Compton pair production cascades (Levinson & Rieger 2011) or proton-induced synchrotron cascades (Mannheim 1993). Further, the log-normal distribution could also be traced back to the acceleration process itself, for example, with linear Gaussian fluctuations in the particle acceleration rate inside the region of acceleration (Sinha et al. 2018). Finally, it should be men- \ntioned that additive processes in specific scenarios can also lead to flux log-normality such as the overall flux (experienced Doppler boosting) from a large number of mini jets within a jet with random orientations (Biteau & Giebels 2012). \nFor each object, we constructed a flux histogram for the 1-m binned LC chosen over 2-m binned data because it simply provides a statistically more significant fit of the distribution. The probability density function (PDF) of PKS 2155-83 and PKS 2255-282 were fitted by Gaussian G( ϕ ) and log-normal L( ϕ ) distributions given by \nG ( φ \| µ, σ ) = 1 √ 2 πσ exp ( -( φ -µ ) 2 2 σ 2 ) (1) \nand \nL ( φ \| µ, σ ) = 1 √ 2 πσφ exp ( -(log 10 ( φ ) -µ ) 2 2 σ 2 ) , (2) \nwhere µ and σ are the mean and standard deviation, respectively. \nFor PKS 2155-83 PDF, the log-normal distribution ( r 2 1 -m = 0 . 94) seems to be preferred over a Gaussian ( r 2 1 -m = 0 . 88), which invokes a nonlinear, multiplicative process for the underlying variability rather than additive models. On the contrary, the PKS 2255-282 PDF has a comparable degree of fitness with both distributions; r 2 1 -m = 0 . 99 and = 0 . 95 for Gaussian and lognormal distributions, respectively. The Gaussian and log-normal fit parameters, variance, probability, and WStatistic of both sources are listed in Table 1 for 1-m and 2-m cases. The normality test for 1-m and 2-m flux histograms for both objects is rejected.", '3. SEARCH FOR γ -RAY QUASI-PERIODICITY': 'Several approaches have been used to inspect the quasi-periodic variability in blazars (Wang et al. 2022). In the present study, different methods were utilized to search for periodicity in the LC. These are described in the following sub-sections.', '3.1. Auto-correlation Function': "The auto-correlation function (ACF) is a reliable method suitable for detecting non-sinusoidal periodicities. It involves the correlation of the time series with itself, i.e., with the same series lagged by one or more time units. Constant auto-correlation is associated with a system remaining in the same state from one observation to the next; rapidly decaying ACF indicates a high degree of randomness in the time series, while periodicity in the ACF reflects corresponding periodicity in the data. \nTable 1. Fit parameters of PKS 2155-83 and PKS 2255-282 PDFs associated with the log-normal and Gaussian distributions, and flux normality tests. \n*In units of × 10 - 7 photon cm - 2 s - 1 . \nThe ACF between observations ( f i ( t )) separated by τ (= 0, 1, 2, 3,. . . , N ) time steps is given by \nACF ( τ ) = ∑ N -τ i =1 { f i ( t ) -ˆ f 1 → N -τ }{ f i + τ ( t ) -ˆ f 1+ τ → N } ( N -τ ) σ 2 , (3) \nwhere ˆ f 1 → N -τ and ˆ f 1+ τ → N are the means of the first (from the first to N -τ observations), and the last (from 1 + τ to N observations) N -τ of the data points, respectively. \nWe used pyzdcf 3 , a Python module that is utilized for robustly estimating cross-correlation functions of astronomical time-series data that are sparse and unevenly sampled (Alexander 1997). A Savitzky-Golay filter 4 was also applied to smooth the ACF, which effectively decreases the low-frequency fluctuations while preserving the overall tendency of the signal (Press & Teukolsky 1990). The signal's period is the median of a list of periods computed from the intervals between successive maxima and minima. The uncertainty is determined using the equation proposed by McQuillan et al. (2013) as \nσ P = 1 . 483 × MAD √ N -1 , (4) \nwhere MAD is the periods' median of the absolute deviations implied from the peaks list and N is the number of peaks in the correlation. To determine the significance, we simulated 10 5 LCs, using Emmanoulopoulos' method (Emmanoulopoulos et al. 2013) as coded in Python in Connolly (2015), that match both the power spectral density and probability density function of the object's LC. For each simulated LC, the ACF was applied, and the percentile was computed for each period to estimate the power confidence level. \nThe obtained ACFs are shown in Fig. 3, for 1-m binned LCs. For the 1-m and 2-m binned LCs of \nPKS 2155-83, the estimated period is at T = 4 . 79 ± 0 . 35 yr (2 . 8 σ ) and at T = 4 . 68 ± 0 . 24 yr (1 . 8 σ ), respectively; and for the 1-m and 2-m binned LCs of PKS 2255-282 is at T = 6 . 53 ± 1 . 34 yr (3 . 3 σ ) and at T = 6 . 27 ± 0 . 16 yr (3 σ ), respectively.", '3.2. Date-Compensated Discrete Fourier Transform': "Another powerful method is the date-compensated discrete Fourier transform (DCDFT), proposed by Ferraz-Mello (1981). It tailors to unevenly spaced data, utilizing the notion of function space projection to realize a Fourier transform. For a given test frequency, the power and amplitude of the DCDFT of unequally spaced data are given by \nP ( w, \| x ⟩ ) = N [ ⟨ y \| y ⟩ - ⟨ 1 \| y ⟩ 2 ] 2 S 2 (5) \nand \nA ( w, \| x ⟩ ) = √ 2( ⟨ y \| y ⟩ - ⟨ 1 \| y ⟩ 2 ) , (6) \nwhere N is the number of data points, y is the time simulation function, and S 2 is the variance of the time series. The existence of gaps in the data produces spurious peaks in the power spectrum. The CLEANest algorithm can remove spurious peaks, which can be implemented as explained in Foster (1995). We used the AAVSO VStar software 5 (Benn 2012) to perform the DCDFT and to run CLEANest period analysis refinement algorithm. In what follows, we enclose the DCDFT values obtained via the DCDFT+CLEANest method by parentheses. The timescales quoted were estimated by fitting the power peak to a Gaussian curve, and uncertainty of the signal is the fitting's half width at half maximum (HWHM). \nThe PKS 2155-83 1-m binned LC showed two clear peaks at 4 . 84 ± 0 . 53 yr (4.71 yr) and 1 . 38 ± 0 . 05 yr (1.38 yr). Whereas for the 2-m binned LC, it showed only one such peak at 5 . 00 ± 0 . 69 yr (4.68 yr), as shown in Fig. 4. In the case of PKS 2255-282, the 1-m binned LC showed three peaks at 1 . 42 ± 0 . 05 yr (1.42 yr), 2 . 79 ± 0 . 17 yr (2.77 yr), and 5 . 88 ± 0 . 82 yr (5.64 yr). Comparable results were obtained for the 2-m binned LC. \n<!-- image --> \nFigure 3. Auto-correlation function for PKS 2155-83 (top panel) and PKS 2255-282 (bottom panel) 1-m binned LCs. The blue lines represent the smoothed correlation, using a Savitzky-Golay filter. The green, red, indigo, and brown lines represent the local 1 σ , 2 σ , 3 σ , and 4 σ chances of observing the auto-correlation levels indicated by the corresponding lines. These were estimated through simulations of 10 5 LCs, utilizing Emmanoulopoulos' method (Emmanoulopoulos et al. 2013). \n<!-- image --> \nFigure 4. DCDFT of the 1-m (blue line) and 2-m (red line) binned LCs of PKS 2155-83 (top panel) and PKS 2255-282 (bottom panel). \n<!-- image -->", '3.3. The Lomb-Scargle Periodogram': "Further method used to search for quasi-periodicity is the Lomb-Scargle periodogram (LSP; Lomb 1976; Scargle 1982), which is a widely used algorithm to establish and characterize periodicity in astronomy, even when the LC has gaps and irregularities. The standard normalized LSP is obtained by fitting the LC to sinusoidal waves of the form y ( t ) = A cos( ωt ) + B sin( ωt ). It is defined for a time series ( t i , y i ) as \nP ( ω ) = 1 2 { ( ∑ i y i cos ω ( t i -τ )) 2 ∑ i cos 2 ω ( t i -τ ) + ( ∑ i y i sin ω ( t i -τ )) 2 ∑ i sin 2 ω ( t i -τ ) } \nwhere, τ is specified for each frequency to ensure timeshift invariance, such that \nτ = 1 2 ω tan -1 (∑ i sin(2 ωt i ) ∑ i cos(2 ωt i ) ) . (7) \nThe generalized Lomb-Scargle periodogram (GLSP) is superior to the standard LSP (Ackermann et al. 2015; Prokhorov & Moraghan 2017). Unlike the LSP, the GLSP does not assume that the fitted sine function's mean is the same as the mean of the data. Instead, it accounts for an offset, c , to the fitting sinusoidal function, i.e., y(t) = A cos( ωt ) + B cos( ωt ) + c . For γ -ray \nblazars, this term may come from the isotropic diffuse γ -ray background. In addition, the GLSP takes measurement errors into consideration. \nThe GLSP powers of the 1-m binned LC of PKS 215583 is shown in Fig. 5. In astronomical observations, spurious peaks can spike up due to various contributing factors (VanderPlas 2018; Vaughan et al. 2016), e.g., window function aliasing and red noise variability background (Vaughan 2005). Therefore, the estimated period uncertainty is an essential aspect of reporting the periodogram's results. The false alarm probability (FAP) is one way to quantify peak significance. In the periodograms of PKS 2155-83, the best periods of maximum powers were found to be T = 4 . 45 ± 0 . 13 yr with a FAP of 4 . 51 × 10 -11 and T = 4 . 42 ± 0 . 08 yr with a FAP of 3 . 64 × 10 -8 for the 1-m and 2-m binned LCs, respectively. \nNot only the gaps and observation errors influence the periodogram result, but also the bin size has a remarkable impact, particularly on the high-frequency signal. PKS 2155-83 was reported with QPO signal at T = 1 . 4 ± 0 . 1 yr (2 . 8 σ ) (Pe˜nil et al. 2020) in agreement with the present result of T = 1 . 43 ± 0 . 05 yr (2 . 5 σ ) for the 1-m binned LC. This high-frequency signal disappeared in the 2-m binning LC's periodogram, as the variation details decrease. \nTo further examine the significance of the power peaks, the 10 5 simulated LCs were used. The period was estimated by fitting the power peak to a Gaussian curve, and its uncertainty is the fitting's HWHM. In this way, a peak was identified with a period of 4 . 69 ± 0 . 79 yr for the 1-m binned LC and 4 . 55 ± 0 . 84 yr for the 2-m binned LC at a significance of (3 σ ) for both of them (Table 2). \nThe estimated periods corresponding to the maximum powers in the periodograms of the 1-m and 2-m binned LCs of PKS 2255-282 were T = 6 . 16 ± 0 . 27 yr with a FAP of 1 . 95 × 10 -8 and T = 6 . 12 ± 0 . 41 yr with a FAP of 8 . 26 × 10 -3 , respectively. The GLSP of the PKS 2255-282 1-m binned LC showed periods (in years) of 1 . 43 ± 0 . 05 (2 . 5 σ ) and 6 . 82 ± 2 . 25 (2 . 8 σ ) (as shown in Fig. 5), while the GLSP of the 2-m binned LC showed only the low frequency period at T = 6 . 73 ± 1 . 35 yr with a significance of 2 . 6 σ (Table 2).", '3.4. The Weighted Wavelet Z-transform': "An additional efficient method for detecting periodicity associated with LCs of uneven data sampling is the weighted wavelet z-transform (WWZ; Foster 1996). It is based on a similar notion as the LSP, where the data is fitted by sinusoidal waves. The WWZ can record the possible existence of quasi-periodic variability with \nFigure 5. The GLSPs and WWZs of PKS 2155-83 (top panel) and PKS 2255-282 (bottom panel) γ -ray 1-m binned LCs. The GLSPs are represented in black lines and the WWZs are represented in blue lines. The green, red, and magenta lines represent the 2 σ , 3 σ , and 4 σ confidence levels, respectively, of the GLSPs of 10 5 simulated LCs utilizing Emmanoulopoulos' method (Emmanoulopoulos et al. 2013). \n<!-- image --> \na transient nature, where the waves are localized in both frequency and time domains (Bhatta et al. 2016; Mohan & Mangalam 2015; Benkhali et al. 2020). \nFor the PKS 2155-83 LC, the WWZ gave (see Figs. 5 and 6) a peak at a period of 4 . 88 ± 0 . 61 yr and 4 . 93 ± 0 . 74 yr for the 1-m and 2-m binned LCs, respectively. The WWZ of PKS 2255-282 gave a period of 5 . 87 ± 0 . 85 yr and 5 . 81 ± 0 . 85 yr for the 1-m and 2-m binned LCs, respectively. In addition, the WWZ of the PKS 2255282 1-m binned LC displays a notable peak at a period of about 1000 days.", '3.5. REDFIT': 'The LSP and WWZ are strongly affected by red noise at low frequencies, where they place peaks that mimic a real periodicity. The REDFIT method is a suitable tool for detecting periodicity in the dominant red noise LCs of blazars. It simply uses a first-order autoregressive (AR1) model (Hasselmann 1976) to precisely assess the periodogram peaks against stochastic fluctuations (Zhang et al. 2021). The method was coded in \nTable 2. The estimated periods of the ∼ 15.6 year 1-m and 2-m binned LCs of PKS 2155-83 and PKS 2255-282 with their estimated significance. \n<!-- image --> \nFigure 6. Two-dimensional contour plots of the WWZ power for the 1-m binned LCs of PKS 2155-83 (top panel) and PKS 2255-282 (bottom panel). \n<!-- image --> \nFortran 90 by Schulz & Mudelsee (2002). We used the REDFIT3.8e3 package in the present analysis 6 . \nCompatible with previous estimates, the results for PKS 2155-83 suggest a period of 4 . 82 ± 0 . 55 yr at significance exceeding 99% and 4 . 63 ± 0 . 62 yr at 2 . 2 σ (Fig. 7) for the 1-m and 2-m binned LCs, respectively. The results furthermore suggest another peak of T = 1 . 37 yr at 2 . 5 σ and 2 σ for the 1-m and 2-m binned LCs, respectively. For the 1-m and 2-m binned LCs of PKS \n2255-282, no significant signal was detected except for a period of about 550 days, as previously reported (Pe˜nil et al. 2020), at significance of 2 . 3 σ and 2 . 4 σ for the 1-m and 2-m binned LCs, respectively. It should be noted that the maximum significance provided by REDFIT is limited to 2 . 5 σ .', '4. SUMMARY AND DISCUSSION': "The growing number of reports regarding the possible existence of quasi-periodicities in the light curves of blazars is interesting; in terms of the information it may embody on the emission processes of these systems, as well as the possibility of detecting binary SMBH systems in the process of merging. A larger sample of candidate objects should include going beyond the well-sampled light curve, where the majority of its time bins have significant observations and only a minority constitute observations with upper limits or gaps. \nHere, we studied two such cases of FSRQs from the Fermi -LAT 4th catalog. The 1-month and 2-months binned γ -ray LCs were then generated via the maximum likelihood technique in the energy range 100 MeV500 GeV for about 15.6 years. The selected time bins provided data points with a signal-to-noise ratio above 2 σ for a bin size of 1-month and 3 σ for a bin size of 2-months. The LCs of the two sources showed distinctive behaviors of high and low state alternation. PKS 2155-83 displayed four high states in 2010, 2014, 2019, and 2023, interspersing its otherwise low state. PKS 2255-282 showed three high states in 2009-2013, 20172021, and 2023-up to the end of data interrupting its low state. \nThe probability density function of PKS 2155-83 tended toward log-normality, especially in the case of the 1-month binned LC, indicative of a nonlinear, multiplicative processes underlying the variability, rather than an additive stochastic process (e.g., associated with many superposed 'mini-jets' in an AGN engine or simple shot-noise). The probability density function of PKS 2255-282, on the other hand, revealed a comparable de- \n<!-- image --> \nFigure 7. REDFIT periodicity analysis for the 1-m binned LCs of PKS 2155-83 (top panel) and PKS 2255-282 (bottom right). The bias-corrected power spectrum is represented by the black line. Spectrum of the theoretical red noise (red line), 95% confidence level (blue dashed line), and 99% confidence level (orange dashed line) were estimated by fitting the data with AR1 process. \n<!-- image --> \ngree of fitness with both log-normal and Gaussian distributions. \nVarious methods were applied to the LCs to assess the possible existence of quasi-periodicity. Namely, the auto-correlation function, the date-compensated discrete Fourier transform, the generalized Lomb-Scargle periodogram, the weighted wavelet z-transform, and the REDFIT algorithm. Periodicity peaks were found, using these methods, for PKS 2155-83 in the ranges (in year) 4 . 45 ± 0 . 13-5 . 03 ± 0 . 68 and 4 . 42 ± 0 . 08-4 . 93 ± 0 . 74 for 1-month and 2-months binned LCs, respectively. While, for PKS 2255-282, they were found in the ranges of 5 . 87 ± 0 . 85-6 . 82 ± 2 . 25 and 5 . 81 ± 0 . 85-6 . 73 ± 1 . 35 for 1 month and 2 months binned LCs, respectively. For \nexample, the GLSP of PKS 2155-83 showed periods at 4 . 69 ± 0 . 79 yr and at 4 . 55 ± 0 . 84 yr for the 1-month and 2months binned LCs, respectively. Both with significance 3 σ . The PKS 2255-282 1-month binned LC showed two possible periods at 1 . 43 ± 0 . 05 yr (2 . 5 σ ) and 6 . 82 ± 2 . 25 yr (2 . 8 σ ), while the 2-months binned LC showed a period at 6 . 73 ± 1 . 35 yr (2 . 6 σ ) and no noticeable high frequency signal. Recently, results of 19 blazars were reported using the first 12 years of data from the Fermi -LAT and multiwavelength archival data from radio, infrared, and optical bands (Pe˜nil et al. 2024). This study reported no periodic modulations from PKS 2255-282 except of 1 . 4 ± 0 . 1 yr (2-3 σ ) from the cross-correlation between γ -rays and the V-band. The disagreement between the suggested periodicity in the present work and the results from the optical observation may argue for the difference between the optical emission region and/or mechanism and the corresponding ones for gamma radiation. \nThe existence and origins of QPOs in blazars are still controversial (Sobacchi et al. 2016; Sandrinelli et al. 2018; Tavani et al. 2018). If confirmed, periodicities may be linked to the process feeding the jet and/or to the relativistic jet itself (Ackermann et al. 2015). They may in general involve scenarios invoking a binary SMBH AGN system (as, for example, discussed in Zhou et al. 2018). Intrinsic origins include possible oscillations associated with instability in the accretion disk or jet formation region (Tchekhovskoy et al. 2011). The characteristic timescales of the pulsational accretion flow instabilities can range from minutes to hours (Honma et al. 1992; Tchekhovskoy & McKinney 2012). This is outside the periods suggested in this study. Nonetheless, in the case of slow-spinning supermassive black holes, magnetohydrodynamics simulations of magnetically choked accretion flow produce longer frequencies (Tchekhovskoy & McKinney 2012). \nAnother possible origin of QPOs in blazars could also be associated with apparent geometrical effects (Rieger 2004) e.g., jet precession/helical jet (Caproni et al. 2013; Sobacchi et al. 2016; Vlahakis & Tsinganos 1998; Hardee & Rosen 1999; Villata & Raiteri 1999; Nakamura & Meier 2004; Ostorero et al. 2004). In such cases, the observed flux will undergo periodic modulation due to the periodic variation of the Doppler magnification factor (Ackermann et al. 2015). This scenario does not need intrinsic flux modulation and does not induce oscillations in the spectral index. PKS 2155-83 and PKS 2255-282 showed no flux-photon index correlation along the full time range of observation ( ∼ 15.6 yr; with time binning of 1-month and 2-months). Therefore, both objects may favor this second origin. \nMerging supermassive black holes (Begelman et al. 1980; Barnes & Hernquist 1992) may induce both types of (intrinsic and geometrical) quasi-periodicities. In particular, a SMBH binary system with a milli-pc separation and a total mass of ∼ 10 8 M ⊙ in the early inspiral gravitational-wave driven regime would induce jet precession with timescales of SMBH binary-induced periodicities are ranging from ∼ 1 to ∼ 25 years (Sobacchi et al. 2016; Komossa & Zensus 2014; Rieger 2007). Given the redshifts of PKS 2155-83 (z = 1 . 865) and PKS 2255-282 (z = 0 . 926), reflecting cosmological epochs when merging between galaxies and their embedded black holes were still relatively frequent, the binary black hole scenario may be, in principle, especially relevant. \nWe considered a simple model within this general framework Sobacchi et al. (2016). It assumes a binary system of SMBHs on circular orbits. The direction of the jet, carried by one SMBH, in the center of mass frame is perpendicular to the orbital plane. The jet deviates with an angle ∆ α , due to the imprint of the orbital velocity, v , of the jet-carrying black hole on the highly relativistic jet. Consequently, the angle θ obs between the jet and the distant observer oscillates with an amplitude ∆ θ obs , with the same period T of the orbital motion. 7 This simple model thus incorporates features (particularly, a characteristic timescale) that may be generic to potential blazar quasi-periodic signals originating from binary SMBH systems. \nBased on this model and the periodicities suggested by the GLSP, where T int is 1.64 yr and 3.63 yr, for PKS 2155-83 and PKS 2255-282, respectively, one can estimate the total BH masses. In this context, the total masses (as a function of ∆ θ obs and mass ratio q ) are \nM = 1 . 4 × 10 8 ( 1 + q q ) 3 ( ∆ θ obs 5 · ) 3 M ⊙ (8) \nand \nM = 3 . 0 × 10 8 ( 1 + q q ) 3 ( ∆ θ obs 5 · ) 3 M ⊙ (9) \nfor PKS 2155-83 and PKS 2255-282, respectively. The separation of the binary is found to be, with ∆ θ obs = 5 · , R = 0 . 0035(1 + q ) /q pc and the SMBHs merging timescale is T GW = 3 . 2 × 10 4 q ( q/ (1 + q )) 3 yr, for PKS \n2155-83. A similar exercise gives a separation of the binary R = 0 . 0104(1 + q ) /q pc and SMBHs merging timescale is T GW = 6 . 9 × 10 4 q ( q/ (1 + q )) 3 yr, for PKS 2255-282. Note that the jet has to be assumed to be carried by the secondary SMBH (i.e. q ≳ 1) to avoid extremely short timescales of orbital decay. \nAlthough the results are quite sensitive to ∆ θ obs , the deflection angle of the highly relativistic jet ∆ α , and therefore the amplitude ∆ θ obs , are constrained to a small angle, where ∆ α ≃ v/c = ( q/ (1+ q ))( GM/Rc 2 ) 1 / 2 and ∆ θ obs = 2∆ α . In order to have observable consequences on the QPO timescales inferred one needs this angle to be of the order of a few degrees (Sobacchi et al. 2016). \nThe inferred combined black hole masses are comparable, with reasonable estimation, in both cases considered with independent estimates of the compact central masses estimated in the respective galaxies (e.g., via dynamical virial equilibrium calculations reported in literature Xiong & Zhang 2014, where it is found that log 10 ( M BH /M ⊙ ) were 9.02 and 9.16 for PKS 2155-83 Shaw et al. 2012, and 8.92 and 9.16 for PKS 2255-282 Gu et al. 2001). Such supermassive black holes are inferred to be abundant at much higher redshifts. Further searches for high redshift blazars with potential periodicities, in conjunction with gravitational wave signals from SMBH systems from future detectors, such as LISA, may thus serve to test the consistency of the merging black hole scenario as an origin of such signals. This may, in turn, potentially shed light on the SMBH merger rate in the context of standard hierarchical galaxy formation in standard cosmology, ultimately providing a test of the model itself.", 'ACKNOWLEDGEMENTS': "We thank the referee for a careful reading and insightful suggestions that helped improve our manuscript. We also acknowledge the use of Fermitools-conda, DELCgen-Simulating light curves (Connolly 2015), Matplotlib (Hunter 2007), Savgol filter (Virtanen et al. 2020), pyZDCF, PyAstronomy (Czesla et al. 2019), NumPy (Harris et al. 2020), AAVSO VStar software (Benn 2012), REDFIT (Schulz & Mudelsee 2002), astroML (Ivezi'c et al. 2014).", 'REFERENCES': 'Abdollahi, S., Acero, F., Baldini, L., et al. 2022, The \nAstrophysical Journal Supplement Series, 260, 53, \ndoi: 10.3847/1538-4365/ac6751'}
2024M&PS...59.3087A	Over the Nullarbor Plain in South Australia the Desert Fireball Network detected a fireball on the night of June 1 2019 730 pm local time and 6 weeks later recovered a single meteorite 42 g named Arpu Kuilpu. This meteorite was then distributed to a consortium of collaborating institutions to be measured and analyzed by a number of methodologies including SEMEDS EPMA ICPMS gammaray spectrometry ideal gas pycnometry magnetic susceptibility measurement CT optical microscopy and accelerator and noble gas mass spectrometry techniques. These analyses revealed that Arpu Kuilpu is an unbrecciated H5 ordinary chondrite with minimal weathering W01 and minimal shock S2. The olivine and pyroxene mineral compositions in mole are Fa 19.2 0.2 and Fs 16.8 0.2 further supporting the H5 type and class. The measured oxygen isotopes are also consistent with an H chondrite SUP17SUPO 2.904 0.177 SUP18SUPO 4.163 0.336 SUP17SUPO 0.740 0.002. Ideal gas pycnometry measured bulk and grain densities of 3.66 0.02 and 3.77 0.02 g cmSUP3SUP respectively yielding a porosity of 3.0 0.7. The magnetic susceptibility of this meteorite is log 5.16 0.08. The most recent impactrelated heating event experienced by Arpu Kuilpu was measured by SUP40SUPArSUP39SUPAr chronology to be 4467 16 Ma while the cosmic ray exposure age is estimated to be between 6 and 8 Ma. The noble gas isotopes radionuclides and fireball observations all indicate that Arpu Kuilpus meteoroid was quite small maximum radius of 10 cm though more likely between 1 and 5 cm. Although this meteorite is a rather ordinary ordinary chondrite its prior orbit resembled that of a Jupiter Family Comet JFC further lending support to the assertion that many cm to msized objects on JFC orbits are asteroidal rather than cometary in origin.	2024-11-01T00:00:00Z	['10.48550/arXiv.2409.10382', '2024arXiv240910382A', '2024M&PS...59.3087A', 'arXiv:2409.10382', '10.1111/maps.14268']	['Astrophysics - Earth and Planetary Astrophysics', 'Physics - Geophysics']	The Arpu Kuilpu meteorite Indepth characterization of an H5 chondrite delivered from a Jupiter Family Comet orbit	2,024	190	0.41	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	0	https://arxiv.org/pdf/2409.10382.pdf	{'The Arpu Kuilpu Meteorite: In-depth characterization of an H5 chondrite delivered from a Jupiter Family Comet orbit.': 'Seamus L. Anderson* 1 , Gretchen K. Benedix 1,2 , Belinda Godel 3 , Romain M. L. Alosius 4 , Daniela Krietsch 4 , Henner Busemann 4 , Colin Maden 4 , Jon M. Friedrich 5,6 , Lara R. McMonigal 5 , Kees C. Welten 7 , Marc W. Caffee 8,9 , Robert J. Macke 10 , Seán Cadogan 11 , Dominic H. Ryan 12 Fred Jourdan 13 , Celia Mayers 13 , Matthias Laubenstein 14 , Richard C. Greenwood 15 , Malcom P. Roberts 16 , Hadrien A. R. Devillepoix 1,17 , Eleanor K. Sansom 1,17 , Martin C. Towner 1 , Martin Cupák 1,17,18 , Philip A. Bland 1 , Lucy V. Forman 1 , John H. Fairweather 1 , Ashley F. Rogers 1 , Nicholas E. Timms 1 \n- 1 Space Science and Technology Centre, School of Earth and Planetary Sciences, Curtin University, GPO Box U1987, Perth, WA, Australia.\n- 2 Dept. Earth and Planetary Sciences, Western Australian Museum, Locked Bag 49, Welshpool, WA, Australia.\n- 3 CSIRO Mineral Resources, Kensington, WA, Australia.\n- 4 Institute of Geochemistry and Petrology, ETH Zurich, CH-8092 Zurich, Switzerland. \n- 6 Dept. of Earth and Planetary Sciences, American Museum of Natural History, 79 th Street at Central Park West, New York, NY 10024, USA.\n- 7 Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, USA. \nAccepted for publication in: Meteoritics and Planetary Science (13 September 2024).', 'ABSTRACT': "Over the Nullarbor Plain in South Australia, the Desert Fireball Network detected a fireball on the night of 1 June 2019 (7:30 pm local time), and six weeks later recovered a single meteorite (42 g) named Arpu Kuilpu. This meteorite was then distributed to a consortium of collaborating institutions to be measured and analyzed by a number of methodologies including: SEM-EDS, EPMA, ICP-MS, gamma-ray spectrometry, ideal gas pycnometry, magnetic susceptibility measurement , μCT, optical microscopy, and accelerator and noble gas mass spectrometry techniques. These analyses revealed that Arpu Kuilpu is an unbrecciated H5 ordinary chondrite, with minimal weathering (W0-1) and minimal shock (S2). The olivine and pyroxene mineral compositions (in mol%) are Fa: 19.2 ± 0.2, and Fs: 16.8 ± 0.2, further supporting the H5 type and class. The measured oxygen isotopes are also consistent with an H chondrite (δ 17 O% = 2.904 ± 0.177; δ 18 O% = 4.163 ± 0.336; Δ 17 O% = 0.740 ± 0.002). Ideal gas pycnometry measured bulk and grain densities of 3.66 ± 0.02 and 3.77 ± 0.02 g cm -3 , respectively, yielding a porosity of 3.0 % ± 0.7. The magnetic susceptibility of this meteorite is log χ = 5.16 ± 0.08. The most recent impact-related heating event experienced by Arpu Kuilpu was measured by 40 Ar/ 39 Ar chronology to be 4467 ± 16 Ma, while the cosmic ray exposure age is estimated to be between 6-8 Ma. The noble gas isotopes, radionuclides, and fireball observations all indicate that Arpu Kuilpu's meteoroid was quite small (maximum radius of 10 cm, though more likely between 1-5 cm). Although this meteorite is a rather ordinary ordinary chondrite, its prior orbit resembled that of a Jupiter Family Comet (JFC) further lending support to the assertion that many cm- to m-sized objects on JFC orbits are asteroidal rather than cometary in origin.", 'INTRODUCTION': 'Although meteorites continuously fall to Earth, most lack any spatial context with regards to their orbital origin. Fireball observatory networks (Devillepoix et al. 2020; Oberst et al. 1998; TrigoRodríguez et al. 2006; and others) such as the Desert Fireball Network (DFN; Bland et al. 2012) are uniquely enabled to observe fireballs in order to calculate both their prior orbit, and the approximate landing site of the resulting meteorite. To date, 54 meteorites with associated orbits have been recovered and analyzed (Meier, 2017; see: https://www.meteoriteorbits.info/) in this effort to better understand how cm- to m-sized material is transferred from their asteroidal source bodies across the inner solar system. One such meteorite, Arpu Kuilpu (H5; Gattacecca et al. 2022), was observed by the DFN to fall in South Australia on 1 st June 2019 at approximately 7:30 pm local time, near the Hughes train siding close to the border with Western Australia (Shober et al. 2022). This meteorite was a particularly high-priority target for recovery as it impacted Earth from a Jupiter family Comet (JFC) orbit (a=2.75 ± 0.03 AU; e=0.671 ± 0.003; i=2.03 ± 0.01 deg; Tj=2.97 ± 0.02 (Shober et al. 2022)), a rare origin for most orbitally constrained meteorites. A meteorite searching team was sent to the fall site six weeks later, recovering a single rock (42 g) on the 2 nd day. The name Arpu Kuilpu (Are-puh Kooill-puh) was provided by the traditional custodians of the area, the Maralinga people. \nWhile the work by Shober et al. (2022) details the astronomical observations (see their Fig. 1 and Fig. 3) and modelling of the fireball event as well as the recovery and classification of the meteorite as an H5 chondrite, this work focuses primarily on the physical, chemical, geological, and chronological aspects of the meteorite itself to provide a combined history for the formation of Arpu Kuilpu within its asteroidal parent body in the early solar system, as well as its more recent ejection as a meteoroid from its contemporary source body, and how it may relate to other orbital meteorites.', 'METHODS': 'We assembled a consortium ( Table 1 ) of scientists from nearly a dozen institutions to perform a wide array of analyses to properly characterize many aspects of this meteorite. \nTable 1 . The Arpu Kuilpu Meteorite Consortium \n*SEM-EDS: Scanning Electron Microscope-Energy Dispersive X-ray Spectroscopy; ICP-MS: Inductively Coupled Plasma Mass Spectrometry; AMS: Accelerator Mass Spectrometry; ICP-OES: Inductively Coupled Plasma Optical Emission Spectroscopy; HPGe: HyperPure Germanium; CRE: Cosmic Ray Exposure; AMNH: American Museum of Natural History; INFN: Italian National Institute for Nuclear Physics; JdLC: John de Laeter Centre; UWA: University of Western Australia; CSIRO: Commonwealth Scientific and Industrial Research Organisation; ETH: Swiss Federal Institute of Technology; CMCA: Centre for Microscopy, Characterization and Analysis.', 'HPGe Gamma-Ray Spectrometry (Short-lived Radionuclides)': 'Cosmic-ray-produced (cosmogenic) radionuclide concentrations were analyzed by means of non-destructive high purity germanium (HPGe) gamma-ray spectrometry, and the counting efficiencies have been calculated using thoroughly tested Monte Carlo codes. One specimen of Arpu Kuilpu was measured in the underground laboratories at the Laboratori Nazionali del Gran Sasso (LNGS) (Arpesella, 1996; Laubenstein, 2017) for 40.63 days (160 days after the fall date of 1 June 2019).', 'Micro XRay Computed Tomography (μCT; Bulk Density and Petrofabric Analysis)': 'The entire 42.24 g meteorite was imaged in 3D via X-ray computed tomography at CSIRO Mineral Resources (Kensington, WA) using a Zeiss Versa 520 3D X-ray microscope. The instrument settings were120 kV and 10 W, producing an 11 µm voxel -1 size (3D pixel). Additionally, a 430.3 mg sub-sample was imaged at the AMNH at a resolution of 12.8 µm voxel -1 edge using polychromatic Xrays with the GE phoenix v\|tome\|x s μCT system operating a tungsten X-ray tube at 110 kV and 13.2 W. The AMNH µCT data were intended to be used to verify the homogeneity of the sample prior to further sub-sampling for elemental analysis; however, we examined the petrofabric texture of this subsample too. \nTo quantitatively examine the petrofabric results, we used the same techniques as described in Friedrich et al. (2008; 2017). In short, we digitally separated the high-density Fe-Ni metal grains within the µCT volume using the program Blob3D (Ketcham, 2005). Best fit ellipsoids were constructed around the largest 5000 metal grains; next, the intersection of a line passing through the long axis of the best fit ellipsoid and a hemisphere enclosing the ellipsoid are plotted on a lower hemisphere equal area stereographic projection. Statistical descriptors of the fabric tensor eigenvalues (Woodcock, 1977; Woodcock and Naylor, 1983; Jelinek, 1981) generated from the collective metal grain projection data and examination of the individual best fit ellipsoid shapes using ratios of the ellipsoid long, intermediate, and short axes (Zingg, 1935; Blott and Pye, 2008) allows for a quantitative and thorough examination of the 3D petrofabric defined by metal grain shape preferred orientations within the chondrite.', 'Scanning Electron Microscopy Energy Dispersive Spectroscopy (SEM-EDS; Meteorite Type via Mineral Phase Abundance)': 'Using the TESCAN TIMA Scanning Electron Microscope (SEM), located at the John de Laeter Centre at Curtin University, we obtained element maps of our 25 mm-diameter epoxy-mounted sample via energy dispersive spectroscopy (EDS), which used four PulsTor Energy, silicon drift, Peltier cooled detectors to collect data. Prior to this, the sample mount was polished and carbon-coated to mitigate surface charging. TIMA maps were collected using a 25 kV acceleration voltage, 6.0 nA beam current, 15 mm working distance, and 90 nm spot size. Seven element maps (Al, Ca, Fe, Cr, Mg, Si, S) were combined to create a mineral phase map using a python script (Anderson, 2023).', 'Electron Probe MicroAnalysis (EPMA; Petrologic Type via Mineral Compositions)': 'The major element compositions of olivine, orthopyroxene, and chromite in Arpu Kuilpu were measured using the JEOL 8530F Electron Probe MicroAnalyser (EPMA) at the Centre for Microscopy and Microanalysis, University of Western Australia. On the same polished sample mount used for SEM-EDS, we selected 20 olivine, 20 orthopyroxene, and 11 chromite grains for elemental measurements using a beam current of 20 nA and an accelerating voltage of 15 kV. We also collected calibration measurements of our mineral standards: wollastonite (Si and Ca), periclase (Mg), magnetite (Fe), manganese metal (Mn), corundum (Al), jadeite (Na), and Durango apatite (P).', 'Optical Microscopy': 'The optical mosaic photomicrographs of the thin section, taken using the Zeiss Axio are shown in Figure 2. Further manual investigation using a rotating stage revealed both sharp and undulose extinction features in olivine and plagioclase grains indicating this meteorite experienced very low shock-induced pressures or temperatures, equating to S1-S2 in the Stöffler et al. (2018) classification scheme. No opaque shock veins or mosaicism were observed, eliminating the possibility for localized S3 shock or higher. \nFigure 2. Cross-polarized image of our Arpu Kuilpu thin section, which clearly shows a texture consistent with an equilibrated H chondrite. Further investigation of this thin section under a rotating stage reveals undulose light extinction in olivine and plagioclase grains, indicating a low-pressure shock history (S 1-2; Stöffler et al. 2018). \n<!-- image -->', 'Mössbauer Spectroscopy (Weathering State)': 'The spectrum of the Arpu Kuilpu meteorite sample shown in Figure 3 is dominated by a pair of peaks at ∼ 0 mm/s and +2.5 mm/s. This wide split doublet, offset to positive velocities, is characteristic of a ferrous (Fe2+) mineral. The distance between the lines is the quadrupole splitting (QS) and is associated with an electric field gradient at the iron site in the mineral, while the center of the doublet (arithmetic mean of the two peak positions) is the isomer shift (IS). Closer inspection of the two peaks reveals that each of the lines has a second, weaker line associated with it and these two weaker lines are due to a second Fe2+ mineral. The fitted values of the QS and IS for these components, given in Table 3, allow them to be identified as arising from Olivine and Pyroxene, with the former mineral being about twice as abundant as the latter (Dyar et al., 2006). \nTo the left and right of the primary doublets there are a series of weaker peaks. These are due to the presence of two magnetic minerals: Fe-Ni metal, and Troilite. As with Olivine and Pyroxene, the two magnetic minerals are identified using the fitted parameters (IS, QS and hyperfine field) shown in \nTable 3. Finally, we observed a weak ferric component in the spectrum that accounted for ∼ 2% of the total area and is attributed to paramagnetic nanoparticles. The results of the Mössbauer analysis show little terrestrial alteration, as any Fe 3+ signal in an ordinary chondrite (which is quite low for Arpu Kuilpu) would originate only from terrestrial alteration. \nTable 3. The values of Isomer Shift (I.S.), and Quadrupole Splitter (Q.S). for subspectra #1 and #2 indicate Fe 2+ and are assigned to olivine and pyroxene, respectively. Subspectrum #3 indicates Fe 3+ paramagnetic nanoparticles (n-p-3+). Component #4 is a magnetic sextet with small I.S. and Q.S. values, and a magnetic Hyperfine Field (H.F.F.) of 33.59 T indicating Fe metal. Component #5 is a magnetic sextet whose I.S. and H.F.F. values indicate Troilite.Kuilpu meteorite RT 98 Arpu \nArpu Kuilpu meteorite RT \n<!-- image --> \nFigure 3 . The Mössbauer spectrum of Arpu Kuilpu (top) and the fitting difference (bottom), note the dramatically different vertical scales. \n<!-- image -->', 'Inductively Coupled Plasma Mass Spectrometry (ICP-MS; Bulk Major and Trace Element Geochemistry)': 'We performed elemental analysis on three chips of Arpu Kuilpu (103.3, 107.8, 128.1 mg) at the Chemistry Department of Fordham University. We first ground each chip in an agate mortar and pestle. To dissolve the powders, we used a combination of HF and HNO3 in a high pressure CEM Mars 5 microwave digestion system, dried the resulting mixture to incipient dryness on a hotplate, treated the residue with HClO4, and again brought it to incipient dryness to dissolve the samples. In addition to the Arpu Kuilpu samples, a procedural blank and the Allende Standard Reference Meteorite were taken up in 1% HNO3 and analyzed with a ThermoElemental X-Series II ICPMS with the methods outlined in Friedrich et al. (2003) for trace elements, and Wolf et al. (2012) for major and minor elements. For examination of the data, we compare them with Orgueil CI chondrite values from Friedrich et al. (2002) for trace elements and the compiled Orgueil results of Anders and Grevesse (1989) for major and minor elements.', 'Oxygen Isotopes': 'The oxygen isotope ratios measured in Arpu Kuilpu can be seen in Figure 5 and are as follows (all with 1σ uncertainties): δ 17 O% = 2.904 ± 0.177; δ 18 O% = 4.163 ± 0.336; Δ 17 O% = 0.740 ± 0.002. \nFigure 5. The oxygen isotope composition of Arpu Kuilpu plotted in the context of other ordinary chondrites (data sourced from Clayton et al. (1991)), clearly showing that this meteorite is an H chondrite. \n<!-- image -->', 'Laser Scanning and Ideal Gas Pycnometry (Density and Porosity)': 'A 14.64 g piece of Arpu Kuilpu was used for the determination of bulk and grain density, as well as porosity. Bulk volume was measured with NextEngine model 2020i ScannerHDPro laser scanner with a rotating platform (Macke et al. 2015). Scans were performed at 16k dots per inch (dpi). For each of the three possible orientations of the meteorite, 16 scans were performed, with the platform rotated 1/16 of a complete circle between each scan. Scans were trimmed of artifacts and aligned to construct a "watertight" computer model of the specimen, from which we calculated the bulk volume. Grain volume was measured via ideal gas pycnometry with a Quantachrome Ultrapycnometer 1000, using gaseous nitrogen. The measurement procedure was run 15 times, of which the average of the last 6 were taken as the grain volume of the specimen, consistent with the technique described in Macke (2010). Porosity (P) was calculated from the bulk and grain densities (ρ): P = 1 ρ bulk /ρ grain', 'Magnetic Susceptibility': 'The magnetic susceptibility measured from Arpu Kuilpu is log χ = 5.16 ± 0.08 (log SI units), consistent with other H chondrites (Consolmagno et al. 2006; Macke et al. 2011).', 'Cosmogenic Radionuclides (Cosmic Ray Exposure)': "We prepared samples by using a ~0.24 g chip of Arpu Kuilpu for analysis of the cosmogenic radionuclides 10 Be (half-life=1.36 × 10 6 yr), 26 Al (7.05 × 10 5 yr) and 36 Cl (3.01 × 10 5 yr), following the procedures described in Welten et al. (2011). We crushed the sample in an agate mortar and separated the magnetic (metal) from the non-magnetic (stone) fraction. The magnetic fraction was purified by ultrasonic agitation in 0.2N HCl to remove attached troilite. After rinsing the metal four times with MilliQ water and once with ethanol, we obtained 33 mg of relatively clean metal, corresponding to 13.9 wt% bulk metal. The metal fraction was further purified by ultrasonic agitation in concentrated HF for 15 min to dissolve attached silicates. We dissolved 32 mg of the purified metal in ~10% HNO3 along with a carrier solution containing 0.90 mg Be, 1.01 mg Al, and 2.99 mg Cl. After dissolution, we took a small aliquot (~5.5%) of the dissolved sample for chemical analysis and used the remaining solution for cosmogenic radionuclide analysis. We also dissolved 96 mg of the stone fraction of Arpu Kuilpu, along with 3.19 mg of Be carrier and 2.77 mg of Cl carrier, in concentrated HF/HNO3 by heating the mixture for 24 h inside a Parr Teflon digestion bomb at 125 °C. After cooling the sample to room temperature, we separated the Cl fraction as AgCl, and removed Si in the solution by repeatedly fuming the sample to dryness with HClO4. The residue was dissolved in dilute HCl and a small aliquot (3.4%) was taken for chemical analysis. After adding ~5.0 mg of additional Al carrier to the remaining solution, we separated the Be and Al fraction for radionuclide analysis. \nAfter separating and purifying the Be, Al and Cl fractions, the 10 Be/Be, 26 Al/Al and 36 Cl/Cl ratios of the samples, blanks and standards were measured by Accelerator Mass Spectrometry (AMS) at Purdue University's PRIME Lab (Sharma et al. 2000). The measured ratios were normalized to those of well-known AMS standards (Nishiizumi 2004; Nishiizumi et al. 2007; Sharma et al. 1990) and converted to concentrations in disintegrations per minute per kg (dpm/kg). \nFor the chemical analysis, we made two consecutive dilutions of the aliquots of the dissolved metal and stone fractions for characterization by ICP-OES. All dilutions were measured on a Thermo Fisher iCAP 6300 duo instrument.", 'Noble Gas Composition (Cosmic Ray Exposure, Meteoroid size and Sample Shielding, Gas Retention Age)': "Measurements of all stable noble gas isotopes (He, Ne, Ar, Kr, Xe) were performed on two aliquots (29.267 ± 0.015 mg (AKL) and 27.178 ± 0.014 mg (AKS)) of Arpu Kuilpu. The samples were first wrapped in aluminum foil and heated at 110°C in ultra-high vacuum for several days to remove adsorbed atmospheric gases. The measurements were carried out on the in-house-built noble gas mass spectrometer 'Albatros' at ETH Zürich, using the procedures described in detail in Riebe et al. (2017). Gas extraction was achieved by melting the samples in a Mo crucible at ~1700 °C for ~25 min. The blank corrections for both aliquots are <1.5% of the He and Ne isotope signals, and <2.5% for Ar. The corrections for all Kr isotopes were < 7% of the signals, and < 2% for all Xe isotopes. \nWe numerically separated the cosmogenic (cos) and trapped (tr) components using a twocomponent deconvolution between ( 36 Ar/ 38 Ar)cos (0.63-0.67; Wieler, 2002) and ( 36 Ar/ 38 Ar)tr. The trapped component used for the deconvolution was delimited by the values for Q and air (5.32-5.34; Busemann et al. 2000; Nier, 1950) since no solar wind component was detected for He and Ne. \nTo constrain the CRE age from noble gas data, we calculated the production rates of cosmogenic 3 He, 21 Ne, and 38 Ar based on the model for ordinary chondrites from Leya & Masarik (2009), which considers the pre-atmospheric size of the meteoroid, the depth of the sample within the meteoroid, and the bulk chemical composition of the sample. Where possible, we used the elemental concentrations reported by the ICP-MS portion of this study in these model calculations. For elements not measured in this study (via ICP-MS) we used data for H chondrites from the literature (Si and S from Alexander, 2019; C from Lodders and Fegley, 1998; while O was calculated for the sum of all concentrations to reach 100 wt%). Since the original observations of the fireball (Shober et al. 2022) suggest a pre-atmospheric meteoroid size of ~5 cm, we used a modified version (Wieler et al. 2016) of the Leya and Masarik (2009) model, which considers small (<7 cm) H chondrite meteoroids. Ratios of cosmogenic noble gases, specifically the cosmogenic 22 Ne/ 21 Ne ratio can be used as a shielding indicator to then calculate the size of the original meteoroid and the meteorite's burial depth within it, which is required to determine production rates and CRE age. Consistent with the small preatmospheric meteoroid size assessed independently, only the modified model for small H chondritic meteoroids (Wieler et al. 2016) revealed matches between prediction and the measured high cosmogenic 22 Ne/ 21 Ne ratios. \nGas retention ages for both aliquots were calculated separately using the U/Th-He and K-Ar chronometers with the U, Th, and K concentrations from the Gamma-Ray Spectrometry portion of this study. The 4 Herad values for the U/Th-He chronometer were calculated by assuming that the 3 He concentrations were cosmogenic, and applying a ( 4 He/ 3 He)cos ratio between 5.2 and 6.1 (Wieler, 2002). The 4 Hecos concentration was then subtracted from our measured 4 He concentration (with 4 Hetr being negligible). The 40 Arrad values for the K-Ar chronometer were calculated with the deconvoluted 36 Artr, adopting a ratio ( 40 Ar/ 36 Ar)tr between 0 and 295.5 (covering Q and air composition; Busemann et al. 2000; Steiger and Jäger, 1977).", '40 Ar/ 39 Ar Chronology (Thermal and Impact History)': 'To determine the 40 Ar/ 39 Ar chronology, we crushed a whole rock fragment from Arpu Kuilpu and selected nine pyroxene aliquots each including between 1 and 30 grains, and with each grain ranging in size from 150 to 350 µm in diameter. We also selected a single plagioclase grain . The samples were irradiated for 40 hours, then analyzed at the Western Australian Argon Isotope Facility at Curtin University using a ARGUS VI Mass Spectrometer, using a 10.4 µm CO2 laser to affect step heating for 60 sec. The complete procedure is similar to the one described by Jourdan et al. (2020) for other meteorite samples.', 'SEM-EDS (Meteorite Class and Texture)': "The mineral map produced by SEM-EDS imaging is shown in Figure 1 , and clearly reveals a texture consistent with an equilibrated ordinary chondrite, consisting of Fe-Ni metal (kamacite and taenite), chromite, troilite, plagioclase and Ca-phosphate grains, as well as olivine and pyroxene chondrules, all set within a recrystallized, mostly silicate matrix. There are recognizable chondrules, though they have fairly indistinct boundaries separating them from the matrix. \nFigure 1. A false color mineral map of Arpu Kuilpu's epoxy -mounted thick section taken from SEMEDS data (initially appearing as Fig. 9 in Shober et al. 2022), which clearly shows a texture typical of an equilibrated ordinary chondrite. The large void on the right is an artifact of the polishing step when a remnant chondrule was likely and unintentionally removed. \n<!-- image -->", 'EPMA (Electron Probe MicroAnalysis; Mineral Compositions)': "The full dataset from the EPMA is displayed in Table 2. The olivine compositions of Arpu Kuilpu average at fayalite = 19.2 ± 0.2 mol% (n=20), while the orthopyroxenes reside at ferrosilite = 16.8 ± 0.2 mol% and wollastonite = 1.4 ± 0.2 mol% (n=20). The chromite compositions are: Cr/(Cr + Al) = 85.5 ± 0.3, and Fe/(Fe + Mg) = 84.4 ± 1.2 (n=11). The olivine and pyroxene compositions reveal that Arpu Kuilpu is an H chondrite (Van Schmus and Wood, 1967), while the Wo value of 1.4 indicates a petrologic type 5 (Scott et al. 1986). All uncertainties listed above are 1σ. \nTable 2. EPMA measurements of Arpu Kuilpu's Olivines, Orthopyroxenes, and Chromites . Oxides concentrations are displayed in wt%.", 'ICP-MS (Bulk Major and Trace Element Analysis)': "The 50 major, minor, and trace elements measured via ICP-MS are shown in Table 4 . The higher inter-aliquot variability of the light REE may be due to one of the aliquots being very slightly enriched in phosphates or having an unrepresentative pyroxene/plagioclase ratio relative to the whole sample (Mason and Graham, 1970). The Fe, Ni, and Co abundances normalized to CI and Mg ( Figure 4 ) of Arpu Kuilpu are more consistent with what would be expected for an H chondrite than the \nTable 4 . Abundances of 50 elements in the Arpu Kuilpu chondrite. Errors, given as percent relative standard deviation (%RSD), are ≤14.0% for all elements except for the light Rare Earth Elements La, Ce, Pr, Nd, and Sm, where all had %RSD values between 16.5-17.6 based on the three replicate aliquots analyzed. Other more variable elements include Re (17.6%), Ir (25.0%), Sn (20.0%), Cu (21.2%), As (16.6%), Sb (37.9%). \nsiderophile trace elements Re, Ir, Mo, Pt, and Pd which are all nearly 40% higher in abundance (normalized to CI and Mg) than the major elements Fe and Ni. Perhaps we are seeing a refractory siderophile nugget affecting enrichment of the refractory siderophile elements within our analyzed aliquots. \nFigure 4 . The bulk elemental concentrations of the Arpu Kuilpu meteorite, measured via ICP-MS, normalized to Magnesium and CI chondrite abundances. The top plot shows most of the lithophile elements, which have a mean and 1 σ normalized abundance of 0.91 ± 0.07, with the dotted line representing the 0.9 mean lithophile abundance observed in ordinary chondrites (Kallemeyn et al., 1989). The bottom plot displays Arpu Kuilpu's siderophile elemental concentrations (left, Re -Pd), as well as a few other lithophile and chalcophile abundances. The three dotted lines, represent the normalized mean siderophile abundances for H, L, and LL chondrites reported in Kallemeyn et al. (1989), which for Arpu Kuilpu is 1.42 ± 0.23, closest to that of H chondrites. \n<!-- image -->", 'Laser Scanning and Ideal Gas Pycnometry': 'We measured the bulk density of the 14.64 g sample of Arpu Kuilpu to be 3.66 ± 0.02 g cm -3 , and a grain density of 3.77 ± 0.02 g cm -3 . These data taken together yield a porosity of 3.0 % ± 0.7, which is typical of ordinary chondrites (Britt and Consolmagno, 2003).', 'Micro X-Ray Computed Tomography Bulk Density': "Using the μCT data from the 430.3 mg piece, we measured the subsample's volume to be 118.5 mm 3 , yielding a bulk density of 3.63 g cm -3 which is in good agreement with the 3.66 g cm -3 found by laser scanning and Ideal Gas Pycnometry of a separate aliquot (see above). This indicates that the sample is relatively homogeneous at the 0.1-10 cm 3 scale. Visual inspection of the tomography volumes confirms this observation. These bulk densities are consistent with the average H chondrite fall bulk density of 3.42 ± 0.19 g cm -3 (Consolmagno et al. 2008).", 'Petrofabric and Shock State': "A typical µCT 'slice' of the Arpu Kuilpu ordinary chondrite is shown in Figure 6 . The petrofabric, as defined by the metal grain shape preferred orientations (see methods), of Arpu Kuilpu shows the major axes of the grains define a relatively indistinct girdle and the minor axes cluster weakly at high angles to the major axis girdle ( Figure 7) . Together these features are the signature of a foliation petrofabric. It is generally agreed that foliation petrofabrics in chondrites are the result of grain rotation and alignment during uniaxial hypervelocity impact deformation (Gattacceca et al. 2005; Friedrich et al. 2008). \nFigure 6 . μCT 'slice' of a chip of the Arpu Kuilpu ordinary chondrite. The higher the greyscale value, the higher the average atomic weight of the material. The bright white Fe-Ni metal grains and pale grey Fe sulfides can be easily distinguished from the moderate grey silicates and the air around the chip represented as black. Although small cracks can occasionally be seen, there is no evidence of brecciation. \n<!-- image --> \nExamination of the eigenvalue tensor fabric descriptors shows that the major axis shape parameter (Kmajor = 0.73) indicates a girdle distribution (K<1) (also see Figure 7 ). The minor axis shape parameter (Kminor = 1.401) can be classified as a cluster distribution (K>1). Both of these parameters agree with visual inspection of the fabric. The major axis anisotropy parameter has a value of 0.071. It is known that the fabric anisotropy varies in intensity with increasing petrographic shock stage (Friedrich et al. 2008; Friedrich et al. 2017). Based on an anisotropy of 0.071, a best estimate for the shock stage (Stöffler et al. 2018) of the sample may be S1 to S2, which matches with other microstructural features observed in the thin section. We also examined the anisotropy of the fabric of the largest 1000 grains in the smaller 435 mg sample collected with a different CT instrument and at a different spatial resolution. The anisotropy parameter for this sample was strikingly similar with a value of 0.073. \nFigure 7 . Stereoplots and densities of orientations of the major and minor axis of best fit ellipsoids around the largest 5000 metal grains in the Arpu Kuilpu ordinary chondrite. Arpu Kuilpu ' s major axis fabric is a relatively indistinct girdle and the minor axis is a cluster, together being the signature of a very weak foliation petrofabric. \n<!-- image --> \nThe shapes of the digitally constructed best-fit ellipsoids digitally constructed around the largest 5000 metal grains in Arpu Kuilpu are shown in Table 5 . Of the metallic grains, 35.7 % have sub-equant spheroid shapes (see Blott and Pye, 2008 for shape descriptors) with the next most abundant (31.6 %) being prolate spheroids. Oblate spheroid grains (16.4%) make up the largest remaining category of metal grain shapes. \nTable 5. Shapes of largest 5000 metal grains in the Arpu Kuilpu chondrite \nIn Figure 8 , we place the physical properties porosity and fabric anisotropy of the Arpu Kuilpu into the context of other ordinary chondrites. Arpu Kuilpu falls into a region of very low porosity, but also very low fabric anisotropy that is indicative of high ambient heat during or after the minor shock pressure that Arpu Kuilpu experienced (see Friedrich et al. 2017). \nFigure 8. Plot of porosity versus petrofabric anisotropy for typical ordinary chondrites and a suite of low-shock, low-porosity, low-foliation ordinary chondrites (data from Friedrich et al. 2017) and the Arpu Kuilpu ordinary chondrite. Errors of these measured values are shown for selected samples to give the reader an idea of the typical magnitude of the associated errors. Arpu Kuilpu falls into a region of very low porosity, but also very low fabric anisotropy, indicative of the sample having experienced high ambient heat during or after the impact-related shock that has affected this meteorite. \n<!-- image -->", 'Gamma-Ray Spectrometry': 'The results from the gamma-ray spectrometry measurements displayed in Table 6 were briefly mentioned by Shober et al. (2022) but are re-presented here for clarity. The main conclusion of Shober et al. (2022) was that the low 60 Co activity (0.8 ± 0.3 dpm kg -1 ) indicated that the original meteoroid was small enough (~10 cm; Bonino et al. 2001) that its recent cosmic-ray exposure in space did not produce a significant flux of thermal neutrons. The measured 26 Al activity using this method is consistent with a small H chondrite (Bhandari et al. 1989; Bonino et al. 2001; Leya and Masarik, 2009), and with the AMS measurements (see below) of 26 Al. Considering the time of measurement (183 days after the fireball event) the activities are consistent with a rock that stopped accumulating cosmogenic radionuclides on 1 June 2019. \nTable 6. Massic activities (corrected to date of fall of the meteorite 1 st June 2019) of cosmogenic and primordial radionuclides in the specimen of the Arpu Kuilpu stone measured by non-destructive gamma-ray spectrometry. Errors include a 1 σ uncertainty of 10% in the detector efficiency calibration. \nWhen we compare the radionuclide concentrations with cosmic ray production estimations for 26 Al (Leya and Masarik, 2009), 60 Co (Eberhardt et al. 1963, Spergel et al. 1986), 54 Mn (Kohman and Bender, 1967), and 22 Na (Bhandari et al. 1993), and assume the specimen is from the central part (which may not be a reliable assumption, see Discussion), the best agreements are obtained (in the sequence of the given isotopes) for radii of < 10 cm, < 20 cm, < 5 cm and 5-8 cm. The 22 Na/ 26 Al ratio for this specimen is 1.9 ± 0.2. Combining these results of these radionuclides, we infer a radius of 5 to 10 cm, for a roughly spherical meteoroid. Alternatively, it can be from the surface (<5 cm) of a larger meteoroid. \nThe concentrations of U, Th, and K, derived from the activities of the naturally occurring nuclides ( 235 U, 238 U, 232 Th, 40 K; Table 6) are consistent with average concentrations in H chondrites (Wasson and Kallemeyn, 1988).', 'Long-lived Cosmogenic Radionuclides (AMS and ICP-OES)': "In general, the bulk composition of the Arpu Kuilpu sample that was used for cosmogenic radionuclide analysis (measured via ICP-OES, see Table 7 ) is consistent with both the average H chondrite composition of Wasson and Kallemeyn (1988), and with the measurements made via ICP-MS (see above). The Co and Ni concentrations in the metal fraction of Arpu Kuilpu are consistent with H chondrite classification, while the Mg concentration of <10 ppm indicates that the purified metal contained no detectable (<0.01 wt%) silicate contamination. This implies that 10 Be and 26 Al contributions from silicates are negligible and thus that the measured 10 Be and 26 Al concentrations require no correction. \nTable 7. Chemical composition of metal and stone fractions of Arpu Kuilpu. All values are in wt %. The bulk composition (ICP-OES) is calculated from measured values in metal and stone fractions, which are 13.9 and 86.1 wt%, respectively. The ICP-MS measurements (see Table 4) are repeated here for comparison. The last column shows average H chondrite composition from Wasson and Kallemeyn (1988). Entries 'nm' indicate not measured. \nThe results of the AMS measurements of Arpu Kuilpu are shown in Table 8. The high concentrations of 10 Be and 26 Al in the metal phase as well as the 26 Al/ 10 Be ratio of 0.77 ± 0.03 indicate that the Arpu Kuilpu meteorite had a minimum CRE age of 4-6 Myr in a relatively small object (radius < 10 cm). \nTable 8 . Cosmogenic radionuclide concentrations (in dpm/kg) in the metal and stone fractions of Arpu Kuilpu. The bulk values are calculated from the metal and stone based on measured metal and stone proportions of 13.9 and 86.1 wt%, respectively. The last column shows the 10 Be(sto)/ 10 Be(met) and 26 Al(sto)/ 26 Al(met) ratios, which can be used as a shielding indicator (see Discussion).", 'Noble Gas Isotopes': "The measured concentrations and ratios of the noble gases in Arpu Kuilpu are listed in Table 9. Both He and Ne are purely cosmogenic (cos) apart from 4 He. The non-cosmogenic fraction of the 4 He signal is assumed to be radiogenic (rad), which is supported by the Ne isotope values that do not indicate the presence of a trapped (tr) component. A solar wind component could therefore be excluded, which also rules out the possibility of this meteorite being a regolith breccia. The absence of primordially trapped He is likely caused by thermal processing at temperatures going up to ~740 °C for H5 chondrites (olivine -spinel equilibrium temperature, Kessel et al. 2007). \nThe 36 Ar/ 38 Ar values for both samples (3.28 ± 0.02 for AKS and 2.73 ± 0.02 for AKL) do not match pure cosmogenic compositions (0.63-0.67; Wieler, 2002) and imply mixing between trapped (tr) and cosmogenic (cos) Ar. The trapped component cannot be identified since the isotopic compositions Q and air 36 Ar/ 38 Ar ratios are too close to each other. Although a small cosmogenic component can be observed in 38 Ar, it can neither be recognized in the 78 Kr/ 84 Kr and 80 Kr/ 84 Kr ratios nor in the light Xe \nisotopes. Both measured aliquots display high 129 Xe/ 132 Xe ratios resulting from a short-lived 129 Iderived 129 Xe excess, a feature commonly observed in OCs of types 5 or 6 (e.g. Alaerts et al. 1979; Moniot, 1980), fitting with Arpu Kuilpu's H5 petrologic type. \nTable 9. The noble gas isotopic concentrations and ratios of the two measured aliquots from Arpu Kuilpu (AKL and AKS; 84 Kr = 100; 132 Xe = 100). \nTable 10. Radiogenic concentrations of He and Ar (in 10 -8 cm 3 STP/g), and gas retention ages (T4 and T40) calculated using U, K, and Th concentrations from the Gamma-ray spectroscopy section of this study (see Table 6 ). \nThe U/Th-He and K-Ar chronometers for both aliquots yield old gas retention ages around 4.04.6 Gyr essentially within the same range ( Table 10 ), suggesting no major resetting events in Arpu Kuilpu's history. \nThe production rates of cosmogenic 3 He, 21 Ne and 38 Ar calculated with the above-mentioned model and the resulting CRE ages of the two aliquots are given in Table 11 . The T3, T21 and T38 ages are comparable, with the T21 and T38 ages overlapping within their ranges for AKS. The T3 ages for both aliquots overlap with each other and are lower than the T21 and T38 ages. The T21 age for AKL is slightly higher than for AKS. This discrepancy could stem from a difference in exposure to solar \ncosmic rays (SCR) which can affect material in the upper few cm of a meteoroid. While the 22 Ne/ 21 Ne ratios of both aliquots suggest a pre-atmospheric radius of the meteoroid of 1 cm according to the model, the sample depth of AKL is shallower, which might have resulted in a slightly higher SCRderived 21 Necos production ( Table 11 ). Due to this possibility, the T21 age for AKL of ~10 Myr is not used as the maximum for our preferred CRE age. The ( 3 He/ 21 Ne)cos and ( 21 Ne/ 22 Ne)cos ratios of AKS and AKL do not indicate significant 3 He loss. According to the cosmogenic noble gas concentrations, the CRE age for Arpu Kuilpu is ~7-9 Myr ( Table 11 ). \nTable 11. Suitable shielding conditions matching our cosmogenic 21 Ne/ 22 Ne value (average for AKS and AKL: 0.772 ± 0.006), determined with the model by Leya and Masarik (2009) updated for small H chondrite meteoroids (Wieler et al. 2016), cosmogenic 3 He, 21 Ne and 38 Ar isotope concentrations (in 10 -8 cm 3 STP/g), production rates Px (in 10 -8 cm 3 /(g × Myr)), and calculated CRE ages Tx (in Myr).", '40 Ar/ 39 Ar Chronology': 'We selected nine pyroxene aliquots including between 1 and 30 grains, and with each grain ranging in size from 150-350 µm in diameter, along with one plagioclase grain for 40 Ar/ 39 Ar chronology. Seven pyroxene aliquots yielded statistically indistinguishable plateau ages ranging from 4409 ± 71 Ma and 4501 ± 178 Ma ( Figure 9B ) and their combined ages result in a weighted mean age of 4467 ± 16 Ma ( Figure 9A ; P = 0.16; 2σ). Two single grain pyroxene analysis yielded plateau age of 4251 ± 163 Ma and 4310 ± 135 Ma slightly younger than the pyroxene bulk population. The plagioclase analysis did not yield enough gas for a successful analysis. \nFigure 9. (A) Example of a plateau age (2σ) obtained for one (Hughes 30GRN 150u pyxB) of the seven pyroxene aliquots extracted from the Arpu Kuilpu meteorite. (B) Weighted mean age of the seven plateau ages obtained on pyroxene populations. \n<!-- image -->', 'Meteorite Class and Type': 'Every relevant analysis we have performed on this meteorite confirms that Arpu Kuilpu is indeed an H5 ordinary chondrite. Gamma-Ray spectrometry, µCT, oxygen isotopes, ICP-MS, ICPOES, Mössbauer spectroscopy, ideal gas pycnometry, and magnetic susceptibility, each indicate the general H chondrite meteorite class, while results from the SEM-EDS ( Figure 1) and optical microscopy ( Figure 2) analyses reveal a texture consistent with a petrologic type 5, further confirmed by the EPMA measurements of individual mineral compositions. The lack of trapped noble gasses also excludes the possibility for lower (3 or 4) petrologic types.', 'Weathering': 'Arpu Kuilpu was recovered approximately six weeks after its fall, and although 8.4 mm of rainfall was recorded at the nearest weather station at Forrest Airport in Western Australia between its fall and recovery (Shober et al. 2022), visual inspection of the meteorite did not reveal significant weathering. Inspection of the thin section via optical microscopy also reveals little terrestrial alteration, all of which is also supported by a very minor Fe 3+ signal from the Mössbauer spectroscopy measurements ( Table 3; Figure 3 ).', 'Shock State and Impact History': 'The traditional method of determining the shock stage in meteorites, optical microscopy, has revealed that Arpu Kuilpu experienced relatively little shock (S1-S2; Stöffler et al. 2018). This is further supported by petrofabric measurements taken from the μCT data (Friedrich et al. 2008; 2017), seen in Figure 7 . By combining the major axis anisotropy value of C = 0.071 with the porosity of 3.0 ± 0.7% (measured via ideal gas pycnometry), as seen in Figure 8 , Arpu Kuilpu likely experienced high ambient heat at approximately the same time that it underwent its minor impact induced shock. The results of the 40 Ar/ 39 Ar experiments precisely constrain the latest possible time for such an impact event \nto the very early solar system at 4467 ± 16 Ma, which is also supported by old ages (> 4.0 Gyr) from the noble gas analysis via U/Th-He dating, further ruling out the possibility for recent impacts (see Table 10 ). \nNote that the 40 Ar/ 39 Ar systematic in pyroxene is relatively easy to reset and even minor impacts can produce enough energy to reset pyroxene provided that the shock wave creates a spike of high temperature for duration as short as few micro-seconds (Cassata et al., 2011; Kennedy et al., 2019; Jourdan et al., 2020), which in the present case, would be exacerbated by a high ambient heat. Minor subsequent impacts might have been recorded by individual crystals of pyroxene at 4251 ± 163 Ma and 4310 ± 135 Ma, but considering that these two ages are isolated, the shock event might have been very small with very focused energy, not sufficient to affect the majority of the pyroxene crystals.', 'Meteoroid Size, Burial Depth, and CRE Age': "Determining the irradiation depth of Arpu Kuilpu within its original meteoroid, as well as the meteoroid's size, and CRE age, requires the insight of three methodologies employed in this study: gamma-ray spectrometry, AMS, and noble gas analysis, as well as the fireball observations presented in Shober et al. (2022). For the pre-atmospheric size of the meteorite, each method provides an estimate based on a different time interval, with the fireball data representing the size at the time of atmospheric entry, the short-lived radionuclides the size during the past few years before entry, the long-lived radionuclide the size of the past few million years before entry and the noble gases the average size during its entire cosmic-ray exposure. For a meteorite with a simple CRE history, i.e., with no significant changes in size during CRE, all four estimates should yield the same size within the uncertainties of each method and the model calculations associated with it. \nThe first calculation of Arpu Kuilpu's original size as a meteoroid is given in Shober et al. (2022), where a pre-atmospheric size of approximately 5 cm (radius) is derived from the fireball observations, though this size estimate is prone to variation due to uncertainties in model parameters such as shape and spin of the meteoroid which are assumed values. Arpu Kuilpu's atmospheric entry was also observed to have experienced a fragmentation event near the end of the fireball (Shober et al. 2022), further opening the possibility that the meteorite could have come from a non-central part of the meteoroid. \nThe results of the non-destructive HPGe Gamma-Ray Spectrometry measurements ( Table 6) also support a small meteoroid size between 5-10 cm (assuming a spherical meteoroid). The activities of the short-lived radioisotopes, with half-lives that are less than the orbital period ( Table 6 ), represent the production integrated over the last segment of the orbit. The fall of Arpu Kuilpu occurred during a minimum at the end of solar cycle 24, as indicated by the neutron monitor data (Bartol, 2020). The cosmic ray flux was high in the six months before the fall, so the activities for the very short-lived radionuclides are expected to be high, as earlier reported (Jenniskens et al. 2014 and references cited therein; see Table 6 ). \nAs discussed in the methods, the 22 Ne/ 21 Ne ratio can be used as a shielding factor to estimate the size of the original meteoroid, the cosmogenic noble gas production rates, and the meteorite's depth within the meteoroid. For Arpu Kuilpu's two analyzed aliquots, the 22 Ne/ 21 Ne ratios nearly agree within error (~1.28; see Table 9 ), and are relatively high, suggesting a solar cosmic ray (SCR) component. As stated in the Methods and Results sections, these high 22 Ne/ 21 Ne ratios yielded no matches for meteoroids 10-500 cm when using the model from Leya and Masarik (2009), so we instead used a modified version of this model (Wieler et al. 2016) intended for a meteoroid radius up to 7 cm, which yields a very small meteoroid radius (~1 cm), and an even shallower burial depth (<1 cm) for Arpu Kuilpu. Considering the modeled production rates of 3 He, 21 Ne, and 38 Ar ( Table 11 ), the noble gas content suggests a CRE age of ~7-9 Ma. \nThe AMS-measured concentrations of 10 Be (16.1 ± 0.2 dpm kg -1 ) and 26 Al (42.6 ± 0.8 dpm kg -1 ) (see Table 7) in the stone fraction of Arpu Kuilpu are on the low end of the range of 16-25 dpm kg -1 \n( 10 Be) and 40-85 dpm kg -1 ( 26 Al) predicted by the model of Leya and Masarik (2009), suggesting that Arpu Kuilpu's meteoroid was small with maximum radius of 10 cm ( Figure 10 ). Unfortunately, this model did not provide production rates for objects smaller than 10 cm, though these concentrations of 10 Be, 26 Al and 36 Cl are consistent with irradiation at a depth of <1 cm in a 10 cm object ( Figure 10 1b1d). The very low shielding conditions are also supported by the relatively high 10 Be and 26 Al concentrations in the metal phase and the low 10 Be(sto)/ 10 Be(met) and 26 Al(sto)/ 26 Al(met) ratios of 2.9 and 9.9, respectively ( Table 7 ). These ratios are strongly dependent on shielding conditions (Nagai et al. 1990), with 10 Be(sto)/ 10 Be(met) increasing from values of ~3 in small chondrites to values of 5-13 in large chondrites, like FRO 90174 and Gold Basin (Welten et al. 2001; 2003). The 26 Al(sto)/ 26 Al(met) ratios show a range of 9-70 for the same chondrites. The values found in Arpu Kuilpu are among the lowest values reported in ordinary chondrites, similar to those found in several small Antarctic Hchondrites from the Frontier Mountain (FRO) icefield (Welten et al. 2001), including FRO 90037 (13.1 g) and 90151 (33.6 g). These small H-chondrites also had very high 22 Ne/ 21 Ne ratios of 1.27-1.32, similar to Arpu Kuilpu (1.28, see Table 9) . Such little shielding appears unlikely, as the outer >1 cm of the meteoroid is often lost during atmospheric ablation. However, since fragmentation was observed during the fireball event, it is quite possible that the meteorite came from a location close to the surface of the meteoroid, and survived to the ground, experiencing non-uniform ablation. Wieler et al. (2016) and Roth et al. (2017) have shown that SCR-derived Ne can commonly be observed in small ordinary chondrites and martian meteorites. An alternative approach to calculating the CRE age, which does not rely on the 22 Ne/ 21 Ne ratio, is to use the correlation between the 3 He, 21 Ne, and 38 Ar production rates as well as the bulk 26 Al production rate from the Leya and Masarik (2009) model albeit only for meteoroids with radii ≥ 10 cm. Arpu Kuilpu's bulk 26 Al concentration of 37.2 dpm kg -1 is just within the 35-70 dpm kg -1 range found in this model. Assuming that these correlations can be extended to slightly smaller objects with ~5 cm radius, we can estimate 3 He, 21 Ne and 38 Ar production rates (in units of 10 -8 cm 3 STP/g/Ma) of 1.40, 0.176 and 0.035, respectively (note that these differ from those presented in Table 11, which used the modified model from Leya and Masarik (2009) extended to radii ≤ 10 cm (Wieler et al. 2016) and 22 Ne/ 21 Ne ratios). Using the cosmogenic 3 He, 21 Ne and 38 Ar concentrations from Table 11 , this approach yields CRE ages of 6.2, 5.5 and 5.7 Ma, with an average value of 5.8 ± 0.6 Ma (based on an uncertainty of ~10% for the production rate calculations). \nThis slight discrepancy in results between the noble gas only-informed model for small radii ≤ 10 cm (Wieler et al. 2016; 1 cm radius, <1 cm burial depth, ~7-9 Ma CRE age), and the 26 Al-informed Leya and Masarik (2009) approach using a model for radii ≥ 10 cm, extending it to ≤ 10 cm (adopting a 5-10 cm radius (based on gamma-ray spectrometry and short-lived radionuclides), <1 cm burial depth, ~6 Ma CRE age), could be explained in two ways: sample heterogeneity and model uncertainty. Although the aliquots for the AMS, noble gas, and ICP-MS analyses all originated from the same physical location, approximately ~0.5 cm within the meteorite, they sometimes show minor, yet noticeable deviation from each other and from typical H chondrite elemental concentrations. The ICPMS results, for instance, show an enrichment in trace siderophile elements which we attribute to an embedded refractory siderophile nugget ( Figure 4; Table 4 ). While the noble gas concentrations for the two aliquots measured (AKS and AKL) are close to one another, they often do not agree within error ( Table 9 ). Considering these aliquot-to-aliquot variations, and model uncertainties in both Wieler et al. (2016), and Leya and Masarik (2009), particularly concerning the covered meteoroid size, we assert these differences to be at the edge of our ability to infer the exact meteoroid history, and that future work may be needed to supplant or expand on existing models to comprehensively extend our understanding of CRE to very small (<10 cm) meteoroids and their surviving meteorites. \nCombining these methods and their insights, we conclude that Arpu Kuilpu came from a small (<5 cm) meteoroid, very near to its surface (<1 cm), and that its meteoroid spent ~6-8 Ma in interplanetary space, separate from its contemporary asteroidal source body. \nFigure 10. (a) Comparison of the measured concentrations of 10 Be and 26 Al in the stone fraction of Arpu Kuilpu with calculated 10 Be and 26 Al production rates in the stone fraction of ordinary chondrites with radii of 10-100 cm. Panels (b-d) shows a comparison of the measured concentrations of 10 Be (b) 26 Al (c) and 36 Cl (d) in the stone fraction of Arpu Kuilpu with calculated depth profiles in the stone fraction of H-chondrites with radii of 10-30 cm. All calculations are based on the model of Leya and Masarik (2009). \n<!-- image -->", 'Relation to Other Orbital Meteorites': "Considering the 53 other meteorites with associated orbits that have been recovered so far (at the time of this writing; Meier, 2017), the equilibrated H chondrites (H4s, H5s, and H6s) account for more than 30% (excluding multi-lithological falls), to which we will compare Arpu Kuilpu based on two main criteria: meteoroid orbit and CRE age ( Table 12) . We would also like to note that although the H5/6 chondrites Al-Khadhaf (MB 112; Gattacceca et al., 2024) and Santa Filomena (Tosi et al., 2023) are both falls with known orbits, their CRE ages and probable source resonances are currently unknown, therefore we did not include them in our comparisons to Arpu Kuilpu. For likeness in orbital parameters, Ejby (Spurný et al. 2017) and Košice (Borovička et al. 2013) are the closest matches as they both have a high probability of being sourced from a JFC orbit (although with relatively high associated uncertainties), with Hamburg (Brown et al. 2019), then Murrili (Sansom et al. 2020) being next closest with Tisserand parameters near the border of being considered a JFC orbit. Interestingly, the CRE ages of Murrili (7 ± 1 Myr; Anderson et al. 2021) and Košice (5 -7 Myr; Povinec et al. 2015) are similar to Arpu Kuilpu's ~6 -8 Myr, falling within the broad peak of 6-8 Myr for H5 chondrites identified by Marti and Graf (1992). Although the Lost City and Morávka meteorites also have CRE ages falling into this broad peak (Bogard et al. 1971; Borovička et al. 2003b), their orbits are distinctly dissimilar, originating from the inner main belt, on the other side of the 3:1 mean motion resonance with Jupiter (McCrosky et al. 1971; Borovička et al. 2003a). Therefore, at first assessment, the most likely orbital meteoritic sibling to Arpu Kuilpu is Košice, since it has the same CRE age and orbit, within errors (which are comparatively high for the latter). \nThough to properly compare Arpu Kuilpu to other orbital H chondrites, we must also consider how orbits can evolve over time. Although the semi major axes of most small meteoroids change \ngradually over many orbital periods, due to the Yarkovsky effect (Bottke et al., 2001), eventually wandering into a mean motion resonance with Jupiter which then increases its eccentricity ultimately causing a collision with one of the inner planets, the work by Shober et al. (2020a; 2020b) demonstrates an additional mechanism for orbital evolution. They show how a meteoroid with a nearEarth orbit (a < 2 AU) can have a close encounter with an inner planet, immediately altering its path to a JFC-like orbit. This is further supported by a later study (Shober et al. 2021) that shows how most objects in the cm-m size range on JFC orbits are asteroidal in nature, and not composed of volatile-rich material (originating from trans-Neptunian space (Nesvorný et al. 2010; Fernández and Sosa, 2015)) typical of dust and km sized objects existing on similar orbits. Using this mechanism, we can speculate that Arpu Kuilpu, Košice, Murrili, and Lost City were all liberated from the same source body in the same ejection event (~68 Ma) in the inner solar system, but that only Arpu Kuilpu and Košice underwent close encounters to transfer into a JFC-like orbit. Although Morávka has a similar CRE age to Arpu Kuilpu, its high inclination (i = 32; Table 12 ), especially compared to the other H5's listed here, make a close orbital link unlikely. For Ejby and Hamburg, it is possible that they were ejected at different times, from either the same or different source bodies in the inner solar system, then also independently experienced this change in orbit, which eventually delivered them to Earth. It is also entirely possible that none of these rocks are related to each other and that each one represents a discrete source body within the solar system. Only as more orbital meteorites are collected and analyzed, can we more confidently infer about the nature of contemporary H chondrite source bodies. \nTable 12. Other orbital meteorites that may be related to Arpu Kuilpu, based on orbital parameters and CRE age. \na: Shober et al. (2022); b: Borovička et al. (2013); c: Povinec et al. (2015); d: Borovička et al. (2003a); e: Borovička et al. (2003b); f: Ceplecha and ReVelle , (2005); g: McCrosky et al. (1971); h: Bogard et al. (1971); i: Brown et al. (2019); j: Heck et al. (2020); k: Sansom et al. (2020); l: Anderson et al. (2021); m: Spurný et al. (2017); n: Haack et al. (2019);", 'CONCLUSIONS': "The Arpu Kuilpu orbital meteorite is a minimally-shocked, lightly-weathered, unbrecciated, H5 chondrite that was delivered to Earth on a JFC-like orbit. The very minor impact-related shock pressure it experienced occurred very early in the solar system's history. It was likely separated from its contemporary, asteroidal source body approximately 6-8 Ma ago, and may be related to other orbital H chondrites, with Košice being the closest match when considering prior orbits and CRE ages (see Discussion). Combining Arpu Kuilpu's asteroidal composition with its JFC -like orbit prior to impact further confirms the observation that small meteoroids (cm-m size) on JFC orbits can originate from the asteroid belt. Below ( Table 13 ) we list the major features of this meteorite, and the methods used to determine them. \nTable 13. The main features of the Arpu Kuilpu meteorite.", 'ACKNOWLEDGMENTS': "We would like to thank the Maralinga Tjarutja community for permission to search on their land, as well as providing the name for this meteorite. We also thank C. Alexander and J. Bridges for their careful and constructive reviews. This work was funded by the Australian Research Council's Discovery Project Scheme (DP170102529), and has been partially carried out within the framework of the Swiss NSF (NCCR PlanentS) under grant 51NF40 205606.", 'REFERENCES': "Alaerts L., Lewis R. S., Anders, E. 1979. Isotopic anomalies of noble gases in meteorites and their origins -III. LL-chondrites. Geochimica et Cosmochimica Acta 43: 1399-1415. https://doi.org/https://doi.org/10.1016/0016-7037(79)90134-0 \nAlexander C. M. O'D. 2019. 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2023A&A...673A.114H	Context. Data from the Gaia satellite are revolutionising our understanding of the Milky Way. With every new data release there is a need to update the census of open clusters. BR Aims We aim to conduct a blind allsky search for open clusters using 729 million sources from Gaia DR3 down to magnitude G 20 creating a homogeneous catalogue of clusters including many new objects. BR Methods We used the Hierarchical DensityBased Spatial Clustering of Applications with Noise HDBSCAN algorithm to recover clusters. We validated our clusters using a statistical density test and a Bayesian convolutional neural network for colourmagnitude diagram classification. We inferred basic astrometric parameters ages extinctions and distances for the clusters in the catalogue. BR Results We recovered 7167 clusters 2387 of which are candidate new objects and 4782 of which crossmatch to objects in the literature including 134 globular clusters. A more stringent cut of our catalogue contains 4105 highly reliable clusters 739 of which are new. Owing to the scope of our methodology we are able to tentatively suggest that many of the clusters we are unable to detect may not be real including 1152 clusters from the Milky Way Star Cluster MWSC catalogue that should have been detectable in Gaia data. Our cluster membership lists include many new members and often include tidal tails. Our catalogues distribution traces the galactic warp the spiral arm structure and the dust distribution of the Milky Way. While much of the content of our catalogue contains bound open and globular clusters as many as a few thousand of our clusters are more compatible with unbound moving groups which we will classify in an upcoming work. BR Conclusions We have conducted the largest search for open clusters to date producing a single homogeneous star cluster catalogue which we make available with this paper. P Full Tables 3 B.1 and the cluster members Appendix A are only available at the CDS via anonymous ftp to A hrefhttpscdsarc.cds.unistra.frcdsarc.cds.unistra.frA ftp130.79.128.5 or via A hrefhttpscdsarc.cds.unistra.frvizbincatJAA673A114httpscdsarc.cds.unistra.frvizbincatJAA673A114A	2023-05-01T00:00:00Z	['10.1051/0004-6361/202346285', '10.48550/arXiv.2303.13424', 'arXiv:2303.13424', '2023arXiv230313424H', '2023A&A...673A.114H']	['open clusters and associations: general', 'methods: data analysis', 'catalogs', 'astrometry', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Instrumentation and Methods for Astrophysics']	Improving the open cluster census. II. An allsky cluster catalogue with Gaia DR3	2,023	190	0.74	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	119	https://arxiv.org/pdf/2303.13424.pdf	{'II. An all-sky cluster catalogue with Gaia DR3 ?': 'Emily L. Hunt 1, ?? and Sabine Re GLYPH<11> ert 1 \nLandessternwarte, Zentrum für Astronomie der Universität Heidelberg, Königstuhl 12, 69117 Heidelberg, Germany e-mail: [email protected] \nReceived 1 st March 2023; accepted 21 st March 2023', 'ABSTRACT': "Context. Data from the Gaia satellite are revolutionising our understanding of the Milky Way. With every new data release, there is a need to update the census of open clusters. \nAims. We aim to conduct a blind, all-sky search for open clusters using 729 million sources from Gaia DR3 down to magnitude G GLYPH<24> 20, creating a homogeneous catalogue of clusters including many new objects. \nMethods. We used the Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) algorithm to recover clusters. We validated our clusters using a statistical density test and a Bayesian convolutional neural network for colour-magnitude diagram classification. We inferred basic astrometric parameters, ages, extinctions, and distances for the clusters in the catalogue. \nResults. We recovered 7167 clusters, 2387 of which are candidate new objects and 4782 of which crossmatch to objects in the literature, including 134 globular clusters. A more stringent cut of our catalogue contains 4105 highly reliable clusters, 739 of which are new. Owing to the scope of our methodology, we are able to tentatively suggest that many of the clusters we are unable to detect may not be real, including 1152 clusters from the Milky Way Star Cluster (MWSC) catalogue that should have been detectable in Gaia data. Our cluster membership lists include many new members and often include tidal tails. Our catalogue's distribution traces the galactic warp, the spiral arm structure, and the dust distribution of the Milky Way. While much of the content of our catalogue contains bound open and globular clusters, as many as a few thousand of our clusters are more compatible with unbound moving groups, which we will classify in an upcoming work. \nConclusions. We have conducted the largest search for open clusters to date, producing a single homogeneous star cluster catalogue which we make available with this paper. \nKey words. open clusters and associations: general - Methods: data analysis - Catalogs - Astrometry", '1. Introduction': 'The Milky Way galaxy is an intricate ecosystem of ongoing star formation, evolution, and destruction. Open clusters (OCs) are one such part of this system, which form when molecular clouds condense into stars and may further condense into gravitationally bound groups of a few dozen to a few thousand stars. Hence, OCs o GLYPH<11> er an important way to study the immediate aftermath of star formation, as well as the ongoing evolution of stars up to an age of around GLYPH<24> 1 Gyr, after which most OCs will have been broken up, with their member stars dissolving back into the galactic disk (Portegies Zwart et al. 2010; Krumholz et al. 2019; Krause et al. 2020). \nOur view of OCs has always been complicated by their sparsity and their typical location in the galactic disk, making them challenging to isolate from field stars along the line of sight (Cantat-Gaudin 2022). However, dramatically improved astrometric and photometric data from the Gaia satellite (Gaia Collaboration et al. 2016) are revolutionising our understanding of OCs and the overall Milky Way. Compared with the Hipparcos \nmission (Perryman et al. 1997), Gaia provides order of magnitude improvements in proper motion and parallax accuracy for around 10 4 times as many stars, with over 1 billion sources in total. \nBecause of these improvements, Gaia has enabled many new insights into all properties of OCs. Works such as Meingast et al. (2021) and Tarricq et al. (2022) have shown that many nearby OCs have tidal tails or comas of ejected member stars indicative of their ongoing tidal disruption by the Milky Way. Other works such as Bossini et al. (2019) and Cantat-Gaudin et al. (2020) have used Gaia photometry to infer cluster ages, extinctions, and distances, which can then be used to make wider inferences about the Milky Way, such as in Castro-Ginard et al. (2021) who used OCs to trace the spiral arms of the galaxy. Cleaned Gaia cluster membership lists also improve spectroscopic studies such as Baratella et al. (2020), who combined Gaia data with ground-based spectroscopic measurements to study the chemistry of OCs. \nAt the heart of all science with OCs, however, is the census of OCs itself. Particularly in the four years since Gaia Data Release 2 (DR2, Brown et al. 2018), many works have contributed major new insights into the census of OCs. Works such as Cantat-Gaudin et al. (2018), Cantat-Gaudin & Anders (2020), and Jaehnig et al. (2021) provide new membership lists for OCs with a significantly higher number of stars and reduced outliers from the field when compared to preGaia works. Thousands of \nnew OCs have been reported using a range of unsupervised machine learning techniques, such as in Castro-Ginard et al. (2018, 2019, 2020, 2022), Cantat-Gaudin et al. (2019), or Liu & Pang (2019). The reliability of the census has also been improved, with works such as Cantat-Gaudin & Anders (2020) finding that a number of OCs discovered before Gaia are likely to be asterisms. \nOne might wonder how much further Gaia can improve the census of OCs, and what these improvements could reveal. In Hunt & Re GLYPH<11> ert (2021) (hereafter Paper 1), we compare three different approaches for recovering OCs in Gaia DR2data, and find that the HDBSCAN clustering algorithm (Hierarchical DensityBased Spatial Clustering of Applications with Noise, Campello et al. 2013) is the most sensitive approach, although it is essential to reduce false positives with additional post-processing. In this work, we conduct the largest blind search for star clusters to date in Gaia data, using Gaia DR3 (Gaia Collaboration et al. 2021), methods developed in Paper 1, and additional validation criteria based on the photometry of every detected cluster. \nIn Sect. 2, we describe the Gaia DR3 data used in this work and the quality cuts we adopted to filter out unreliable sources. In Sect. 3, we briefly recap our clustering method from Paper 1 and tweaks made to improve cluster recovery within 1 kpc. We then outline a method to validate cluster candidates using their photometry in Sect. 4, which we generalise to additionally infer ages, extinctions, and photometric distances to our clusters in Sect. 5. In Sect. 6, we crossmatch our catalogue against literature works. Section 7 presents an overview of our catalogue. We discuss the non-detections of some literature clusters in Sect. 8, and discuss required steps a future work will take to improve the reliability of our new cluster candidates in Sect. 9. Section 10 summarises this work. \nDuring the preparation of this work, we found that many of the star clusters we detect appear much more compatible with unbound moving groups than bound OCs, regardless of the quality of their photometry or how strong of an overdensity they are. In an upcoming third paper, we will classify the clusters resulting from this work into bound and unbound clusters, which will result in our final catalogue. This work will follow shortly (Hunt & Re GLYPH<11> ert, in prep. ).', '2. Data': 'In this section, we present a brief overview of Gaia DR3 data and the preprocessing steps applied to prepare it for clustering analysis.', '2.1. Gaia DR3': 'The latest release of Gaia (Gaia Collaboration et al. 2016) astrometry and photometry, Gaia DR3, presents an update to Gaia DR2, based on an extra 12 months of data and various improvements to data processing. Astrometric and photometric data were released early in Gaia EDR3 (Gaia Collaboration et al. 2021), with the full DR3 release containing other data products such as low-resolution spectra and updated radial velocities that we also make limited use of in this work Gaia Collaboration et al. (2022). In total, DR3 contains 1.47 billion sources with 5- or 6parameter astrometry, with a 30% improvement in parallax precisions and a roughly doubled accuracy in proper motions. These improvements have a large impact on the detectability of OCs in Gaia - particularly for proper motions, where distant OCs have a signal-to-noise ratio (S / N) increased by a factor of GLYPH<24> 4 in Gaia \nFig. 1. Comparison of cluster membership lists detected using Gaia DR3 data cut at G < 18 (black empty circles) and a Rybizki et al. (2022) v1 criterion greater than 0.5 (blue filled circles) using separate runs of HDBSCAN and our pipeline for each cut, shown for Auner 1 (left) and Ruprecht 134 (right). \n<!-- image --> \nDR3 proper motion diagrams, owing to the halving in size of the Gaussian distribution of stars in both axes for distant clusters with proper motion dispersions smaller than Gaia errors. \nIn addition, many improvements have been made to the processing and understanding of Gaia data and systematics for Gaia DR3. Most notably for OCs, Lindegren et al. (2021b) provide a recipe for greatly reducing remaining parallax systematics for most sources in Gaia DR3 down to a few GLYPH<22> as in the best cases, which should significantly improve the accuracy of distances to the most distant clusters. Cantat-Gaudin & Brandt (2021) provide a recipe for correcting the proper motions of certain bright stars around G GLYPH<24> 13. While both of these corrections are too small to make a di GLYPH<11> erence in unsupervised cluster searches, they are included in later cluster parameter determinations to improve the accuracy of final catalogue values.', '2.2. Outlier removal': "Despite improvements between Gaia DR2 and DR3, many sources in the catalogue are still unreliable due to a number of reasons. For instance, blending in crowded fields can cause both astrometric and photometric errors, with sources being erroneously combined or split for any or all Gaia measurements of the source. This is a particular issue in regions of the galactic disk with high numbers of sources. In addition, resolved and unresolved binary stars in DR3 may contribute significant errors to derived astrometric measurements for these sources, especially when their period is close to the one year baseline used to measure parallaxes (Penoyre et al. 2022; Lindegren et al. 2021a), as well as causing issues with photometric measurements due to blending (Riello et al. 2021; Golovin et al. 2023). \nTo remove unreliable sources, a number of di GLYPH<11> erent quality cuts were investigated, both in isolation and combined: firstly, simple magnitude cuts, including G < 18 as adopted in works such as Paper 1 and Cantat-Gaudin et al. (2018), G < 19, and G < 20; secondly, a cut on renormalised unit weight error (RUWE) values in the main Gaia source table; and finally, a cut presented in Rybizki et al. (2022), which uses a neural network and 17 diagnostic columns in the Gaia EDR3 data release to classify astrometric solutions as reliable and unreliable, where we required a quality value of at least 0.5. \nTo evaluate the performance of these cuts, the reliability of cluster recovery with HDBSCAN (Hierarchical Density-Based Spatial Clustering of Applications with Noise, Campello et al. 2013; McInnes et al. 2017) was inspected manually for 15 challenging to detect clusters given di GLYPH<11> erent combinations of these cuts. Notable clusters in this process include Ruprecht 134, a di GLYPH<14> cult to recover cluster located in the most crowded region of the galactic disk at l ; b = (0 : 28 GLYPH<14> ; GLYPH<0> 1 : 63 GLYPH<14> ) and at a distance of GLYPH<24> 3 kpc, in addition to a number of clusters reported in CantatGaudin & Anders (2020) but not detected in Paper 1 in Gaia DR2, such as Berkeley 91 and Auner 1. \nAsingle, magnitude-independent cut based only on the quality flag of Rybizki et al. (2022) was found to outperform all other cuts trialed for cluster recovery. On average, for the trial set of 15 clusters, clusters recovered using this cut had the highest S / N of any recovered by any of the trialed cuts, with S / Ns being an average of 65% higher than clusters recovered using the G < 18 cut common in the literature (see e.g. Cantat-Gaudin et al. 2018; Castro-Ginard et al. 2022). Clusters almost always had more member stars than a simple G < 18 cut, with up to around twice as many member stars for distant, faint clusters where only giant stars can be resolved for magnitudes G < 18, such as for the distant cluster Auner 1 at a distance of 6.8 kpc. Inevitably, this cut should result in more complete membership lists and a more complete overall catalogue of clusters. \nAs a visual example, the CMDs of Auner 1 and Ruprecht 134 from clustering analyses using this cut and a G < 18 cut are compared in Fig. 1. Auner 1 is a distant and di GLYPH<14> cult to detect cluster, for which only 51 stars are detected in the G < 18 trial for a cluster S / N of 10.8 GLYPH<27> . However, the Rybizki cut cluster includes many additional faint sources, for a total of 139 member stars and an improved S / N of 17.9 GLYPH<27> . In the case of Ruprecht 134, a massive cluster in a crowded region near the galactic centre, the Rybizki cut cluster has fewer sources than the G < 18 cut (277 to 355) but a higher S / N (24.7 GLYPH<27> to 16.6 GLYPH<27> ), with the Rybizki cut removing a number of spurious sources from the cluster membership and the field - improving the cluster membership list and the cluster's contrast against field stars. \nCompared to having no cut at all, adoption of this cut typically has a minimal impact on the number of member stars for all clusters - it appears that sources with unreliable astrometry are already so unreliable that their position in 5D Gaia astrometry is too far from the bulk cluster position to be tagged as members, and few outliers are removed from cluster CMDs by this (or any) cut. Instead, in the crowded region at the galactic centre around Ruprecht 134, 85% of the sources in this field were removed by the cut, yet all reliable clusters in this field (including the nearby UFMG 88 reported by Ferreira et al. 2021) remained with a similar membership list to with no cut at all. In addition, the lack of a magnitude cut means that in sparse fields where faint sources have reliable astrometry, clusters such as the high galactic latitude Blanco 1 have membership lists down to fainter than G GLYPH<24> 20, two magnitudes fainter than the membership list of Cantat-Gaudin & Anders (2020) for this cluster. \nOnly the v1 version of the Rybizki et al. (2022) quality flag was available during preparation of cluster membership lists in this work, for which a minimum value of 0.5 was adopted. Later versions of the initial Rybizki et al. (2022) pre-print and eventual published paper have a slightly improved version of the quality flag, although in practice it was found to make a negligible difference to the final results of this work and so clustering analysis was not revised to include it. \nIn total, 729.7 million sources in Gaia DR3 have a Rybizki et al. (2022) v1 quality flag of at least 0.5 and were selected for \nfurther clustering analysis in this work. This represents significantly more sources than the 301.7 million sources with G < 18, a cut adopted in works such as Castro-Ginard et al. (2022) or Cantat-Gaudin & Anders (2020), and should result in a greater total number of both detected clusters and member stars.", '2.3. Data partitioning': 'Finally, due to computational reasons, we partition the Gaia dataset into three separate collections for further analysis, as it is not possible to e GLYPH<14> ciently perform clustering analysis with 729.7 million sources at once. We aim to divide the Gaia dataset in such a way so that no more than 20 million sources are in any one field and so that a cluster of around 20 pc tidal radius can always be reliably detected regardless of its distance or location within adopted fields, which should be a reasonable upper size limit for almost all OCs based on Kharchenko et al. (2013) and Cantat-Gaudin & Anders (2020). \nAs in Paper 1, the HEALPix (Hierarchical Equal Area isoLatitude Pixelation) tessellation scheme was used to segment the entire Gaia dataset (Górski et al. 2005), with calculations performed by the Python package Healpy (Zonca et al. 2019). This has advantages over other methods to subdivide spheres into a finite number of regions, in that all regions at a given tessellation level have the same area, and spherical distortions are minimised. However, unlike in Paper 1, the origin of the HEALPix grid was set at the origin of galactic coordinates ( l ; b = (0 GLYPH<14> ; 0 GLYPH<14> )), instead of the default ICRS origin at right ascension and declination values of GLYPH<11>; GLYPH<14> = (0 GLYPH<14> ; 0 GLYPH<14> ) used in Gaia data releases, as this places most remaining spherical distortions at high galactic latitudes where we expect to find few clusters, meaning that all fields on the most important regions of the galactic disk are simple quadrilaterals. \nWe adopted three di GLYPH<11> erent partitioning schemes to detect clusters in three di GLYPH<11> erent distance ranges: those more distant than 750 pc, those closer than 750 pc, and those closer than 150 pc. Each scheme used large enough fields to detect clusters at each di GLYPH<11> erent distance range, but while minimising the number of stars in each field to keep the fields feasible to perform clustering analysis on. Firstly, for the most distant clusters, we adopted the same methodology as in Paper 1, dividing the entire Gaia dataset into 12288 HEALPix level five pixels. To avoid losing clusters on the edge of each pixel, each pixel is grouped into fields containing the pixel itself and its eight nearest neighbours, e GLYPH<11> ectively overlapping each GLYPH<25> 5 : 5 GLYPH<14> GLYPH<2> 5 : 5 GLYPH<14> field by 1 : 8 GLYPH<14> with all surrounding neighbours, with every pixel appearing in nine separate fields and in the centre of one. Next, to detect clusters closer than 750 pc, a HEALPix level two scheme with 192 pixels was adopted, containing only sources with $ > 1 mas, using the same nine pixels per field system and resulting in overlapping fields of size GLYPH<25> 44 GLYPH<14> GLYPH<2> 44 GLYPH<14> . Finally, for clusters closer than 150 pc, which can have large extents on the sky, a single field containing all stars closer than 250 pc was used, based on photo-geometric distances to sources in Bailer-Jones et al. (2021). \nBetween these three systems, all bound members of all open clusters of size 20 pc or smaller should be contained within these fields - although in reality, this is only a worst-case constraint at the 750 pc and 150 pc crossover points and for a cluster in the worst possible location in a field, and many significantly larger clusters (including tidal tails many times their size) would be detectable in other regions.', '3. Cluster recovery': 'Next, we discuss the methodology we adopted to recover clusters in Gaia data, assign basic parameters, and crossmatch to existing cluster catalogues in the literature.', '3.1. HDBSCAN': 'Many di GLYPH<11> erent algorithms have been used to date to recover clusters in Gaia data. We present a review and full explanation of these algorithms in Paper 1, in which we found that the HDBSCAN algorithm (Campello et al. 2013; McInnes et al. 2017) is the most sensitive for recovering OCs in Gaia data. \nBriefly, HDBSCAN is an updated version of the DBSCAN algorithm (Ester et al. 1996), for which only a minimum cluster size mclSize and minimum number of points in the neighbourhood of a cluster core point mPts must be specified, unlike DBSCAN which instead uses mPts and a minimum, global distance between points in a cluster GLYPH<15> . DBSCAN has seen much use in the literature so far for OC recovery, such as in Castro-Ginard et al. (2018, 2019, 2020, 2022) or He et al. (2021, 2022a). The main challenge of DBSCAN is that GLYPH<15> must be set globally for an entire dataset, which can limit the sensitivity of the algorithm for datasets of varying density - such as the Gaia dataset, which has di GLYPH<11> erent densities at di GLYPH<11> erent distances and locations within the galaxy. \nInstead, HDBSCAN copes with varying density datasets by e GLYPH<11> ectively considering all possible DBSCAN GLYPH<15> solutions for all regions of a dataset, selecting the best clusters based on the lower limit of cluster size mclSize . HDBSCAN has so far been used to detect moving groups in Gaia data by Kounkel & Covey (2019) and Kounkel et al. (2020), as well as being used to find 41 new OCs in Paper 1, and being used by Tarricq et al. (2022) to reveal new tidal tails and comas of numerous OCs within 1.5 kpc. HDBSCAN has not yet been used to conduct a search through all Gaia data for OCs. \nA major flaw of HDBSCAN, however, is its high false positive rate. In Paper 1, we show that this is due to the algorithm being overconfident, reporting dense random fluctuations of a given dataset as clusters. To mitigate this, we adopt the cluster significance test (CST) from Paper 1, which searches for field stars surrounding a cluster and compares the nearest neighbour distribution of cluster stars with that of field stars. This then produces a signal-to-noise ratio (S / N), with CST scores greater than 5 GLYPH<27> corresponding to highly likely clusters. \nThe issue of how to convert the five dimensions of Gaia astrometry into a form best usable by a clustering algorithm is an open problem. Converting proper motions and parallaxes to velocities and distances respectively is one such approach (e.g. as in Kounkel et al. 2020; He et al. 2022a), although a major issue is that converting Gaia parallaxes to distances is nontrivial and results in asymmetric errors and non-Gaussian parameter distributions (Bailer-Jones et al. 2021). Instead, we use the approach adopted in Paper 1, similar to that of works such as Castro-Ginard et al. (2018) and Liu & Pang (2019). We use Gaia positions, proper motions, and parallaxes directly, but with two preprocessing steps: firstly, recentring them into a coordinate frame with an origin at the centre of each respective field, which removes spherical distortions present at high declinations; secondly, rescaling all five axes of the dataset to have the same median and interquartile range, e GLYPH<11> ectively removing the units of each axis of the data. Particularly for HDBSCAN, which can cope with varying density datasets, the choice to use these five simple recentred and rescaled features was found to have no im- \npact on the detectability and membership lists of nearby clusters, while having great benefits for clusters more distant than GLYPH<24> 2 kpc, for which a distance-based approach causes many clusters to have sparser, non-Gaussian, and more challenging to detect distributions. \nThe one exception to this in this work is for the single field of all stars within 250 pc, which was adopted to help improve the accuracy of cluster membership lists for very nearby clusters with large angular extents on the sky such as the Hyades. Given that this field covers the entire sky, it is not possible to avoid high latitude spherical distortions with a simple recentring; instead, photo-geometric distances from Bailer-Jones et al. (2021) were used to convert positions and parallaxes to a Cartesian coordinate frame, with proper motions converted to tangential velocities. At such small distances, the uncertainties in Bailer-Jones et al. (2021) are small and not prior-dominated, and so reliance on Gaia -derived distances for the single nearby field should not cause any issues.', '3.2. Clustering analysis and catalogue merging': "Using HDBSCAN and the same range of parameter choices as in Paper 1 ( mclSize GLYPH<15> f 10 ; 20 ; 40 ; 80 g , mPts = 10), clustering analysis on all HEALPix level two and five fields was completed in around eight days of runtime on a machine with a 48 core Intel(R) Xeon(R) E5-2650 CPU with 48 GB of RAM. This run was mostly RAM-limited due to the worst-case O ( n 3 ) memory use of the HDBSCAN implementation used for the largest fields. Given that fields overlap and that di GLYPH<11> erent parameter choices can detect the same cluster, each cluster can be duplicated up to four times within a single field, up to nine times by appearing in all neighbouring fields and a further time by appearing in di GLYPH<11> erent distance ranges (if the cluster has a distance between 0.7 to 1 kpc, or less than 250 pc). Hence, in the worst case, a single cluster could be duplicated 72 times. It is essential and non-trivial to merge the results of all fields accurately and without losing or duplicating any one individual cluster. \nIn total, 7.1 million di GLYPH<11> erent clusters were detected (including duplicates), almost all of which are astrometric false positives due to the oversensitivity flaws of HDBSCAN discussed in Paper 1. These clusters can be removed by using their astrometric S / N, as derived by the CST. Figure 2 shows histograms of the S / Ns of detected clusters, showing a clear spike in count for S / N < 0 : 5 and an increasing trend in S / N for S / N . 3 that deviates from the relatively straight log-linear relation in S / Npresent for S / N > 3, suggesting that an additional component of false positives is contributing to the otherwise log-linear component of reliable astrometric clusters at low S / Ns. This figure, our results from Paper 1, and the poor quality of the low-S / N clusters we detect strongly support that most low-S / N clusters are false positives; however, exactly where to set an S / N threshold is a non-trivial decision that has a large e GLYPH<11> ect on the rest of the catalogue. A catalogue can choose to prioritise completeness, having a low threshold and including as many true positives as possible, but while inevitably including many false positives and sacrificing precision; or, a catalogue can do the opposite, having a lower completeness but also minimal false positives and maximised reliability of all objects in the catalogue. \nFor the purposes of this work, we chose to prioritise the precision and reliability of the catalogue, adopting a higher threshold on the minimum S / N of clusters. This sacrifices some completeness so that all final catalogue entries are likely to be real astrometric overdensities and not mere statistical fluctuations. This approach also comes with a key advantage. Our field tiling strat- \n<!-- image --> \nFig. 2. Statistics of all detected clusters compared against the final catalogue. Top : distribution of the number of member stars of detected clusters, n stars, for all detected clusters in all fields before catalogue merging and duplicate removal (solid blue line), for the final catalogue (solid orange line), and amongst clusters in the final catalogue that crossmatch to clusters in the literature, for all literature clusters (solid red line) and for only those detected before the release of Gaia EDR3 (dotted purple line). Bottom: as above, but for the astrometric S / N (CST score) for all clusters in these sets. S / Ns have a maximum value of 38 due to numerical reasons. \n<!-- image --> \negy aimed to prevent any real clusters from being 'lost', aiming to recover > 99%of real, good-quality OCs in a single catalogue. However, merging the results of so many separate clustering runs is a di GLYPH<14> cult and non-trivial task, and early experiments showed that the inclusion of false positives in the catalogue had a severe e GLYPH<11> ect on the reliability and accuracy of the catalogue merging process. It was common that false positives and clear real OCs would share members in di GLYPH<11> erent clustering runs, meaning that low S / Nthresholds on the final catalogue would adversely a GLYPH<11> ect the catalogue's completeness at higher S / Ns. For the purposes of this work, we set a higher threshold on the minimum S / N, requiring S / N > 3 GLYPH<27> . This cut was found to maximise the quality of later catalogue merging steps, while removing a high number of false positives and retaining reliable clusters. Many false positives share member stars with real OCs, which greatly complicated the merging process and made the choice of which cluster \nto keep challenging. A single S / N cut means that our incompleteness is well characterised and easy to understand, whereas lower cuts were found to adversely a GLYPH<11> ect catalogue completeness even at high S / Ns in a di GLYPH<14> cult to characterise way. In addition, while our adopted cut is at an S / N of 3 GLYPH<27> , clusters with an S / Nlower than even 5 GLYPH<27> may have minimal scientific usefulness, as they cannot be asserted as being real astrometric overdensities beyond any reasonable doubt; as such, it is not worth including such clusters in the catalogue at the expense of the recovery of better, real objects. \nInevitably, some low-S / N real OCs are likely to be lost in this process. We discuss the number of literature objects that are lost due to this cut in Sect. 8.1.3, and we briefly discuss some of the improvements to clustering algorithms that could be used to simplify the merging process and entirely remove the need for an S / N cut to ensure the catalogue's reliability in Sect. 10. \nAfter dropping unreliable low S / N clusters, the results of each parameter run in every field were merged. For clusters where every mclSize detected an identical object, duplicates were simply dropped. In some cases (such as for the largest OCs and GCs), smaller mclSize runs may split the cluster into two subclusters. Generally, it was possible to remove duplicate small subclusters by only keeping the single largest cluster. This process was extensively checked by hand, keeping smaller clusters instead in the case of some binary and coincident clusters which are better selected as being split, which was aided by fitting Gaussian mixture models to every cluster and evaluating the Bayesian information criterion of one and two-component fits, flagging clusters where a two component fit was preferred for potential splitting. \nSecondly, cluster duplicates between fields must be removed. Using maximum likelihood distances calculated with the method presented in Cantat-Gaudin et al. (2018), clusters likely to be affected by edge e GLYPH<11> ects or likely to be better detected at a di GLYPH<11> erent HEALPix level were removed. Clusters from the 250 pc run were only kept if they were closer than 175 pc. Clusters from the HEALPix level 2 run were only kept with distances between 150 and 750 pc. Finally, clusters from the HEALPix level 5 run were only kept if they had distances greater than 700 pc. The small overlaps in these distance ranges allow the best cluster to be selected later for clusters on the boundaries. \nNext, duplicate clusters due to the overlap between fields must be removed. As each field is composed of nine pixels, a cluster can appear in up to nine separate fields. Keeping only clusters in the central pixel of every field is su GLYPH<14> cient to mostly remove duplicates, retaining only the best cluster detection in the central pixel where edge e GLYPH<11> ects are minimised. However, cluster membership lists are often not identical between fields, and it is hence possible that a cluster's mean position could be di GLYPH<11> erent enough between runs to appear in the central pixel of multiple fields or to never appear in the central pixel of any field. Particularly for small clusters of 20 stars or less, the inclusion or removal of even a single star can have a reasonable impact on the mean position of the cluster. This e GLYPH<11> ect is worst for the nearest clusters with the largest angular extents on the sky relative to the field they are in. While this e GLYPH<11> ect only impacts a small number of clusters (causing around GLYPH<24> 1% of clusters reported in Cantat-Gaudin & Anders (2020) to be lost), it is nevertheless important to address to ensure the final catalogue is as complete as possible. \nTo mitigate this e GLYPH<11> ect, clusters near to the edge of a central pixel were also kept. After extensive testing, it was found that cluster positions generally vary by no more than GLYPH<24> 1 pc at the distance of the cluster between di GLYPH<11> erent fields. We adopt a more \ntolerant cut corresponding to GLYPH<24> 5 pc for a cluster at a worstcase distance, such that clusters within 1.91 GLYPH<14> (HEALPix level 2) or 0.41 GLYPH<14> (HEALPix level 5) of the edge of a central pixel were also kept. This is small compared to the overall field sizes of GLYPH<25> 44 GLYPH<14> GLYPH<2> 44 GLYPH<14> (HEALPix level 2) or GLYPH<25> 5 : 5 GLYPH<14> GLYPH<2> 5 : 5 GLYPH<14> (HEALPix level 5), but was nevertheless found to be su GLYPH<14> cient to avoid losing any genuine clusters. \nThese processes removed most duplicated clusters while minimising the number of clusters lost during the merging process, although some duplicates still remained within the allowed overlaps between fields. These clusters were removed by looking for clusters with similar membership lists, mean positions, mean proper motions, and mean parallaxes, and selecting the cluster in each case with only the highest distance from any field edge. This process was also verified extensively by hand. For 23 large clusters (typically with tidal tails larger than the field they are in), duplicate clusters were similar but with both having additional members. In these cases, the clusters were merged into single clusters. \nFinally, the catalogue was checked for clear, known binary clusters that were not correctly split by HDBSCAN. Four probable cases were identified, including the close binary Collinder 394 / NGC6716as well as UBC 76 / UBC77. Generally, these binary clusters had very similar proper motion and parallax distributions, making them di GLYPH<14> cult or impossible for the HDBSCAN algorithm to split - particularly since HDBSCAN cannot assign members to two clusters at once, although this is necessary for such close and di GLYPH<14> cult to separate objects. These clusters were split with Gaussian mixture models by selecting the number of components with the highest Bayesian information criterion. In all four cases, multiple components were preferred over a single component. It is likely that some other objects in the catalogue may also be better described as binary clusters, although this would need to be investigated carefully on a caseby-case basis (see e.g. Kovaleva et al. 2020; Anders et al. 2022) or with analysis using improved astrometry of a future Gaia data release. This resulted in a list of 7788 clusters for further analysis.", '3.3. Additional parameters and membership determination': "Cluster parameters were mostly determined following the same approach as in Paper 1. However, it was noticed that many clusters are detected with tidal tails or comas, despite this study not being initially designed to detect cluster tidal tails. This is particularly common for clusters within GLYPH<24> 2 kpc. In many cases, this can cause clusters to have strongly biased mean parameters, such as for the cluster Mamajek 4 at a distance of 444 pc. Mamajek 4 has a tidal tail that stretches for 15 GLYPH<14> or 100 pc from its core, although only one side of the tail is detected due to limitations of the size of the field it was detected in. Using a simple mean position and proper motion for such clusters is hence a GLYPH<11> ected by this asymmetry and is strongly biased. \nInstead, we aim to derive cluster parameters for the central part of clusters only. In practice, particularly for dissolving clusters with a majority of their mass in their tidal tails, it can be di GLYPH<14> cult to decide where stars should be called members of the cluster or members of the field. For instance, Tarricq et al. (2022) attempted to derive structural parameters for 467 OCs within 1.5 kpc, but their method (based on fitting King (1962) profiles) only succeeded on 389 clusters. To allow for accurate parameters to be inferred for all clusters homogeneously, we adopt a simple methodology comparing the density of cluster members with that of the field. \nFirstly, cluster members with a HDBSCAN membership probability of less than 50% were discarded. HDBSCAN membership probabilities are not based on Gaia uncertainties, but rather only on the proximity of a given member to the bulk of the cluster. It was noticed that membership probabilities lower than this limit always correspond to low-quality cluster members or members of tidal tails, and are hence not worth including in the determination of reliable parameters of clusters. \nNext, using these members, cluster centres are derived in a way insensitive to asymmetries. Kernel density estimation was used to select the modal point of the cluster stellar distribution, with a bandwidth set to 1 pc at the distance of the cluster. \nFinally, using this cluster centre, the radius at which the overall cluster has the best contrast to field stars was selected. In practice, this is similar to the King (1962) definition of tidal radius as the radius at which a cluster's density begins to exceed that of the density of the field, but is model-independent and can be easily and e GLYPH<14> ciently computed for the entire catalogue by selecting the radius at which a cluster has the highest CST against field stars. For instance, for well-defined clusters such as the Pleiades and Blanco 1, this radius was found to exclude cluster tidal tails while corresponding well with literature tidal radius values in Kharchenko et al. (2013) (see Sect. 7 for a discussion of our cluster radii.) \nMean parameters such as mean proper motion and parallax were then calculated given the members within the cluster's estimated tidal radius, in addition to maximum likelihood cluster distances calculated using the method of Cantat-Gaudin et al. (2018). To calculate more accurate distances, the parallax bias of member stars was corrected using the method in Lindegren et al. (2021b), which improved the accuracy of cluster distances particularly for distant clusters. As the Lindegren et al. (2021b) parallax correction can only be applied for certain parameter ranges, for six clusters, too few sources (or no sources) had available corrections, and so we applied a simple global o GLYPH<11> set of $ 0 = GLYPH<0> 17 GLYPH<22> as as derived in Lindegren et al. (2021b). These six clusters are flagged in the final catalogue as having less accurate distances. Overall, although the Cantat-Gaudin et al. (2018) distance method assumes that the size of clusters is negligible compared to their distance, which introduces a bias for nearby clusters, our astrometric cluster distances were nevertheless found to agree well with the literature. For instance, we derive a distance of 47 : 19 + 0 : 004 GLYPH<0> 0 : 005 pc to the Hyades, which is comparable to the 47 : 34 GLYPH<6> 0 : 21 pc distance in McArthur et al. (2011), who use Hubble Space Telescope parallaxes to a subset of Hyades member stars to derive its distance. \nIn addition, King (1962) core radii were estimated given our estimated tidal radius rt and radius containing 50% of members of the core r 50, since there exists only one solution to the number density equation in King (1962) (Eqn. 18) given n ( r 50) and rt . While approximate and less accurate than full Markov chain Monte-Carlo (MCMC) fits such as those performed in Tarricq et al. (2022), these core radii still provide a good approximation of a King (1962) model fit and compared well to literature values for well-defined clusters for which di GLYPH<11> erent works have similar membership lists. Having calculated basic astrometric parameters for our clusters, we next calculate photometric parameters for our clusters using convolutional neural networks.", '4. Photometric validation': "In this section, we use photometry to validate members of the cluster catalogue as being compatible with single-population OCs and infer basic parameters, entirely using neural networks \nTable 1. Probability distributions used for simulated clusters for training of the CMD classifier. \nNotes. Distributions of parameters are quoted as uniform distributions U ( a ; b ) between a and b , beta distributions B ( a ; b ) with parameters a and b , truncated exponential distributions T ( a ) truncated at a , R ( a ; b ; x ) which is a weighted choice with probability x of choosing value a and probability 1 GLYPH<0> x of choosing value b , and S which is a random sign with value + 1 or GLYPH<0> 1. ( a ) Distances d in kpc. \nand simulated data. While Castro-Ginard et al. (2018, 2019, 2020, 2022) successfully use neural networks to classify candidate clusters as real or false with their photometry, and while Cantat-Gaudin et al. (2020) and Kounkel et al. (2020) use neural networks to infer the ages, extinctions, and distances of their catalogued clusters, all of these works rely partially or entirely on existing examples of OCs detected in Gaia . \nWhile such an approach mitigates issues with simulated training data, namely that stellar isochrones such as Bressan et al. (2012) are typically an imperfect fit to the observed CMDs of OCs (Cantat-Gaudin et al. 2020), it is di GLYPH<14> cult to guarantee that a small training dataset that relies mostly or entirely on examples of OCs from Gaia accurately covers a full range in parameters such as absolute extinction, di GLYPH<11> erential extinction, distance, metallicity, and age. In particular, due to the di GLYPH<11> erent cuts on Gaia data used in this work, we often detect significantly more member stars for many clusters and up to two magnitudes fainter than the membership lists of Cantat-Gaudin & Anders (2020); hence, particularly for more distant OCs, our membership di GLYPH<11> erences have a significant impact on inferred parameters, making existing literature catalogues inappropriate to use as training data. Simulated data, if it can be simulated accurately enough, would o GLYPH<11> er an attractive way to quickly generate new training data applicable to new methodologies and new Gaia datasets or even other instruments, entirely based on a ground truth or 'best estimate' of how OCs should appear based on prior knowledge from stellar evolution models. Additionally, training data based on real clusters are biased towards an unknown selection e GLYPH<11> ect of how a human defines a real cluster - whereas for simulated data, we are able to exactly state the distributions we assume real OCs are drawn from, hence giving more knowledge of any selection biases this may cause. \nA key issue found in early experiments is that typical machine learning approaches are deterministic, and hence do not quantify the underlying uncertainties on their predictions. To aid with the use of simulated data, we adopt an approximate Bayesian neural network (BNN) framework using variational inference. In practice, true Bayesian machine learning is impractical to achieve with current methods; however, variational inference-based approaches o GLYPH<11> er an approximate and fast way to estimate the uncertainty of a neural network model by approximating parameters with simple probability distributions (Goan &Fookes 2020; Jospin et al. 2022), of which networks can then be sampled multiple times to produce a probability distribution \nfor their output. The BNN approach we trialed had similar accuracy to a purely deterministic one except while also outputting uncertainties, allowing us to estimate the uncertainty of our classifier. We provide a broader overview of our adopted variational inference-based approach in Appendix C. Next, we discuss the creation of training data for our CMD classifier.", '4.1. Simulated real OCs': 'A number of steps were used to generate examples of real OCs to train our CMD classifier. Basic OC generation was conducted using SPISEA (Hosek Jr et al. 2020) to simulate singlepopulation clusters from PARSEC evolution models (Marigo et al. 2017), with extinction calculated star-by-star using a Cardelli et al. (1989) extinction law with RV = 3 : 1. Stars were sampled from these isochrones with SPISEA using a Kroupa (2001) IMF. In addition, SPISEA was used to supplement simulated OC CMDs with unresolved binary stars based on general relations derived in Lu et al. (2013) for zero-age star clusters. The values in this work were found to correspond relatively well to Gaia observations, with a mass-dependent multiplicity frequency peaking at 100% for clusters of masses above 5 M GLYPH<12> . In practice, unresolved binary stars have negligible impact on the final cluster CMDs fed to the network, as typical binary sequences observed in Gaia photometry are smaller than the size of the pixels in input CMD images. SPISEA was also used to apply Gaussian-distributed di GLYPH<11> erential reddening, with values up to a standard deviation of 0.6 in the highest cases, reflecting the most extreme examples of di GLYPH<11> erentially reddened reliable clusters found in Cantat-Gaudin & Anders (2020). \nNext, a random location on the galactic disk was selected for each cluster, which was used to simulate a realistic selection function and photometric errors. The magnitude-dependent selection function of Gaia DR3 at each given location was queried using the selectionfunctions package presented in Boubert &Everall (2020) and Boubert et al. (2020), which gives the basic probability that a source appears in Gaia as a function of position and G-band magnitude. We use the online version of their package updated for Gaia DR3. The selectionfunctions package is based on the dustmaps package from Green (2018). In addition, the selection function of every cluster was also corrected for the cuts to Gaia data applied in Sect. 2.2. During the preparation of this work, Cantat-Gaudin et al. (2023) released a new selection function for Gaia DR3 which suggested that the earlier work of Boubert & Everall (2020); Boubert et al. (2020) can be over-confident at the faint end; however, given that our cluster membership lists are overwhelmingly dominated by the selection function of our cuts on Gaia data at magnitudes G > 18, and not the pure selection function of Gaia , we found that it made too small of a di GLYPH<11> erence to our simulated clusters to be worth updating our training data for, although we will adopt their work in future works. Realistic photometric uncertainties were added to sources based on the distribution of source uncertainties at the selected location, which are generally larger in crowded fields. We added systematic o GLYPH<11> sets in simulated BP and RP Gaia photometry for faint sources using relations in Riello et al. (2021). \nOutliers were not added to simulated cluster CMDs, as most clusters are already detected with very few or no outliers; instead, we wish the CMD classifier to quantify the evidence for a cluster being real based on its photometry alone, which photometric outliers inherently reduce. In this way, CMDs of clusters with a high number of outliers are scored more negatively by the network as they have less photometric evidence supporting them being real. Blue stragglers were also not added to cluster CMDs \nas they are indistinguishable from photometric outliers, although in practice, real OCs with blue straggler stars were not found to be scored significantly lower by the trained network. \n10 000 examples of simulated real clusters were generated to use as one half of the simulated cluster dataset. Distributions of parameters such as age log t , extinction AV , di GLYPH<11> erential extinction GLYPH<1> AV and distance modulus m GLYPH<0> M were carefully chosen after many iterations to minimise systematics deriving from the overall distribution of training data in the dataset, while ensuring that the CMD classifier was trained on a representative set of simulated real OCs. Fundamentally, the objective of the training data are not to match the real distribution of OCs, but rather to yield an unbiased and representative sample of OCs to train the BNN on, such that the BNN can provide an unbiased classification of any object. For instance, while a distribution of the number of visible stars n based on the distribution of stars in Cantat-Gaudin & Anders (2020) (corrected for our deeper magnitude limit) was found to work well to produce an unbiased classifier, in other cases, such as for log t and m GLYPH<0> M , the use of a uniform distribution (instead of one based on the expected distribution of clusters) was essential to avoid biasing the classifier towards certain ages or distances. These distributions are listed in Table 1.', '4.2. Simulated fake OCs': "A number of methods to simulate fake OCs reminiscent of false positives sometimes reported by HDBSCAN were trialed. As a clustering algorithm, the member stars of each cluster reported by the algorithm are spatially correlated, with a similar position, proper motion, and parallax. Hence, it is important that false positives contain member stars with similar astrometric parameters. Simply randomly selecting stars from Gaia data to construct each false positive was found to result in clusters that were too pessimistic. \nInstead, to generate false positives with spatially correlated member stars, a star was first selected randomly from the entire Gaia dataset as an origin point. This ensures inherently that false positives are more likely to occur in the densest regions of the Gaia dataset, which was a behaviour observed inherently for HDBSCAN in Paper 1. A total number of stars for the cluster was selected from the same distribution as used for simulated real OCs. Then, a 5D hypersphere in position, proper motion, and parallax was expanded randomly around this star until the hypersphere contained the required number of stars. In this way, false positives with spatially correlated member stars were generated. Actual OCs make up a small enough portion of the Gaia dataset - 610 000 in the final version of the catalogue, or fewer than 0.1% - that it was not found to be necessary to first remove them from data used to generate false positives. This is similar to the false positive generation method used in Castro-Ginard et al. (2022). \n10 000 false positives were generated using this methodology to provide the other half of the training dataset. While most false positives have obviously poor quality CMDs, false positives generated from regions of field stars with roughly homogeneous ages and composition (such as from the galactic halo) often had more homogeneous CMDs, that could be compatible with highly di GLYPH<11> erentially reddened OCs. However, this is a useful property of the training dataset, given the variational inference approach used in the network: this 'overlap' between highly di GLYPH<11> erentially reddened true positives and chance alignments of somewhat-similar field stars reflects on the real distributions of field stars in the galactic disk. Real Gaia cluster candidates with \nTable 2. Human classifier performance. \nNotes. Results of human classification when applied to a test dataset of 2000 clusters detected by HDBSCAN in this work as well as two datasets of simulated real and fake clusters. \nworse-quality CMDs making them compatible with both a real OC or a chance clustering of field stars hence have broad or bimodal PDFs from the BNN CMD classifier, reflecting how photometry alone o GLYPH<11> ers only poor evidence of whether or not these objects are real or fake star clusters.", '4.3. Test dataset': "In order to test the trained networks against real Gaia data and ensure that they can be generalised from their training on simulated data to use on real data, a test dataset of 2000 clusters randomly selected from the initial HDBSCAN clustering was selected and classified by hand, in addition to 250 simulated real clusters and 250 simulated fake ones to estimate the accuracy of human classification. These di GLYPH<11> erent datasets were classified in one classification run to avoid biasing the human classifier. Clusters were classified into 'true positive' (TP) and 'false positive' (FP) categories, in addition to two other categories for clusters that are most likely to be true or false clusters but are somewhat uncertain (abbreviated as 'TP?' or 'FP?'), due to the presence of outliers, a small number of stars, or very high di GLYPH<11> erential reddening that is compatible with both an association of field stars or a highly di GLYPH<11> erentially reddened OC. The results of this classification are shown in Table 2. \nOf clusters reported by HDBSCAN, 53.6% were handclassified as being highly likely to be real, with a further 26.5% being potentially real, suggesting that most clusters we detect have a reliable CMD. Only 8.9% were highly unlikely to be real with a further 11.0% classified as probably not real, suggesting that around 80% of clusters reported by HDBSCAN are likely to have single stellar populations based on human classifications. \nIn testing the human classifier, 92.0% of simulated real clusters were correctly classified as real or potentially real, although only 59.2% of simulated fake clusters were classified as false or potentially false. 14.0% of simulated fake clusters were in fact classified as highly likely to be real. This shows the inherent limitations of using photometry to validate OCs, as spatially correlated groups of field stars can often have somewhathomogeneous CMDs when all field stars in a given region have a similar age and chemistry (see Sect. 4.2), which can even fool a human classifier. This is particularly common in the halo and thick disk where most stars have a similar, old age. This is an important limitation of the human-classified test data to bear in mind, as a small fraction of clusters classified by hand as true positives will always in fact be false positives. Nevertheless, CMD classification is still a necessary validation tool to help ensure that detected cluster candidates are reliable, as many of the worst quality clusters can still be removed with this method. \nFig. 3. Performance of the CMD classifier on the independent test dataset of 2000 clusters detected by HDBSCAN in Gaia data and labelled by hand. Clusters are labelled as true positives or false positives, with clusters where the human classifier was less certain being additionally flagged. \n<!-- image -->", '4.4. Network training and validation': 'The 20 000 simulated real and fake OCs were split randomly into a training set of 16 000 clusters and a validation dataset of 4 000 clusters to assess network overfitting. As the simulated fake OCs have a di GLYPH<11> erent distribution of distance moduli to the simulated real OCs, fake OCs at undersampled and oversampled distances were weighted to be emphasised more or less strongly during training, preventing systematics due to di GLYPH<11> erences in distance distributions. \nWe used the implementations of neural networks and probabilistic layers in TensorFlow (Abadi et al. 2015, 2016) and TensorFlow Probability (Dillon et al. 2017) for all networks used in this work. Networks were trained with the Adam optimisation algorithm (Kingma & Ba 2017). A number of di GLYPH<11> erent neural network structures were trialed. Convolutional neural networks (CNNs), which convolve two-dimensional input with learnt filters, were found to perform ideally for the problem at hand, and have seen extensive use in the astronomical literature (e.g. Castro-Ginard et al. 2022; Becker et al. 2021; Killestein et al. 2021). \nAs input, the optimal network trialed used cluster CMDs converted to absolute magnitudes, with stars of absolute G magnitudes greater than 10 or lower than GLYPH<0> 2 cut away. Generally, this cuts certain very low mass M stars and bright O stars from cluster CMDs, which were found to be poorly simulated by PARSEC isochrones with their inclusion only worsening network performance on real data. In practice, very few stars are cut due to this limitation, with O stars making up only a very small proportion of sources in young clusters and M dwarfs fainter than MG = 10 only being brighter than G = 20 for clusters within 1 kpc, at which point the rest of the cluster CMD can be resolved well. In addition, BP GLYPH<0> RP colours were cut between -0.4 to 4, which in practice is a wide enough colour range to include almost all sources but while providing a good range to discretise cluster CMDs between. Sources with very low BP and RP fluxes that have overestimated BP or RP magnitudes were removed using cuts from Riello et al. (2021), as these also only confused the network, despite these systematics being simulated in the training data. Finally, in terms of structure, the optimal net- \nwork trialed was trained on CMDs discretised into 32 GLYPH<2> 32 pixel images, corresponding to pixels of size 0 : 38 GLYPH<2> 0 : 11 mag. These images were first processed by three convolutional layers with 5 GLYPH<2> 5 pixel kernels of 6, 16, and 120 filters respectively. Max pooling layers were placed between these convolutional layers to speed up training and inference. Convolution layer output was connected to a single densely connected layer of 128 nodes, with a final single node for output. The distance modulus of the cluster based on the parallax-derived cluster distances was also fed to the network as an auxiliary input into the 128 node dense layer, in a similar way to the network of Cantat-Gaudin et al. (2020) which also uses both photometric and astrometric input simultaneously. All layers used Rectified Linear Unit (ReLU) activation other than a sigmoid activation function applied to the final output to constrain network output in the range [0 ; 1] as a probability distribution. \nThe final network had binary accuracies (the percentage of clusters given the correct true or false label) of 95% for both training and validation data, indicating that the network did not overfit to training samples when compared with other simulated data. Fig. 3 shows the performance of the network compared to the human-labelled test dataset of real clusters detected by HDBSCAN in Gaia after sampling the network 1000 times to generate PDFs for every object, with 85.5% of clusters labelled highly likely to be real and 91.3% of clusters labelled highly unlikely to be real having a median predicted probability greater or less than 0.5 respectively. Clusters where the human classifier was less certain have a much broader distribution, although this also reflects inherent uncertainties in the test dataset discussed in Sect. 4.3. Finally, only 4.3% and 2.5% of highly likely real and highly likely false clusters had predicted labels that disagree with human labels at more than the 2 GLYPH<27> level - namely, that 97.5% of their PDF is below or above 0.5 respectively. It is important to recall that these quantities merely validate the general agreement between two independent classifiers (the human classifier and the automated CMD classifier) on the same dataset, and do not exactly measure the ground truth sensitivity or accuracy of the CMDclassifier, as the human class labels themself are uncertain Sect. 4.3. Instead, these data show that the CMD classifier can perform comparably well to human classification, except with the added bonuses of speed and reproducibility. \nFig. 4 shows CMD classifier PDFs for four clusters from all human classes, including the names of any clusters that crossmatched to real objects. In general, CMD classifier predictions generally agreed well with the human-assigned labels, also generally with higher uncertainty and a broader PDF in cases where the human classifier was less certain. For clusters with clear, high-quality CMDs such as UPK 282, the CMD classifier outputs PDFs that strongly suggest they are real. Teutsch 110 is a less well-defined cluster that, if real, must have di GLYPH<11> erential reddening and a few outliers, and is hence not classified as strongly. The candidate new cluster shown is a similar case albeit with a worse CMD, making it relatively unlikely to be real given this HDBSCAN detection. Finally, Theia 4560 is visible as a large and statistically significant overdensity in Gaia data as detected by Kounkel et al. (2020), although the overdensity as detected in this work does not appear to contain a homogeneous population of stars and is hence classified weakly. CMD classifier median probabilities and confidence intervals for all clusters are listed in Table 4, based on 1000 samples of the network for each cluster. \nFig. 4. Four examples of classified cluster CMDs from the test dataset, with cluster CMDs on the top row and their PDFs of predicted probabilities on the bottom row. Cluster names and human-assigned labels are indicated on the figures. PDFs are generated by sampling the CMD classifier 1000 times for every cluster. \n<!-- image -->', '5.1. CMD classifier modifications': "While not a main focus of this work, we also show that the approach based on simulated data and an approximate BNN using variational inference is also applicable for age log t , extinction AV , di GLYPH<11> erential extinction GLYPH<1> AV and distance modulus m GLYPH<0> M inference. Recently, Cantat-Gaudin et al. (2020) use a neural network to infer log t , AV and m GLYPH<0> M for around 2000 OCs. In their work, a training dataset based on simulated OCs alone is not found to be su GLYPH<14> ciently accurate to train a neural network. While simulated data were found to be accurate enough for the CMD classifier in Sect. 4, parameter inference is more challenging, as a network must learn to infer multiple parameters from a CMD alone and generalise this accurately to real data. However, our approach has a number of di GLYPH<11> erences to theirs: firstly, we use a convolutional neural network, which may be better able to capture structure in CMDs due to its 2D approach, which may also reduce training data overfitting; secondly, our network is approximately Bayesian, and includes uncertainty estimates that quantify when it may have failed; finally, although Cantat-Gaudin et al. (2020) do not elaborate on how they simulate clusters in their work, our methodology is be di GLYPH<11> erent and may produce di GLYPH<11> erent results. Hence, despite recent literature suggesting that using purely simulated data is not possible for parameter inference with CMDs, it is still worth attempting, as training on simulated data is attractive for reasons discussed in Sect. 4. \nTo create a parameter inference network, we used a similar network structure to that of Sect. 4.4, except with some tweaks to the network output to infer parameters. To better predict the aleatoric uncertainty of network output for this multipleparameter network, network output was changed to a beta distribution for each parameter. These distributions can take any shape \nfrom a uniform (completely uncertain) distribution to a single point-like estimate. The output was then scaled to be within the minimum and maximum ranges of the training data. To train the network, 50 000 simulated clusters were created using the same methodology as in Sect. 4.1, changing the distribution of cluster extinctions AV (as defined in Table 1) to simply be uniform between 0 and 7. \nIn initial comparisons with literature results, di GLYPH<11> erential reddening was found to strongly correlate with disagreements in extinction (and to a lesser extent, age) between this work and others. A primary cause of this is that while many works (e.g. Cantat-Gaudin et al. 2020; Bossini et al. 2019) use the so-called 'blue edge' of a CMD for isochrone fitting, meaning that GLYPH<1> AV is only positive. This contrasts to SPISEA's default GLYPH<1> AV model, which is Gaussian - with cluster stars having both positive and negative GLYPH<1> AV values. \nHowever, changing SPISEA's GLYPH<1> AV model to also only be positive (and hence defining GLYPH<1> AV in terms of the blue edge of cluster CMDs) was not found to be helpful. Owing to HDBSCAN's high sensitivity, we detect a higher number of stars outside of the core of clusters than in the membership lists of Cantat-Gaudin & Anders (2020), which are constructed with the UPMASK algorithm (Krone-Martins & Moitinho 2014) and for many clusters only select stars in the core. This means that our CMDs are constructed from clusters with significantly larger angular extents on the sky and are hence often more strongly differentially reddened than in Cantat-Gaudin & Anders (2020), with many clusters having a blue edge at an extinction value up to 1 magnitude lower than in Cantat-Gaudin & Anders (2020). For instance, NGC 884 is an example of this, with our membership list being larger and more strongly di GLYPH<11> erentially reddened. A blue-edge based definition of AV means that di GLYPH<11> erent works \nFig. 5. Photometric parameters derived in this work compared against test datasets. Top row : 2D histograms showing the performance of the trained photometric parameter inference network on all 10 000 clusters from the validation dataset. The mean output uncertainty is shown with white error bars. As indicated by the dashed lines, predicted values on the y axis should be equal to true values on the x axis. The root mean square error (RMSE) and mean absolute error in terms of output network uncertainty (MAE) are given in the top left. All plots and the RMSE are in units of magnitude other than on age plots which are logarithms of cluster age in years. Other rows : comparison between network predicted parameters and ages, extinctions, and distance moduli for 247, 1753, and 1206 clusters in common with the catalogues of Bossini et al. (2019), Cantat-Gaudin et al. (2020), and Kharchenko et al. (2013) respectively. Points are shaded based on the di GLYPH<11> erential extinction we infer for each cluster. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nLiterature log \nt \nValidation data \n<!-- image --> \n<!-- image --> \n- \nCantat-Gaudin et al. 2020 \n- \n<!-- image --> \nV \n<!-- image --> \n<!-- image --> \nV \n<!-- image --> \n- \n- \n- \n3 \n. \n0 \n2 \n. \n5 \n2 \n. \n0 \n1 \n. \n5 \n1 \n. \n0 \n<!-- image --> \n- \nproduce di GLYPH<11> erent values of AV depending on how sensitive their membership recovery process is. \nInstead, we continue using the default SPISEA GLYPH<1> AV definition centred on the mean cluster AV , but while also using the network to infer GLYPH<1> AV for every cluster, which can then be used \nas a correction to convert between extinctions in this work and others that use a blue-edge definition. In practice, GLYPH<1> AV is very di GLYPH<14> cult to measure, as it is degenerate with other e GLYPH<11> ects that broaden cluster CMDs, including unresolved binary stars and outliers. Against validation and test data, our median GLYPH<1> AV values \n- \nV \nA \n∆ \nFig. 6. Extinction values from Cantat-Gaudin et al. (2020) compared against this work when corrected for di GLYPH<11> erential extinction with an estimate of cluster di GLYPH<11> erential extinction, plotted in the same style as Fig. 5. The dashed black line shows where y values equal x ones; the dashed grey line shows the same but o GLYPH<11> set by -0.4. \n<!-- image --> \nare found to be o GLYPH<11> set by around 0.4 due to unresolved binaries. Nevertheless, this parameter is helpful to aid comparisons with literature works. \nFinally, we also updated our GLYPH<1> AV model from the Gaussian default model in SPISEA to instead use the di GLYPH<11> erential reddening as would be expected from stars sampled from a King profile (King 1962), assuming a first order (linear) gradient in di GLYPH<11> erential extinction across a cluster. This model is narrower than the Gaussian model while retaining highly di GLYPH<11> erentially reddened stars (which would be at the outskirts of a cluster), and was found to slightly improve GLYPH<1> AV inference. This model depends on two parameters: the total di GLYPH<11> erential extinction across a cluster, which was matched to have the same range as the previous Gaussian model at a 3 GLYPH<27> level; and the ratio between core and tidal radius, which was set to the median value for open clusters from Kharchenko et al. (2013). \nAgainst our validation dataset of 10 000 simulated clusters, the network performs well with no clear systematics in log t , AV or m GLYPH<0> M . However, owing to the degeneracy between GLYPH<1> AV and other e GLYPH<11> ects such as unresolved binary stars, outliers, and photometric uncertainties, values of GLYPH<1> AV smaller than 0.4 are not typically correctly predicted, although the true value is typically still within 1 GLYPH<27> uncertainty of the predicted value. These results are plotted on the top row of Fig. 5. \nUsing the best trained network after a number of experiments, all clusters in our catalogue closer than a maximum distance of 15 kpc have ages, extinctions, di GLYPH<11> erential extinctions, and distance moduli listed in Table 4. These parameters are based on 1000 samples of the network for each cluster.", '5.2. Comparison with other works': 'We briefly compare our photometric parameters to other works in the literature. Firstly, Fig. 7 shows example predicted isochrones for four OCs in this work. In the first case, NGC 2910 is a cluster with a well-behaved isochrone where all works agree \nrelatively well. On the other hand, Ha GLYPH<11> ner 14 shows relatively strong di GLYPH<11> erential reddening, and di GLYPH<11> erent definitions of di GLYPH<11> erential reddening between di GLYPH<11> erent works cause isochrone fits to disagree. Berkeley 15 is a sparse cluster where both di GLYPH<11> erential reddening and field star outliers a GLYPH<11> ect di GLYPH<11> erent works in di GLYPH<11> erent ways, with our updated Gaia DR3 membership list having fewer outliers than that of Cantat-Gaudin et al. (2018). Ruprecht 147 is a nearby and particularly old cluster ( GLYPH<24> 1 Gyr), where blue straggler stars systematically a GLYPH<11> ected our network and caused an incorrect younger age value to be predicted for this cluster. It is clear from these plots that for all but the most well-behaved OCs, di GLYPH<11> erent works can have di GLYPH<11> erent photometric parameters. \nFig. 5 compares all network predictions with values from four test datasets. An advantage of our simulated training approach is that network predictions can now be compared to other literature works, which act as independent test datasets which can verify the accuracy of our network. It is important to note that our results never agree perfectly, however, particularly since all works we compare to are based on Gaia DR2 or preGaia OCmembership lists that may be significantly less clean or have significantly fewer stars than our Gaia DR3 membership lists. \nBossini et al. (2019) provide a catalogue of precise OC parameters from Bayesian isochrone fitting using the BASE-9 algorithm (von Hippel et al. 2006). A key di GLYPH<11> erence is that their work uses metallicity estimates from the literature where available, whereas our approach is based entirely on Gaia DR3 parameters and assumes a given cluster can have any metallicity as drawn from a broad probability distribution based on literature values (Table 1). Nevertheless, our results still agree well with theirs in log t , AV and m GLYPH<0> M . In cases where our log t estimates disagree most strongly, this is typically due to di GLYPH<11> erences in OC membership list. There is however a possible minor systematic between our two works for OCs with extinctions below 0.6, many of which we infer smaller extinctions for than them; this may be as a result of AV vs. metallicity degeneracies. However, their values are typically only 1 to 2 GLYPH<27> from ours. \nOur parameters agree less strongly with the results of CantatGaudin et al. (2020), which are derived from a neural network trained on isochrone fits from a variety of works (including Bossini et al. 2019). This is to be expected to some extent, as while Bossini et al. (2019) only fit isochrones to a subset of OCs with clean membership lists and the least di GLYPH<11> erential reddening, Cantat-Gaudin et al. (2020) fit isochrones to all known OCs at the time, including many sparse objects which may now have significantly di GLYPH<11> erent membership lists in our current Gaia DR3 work. However, some di GLYPH<11> erences persist. A clear systematic in our and their AV values is clear, although this is likely due to their di GLYPH<11> erent blue edge definition of extinction (whereas our network fits to the mean extinction in a cluster.) Figure 6 shows a crude conversion between our AV values and their blue-edge AV values. While this removes the systematic di GLYPH<11> erence in gradient, our converted AV values are still generally smaller than theirs by around 0.4 to 0.5 on average. This is likely due to two e GLYPH<11> ects; firstly, as shown by the results on validation data, GLYPH<1> AV is generally overestimated for our validation data by around GLYPH<24> 0 : 4 due to degeneracies with unresolved binary stars, outlier non-member stars, and photometric uncertainties, which may explain some of this discrepancy, particularly for clusters with lower GLYPH<1> AV values. Secondly, our membership lists generally cover a wider extent on the sky than those used in Cantat-Gaudin et al. (2020), meaning that our clusters are often larger and hence are more extremely di GLYPH<11> erentially reddened between separate sides of the cluster; hence, a conversion between the works based on our GLYPH<1> AV values is likely to frequently over-correct for the di GLYPH<11> erence \nFig. 7. Predicted cluster isochrones from this work (solid blue line) compared with those from other works. Cluster members are plotted in black and shaded according to their membership probability. \n<!-- image --> \nin AV definition. Finally, some of our ages for the oldest clusters (log t > 9) appear systematically younger, on average by around 2 GLYPH<27> ; in some cases, this may be due to our fits being disrupted by blue straggler stars (Fig. 7, see Ruprecht 147.) The training data we use for our photometric parameter inference are adapted from our CMD classifier in Sect. 4, for which blue straggler stars were not found to have a negative impact on the accuracy of our network and were hence not included. Future works using purely simulated data to train a photometric parameter inference neural network would benefit from inclusion of blue straggler stars in their training data, although in practice the origin of blue stragglers is still disputed, and these stars may hence be challenging to simulate accurate photometry for (Bo GLYPH<14> n et al. 2015; CantatGaudin 2022). \nFinally, our results have limited agreement with those of Kharchenko et al. (2013). While some clusters have similar values between their work and ours, particularly for AV and particularly for the largest and most clearly defined clusters (Fig. 7), many sparse clusters that were di GLYPH<14> cult to detect before Gaia have very di GLYPH<11> erent photometric parameters. This typically appears to be caused by extremely di GLYPH<11> erent cluster membership lists. Before Gaia , OCs were often challenging to separate from field stars (Cantat-Gaudin 2022), requiring that suspected outliers be removed iteratively to improve CMD quality (Kharchenko et al. 2012). However, this process can also remove true cluster members, which can cause resulting cluster membership lists to be incorrect (Cantat-Gaudin & Anders 2020). This discrepancy with the results of Kharchenko et al. (2013) is also reported by Cantat-Gaudin et al. (2020), who also find that many photometric parameters derived before Gaia are strongly discrepant with current results. In addition, while the number of member stars reported in Kharchenko et al. (2013) is generally a poor predictor for whether or not a given cluster in their work has very di GLYPH<11> erent parameters to ours, there are some cases (such as clusters in their work with AV > 5 that we derive much smaller values for) where the most discrepant clusters were also the smallest, with fewer than 20 member stars in reported in Kharchenko et al. (2013). \nAlthough approximate, these results still agree well within the sample-limited but accurate Bayesian isochrone fits of Bossini et al. (2019) and agree relatively well (albeit with some caveats) with the machine learning derived parameters of CantatGaudin & Anders (2020). This work o GLYPH<11> ers a large and homogeneously derived catalogue of photometric parameters with su GLYPH<14> - \ncient accuracy for basic analysis. In the next section, we use the ages and extinctions we derived here to aid with discussion of our cluster sample.', '6.1. Crossmatch strategy': "Before conducting further analysis on the cluster catalogue, such as restricting it to only clusters with reliable colour-magnitude diagrams or removing moving groups, it is helpful to crossmatch our results to literature catalogues to allow for easier comparisons between derived parameters and other works. In particular, this makes it possible to compare whether clusters reported in other works are compatible with real open clusters given further parameters derived in Sect. 4 and the third paper in this series, Hunt & Re GLYPH<11> ert, in prep. , where we will derive dynamical parameters for our census of star clusters. \nIn Paper 1, we crossmatched by assigning matches to clusters when their mean positions were compatible to within their tidal radii and when their mean proper motions and parallaxes were compatible within five standard errors. In initial testing, the crossmatch strategy of Paper 1 was found to be insu GLYPH<14> cient for two reasons when comparing between Gaia DR3 astrometry and Gaia DR2 astrometry, in addition to a further issue with the positional strategy used. \nFirstly, the standard errors on mean proper motions and parallaxes in Gaia DR2 can be as small as 5 to 10 GLYPH<22> as for the largest clusters in catalogues such as Cantat-Gaudin & Anders (2020), although this is smaller than estimated upper limits on systematics in Gaia DR2 of 50 GLYPH<22> as (Lindegren et al. 2018). Many reliable clusters are hence missed when treating DR2 positions exactly, as they have systematics significantly larger than their standard errors, with positions in DR3 that can deviate systematically from their DR2 positions by 50 GLYPH<22> as or more. \nSecondly, membership lists can di GLYPH<11> er between works and can be significantly di GLYPH<11> erent for the same cluster - for instance, works such as Castro-Ginard et al. (2020) only used stars down to G = 17, whereas this work often has membership lists down to G GLYPH<24> 20. Many clusters hence have significantly di GLYPH<11> erent membership lists that can result in di GLYPH<11> erent mean parameters, particularly for asymmetric clusters. \nOur positional crossmatch strategy was also revised and improved. Paper 1 used a conservative strategy for matching on position, which assumed that a cluster is a positional match if the \ncentre of the literature cluster is closer than either the Paper 1 or literature radius for a given cluster. However, in practice, this strategy appears almost always too conservative, as many distant, compact clusters reported in catalogues such as Froebrich et al. (2007) would match to large, nearby clusters that happen to contain the distant object within one radius, despite the cluster centres being strongly incompatible given the smaller (literature) radius. \nTo improve positional crossmatching, we instead define a positional match to require that the centre of the literature cluster is closer than both the current and literature radius, which in almost all cases still recovers reliable matches but while not erroneously matching to compact, distant objects with significantly di GLYPH<11> erent sizes and cluster centres. Then, for catalogues with Gaia astrometry available, we also match on proper motions and parallaxes, requiring that the new mean proper motion and parallax are within two standard deviations of the literature value (with both current and literature standard deviations summed in quadrature.) This approach with standard deviations matches clusters if a new cluster is within allowed ranges of the dispersion of the current and literature entries, with the principles that exact statistical matching based on standard errors is not possible as unknown systematic errors dominate, and that a cluster within the dispersion of a literature entry is likely to be the same object. Using a higher maximum value of the dispersion was not found to significantly increase the number of literature clusters recovered by more than 1%, but while adding many false crossmatches to other nearby objects that greatly worsen the reliability of the overall crossmatching process. \nSome special cases are also worth mentioning: the catalogue of Kharchenko et al. (2013) is based on PPMXL proper motions and distances from isochrone fitting by hand, which are generally significantly less accurate than Gaia astrometry. Hence, we crossmatch to Kharchenko et al. (2013) with both a position-only and a second positions, proper motions, and distances crossmatch which can more strongly confirm the most reliable matches. Some catalogues list only a radius containing 50% of members for entries (e.g. Cantat-Gaudin & Anders 2020); for these catalogues, we use twice this radius to approximate the total size of the cluster. Other works (e.g. Castro-Ginard et al. 2020; He et al. 2022a) list only standard deviations of the mean position; for these catalogues, we use twice the geometric mean of this standard deviation on position to approximate the total size of the cluster. Finally, Kounkel et al. (2020) does not list uncertainties or dispersions on mean parameters, and so these were manually recalculated with our own pipeline using their lists of members. \nAfter an extensive search of the literature for recent catalogues, excluding works already listed entirely in other catalogues (such as Froebrich et al. (2007), which appears in its complete form within Bica et al. (2018)), we crossmatch against 26 di GLYPH<11> erent works listed in Table 3. In addition, as our catalogue contains many moving groups, globular clusters, and a handful of clusters associated with the Magellanic clouds, we also crossmatch against the Kounkel et al. (2020) catalogue of predominantly moving groups, the Vasiliev & Baumgardt (2021) Gaia DR3 catalogue of globular clusters and the Bica et al. (2008) catalogue of star clusters in the Magellanic clouds. Names between catalogues were standardised as much as possible to facilitate easier comparison and remove duplicated clusters. One such example are ESO clusters, which are numbered based on their position in the form 'ESO XXX-XX' in the original work and Kharchenko et al. (2013), but with numbers that are separated by a space instead of a dash in Cantat-Gaudin & Anders \nTable 3. Results of crossmatching against literature catalogues sorted by n clusters. \nNotes. 32 catalogues of OCs are listed in the first section of the table, in addition to three catalogues at the bottom of other star clusters. ( a ) Original work and this work uses the acronym 'FoF' to name clusters, although others list with acronym 'LP'. ( b ) Cluster(s) in these works were unnamed, and so cluster acronyms were adopted based on first letters of surnames of authors. ( c ) Catalogue of predominantly moving groups, although many are also open clusters. ( d ) Position-only catalogue of objects in the Magellanic clouds. ( e ) Catalogue of globular clusters. \n(2020) and Dias et al. (2002), or often miss leading zeroes in Bica et al. (2018).", '6.2. Recovery of clusters from prior works': 'Table 3 shows that this work has a high recovery rate of OCs from other works. As shown in Table 3, we recover 96.6% of clusters from Cantat-Gaudin & Anders (2020), higher than the 86.4% of clusters recovered in Paper 1. Generally, clusters not recovered in Paper 1 were sparse, barely-visible overdensities in Gaia DR2 which often now stand out strongly in Gaia DR3, including clusters such as Berkeley 91 and Auner 1, which we now detect reliably at S / Ns of 9.7 GLYPH<27> and 12.5 GLYPH<27> respectively. The fact that only Cantat-Gaudin & Anders (2020) was able to detect these clusters in DR2 is likely due to a di GLYPH<11> erence in methodology \n- by starting with prior cluster positions, their search regions for these clusters are smaller and may help the clusters to stand out. However, the disadvantage of such an approach is that it may also introduce a handful of false positives, due to poor statistics inherent in such small search regions - in Paper 1, we comment that a handful of clusters in Cantat-Gaudin et al. (2020) may not exist, which may be the case for some of the 3.4% of clusters we are still not able to recover in Gaia DR3 despite the greatly improved astrometry and clear benefits to the S / N of other previously undetected clusters. \nWe recover most of the new clusters reported in CastroGinard et al. (2020) (a work based on Gaia DR2) and CastroGinard et al. (2022) (a work based on Gaia EDR3), recovering almost exactly 89% of both catalogues, showing that a majority of these objects can be confirmed independently. The reason for the non-recovery of around 11% of clusters in both cases is not clear, although the fact that this amount is similar between both clusters detected with Gaia DR2 and EDR3 suggests that it is a fundamental methodological di GLYPH<11> erence (their works use the DBSCAN algorithm, see Paper 1 for a review) rather than a data one. \nHowever, we recover fewer of the new clusters reported by other DBSCAN-based works such as Hao et al. (2020, 2022) and He et al. (2021, 2022c,a,b), recovering fewer than 50% of the clusters reported in He et al. (2022c,b) using Gaia EDR3 data. \nAdditionally, while a large fraction of clusters reported before Gaia and catalogued in works such as Dias et al. (2002), Kharchenko et al. (2013), and Bica et al. (2018) still do not appear in Gaia DR3, we are able to reliably detect an additional 277 clusters from Dias et al. (2002), 292 clusters from Kharchenko et al. (2013), and 127 clusters from Bica et al. (2018) that do not appear in the Gaia DR2 catalogue of CantatGaudin & Anders (2020) (excluding GCs in all cases, as the catalogue of Cantat-Gaudin & Anders (2020) does not contain them.) \nNotably, we are unable to detect any of the high galactic latitude OCs that have been reported recently in Li & Mao (2023), despite the fact that OCs at such high latitudes should stand out clearly against the low number of field stars in the galactic halo. This echoes the results of Cantat-Gaudin et al. (2018) and Cantat-Gaudin & Anders (2020), who also find that high latitude OCs that have been reported in works such as Schmeja et al. (2014) are undetectable in Gaia data. \nWe discuss possible reasons for the non-detection of many literature OCs further in Sect. 8. \nFinally, it is worth commenting on our detections of moving groups, globular clusters, and Magellanic cloud objects. We are only able to detect 18.1% of moving groups and clusters from the catalogue of Kounkel et al. (2020), despite this work using the same algorithm (HDBSCAN). Many of the groups reported in Kounkel et al. (2020) have large on-sky extents that are larger than the fields used in this work. However, although 2276 of their 8281 clusters are compact enough to be easily detectable in our fields, we only recover 622 (27.3%) of these compact groups, many of which correspond anyway to known nearby OCs. In Paper 1, we found that while HDBSCAN is the most sensitive clustering algorithm for application to Gaia data, it also reports a large number of false positives without additional postprocessing to remove clusters based on their statistical significance. It may be that these clusters are false positives, although this should be investigated further in detail (see e.g. Zucker et al. 2022). \nThe recovery of a large fraction of GCs in Vasiliev & Baumgardt (2021) shows that HDBSCAN can be used to e GLYPH<11> ectively recover GCs. The non-recovered objects are mostly distant and \nFig. 8. Member stars for the candidate new cluster HSC 2384 (red squares) compared against the nearby cluster IC 2602 (black circles). Four plots of are shown, comparing positions (top left), proper motions (top right) and photometry (bottom right). The bottom left plot shows a histogram of all distances to individual member stars. \n<!-- image --> \n- \nheavily reddened GCs whose member stars can only be recovered with a prior position and distance to narrow the search region. Finally, while not a focus of this work, the recovery of 22 Magellanic cloud star clusters from Bica et al. (2008) shows that Gaia data could be used to make limited inferences on existing Magellanic cloud clusters in a future work, although we do not appear to detect any new clusters in the Magellanic clouds as their distance is too high.', '6.3. Assignment of names': 'As many of the objects we detect crossmatch to multiple entries in the literature (or vice-versa), assigning detected clusters to literature names can be non-trivial. A total of 7022 literature clusters crossmatch to 4944 of the entries in our catalogue, of which only 2749 matches are direct one-to-one matches where a single detected cluster can be easily assigned a single name. \n1396 detected clusters each match to multiple literature entries. In these cases, the main cluster name was assigned based on the date of submission to a journal, with other names recorded in a separate column of alternative names for this object. \nIn 64 cases, multiple detected clusters crossmatched to the same literature object. The best match was selected based on position (or proper motions and distances, if available), with other objects instead recorded as new clusters. \nFinally, there were 265 groups of crossmatches where multiple detected clusters crossmatched to multiple literature clusters, where assigning one match a GLYPH<11> ects other matches. This is common in regions where many clusters are in a small area, such as in star formation regions like the Carina nebula. For simplicity, and since many of these groups contain literature entries with only positions available, we assign the best match on cluster positions only, iterating over all matches within a group accepting the match with the smallest positional separation and then removing all other literature entries with the same name within \nTable 4. Mean parameters for the clusters detected in this study. \nNotes. Standard errors for mean proper motions and parallaxes are shown in the brackets. The full version of this table with 7167 rows and many extra columns is available at the CDS only, with a complete description of the included additional data in Appendix A. ( a ) Internal designation used to link final catalogue entries to their crossmatching results in Table B.1.', 'Comparison with other catalogues': "<!-- image --> \nFig. 9. Distance and spatial distributions of clusters in this work. Left : the distance distribution of all clusters in this work that do not crossmatch to known GCs compared to other catalogues. Right : The distribution of clusters in this work in Cartesian coordinates centred on the Sun, cut to only those within 5 kpc in the X or Y directions. All previously reported clusters that we redetect are shown as blue triangles, and all objects new in this work shown as orange circles. \n<!-- image --> \nthis group. All valid matches for every cluster are recorded in a separate column, and as these crossmatches represent the most di GLYPH<14> cult to assign reliably, clusters where their name has been assigned in this way are flagged in the catalogue as crossmatches that were particularly di GLYPH<14> cult to assign. \nAfter assigning names to clusters, removing 22 objects associated with the Magellanic clouds, 17 objects associated with galaxies or dwarf galaxies, and 582 objects clearly associated with stellar streams in the galactic halo, our catalogue contains 7167 clusters, and is listed in Table 4 and online at the CDS, with tables of member stars and the rejected Magellanic cloud objects, galaxies, and stellar streams available online only. 2387 of these clusters are unreported in the literature and are candidate new objects, which we label with the acronym 'HSC' (standing for HDBSCAN Star Cluster.) Most of these objects have goodquality CMDs, and some are likely to be new OCs. For instance, HSC 2384 is a nearby new OC candidate at a distance of only \n551 pc with 273 member stars and a high astrometric S / N of 23 : 6 GLYPH<27> , which likely avoided prior detection due to being obscured by IC 2602 and mis-crossmatched to it (shown in Fig. 8.) However, many appear to be more consistent with unbound moving groups, and will require further classification based on their structure and dynamics. In addition, we provide a table of all crossmatches and non-crossmatches against the clusters in this work in Table B.1. \nIn the next sections, we discuss multiple aspects of the overall catalogue. Firstly, we discuss the overall catalogue of existing clusters in Sect. 7, including its distribution and the quality of its membership lists. Section 8 discusses why some literature clusters are undetected. Finally, Sect. 9 discusses why existing approaches to di GLYPH<11> erentiate between moving groups and OCs are inadequate to classify the new clusters detected in this work, a topic that will be explored further in a future work (Hunt & Reffert, in prep. ). \n<!-- image --> \nFig. 10. Spatial distributions of clusters detected in this work shaded on our derived log t and AV values. Left: side-on and top-down distribution of clusters in heliocentric coordinates that do not crossmatch to known GCs. The galactic centre is to the right, with the Sun at (0 ; 0). Only clusters passing two quality cuts are plotted: firstly, those with a CST score above 5 GLYPH<27> , meaning they are highly probable astrometric overdensities; and secondly, a median CMD class above 0.5, which are those compatible with single population star clusters. Clusters are plotted in descending age order, meaning points representing young clusters are most visible in crowded regions. Right: as left, except clusters are colour-coded by extinction AV . Clusters are plotted in ascending order of extinction. \n<!-- image --> \nFig. 11. Histogram of ages of all clusters in this work with median CMD classes greater than 0.5 - specifically, all clusters with photometry that is compatible with a single population of stars. These are compared to the ages of all clusters in the catalogues of Kharchenko et al. (2013), Kounkel et al. (2020), and Cantat-Gaudin et al. (2020). Known GCs are excluded from the results of this work and the results of previous works for this plot. \n<!-- image -->", '7. Overall results': 'In this section, we briefly discuss the structure and characteristics of the overall catalogue of 7167 clusters.', '7.1. Suggested cuts on the catalogue for a high-quality cluster sample': 'Our catalogue also includes objects that we detect with CST scores as low as 3 GLYPH<27> , and objects with low-quality CMDs given the results of our classifier in Sect. 4. Such clusters are included in our catalogue for completeness, as a low-quality CMD may be caused by a poor detection of a real OC by our cluster recovery method, and a cluster with a low CST that is not a guaranteed astrometric overdensity may still be a real cluster that could be validated by a future Gaia data release. However, these clusters are not particularly scientifically useful for studies of star clusters, as they cannot be validated as real within this work, or even with any currently available data. \nHence, in discussions of the overall structure of our results, we predominantly discuss the most reliable sample of 4105 clusters within the catalogue: those with a median CMD class greater than 0.5, meaning that they are likely to be a largely homogeneous single population of stars as in OCs and moving groups, allowing some tolerance for blue stragglers and extended mainsequence turno GLYPH<11> s; and a CST of greater than 5 GLYPH<27> , corresponding to clusters with a high likelihood of being real overdensities within Gaia data and not simply a statistical fluctuation. The \nFig. 12. Cluster radii derived in this work (dashed black line) compared against the distributions of cluster radii in various literature works. Top row: rc (top left) and rt (top right) of 1446 clusters from Kharchenko et al. (2013) that we redetect in this work (solid orange curve) compared against our approximately estimated King (1962) radii for these 1446 clusters. Middle row: same as top, except for radii of 202 clusters from Tarricq et al. (2022) that have derived King radii (solid purple curve). Bottom: r 50 measurements from Cantat-Gaudin & Anders (2020) compared against our r 50 measurements for the 1343 clusters from their work that we redetect. \n<!-- image --> \nmore tenuous 3062 objects excluded by this cut may still be used in some analyses, although with the caveat that these objects are less likely to be real star clusters.', '7.2. General distribution': 'The distribution of clusters in our catalogue is generally similar to that of other Gaia -based works such as Cantat-Gaudin & Anders (2020), albeit with more stark di GLYPH<11> erences when compared to those compiled before Gaia , such as Kharchenko et al. (2013). Comparisons are also useful to the catalogue of structures, moving groups, and star clusters of Kounkel et al. (2020) and papers based on Gaia DR3 data that report new clusters, such as CastroGinard et al. (2022). \nFigure 9 shows the distance distribution of clusters in this work, as well as the X ; Y distribution of clusters we re-detect and objects new to this work. Owing to the improved astrometry of Gaia DR3 and the clustering method we use (see Paper 1), our \ncatalogue has a high total number of clusters in most distance bins relative to other catalogues. As expected from the results in Paper 1, HDBSCAN is a cluster recovery technique sensitive across all distance ranges. However, HDBSCAN is sensitive to all clusters within Gaia data, as it is unbiased on the shape of clusters it reports; hence, the catalogue contains a large number of moving groups, which are generally detected near to the Sun. The catalogue contains around 8x as many objects as the open cluster catalogue of Cantat-Gaudin & Anders (2020) within 500 pc, clearly visible as an overdensity of new objects and in the distance distribution of Fig. 9. These objects are often di GLYPH<14> cult to classify as being OCs or moving groups (see Sect. 9). \nThe age and extinction distribution of Fig. 10 is similar to that of Cantat-Gaudin et al. (2020). A number of structures stand out, including: the imprint of the galactic warp in X ; Z plots for X < GLYPH<0> 2 kpc; the presence of spiral arm structure amongst young clusters very similar to that reported in works such as CastroGinard et al. (2021); and the general flatness of the distribution of compact star clusters in the Milky Way other than GCs, with few existing at heights of j Z j > 250 pc. Additionally, clusters towards the galactic centre generally have high AV values of 5 or greater, suggesting that extinction may be a limiting factor in the detection of clusters in this direction. \nDi GLYPH<11> erences to preGaia works are most apparent in the age histogram of Fig. 11, however. Our combined age distribution is relatively similar to that of Cantat-Gaudin et al. (2020), albeit with a slightly lower median age around log t GLYPH<25> 8 and no additional bump between 7 < log t < 8. However, the star cluster catalogue of Kharchenko et al. (2013) skews significantly older, with the most common (modal) age for clusters being around log t GLYPH<25> 9, an age range where we detect few clusters. A similar pattern is also visible for the catalogue of Kounkel et al. (2020), whose moving group and star cluster catalogue contains many unbound, old structures. Many of these objects have similar ages to the typical ages of unclustered stars in the Milky Way disk. In Sect. 8, we elaborate on how some of these age di GLYPH<11> erences may be caused by these catalogues containing a number of old false positive clusters. \nFinally, Fig. 12 shows the distribution of cluster radii compared between this work and the works of Kharchenko et al. (2013), Tarricq et al. (2022), and Cantat-Gaudin & Anders (2020). Our cluster radii agree most strongly with those in Cantat-Gaudin & Anders (2020), with a similar distribution of cluster radii containing 50% of members r 50. The King (1962) core radii rc that we derive, when compared against those in Kharchenko et al. (2013) and Tarricq et al. (2022), are generally larger. This may be due to our more populated membership lists, particularly for faint stars, due to our lack of a magnitude cut in our clustering analysis. Particularly for clusters with a high degree of mass segregation, this di GLYPH<11> erence in memberships would cause our clusters to have larger observed cores. Our tidal radii rt are slightly larger than those in Kharchenko et al. (2013), but much smaller than those in Tarricq et al. (2022). In the first case, the di GLYPH<11> erence may be due to the improved precision of Gaia data compared to preGaia works, causing us to detect more member stars at the outskirts of clusters and hence derive larger cluster tidal radii, with this e GLYPH<11> ect again being stronger for mass segregated clusters. In the second case, since Tarricq et al. (2022) also explicitly searched for cluster tidal tails and comas in their work, it may be that their extended cluster membership lists mean that they report higher cluster tidal radii. \n· \nFig. 13. Membership list comparisons between this work and the catalogue of Cantat-Gaudin & Anders (2020), using three clusters selected at random (upper three) and two clusters selected at random that were detected in Castro-Ginard et al. (2018) using Gaia DR1 data. Stars assigned as members by this work are plotted with filled blue circles, while members reported by Cantat-Gaudin & Anders (2020) are plotted with empty black circles. The first three columns compare the astrometry of cluster members in galactic coordinates, proper motions, and parallax as a function of l . The final column compares colour-magnitude diagrams of each resulting membership list. For every cluster, various parameters are labelled on the plots: number of member stars in Cantat-Gaudin & Anders (2020) N TCG, number of member stars in this work N , astrometric S / Nas estimated by the CST, distance d , and probability of being a single stellar population given the neural network in Sect. 4. \n<!-- image --> \nFig. 14. Two examples of clusters in the catalogue that have detected tidal structures. The spatial distribution of the clusters Blanco 1 (top row) and Mamajek 4 (bottom row) are plotted on the left, with member stars reported in this work shown as filled blue circles and compared against member stars from Cantat-Gaudin & Anders (2020) which are plotted as empty black circles. CMDs are shown in the two plots on the right for both clusters. \n<!-- image --> \n-', '7.3. Membership lists for individual clusters': "Owing to the improved quality of Gaia DR3 data and the expanded selection of 729 million stars from Gaia data used as input into our cluster recovery pipeline, clusters in this work generally have more populated membership lists than in previous catalogues. Fig. 13 compares our membership lists with those from Cantat-Gaudin & Anders (2020) for five clusters randomly selected from our catalogue. Our membership lists typically have a higher total number of stars, with virtually all new member stars being compatible with the existing cluster CMD. This is particularly the case for clusters in regions with minimal crowding, where Gaia has a high completeness of stars with 5-parameter astrometry down to G GLYPH<24> 20, with our membership lists containing stars down to approximately this limit. For more distant clusters such as Kronberger 4, membership lists are comparable in quality to those of Cantat-Gaudin & Anders (2020), as Gaia DR3 data does not present a large improvement in the astrometric quality of these distant sources compared to DR2. On average, our work contains 2 : 1 times as many member stars as the clusters we have in common with Cantat-Gaudin & Anders (2020), and 4 : 1 times as many member stars as the clusters we have in common with Kharchenko et al. (2013). \nA second major advantage of our pipeline is that clusters are not forced to take a spherical shape, as with other methods such as Gaussian mixture models (Paper 1). Hence, we are able to detect tidal tails for many of the clusters in the catalogue, especially for those that are nearby and within 1 GLYPH<0> 2 kpc. Tarricq et al. (2022) use HDBSCAN to detect tidal tails for 71 nearby OCs, many of which we are also able to detect. Figure 14 shows two examples of nearby clusters with well-resolved tidal tails using our methodology, Blanco 1 and Mamajek 4, both of which have reported tidal tails stretching around 50 pc from the centre of the cluster. Virtually all stars within the tidal structures appear \ncompatible with the isochrone of the cluster core, suggesting that they are stars with the same age, composition, and origin as the stars in the cluster cores. Particularly for clusters within 1 kpc, many of the clusters in our catalogue have tidal tails or comas. \nHowever, as no current methodology for star cluster recovery from Gaia data is perfect (Paper 1), our membership lists are not without caveats - both of which are consistent with our results from Paper 1, but that are still worth mentioning in the main work of this catalogue. \nFirstly, for distant OCs, our method may return fewer members than some other approaches. At high distances ( d & 5 kpc), the errors on Gaia parallaxes and proper motions generally become much higher than the intrinsic dispersion of OCs, meaning that many members have low membership probabilities and can only be reliably assigned as members by incorporating error information. Our methodology does not use error information in the clustering analysis for reasons of speed and the fact that HDBSCAN does not directly include a way to consider errors on data in clustering analysis, although other methods such as UPMASK(Krone-Martins & Moitinho 2014) which do consider error information could return better membership lists for these distant clusters. This is visible for Kronberger 4 in Fig. 13, where the membership list of Cantat-Gaudin & Anders (2020) (which was compiled using UPMASK) has a slightly higher number of sources than our membership list, even though our list was compiled from a greater number of input sources due to our lack of a G -magnitude cut. \nSecondly, HDBSCAN may sometimes return too many members, selecting regions larger than just an OC's core and tidal tails. This is particularly common for young clusters, which are often embedded in regions of high stellar density where recent hierarchical star formation has occurred (Portegies Zwart et al. 2010). These clusters can be di GLYPH<14> cult for HDBSCAN to isolate from other surrounding stars and sub-clusters. One particular example can be seen for UPK 545 in Fig. 13. Although the tail emerging from the cluster core in the upper-left of the ( l ; b ) plot appears compatible with a tidal tail, the connected structure to the right of the cluster is not. It appears to have the same age and composition as the cluster core, with all members of the tail being photometrically consistent with it. However, this 'o GLYPH<11> shoot' from the cluster may be better described as a separate cluster, which may also be bound to the core of UPK 545 in a binary pair of clusters, due to their proximity. Edge cases such as these are impossible to deal with autonomously with our current methodology and HDBSCAN alone, and require manual selection and separation of certain clusters in the catalogue into multiple separate components. \nOn a whole, the primary advantage of our catalogue is its completeness, generally reporting more member stars than previous works in the literature and doing so with a homogeneous methodology for a high number of total clusters. However, this is also the primary disadvantage of our catalogue: there are too many clusters and too many edge cases for all membership lists to be perfect, given only one clustering methodology. Hence, users of the catalogue who work with a small enough number of clusters are encouraged to manually check cluster membership lists and refine them depending on their application. To give one example, a user who wishes to only study cluster cores could refine our cluster membership lists by selecting a subset of them with Gaussian mixture models. With careful manual tweaking of the parameters of the mixture models, such a method could be used to remove tidal tails or possible other cluster components from our membership lists where necessary. Having discussed the general results of clusters in our catalogue, we next discuss \nthe reasons why many clusters reported in the literature may not appear in our catalogue.", '8. Reasons for the non-detection of some literature objects': 'Thousands of new OCs and moving groups have been reported since the release of Gaia DR2 (Brown et al. 2018), with over 2000 reported in the last two years using Gaia DR3 data alone (Gaia Collaboration et al. 2021). While multiple works have commented on the reliability of individual clusters in the literature at-length (e.g. Cantat-Gaudin & Anders 2020; Piatti et al. 2023), as an unbiased search for all clusters within all of Gaia DR3, the results of this work o GLYPH<11> er a unique way to review the reliability of recently detected OCs on a large scale. In addition, with hundreds of literature OCs newly redetected in this work, this work also o GLYPH<11> ers a chance to update the status of many older clusters reported in the preGaia era. \nThe non-detection of a cluster by this work can be a result of multiple di GLYPH<11> erent factors. It is important to first rule out any possible methodological reasons before claiming that a given cluster does not exist. In Paper 1, we showed that our methodology has a high sensitivity, and hence a literature cluster being non-detected in this work can nevertheless raise strong doubts about whether or not it is real. With thousands of non-detected clusters, there are far too many to review all clusters individually, and hence we do not aim to decisively prove that some literature clusters are not real. We discuss the six main methodological and datarelated reasons why a cluster may not appear in this work, concluding with questioning the existence of many objects reported in existing literature works.', '8.1.1. Limitations of the clustering algorithm used': 'An obvious reason why we may not detect a given literature OC is due to limitations of the HDBSCAN algorithm that we use in this work. While we found in Paper 1 that HDBSCAN is the most sensitive clustering algorithm overall, DBSCAN was slightly more sensitive for clusters at distances greater than 5 kpc when applied to Gaia DR2 data. On the other hand, with respect to cluster size, HDBSCAN was the most sensitive algorithm for all sizes of cluster, although HDBSCAN and DBSCAN had similar or identical sensitivity for clusters with a number of members stars of n stars = 10. Age and extinction were not found to have any significant di GLYPH<11> erential impact on the sensitivity of the algorithms trialed, with all algorithms being more or less equally a GLYPH<11> ected by older and / or heavily reddened clusters having fewer visible member stars, and hence being harder to detect. \nThe main limitation of HDBSCAN should be for clusters at distances greater than 5 kpc. However, only 6% and 21% of clusters from the DBSCAN-based works of Castro-Ginard et al. (2020) and Castro-Ginard et al. (2022) respectively that we are unable to detect have reported parallaxes of less than 0.2 mas, suggesting that distance-related detection issues alone are not enough to explain why certain clusters from these works are not detected. Additionally, we note that Castro-Ginard et al. (2022) using Gaia EDR3 were only able to recover & 80% of clusters they found in DR2 in Castro-Ginard et al. (2020), and so DBSCAN itself between Gaia data releases is not able to reliably reconfirm all clusters it detected previously. \nNevertheless, Fig. 15 shows that our chance of recovering clusters at high distances can be lower for certain works. In particular, although we are unable to recover only 3.4% of clusters reported in Cantat-Gaudin & Anders (2020), most of the clusters from their work that we are unable to recover are small clusters at distances above 5 kpc, suggesting that an algorithmic limitation may contribute to why we are unable to recover remaining objects from Cantat-Gaudin & Anders (2020). A key di GLYPH<11> erence between our work and Cantat-Gaudin & Anders (2020) is that their work used locations of clusters reported in the literature to narrow their search regions, which may in some cases be enough to make very distant clusters at the absolute limit of detectability in Gaia stand out. Future Gaia data releases with better data should provide additional clarity on whether or not such objects are real.', '8.1.2. Differences in the definition of an OC': "There is no single agreed upon definition of an OC in the literature, and the slight di GLYPH<11> erences in definition between works could cause some clusters to be detected or missed. \nPrinciple amongst these definitions is the minimum number of observed member stars for a valid cluster, n stars, min, which is important to distinguish star clusters from multiple star systems, also being used by some works as a proxy for the significance of a cluster relative to the field. In the literature, values of n stars, min range from 8 in Castro-Ginard et al. (2022) to as high as 50 in Liu & Pang (2019), with most works coalescing around a value of between 10 and 12 (Krumholz et al. 2019). For the purposes of this work, we adopt a value of 10, and we should hence miss very few literature clusters due to this constraint alone. \nSecondly, OCs generally have a population of stars with the same age and chemical composition, due to forming at the same time from the same molecular cloud (Cantat-Gaudin 2022). In practice, this is a di GLYPH<14> cult definition to constrain observationally, with the CMDs of OCs being broadened by e GLYPH<11> ects such as differential extinction or outliers which are not true member stars, with these e GLYPH<11> ects being worse with increasing distance and field star density. In addition, many OCs are not perfect single populations, with some hosting blue stragglers or having a clear second population in the form of an extended main-sequence turno GLYPH<11> (Cantat-Gaudin 2022). For the purposes of this work, we classify our clusters with our CMD classifier (see Sect. 4) and include all clusters in the final catalogue, instead leaving the task of removing clusters with poor photometry to the end user (recommending a minimum class value of 0.5). This means that no clusters are missing from the catalogue due to photometric reasons. \nFinally, OCs must be distinguished from other types of single-population stellar overdensities. Star clusters can be divided into bound clusters (such as OCs and GCs) and unbound clusters (typically referred to as moving groups). Some works, such as Cantat-Gaudin & Anders (2020), use basic cuts on mean parameters to remove clear moving groups from their catalogue; we leave the classification of moving groups in our catalogue to a future work (Hunt & Re GLYPH<11> ert, in prep. ) for reasons discussed in Sect. 9, and hence, no OCs are be missing from this work due to being catalogued as moving groups. We do, however, flag known GCs in our catalogue by crossmatching against the catalogue of GCs of Vasiliev & Baumgardt (2021), with GCs in the Milky Way being distinguished from OCs by their age, which is typically greater than GLYPH<24> 6 Gyr , and their mass, which is typically greater than GLYPH<24> 10 4 M GLYPH<12> , whereas most OCs have masses no higher than GLYPH<24> 5000 M GLYPH<12> (Kharchenko et al. 2013). In total, differences in the fundamental definition of an OC between works \nFig. 15. Plots showing the fraction of clusters undetected by this work when compared to various literature works or series of literature works, shown as a histogram of various parameters as a solid blue line for all clusters in the catalogue, and a dashed grey line for clusters in the high quality sample defined in Sect. 7.1. The dashed orange lines show the number of clusters in each bin. Optimum histogram bin widths were selected automatically using numpy (Harris et al. 2020). From left to right, each column shows the number of stars N , distance d , age log t and extinction Ax reported in each catalogue. For the top four groups of catalogues, extinctions were given in the V band. For the lower two, extinctions were given in Gaia's G band, which are generally slightly lower. \n<!-- image --> \nshould have a small impact on the inclusion of OCs in this work when compared to others.", '8.1.3. Different quality cuts between different works': 'Di GLYPH<11> erent works in the literature often place di GLYPH<11> erent quality cuts on their catalogues, meaning that another possible reason why a given literature cluster does not appear in this catalogue would \nbe if it has been cut for quality reasons. Our catalogue adopts a philosophy of allowing users to decide their own quality cuts as much as possible, and hence includes all objects with bad photometry as well as moving groups that are unlikely to be bound OCs. The approach of allowing end users of the catalogue to define their own quality cuts is a similar philosophy to how Gaia data releases include many poor-quality sources, instead allowing users decide how strongly they wish to cut the Gaia catalogue (Gaia Collaboration et al. 2021). Poor photometry and the \nbound or unbound status hence do not impact our recoverability of clusters in Fig. 15. \nHowever, the sole quality cut applied to the catalogue that would a GLYPH<11> ect its sensitivity is a cut on the astrometric S / N of detected clusters (derived using the CST) at 3 GLYPH<27> . This was performed because clusters with an S / N below this threshold are likely to be false positives, and because the high number of clusters below this threshold greatly complicated the process of merging results between di GLYPH<11> erent runs (see Sect. 3). Including such a quality cut dramatically improved the run merging process and hence our membership lists and completeness for reliable clusters, which is a more important scientific product than a list of low quality clusters that we cannot deem likely to be real clusters based on their S / N alone. \nWhile we believe this is a fair trade-o GLYPH<11> to produce a catalogue that is as reliable as possible overall, it is likely that some real clusters are be missed due to this cut on S / N. For instance, in Paper 1 using Gaia DR2 data, we tentatively detected Teutsch 156 with an S / N of 0.68 GLYPH<27> , which counted as a non-detection; however, using Gaia DR3, we clearly detect Teutsch 156 with an S / N of 16.3 GLYPH<27> . It is di GLYPH<14> cult to know exactly how many real literature clusters are missed due to this cut, particularly since some clusters in the literature with an S / N below 3 GLYPH<27> are likely to be statistical fluctuations and not real clusters, especially for S / Ns below 1 GLYPH<27> . This can be approximately estimated using the histogram of detected cluster S / Ns in Fig. 2. Since the distribution of literature cluster S / Ns is roughly flat for S / Ns below 10 GLYPH<27> , assuming that this trend continues for S / Ns below 3 GLYPH<27> , we may have missed approximately GLYPH<24> 300 crossmatches to clusters reported before Gaia DR3 and an additional GLYPH<24> 400 reported using Gaia DR3 data - although, owing to the low S / Ns that such objects would inevitably have, it is also likely that a number of these crossmatches would be false positives. \nInevitably, a repeat of this work with better data (such as Gaia DR4) would likely detect more of the objects that we do not recover with a su GLYPH<14> cient statistical significance using Gaia DR3 data. In the future, further development of clustering algorithms that produce fewer false positives and can be ran on more data at once (both of which would tremendously simplify the runmerging process) would allow the minimum S / N threshold to be lowered.', '8.1.4. When two clusters are catalogued as one cluster': 'Certain other non-detections can be explained by further methodological di GLYPH<11> erences. Sometimes, clusters reported as multiples in the literature are reported as a single object by HDBSCAN, even across all of its mclSize runs. A notable example is UPK 533 from Sim et al. (2019), which was re-detected by Cantat-Gaudin & Anders (2020), but which HDBSCAN assigns as simply being a member of a tidal tail of a di GLYPH<11> erent and significantly larger nearby cluster, UPK 545, with no HDBSCAN mclSize run separating the two objects. UPK 545 is shown in Fig. 13 on the third row. In this and other edge cases, our catalogue merges the two objects. An improved clustering algorithm that can separate edge-case binary clusters such as these autonomously would be helpful. However, only a small fraction of clusters (fewer than 1%) are a GLYPH<11> ected by this issue.', "8.1.5. When a literature catalogue's parameters deviate too strongly from a detected cluster": 'While our crossmatching procedure as outlined in Sect. 6 aims to be as fair as possible, generally giving the benefit of the doubt to potential crossmatches, there are nevertheless cases where clusters reported in the literature still remain outside of our bounds for an accepted match. Generally, in all cases where this occurs, our detected cluster is significantly di GLYPH<11> erent to the literature object in at least one of the parameters considered for crossmatching, with these clusters representing ambiguous cases where it is not clear that the reported literature cluster is truly the same object. \nCWNU 528 as reported in He et al. (2022a) is one example of a cluster reported in the literature that we are unable to detect within our crossmatching criteria. CWNU 528 is reported in He et al. (2022a) with 24 member stars, but appears to be a small o GLYPH<11> shoot of the recently reported new cluster OCSN 82 from Qin et al. (2023), which has an overall position di GLYPH<11> erent by around 3 GLYPH<14> and a total of 157 member stars. CWNU 528 is so much smaller than OCSN 82 and at such a di GLYPH<11> erent location that it does not crossmatch to it given our adopted crossmatching scheme, even though a few of the member stars in our detection of OCSN 82 are in common with CWNU 528 and they have similar proper motions and parallaxes. \nThis case is likely to have been repeated a few times, and appears particularly common with clusters detected in Gaia data using the DBSCAN algorithm (as in He et al. 2022a). In Paper 1, we commented that while DBSCAN has an excellent sensitivity and low false positive rate (depending strongly how the GLYPH<15> parameter is chosen), it often had the sparsest and most incomplete membership lists of all algorithms we studied. Hence, detections of clusters may be so di GLYPH<11> erent or poor compared to what another algorithm recovers that crossmatch criteria may not be fulfilled, even when using a very permissive crossmatching scheme. In these cases, it is debatable whether the literature cluster is even the same object as the newly detected one.', '8.1.6. Limitations of Gaia data': 'Finally, it is worth considering the limitations of Gaia data itself, particularly when comparing our catalogue to works created from di GLYPH<11> erent data sources. Notably, the catalogue of Kharchenko et al. (2013) was compiled before Gaia and used infrared data from 2MASS (Skrutskie et al. 2006). Cantat-Gaudin &Anders (2020) are unable to recover a majority of the clusters from Kharchenko et al. (2013) using Gaia DR2 data, and we are unable to recover 48.4% of the clusters reported in their catalogue in Gaia DR3 data. Given that infrared light is significantly less a GLYPH<11> ected by extinction than the visual light used to compile Gaia data, it begs the question of whether many clusters from Kharchenko et al. (2013) may still be missing from Gaia -based catalogues due to extinction limits. \nHowever, Fig. 15 shows that extinction does not appear to play a major role in the non-detection of many clusters from Kharchenko et al. (2013). If extinction was a major contributor to why we are unable to detect so many of the clusters in their catalogue, then one would expect to see a linear trend in f undetected; all of their low-extinction clusters would be easily detected in Gaia , until some cut-o GLYPH<11> value beyond which Gaia detects no further clusters. On the contrary, most of their clusters have AV < 5, and we are unable to detect around 50% of all clusters in this range with an approximately flat and uncorrelated distribution in the fraction of clusters recovered. \nA few dozen of their reported clusters may be genuinely challenging to detect in Gaia data, since some of their clusters have AV > 5 and are at high distances of greater than 10 kpc. However, the majority of their clusters are within 10 kpc and have AV < 5. Given that Gaia data have GLYPH<24> 10 3 times greater astrometric precision than Hipparcos data for GLYPH<24> 10 5 times as many stars (Gaia Collaboration et al. 2021), and given that our chance of detecting a cluster reported in Kharchenko et al. (2013) is uncorrelated with extinction for AV < 5, limitations of Gaia data do not appear to be responsible for the bulk of non-detections of clusters from preGaia works, despite assertions in recent works that Gaia data may be extinction-limited and unable to recover many highly reddened OCs from infrared datasets. Nevertheless, a handful of high-extinction clusters with AV > 5 reported in the literature may still be challenging to recover in Gaia data.', '8.2. The cluster does not exist': "Having exhausted all other major possibilities for why a cluster may not appear in our catalogue, the final potential reason would be that the cluster simply does not exist. As stated in the introduction to this section, far too many clusters are nondetected in this work for us to individually review them all and decisively prove that they are not real; however, we can give a broad overview of the typical characteristics of non-detected clusters, and contrast the similarities and di GLYPH<11> erences between non-detected clusters in this work. \nFigure 15 shows that the parameter most strongly correlated with f undetected is the number of member stars N , with the smallest clusters from all papers being the least likely to be redetected. Few works report the statistical likelihood of a cluster being real in a way similar to the CST used in this work; however, N can be thought of as a good proxy for the statistical significance of a cluster, as it stands that a cluster with fewer member stars is probably less likely to be real. Clusters with fewer than 20 reported sources are often the most di GLYPH<14> cult to redetect. \nIn general, since most works in Fig. 15 use Gaia DR2 data or stronger cuts on Gaia data than our methodology, there are many cases where we should be able to detect their reported clusters easily and with a higher number of member stars and statistical significance. The fact that we cannot suggests that some of these clusters may have been statistically insignificant associations of a small number of member stars. \nThe distance of undetected reported literature clusters is similarly revealing. In Sect. 8.1.1, we suggest that some clusters may be undetected in this work at high distances due to limitations of the HDBSCAN algorithm. However, given that HDBSCAN should be the most sensitive algorithm for recovery of nearby clusters (Paper 1), it makes little sense that we are unable to recover a number of nearby clusters within 1 kpc for most of the works in Fig. 15. Many of these nearby and undetected objects may not be real, as there is no reason why we should not be able to detect them using the improved data of Gaia DR3 and the most sensitive algorithm for recovery of nearby OCs. \nThe age of undetected clusters paints a complicated picture. In principle, detecting an old cluster has two challenges. Firstly, as the cluster ages, the brightest stars in the cluster evolve into faint remnants, which reduces the number of stars visible in the cluster. This is a particular issue for distant old clusters, as the remaining fainter and longer-lived stars in a cluster may be below a survey's magnitude limit. In the case of Gaia , stars near to its magnitude limit have the lowest accuracy astrometry, reducing the signal-to-noise ratio of a given old, distant cluster in proper motion and parallax space - further complicating its detection. \nSecondly, as clusters age, they are theorised to take a sparser and less centrally concentrated distribution (Portegies Zwart et al. 2010), reducing their signal-to-noise ratio relative to background field stars in positional data. \nAlthough old clusters are likely to be harder to detect, in Paper 1, we found that the age of a reported cluster generally has the same e GLYPH<11> ect on all algorithms: their lower number counts and sparsity a GLYPH<11> ect all algorithms more or less equally in making them harder to detect. However, there are correlations between f undetected and log t for almost all papers in Fig. 15, despite all of them other than Kharchenko et al. (2013) being based on Gaia data and using methods found in Paper 1 to be equally a GLYPH<11> ected by cluster age. Hence, these correlations may be more informative about the types of cluster in other catalogues that are false positives than on whether or not a given catalogue used a better method. \nFor all works other than Cantat-Gaudin & Anders (2020), clusters older than an age of around 1 Gyr (log t > 9) are much less likely to be redetected. Zucker et al. (2022) have recently investigated the nature of the groups reported in Kounkel et al. (2020), and find that many of them have ages GLYPH<24> 120 times larger than their dispersal times while being unbound and chemically homogeneous with their surrounding field stars - strongly suggesting that they are merely associations of field stars and not physical groupings. The fact that we are unable to redetect almost any of the groups older than 1 Gyr reported in Kounkel et al. (2020) supports this conclusion, with it being plausible that many of their oldest groups are instead associations of field stars, consistent with the mean ages of field stars in the galactic thin and thick disks of a few Gyr. The similar correlations with old clusters being undetected for other works may also suggest that a number of other old clusters reported in the literature are also associations of field stars with mean ages similar to that of the typical ages of unclustered field stars in the galactic disk. \nThe reasons for the non-detection of some young clusters are less clear, and are more surprising given that young clusters should be easier to detect. In the case of Cantat-Gaudin & Anders (2020), the handful of young clusters that we are unable to detect are also at high distances, which may mean that their non-detection is entirely a result of our own methodological limitations (see Sect. 8.1.1.) On the other hand, these distant, young clusters may have originally been detected by hand-searching for OB stars in preGaia works and cataloguing them as OCs, but without a test of their physical nature, which could mean that they are associations. Similar reasoning could also be applied to the non-detected young clusters from Kharchenko et al. (2013). Both possibilities are plausible, and this should be investigated further in another work. \nFinally, the reasons for the spikes in non-detected clusters between 7 < log t < 8 for Castro-Ginard et al. (2020, 2022) and between 6 < log t < 7 in Hao et al. (2022) remain unclear. These works are entirely compiled from Gaia DR2 and EDR3 data using the DBSCAN algorithm. Given that our results in Paper 1 suggest that clustering algorithms applied to Gaia data have no di GLYPH<11> erences between themselves in their ability to detect clusters based on their age, there is no clear reason why these clusters would be undetectable. The non-detection of these clusters should be investigated further. \nFor most works, extinction AV does not predict the chance of redetecting a given cluster. In Sect. 8.1.6, we discuss that AV values of greater than GLYPH<24> 5 appear to reduce the chance of a cluster being recovered in Gaia data. The increasing trend in f undetected for Cantat-Gaudin & Anders (2020) as a function of AV appears to entirely be due to our lower chance of detecting clusters with \n<!-- image --> \nFig. 16. Geometric mean of the proper motion dispersion (left) and radius containing 50% of members (right) for the clusters reported in this work, as a function of distance. Clusters are split between those detected in previous works (blue circles) and those newly reported in this work, divided between the high quality (orange squares) and low quality (grey triangles) samples defined in Sect. 7.1. The cuts on cluster parameters to distinguish between bound OCs and unbound moving groups or associations proposed in Cantat-Gaudin & Anders (2020) are shown as a dashed black line. \n<!-- image --> \nd > 10 kpc, since distant clusters also often have a high AV . No other clear correlations exist for other works in Fig. 15 with respect to extinction, other than for a few dozen preGaia clusters from the infra-red catalogue of Kharchenko et al. (2013) with AV & 5 that we are unable to redetect with Gaia data. \nIn summary, we find that there are many potential reasons for the non-detection of given clusters from the literature, all of which should be investigated in more depth in future works. Verifying that new clusters reported in the literature are real is arguably as important as reporting them. While we cannot provide conclusive reasons for the non-detection of given clusters, given the scope of this survey, the overall trends we have identified should still be helpful and suggestive in whether or not given objects are real. We provide a table of all clusters non-detected by this work in Table B.1 and at the CDS.", '9. The difficulties of distinguishing between open clusters and moving groups': "Having discussed the catalogue's overall quality for the verification and study of clusters reported previously in the literature, it is worth discussing the 2387 new objects reported in this work 739 of which have a median CMD class above 0.5 and a CST of greater than 5 GLYPH<27> , and are hence the most reliable new objects that we report.", '9.1. The case against many of our new clusters being OCs': "On first inspection, despite having reliable CMDs and being statistically significant astrometric overdensities, many of our most reliable new objects have sparse density and proper motion distributions that appear more compatible with moving groups than spherically symmetric OCs with King (King 1962) or Plummerlike (Plummer 1911) profiles. Figure 17 shows three clusters randomly selected from the 739 most reliable objects. HSC 1131 is a sparse, elongated grouping of stars in the thin disk, with a \nstringy nature much more compatible with a moving group than an OC. HSC 2376 is less clear, showing a more Gaussian clumping reminiscent of an OC within proper motion space but while still being relatively sparse, with r 50 = 8 : 9 pc. HSC 1185 appears visually to be the most OC-like cluster, with its distribution of member stars forming compacter Gaussian-like overdensities in spatial and proper motion plots. \nWhile we have used tests on statistical significance and cluster CMDs to determine the reliability of clusters in the catalogue, it is clear that a further test on the astrometric parameters of clusters (such as sparsity and proper motion dispersion) is necessary. Cantat-Gaudin & Anders (2020) propose two tolerant cuts on cluster parameters, finding that requiring the geometric mean of proper motion dispersion to be less than a criterion (corresponding to GLYPH<24> 5 kms GLYPH<0> 1 ) and r 50 < 20 pc removed objects highly unlikely to be OCs from their sample. \nHowever, Fig. 16 shows that with the exception of some clusters that are clearly associated with stellar streams (based on their location, CMD, and sparsity at distances greater than GLYPH<24> 3 kpc), most new clusters detected in this work are compatible with OCs given the tolerant cuts in Cantat-Gaudin & Anders (2020). \nIf almost all of the new clusters that we detect within 1 kpc of the Sun are in fact OCs, then this would represent a total paradigm shift in the census of OCs - with a large number of previously unseen low number count, low mass, and sparse clusters being detectable nearby with Gaia data. In reality, there are good reasons for this not being the case, and a more stringent cut on the astrometric parameters of candidate OCs is necessary. \nIn the preparation of this work, much e GLYPH<11> ort was put in to attempting to find a more stringent cut on basic astrometric parameters (or some combination of them) to distinguish OCs from moving groups. We found that whether or not a cluster is a bound OC cannot be decided accurately based on individual cuts on r 50 or proper motion dispersions alone, and instead requires at least some modelling of the cluster's spatial profile, its velocity pro- \n) \n· \n( \nb \nFig. 17. Three newly reported clusters randomly selected from the cluster catalogue and ordered by increasing distance, with member stars plotted as a function of their astrometric and photometric data as in Fig. 13. All clusters pass the cuts proposed in Cantat-Gaudin & Anders (2020), have good-quality CMDs passing the cuts from Sect. 4, and have astrometric significances of greater than 5 GLYPH<27> , meaning they are almost certainly real overdensities in Gaia data. \n<!-- image --> \nfile, and its mass. In the next section, we discuss the di GLYPH<14> culties of such a method, which will be applied in the next paper in this series.", '9.2. A test for if our OC candidates are bound': "A given system is said to be in virial equilibrium if the absolute value of its potential energy j V j is equal to twice its kinetic energy T . A number of works have recently used a relationship derived from the virial theorem, which predicts a velocity dispersion that a cluster should have if it is bound, GLYPH<27> vir, based on its mass and radius. This can be compared to the cluster's measured 1D velocity dispersion GLYPH<27> 1 D , which should equal GLYPH<27> vir if the cluster is bound: \nGLYPH<27> vir = s GM GLYPH<17> r hm GLYPH<25> GLYPH<27> 1 D for a bound cluster, (1) \nwhere r hm is the cluster's half-mass radius, M is the cluster's mass, G is the gravitational constant and GLYPH<17> is a constant depending on the cluster's density profile that is usually set to 10 (Porte- \ngies Zwart et al. 2010). In the case when GLYPH<27> 1 D >> GLYPH<27> vir, the cluster is likely to be unbound. This relationship has been used by works such as Bravi et al. (2018), Kuhn et al. (2019), and Pang et al. (2021) to test the virial nature of OCs using Gaia data, albeit in limited studies of no more than 28 clusters in one work. \nWhile this relation is a promising way to distinguish between bound OCs and unbound moving groups, scaling this methodology to apply across our entire catalogue is extremely challenging. There are many systematics that can enter velocity dispersion, mass, and radius measurements, all of which must be reduced as much as possible to produce meaningful classifications. The clusters in our catalogue range across two orders of magnitude in distance, many orders of magnitude in mass, and two orders of magnitude in radius, with clusters of di GLYPH<11> erent parameters having fundamentally di GLYPH<11> erent challenges. For instance, nearby clusters may have tidal tails that must be removed from membership lists and may su GLYPH<11> er from projection e GLYPH<11> ects due to their radial velocity that would skew the measurement of their velocity dispersion with proper motions. On the other hand, distant clusters will push the limits of Gaia's astrometric measurements, with velocity dispersions being di GLYPH<14> cult to measure precisely. \nGiven the scope of such a method, we leave its implementation to a future work. To restrict our catalogue to a reliable sample of OCs, users of our catalogue may for now use our CST scores, CMD classifications, and the criteria from Cantat-Gaudin & Anders (2020) to remove objects highly unlikely to be OCs. The next work to follow this one will provide a more accurate way to separate OCs from moving groups, and is anticipated to be submitted soon (Hunt & Re GLYPH<11> ert, in prep. ).", '10. Conclusions and future prospects': "In this work, we conducted a blind all-sky search for Milky Way star clusters using Gaia DR3 data. We show that a single blind search can be used to produce a homogeneous star cluster catalogue in the Gaia era. We used the HDBSCAN algorithm, a density-based test of cluster significance, and a data partitioning scheme to detect as many reliable clusters as possible, producing a catalogue that is as complete and reliable as possible given current data. In total, the catalogue contains 7167 clusters, of which 4105 clusters form the most reliable sub-sample of objects with median CMD classifications greater than 0.5 and S / Ns greater than 5 GLYPH<27> . \nWeprovide a wide range of parameters for clusters in the catalogue, including: basic astrometric parameters, S / Ns that correspond to their statistical significance given Gaia astrometry, CMD quality classifications, ages, extinctions, distances, and Gaia DR3 radial velocities. We recover large, expansive membership lists for many OCs, often including tidal tails for clusters within GLYPH<24> 1 kpc. Membership lists for all of our clusters are also available as a part of the catalogue (see Appendix A and the CDS). \nExtensive care was taken to crossmatch our catalogue against 35 other works. To the best of the authors' knowledge, these works catalogue all OCs reported in the literature, including many thousands of OCs recently reported in the literature using Gaia data that are yet to be verified independently. 7022 clusters reported in the literature crossmatch against 4944 of the entries in our catalogue, including around 2000 of which we are able to independently verify for the first time. The spatial and age distribution of our catalogue traces the spiral arms in a similar way to many other recent works (e.g. Cantat-Gaudin et al. 2020; Castro-Ginard et al. 2021). \nHowever, we are unable to recover many of the clusters reported in the literature, despite our methodology having the highest sensitivity for OC recovery of all methods we trialed in Paper 1. We discuss reasons why we may be unable to detect an OC and are able to tentatively suggest that many thousands of clusters reported in the literature may not be real, including calling into question the common assertion that Gaia is unable to recover a large fraction of OCs reported before Gaia due to being extinction-limited. Further investigations into whether or not many of the OCs we are unable to detect are real would be helpful to improve the accuracy of the OC census. \nOur catalogue contains 2387 new objects as yet unreported in the literature, 739 of which are a part of our most reliable sample of clusters with median CMD classifications of greater than 0.5 and an S / N of greater than 5 GLYPH<27> . While some of these objects are likely to be new OCs, we find that many are more compatible with unbound moving groups, as our methodology is sensitive to all kinds of stellar overdensity in Gaia data. We find there is often no simple way to distinguish between the sparse, compact moving groups we detect and OCs, with the cuts on basic parameters proposed in Cantat-Gaudin & Anders (2020) being too lenient. In an upcoming work, we will use the virial \ntheorem to distinguish between bound and unbound clusters with a probabilistic methodology (Hunt & Re GLYPH<11> ert, in prep. ). \nThe coming decade of Milky Way star cluster research is likely to continue to be exciting and fast-paced. Firstly, the quality of available data will increase ever-higher. Gaia DR4 will be produced from GLYPH<24> 66 months of data, almost double that of Gaia DR3, which will result in a large jump in the accuracy of available astrometric and photometric data. DR4 is currently slated for release no sooner than the end of 2025. The current planned final Gaia data release, DR5, may be based on around ten years of data, again roughly doubling the amount of input data used (Gaia Collaboration et al. 2021). Such large improvements in the accuracy of available astrometric data will inevitably result in more new clusters and improvements in the S / Nand membership lists of existing clusters, further increasing the completeness and purity of the OC census. \nSecondly, methodological improvements will continue to ease the process of star cluster recovery and characterisation. In the preparation of this work, it was still necessary to extensively verify many results by hand and develop postprocessing techniques to clean false positives from our catalogue. Improvements in clustering algorithms and techniques over the coming decade could make the process of cluster recovery more straightforward, accurate, and sensitive, with new methodologies such as Significance Mode Analysis (SigMA) methodology (Ratzenböck et al. 2022) showing promise in this area. As we discussed in Paper 1, there is currently no known perfect way to recover OCs from Gaia data; much work remains to be done to try and find one. \nAcknowledgements. Wethank the anonymous referee for their helpful comments that improved the clarity of this work. We thank Tristan Cantat-Gaudin for reporting crossmatch issues to dwarf galaxies and NGC 6656 in an earlier version of this paper. E.L.H. and S.R. gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 138713538 - SFB 881 ('The Milky Way System', subproject B5). This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/ gaia/dpac/consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of NASA's Astrophysics Data System Bibliographic Services. This research also made use of the SIMBAD database, operated at CDS, Strasbourg, France (Wenger, M. et al. 2000). In addition to those cited in the main body of the text, this work made use of the open source Python packages NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), IPython (Pérez & Granger 2007), Jupyter (Kluyver et al. 2016), Matplotlib (Hunter 2007), pandas (McKinney 2010; Reback et al. 2020), Astropy (Robitaille et al. 2013; Astropy Collaboration et al. 2018), healpy (Zonca et al. 2019), and scikit-learn Pedregosa et al. 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Ser., 143, 9 \nZari, E., Hashemi, H., Brown, A. G. A., Jardine, K., & de Zeeuw, P. T. 2018, Astronomy & Astrophysics, 620, A172 \n- Zonca, A., Singer, L. P., Lenz, D., et al. 2019, Journal of Open Source Software, 4, 1298\n- Zucker, C., Peek, J. E. G., & Loebman, S. 2022, The Astrophysical Journal, 936, 160', 'Appendix A: Description of contents of online tables': 'We provide tables of clusters, rejected clusters, member stars, and members stars for rejected clusters at the CDS. Tables of clusters follow the table format in Table A. Tables of members follow the same columns and column naming scheme as in Gaia DR3 (Gaia Collaboration et al. 2022), except while also having columns referencing the cluster name and cluster ID we assign them to, the cluster membership probability, and a flag for if the star is a member within our estimated tidal radius rt .', 'Appendix B: Table of crossmatch results': 'Here we provide a table of all crossmatches to all literature clusters that meet our adopted crossmatch criteria from Sect. 6 in Table B.1. For every cluster in the literature that we detect in this work, the table lists the internal cluster ID corresponding to our table of clusters in Table 4 that corresponds to this object. For clusters that we do not redetect, only a blank row with the cluster name, source paper, and type of crossmatch is shown.', 'Appendix C: Bayesian neural networks': "Given that Bayesian neural networks (BNNs) are only just beginning to see use in the astronomical literature (e.g. HuertasCompany et al. 2019), here we provide a brief background overview of the advantages and caveats of the approximate BNN methodology we adopted in Sect. 4 and Sect. 5. \nBNNs are a somewhat elusive area of open research in machine learning. Their appeal is clear: unlike a deterministic approach or an approach based on simply perturbing network inputs, a perfect BNN would be able to estimate both aleatoric uncertainties, which are uncertainties that result from random phenomena, such as uncertainty on photometric measurements; and epistemic uncertainties, which are uncertainties that result from a lack of knowledge about the underlying processes being modelled. For instance, any remaining gaps or issues in the simulated training data we use would cause a traditional deterministic neural network to always output an incorrect answer, whereas a probabilistic neural network should at least output a wide range of answers that demonstrate its uncertainty in such di GLYPH<14> cult cases (Goan & Fookes 2020; Jospin et al. 2022). \nIn practice, there is currently no perfect BNN architecture, with all approaches having some flaws (Goan & Fookes 2020; Jospin et al. 2022). While a Monte-Carlo Markov chain (MCMC)-based approach should in theory be superior, where every network weight has an arbitrary posterior distribution, MCMC-based BNNs are extremely di GLYPH<14> cult or impossible to train accurately, with current sampling techniques being inadequate (Goan & Fookes 2020). In addition, BNNs are often time consuming to train. Instead, 'variational inference' is widely used to approximate BNNs. In this technique, an ideal BNN is approximated by perturbing network features, approximating a BNN by 'emphasising or de-emphasising' certain parts of a trained model when the model is sampled. This can then be used to estimate the epistemic uncertainty of a model by sampling a variational network multiple times. \nMany approaches for variational inference exist in the literature, with a common approach being dropout regularisation as an approximation of a BNN (Gal & Ghahramani 2015), having also been used within astronomy (e.g. Huertas-Company et al. 2019; Leung & Bovy 2019). However, this approximation is not inherently Bayesian (Hron et al. 2017), and may be improved \nTable A.1. Description of the columns in the tables of detected clusters. \nNotes. The full version is available at the CDS. ( a ) Mean value, standard deviation GLYPH<27> , and standard error GLYPH<27> = p n are given. ( b ) Median value and various confidence intervals are given. ( c ) g for objects in the Vasiliev & Baumgardt (2021) GC catalogue, otherwise o (OC) or m (moving group) for clusters according to the empirical cuts in Cantat-Gaudin & Anders (2020). ( d ) Flag indicating six clusters for which parallax bias correction using the method of Lindegren et al. (2021b) was not possible, and a global o GLYPH<11> set was used instead (see Sect. 3.3). ( e ) Corrected using cluster distances to be relative to cluster centre. ( f ) Cluster CMD classes (the probability of a given cluster being a single coeval population of stars) derived using the neural network in Sect. 4. Some clusters also appeared in our human-labelled test dataset, for which human classes are also listed. ( g ) Indicates 25 clusters merged by hand where initial HDBSCAN clustering was unsatisfactory (see Sect. 3). ( h ) Indicates nine clusters with members from an additional Gaussian mixture model clustering step, typically applied to di GLYPH<14> cult to separate binary clusters. ( i ) Method used to assign name to cluster. In particular, 'many to many' crossmatches are the most di GLYPH<14> cult to assign due to multiple objects crossmatching to multiple literature entries co-dependently (see Sect. 6.3 for full discussion of final cluster name assignments.) \nupon with recent developments in the literature. Another common approximation is to assume that all layer kernel and bias weights are drawn from simple distributions, such as independent Gaussian distributions. This allows for gradients during network training to be calculated straightforwardly using Bayes by backpropagation (Blundell et al. 2015). This approximation can hold relatively well for (simple) neural networks, which often have normally distributed weights, but may cause underfitting on \nTable B.1. All cluster crossmatches, including literature clusters that have no match. \nNotes. The full version is available at the CDS; the above only shows crossmatches against a selection of Basel clusters. Depending on the type of work crossmatched against, only separations in terms of position GLYPH<18> may be listed. For works with astrometry, separations s with respect to GLYPH<22>GLYPH<11> GLYPH<3> , GLYPH<22>GLYPH<14> , and $ are shown, in addition to separations GLYPH<27> which are in terms of standard deviations about the mean of the astrometry of these clusters added together in quadrature, after accounting for worst-case systematics. Cluster entries in the literature that did not have a valid crossmatch against any cluster detected in this study are listed with only the name, source, and source type columns filled. Recalling Sect. 6, for a valid crossmatch, we require GLYPH<18> r < 1, and additionally, when crossmatching to a work with full five parameter astrometry, all GLYPH<27> values to be less than two. ( a ) The separation between cluster centres in terms of the largest cluster radius available, GLYPH<18> r = GLYPH<18>= max( rt ; rt ; lit) \nmore complicated problems (Goan & Fookes 2020). Due to the time-consuming nature of repeated samples of all kernel and bias posterior distributions, we also apply an approximation known as Flipout to more e GLYPH<14> ciently sample them with a lower runtime while preserving good training characteristics (Wen et al. 2018). Similar approaches using Bayes by backpropagation and Flipout have seen some use in the astronomy literature (e.g. Lin & Wu 2021). We use the implementations of DenseFlipout and Convolution2DFlipout layers in TensorFlow Probability (Dillon et al. 2017), minimising the evidence lower bound (ELBO) loss (Blundell et al. 2015). \nIn initial tests, these approximations produced network outputs with reliable uncertainty estimates that correspond well to the uncertainty inherent to classifying star cluster CMDs. It is \nworth noting from the literature that variational-inference based approaches are still more overconfident than a true BNN when applied to unseen data (Goan & Fookes 2020), and that this approach is still an imperfect estimator of the true uncertainty of our model; nevertheless, our adopted method was found to be as accurate as a traditional deterministic network architecture of the same configuration when applied to our training data, but while providing an estimate of its uncertainty and without dramatically increasing runtime during training or sampling."}
2024arXiv240902164B	The Atacama Large Millimetresubmillimetre Array ALMA receivers and technical papers are cited fewer than once in every six publications. This citation shortage is impeding the development of future submillimetre instruments. In an effort to facilitate the correct citations of ALMA receivers and technical papers this memo provides a comprehensive list of papers for the scientific community. This list was produced in discussion with the scientific and instrumentalist community based on a June 2024 survey at the European Southern Observatory workshop on the ALMA Wideband Sensitivity Upgrade as well as with the ALMA technical staff. The authors now encourage the community to enhance their alreadyexcellent ALMA science with the appropriate references to ensure future submillimetre instrumentation can keep addressing the key questions about our Universe.	2024-09-01T00:00:00Z	['arXiv:2409.02164', '2024arXiv240902164B', '10.48550/arXiv.2409.02164']	['Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics']	Recommended receiver papers for ALMA users	2,024	190	0.5	['EPRINT_HTML', 'EPRINT_PDF']	1	https://arxiv.org/pdf/2409.02164.pdf	{'Recommended receiver papers for ALMA users': 'Tom Bakx 1 and John Conway 2 , Chalmers University of Technology, Sweden', '5th September 2024': "1. Introduction. In an ideal world, observational astronomers and instrumentalists are in a natural symbiosis, where state-of-the-art instruments allow for observations that challenge our understanding of the fundamental properties of our Universe and everything within it. In this symbiosis, astronomers acknowledge the contributions of the instrumentalists, in particular via citing instrumental academic papers, which in turn enables the further development of science-guided instruments via increased access to research funding. In contrast to this ideal, the total number of ALMA science papers that make use of a given ALMA receiver band currently outstrips the number that cites that band's instrument paper by a factor of between 6 and 300 (see Figure 1). \nFigure 1: The number of publications (as of May 2024) associated with each band according to the ALMA science archive after accounting for duplicates and multi-band observations. The number of citations of the band instrument description paper is overlaid, showing the discrepancy between the scientific usage of the ALMA receivers and the academic recognition of the instrumentalists who have built these receivers. \n<!-- image --> \nNote - this memo solely contains the opinion of the authors formed in discussion with the astronomer and instrument builder communities, and does not reflect the official policy of the ALMA telescope.", '2. Recommended Receiver Papers.': 'In this memo, we aim to facilitate the access to recommended receiver instrument references (see Table 1) in an effort to boost citation numbers. Such community-wide recognition will greatly assist instrument researchers in delivering future projects, including the development of new receivers for the Wide-band Sensitivity Upgrade (WSU; Carpenter et al. 2023). Although receiver citation information is already (partially) available in Table 4.2 (page 31 of the current version) of the ALMA Technical Handbook at \nhttps://almascience.eso.org/documents-and-tools/cycle11/alma-technical-handbook, this memo updates this receiver citation information and gives direct ADS hyperlinks to the relevant papers to facilitate citing these receiver publications in future ALMA astronomy papers. \nTable 1 lists for each receiver band the recommended references, and has been produced through direct communication with each relevant receiver group. For Bands 1 and 2, the final papers describing these receivers are in preparation with likely completion within approximately 6 months; in the meantime the relevant receiver groups have confirmed that the references given in Table 1 should be used. When final papers are available for these two bands an amendment to this memo will be submitted. For some receiver bands, there are multiple references that together give the full technical description of the receiver; these additional references, including a reference to the local oscillator development common to all receiver bands (excluding future Band 2), are given in the Appendix Table 2. \nAlthough this memo concentrates on the recommended citations for ALMA receivers, there are many other papers relevant to the technical development of ALMA that astronomers may also wish to cite. An incomplete list of such non-receiver technical papers is given in Appendix Table 3. It should be noted that, unlike for the receiver papers, the authors of this memo have not made contact with the development groups to confirm these references. The authors have taken efforts to produce a complete and verified list of technical references in this memo, and invite missed or further references for updated versions of this memo. \nTable 1: Recommended paper references for ALMA receivers (Cit. indicates the total number of citations to the listed paper in both the astronomy and instrumentation literature, and Pub. gives the number of astronomy papers that have made use of that receiver as of May 2024). \n- Note * The proposed papers to cite for Band 1 and 2 are preliminary pending the publication of final instrument papers that are currently in preparation, (expected within 6 to 8 months).\n- 3. Community feedback. At the June 2024 ESO Workshop on the ALMA Wideband Sensitivity Upgrade, which was well attended by both astronomers and instrumentalists, discussions were held on the issue of instrument citations. In addition, during the meeting thirty participants participated in the open questionnaire which provided useful context and suggestions around the instrument citation issue. \nThe discussion and survey showed in general strong support amongst astronomers for acknowledging the work of instrumentalists via citations. Two-thirds of those responding to the survey noted that a citation policy within the ALMA consortium would be desirable to give guidance as to which technical citations should be included in papers. The replies were divided on whether instrumental references are best placed at the first mention of the instrument in the observation section of a paper or within the typical acknowledgement to the ALMA datasets at the end of a paper. Given there is currently no official policy on this we suggest here that astronomers, if they wish to acknowledge the contribution of their instrumentation colleagues, make their own choice of where in the paper to add such a citation given that both options would help alleviate the citation issue highlighted in Figure 1. Experience with our discussions with instrumentalists strongly indicates that citations, wherever they are placed in a paper, are much more valuable than giving written \n- acknowledgements to receiver groups. \nAn important point brought up in the survey was the difference between the journals that instrumentalists (e.g., IEEE) and observers (e.g., A&A, ApJ) use to publish and the fact that astronomers may not have access through their institute to the full text of the instrument paper they would be citing. One recommendation to alleviate this issue is to publish the most recent version of the instrument paper to a preprint server (or other open web site) accessible to astronomers so everyone can read the references. Beyond efforts from the community, and from within ALMA, to encourage the appropriate referencing of receivers, two participants point out that the astronomical journals should also consider reminding the observers for adequate referencing of the receivers used in submitted manuscripts. \nTwo participants noted that the citations should liberally include other ALMA technical aspects and operations. To accommodate these comments, and to celebrate the associated efforts of the researchers that enable the operation of ALMA as a facility, we provide a partial list of such references in Table 3. In the future, an online tool to facilitate the search for the appropriate references (hosted, for example, on the ALMA web page as suggested by three participants) would mark the ideal solution to the citation issue, and will furthermore help the citation of updated papers in the future.', 'Appendix - Additional references': 'This appendix gives additional receiver references (Table 2) and a partial list (Table 3) of suggested non-receiver references to other technical aspects of the current ALMA system. \nTable 2: Additional receiver references \nTable 3: Additional ALMA non-receiver technical references'}
2024ApJ...975..178K	The James Webb Space Telescope JWST has uncovered a ubiquitous population of dustobscured compact sources at z 4. Many of these objects exhibit signs of active galactic nucleus AGN activity making their study crucial for understanding the formation of supermassive black holes SMBHs and their growth with host galaxies. In this work we examine low and mediumresolution JWSTNIRSpec spectra from the JADES GTO public data release in the GOODSN field of a red luminous M SUBBSUB 22.2 mag and compact lt500 pc source at z 4.13. The restoptical SUBrestSUB gt 4000 continuum of this source is strongly dominated by a massive logSUB10SUBM SUBSUBM SUBSUB 10.6 quenched logSUB10SUBsSFRyrSUP1SUP lt 11 galaxy as indicated by the clear presence of a Balmer break and stellar absorption lines. Star formation history modeling reveals a starburst episode followed by rapid quenching about 200 Myr ago. The spectrum shows extremely broad FWHM 2500 km sSUP1SUP H emission and elevated optical line ratios indicating an actively accreting SMBH. Moreover our work has potentially revealed clear AGN signatures in the restUV in little red dots for the first time via the detection of a strong Ly emission and a broad Mg II doublet. The derived black hole mass of logSUB10SUBM SUBBHSUBM SUBSUB 7.3 results in M SUBBHSUBM SUBSUB 0.04 consistent with the local relations unlike the elevated ratios in other highz reddened AGN. Finally we use JWST data from AGN at z 410 to explore an evolutionary link between highz reddened AGN early quiescent galaxies and local ellipticals.	2024-11-01T00:00:00Z	['2024ApJ...975..178K', '10.3847/1538-4357/ad7d03', 'arXiv:2407.20320', '2024arXiv240720320K', '10.48550/arXiv.2407.20320']	['Active galactic nuclei', 'High-redshift galaxies', 'Early universe', 'Galaxy quenching', 'Quenched galaxies', '16', '734', '435', '2040', '2016', 'Astrophysics - Astrophysics of Galaxies']	Silencing the Giant Evidence of Active Galactic Nucleus Feedback and Quenching in a Little Red Dot at z 4.13	2,024	190	0.65	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	17	https://arxiv.org/pdf/2407.20320.pdf	{'Silencing the Giant: Evidence of AGN Feedback and Quenching in a Little Red Dot at z = 4 . 13': "Vasily Kokorev, 1 John Chisholm, 1 Ryan Endsley, 1 Steven L. Finkelstein, 1 Jenny E. Greene, 2 Hollis B. Akins, 1 Volker Bromm, 1 Caitlin M. Casey, 1 Seiji Fujimoto, 1, 3 Ivo Labb'e, 4 and Rebecca L. Larson 5 \n1 \nDepartment of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 2 Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544 3 Cosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen, Jagtvej 128, København N, DK-2200, Denmark 4 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Melbourne, VIC 3122, Australia 5 School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623, USA \n(Received n/a; Revised n/a; Accepted n/a) \nSubmitted to ApJ", 'ABSTRACT': 'The James Webb Space Telescope ( JWST ) has uncovered a ubiquitous population of dust-obscured compact sources at z ≳ 4. Many of these objects exhibit signs of active galactic nucleus (AGN) activity, making their study crucial for understanding the formation of supermassive black holes (SMBHs) and their growth with host galaxies. In this work, we examine low and medium resolution JWST /NIRSpec spectra from the JADES GTO public data release in the GOODS-N field of a red, luminous ( M B ∼ -22 . 2 mag) and compact ( < 500 pc) source at z = 4 . 13. The rest-optical ( λ rest > 4000 ˚ A) continuum of this source is strongly dominated by a massive (log 10 [ M ∗ /M ⊙ ] ∼ 10 . 6), quenched (log 10 [sSFR/yr -1 ] < -11) galaxy, as indicated by the clear presence of a Balmer break and stellar absorption lines. Starformation history modeling reveals a starburst episode followed by rapid quenching about 200 Myr ago. The spectrum shows extremely broad (FWHM ∼ 2500 km/s) H α emission and elevated optical line ratios, indicating an actively accreting SMBH. Moreover, our work has potentially revealed clear AGN signatures in the rest-UV in LRDs for the first time, via a detection of a strong Ly α emission and a broad Mg ii doublet. The derived black hole mass of log 10 ( M BH /M ⊙ ) ∼ 7 . 3 results in M BH /M ∗ ∼ 0 . 04 %, consistent with the local relations, unlike the elevated ratios in other highz reddened AGN. Finally, we use JWST data from AGN at z = 4 -10 to explore an evolutionary link between highz reddened AGN, early quiescent galaxies, and local ellipticals. \nKeywords: Active galactic nuclei (16), High-redshift galaxies (734), Early universe (435), Galaxy quenching (2040), Quenched galaxies (2016),', '1. INTRODUCTION': "Observations over the past few decades have established that super massive black holes (SMBHs) exist in the cores of nearly all massive galaxies locally (Kormendy et al. 1997; Richstone et al. 1998; Kormendy & Ho 2013). The growth of SMBHs is driven by gas accretion, designating such black holes as an active galactic nucleus (AGN). Powerful AGN drive the energy and momentum in the interstellar medium (ISM) of galaxies, \nCorresponding author: Vasily Kokorev \[email protected] \nheating up the gas and slowing down star formation. As such the growth of stars in galaxies and the physics of their SMBHs are tightly linked (Heckman & Best 2014). \nThe James Webb Space Telescope ( JWST ) is at the center stage of discovery of highz AGN, getting ever closer to their epoch of formation. The first two years of its operation have marked a discovery of numerous, previously-missing, UV-faint AGN, shifting our paradigm of SMBH formation (Smith & Bromm 2019). These have been identified through a combination of directly observing broad lines (Type I; Goulding et al. 2023; Harikane et al. 2023; Juodˇzbalis et al. 2024; Kocevski et al. 2023; Larson et al. 2023; Maiolino et al. 2023a,b; Matthee et al. 2023; Ubler et al. 2023), inferring \ntheir AGN nature through ionization (Type II; Chisholm et al. 2024; Scholtz et al. 2023), or some combination of color and morphology (Barro et al. 2023; Iani et al. 2024; Yang et al. 2023). \nAmong these, an enigmatic population of compact red sources, so called 'little red dots' (LRDs; Matthee et al. 2023), stands out in particular. These show a characteristic double-break feature in their spectral energy distributions (SEDs), with seemingly dust-free UV emission and a steep red slope in the rest-optical (Furtak et al. 2023; Kocevski et al. 2023; Labb'e et al. 2023a). These were initially speculated to be massive compact galaxies at early epochs (Barro et al. 2023; Labb'e et al. 2023b), however spectroscopic follow-up observations in many of these sources have shown clear evidence of broad Balmer emission and high-ionization, all implicating them as actively accreting SMBHs (Fujimoto et al. 2023; Furtak et al. 2024; Greene et al. 2024; Killi et al. 2023; Kocevski et al. 2023; Kokorev et al. 2023a; Matthee et al. 2023; Ubler et al. 2023). \nStaggeringly, SMBHs in a large number of highz AGN, including LRDs, are overmassive, with the ratio between their black hole and stellar mass ( M BH /M ∗ ) reaching upwards of ∼ 30 -50 % (Bogdan et al. 2023; Goulding et al. 2023; Juodˇzbalis et al. 2024; Kokorev et al. 2023a) which significantly deviates from the local relations (e.g. Greene et al. 2016) by more than a few orders of magnitude (Pacucci et al. 2023; Pacucci & Loeb 2024). Seemingly the growth of these black holes completely outpaces that of their host galaxies, at least at early times (e.g., Jeon et al. 2024). In contrast, quite a few of these systems have been shown to be fully consistent with the local M BH -σ relations (Maiolino et al. 2023b), which could point toward the presence of gas reservoirs that sustain the growth of the central AGN, but which are inefficient at forming stars. \nDetailed spectroscopic (Greene et al. 2024; Maiolino et al. 2023b; Matthee et al. 2023) and photometric (Akins et al. 2024; Kocevski et al. 2024; Kokorev et al. 2024) forays into their number densities reveal LRDs to be much more abundant compared to the UV-selected quasars at z ∼ 4 -6. They appear to account for a significant fraction of all broad-line selected AGN (Harikane et al. 2023) and even for a few percent of the total galaxy population at z > 6. The over-abundance of AGN with large BH masses relative to local scaling relations in the first few billion years of cosmic evolution can therefore truly test our models and preconceived vision of BH seeding as well as their evolution alongside their host galaxies (Dayal et al. 2024; Dayal 2024; Natarajan et al. 2023). \nWhat has yet remained unclear is the origin of the observed rest-UV emission in LRDs. So far, the interpretations range from scattered light from the AGN itself or an unobscured host galaxy component (Akins et al. 2023; Barro et al. 2023; Greene et al. 2024; Kocevski et al. 2023, 2024; Labb'e et al. 2023a). Given the similarities between the observed UV slopes of quasars and starforming galaxies, neither deep JWST spectra (Greene et al. 2024; Kokorev et al. 2023a), nor vast photometric samples of LRDs (Akins et al. 2024; Kocevski et al. 2024; Kokorev et al. 2024) have been able to make a conclusive determination. Lacking a complete understanding of the rest-UV side of LRDs prevents us from robustly deriving properties such as the stellar mass ( M ∗ ), star formation rate (SFR), dust attenuation ( A V ) and ionization state in these sources. This in turn limits our ability to attain an accurate view of the impact the AGN has on its host. \nFinally, despite hiding behind a thick veil of dust, these objects can be quite bright, sometimes reaching observed F444W magnitudes of 22 -23 (Akins et al. 2024; Kokorev et al. 2024; Wang et al. 2024; Labb'e et al. in prep.), and yet a similar class of objects has not been detected at either intermediate ( z ≲ 4) redshifts or locally (Kocevski et al. 2024). Seemingly this complete absence of overmassive SMBHs at late times implies that such AGN only exist at early times and on short timescales, followed by an uncertain descendant population. Recent works by Wang et al. (2024) and Labb'e et al. (in prep.) show that some reddened broadline (BL) AGN already have quite evolved stellar populations by z ∼ 4, as implied by their strong Balmer breaks and absorption lines. Therefore, older LRDs may be positioned more in line with quiescent galaxies found at similar redshifts (Carnall et al. 2023; de Graaff et al. 2024), rather than extreme AGN at highz . Could this be the next step of overmassive SMBH evolution? \nIn this work, we report the discovery of BL AGN emission in an otherwise stellar dominated galaxy at z = 4 . 13. The deep JWST /NIRSpec Micro-Shutter Assembly (MSA) PRISM and medium resolution grating spectra in the GOODS-N field (D'Eugenio et al. 2024), allow us to identify the presence of a broad H α emission alongside an an array of [O iii ], [N ii ] and [S ii ] lines, all showing line ratios typical for AGN activity. At the same time, we also observe a strong Balmer break and a multitude of Balmer absorption lines indicative of an evolved stellar population. While this galaxy is completely consistent with the little red dot (LRD) color and compactness criteria, the observed red rest-optical color likely originates from an evolved stellar population, rather than from the reddened AGN continuum emission. Through the examination of black hole to \nhost mass ratio we conclude that GN-72127 has more in common with quiescent galaxies (Carnall et al. 2023; de Graaff et al. 2024) and local ellipticals rather than the extreme highz AGN observed with JWST . Using that information we attempt to draw an evolutionary sequence that connects objects with elevated M BH /M ∗ at highz to the evolved galaxies with residual AGN activity we observe later on. Moreover, our work has potentially revealed clear AGN signatures in the rest-UV in LRDs for the first time, via a detection of a strong Ly α emission and a broad Mg ii doublet. We present the data in Section 2, line and continuum fitting as well as size measurements in Section 3, galaxy and black hole properties in Section 4, and the final discussion in Section 5. \nThroughout this work we assume a flat ΛCDM cosmology (e.g. Planck Collaboration et al. 2020) with Ω m , 0 = 0 . 3, Ω Λ , 0 = 0 . 7 and H 0 = 70 km s -1 Mpc -1 , and a Chabrier (2003) initial mass function (IMF) between 0 . 1 -100 M ⊙ . All magnitudes are expressed in the AB system (Oke 1974).", '2. OBSERVATIONS AND DATA': "In our work we use the data obtained as a part of the JWST Advanced Deep Extragalactic Survey (JADES; Bunker et al. 2023; Eisenstein et al. 2023a,b), which is based on more than 770 hours of the GTO time with the NIRCam and NIRSpec instruments. In particular, we focus our efforts on the public JADES Data Release 3 (DR3), which covers the GOODS-North field (Giavalisco et al. 2004) and includes the NIRCam coverage in 7 broad-band (F090W, F115W, F150W, F200W, F277W, F356W, F444W) and 2 medium-band (F335M, F410M) filters. The data are fully publicly available, including the reduced spectra and catalogs 1 . \nMost pertinent to our work, this field was targeted by NIRSpec Multi-Shutter Array (MSA) over 4 separate MSA observations. Briefly, each observation consisted of 3 pointings, each with a small spatial offset and used the PRISM ( R ∼ 100), G140M, G235M and G395M ( R ∼ 1000) gratings. Observations used in our work were completed between April and May 2023. Details concerning both the photometric and spectroscopic data reduction, calibration and extraction of the spectra can be found in Bunker et al. (2023) and D'Eugenio et al. (2024). \nWe, however, would like to note a few crucial details. Factors such as path-loss correction, flux calibration uncertainty, slit position, source location within the slit, \nand source self-subtraction can affect the normalization, especially when combining observations from multiple MSA pointings. To address this, the JADES team has rescaled the spectra by convolving the 1D spectra with all NIRCam filters and correcting them by fitting a firstorder polynomial to the ratio between NIRCam and convolved NIRSpec flux densities (similar to Greene et al. 2024; Kokorev et al. 2023a; Price et al., in prep). We verified the consistency between photometry (D'Eugenio et al. 2024) and spectra for all dispersers, and find the results satisfactory. \nSecond, the MSA configuration employed by the JADES team allows for the contamination of the spectra by other objects in the same detector row. However, for bright objects with multiple line detections, and robust spectroscopic redshifts, the contaminant lines are easy to identify and mask out.", '3. DATA ANALYSIS': 'The data from JADES in the GOODS-N field are excellently suited to explore the enigmatic rest-frame UV emission of compact obscured AGN. The available JWST photometry covers a complete wavelength range from 0.4 - 5 µ m, in 7 broad and 2 medium bands, reaching a median 5 σ depth of 29.4 AB mag in F444W filter allowing for robust target pre-selection. Moreover, the exquisite coverage provided by all 3 medium resolution dispersers, especially by the bluest G140M grating, and the PRISM allows for robust identification of spectral features such as broad and narrow lines in both rest-UV and rest-optical parts of the spectrum.', '3.1. Target Selection': "In the last year many BL AGN have been successfully identified via their broad band colors and restframe optical compactness, and subsequently verified via the presence of broad lines in their rest-optical spectra (Greene et al. 2024; Furtak et al. 2024; Killi et al. 2023; Kocevski et al. 2023; Kokorev et al. 2023a; Wang et al. 2024). Namely, using a combination of broad-band NIRCam colors covering the characteristic 'break' feature present in LRDs and a compactness criterion was shown to be extremely successful ( > 80%) at identifying broad line AGN with photometry and size alone. \nTo identify our targets we use the same criteria as Labb'e et al. (2023a), Greene et al. (2024), and Kokorev et al. (2024), aiming to maintain a consistent approach in selecting LRDs, particularly regarding aperture sizes. These are as follows: \nred1 = F115W -F150W < 0 . 8 & \nF200W -F277W > 0 . 7 & \nF200W -F356W > 1 . 0 \nFigure 1. Top: JWST /NIRCam 2 . '' 0 stamps and the RGB short and long wavelength color images comprised of the F115W, F150W, F200W and F277W, F356W, and F444W bands, respectively. The MSA slitlet layout (white) is overlaid on a separate LW color image, for clarity. The source has a very clear PSF-dominated morphology present in all filters. Each panel shows the total magnitude as presented in the JADES DR3 catalog (D'Eugenio et al. 2024). The source is extremely bright and is detected in all bands at > 50 σ . Middle: 2D MSA PRISM spectrum. Bottom: Optimally extracted 1D spectrum of the galaxy in the observed frame. We show the data in black, while the uncertainty on the spectrum is in dashed red. Best-fit spline continuum and line msaexp model to the data is shown in solid red. Assuming the best-fit to the PRISM with z spec = 4 . 129 ± 0 . 035, we show the positions and label the prominent emission with significant ( > 3 σ ) detections as solid vertical lines. Emission lines for which we only obtain an upper limit are shown with dashed lines. The position of the Balmer/4000 ˚ A break is shown with a shaded region. \n<!-- image --> \nor \nred2 = F150W -F200W < 0 . 8 & F277W -F356W > 0 . 6 & F277W -F444W > 0 . 7 , \nplus the compactness cut given by: \ncompact = f f444w (0 . '' 4) /f f444w (0 . '' 2) < 1 . 7 . \nThe final selection is then ( red1 \| red2 ) & compact . \nThe JADES GOODS-N DR3 public photometric catalog includes both circular aperture and total (i.e., Kron) photometry for 85,709 sources in the GOODS-N field (D'Eugenio et al. 2024). Since we are focusing on compact sources, we specifically use D =0 . '' 3 apertures ( CIRC2 ) and convert them to total fluxes using the aperture corrections provided by the JADES team. These smaller apertures offer a good balance for reliably measuring fluxes of point-like objects (e.g., see Kokorev et al. 2023a; Kocevski et al. 2024; Labb'e et al. 2023a; Trus- \net al. 2024), while keeping aperture corrections relatively low. Finally, we also require for the object to be covered by all 3 medium resolution gratings and the PRISM, and also have a valid redshift from the JADES catalog (D'Eugenio et al. 2024). \nUsing this selection method, we first identify 11 LRDs in the data ranging from z ∼ 4 -7, all targets have clear rest-optical line detections in the PRISM, as well as the G235M and G395M gratings. Moreover, 4/11 LRDs that we find were already confirmed and presented as BL AGN in a work by Maiolino et al. (2023b). Not all however have significant line detections in the restUV, covered by the G140M grating and PRISM. This is imperative to our study as we aim to simultaneously examine the rest-UV and rest-optical properties of LRDs. Despite that, we find that a single source from the initial 11 stands out in particular, owed to its broad lines in the rest-optical and multiple line detections in the rest-UV. \n. \n. \nTable 1. Source Properties \nWe thus identify GN-72127, a potential AGN candidate at z = 4 . 1288, located at R.A. = 189 · .265718, decl.= 62 · .168393 (Table 1). Consistent with LRDs selected in previous studies, GN-72127 is red in the restoptical but remains visible, albeit moderately red, in the rest-UV. It is also very bright in the F444W broad-band filter ( ∼ 23.1 mag) while remaining completely point-like (see Section 3.9). Of prime interest to our work, however, is the spectral coverage of GN-72127. A combination of NIRSpec/PRISM observations and, most crucially, the exquisite coverage by three medium-resolution dispersers ( t int =3107 s each) provides the most comprehensive set of data to date for investigating the physics of these reddened AGN. We present 2 . '' 0 cutouts of the source in each medium and broad-band NIRCam filter alongside a PRISM spectrum in Figure 1. \nTable 2. Measured emission line fluxes † and rest-frame equivalent widths of absorption lines (EW 0 )Absorption Lines \n- † No dust correction was applied.\n- 1 Derived from the best-fit BAGPIPES stellar model. \nDespite clear AGN signatures in LRDs, a vast majority of them remains X-ray undetected (e.g. see Akins et al. 2024; Greene et al. 2024; Maiolino et al. 2024; Pacucci & Narayan 2024). In line with this, we also find nothing at the position of the source in the deep ( ∼ 170 . 43 ks) Chandra data (Brandt 2001).", '3.2. Initial Line Identification': 'We start our investigation of all available low and medium resolution spectra by running a heavily modified version of msaexp (Brammer 2022). Briefly, msaexp models emission and absorption lines as Gaus- \nsians and fits the continuum as a series of cubic splines, albeit at a fixed (and user defined) velocity width for all lines. Our key modifications 2 to the code include making the line width a free parameter, and also adding an option to fit the same line with multiple components (i.e., narrow and broad). For each available spectrum of the source, we allow the fit to search for the best fit within a narrow redshift range (∆ z ∼ 0 . 02), with the prior set by drawing from the JADES spectroscopic catalog. We overlay the best fit to the PRISM in Figure 1. \nWe find that the z spec derived from the PRISM and 3 medium resolution dispersers match up quite well, without any significant offsets ( < 150 km/s). With this approach we identify a number of the key features. These are - a potential broad Mg ii and H α emission, strong [O iii ] and [N ii ] doublets, a strong Balmer break as well as the higher order (H γ through H η ) Balmer absorption features (Figure 2). The latter two imply that the rest-optical continuum could be dominated by an older stellar population. All of the above are securely identified in PRISM as well as the gratings. We also note a very prominent Ly α emission, albeit only in the PRISM, which has insufficient spectral resolution to compute any systemic velocity offset. \nWe note that not all spectral lines can be modeled well by our routine. While it is adequate at fitting the majority of narrow and broad spectral features rather quickly, which is useful for large samples of spectra, its reliability is quickly diminished by intricacies of certain line complexes. This is especially apparent when uncommon line ratios are required, in the regions where S/N is low or the continuum is simply too noisy. A robust identification of certain spectral features, such as broad lines, is imperative to properly classify the object. Therefore, we will focus on certain parts of the spectrum with a more sophisticated procedure, which we will describe in subsequent sections.', '3.3. G395M: Broad H α +[N ii ]': 'Compared to the PRISM the G395M grating has a much higher spectral resolution which allows us to resolve the, otherwise blended, emission from the [N ii ] doublet and H α . The [S ii ] doublet remains blended however. Crucially, the availability of multiple spectral pixels sampling the line complex enables us to try and fit both broad and narrow components to the Balmer emission. While we do not detect any contaminant lines in the spectrum we note the presence of corrupted data (NaN values) in the spectrum, likely a result of issues during data reduction or 1D spectrum extraction. These \nfeatures are, however, minor and do not affect our fitting routine. In addition, as we already mentioned in Section 2, due to the MSA configuration employed by the JADES team, the spectra can overlap, resulting in additional lines from neighboring sources. However, as the redshift for the source is known precisely, these contaminant lines are easily masked out. \nWe fit each line complex simultaneously alongside the continuum. Each emission line is modeled with a standard Gaussian profile, with the position allowed to vary within a narrow redshift range around the bestfit msaexp solution. We fix the ratio between the [N ii λ 6549 ] and [N ii λ 6585 ] to 1:3, respectively, as is normally done in the literature (e.g. see de Graaff et al. 2024). We also fit the H α emission with both narrow and broad components. The ratio between lines in the [S ii ] doublet is allowed to vary freely. All narrow line velocities are tied together and we assume the same redshift for all lines. Finally, we model the local continuum with a first order polynomial. Given the limited number of spectral pixels we do not include an absorption component when we fit the H α , as the fit would become too degenerate. Instead, to account for H α absorption, we use the best-fit stellar continuum model fitted to the host galaxy as described in Section 3.8. The derived line fluxes of H α and [N ii ] lines are then corrected for the stellar absorption. Generally however, the EW of the stellar absorption is comparable for all Balmer lines, meaning that the redder (i.e. H α ) absorption lines have a weaker contribution to the spectra, compared to the higher order lines, and are thus less important to take into account. \nThe fit is initialized by creating all of the models on an oversampled grid, which is then interpolated onto the wavelength axis that mimics the heterogeneous grating resolution. Finally, we allow for a custom over sampled spectral resolution scaled by a factor of 1.3. It was shown that a spectral resolution of a point-like source can be higher than that of a source uniformly illuminating the slitlet (see e.g. de Graaff et al. 2023; Greene et al. 2024; Kokorev et al. 2023a). The best fit is a found via a nonlinear least-squares minimization, with the MCMC uncertainty derived from the covariance matrix. \nFrom the fit we derive a redshift of z spec = 4 . 1294 ± 0 . 004, FWHM of the narrow lines (H α , [N ii ], [S ii ]) equal to 510 ± 72 km/s and a broad H α component with a width of 2573 ± 585 km/s. We strongly detect the [N ii ] and [S ii ] doublets, and recover both narrow and broad Hα components at 3 σ level. To verify whether a broad line fit is required at all, we perform additional modeling without allowing for a broad H α emission. To verify the significance of this broad-line we calculate the Bayesian \nInformation Criterion (BIC) difference between the narrow+broad and narrow lines-only fits. We find the narrow only fit to be an inadequate representation of the data (BIC narrow -BIC narrow+broad =∆BIC > 20), which indicates a very strong evidence (using the criteria defined in Jeffreys 1961) for the broad component being present in H α . \nWe measure log 10 ([N ii ] λ 6586 /H α )=0 . 15 +0 . 24 -0 . 16 , which is too high to have originated from star-formation (Kewley et al. 2001; Grossi et al. 2009). The [S ii ] doublet at λλ 6717,6731 ˚ A is another useful diagnostic for gas ionization when coupled with the adjacent H α line. We also find it to be elevated, with log 10 ([S ii ] λλ 6717 , 6731 ]/H α )= -0 . 38 +0 . 21 -0 . 13 . While it is possible to reach these high line ratios via e.g. shocks, the presence of a robust broad H α emission exceeding 2000 km/s, much faster than could be explained by shocks (Allen et al. 2008), gives us the first hint as to the presence of an AGN (Kewley et al. 2006) in this otherwise seemingly stellar-light dominated galaxy. The best-fit is presented in the rightmost panel of Figure 2. For now we will refrain from making a definitive statement regarding the AGN presence and will explore other gratings.', '3.4. G235M: H β and [O iii ]': 'We continue our investigation by shifting to a bluer medium resolution disperser - G235M (middle panel of Figure 2). Following the same fitting procedure as before, we now fit the H β line with two components (emission and absorption), and the [O iii ] where we do not fix the ratio between the lines. We assume the same velocity dispersion for all emission lines and leave the velocity of the absorption line to be independent. The H β is fit with only a single narrow component. \nFirstly, we derive a z spec = 4 . 1307 ± 0 . 005, consistent with the G395M, FWHM of the emission lines equal to 529 ± 81 km/s and a largely unconstrained broad absorption component with a width of 2429 ± 835 km/s. We strongly detect the narrow [O iii ] doublet and find the FWHM to match that of the narrow lines in the G395M grating. Therefore it is likely that H α , H β , [N ii ], [O iii ] and [S ii ] originate from the same region in the galaxy. We also confirm a very weak H β emission, marginally detected at ∼ 1 . 5 σ level. At the same time we also can clearly detect an H β line in absorption, however the noise present around that feature does not allow us to securely identify the width of that feature. While uncertain, what is clear is that the absorption profile is wider compared to the emission lines. We will focus on all available absorption features and their widths in the next sections. Notably, while the narrow H β is largely undetected we can still use this information and de- \nwer limit on the log 10 ([O iii ] λ 5007 /H β ) > 0 . 50. This high value again puts us at odds with having originated purely from star formation.', '3.5. G140M: Mg ii': 'Finally we focus on the last available grating the G140M. As is typical of LRDs, the object is rather faint in the rest-UV as such we do not detect a large array of lines in this disperser compared to the other ones, however one feature stands out. We perform a fit to the Mg ii doublet ( λλ 2796 , 2803) with a narrow and broad component for each, as shown in the left panel of Figure 2. During the fit we allow the Mg ii 2796 /Mg ii 2803 ratio to vary freely. We find the FWHM of the narrow component to be equal to 482 ± 371 km/s, not constrained as well as for the other narrow lines in the spectrum, but still consistent within the uncertainty. Including the broad component, we find its width to be equal to 1100 ± 535 km/s, marginally detected at ∼ 2 σ level. As before we attempt the same fit with a narrow only Mg ii doublet, and find it to be moderately worse (∆BIC=2.5), compared to the narrow+broad fit. We also find that the Mg ii 2796 /Mg ii 2803 for both narrow and broad lines is consistent with the accepted ranges ∼ 0 . 5 -1 within the uncertainty (Laor et al. 1997; Wang et al. 2009). \nWhile the evidence is not as strong as for the broad H α , we will refrain from making a definitive statement on whether the broad Mg ii emission is present in the rest-UV spectrum of the source. This possibility however can not be ruled out either, as it does appear that a broad-line component makes the fit better. The potential presence of broad Mg ii emission provides us with another piece of the evidence that an active black hole might be present in GN-72127. \nJust like the broad hydrogen features, broad Mg ii lines are also formed in the BLR, but typically originate slightly farther out from the central black hole compared to e.g. H α . This difference in location can result in somewhat lower velocities for Mg ii -emitting gas clouds (see e.g. Wang et al. 2009), which is what we find in our fit. In addition, the S/N of the G140M at the position of Mg ii barely allows us to model the continuum, as such broader Mg ii wings are obscured by the noise, making our derived line width lower than what it is in reality.', '3.6. Balmer Decrement': 'It is well established that the ratio between observed fluxes of Balmer series lines can be used to determine the dust extinction. Given the quiescent nature of the source we have a strong reason to believe that all observed emission lines originate from either the NLR or the BLR around an AGN. As such applying the dust at- \nFigure 2. Top: A series of panels showing fits to the main line series of interest. From left to right these are, a tentative broad line fit to the Mg ii doublet; H β and [O iii ] doublet; H α , [N ii ] and [S ii ] doublets. We show best-fit narrow lines in blue, apart from the [N ii ] which is shown in orange for clarity. The broad emission (or absorption) best fits are shown in green. The total combined model including the continuum is plotted in red. The original spectrum is in black and the error spectrum is dashed red. Bottom: Full collection of all available medium and low resolution MSA spectra. These are as follows, PRISM (black), G140M (violet), G235M (blue) and G395M (red). The presence of broad lines in the G140M and G395M spectra is our first hint of AGN activity. \n<!-- image --> \ntenuation derived from the continuum would not be adequate to derive AGN properties. To compute the A V we use the narrow line ratio - H α /H β computed from our fits to the G395M and G235M dispersers. As mentioned in Section 3.4, the narrow H β line is only detected at 2 σ , so our A V would only be an approximation. By doing this we find H α /H β ≈ 4 . 28. Provided the same ratio holds between narrow and broad emission lines, we also confirm that the broad component in H β would not be detectable in our data (EW 0 ∼ 5 ˚ A), thus justifying our narrow only fit to H β in Section 3.4. \nIn line with other works on reddened quasars (e.g. see Hopkins et al. 2004), highz galaxies (Capak et al. 2015; Reddy et al. 2015, 2018), and LRDs themselves (e.g. Furtak et al. 2024; Greene et al. 2024; Kokorev et al. 2023a) we use the Small Magelannic Cloud (SMC) reddening law (Gordon et al. 2003). Assuming that case \nB recombination applies in the NLR of an AGN (Osterbrock 1989), we find that the observed line ratio gives us a moderate attenuation of the region around the central black hole equal to A V = 1 . 1 +0 . 8 -0 . 1 .', '3.7. Other Notable Lines': 'Beyond the lines discussed in earlier sections, the spectra reveal a multitude of other emission and absorption features. As these are quite numerous, considering the volume of data for this object alone, we will not focus on each one individually, but rather point out some that could be of particular interest to our study. The derived fluxes for all lines in our work can be found in Table 2. \nWe fit the Ly α emission in the PRISM spectrum using a single Gaussian model, allowing both the FWHM and redshift to vary. Due to a limited spectral resolution of the PRISM at this wavelength ( R ∼ 60), we are un- \nFigure 3. A fit to the H γ and [O iii ] λ 4363 lines. The spectrum is in black and the total fit is in red. The error spectrum is shown in dashed red. The velocity widths and redshift were fixed to the values form the previous fits. The derived intrinsic [O iii ] λ 4363/[O iii ] λ 5007 of ∼ 0 . 5 is indicative of extremely high ionization, likely induced by AGN activity. \n<!-- image --> \ne to precisely measure the width or offset of this line from the systematic redshift. However, the line spans 3 spectral pixels, each approximately 900 km/s wide in the rest frame, suggesting that the Ly α emission in GN72127 could be broad. Notably, the line is quite strong, with a measured rest-frame equivalent width (EW 0 ) of 400 ± 50 ˚ A, significantly exceeding the typical values found in star-forming galaxies at z ∼ 4 . 5 (Zheng et al. 2014; Hashimoto et al. 2017). \nIn the G235M spectrum we identify the auroral [O iii ] λ 4363 line, which can be used to trace both ionization and metallicity. It has been suggested that at higher metallicities [O iii ] can blended with the adjacent [Fe iii ] λ 4360 line (e.g. see Curti et al. 2017). We attempt two fits, one including the [Fe iii ] line and one without it. In both cases we fix the velocity widths and redshift to the other narrow lines. Our results indicate that the [O iii ] line by itself better represents the data, as opposed to fitting it alongside [Fe iii ] (∆BIC ∼ 3). Given that there are only so many spectral pixels covering the line, we believe that the fit with a λ 4360 feature line is simply too degenerate to yield an adequate BIC. Therefore, we cannot entirely rule out the possibility that [O iii ] λ 4363 is free from contamination. This is well reflected in our uncertainties on the derived line flux. We show the best fit to the line in Figure 3. \nFinally, we note the presence of the O i λ 8446 line detected in the PRISM spectrum at ∼ 3 σ significance Table 2. The O i line is produced through the Ly β fluorescence process which involves the absorption of Ly β photons by neutral oxygen, which is then re-emitted at 8466 ˚ A. This process requires dense, highly ionized environments, such as the BLR of an AGN (Osterbrock & Ferland 2006).', '3.8. Exploring the Host Galaxy': "So far we have identified broad line emission, enhanced emission line ratios and a high Ly α EW which all point toward a presence of an AGN in GN-72127. Despite that, the unambiguous presence of the Balmer break and associated absorption lines in the spectrum clearly indicates that the majority of the continuum emission in GN-72127 is still likely dominated by older stars. To this end we would like to perform a dedicated fit to the host galaxy in order to derive the properties of the stellar population in GN-72127. \nThe available JWST PRISM and medium grating data are jointly fit with the BAGPIPES SED fitting code (Carnall et al. 2018, 2019). The setup for the fit is as follows. We adopt a non-parametric SFH with the 'bursty continuity' prior from Tacchella et al. (2022), using eight time bins where the SFR is fit to a constant value in each bin. The first four bins are set to lookback times of 0-3 Myr, 3-10 Myr, 10-30 Myr, and 30-100 Myr, while the last four bins are logarithmically spaced between 100 Myr and t max = t universe ( z = z spec ) -t universe ( z = 20). Dust attenuation is assumed to follow the Calzetti et al. (2000) law and the rest-frame V-band attenuation is fit (log-uniform prior) in the range A V = 0 . 001 -5 mag. Stellar nebular emission is included with an ionization parameter in the range -4 ≤ log U ≤ -2. We allow for a stellar velocity dispersion in the range 50-500 km/s and metallicity between 0.1-2.5 Z ⊙ (log-uniform prior for both). Finally, we allow BAGPIPES to modify the input spectral noise array by a factor between 0.5-2. As mentioned above, the AGN-like ratios of the narrow emission lines imply that they potentially originate in the NLR, rather than the host. We therefore have removed the contribution of the emission lines to our fit by masking them out. \nFrom the BAGPIPES fit we find the galaxy to be very massive log 10 ( M ∗ ) = 10 . 63 ± 0 . 02, with little dust attenuation ( A V ∼ 0 . 4) and essentially devoid of all starformation activity (log 10 (sSFR) < -11 . 5). The bestfit SFH reveals a short burst of star-formation some ∼ 300 Myr ago, which lasted for ∼ 100 Myr, forming at roughly 40 -60 M ⊙ /yr. The galaxy then appears to have quenched rapidly at z ∼ 5 . 5, remaining largely \nFigure 4. Examination of the absorption features in the G235M spectrum of the quiescent host galaxy. We show our best fit to the absorption lines+[Ne iii ] in blue, while the non-parametric SFH BAGPIPES fit is in red. The regions of the spectrum (black) that were masked out due to contaminant lines are shown as a shaded area. The error array is in dashed red. This galaxy exhibits deep Balmer absorption lines with the spectrum resembling that of an A-type star, common in lower-redshift post-starburst galaxies (Goto 2007; Wild et al. 2009, 2020), as well as most recent exploration of z > 4 quiescent galaxies (Carnall et al. 2023; de Graaff et al. 2024). On the inset we show the best-fit non-parametric SFH, which clearly shows that GN-72127 experienced a significant, rapid drop in the SFR within the last ∼ 200 -300 Myr ( z ∼ 5 . 8). \n<!-- image --> \ndormant all the way until the time we observed it. In Figure 4 we present the best-fit BAGPIPES SED, alongside a line-specific fit to the relevant part of the spectrum containing absorption lines. An array of high order Balmer absorption lines from H η to H γ , alongside a tentative detection of the Ca K feature is reminiscent of the highestz quiescent galaxies found with JWST (Carnall et al. 2023; de Graaff et al. 2024). \nWhile BAGPIPES measures the width of the absorption features alongside other parameters, that fit was done on data that span multiple spectral resolutions, thus the derived widths might be inadvertently broadened. To keep all of our velocity measurements consistent, we have re-fit all of the absorption features, as well as the Ne iii emission with the same method as in Section 3.3. We show our final result in Figure 4. For the absorption lines we derive a FWHM of 959 ± 162 km/s. This corresponds to the velocity of the stars in the galaxy, rather than the gas, as was done in previous works concerning BL AGN (Maiolino et al. 2023b). The absorption lines from stars trace the dynamical mass, and are broader than the narrow lines (FWHM/2.355 = σ ∗ = 407 ± 71 km/s), consistent with the notion that all \nour narrow emission lines likely originate from the NLR, rather than the host galaxy.", '3.9. Size Measurements': "As the cutouts in Figure 1 indicate, the source is very compact, bordering on unresolved in the redder bands. We now would like to measure the effective radius ( r eff ) to confirm whether that is indeed the case by modeling the object with GALFIT (Peng et al. 2002, 2010). When fitting we take into account the effects of the PSF, which we have measured empirically from the bright stars in the field. We model the object with a S'ersic (S'ersic 1963) profile where the source position, brightness, effective radius, S'ersic index, and axis ratio as allowed to vary. \nIn all LW bands (F277W, F356W, F410M, F444W) the measured effective radius varies from 0 . '' 034 -0 . '' 04. In this case a source can be considered to be point-like and unresolved since its effective radius is smaller than the empirical PSF HWHM ( ∼ 0 . '' 07 for F444W). It is however possible to measure sizes of objects that are smaller that the PSF as shown was shown by other works on compact objects (van der Wel et al. 2014; Labb'e et al. \nFigure 5. BPT ( left ) and BPT-[S ii ] ( right ) ionization diagnostics. The [N ii ] and [S ii ] to H α line ratios found in the object (red) appear to vastly exceed those found in highz SFGs (orange; Cameron et al. 2023; Sanders et al. 2023), Type I (blue; Maiolino et al. 2023b) and Type II AGN (grey; Chisholm et al. 2024). On the other hand we find a comparable ionization to the highestz QG from de Graaff et al. (2024) (green) and Carnall et al. (2023) (vertical blue line). The line ratios we find are indicative of AGN-dominated ionization as the tracks from Kewley et al. (2001) (solid) and Kauffmann et al. (2003) (dashed) appear to suggest. \n<!-- image --> \n2023a). Taking the redshift into account, we derive an upper limit on the physical effective radius to be ≲ 300 pc. The point-like nature and size of this source in restoptical is consistent with a highly concentrated, PSF dominated objects explored in the recent exploration of highz reddened AGN (Furtak et al. 2024; Kocevski et al. 2024; Kokorev et al. 2023a, 2024; Labb'e et al. 2023a) as well as the similarly compact highestz quiescent galaxies (Carnall et al. 2023; de Graaff et al. 2024). \nOn the other hand, we marginally resolve the source in the rest-UV, sampled by all other bluer filters, with the median size across all SW filters equal to 0 . '' 07 ± 0 . '' 01, roughly × 2 . 5 larger than the PSF (HWHM ∼ 0 . '' 025), resulting in r eff = 490 ± 50 pc. We also derive a S'ersic index n ≈ 1 . 15, implying a lower degree of central concentration in the rest-UV. As GN-72127 is not resolved in rest-optical it is difficult to say with a high degree of certainty if GN-72127 has a more compact morphology in the redder filters. Although it is entirely feasible that the extended emission present in the rest-UV might imply that this part of the spectrum is dominated by stars, rather than the AGN light. A similar finding has already been reported in a stellar-light dominated LRD with extremely strong Balmer emission (Labb'e et al., in prep.). \nNext, we would like to use the effective radius to gauge the dynamical mass as well as the stellar surface density (Σ ∗ ). As the stellar light will be dominant at longer, rest-optical wavelengths, this would make the F444W band the most physically constraining for this type of study, as was done in Labb'e et al. (2023a). The F444W band should also trace the region with the lowest amount of dust obscuration. \nUsing the derived velocity widths of the absorption lines and the F444W size of the object we derive the dynamical stellar mass by following M dyn = 5 r eff × σ 2 ∗ /G , where G is the gravitational constant, (Cappellari et al. 2006; de Graaff et al. 2024). The σ we derive in Section 3.8 is explicitly that of the stellar population itself, and should better trace the true M ∗ (plus gas and dark matter) as opposed to the widths of narrow nebular lines. Given the r eff < 300 pc and the stellar velocity dispersion σ ∗ = 407 ± 71 km s -1 - we obtain an upper limit on M dyn < 10 . 76 M ⊙ , fully consistent with the M ∗ we measure from SED fitting. Finally, we derive an upper limit on the stellar mass surface of log 10 (Σ ∗ /M ⊙ kpc -2 ) < 11 . 01. This is high but does not exceed the maximum limits attainable during an intense burst of star formation (Hopkins et al. 2010; Grudi'c et al. 2019), and is consistent with some of the densest \nFigure 6. An exploration of z ∼ 4 -7 AGN on the fundamental Faber-Jackson relation (solid line; Faber & Jackson 1976; Shen et al. 2008; Cappellari 2016). We show GN-72127 in red, Type I AGN from Maiolino et al. (2023b) in blue, and Type II AGN from (Chisholm et al. 2024) in grey. For comparison we additionally overlay results for various local elliptical, spherical and bulge galaxies (Bender et al. 1992). The source is located exactly on the Faber-Jackson relation, signifying how it is already similar to local ellipticals at z ∼ 4, and further implying that the continuum is indeed stellar dominated. \n<!-- image --> \nrecently discovered quiescent galaxies at highz (Carnall et al. 2023; de Graaff et al. 2024).", '4.1. Black Hole Properties': 'Time domain observations of local quasars draw a correlation between the width of the broad Balmer series lines and the size of the BLR (e.g., Kaspi et al. 2000; Greene & Ho 2005), allowing for the BH mass to be estimated from the width of the emission lines and their luminosities. We compute the BH mass ( M BH ) from the luminosity and the width of the broad H α line, and restframe 5100 ˚ A luminosity of the underlying continuum. Since we have established that the Balmer emission most likely all comes from the region around the AGN itself, i.e. the narrow and broad line regions (NLR and BLR), we will use the A V value derived from the Balmer decrement, and assume that NLR and BLR have similar amounts of dust extinction, to correct the H α . On the other hand, the rest-optical continuum is likely to be stellar light dominated, so we will use an A V ∼ 0 . 4 value \nderived from the BAGPIPES fit (rather than A V ∼ 1 . 1 from the decrement) to the host galaxy when correcting the L 5100 . \nWith the above considerations in mind, from the H α we derive log 10 ( M BH , H α /M ⊙ )=7.31 ± 0.21 and from the continuum we get log 10 ( M BH , 5100 /M ⊙ )=7.90 ± 0.15. The uncertainty on both of these estimates is largely dominated by the scatter in the relations used to derive the masses, rather than the errors on the line flux and/or dust extinction (Kollmeier et al. 2006). If the rest-optical was mostly dominated by the AGN emission, one would expect both of these results to be consistent (e.g. see Kokorev et al. 2023a), however we find that the 5100 ˚ A estimate, even with a comparably modest dust obscuration, returns a mass that is × 5 higher. This in return might imply that despite the H α line being clearly detectable, the continuum around it is still dominated by the stellar light. The degree to which this is happening is highly uncertain, and would require dedicated modeling of the continuum to determine the fractional contribution of the AGN light to the total SED, which is made difficult by the lack of data redder of the F444W band. As such, for the rest of this work we will adopt the M BH derived from the H α luminosity as our final result. \nIn addition to the mass, we also use the broad H α to derive the bolometric luminosity ( L bol ) of the AGN. We use L bol = 130 × L H α (Richards et al. 2006; Stern & Laor 2012) and obtain L bol = (3 . 46 ± 1 . 10) × 10 44 erg/s. Using the above values, the L bol /L edd is then ≈ 0 . 12, implying that the object is accreting at a sub-Eddington rate. This is much lower compared to the extreme AGN found at highz (Furtak et al. 2024; Larson et al. 2023; Kokorev et al. 2023a; Maiolino et al. 2023b), however some AGN at highz also appear to be less active (Kocevski et al. 2023; Matthee et al. 2023) or even completely dormant (Juodˇzbalis et al. 2024).', '4.2. Ionization': 'As briefly stated in the previous sections, some of the nebular narrow lines ratios in the source look elevated when compared to massive star-forming galaxies. In this section we would like to explore this further and investigate the potential ionization mechanisms in the galaxy. \nIn Figure 5 we show the [O iii ]/H β ratio as a function of [N ii ]/H α and [S ii ]/H α . Given a strong absorption feature present in H β (Figure 2) we only report these ratios as upper limits. Both [N ii ] and [S ii ] doublets are however detected at S/N > 3 (Table 2) and should provide a more robust diagnostic. Our derived line ratios place us securely in the AGN locus of the BPT (Baldwin et al. 1981) diagram, with regions delimited by Kewley \net al. (2001) and Kauffmann et al. (2003). In addition we also derive log 10 ([OI] λ 6302/H α ) ∼ -0 . 42, which again securely places us in the AGN portion of the ionization diagnostic (Kewley et al. 2006). \nWe also observe an extremely high ratio between the auroral [O iii ] λ 4363 line and [O iii ] λ 5007 (RO3) of 0 . 35 +0 . 17 -0 . 15 , and 0 . 54 +0 . 35 -0 . 25 when applying the NLR dust correction we derive in Section 3.6, respectively. This is quite extreme, and does exceed the dust-corrected RO3=0.32 found in z ∼ 8 . 5 LRD by Kokorev et al. (2023a). The object also appears to be highly ionized when compared to highz SFGs, with RO3=0.048 presented by (Katz et al. 2023). Even if our dust-correction is overly aggressive, the observed ratio we find is hard to reconcile with ranges laid out in Nicholls et al. (2020) and will lead to extreme electron temperatures and densities. However, photoionization models presented in Baskin & Laor (2005) alongside RO3 studies from decades of lowz Seyfert investigations (Koski & Osterbrock 1976; Osterbrock 1978; Ferland & Netzer 1983; Dopita & Sutherland 1995; Nagao et al. 2001; Binette et al. 2022, just to name a few) do imply that the necessary high temperatures and densities can be reached in the NLR around an AGN. As mentioned in the previous section, high metallicities can cause the [Fe iii ] λ 4360 line to contaminate the [O iii ] λ 4363 flux (Curti et al. 2017). Although our fit to the spectrum does not show evidence of [Fe iii ] emission, the NLR can be easily enriched with metals due to its relatively low mass, resulting in the contribution of this line to [O iii ] λ 4363 to be potentially non-negligible. This could mean that our RO3 is overestimated, but only higher resolution data from the Hgratings can be used to definitely answer this question. Another possibility for such a high ratio is a highly dense NLR. The critical density of [O iii ] λ 4363 is roughly 50 × higher than that of [O iii ] λ 5007, so if the density of some regions within the NLR exceeds 10 5 . 8 cm -3 , then RO3 can be artificially boosted as well.', '4.3. AGN Quenching in a Little Red Dot': "With all the pieces in place, we now would like to examine the full picture and reflect on the potential nature of the source. While GN-72127 fulfills color and compactness criteria for LRDs, the colors themselves seem to originate from the evolved stellar population, rather than AGN continuum. All signs point toward GN-72127 being a massive (log 10 ( M ∗ ) ∼ 10 . 6) quiescent galaxy. The SFH inferred from a multitude of absorption lines as well as a strong Balmer break indicate that this galaxy has ceased, at least temporarily, its star formation. In the rest-optical, a combination of broad lines and AGN-like line ratios, betrays a moder- \ny active AGN ( L bol /L edd ∼ 0 . 1) with a black hole mass of log 10 ( M BH ) ∼ 7 . 3, embedded within this massive, quenched galaxy. At the same time, in the rest-UV we detect a strong (EW ∼ 400 ˚ A) Ly α , inconsistent with the low SFR of the host, plus a Mg ii doublet in emission. While the evidence for the broad component in Mg ii is itself tentative, the mere fact that we find this line in emission in a solar metallicity quiescent galaxy, already strongly suggests it originates from an AGN and not star formation (Burchett et al. 2021; Witstok et al. 2021). \nWhile the LRD we study has clear AGN signatures, such as broad lines and AGN-like ionization, it does appear to have completely settled on the local relations linking black holes and their host galaxies. For example, we find the M BH /M ∗ in GN-72127 to be completely consistent with the local relations (Kormendy & Ho 2013; Greene et al. 2016, 2020). While the narrow lines in the source likely originate from the NLR around an AGN, the availability of clear stellar absorption features allows us to robustly measure the stellar velocity dispersion. We place GN-72127 on the FaberJackson sequence (Faber & Jackson 1976) in Figure 6 and find that the object has largely 'settled' onto the same evolutionary trajectory as local elliptical galaxies that exist on the fundamental plane (Djorgovski & Davis 1987). This is unsurprising as we find that the source does in fact have a lot in common with the massive quiescent galaxies recently uncovered by JWST (Carnall et al. 2023; de Graaff et al. 2024), rather than the record breaking highz LRDs. In the final section we would like to reflect on how this source potentially came to be, and where it might be headed.", '5.1. Recently Discovered AGN and Their Hosts': "The last two years of observations with JWST have provided us with constraints on M BH /M ∗ for a large sample of AGN across a wide epoch of early cosmic history ( z = 4 -9). We aim to explore these data to determine if there is an evolutionary link between AGNdominated highz objects with high M BH /M ∗ (Bogdan et al. 2023; Endsley et al. 2023; Goulding et al. 2023; Kokorev et al. 2023a; Juodˇzbalis et al. 2024), reddened AGNs at z ∼ 4 which have grown their stellar populations significantly over the last ≲ 1 Gyr (Labb'e et al., in prep, Wang et al. 2024), and the earliest quiescent galaxies at z > 4 (Carnall et al. 2023; de Graaff et al. 2024). \nIn Figure 7, we show how the M BH /M ∗ relation evolves for JWST -detected AGN and quiescent galaxies containing AGN signatures, compared to the local \nFigure 7. Potential evolution of the offset between derived M BH /M ∗ and a local relation (Greene et al. 2024) across time (left) and stellar mass (right). We show an array of highz AGN (Bogdan et al. 2023; Goulding et al. 2023; Chisholm et al. 2024; Fujimoto et al. 2022; Furtak et al. 2024; Juodˇzbalis et al. 2024; Kokorev et al. 2023a; Larson et al. 2023; Maiolino et al. 2023a,b; Wang et al. 2024). We specifically highlight objects that would be selected as LRDs with a red envelope. In gray and maroon we also show tracks of M BH /M ∗ relations from Greene et al. (2024) and Pacucci et al. (2023), respectively. We note the emergent downward trend of M BH /M ∗ toward the local relations. \n<!-- image --> \nrelation (Greene et al. 2024), as a function of both cosmic age (redshift) and stellar mass. Starting with the epoch just ∼ 400 Myr ( z ∼ 11) after the Big Bang, we cover roughly 1 Gyr of cosmic history in order to reflect on a potential evolutionary path that can tie this picture together. \nFormation mechanisms of AGN with elevated M BH /M ∗ ratios remain elusive. Proposed solutions range from super-Eddington accretion on remnants of Population III stars (Furtak et al. 2024; Larson et al. 2023) and compact starbursts (Kroupa et al. 2020), to direct collapse seeds (Bogdan et al. 2023; Dayal et al. 2024; Goulding et al. 2023; Kokorev et al. 2023a; Natarajan et al. 2023; Maiolino et al. 2023b) and primordial black holes (Liu & Bromm 2022; Dayal 2024). These overmassive black holes could also be explained by by the existence of a new M BH -M ∗ sequence (Pacucci et al. 2023; Pacucci & Loeb 2024), or a high covering fraction, implying that locally-calibrated virial relations which link black hole mass to H α luminosity are not applicable at highz (Maiolino et al. 2024; Pacucci & Narayan 2024). The perceived excess of these high- \nBHs could also simply stem from a selection bias in luminosity-limited BL samples where the outliers are the only detectable sources and a change to the underlying scaling relations is not necessary (Li et al. 2024). Formation pathway aside, AGN with very little to no detectable host galaxy component have been found across a multitude of extragalactic fields, too many to be ignored. \nWhile the gas in these systems might not be fueling star formation, a considerable amount is still required to sustain AGN accretion. In their work Maiolino et al. (2023b) present a sample of highz AGN (45% being LRDs, using the same criteria as in Section 3.1) which do not follow the local M BH -M ∗ relation but align with local dynamical black hole mass relations. This would imply that these systems do contain gas but are inefficient at forming stars. The initial view of the farinfrared (FIR) emission in LRDs provided by ALMA reveals a dearth of SF activity, albeit these observations still remain relatively shallow. So far, not a single LRD has been detected in the FIR (Akins et al. 2024; Casey et al. 2024; Labb'e et al. 2023a). It has even been sug- \nted that most LRDs are not AGN, but rather compact starburst galaxies instead (P'erez-Gonz'alez et al. 2024a; Williams et al. 2023), as they lack a rising powerlaw in the mid-infrared (MIR). There is no evidence so far from the MIR/FIR to support or refute that notion, and objects with clear AGN signatures have been found to also show a lack of a dusty torus emission (Wang et al. 2024), which could be a result of a patchy BLR (Maiolino et al. 2024). \nAt lower redshifts ( z ∼ 4 -5), we are presented with another curiosity: some objects that fulfill the LRD criteria (Labb'e et al. 2023a; Greene et al. 2024) also appear to contain evolved stellar populations. Model fits to these systems indicate very early and rapid star formation, followed by a significant reduction (Wang et al. 2024; Labb'e et al., in prep.) or near cessation of SF activity, as in GN-72127, as they approach the local M BH -M ∗ relation. This source, in particular, provides a unique opportunity to examine the M BH -M ∗ ratio and the SFH in reddened AGN with unprecedented detail. The clearly stellar-dominated continuum and broad H α emission allow us to extract information concerning both the central black hole and detailed properties of the stellar population in the host galaxy. \nContrasting the SFH of GN-72127, with the objects in Wang et al. (2024), reveals a scenario where compact, reddened AGN enter a starburst phase 300 -400 Myr before the epoch of observation and then appear to quench. This indicates that there is a stage between elevated and local-like M BH -M ∗ where these AGN experience rapid star formation, possibly due to a shutdown of AGN feedback, gas virialization, a merger, or a combination of multiple factors. During this stage, we expect that these AGN would be still identifiable (e.g. through high ionization or broad lines), but would also shine brightly in the FIR as they are now actively (and rapidly) forming stars. Given that this phase is shortlived ( ∼ 100 Myr), catching these objects in the act is challenging, although it may have already been accomplished. \nIn their preJWST work, Fujimoto et al. (2022) presented GNz7q, a compact object at z ∼ 7 . 2, which potentially bridges quasars at cosmic dawn with galaxies at cosmic noon. The spectral shape and extremely high UV-luminosity surface density of GNz7q indicate it cannot be explained by extreme star formation alone, suggesting its AGN nature. At the same time it is embedded in an extremely FIR luminous host starburst galaxy (SFR IR > 1600 M ⊙ /yr), all within ∼ 480 pc, precisely the type of transitioning, short-lived object we were looking for. Moreover, GNz7q is extremely faint in X-rays, suggesting a Compton-thick, super-Eddington \nblack hole accretion disk, a characteristic observed in highz AGN (Endsley et al. 2023; Furtak et al. 2024; Greene et al. 2024; Kokorev et al. 2023a; Maiolino et al. 2023a). It does appear that star-formation can take place in galaxies hosting massive AGN and can do so at quite an accelerated rate. Notably, GNz7q is a very extreme example but there likely exists a small population of LRD hosts in the starburst phase, that is yet to be identified. It is very possible that these are NIRCam dark, due to extreme dust obscuration ( A V > 4 -5) and low surface brightness, making the detection of rest-UV/optical emission extremely challenging. The existence of such dusty systems at high redshift was briefly explored in Kokorev et al. (2023b), and recently, a NIRCam-dark source was identified by P'erez-Gonz'alez et al. (2024b), yet the exact number densities of these objects are still unclear.", '5.2. Evolution and Link to the Quiescent Galaxies': "A multitude of studies preceding the launch of JWST have laid out the possibility that cores of the most massive galaxies in our local Universe have, in fact, formed very rapidly. This is generally thought to happen via a burst of star-formation at z > 5 -6, followed by a rapid quenching (Thomas et al. 2005; Conroy et al. 2014; van Dokkum et al. 2015). \nThe identification and detailed analysis of z = 3 -5 quiescent galaxies, made possible with NIRSpec, have further supported the notion of rapid star-formation and subsequent quenching (Carnall et al. 2023; de Graaff et al. 2024; Glazebrook et al. 2024). Some of these galaxies seemingly retain their AGN signatures as either broad lines (Carnall et al. 2023) or high ionization (de Graaff et al. 2024). The sample presented recently in Wang et al. (2024) seems to show the progenitors of some of these quiescent galaxies as their star-formation undergo a truncation. The authors propose a scenario where massive galaxies have simultaneously grown alongside their SMBHs and then proceed to quench rapidly, however we would like to offer an alternative interpretation. \nIt is possible that the systems with elevated M BH /M ∗ we find with JWST will undergo episodes of intense star formation. These are likely induced through merging, as many LRDs were found to exist in overdensities (Fujimoto et al. 2023; Greene et al. 2024; Kokorev et al. 2023a; Labb'e et al., in prep.). Following a short-lived yet intense starburst, like the one observed in Fujimoto et al. (2022) or predicted by Carnall et al. (2023) and de Graaff et al. (2024), the stellar mass of the host galaxy catches up to its over-massive BH and settles onto the local BH-host relations. The next likely step of their \nevolution is further expansion and transition toward the giant ellipticals we see locally. \nHaving undergone a burst and then a rapid quenching (Figure 4), GN-72127 appears to agree with this picture. Dominated by stellar light the object still contains clear AGN signatures, which suggests that rapid quenching could have been driven by AGN feedback rather than gas depletion. We propose that while some highz AGN may not initially follow concurrent growth of BHs and their stellar populations, they will eventually 'settle' onto the local relations, as time progresses. \nGenerally, elliptical galaxies are thought to form through processes such as mergers of smaller galaxies. During these mergers, the system undergoes violent relaxation, redistributing energy and leading to a state of equilibrium. The duration of such a process will be longer for extended and less massive systems, and shorter for the most massive and compact ones. We can see how close the source is to a 'settled' virialized state by examining its position on the Faber-Jackson relation Figure 6. We find that GN-72127 occupies a position very similar to the local ellipticals, implying that such virialization may already take place quite early, at z ∼ 4, potentially for the most massive objects. On the other hand BHs with measured σ from Maiolino et al. (2023b), especially the ones with similar stellar mass, are at higher redshift compared to GN-72127 and are likely still evolving into their equilibrium state. The achievement of a virial equilibrium is crucial to our argument, as then it would make it difficult for galaxies to have significant M BH /M ∗ variations as their stellar masses exceed ∼ 10 10 M ⊙ , ultimately aligning them with local relations. \nDespite this, we find that our source is offset from the local M BH -σ relation (e.g. Kormendy & Ho 2013; Greene et al. 2020). Our derived σ ∗ suggests a M BH ∼ 10 9 M ⊙ , ∼ 2 dex higher than our current measurement. It is therefore possible that the stellar component of GN-72127 still requires more time to relax and expand from its very compact configuration to align with the local M BH -σ trend, due to the conservation of angular momentum, driving down the σ . Importantly, we do not observe any compact objects with LRD colors below z ∼ 4 (Kocevski et al. 2024), supporting the idea that these objects have begun to expand and dynamically relax into structures resembling lowerz ellipticals. Furthermore, studies of elliptical galaxies from z = 2 to 0 show that their effective radii grow by a factor of 2 -4 over ∼ 10 Gyr timescales (van der Wel et al. 2008; Cassata et al. 2011; van der Wel et al. 2014; Xie et al. 2015). This observed growth and relaxation process is consis- \nansformation of LRDs into the quiescent elliptical galaxies we see at lower redshifts. \nIt is worth noting that objects presented in Figure 7 do not necessarily follow these exact evolutionary paths. A large fraction of highz AGN have poorly measured M ∗ as the AGN continuum dominates the emission making it difficult to separate the two (see discussion in Greene et al. 2024; Wang et al. 2024). Dynamical masses can be employed instead, however the same problem manifests itself when trying to decide whether the narrow nebular lines originate from the NLR or the host galaxy itself. In the case of GN-72127, however, the stellar-dominated continuum and the presence of absorption lines allows for robust measurements of both M ∗ and M dyn .", '5.3. Final Remarks': "Using the NIRSpec PRISM and medium resolution G140M, G235M, G395M observations of the GOODS-N field (D'Eugenio et al. 2024), we present the investigation of a little red dot that shows signs of both an evolved stellar population and AGN activity at z = 4 . 13. A careful examination of the spectrum of the source shows a significant ( > 4 σ ) BL component present in the H α line with a FWHM of ∼ 2500 km/s. As we do not observe similar broad features in the strongly detected [N ii ], [S ii ] or the [O iii ] lines, we suggest that the broadening exhibited by H α is a sign of an actively accreting black hole, rather than large-scale outflows. A further evidence for the AGN activity are high ionization inferred from the BPT diagram, the RO3 line diagnostic, as well as a strong Ly α emission with an EW 0 ∼ 400 ˚ A. In the rest-UV we also report a tentative broad component in the Mg ii doublet, which could further strengthen our AGN interpretation. \nDespite being compact and showing clear AGN signatures, GN-72127 appears to be different from the highz LRDs explored with NIRSpec (Furtak et al. 2024; Kokorev et al. 2023a; Greene et al. 2024). We detect a massive (log 10 ( M ∗ ) ∼ 10 . 6) and evolved host, which dominates the continuum of our object, as evidenced by a strong Balmer break as well as a multitude of absorption lines. Detailed SFH fitting reveals that our object has formed most of its stellar population in a short (100 -200 Myr) burst, and then quenched rapidly roughly 200 Myr ago. It is worth noting that stellar masses of LRDs are generally poorly constrained due to difficulties separating the AGN and stellar continuum. Despite that, a strong stellar that we detect in GN-72127 would be clearly visible even at highz , provided it was there. \nWe contrast the position of GN-72127 on the M BH /M ∗ vs z and M ∗ plane to all other highz LRDs and BL AGN uncovered with JWST . This examination \nreveals a potential evolutionary path that starts with overmassive BH in the early Universe, which then form stars in a rapid burst and slowly descend onto the local M BH -M ∗ relation occupied by local ellipticals. \nFinally, we detect very strong Ly α emission, inconsistent to have come solely from star formation, and a tentative broad line in Mg ii doublet. This would mean that, for the first time, we now see clear AGN signatures in the rest-UV spectra of LRDs. Despite that we can not yet ascertain the full origins of blue light in LRDs. While we detect emission lines with strong AGN signatures in the rest-UV, we also find it to be marginally resolved ( ∼ 500 pc), too large to have come just from the central AGN alone. It is likely that some AGN light indeed escaped the thick dust cover through either scattering and some patchy coverage, but the continuum itself is likely to be stellar in origin. Rest-UV emission in GN72127 appears to originate both from stars and AGN, however this still might not be true for the entire population, especially as we move to higher redshifts where AGN seem to be much more dominant. Although it is worth noting still, that observational bias can drive the high incidence of objects with elevated M BH /M ∗ (e.g. Li et al. 2024). Only dedicated deep studies of the restUV emission of LRDs with medium and high resolution NIRSpec gratings onboard JWST can shine more light on this mystery.", 'ACKNOWLEDGMENTS': 'VK acknowledges support from the University of Texas at Austin Cosmic Frontier Center. We thank the JADES team for providing the scientific community with these beautiful JWST data. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope . The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST . Some of the data products presented herein were retrieved from the Dawn JWST Archive (DJA). DJA is an initiative of the Cosmic Dawn Center, which is funded by the Danish National Research Foundation under grant No. 140. 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2024ApJ...973L...8X	We present thermal emission measurements of GJ 1132b spanning 512 m obtained with the MidInfrared Instrument LowResolution Spectrometer on the James Webb Space Telescope. GJ 1132b is an M dwarf rocky planet with T SUBeqSUB 584 K and an orbital period of 1.6 days. We measure a whitelight secondary eclipse depth of 140 17 ppm which corresponds to a dayside brightness temperature of T SUB pdaysideSUB 709 31 K using improved star and planet parameters. This measured temperature is only 1 below the maximum possible dayside temperature of a bare rock i.e. assuming a zeroalbedo planet with no heat redistribution inlineformula mmlmath overflowscrollmmlmsubmmlmrowmmlmiTmmlmimmlmrowmmlmrowmmlmimaxmmlmimmlmrowmmlmsubmmlmath inlineformula inlineformula mmlmath overflowscrollmmlmsubsupmmlmrowmmlmn746mmlmnmmlmrowmmlmrowmmlmommlmommlmn11mmlmnmmlmrowmmlmrowmmlmommlmommlmn14mmlmnmmlmrowmmlmsubsupmmlmath inlineformula K. The emission spectrum is consistent with a featureless blackbody which agrees with a wide range of possible surface compositions. By comparing forward models to the dayside emission spectrum we rule out Earththickness P 1 bar atmospheres with at least 1 HSUB2SUBO atmospheres of any modeled thickness 10SUP4SUP to 10SUP2SUP bars that contain at least 1 COSUB2SUB and thick Venuslike atmospheres P 100 bars with at least 1 ppm COSUB2SUB or HSUB2SUBO. We therefore conclude that GJ 1132b likely does not have a significant atmosphere. This finding supports the concept of a universal cosmic shoreline given the high level of bolometric and extreme ultraviolet EUV and Xrays collectively XUV irradiation received by the planet.	2024-09-01T00:00:00Z	['2024ApJ...973L...8X', '10.48550/arXiv.2408.13340', '2024arXiv240813340X', '10.3847/2041-8213/ad72e9', 'arXiv:2408.13340']	['Exoplanet astronomy', 'Exoplanet atmospheres', 'Exoplanet atmospheric composition', 'Exoplanet surface characteristics', 'Exoplanet surface composition', 'Astrobiology', 'Biosignatures', 'Interdisciplinary astronomy', '486', '487', '2021', '496', '2022', '74', '2018', '804', 'Astrophysics - Earth and Planetary Astrophysics']	JWST Thermal Emission of the Terrestrial Exoplanet GJ 1132b	2,024	190	0.64	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	8	https://arxiv.org/pdf/2408.13340.pdf	{'JWST Thermal Emission of the Terrestrial Exoplanet GJ 1132b': "Qiao Xue , 1 Jacob L. Bean , 1 Michael Zhang , 1 Alexandra Mahajan , 2 Jegug Ih , 3 Jason D. Eastman , 2 Jonathan Lunine , 4 Megan Weiner Mansfield , 5, 6 Brandon Park Coy , 7 Eliza M.-R. Kempton , 3 Daniel Koll , 8 and Edwin Kite 7 \n1 \n7 8 Peking University, Beijing, People's Republic of China \nDepartment of Astronomy & Astrophysics, University of Chicago, Chicago, IL, USA 2 Center for Astrophysics \| Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA 3 Department of Astronomy, University of Maryland, College Park, MD, USA 4 Department of Astronomy, Cornell University, Ithaca, NY, USA 5 Steward Observatory, University of Arizona, Tucson, AZ, USA 6 NHFP Sagan Fellow Department of the Geophysical Sciences, University of Chicago, Chicago, IL, 60637, USA", 'ABSTRACT': "We present thermal emission measurements of GJ 1132b spanning 5 - 12 µm obtained with the Mid-Infrared Instrument Low-Resolution Spectrometer (MIRI/LRS) on the James Webb Space Telescope (JWST). GJ1132b is an M-dwarf rocky planet with T eq = 584K and an orbital period of 1.6 days. We measure a white-light secondary eclipse depth of 140 ± 17 ppm, which corresponds to a dayside brightness temperature of T p, dayside = 709 ± 31 K using improved star and planet parameters. This measured temperature is only 1 σ below the maximum possible dayside temperature of a bare rock (i.e., assuming a zero albedo planet with no heat redistribution, T max = 746 +14 -11 K). The emission spectrum is consistent with a featureless blackbody, which agrees with a wide range of possible surface compositions. By comparing forward models to the dayside emission spectrum, we rule out Earth-thickness (P ∼ 1 bar) atmospheres with at least 1% H 2 O, atmospheres of any modeled thickness (10 -4 - 10 2 bar) that contain at least 1% CO 2 , and thick, Venus-like atmospheres ( P ≳ 100 bar) with at least 1 ppm CO 2 or H 2 O. We therefore conclude that GJ 1132b likely does not have a significant atmosphere. This finding supports the concept of a universal 'Cosmic Shoreline' given the high level of bolometric and XUV irradiation received by the planet. \nKeywords: Exoplanet atmospheres (487), Extrasolar rocky planets (511), Exoplanet atmospheric composition (2021), Exoplanet atmospheric structure (2310)", '1. INTRODUCTION': "One of the most exciting questions in astronomy right now is whether small rocky planets orbiting M dwarfs can host atmospheres, which is directly related to their habitability. About 61% of the stars within 10 pc around us are M \nstars (Henry et al. 2018; Reyl'e et al. 2021), and it has become clear from the Kepler mission (Mulders et al. 2015; Dressing & Charbonneau 2015) and radial velocity surveys (Sabotta et al. 2021) that these stars preferentially host rocky planets. More importantly, their lower luminos- \nty and smaller radius relative to Sun-like stars open a window for studying the atmospheres of transiting habitable M-dwarf planets using existing facilities. \nHowever, M dwarfs are also known to be very active (West et al. 2008). UV emission during the pre-main sequence stage of M stars drives water loss and O 2 buildup (Luger & Barnes 2015). On the other side, powerful flares and strong UV radiation may destroy the atmosphere of the planets through photoevaporation or stellar wind erosion (Zendejas et al. 2010; Do Amaral et al. 2022; Affolter et al. 2023). \nNo previous measurements have found strong evidence for atmospheres on rocky M-dwarf planets, and recent JWST results are highly constraining non-detections or at most inconclusive. Mid-Infrared Instrument (MIRI) emission observations of three M-dwarf rocky planets, TRAPPIST-1b, c (both 15 µ m filter photometry), and GJ 367b (5 - 12 µ mlow resolution spectroscopy), have shown no evidence of substantial atmospheres (Greene et al. 2023; Zieba et al. 2023; Ih et al. 2023; Lincowski et al. 2023; Zhang et al. 2024). On the other hand, with transmission spectroscopy, the results are unclear either due to possible stellar contamination (Gl 486b: Moran et al. 2023; TRAPPIST1 b: Lim et al. 2023; GJ 1132b: May et al. 2023; LHS1140b: Cadieux et al. 2024) or degeneracy of different scenarios that could cause featureless spectra, e.g., a high mean molecular weight atmosphere, a high-altitude cloud atmosphere, no or very thin atmosphere (LHS 475b, LustigYaeger et al. 2023; GJ 341b, Kirk et al. 2024; TOI-836b, Alderson et al. 2024). \nGJ1132b, first detected by the MEarth-South telescope array (Berta-Thompson et al. 2015), is a super-Earth with a relatively low equilibrium temperature T eq ∼ 584 K orbiting a nearby M4type dwarf star. Previous efforts to detect an atmosphere on GJ1132b using transmission spectroscopy with HST/WFC3 (Swain et al. 2021; \nMugnai et al. 2021; Libby-Roberts et al. 2022) and JWST NIRSpec G395H (May et al. 2023) have all been inconclusive about the presence of secondary atmospheres. GJ 1132 is considered to be a weakly active M-star with slow rotation velocity < 2 kms -1 , very long rotation period > 100 days and age > 5 Gyr (Berta-Thompson et al. 2015; Bonfils et al. 2018). GJ1132b is considered to be one of the most suitable rocky planets for thermal emission measurements with JWST according to the emission spectroscopy metric (ESM ∼ 10) of Kempton et al. (2018) 1 . \nIn this study, we use the technique of secondary eclipse thermal emission to probe the rocky planet GJ 1132b for the presence of an atmosphere. This technique was originally developed by Koll et al. (2019) and Mansfield et al. (2019) building off the ideas of Deming et al. (2009) and Selsis et al. (2011). In addition to the recent JWST results described above, this technique has previously been applied using Spitzer Space Telescope observations (LHS 3844b: Kreidberg et al. 2019; Whittaker et al. 2022; GJ 1252b: Crossfield et al. 2022). However, prior Spitzer observations of GJ1132b were not able to detect the secondary eclipse due to instrumental limits (Dittmann et al. 2017). \nIn this work, we present JWST/MIRI thermal emission measurements of GJ 1132b. We describe the observation details in § 2, and provide an overview of the data reduction in § 3. Refined stellar and planet parameter measurements are described in 4. We present the results in § 5 and discussion in § 6.", '2. OBSERVATION': 'We observed one secondary eclipse of GJ1132b using JWST MIRI on July 1, 2023 (program GTO 1274, J. Lunine PI), with an exposure length of 4.5 hours. A single eclipse \nwas planned because Koll et al. (2019) showed that this would give sufficient signal-to-noise to be highly constraining for potential atmospheres. The observation was taken in the Low Resolution Spectrometer (LRS) slitless timeseries mode (Kendrew et al. 2015), with subarray SLITLESSPRISM , which covers a wavelength range of 5 - 12 µ m. The visit was scheduled around the time expected for the secondary eclipse based on the transit ephemeris and assuming a circular orbit. The eclipse was detected at the expected time at 8 σ confidence in the white light curve, as described below. The visit began 2.6 hours before the eclipse, continued for the 46-minute eclipse duration, and concluded 1.2 hours after the end of the eclipse. A total of 3,594 integrations with 28 groups per integration (4.6 s per integration) were obtained.', '3. DATA ANALYSIS': 'To ensure the reproducibility of the results, we performed data analyses with both the SPARTA (first described in Kempton et al. 2023) 2 and Eureka! codes (Bell et al. 2022). These two JWST pipelines have shown remarkable agreement for MIRI and NIRCam data (Kempton et al. 2023; Xue et al. 2024a; Zhang et al. 2024; Powell et al. 2024; Bell et al. 2024). We present an overview of the analyses conducted by each pipeline in the following sections.', '3.1. SPARTA': "Starting from the uncal.fits files, SPARTA provides completely independent data reduction procedures, without using any codes from other existing pipelines. Our reduction is based on the most recent release of SPARTA , details of which can be found in Zhang et al. (Appendix A, 2024). Figure 1 shows the raw 2D light curves in the wavelength-integration plane and the systematics-corrected white light curve \noverplotted by the best eclipse model from our preferred reduction SPARTA gr5 (see § 5). \nAn anomalous downward offset of the last group in almost all the integrations was found in our data (more details can be found in Morrison et al. 2023), thus we excluded the last group from the up-the-ramp fitting. We also observed non-linear behaviors of the first ∼ 11 groups (appendix Figure A8, Morrison et al. 2023; Dyrek et al. 2024). To test the impact of this non-linearity, we performed three independent analyses with SPARTA , removing zero, 5, and 11 groups from the up-the-ramp fitting, which we refer to as gr0, gr5, and gr11, respectively throughout the rest of this Letter. \nThe center of the spectral trace is fixed at column 36 (determined by fitting it with a Gaussian curve) and a window with a half-width of 3 pixels is chosen for extraction. We adopted the result by simple extraction instead of optimal extraction because the latter introduced more scatter and inflated the median absolute deviation of the light curves. \nWe identified a variety of systematics in the data. The two most evident systematics are the exponential downward ramp in the first ∼ 400 integrations and the odd-even effect (alternating bright/dark columns). Both of these have been seen in all the published MIRI/LRS transiting planet datasets (e.g., Bouwman et al. 2023; Kempton et al. 2023). However, in our dataset, we do not see the so-called 'shadowed region effect' spanning 10.5 - 12 µm that has been discovered in some of the published datasets (Bell et al. 2024). On the other hand, we found anomalous bright strips at 10.7861 and 10.8060 µ m (indicated by the red arrow in Figure 1(a)), the cause of which has not been determined. The pixel-level light curves at these two wavelengths are shown in Appendix Figure A9. It is unlikely that they are caused by stellar activity or mirror tilt events (Schlawin et al. 2023). We ultimately removed the data \nat these two wavelengths. We performed a 4 σ rejection to the data based on fluxes and positions of the spectral traces, gathered data from 5 to 12 µ m, and binned the fluxes into 0.5 µ m bins to get spectroscopic light curves. \nThe systematics were detrended with: \nF sys = F star (1 + Ae -t/τ + c x x + c y y + m ( t -t mean )) (1) \nwhere A is the exponential ramp amplitude, τ is the decay timescale; x and y are the positions of the trace in the dispersion and spatial directions, respectively (after rotating the raw images 90 · ); and c x , c y and m are the linear terms to decorrelate drifts. The secondary eclipse was modeled with batman (Kreidberg 2015). We adopted the orbital period P =1.62892911 days, semi-major axis a/R ⋆ = 15.2601, eccentricity e = 0.0118, inclination i = 88.16 · , and argument of periapsis ω = -95.8 · . These parameters were fixed for all the fitting procedures; see § 4 and Table A1 for how we obtained them. The eclipse time was determined by fitting the white light curve, and the best-fit value was used for the spectroscopic light curve fittings. We implemented the Markov chain Monte Carlo (MCMC) with emcee (Foreman-Mackey et al. 2012) to fit both the white light curve and the spectroscopic light curves in SPARTA .", '3.2. Eureka!': 'Stages 1 and 2 in Eureka! are identical to those in the jwst pipeline (Bushouse et al. 2022). We followed the default jump detection = 4 . 0 because changing it to 6 . 0, 8 . 0 or 12 . 0 yielded the same white light secondary eclipse depth. In Stage 3, we defined the subarray region of interest as [80, 393] in the x direction and [10, 62] in the y direction. We applied the outlier rejection routine to the full frame with threshold [5,5] along the time axis, resulting in 0.01% of the pixels flagged as bad. Optimal spectral extraction was performed with \nspec hw = 5 pixels and bg hw = 10 pixels. The spatial profiles were constructed by the median frame with an outlier rejection threshold = 10 σ . In Stage 4, we collected data from 5 to 12 µm and binned them into 14 spectroscopic light curves. Then we performed 4 σ outlier rejection with a box-car filter with box width = 15. For the fitting, we utilized the dynamic nested sampling algorithms with the dynesty package 3 . The light curves were fitted with a joint systematic model: \nF sys =( c 0 + c 1 ( t -t mean )) × (1 + r 0 e -( t min -t min, 0 ) /r 1 ) × (1 + c y y ) × (1 + c ∆ σ y ∆ σ y ) (2) \nwhere c 0 and c 1 are the linear decorrelation terms; r 0 and r 1 are the amplitude and the timescale of the exponential; y is the drift or jitter in the spatial direction and ∆ σ y is the change in the PSF width in the spatial direction Decorrelation in the dispersion direction was not included because we did not see strong trends with x and ∆ x . We obtained the secondary eclipse model from batman with the P , a/R ⋆ , i , e , and ω fixed to the same values described above in the SPARTA section.', '4. HOST STAR AND PLANET PARAMETERS': "Accurately assessing whether a planet has an atmosphere using thermal emission measurements hinges critically on knowledge of the planet and star parameters because the technique boils down to an energy balance argument (Koll et al. 2019). We ultimately need to know the amount of energy the planet receives, and we need to be able to convert a measured secondary eclipse depth into a planetary \nFigure 1. Light curves produced by our nominal SPARTA reduction. (a) Raw 2D light curves in the wavelength-integration plane. Anomalies such as the odd-even effect and exponential downward ramps at the beginning ∼ 400 integrations of the observation ( ∼ 30 min) are found. The red arrow indicates the very bright strip (3 × baseline flux) that we eventually removed from our nominal reduction. The start, middle, and end of the eclipse are shown with dashed dark red lines. (b) The systematics-corrected white light curve is shown in the top panel overplotted with the best batman eclipse model (blue line) and binned systematics-corrected fluxes (orange dots). The bottom panel shows the residuals between the best-fit total model( F sys and F batman ) and binned fluxes. We excluded the first 400 integrations (30 minutes) due to the strong exponential ramp. The middle two panels show the drifts of the trace positions in the spatial and dispersion directions, which are decorrelated from the data with linear models. \n<!-- image --> \nflux. The former depends on the stellar effective temperature (we adopt T s for this instead of the usual T eff because we want to distinguish the stellar and planetary temperatures below) and the planet's semi-major axis in units of the stellar radius ( a/R ⋆ ). Converting the secondary eclipse depth into a planetary flux also requires the planet-to-star radius ratio ( R p /R ⋆ ) and the stellar spectrum in the observed bandpass. \nWe therefore followed the procedure of Mahajan et al. (2024) to use EXOFASTv2 (Eastman et al. 2019) to leverage the stellar density derived from JWST transit and secondary eclipse observations to obtain precise stellar and plan- \netary parameters, as shown in Table A1. For our analysis of the GJ 1132 system, we used the two primary transits from JWST published by May et al. (2023), one taken on February 25 2023 and the other on March 5 2023, as well the secondary eclipse presented in this paper (adopting the SPARTA gr5 reduction, see next section). Note that for the GJ 1132b primary transits, there were no processed white light curves available. The data were processed from the Eureka! Stage3 products available online 4 . The data went through Stage4 using the same \nparameters as May et al. (2023). We also used the first sector of data (March 2019) from the Transiting Exoplanet Survey Satellite (TESS) to maximize the baseline between the TESS and JWST observations and therefore provide the best constraint on the period. While there were additional transits that could have constrained the period further, the addition of those data resulted in increased runtime for our models with minimal improvement in our measurement of the period. \nOur results are a dramatic improvement over the previous best analysis of the GJ 1132 system by Bonfils et al. (2018). Notably, we measure the stellar radius 25% more precise, the radius of planet b 2 . 7 × more precise, and the stellar temperature 2 × more precise. Also note that we do not assume a circular orbit. Rather, the timing and duration of the primary and secondary eclipses allow us to measure an eccentricity that is precisely near but not zero ( e = 0 . 0118 +0 . 047 -0 . 0099 ). \nUnlike the analysis in Mahajan et al. (2024), we added the ability to model a ramp simultaneously to EXOFASTv2 , and included that for all JWST light curves. In addition, for this system, the primary and secondaries were not contiguous, which added the uncertainty in the period to our determination of the eccentricity from the eclipse timing. A major reason for the inclusion of the TESS light curves was to render this period uncertainty negligible. Note that Dittmann et al. (2017) found no evidence of significant transit timing variations (TTVs) for the GJ 1132 system, so we assumed a linear ephemeris. \nLike Mahajan et al. (2024), we include radial velocity data (Bonfils et al. 2018) to measure the masses and eccentricity as precisely as possible. For this system, we simultaneously model all three planetary signals, including a radial velocity-only model for the non-transiting c \nplanet and the d candidate, which may be stellar activity (Bonfils et al. 2018). \nTypically, EXOFASTv2 runs until the Gelman-Rubin statistic is less than 1.01 and the number of independent draws is greater than 1000 for each parameter. In this fit, however, a handful of parameters did not pass this strict convergence criteria. In particular, the masses of the two planets and the ramp parameters for the eclipse detrending had Gelman-Rubin statistics of ranging from 1.05-1.09 and independent draws ranging from 800-1200. We expect the impact on the inferred median values and uncertainties to be negligible (Eastman et al. 2019, Eq 35), and these parameters are not strongly covariant with the parameters used in subsequent analyses. \nThe posterior of the MCMC sampling of this global model is subsequently used to account for the correlation between stellar and planet parameters when deriving dayside brightness temperature of the planet, as is discussed in § 5.1.", '5. INTERPRETATION': 'Figure 2 shows a comparison of the results from the different reductions and modeling approaches. We found good agreement in the spectra and white light eclipse depths between all these analyses. The most discrepant result is the white-light eclipse depth from the SPARTA gr0 reduction, which is 10% higher than the average of the other four, but still consistent with the others within the 1 σ range. The average error of the eclipse depths of the SPARTA reductions with more groups trimmed at the beginning is larger due to the loss of information, but we consider the results of these reductions more robust. Notably, the lower bound of Fp/Fs in the gr11 reduction at 11.25 µm is negative, which is unphysical. Due to the significant non-linear trend we observed in the first ∼ 5 groups (appendix Figure 5), and the error inflation on the spectrum in the gr11 reduction, we decided to use the SPARTA gr5 reduction as \nour preferred version. Combining with the § 4 analysis, we report the white-light eclipse depth to be 140 ± 17 ppm. We perform two kinds of modeling below to interpret these results. \nFigure 2. Emission spectra (main panel) and white-light eclipse depths (inset panel) from the different reductions. The results from Eureka! are shown in dark blue, and the results from SPARTA with different up-the-ramp fitting strategies are presented in pink (using all groups), green (with the first 5 groups removed) and yellow (with the first 11 groups removed). The wavelengths are shifted for visualization purposes. The grey dashed line in the inset panel shows eclipse depth corresponding to the maximum dayside temperature of the planet with the shaded area indicating ± 1 σ . The whitelight bandpass is shown with the gray triangle in the main plot. We plot the eclipse depth measured by the global model described in § 4 with the red square. \n<!-- image -->', '5.1. Inferred Albedo and Heat Recirculation Efficiency': "In this section, we derive the dayside temperature of GJ 1132b from the measured white light eclipse depth assuming the planet is a perfect blackbody with a smooth surface. That is, we determine the planet's brightness temperature over the broad 5 - 12 µ m MIRI/LRS band. We \nthen compare this temperature to the maximum possible temperature and use that comparison to put joint constraints on the Bond albedo (but see caveats at the end of this section) and heat recirculation efficiency. \nDeriving the planet's brightness temperature involves inverting the F p /F s determined from the secondary eclipse. This is usually done by fitting the measured F p /F s with a model (e.g., using non-linear least squares). However, we aimed to account for the correlated uncertainties in the eclipse depth and the star and planet parameters. We also aimed to account for the correlated uncertainties between the measured planet temperature and the maximum possible temperature. We took advantage of the posteriors from the MCMC in § 4 to do this. However, it required a different approach than the usual one because there are 557,999 steps in the Markov chains and it would be prohibitively expensive to invert each one by fitting. We therefore follow the procedure below to derive the planet's brightness temperature. \nThe eclipse depth can be modeled by dividing planetary spectra by stellar spectra weighted by instrument throughput in units of photon numbers: \nF p F s = ( R p R s ) 2 · ∫ π · B p ( T p ,λ ) hc/λ · W λ dλ ∫ M s ( T s , log g, [ M/H ] , λ ) hc/λ · W λ dλ (3) \nwhere W λ is the throughput of MIRI/LRS 5 , B p ( T p , λ ) is the planetary blackbody intensity and M s ( T s , log g, [ M/H ] 6 , λ ) is the stellar spectrum. For stellar spectra, we used both models interpolated from the PHOENIX grid (Allard et al. 2012) using the python package pysynphot 7 (STScI Development Team 2013) and models from the SPHINX grid (Iyer et al. \n5 available on Zenodo 10.5281/zenodo.13244543 \n6 [M/H] = log [ n ( Metal ) n ( H ) ] -log [ n ( Metal ) n ( H ) ] ⊙ \n2023, updated on May 30, 2024) interpolated by our own codes. The two model grids gave nearly identical results; we adopted the PHOENIX results below. \nTo take into account instrumental broadening, we convolved the model stellar spectra using a Gaussian kernel with FWHM identical to MIRI/LRS's resolution 8 . We converted blackbody intensity to blackbody flux by multiplying B p ( T p , λ ) with π . Before multiplying by the instrument throughput, the spectra were divided by hc/λ to convert from energy to photon flux. The integrals in Equation 3 are calculated from 5 to 12 µm . \nWe obtain F p F s , R p R s , [M/H], log g , and T s from the MCMC chain sampled when fitting the global model to the white light curve (see § 4). For each sample in the chain, we are able to invert Equation 3 to determine the photon flux emitted by the planet at its dayside. \nFinally, to convert the photon flux emitted by the planet to the dayside brightness temperature without expensive fitting processes, we prepared a photon flux-T p grid calculated with F photon = ∫ π · B p ( T p ,λ ) · W λ hc/λ dλ for T p ranging from 300 to 1,000 K. We then interpolate T p, dayside from this grid. This thus gives us posterior samples for T p that are correlated with the star and planet parameters. From this analysis we find that the brightness temperature of the planet's dayside is 709 ± 31 K. \nTo compare the determined T p, dayside with the maximum possible dayside brightness temperature T max , we define the temperature scaling factor R by \nT p,dayside = T max · R = ( 2 3 ) 1 4 · T s √ a/R s · R (4) \nFollowing Cowan & Agol (2011), R is defined as \nR = ( 2 3 ) -1 4 · (1 -A B ) 1 4 · ( 2 3 -5 12 ε ) 1 4 (5) \nwhere A B is the Bond albedo and ε is the heat recirculation efficiency. Note the ( 2 3 ) 1 4 factors in Equations 4 and 5 are for normalization purpose, so when A B = 0 and ϵ = 0 (zero reflection and no heat redistribution), R = 1, representing the maximum possible dayside temperature, T max = 746 +14 -11 K. \nWe calculate R for each sample in the MCMC chain derived in § 4. We present the corner plots of relevant stellar parameters, planetary dayside temperature and associated R in Figure 3. From the measured white light eclipse depth, we derive a temperature scaling factor of R = 0 . 95 ± 0 . 04 \nCombining equations 4 & 5, we can get T p, dayside as a function of A B and ε . To explore how these two factors affect T p, dayside , we adopted the median values of T s and a/R s from the MCMC (see Table A1), and varied A B and ε from 0 to 1 to calculate T p, dayside as a function of A B and ε . Afterwards, we plotted the map of T p ( A B , ε ) in Figure 4 with the uncertainties of T p, dayside shown with contours. \nBoth the derived R and Figure 4 indicate that our inferred dayside temperature for GJ 1132b is only ∼ 1 σ lower than the maximum possible temperature (746 +14 -11 K). This suggests that the planet has little to no day-to-night heat redistribution and that the surface is relatively dark. Thus the data are consistent with the presence of no atmosphere on the planet. This is reinforced by comparing the joint constraints on the Bond albedo and heat redistribution factor with the values for terrestrial objects in the solar system in Figure 4. The results for GJ 1132b are consistent with the parameters of the airless and nearly-airless bodies (i.e., Mercury, Mars, and Earth's Moon), and they are inconsistent with \nFigure 3. Corner plot of the planet temperature and temperature factor along with key star/planet parameters. The posteriors for T s , a/R ⋆ , and R p /R ⋆ come from the global modeling described in § 4. The posteriors for T p and R come from the modeling described in § 5.1. Black are the results using PHOENIX stellar models, and blue are the results using SPHINX models. \n<!-- image --> \nthe parameters of the planets that have significant atmospheres (i.e., Earth and Venus). \nTo place quantitative limits on the thickness of a possible atmosphere, we followed the approach in Zhang et al. (2024), who used the \nFigure 4. Planetary dayside temperature as a function of Bond albedo A B and heat recirculation efficiency ε . The median and ± 1 σ contours show the 709 ± 31 K dayside temperature derived from the white light curve. The yellow arrow indicates the maximum dayside temperature (746 +14 -11 K) assuming no reflection and no heat recirculation. We adopt A B = 0.119, 0.75, 0.306, 0.250, 0.11 for Mercury, Venus, Earth, Mars, and the Moon. Since the solar system terrestrial planets are not tidally locked, we calculated their heat redistribution efficiency using their polar and equatorial temperatures. Following equation 4 & 5 in Cowan & Agol (2011), we derived ϵ = 8 / 3 (( T equator /T pole ) 4 +5 / 3) . \n<!-- image --> \nscaling relation for the heat redistribution parameter in Koll (2022). We adopted an upper limit on the heat redistribution efficiency of 0.52, which is the 1 σ upper limit from our constraints when fixing A B =0. We find an upper limit of 0.7 bar for a pure H 2 O atmosphere, and 2.4 bar for a pure CO 2 atmosphere. Thus, we can rule out very thick, Venus-like atmospheres solely on energy transport arguments alone. Earth-like, 1 bar atmospheres are marginally consistent with our constraints on the heat redistribution. However, such atmospheres would also likely have a non-zero Bond albedo, as illustrated in Figure 4. \nAcaveat to this analysis is that we do not consider thermal beaming, which may result from \n(e.g.) surface roughness effects (Spencer 1990), and which can increase the observed low-phaseangle brightness temperature of atmospherefree bodies above the maximum stated above (Emery et al. 1998). On the other hand, even a thin atmosphere would likely negate thermal beaming, thus making our atmospheric thickness constraints still valid. \nAnother important caveat to this analysis is that we are actually only constraining the effective albedo over the MIRI/LRS bandpass rather than the true Bond albedo. A blackbody with the dayside temperature we infer for GJ 1132b (709 K) emits 48% of its total energy between 5 and 12 µm . Therefore, we are likely capturing a large fraction of the energy emitted by the planet, and this is a key aspect of estimating the Bond albedo (see discussion in Kempton et al. 2023). On the other hand, even bare surfaces will have wavelength-dependent albedos, and the most likely minerals on GJ 1132b have lower-than-average albedos in the MIRI/LRS bandpass (Hu et al. 2012). As shown by Mansfield et al. (2019), using MIRI/LRS can lead to underestimating the Bond albedo of GJ 1132b by 0.1 - 0.2, while Whittaker et al. (2022) suggested this underestimation is less serious. However, space weathering of close-in planets like GJ 1132b will serve to darken surfaces and make them more blackbody like (Zieba et al. 2023; Lyu et al. 2024, and references therein), thus making our assumptions more valid. With these caveats, if we assume GJ 1132b doesn't have an atmosphere, then our constraints yield A B = 0 . 19 +0 . 12 -0 . 15 .", '5.2. Forward Modeling': "Following the methods in Whittaker et al. (2022) and Ih et al. (2023), we used HELIOS (Malik et al. 2017, 2019a,b) to compute forward models to compare to the dayside thermal emission spectrum of GJ 1132b. Complementing the white light analysis, this allows for incorporating the spectral information to test which \nFigure 5. Goodness-of-fit ( χ 2 ) of the eclipse depths of the suite of atmospheric models, plotted as functions of the modeled surface pressure. The best-fit blackbody (691 K) to the emission spectrum is shown with the black dashed line. The bare surface models are shown in the leftmost column with diamonds. The median and 1σ and 2σ equivalent quantiles of the χ 2 distribution ( k =14, k is the degree of freedom) are shown as the solid black line, the dark gray band, and the light gray band, respectively. The atmosphere models with compositions of CO 2 filled with O 2 are shown with squares and H 2 O filled with O 2 are shown with circles, where colors indicate the percentage of trace gasses. The atmosphere models generally move, from the thinnest (10 -4 bar) to the thickest (10 2 bar) surface pressures, from the χ 2 for the maximally hot temperature (blue dashed line) to the χ 2 for the equilibrium temperature (red dashed line), with deviating inflection points arising due to spectral features in the LRS bandpass and changes in the temperaturepressure profile. Atmospheric models that produce χ 2 values comparable to the best-fit blackbody are plotted in Figure 6. \n<!-- image --> \natmosphere models of varying surface pressure and composition are plausible. We describe the model set up and present which model thermal emission spectra are consistent with the spectra here. \nWe adopted the median system parameters derived in § 4, and we generated the stellar spectrum using the SPHINX model grid (Iyer et al. 2023). We calculated eclipse spectra for atmosphere models of varying surface pressure from 10 -4 to 10 2 bar in 1-dex intervals (assuming a blackbody surface under the atmosphere) and \ncompositions of 1 ppm, 100 ppm, 1%, and 100% CO 2 or H 2 O, with the rest of the composition backfilled with O 2 . We note that the choice of backfilling gas between O 2 and N 2 does not affect our results. We also generated eclipse spectra for bare surface models using the surface albedo spectra from Hu et al. (2012). \nWe determined the brightness temperature in the observed eclipse depth spectrum by using the emcee package to find the temperature that minimizes the least-square fit between the data and a blackbody model. From this we obtain \nFigure 6. Observed brightness temperature spectrum from SPARTA (cyan) and Eureka! (black) plotted against models for bare surfaces (top panel) and atmospheres (bottom) that fit the data. The expected brightness temperatures and their uncertainties propagated from the system parameters are plotted as horizontal lines for varying Bond albedos and redistribution factors. The bandpass functions for the LRS and the 12.8 and 15 µ m MIRI photometry filters are also shown for illustration. As can be seen from 1 ppm and 10 ppm CO 2 models, the presence of CO 2 can be further constrained using 15 µ m MIRI photometry. \n<!-- image --> \nT b = 691 +22 -21 K. This is smaller than the temperature derived from the white light curve because of the difference in how the channels are weighted. In the white light curve, the channels are summed and are thus weighted by photon counts. In the spectrum, the average is computed by weighting by the errors on the different channels. The errors in the spectroscopic channels are relatively larger compared to the stellar photon counts at longer wavelengths because of the increased background in the raw data and the higher levels of red noise in the light curves. Thus the longer wavelength channels, which suggest a higher brightness temperature, are de-weighted in the spectroscopic analysis here. Furthermore, the error bar on the temperature derived from the spectra is smaller than from the white light curve because it doesn't account for uncertainties in the star and planet parameters. Importantly, this is better than 1 σ consistent with the T b from our broadband measurement( § 5.1), further validating our analytical approach. \nFrom the modeled eclipse depth spectra, we computed the goodness-of-fit ( χ 2 ) to the data for each model, which we show in Figure 5. As a function of the surface pressure, the χ 2 values of atmosphere models are controlled by three effects: (1) cooling by redistribution from day to nightside; (2) spectral features from molecular absorption; (3) greenhouse warming. This leads to non-monotonic behavior in χ 2 and potentially multiple inflection points. \nWe find that a broad range of atmospheres are consistent with the data. The atmospheres that are readily ruled out at > 3 σ are either too thick (all of the 100 bar atmospheres and most of the 10 bar atmospheres) or have too much molecular absorption (pure CO 2 atmospheres above 0.006 bar or pure H 2 O atmospheres above 0.16 bar). Compared to the white-light analysis, this sets the upper limit for the pure CO 2 or H 2 O atmospheres at more tenuous surface \npressures, as now the shape of the spectrum is taken into account. As such, the spectrum now rules out even a Mars-like thin atmosphere, if composed entirely of CO 2 . \nThe thinnest atmospheres modelled have χ 2 values very close to that of the blackbody at T max (blue dashed line), while the thickest surface pressure modelled have worse χ 2 values than that of the blackbody at T eq (red dashed line). At intermediate surface pressures of 10 -1 -10 1 bar, the atmosphere models show comparable χ 2 values to the best-fitting blackbody (black dashed line), but do not show an appreciable improvement in goodness-of-fit because the blackbody already explains the data well. \nFor these models, we show the brightness temperature spectra in Figure 6. The brightness temperature of the data is obtained by first multiplying the binned model stellar spectrum by the observed F p /F s spectrum and propagating the system parameter uncertainties to the errors. The few points in the data that have low brightness temperature around 6 µ m align with either CO 2 or H 2 O features, but the errors are large enough to preclude confidently inferring that an absorption feature is present. We also note that the transmission spectra of modeled atmospheres are consistent with the featureless transmission spectrum in May et al. (2023). \nSimilarly, we find that a broad range of bare surface models are consistent with the data. We show the goodness-of-fit for each model in Figure 5 and the brightness temperature spectrum in Figure 6. For these models, the χ 2 values are controlled primarily by their Bond albedo in the shortwave affecting the dayside temperature and secondarily by the spectral emissivity in the MIRI bandpass. As with the atmosphere models, the errors are large enough to preclude inferring a spectral feature.", '6. DISCUSSION': "In this paper, we reported a secondary eclipse observation of the super-Earth GJ 1132b with JWST MIRI/LRS, yielding a measurement of the white-light dayside brightness temperature and dayside emission spectrum. We also refined the star and planet parameters for this benchmark system. Given energy balance, the measured secondary eclipse (140 ± 17 ppm) is very close to the maximum possible depth (164 ± 6 ppm). The dayside emission spectrum exhibits no significant spectral features and is consistent with a blackbody that has T b =691 +22 -21 K. \nWe compared the dayside emission spectrum with a wide range of atmospheric composition models. We found atmospheres with thickness P > 1 bar and containing at least 1% H 2 O are ruled out, and atmospheres of any modeled thickness (10 -4 bar - 10 2 bar), containing at least 1% CO 2 are ruled out. We also found very thick, Venus-like atmospheres ( P ∼ 10 2 bar), even without significant infrared absorbers, are ruled out. A few points around 6 µ m could suggest CO 2 or H 2 Ofeatures, but the errors are too large to infer conclusive results. Due to their high mean molecular weight, the transmission spectra produced by all modeled atmospheres show no significant features, which is consistent with May et al. (2023). From the bare surface models, we found a wide range of possible surface compositions are consistent with the data. Thus we conclude that, given the preponderance of the evidence, the planet likely does not have a significant atmosphere. \nThese results for GJ 1132b provide an additional exoplanet data point with which to test the 'Cosmic Shoreline' theory (Zahnle & Catling 2017). This theory suggests a planet's ability to maintain an atmosphere is based on its predicted escape velocity (v esc ) and its cumulative extreme UV (XUV) irradiation. Figure 7 shows the Cosmic Shoreline for a number of Solar System bodies and rocky M-dwarf \n<!-- image --> \nTitan \nSaturn \nUranus \nNeptune Figure 7. GJ1132b in the context of the Cosmic Shoreline (Zahnle & Catling 2017). In the solar system, the dashed blue line separates bodies that have atmospheres from those that don't. Blue shading illustrates uncertainty on the y-axis. For exoplanets, only rocky planets around M-dwarfs with thermal emission measurements are shown. \nplanets. The normalized cumulative XUV irradiation (normalized to Earth) is estimated using an analytic scaling (Eq. 27 in Zahnle & Catling 2017), and the dashed line is a power law fitted to match Mars. The actual XUV irradiation history of particular exoplanets is unknown, so the blue region shows an illustrative factor of 4 uncertainty. Higher escape velocities and less insolation are more favorable for atmospheric survival. Our target, however, is placed more than 1 σ above the shoreline (Figure 7). Further away from the shoreline are the three M-dwarf planets likely to have no atmospheres based on Spitzer and JWST observations (LHS 3844b, GJ1252b, and GJ367b: Kreidberg et al. 2019; Crossfield et al. 2022; Lyu et al. 2024; Zhang et al. 2024). Recent JWST MIRI observations of TRAPPIST-1 b and c (Greene et al. 2023; Zieba et al. 2023) also suggest that neither of them have thick atmospheres. The relative position of these planets on the Cosmic Shoreline reinforces our conclusion that GJ 1132b is most \nJupiter \nlikely to be a bare rock with no or a thin atmosphere, consistent with what we see in Figure 4. \nThe question of whether M dwarf rocky planets can host atmospheres remains unresolved. Despite their higher risk of losing the atmospheres than for rocky planets around Sun-like stars, as we discussed in § 1, there are still reasons to be optimistic. The planets might be able to accumulate a water-rich envelope beyond the snowline during formation, then migrate inward, or have an H 2 layer that could shield water from loss (e.g. Ribas et al. 2016; Barnes et al. 2018; Kite & Schaefer 2021), but see also Kite & Barnett (2020). Other hypotheses include having magnetic fields that would guard against some loss mechanisms (e.g. Segura et al. 2010; Vidotto et al. 2013), starting rich in carbon derived from refractory organics (Li et al. 2021), renewing their secondary atmospheres from outgassing after an initial loss phase (Kite & Barnett 2020), retaining atmospheres against efficient loss due to atomic line radiative cooling (Nakayama et al. 2022), or being resupplied with volatiles from an external source, such as through cometary bombardment. Although our GJ 1132b observation indicates GJ 1132b likely does not have any atmosphere, our result further constrains the possible location of the Cosmic Shoreline. We hope future observations, such as the proposed 500 hrs JWST DDT program (Redfield et al. 2024), will further refine it.", '7. DATA AVAILABILITY': 'The data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed can be accessed via DOI: 10.17909/br8w-4288. The data that were used to create all of the figures will be freely available on Zenodo (Xue et al. 2024b) DOI: 10.5281/zenodo.13244543. \nAll additional data is available upon request. \nFacilities: \nJWST(MIRI) \nSoftware: Eureka! (Bell et al. 2022), SPARTA (Kempton et al. 2023), Astropy (Astropy Collaboration et al. 2013, 2018, 2022), emcee (Foreman-Mackey et al. 2012), dynesty (Speagle 2019), batman (Kreidberg 2015), HELIOS (Malik et al. 2017, 2019b,a; Whittaker et al. 2022), Matplotlib (Hunter 2007), Numpy (Harris et al. 2020), Scipy (Virtanen et al. 2020), textttEXOFASTv2 (Eastman et al. 2019)', '8. ACKNOWLEDGEMENTS': 'This work is based on observations made with the NASA/ESA/CSA JWST. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 503127 for JWST. The observations are associated with program GTO 1274. MZ is grateful for support from the 51 Pegasi b Fellowship, funded by the Heising-Simons Foundation. MWM acknowledges support through the NASA Hubble Fellowship grant HST-HF251485.001-A awarded by STScI. JL was supported by grant NNX17AL71A from NASA Goddard Spaceflight Center. J. I. and E. M.R. K. acknowledge support from the AEThER Matter-to-Life program, funded by the Alfred P. Sloan Foundation under grant G202114194.', 'A. APPENDICES': 'Figure A8. Median data number counts summed over the data images as a function of group number after linearity correction of the first data segment. Non-linear behavior is seen in the first ∼ 11 groups, and a downward offset in the last group (#28) is seen. Although each pixel behaves differently, this non-linearity is found in all integrations. Two red dashed lines indicate groups #5 and #11. The non-linearity could be caused by the reset-switch charge decay (RSCD; Morrison et al. 2023; Dyrek et al. 2024), which causes a greater increase in the ramp at the start of an integration. Besides our standard Eureka! reduction shown in § 3, we also implemented RscdStep (https://jwst-pipeline.readthedocs.io/en/latest/api/jwst.rscd.RscdStep. html) from the standard jwst pipeline in Eureka! . Similar to what our preferred SPARTA reduction did (i.e., excluding the first 5 groups), RscdStep excluded the first 4 groups from the up-the-ramp fitting for this dataset. The resulted best-fit white light eclipse depth is 147 ± 17 ppm, compared to 143 +17 -19 ppm from our Eureka! standard reduction, 145 ± 20 ppm from our preferred SPARTA gr5 reduction and 141 ± 17 ppm from the global fitting. \n<!-- image --> \nFigure A9. Pixel-level light curves at 10.7861 and 10.8060 µ m. An anomalous rise in flux followed by an exponential decay was found at 10.7861 µm , and a sharp drop at 10.8060 µ m was found at the same timestamp. \n<!-- image --> \nTable A1 . Refined parameters for the GJ 1132 system. \nTable A1 continued \nTable A1 (continued)Table A1 continued \nTable A1 (continued)Table A1 continued \nTable A1 (continued)Table A1 continued \nTable A1 (continued) \nSee Table 3 in Eastman et al. (2019) for a detailed description of all parameters', 'REFERENCES': 'Alderson, L., Batalha, N. E., Wakeford, H. R., et al. 2024, JWST COMPASS: NIRSpec/G395H Transmission Observations of the Super-Earth TOI-836b, arXiv. \nhttp://arxiv.org/abs/2404.00093'}
2024arXiv240910752S	New observational facilities are beginning to enable insights into the threedimensional 3D nature of exoplanets. Transmission spectroscopy is the most widely used method for characterizing transiting temperate exoplanets atmospheres but because it only provides a glimpse of the planets limb and nightside for a typical orbit its ability to probe 3D characteristics is still an active area of research. Here we use the ROCKE3D general circulation model to test the impact of rotation rate a loworder 3D characteristic previously shown to drive differences in planetary phase curves on the transmission spectrum of a representative superEarth across temperatetowarm instellations Sp0.8 1 1.25 1.66 2 2.5 3 4 4.56 Soplus. We find that different rotation regimes do display differences in their transmission spectra primarily driven by clouds and humidity and that the differences shrink or disappear in hotter regimes where water clouds are unable to condense though our simulations do not consider haze formation. The small size of the feature differences and potential for degeneracy with other properties like differing water content or atmospheric structure mean that we do not specifically claim to have identified a single transmission diagnostic for rotation rate but our results can be used for holistic spectrum interpretation and sample creation and suggest the need for more modelling in this area.	2024-09-01T00:00:00Z	['2024arXiv240910752S', '10.48550/arXiv.2409.10752', 'arXiv:2409.10752']	['Astrophysics - Earth and Planetary Astrophysics']	The Spin Zone Synchronously and Asynchronously Rotating Exoplanets Have Spectral Differences in Transmission	2,024	190	0.44	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.10752.pdf	{'The Spin Zone: Synchronously and Asynchronously Rotating Exoplanets Have Spectral Differences in Transmission': 'Nicholas Scarsdale , 1 C. E. Harman, 2 and Thomas J. Fauchez 3, 4, 5 \n1 \nDepartment of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA 2 Planetary Systems Branch, Space Science and Astrobiology Division, NASA Ames Research Center, Moffett Field, CA 94035, USA 3 NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD, 20771, USA 4 Integrated Space Science and Technology Institute, Department of Physics, American University, Washington, DC, USA 5 NASA GSFC Sellers Exoplanet Environments Collaboration', 'ABSTRACT': "New observational facilities are beginning to enable insights into the three-dimensional (3D) nature of exoplanets. Transmission spectroscopy is the most widely used method for characterizing transiting temperate exoplanet's atmospheres, but because it only provides a glimpse of the planet's limb and nightside for a typical orbit, its ability to probe 3D characteristics is still an active area of research. Here, we use the ROCKE-3D general circulation model to test the impact of rotation rate, a 'low-order' 3D characteristic previously shown to drive differences in planetary phase curves, on the transmission spectrum of a representative super-Earth across temperate-to-warm instellations (S p =0.8, 1, 1.25, 1.66, 2, 2.5, 3, 4, 4.56 S ⊕ ). We find that different rotation regimes do display differences in their transmission spectra, primarily driven by clouds and humidity, and that the differences shrink or disappear in hotter regimes where water clouds are unable to condense (though our simulations do not consider haze formation). The small size of the feature differences and potential for degeneracy with other properties, like differing water content or atmospheric structure, mean that we do not specifically claim to have identified a single transmission diagnostic for rotation rate, but our results can be used for holistic spectrum interpretation and sample creation, and suggest the need for more modelling in this area.", '1. INTRODUCTION': "Studies of solar system planets are rich in examples of planets varying in multiple dimensions on disparate timescales: for example, the Earth's clouds, Mars' storms (Shirley 2015), Saturn's polar hexagon (Godfrey 1988; S'anchez-Lavega et al. 2020), etc. These examples are representative of the way that three-dimensional (3D) effects can shift even observations in one-dimension (e.g., a spatially unresolved spectrum of Earth, LustigYaeger et al. 2023). However, although 3D studies of exoplanets have become steadily more common, lower order (typically zero- and one-dimensional) models still predominate in interpreting observations, for two major reasons. First, higher dimensional models are computationally expensive to run, especially when investigating a wide parameter space. Second, data quality has not always been sufficient to investigate the details of higher dimensional effects. By data quality, we mean not only observational constraining power, but also fundamental physical and chemical materials properties (such as molecular line lists; e.g., Tennyson & Yurchenko 2022) needed to simulate a diverse range of targets. Thanks \nto new instruments and computational facilities, the 3D structure of exoplanets is beginning to be understood in more detail. For example, recent observations with the James Webb Space Telescope (JWST) and highresolution ground-based facilities have identified differences between the morning and evening terminators of hot Jupiters (e.g., Tsai et al. 2023; Demangeon et al. 2024). As a computational example, atmospheric retrieval analyses have historically relied on 1D models (Benneke & Seager 2012), but new work is pushing towards pseudo-retrievals in 2D and 3D (e.g., Zingales et al. 2022; Himes et al. 2023). However, there remains much room to explore how three-dimensionality of exoplanets impacts their atmospheres, both from the perspective of underlying physics (and, as an emergent property, habitability), and from our observations of them. In particular, 3D studies of planets larger and hotter than Earth, like we simulate here, are less wellrepresented in the literature than Earth-size, cool-totemperate planets (e.g., Del Genio et al. 2019). \nOutside the domain of hot Jupiters, studies of small exoplanet atmospheres are at the limits of contempo- \nrary observatories' capabilities. Even the new JWST will require significant investment of observing time to detect primary background gases in terrestrial-type atmospheres (e.g., Morley et al. 2017; Fauchez et al. 2019; Pidhorodetska et al. 2021; Fauchez et al. 2022). This challenge will be all the more significant in light of recent observational evidence that stellar contamination may significantly influence the transmission spectra of small planets around M dwarfs (Moran et al. 2023; Lim et al. 2023). Nonetheless, because of their unique combination of potential habitability and observability, the exoplanet community will spend significant observational time in the search for atmospheres around these planets (e.g., the recent Working Group on Strategic Exoplanet Initiatives recommendation to develop a 500 hour JWST Director's Discretionary Time program for this purpose; Redfield et al. 2024). Forward modelling has an important role to play in informing and interpreting these observations. \nTo date, one of the most well-studied characteristics of exoplanets in 3-D is rotation rate. In particular, synchronous rotators, with permanent day- and night-sides, are thought to be dramatically different from planets with rotation regimes like that of Earth (Joshi 2003; Merlis & Schneider 2010; Edson et al. 2011; Showman et al. 2013; Leconte et al. 2013; Yang et al. 2013, 2014; Carone et al. 2014; Bony et al. 2015; Noda et al. 2017; Haqq-Misra et al. 2018; Zhao et al. 2021; Lewis & Hammond 2022). Planetary rotation rate has a variety of potential effects on planetary atmospheric observability. Of particular interest for motivating this work, HaqqMisra et al. (2018) showed that there are observably different phase curves as synchronously rotating planets in the habitable zone cross between dynamic regimes. Wolf et al. (2019) also show that several categories of climate regime in the TRAPPIST-1 planets are mutually discernible by phase curve. JWST has observed full phase curves of TRAPPIST-1 b and c (Program ID#3077; PI Gillon) which may provide an observational perspective on this phenomenon when published. However, phase curves are by definition more time-intensive to obtain than transit or eclipse spectroscopy, and are only feasible for particular systems with sufficient planet-to-star contrast (Shporer 2017). Emission spectroscopy during secondary eclipse is also a useful technique for learning about planetary atmospheres, but becomes relatively more difficult for cooler planets as the planet-to-star flux ratio falls off quickly with temperature (e.g., Morley et al. 2017). Therefore, it is of interest to search for the fingerprints of rotation rate in transmission spectroscopy, which can be performed for a wide range of systems. Along these lines, Cohen et al. (2024) suggest \nthat the rotation rate of synchronously rotating superEarths can have differing effects on the planet's haze distribution. In particular, they identify three circulation regimes, each with distinct haze distributions and corresponding optical depths. The details of the haze behavior is nonlinear, but they broadly find minima in terminator haze optical depth near rotation periods of ∼ 2 and 15 days, and maxima near ∼ 0.25, 7, and > 25 days. Here, we investigate whether differences caused by rotation rate can be observed in exoplanet transmission spectra without the effect of significant hazes (which may not dominate the coolest exoplanet atmospheres; e.g., Yu et al. 2021; Brande et al. 2024; Holmberg & Madhusudhan 2024). \nWe use the general circulation model (GCM) Resolving Orbital and Climate Keys of Earth and Extraterrestrial Environments with Dynamics (ROCKE-3D; Way et al. 2017) to generate a suite of simulations of a representative warm super-Earth exoplanet atmosphere. As we describe below, we tailored our simulations to stay outside the conditions notionally associated with a runaway greenhouse scenario, which prevents convergence for our simulations as constructed. However, we will argue that some of the fundamental physics seen in our simulations can be generalized to other types of planet. In the next section, we describe our modelling setup, along with relevant limitations. In Section 3, we present the differences in spectra between our simulations. In Section 4, we investigate the implications our models present for observations of small planets, and provide concluding remarks in Section 5.", '2.1. Experimental Setup': "ROCKE-3D is a planetary general circulation model developed from (and in parallel with) the NASA Goddard Institute of Space Studies (GISS) ModelE2, and Earth climate model (Schmidt et al. 2014). ROCKE3D has been used to simulate terrestrial, exoplanet, and Martian climate (e.g., Del Genio et al. 2019; Guzewich et al. 2021), including for planets at higher instellations and rotation rates (e.g., Fujii et al. 2017; Kane et al. 2018; Colose et al. 2021; He et al. 2022). More details about ROCKE-3D can be found in Way et al. (2017). We make use of the Planet 1.0 version of the model, which includes the SOCRATES radiation scheme (Edwards 1996; Edwards & Slingo 1996). \nWe created a series of three-dimensional simulations of a terrestrial exoplanet atmosphere. Our model uses the mass and radius of L 98-59 d (Kostov et al. 2019; Cloutier, R. et al. 2019), a super-Earth sized planet that has been identified as a promising target for trans- \nmission spectroscopy (Pidhorodetska et al. 2021). We choose the stellar spectrum for Kepler-1649, an M dwarf with an effective temperature of 3240 K (Angelo et al. 2017), which was the closest match available for the warmer L 98-59 (T eff =3412 K; Cloutier, R. et al. 2019). We scale the total flux to match L 98-59 so that the planet receives the correct irradiation. Our simulations vary rotation rate (P rot = 1 d, P orb ) and irradiation (S p =0.8, 1, 1.25, 1.66, 2, 2.5, 3, 4, 4.56 S ⊕ ), in order to search for differences that might be detectable by transmission spectroscopy. Additional details about our simulation can be found in Table 1. We are motivated to vary rotation rate because, although the circularization timescale is often taken to be short (e.g., Kasting et al. 1993, following Peale 1977), recent work has shown that the dynamical state of small planets is complex (particularly in multi-planet systems like L 98-59, e.g., Vinson et al. 2019) and may induce spin very unlike the classical picture of a synchronous rotator, with a permanent day- and night-side. These differing dynamical states will lead to climatological differences (e.g., Shakespeare & Steffen 2023; Chen et al. 2023). Because a complete dynamical simulation tied to a GCM is computationally unsuited for a parameter space sweep, we use our 1-day rotators as to represent planets which have been excited from their synchronous orbits. Our simulations have zero obliquity and eccentricity, both parameters shown to influence climatology in comparable ways to rotation (e.g., Colose et al. 2021; Olson et al. 2020; He et al. 2022; Jernigan et al. 2023), although given the potential for dynamically complex states, these would be worthwhile to investigate in future work. \nThe simulations presented here are run at 4 · × 5 · latitude-longitude resolution with 40 atmospheric layers from the surface to ∼ 0.1 mbar at the top of the model domain. The surface is initialized as a fully saturated sand, which represents the entire liquid water inventory as we have opted not to run with a large surface water inventory. This helps avoid a transition to a runaway greenhouse scenario (Goldblatt et al. 2013). We make this choice not because a significant water inventory is physically implausible, but because it cannot be investigated with our model. The runaway greenhouse scenario that results on hot terrestrials with large water inventories has only recently been examined with a GCM model for the first time (Chaverot et al. 2023; Turbet et al. 2023). However, we note that the assumption of low water content does have some physical motivations. Super-Earths need not have significant water content (Cowan & Abbot 2014). As well, planets with M-type host stars (like L 98-59) may lose significant amounts of their starting water content due to the host's long \nTable 1. The planet and atmospheric parameters used for our simulations. \npre-main-sequence phase (e.g., Luger & Barnes 2015). Even if a planet avoids these escape scenarios and remains water-rich, the actual amount of atmospherically available water may be low due to cold trapping on the nightside (Lobo et al. 2023). \nThe simulations were run in batches, starting with the lowest instellation cases. The outputs of these simulations were then used as initial conditions for the higher instellation cases, as they were closer to the expected final surface temperatures than the default Earth-like input files. We found that this was necessary to achieve convergence in the hotter simulations. Although we cannot rule out hysteresis, our models lack oceans and except in the coolest two cases have no surface ice, eliminating those drivers of contrast in the planetary albedo. Previous studies also find that climate hysteresis in terrestrial atmospheres is reduced for planets with M-type host stars (Shields et al. 2014) and tidally locked planets (Checlair et al. 2017, 2019). \nWe provide diagnostic maps of surface temperature and wind velocity for two representative irradiation cases in Figure 1 and substellar pressure-temperature profiles in Figure 2. Convergence to a steady state climate is estimated from the imbalance between the incoming and outgoing radiation fluxes (Figure 3), which should be near zero. In our cases, we found that < 1 W/m 2 variations and corresponding temperature variations were common. We attribute this to the limited water inventory and moderate timescale for equilibration with the atmosphere (Figure 4) in all but the hottest cases (see Section 2.2). \nOnce the simulations have converged, we used the Planetary Spectrum Generator (PSG, Villanueva et al. 2018; Villanueva et al. 2022) in order to obtain transmission spectra. PSG and the associated GlobES module 1 uses the full three-dimensional output of the GCM \nTemperature (K) \nTemperature (K) \nto self-consistently perform the 3D radiative transfer calculations of the spectrum in transmission. Because ROCKE-3D only generates atmospheres up to pressures of 0.1 mbar and lower pressures can still impact transmission spectroscopy, we extended the atmospheres up to ∼ 10 nbar, assuming an isothermal profile and the same molecular mixing ratios, mimicking the procedure in Fauchez et al. (2022) (and similarly finding minimal differences). We also applied an additional pre-processor option to obtain the cloud mixing ratio and particle size variables for the liquid and water ice clouds. We followed the recommendations in the PSG handbook (Villanueva et al. 2022) for generating transmission spectra from the .aij* ROCKE3-D output files to ensure that the correct geometry was captured in our spectra.", '2.2. The Limits of ROCKE-3D': "ROCKE-3D was originally developed as a branch of, and in parallel with, Earth climate simulation capabil- \nFigure 2. The pressure-temperature profiles of each simulation at the sub-stellar point (or longitude=0 for the 1-day rotators). The region where the HITRAN line list is inadequate is highlighted in red. \n<!-- image --> \nFigure 1. Maps of ground air temperature and wind speed for the representative 0.8 S ⊕ 1-day (a) and synchronous (b) rotators, and 1.66 S ⊕ 1-day (c) and synchronous (d) rotators. The temperature maps roughly follow expectations, with the synchronous rotators dominated by day- to night-side variation, and 1-day rotators dominated by latitudinal variation. Note that the color bars are different in each subfigure. \n<!-- image --> \n440 \n460 \nFigure 3. Here we show the energy balance of each simulation ordered by instellation. A value of 0 would indicate a perfectly equilibrated simulation. The one-day rotators are almost fully converged. The hottest synchronous rotators display non-zero net fluxes. But, values lower than ∼ 0.75 W m -2 are sufficient to be considered converged (although see Section 2.2 next). \n<!-- image --> \nFigure 4. The median surface temperature of each simulation normalized by final temperature vs. the surface specific humidity. For most simulations, these values track each other closely, but this breaks down in the hottest (4 and 4.56 S ⊕ ) cases. \n<!-- image --> \nities (ModelE2, Schmidt et al. 2014). As a result, it is best suited for modelling small, temperate planets. The parameters of the planet we chose to model are at the limits (and in some cases beyond the limits) of where ROCKE-3D and its underlying correlated-k tables are appropriate given our model setup. We therefore provide here a list of limitations of our model. Only recently has a different publicly available GCM tool that can bypass some of these limitations (particularly those related to the runaway greenhouse effect) been developed (Chaverot et al. 2023; Turbet et al. 2023). \n- 1. Small exoplanets are thought to have a variety of bulk compositions, leading to great diversity in \ntheir atmospheres (see, e.g., Wordsworth & Kreidberg 2022). However, to maintain relative simplicity, our realization of this planet is narrow. Our model is a dry, 'desert' world (i.e., no oceans). It is decidedly not a water world (which this planet may be, per Luque & Pall'e 2022). Its atmospheric composition is entirely N 2 /CO 2 with trace H 2 O. It also does not include significant topography. \n- 2. Our compositional choices result in an atmosphere that likely does not have significant haze particles, photochemical or otherwise. Such hazes are thought to be common in warm exoplanet atmospheres (e.g., Gao et al. 2021). However, cool planets with temperatures similar to Earth appear to have efficient haze removal (Yu et al. 2021), so for our cooler instances, this is a motivated decision. At the same time, different atmospheric compositions (e.g. added methane to serve as a haze precursor) could result in observationally and climatogically significant hazes (Cohen et al. 2024).\n- 3. Some of our hotter simulations, in parts of their atmosphere, exceed the limits at which the HITRAN molecular line list for H 2 O is valid (see Figure 2), which results in problems for the radiative transfer code (Goldblatt et al. 2013). We include them here for completeness. Given our model setup, the physics of these hotter simulations are qualitatively sensible, since in both rotation regimes, they converge on a perpetually sub-saturated atmosphere as the temperature increases. These models produce spectra that resemble the highertemperature simulations that are within the valid HITRAN temperature range. However, we cannot be confident in results outside the valid HITRAN range; thus, the bulk of our conclusions are drawn from the cooler planets, where no line-list issues exist.\n- 4. We self-consistently calculate the orbital distance and rotation rate for each instance, meaning that the synchronous rotators approach the dynamical regime sampled by the 1-day rotators in rotation period as we move to higher instellation (at a 1 day orbital period they would have the same rotation period). Our hottest simulation has an orbital period of ∼ 7 days, which likely crosses into the Rhines rotation regime (Haqq-Misra et al. 2018). However, per the previous point, there is limited information to be gained from the hotter simulations, and the cooler cases resemble the slow rotators (Figure 1). \nFigure 5. The average surface temperature at each time point of our synchronously rotating models, shown for each irradiation step. The hottest of these models, particularly the 4 and 4.56 S ⊕ cases, still have temperature fluctuations at the ∼ few K level. \n<!-- image --> \n- 5. Our simulations are largely within the net flux threshold for convergence ( ≲ 1 W m -2 deviation from 0), and for the most part their temperatures do not fluctuate significantly in the final stages of the model, which is a useful stability indicator (Figure 5). However, particularly in the hottest synchronous cases, there is non-zero net flux (see Figure 3) and persistent temperature fluctuation at the 1-2 K level. Even though the hottest cases still broadly appear converged, there are clear limitations on what information can be extracted from them, and we include them here primarily for completeness.", '3. SOME SYNCHRONOUS AND 1-DAY ROTATORS HAVE DIFFERENT SPECTRA': "The primary result of our work can be summarized as follows: for planets from roughly temperate-to-Venus levels of irradiation (1-2 S ⊕ ), the near-infrared transmission spectra of planets in the two rotation regimes we consider are visibly distinct. This is clear in Figure 6, where we show the synchronous vs 1-day rotator transmission spectra for each model in our instellation grid. Note that this representation differs slightly from the usual representation of transmission spectra. Rather than showing total depth (i.e., including the entire planetary radius), or showing only feature height above the opaque planet + continuum (i.e., minimum of feature height in all plots is 0), we show feature amplitude above the planet surface only. Thus the opaque atmosphere also contributes to the continuum, resulting in 'depths' that have minima ≳ 3ppm. We have selected the 0.55 micron wavelength region specifically because this is \nwhere JWST will be able to obtain precise spectra for small planets. \nIn the simulations up to roughly Venus irradiation, the spectra are distinct in terms of their spectral continuum and features, amplitude of features, and scattering slope, given a spectrum of arbitrarily high precision (see Section 4.2 for a discussion of the implications of these differences for real observations). In the following subsections, we will discuss the characteristics of these differences and the factors that drive them in more detail. We discuss clouds first and then climate factors. We note that these factors are of course intrinsically linked, but are frequently measured separately in atmospheric retrievals, where a deck of grey clouds at arbitrary altitude is a common tool used to match observations (e.g., Welbanks & Madhusudhan 2021).", '3.1. Differences I: Clouds': "ROCKE-3D, and therefore our simulations, considers water (liquid and ice) clouds only. The full treatment of clouds in ROCKE-3D is described in Way et al. (2017), Section 4.2. To briefly summarize, the simulation treats the dynamics, thermodynamics, and microphysics of convection for a mass flux of rising moist air (Del Genio et al. 2015). The simulation maintains sub-saturated grid conditions (in contrast to some other GCMs) in cells where updrafts originate. The sub-gridbox treatment allows nonzero cloud fractions to appear without full saturation of a grid cell and are instead determined from local humidity and stability (see del Genio et al. 1996; Schmidt et al. 2006, 2014). We refer the reader to Way et al. (2017) for the full model details. \nClouds are a well-known and significant driver of the shapes of exoplanet spectra (e.g., Fauchez et al. 2019; Komacek et al. 2020; Suissa et al. 2020; Gao et al. 2021). Our simulations are no different: many spectral differences that we show here are clearly driven by clouds. This is evident by inspection, particularly in the coolest cases where a flat (i.e., spectrally grey) continuum opacity characteristic of clouds is present (top row, Figure 6). \nThese continuum differences become less pronounced and eventually disappear in both rotation cases once we raise the instellation as high as 2 S ⊕ . This occurs because the hotter simulations have little to no cloud cover, and in a cloudy atmosphere, the continuum level is defined by the altitude of the optically thick cloud deck. In Figure 7, we show the total cloud cover in each column of grid cells as a function of longitude to highlight the terminator region, which determines the transmission spectrum. For the synchronous rotators, once instellation rises to 1.66 S ⊕ case, cloud cover becomes very low, and in the hotter cases still, there is \nFigure 6. The spectrum of each simulation in our explored parameter space. Each has the same composition (N 2 -dominated, trace CO 2 and H 2 O; Table 1). Each box shows the synchronous and 1-day rotator cases for different instellation values, which are printed in the legend. Up to 1.66 S ⊕ , the transit spectra are distinguishable. In warmer cases, the differences disappear. \n<!-- image --> \nnegligible cloud cover. In all cases, the cloud cover is almost exclusively stratiform. Convective cloud cover is negligible except in the coolest case (0.8 S ⊕ ), where it makes up a few % of the total cloud coverage. \nBroadly, this result is in agreement with Kopparapu et al. (2016), who find as they increase the instellation of their synchronous rotators, this drives the water vapor greenhouse, increasing absorption at high altitudes. This in turn decreases the lapse rate and suppresses convection, muting cloudiness. The 1-day rotators still re- \ntain appreciable cloud cover in the 1.66 S ⊕ case, being cooler than the synchronous rotators at the same irradiation (Figure 2), but follow the same trend of moving to lower and eventually no cloud cover as instellation increases. \nTo understand why this is the case, we show the specific humidity profiles: by the time the instellation reaches 2 S ⊕ , both rotation regimes have converged on a perpetually sub-saturated atmosphere. The transition occurs at different irradiation thresholds for the syn- \nronous and 1-day cases because of their different climates - the synchronously rotating cases are hotter than the 1-day rotators in the intermediate instellation cases (Figure 2). Note in the 1.66 S ⊕ case (Figure 6, middle left panel), the synchronous rotator's spectrum has already begun to look identical to the hottest cases, but not so for the 1-day rotator. \nWhy do the highest-instellation cases have roughly identical spectra (modulo a small offset), regardless of rotation rate? First, we caution the reader that in these cases (certainly by 2.5 S ⊕ ), we have begun to enter a regime of questionable validity of the HITRAN line lists (Figure 2; Goldblatt et al. 2013), so these results should be treated with caution. However, the behavior can still be interpreted qualitatively. The only spectrally active characteristics of our model atmospheres are CO 2 , H 2 O, N 2 -N 2 collision-induced absorption (CIA), and clouds. Without clouds and their potential associated climate feedback, the hottest cases are composed of perpetually sub-saturated water, N 2 , and trace CO 2 , whose medium resolution spectral signatures are not changing significantly with temperature. The amplitude of spectral features does increase slightly with instellation, as expected given the temperature dependence of the atmospheric scale height. \nRemaining for discussion is the 1.25 S ⊕ case, in which the synchronous rotator displays apparently anomalous behavior. The transit depth above the defined surface (i.e., the continuum height) is consistently ∼ 25 ppm, whereas the other simulations range from 2-10 ppm from coolest to warmest. The reason for this appears to be significantly higher-altitude clouds than in any other simulations, which we show in Figure 8. The 1.25 S ⊕ case synchronous rotator has terminator clouds more than an order of magnitude higher in the atmosphere than either its cooler sychronous rotator counterparts or its 1-day rotator analog, resulting in this 'lifted' continuum. Although the 1.66 S ⊕ case has even higher altitude clouds, their coverage is extremely low (Figure 7), so they do not contribute significantly to the opacity. This can also be seen comparing the corresponding panels in Figures 6 and 9, where eliminating the clouds minimally changes the difference between rotators.", '3.2. Differences II: P-T Profiles and Humidity': "Clouds are a significant driver of the differences in our spectra, but not the only one. In order to test this, we repeated our procedure for generating the spectra with globes and PSG, but with clouds disabled. The results can be seen in Figure 9, where differences persist in some of our spectra despite the absence of clouds when calculating the transmission spectrum. We emphasize that in \nFigure 7. The mean percentage cloud cover of each grid cell plotted by longitude, marginalizing over latitude, for the synchronous rotators (top) and 1-day rotators (bottom). Cloud cover drops precipitously above ∼ 1.25 S ⊕ for the synchronous rotators, but persists to higher instellation for the 1-day rotators. \n<!-- image --> \nthis case, we are only 'turning off' clouds in the spectrum calculation, but the resulting spectrum is created from a simulation whose climatology was sculpted by clouds. This exercise is only a diagnostic of what transmission spectrum feature differences are directly caused by clouds. Qualitatively, the continuum absorption is more similar between rotators in the cloud-free cases (as expected), but some feature heights remain different. In particular, the 1.25 and 1.66 S ⊕ cases show significant differences in the amplitude of the water features from 13 µ m, even in the spectral absence of clouds. As before, in the warmest cases the spectra are indistinguishable. \nWhat is driving these persistent differences? Given the relative simplicity of our models, the atmospheric pressure-temperature and humidity profiles are the two prime suspects. The atmospheric τ = 1 surface will change as the water abundance varies in the atmosphere, represented by these two factors. The 1.25 S ⊕ case, along with the 1.66 S ⊕ case, have the most qualitatively significant differences between the two rotators' cloud-free spectra. Revisiting the pressure-temperature profiles in Figure 2, we can see that the temperature \nFigure 8. The median cloudtop height in each grid cell, marginalized by longitude, for the synchronous (top) and 1day (bottom) rotators. The line widths correspond to the cloudtop height shown in Figure 7. In the 1.25 S ⊕ synchronous rotator case, the cloud is almost an order of magnitude higher at the terminator than for the 1-day case. Note that cases hotter than 1.66 S ⊕ are not shown because they contain no longitude columns of entirely non-zero cloud cover. \n<!-- image --> \ndifference between synchronous and 1-day rotators is largest in these two cases. This will of course drive some differences in the scale height. However, more significantly, these temperature differences are driving the synchronous rotators in these two cases to reach a just sub-saturated state, while the atmospheres of the 1-day rotators are drier (Figure 10). This is likely the dominant driver of spectral differences in the cloud-free cases. \nDriven by the pressure-temperature profile, the water content of the atmosphere also differs significantly between these cases (Figure 10). In the 1.25 S ⊕ case, the hotter synchronous rotator has already moved much of its surface water into atmospheric humidity, where the 1-day rotator remains fairly dry across all latitudes. Tracing the specific humidity across instellation for each rotation case, there is a clear transition from relatively dry to sub-saturated atmospheres. However, the break point occurs earlier (i.e, at lower instellation) for the synchronous rotators than the 1-day. The higher global \ntemperatures (at least in intermediate cases) of the synchronous rotators are the likely driver of this.", '4.1. Comparison To Previous Work': "Before discussing the observational characteristics of our simulations, we continue the thread of Sections 3.1 and 3.2 and compare the climatology of our simulations to previous works which investigated similar scenarios. Much work on super-Earths has focused on water worlds (e.g., Yang et al. 2013; Sergeev et al. 2022). We selected the Hab 1 case (Sergeev et al. 2022) from the TRAPPIST-1 Habitable Atmosphere Intercomparison (THAI) series of papers (Turbet et al. 2022; Sergeev et al. 2022; Fauchez et al. 2022) as a 'standard' water world suitable for comparison to our simulations. Both have an N 2 -dominated atmosphere with 400ppm CO 2 , although our simulations are much drier: the ROCKE-3D implementation of Hab 1 has a 90-meter global ocean, compared to our simulations whose only water is in the moist soil. Hab 1 is also slightly less irradiated than our coolest simulation (900 W m -2 vs 1089 W m -2 in our 0.8 S ⊕ case). \nOur simulations' behavior differs from that of the THAI Hab 1 case, which has substantial cloud cover at the substellar point frequently seen in synchronous rotators in 3-D (e.g., Yang et al. 2013, 2014; Kopparapu et al. 2016). Instead, our simulations qualitatively better resemble the water-limited cases explored in Lobo et al. (2023), with less dayside cloud cover and correspondingly lower dayside albedo. We find similarly large ( ≳ 100 K) day-to-night temperature variations to the water-limited models in Lobo et al. (2023), which they attribute to less energy transport in a drier atmosphere compared to their aquaplanet cases. Our findings of more nightside cloud cover (in the less-irradiated cases; Figure 7) also match the findings in Turbet et al. (2023), who investigated what was required for a 'hot start' case to condense liquid water oceans, analogous to the likely formation conditions of rocky planets. The mechanism that drives the nightside cloud is different between their study and ours. In their case, the highaltitude water vapor is an efficient absorber of incoming stellar radiation, and the resulting heating of the upper atmosphere breaks dayside convection, preventing the formation of convective clouds. Our cooler simulations have no such high-altitude water content (Figure 10), and instead convection is likely inhibited by lack of available atmospheric water. \nWe also investigate potential variability in our simulations, significant in light of the findings of Fauchez et al. (2022) that inter-transit variability was larger than \nFigure 9. The spectra of each instantiation of our simulation with the optical and spectral effects of clouds removed. As in Figure 6, each box shows the synchronous and 1-day rotator cases for different instellation values, which are printed in the legend. Despite the absence of clouds, visible (though smaller) differences persist in the spectra of less-irradiated models. \n<!-- image --> \nspectral differences between, for example, O 2 and H 2 O (Wunderlich et al. 2020). Outputting the simulation instances used to generate Figures 6 at one-month intervals for two years, we measure the variation in transit depth of the 1.8 µ m bandpass, since it and other continuum regions are the most highly varying part of transmission spectra in the near-infrared (Fauchez et al. 2022). We find that the standard deviation of the spectra in this region is at most around 5 ppm (Figure 11). This is roughly consistent with the amplitudes found \nin previous variability studies (e.g., May et al. 2021; Song & Yang 2021; Cohen et al. 2022). The variability peaks for the synchronously rotating simulations in the 1.25 S ⊕ and 1.66 S ⊕ cases, which we observe to have tenuous, high-altitude clouds that are prone to fluctuation. Meanwhile, for the 1-day rotators, the variability increases with irradiation until the hottest case. This may be because the 1-day rotators maintain nonzero cloud cover until the hottest instance (Figure 7), although we reiterate the cautions of Section 2.2 around the results of \nFigure 10. The mean pressure-humidity profile for the 0.8 through 2 S ⊕ cases. The transition to nearly sub-saturated atmospheres occurs at lower irradiation for the synchronous rotators than the 1-day rotators. \n<!-- image --> \nthese hottest instances. In any case, this peak variability amplitude of ∼ 5 and < 1 ppm from the synchronous and 1-day rotators, respectively, in the 1.25 S ⊕ case is not sufficient to wash out the maximum ∼ 20 ppm difference between their spectra, although the difference will be obscured. \nA final point is that the variability itself could be useful as an indirect diagnostic for rotation rate. The scale of this effect is not relevant to JWST observations (for which single transit observations have maximum characteristic precision of around 20ppm), but may be of interest to next generation facilities like the Habitable Worlds Observatory, especially in intermediateinstellation cases where the variability signal differs considerably between rotation regimes. We do reiterate, \nFigure 11. The standard deviation of the transit depth in the 1.8 µ m bandpass for monthly cadence simulations over two years. Probably because of their differing cloud behavior, the synchronous and 1-day rotators have differing patterns of variability. \n<!-- image --> \nhowever, that the exact variability conditions will depend on model choice and setup (Sergeev et al. 2022; Fauchez et al. 2022), so additional testing would be required to confirm this result. This is particularly important given the water-limited nature of our simulations, which as discussed yields important differences when compared to 'aquaplanet' simulations. Overall, this work suggests that the transmission spectra of water-limited super-Earths will be fairly robust to intertransit atmospheric variability, although the variability will make the detection of small effects like those we examine here more difficult.", '4.2. The Differences We Present Are Difficult to Individually Observe': "We have shown above that there are visible differences in the idealized transmission spectrum of a small planet depending on whether it rotates synchronously or not. However, with real observations of an atmosphere identical to the one we model, it would be difficult or impossible to extract these differences. Observations of small planets are already challenging. In this case, we find at best a model difference of roughly 20 ppm, a precision that could only be achieved with large numbers of JWST transits (e.g., Pidhorodetska et al. 2021). Furthermore, the difference would require observations beyond 5 µ mto break the degeneracy with a larger planetary radius (Figure 6), but even JWST's Mid-Infrared Instrument is probably unsuitable for transmission spectroscopy (though not emission, which we do not simulate here; e.g., Greene et al. 2023; Zieba et al. 2023) observations of small planets (Batalha et al. 2018). In any case, this difficulty was to be expected given our model setup: we are considering mostly N 2 /CO 2 atmospheres (i.e., high mean molecular weight). This is a fairly 'pessimistic' (from the observational perspective) case even for a super-Earth atmosphere (e.g., Pidhorodetska et al. 2021). \nEven in a more 'optimistic' (i.e., lower mean molecular weight) atmosphere, this work is not sufficient to claim that planetary rotation rate can be identified from spectra alone. The most significant differences that we present here are driven by clouds, but cloud formation physics is not fully understood (e.g., Gao et al. 2021) and is parameterized within our model. Different and potentially more detailed parameterizations may yield different results (e.g., Sergeev et al. 2020; Lef'evre et al. 2022). Thus, a 'rotation rate signature' will inevitably be degenerate with other physical processes that drive cloud formation and climate such as initial water inventory. As mentioned above, our simulations do not explore the formation of hazes, which are likely a signif- \nicant opacity source in planetary atmospheres (e.g., Gao et al. 2021). That said, this work provides a first pass look at how future observatories, in combination with bespoke forward models, may be able to observationally investigate planetary rotation rate. \nWe must also note that ROCKE-3D is not the only GCM, and there is known to be dispersion in the results that different GCMs produce, depending on the details of their methodology. In particular, the THAI collaboration performed intercomparisons between ROCKE-3D, the Exoplanet Community Atmospheric Model (ExoCAM), The Laboratoire de M'et'eorologie Dynamique-Generic model (LMD-G), and the Met Office Unified Model (UM). Sergeev et al. (2022) treat the moist cases among the THAI models, of which our model is most similar to their Hab 1 state. As described above, Hab 1 has significantly more water than our model (ROCKE-3D's implementation of Hab 1 features a 90 m global ocean) and less irradiated than most of the models we consider, but is otherwise similar (1 bar N 2 atmosphere with ∼ 400ppm CO 2 ). They find that the GCMs have broadly similar cloud behavior (i.e., large cloud decks at the substellar point, following, e.g., Yang et al. 2013, 2014; Kopparapu et al. 2016), but with important differences between them. ROCKE-3D produces atmospheres with the largest fractional cloud cover, while ExoCAM produces clouds with both the highest altitude and highest mixing ratio. Clouds are the main (though not only; Figure 9) driver of spectral differences between our models, so a GCM like LMD-G (the least cloudy of the four studied by Sergeev et al. 2022) may find smaller differences between rotation states. Considering the transmission spectra resulting from the various GCMs' Hab 1 models, Fauchez et al. (2022) find up to a 50% spread in the number of observations required to confidently detect CO 2 , which functions as a proxy for the ability to detect the presence of an atmosphere given the relative magnitude of the 4.3 µ m CO 2 feature. Given the nonlinear cloud behavior we find in our simulations, the 'GCM uncertainty factor' is relevant to our predictions as well, and similar studies with other GCMs would be needed to validate our predictions.", '4.3. Lessons for Wider Parameter Spaces': "Because of the evident non-linearity of 3D simulations (see e.g. the formation of a high-altitude cloud deck in the 1.25 S ⊕ synchronous rotator case), we cannot straightforwardly 'scale up' our simulations to make predictions for planets in other parts of parameter space, like larger sub-Neptunes, H 2 O-rich 'water worlds,' or hotter terrestrials that will form the majority of those \nobserved with JWST. However, some of the physical processes we identify here are likely relevant to those planets. \nFirst, we describe some non-linearity in cloud formation and behavior across our simulations. Although the average percentage cloud cover follows a fairly straightforward pattern from significant clouds at the coolest temperatures to little/no cloud cover in hotter cases (Figure 7), this does not fully capture the cloud behavior. The 1.25 S ⊕ synchronous rotator case exemplifies this best, with high altitude clouds (Figure 8) that affect the behavior of the spectrum (Figure 6, upper right panel) and climate (Figure 2). This result complicates ongoing efforts to understand trends in cloud/haze patterns in exoplanet atmospheres (e.g., Gao et al. 2021; Yu et al. 2021; Brande et al. 2024), and suggests any such relationships will likely have an inherent, effectively stochastic level of scatter due to nonlinear effects like these. Despite the model limitations we describe in Section 2.2, there is no reason to think that effects of this sort are unique to our parameter space. This further suggests that 1D forward models may be insufficient for predicting an individual planet's cloud behavior. \nA related implication for our population-level understanding of exoplanet cloud behavior is what our results suggest of the behavior of condensibles other than water. While more work would be needed to determine how commonly the transition from condensible to too-rarefied occurs for a given species, the result that condensation produces nonlinear and inhomogeneous effects should be broadly applicable. Various species are thought to play the role of condensibles on hot exoplanets that they do not in our solar system, such as KCl and MgSiO 3 , among others (e.g., Gao & Benneke 2018; Herbort et al. 2022). Although the physics and dynamics of atmospheres where these species are the dominant condensibles will be different from the simulations we present here, similar effects could play out, with limited transfer from a lower-atmosphere reservoir broadly analogous to a limited starting water inventory. Our ultimate point is that this further motivates 3D modelling for individual planets to better understand their cloud and climate behavior.", '4.4. Rotation Rate Can Inform Sample Selection': 'In addition to the implications for next-generation observations of individual small planets, where rotation rate could be a parameter inferred by spectroscopic observations, our work here has implications for current observational campaigns. In particular, efforts to infer the presence of a small planet atmosphere from JWST transmission spectroscopy requires finding signals very \nclose to the limit of what it is possible to detect. In constructing a sample of moderately irradiated planets for observation, and subsequently attempting populationlevel inferences, estimated rotation rate will play a role. Although there is no mechanism currently to determine rotation rate directly, system characteristics that impact rotation rate, especially planetary multiplicity (Vinson et al. 2019; Chen et al. 2023; Shakespeare & Steffen 2023), will need to be considered.', '5. CONCLUSIONS': 'The key takeaways of our work are as follows: \n- 1. We created a suite of simulations of a fairly dry super-Earth exoplanet with a N 2 /CO 2 atmosphere and demonstrated that, at Earth-like levels irradiation (up to ∼ 1 . 66S ⊕ ), changing the planetary rotation rate has a corresponding effect on the transmission spectrum. These differences are not observed in hotter planets.\n- 2. We show that the differences in transmission spectra are driven primarily by the presence of nonconvective cloud decks on the terminator. The atmospheric water column profiles also drive some spectral differences at lower temperatures, before the atmosphere becomes perpetually subsaturated.\n- 3. Even though our particular simulations, which have high mean-molecular-weight nitrogen atmospheres, do not show differences at levels that could be detected with reasonable expenditures of JWST time, planets that are larger, have less dense atmospheres, or have even smaller host stars \ncould be laboratories for this phenomenon. However, we reiterate that the differences between transmission spectra of synchronous vs. 1-day rotators should be investigated with multiple GCMs to be considered robust, given previously shown dispersion in outcomes from different models. As well, publicly available GCMs capable of handling larger, hotter planets will be important for confirming the applicability of our results outside of the specific parameter space that we explore.', '6. ACKNOWLEDGEMENTS': "The authors wish to thank Geronimo Villanueva and Eric Wolf for discussions and technical feedback that greatly aided the completion and quality of this manuscript. N.S. wishes to thank Natalie Batalha for mentorship that also greatly enhanced the quality of this manuscript. Support for N.S. was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127. N.S. gratefully acknowledges support from the Heising-Simons Foundation through grant 2021-3197. This material is based upon work supported by NASA'S Interdisciplinary Consortia for Astrobiology Research (NNH19ZDA001N-ICAR) under award number 19-ICAR19 2-0041. This work was supported by NASA's Nexus for Exoplanet System Science (NExSS) and the NASA Interdisciplinary Consortia for Astrobiology Research (ICAR). Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Center for Climate Simulation (NCCS) at Goddard Space Flight Center. C.E.H. acknowledges support through the ROCKE-3D ICAR grant.", 'REFERENCES': '```\nAngelo, I., Rowe, J. F., Howell, S. B., et al. 2017, The Astronomical Journal, 153, 162 Batalha, N. E., Lewis, N. K., Line, M. R., Valenti, J., & Stevenson, K. 2018, ApJL, 856, L34, doi: 10.3847/2041-8213/aab896 Benneke, B., & Seager, S. 2012, The Astrophysical Journal, 753, 100, doi: 10.1088/0004-637X/753/2/100 Bony, S., Stevens, B., Frierson, D. M. W., et al. 2015, Nature Geoscience, 8, 261, doi: 10.1038/ngeo2398 Brande, J., Crossfield, I. J. 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2024MNRAS.534..523C	We identify a lowmetallicity inlineformulatexmath idTM0001 notationLaTeX12log rm Orm H7.59texmathinlineformula Ly inlineformulatexmath idTM0002 notationLaTeXalphatexmathinlineformulaemitting galaxy at inlineformulatexmath idTM0003 notationLaTeXz5.943texmathinlineformula with evidence of a strong Balmer jump arising from nebular continuum. While Balmer jumps are sometimes observed in lowredshift starforming galaxies this galaxy also exhibits a steep turnover in the UV continuum. Such turnovers are typically attributed to absorption by a damped Ly inlineformulatexmath idTM0004 notationLaTeXalphatexmathinlineformula system DLA however the shape of the turnover and the high observed Ly inlineformulatexmath idTM0005 notationLaTeXalphatexmathinlineformula escape fraction inlineformulatexmath idTM0006 notationLaTeXfrm escLyalpha sim 27 rm per centtexmathinlineformula is also consistent with strong nebular twophoton continuum emission. Modelling the UV turnover with a DLA requires extreme column densities inlineformulatexmath idTM0007 notationLaTeXNrm HIgt 1023texmathinlineformula cminlineformulatexmath idTM0008 notationLaTeX2texmathinlineformula and simultaneously explaining the high inlineformulatexmath idTM0009 notationLaTeXfrm escLyalpha texmathinlineformula requires a finetuned geometry. In contrast modelling the spectrum as primarily nebular provides a good fit to both the continuum and emission lines motivating scenarios in which a we are observing only nebular emission or b the ionizing source is powering extreme nebular emission that outshines the stellar emission. The nebularonly scenario could arise if the ionizing source has turned off more recently than the recombination timescale inlineformulatexmath idTM0010 notationLaTeXsimtexmathinlineformula1000 yr hence we may be catching the object at a very specific time. Alternatively hot stars with inlineformulatexmath idTM0011 notationLaTeXTrm effgtrsim 105texmathinlineformula K e.g. WolfRayet or lowmetallicity massive stars produce enough ionizing photons such that the twophoton emission becomes visible. While several stellar SEDs from the literature fit the observed spectrum well the hotstar scenario requires that the number of inlineformulatexmath idTM0012 notationLaTeXgtrsim 50rm Modottexmathinlineformula stars relative to inlineformulatexmath idTM0013 notationLaTeXsim 550rm Modottexmathinlineformula stars is significantly higher than predicted by typical stellar initial mass functions IMFs. The identification of more galaxies with similar spectra may provide evidence for a topheavy IMF at high redshift.	2024-10-01T00:00:00Z	['10.1093/mnras/stae1547', 'arXiv:2311.02051', '2024MNRAS.534..523C', '2023arXiv231102051C', '2024MNRAS.tmp.1525C', '10.48550/arXiv.2311.02051']	['Astrophysics - Astrophysics of Galaxies']	Nebular dominated galaxies insights into the stellar initial mass function at high redshift	2,024	190	0.67	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	74	https://arxiv.org/pdf/2311.02051.pdf	{'No Header': ', 1-22 (2024)', 'Nebular dominated galaxies: insights into the stellar initial mass function at high redshift': "Alex J. Cameron, 1 ★ Harley Katz, 1 , 2 Callum Witten, 3 , 4 Aayush Saxena, 1 Nicolas Laporte, 5 and Andrew J. Bunker, 1 \n- 1 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK\n- 2 Department of Astronomy & Astrophysics, University of Chicago, 5640 S Ellis Avenue, Chicago, IL 60637, USA\n- 3 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\n- 4 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\n- 5 Aix-Marseille Université, CNRS, CNES, LAM (Laboratoire d'Astrophysique de Marseille), UMR 7326, 13388 Marseille, France \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': "We identify a low-metallicity (12 + log ( O / H ) = 7 . 59) Ly 𝛼 -emitting galaxy at 𝑧 = 5 . 943 with evidence of a strong Balmer jump, arising from nebular continuum. While Balmer jumps are sometimes observed in low-redshift star-forming galaxies, this galaxy also exhibits a steep turnover in the UV continuum. Such turnovers are typically attributed to absorption by a damped Ly 𝛼 system (DLA); however, the shape of the turnover and the high observed Ly 𝛼 escape fraction ( 𝑓 esc , Ly 𝛼 ∼ 27%) is also consistent with strong nebular two-photon continuum emission. Modelling the UV turnover with a DLA requires extreme column densities ( 𝑁 HI > 10 23 cm -2 ), and simultaneously explaining the high 𝑓 esc , Ly 𝛼 requires a fine-tuned geometry. In contrast, modelling the spectrum as primarily nebular provides a good fit to both the continuum and emission lines, motivating scenarios in which (a) we are observing only nebular emission or (b) the ionizing source is powering extreme nebular emission that outshines the stellar emission. The nebular-only scenario could arise if the ionising source has 'turned off' more recently than the recombination timescale ( ∼ 1,000 yr), hence we may be catching the object at a very specific time. Alternatively, hot stars with 𝑇 eff ≳ 10 5 K (e.g. Wolf-Rayet or low-metallicity massive stars) produce enough ionizing photons such that the two-photon emission becomes visible. While several stellar SEDs from the literature fit the observed spectrum well, the hot-star scenario requires that the number of ≳ 50 M ⊙ stars relative to ∼ 5 -50 M ⊙ stars is significantly higher than predicted by typical stellar initial mass functions (IMFs). The identification of more galaxies with similar spectra may provide evidence for a top-heavy IMF at high redshift. \nKey words: galaxies: ISM - galaxies: starburst - galaxies: star formation", '1 INTRODUCTION': "The unprecedented sensitivity of JWST in the near-infrared has revolutionised our ability to study the rest-frame ultraviolet to optical spectra of high-redshift galaxies. JWST spectroscopy has already unveiled large samples of emission line galaxies at 𝑧 ≳ 5, providing new insights in to the conditions of the interstellar media (ISM) of these galaxies (Matthee et al. 2023; Cameron et al. 2023b; Mascia et al. 2023; Sanders et al. 2023; Boyett et al. 2024; Roberts-Borsani et al. 2024). Furthermore, the sensitivity of JWST /NIRSpec has even enabled high fidelity spectroscopic continuum measurements, which have provided insights into high-redshift neutral gas (Heintz et al. 2023; Umeda et al. 2023), assembly of massive galaxies in the early Universe (Carnall et al. 2023; Glazebrook et al. 2023), and have also provided the many of the highest spectroscopic redshift confirmations to date (Curtis-Lake et al. 2023; Arrabal Haro et al. 2023). Spectral \nenergy distribution (SED) fitting of many photometric data sets has indicated a need for strong nebular emission, including predictions for strong Balmer jumps (Endsley et al. 2023; Topping et al. 2023). The strong nebular contribution implied by these fits motivates more detailed studies examining the contribution of the nebular continuum to the integrated spectra of high-redshift galaxies. \nThe nebular continuum is comprised of three main components: (1) free-bound emission, (2) free-free emission, and (3) two-photon emission. Free-bound emission is the component that is most commonly observed to make a significant contribution to the integrated optical spectra of galaxies, presenting in the form of the 'Balmer jump', a discontinuity in the spectrum at 𝜆 rest = 3645 Å, corresponding to the Balmer limit. This feature arises due to electron recombinations with ionized hydrogen to the first excited state. Balmer jumps only appear in the spectra of galaxies that contain young stellar populations with high ionizing photon production efficiencies ( 𝜉 i 𝑜𝑛 ). They have been detected in some highly star-forming galaxies at lowredshift (Peimbert & Costero 1969; Guseva et al. 2006, 2007). Many \nFigure 1. Top: 2D Prism/CLEAR spectrum of GS-NDG-9422. Middle: 1D Prism/CLEAR spectrum of GS-NDG-9422 shown in 𝑓 𝜈 (upper middle) and 𝑓 𝜆 (lower middle). Blue squares show 0.15' radius aperture photometry from JWST /NIRCam medium- and broad-band imaging, while orange squares show the predicted photometry obtained by convolving the Prism spectrum with the NIRCam filter transmission profiles. Bottom left: Three-colour image ( 𝐹 090 𝑊 , 𝐹 200 𝑊 , 𝐹 444 𝑊 ) of GS-NDG-9422 showing the positioning of the three NIRSpec micro-shutters across the three nod positions. Bottom centre: Zoom-in of the region surrounding [O /i.pc/i.pc/i.pc] 𝜆 5007 in the G395M spectrum. Bottom right: Zoom-in of the region surrounding H 𝛼 in the G395H spectrum. \n<!-- image --> \nnumerical simulations predict Balmer jumps to be common at high redshift (e.g. Katz et al. 2023a; Wilkins et al. 2023), in line with the early SED fitting results outlined above. In contrast, predictions based on typical stellar population models indicate that the other nebular continuum components, free-free and two-photon emission, are generally expected to be subdominant compared to the stellar and free-bound contributions in the integrated spectra of galaxies. \nThe predicted subdominance of two-photon emission arises because the 𝜉 ion values of typical low-metallicity stellar populations are not high enough to drive nebular continuum that overcomes the steep UV stellar continuum slopes (Leitherer et al. 1999). In contrast, very hot stars, with blackbody temperatures of ∼ 100 , 000 K, produce enough ionizing photons to power nebular continuum emission that outshines their stellar UV emission, and are predicted to exhibit a strong two-photon continuum bump, peaking at 𝜆 rest ≈ 1430 Å (Schaerer 2002; Raiter et al. 2010; Zackrisson et al. 2011; Trussler et al. 2023). Observations of the Lynx arc, a strongly-lensed extreme emission system at 𝑧 = 3 . 357, have suggested that the UV continuum of this system has a strong contribution from two-photon continuum, leading to the suggestion of the presence of hot stars with 𝑇 eff ∼ 80 , 000 K (Fosbury et al. 2003). But this has so far remained an isolated candidate for this type of system, with other examples of two-photon continuum being the domain of unusual nebular-only systems (e.g. Lintott et al. 2009). Nonetheless, the identification of these objects, should they exist, offers powerful insight into the properties of systems with extreme ionising spectra. \nIn this paper, we identify one object, JADES-GS+53.1217527.79763 (GS-NDG-9422 hereafter), at 𝑧 = 5 . 943, that exhibits high equivalent-width emission lines as well as a strong spectral discontinuity near 𝜆 rest = 3645 Å that we interpret as a Balmer jump. The UV continuum exhibits a strong turnover which could be indicative of either a strong damped Lyman𝛼 absorption (DLA) system (e.g. Heintz et al. 2023), or the two-photon continuum introduced above. We present a detailed analysis of this system, outlining a number of scenarios that can potentially explain the physical origin of this intriguing spectrum. \nThe paper is structured as follows: Section 2 outlines the details of data used in this work. In Section 3 we present emission-linebased analysis of the nebular conditions. We perform fitting to the continuum shape in Section 4, comparing the results of DLA and nebular models. Section 5 then considers the nebular case in more detail, exploring the necessary properties of the ionising source. Section 6 outlines the broader implications of the scenarios presented. We then summarise our findings in Section 7. Throughout this paper we adopt the cosmology of Planck Collaboration et al. (2016) with 𝐻 0 = 67 . 31 km s -1 Mpc -1 and Ω m = 0 . 315, which gives a luminosity distance to GS-NDG-9422 of 𝐷 𝑙 = 58 . 5 Gpc.", '2 DATA': "JWST /NIRSpec spectroscopy of GS-NDG-9422 was taken as part of the JADES survey (PID: 1210, PI: Luetzendorf) in five spectral modes, receiving 28 hours integration in Prism/CLEAR and 7 hours integration in each of G140M/F070LP, G235M/F170LP, G395M/F290LP and G395H/F290LP. We use the reduced spectra released as part of the JADES Public Data Release (Eisenstein et al. 2023; Bunker et al. 2023). GS-NDG-9422 falls within the JWST /NIRCam footprint of JADES (PID: 1880, PI: Eisenstein) as well as the JWST Extragalactic Medium-band Survey (JEMS; PID: 1963; PI: Williams; Williams et al. 2023), combining to provide imaging in 14 wide- and medium-band filters. In Figure 1 we show \nphotometry from the publicly released photometric catalogs (Rieke et al. 2023) compared to the observed Prism/CLEAR spectrum. Blue squares in the second-top panel show aperture photometry measured within a 0.15' radius. Orange squares show the predicted values by convolving the observed NIRSpec spectrum with the NIRCam filter transmission profiles. We find there is good agreement across the full spectral range, suggesting the flux calibration of the Prism/CLEAR spectrum is robust. The exception to this is the two filters covering H 𝛼 emission, 𝐹 444 𝑊 and 𝐹 460 𝑀 . In each case, the flux from imaging is 14 % and 17 % higher, respectively, suggesting the flux of H 𝛼 may be marginally underestimated in the Prism/CLEAR flux calibration. We note that the 𝐹 460 𝑀 flux changes by < 2 % when measured within apertures with radii of 0.10' or 0.25', suggesting that this offset is not driven by spatial variations in H 𝛼 .", '2.1 Emission line fitting': 'Where possible, emission lines were fit with a single component Gaussian profile with the continuum modelled as a 1D spline. In cases where lines are sufficiently blended, we fit the whole complex with one component. In some cases, partially blended lines are fit simultaneously with neighbouring lines and fluxes reported separately. These are marked in Table 1. We fit all identifiable lines and report upper limits for notable undetected lines. \nLine fluxes from higher resolution grating spectra of GS-NDG9422 were measured independently. Emission line widths are only spectrally resolved in the high-resolution G395H grating. These show no evidence of a broad component and are well fit with a single component with velocity dispersions < 200 km s -1 (Table 2). \nFluxes derived from different observations are mildly discrepant in some cases. Notably, H 𝛽 , [O/i.pc/i.pc/i.pc] 𝜆 4959, [O/i.pc/i.pc/i.pc] 𝜆 5007 and H 𝛼 lines exhibit higher fluxes in the grating modes. This behaviour is reported in Bunker et al. (2023) who suggest that the Prism flux calibration is more reliable. This conclusion is supported by the good agreement observed in Figure 1 between the Prism spectrum and NIRCam photometry, with the possible exception of H 𝛼 . We note that in low-resolution data, the continuum level for some emission lines can be difficult to determine, especially for Ly 𝛼 , He /i.pc/i.pc + O /i.pc/i.pc/i.pc] and [O /i.pc/i.pc] 𝜆𝜆 3727, which introduces uncertainty into the emission line flux. The clear detection of the continuum across almost the entire Prism coverage, allows us to derive equivalent widths directly from this spectrum (Table 2). Given the noted discrepancy on the H 𝛼 flux between the imaging and spectroscopy, we also derive EW 0 (H 𝛼 ) directly from the imaging. Comparing the measured flux in 𝐹 460 𝑀 (196 ± 5 nJy), shown in Figure 1 to be clearly elevated due to contamination from H 𝛼 , with 𝐹 430 𝑀 (25 ± 4 nJy), which captures clean continuum between He /i.pc 𝜆 5875 and H 𝛼 , we find the imaging implies EW 0 (H 𝛼 ) = 2195 ± 400 Å. Within the uncertainty, the measured difference between EW 0 (H 𝛼 ) Prism and EW 0 (H 𝛼 ) Imaging (Table 2) is consistent with the discrepancy noted above between the observed 𝐹 460 𝑀 flux, and the predicted 𝐹 460 𝑀 flux obtained by convolving the Prism spectrum with the NIRCam filter profile. Throughout our analysis, we adopt the Prism fluxes where possible. However, we ensure that conclusions presented in this work are also consistent with measured grating flux ratios.', '3 DERIVATION OF PHYSICAL CONDITIONS': 'We now explore the physical conditions of the gas in GS-NDG9422 and the basic properties of the ionising source, based on the measured emission line fluxes. Throughout this section, we make \nTable 1. Emission line flux measurements for GS-NDG-9422 across each observed spectrum. Fluxes are reported in units of × 10 -19 erg s -1 cm -2 . \nTable 2. Rest-frame equivalent widths of prominent lines measured from the Prism/CLEAR spectrum, and line widths measured from 𝑅 ∼ 2700 G395H/F290LP grating observations. We also reported the equivalent width of H 𝛼 measured from 𝐹 460 𝑀 and 𝐹 430 𝑀 photometry. \nuse of /p.pc/y.pc/n.pc/e.pc/b.pc (Luridiana et al. 2015) using atomic data from /c.pc/h.pc/i.pc/a.pc/n.pc/t.pc/i.pc (version 10.0.2; Dere et al. 1997; Del Zanna et al. 2021).', '3.1 Diagnostic diagrams': "To explore the properties of the ionising source, in Figure 2, we look at nebular line ratio diagnostic diagrams, comparing measured line ratios from GS-NDG-9422 with photoionisation model predictions for star formation and active galactic nuclei. For star forming models, we adopt the predictions of Gutkin et al. (2016). These use input stellar SEDs based on plane-parallel non-local thermal equilibrium models calculated with Tlusty (e.g. Lanz & Hubeny 2007). Stellar abundance patterns are assumed to follow scaled-solar, which may not be representative of 𝑧 ∼ 6 stellar populations, while nebular abundance patterns allow for some variation in C/O and N/O ratios. Star forming models with 𝑍 = 0 . 001 (the grid value closest to the gasphase oxygen abundance derived for GS-NDG-9422; see Section 3.5) are shown in dark blue, while all other star-forming models are shown in light blue. Model predictions for active galactic nuclei from Feltre et al. (2016) are shown in red ( 𝑍 = 0 . 001) and pink (all other 𝑍 ). \nThe non-detection of [S /i.pc/i.pc] 𝜆𝜆 6716, 6731 places GS-NDG-9422 firmly below the Kewley et al. (2001) maximum-starburst limit of the classical [S /i.pc/i.pc]-VO87 diagram (Veilleux & Osterbrock 1987; top \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 2. GS-NDG-9422 plotted onto several line ratio diagnostic diagrams. Light blue points show model predictions for star-forming regions from Gutkin et al. (2016), while pink points show AGN model predictions from Feltre et al. (2016) across a large range of metallicities. Navy (star-forming) and red (AGN) points show the subsets of these model grids that have 𝑍 = 0 . 001 ( 𝑍 ≈ 0 . 07 𝑍 ⊙ ; 12 + log ( O / H ) ≈ 7 . 54) which is the closest grid value to that measured for GS-NDG-9422 (see Section 3.5). Top left: The strong upper limit on [S /i.pc/i.pc] 𝜆𝜆 6716, 6731 positions GS-NDG-9422 well below the theoretical maximum starburst limit from Kewley et al. (2001) (black line). Top right: The weak detection of He /i.pc/i.pc 𝜆 4686 is also below the maximum starburst limit from Shirazi & Brinchmann (2012) (black line). Strong He /i.pc/i.pc-emitting star-forming galaxies from Shirazi & Brinchmann (2012) and Senchyna et al. (2017) are also shown as orange and green diamonds respectively. We also show measurements from Hanny's Voorwerp, suggested to be a quasar light echo (Lintott et al. 2009; purple line-connected points), which we discuss in Section 5.1. Points of increasing size indicate larger offset from the quasar ranging from 13 to 31 kpc. Bottom left: GS-NDG-9422 lies beyond the AGN region defined by Mingozzi et al. (2024) (black lines), and is more coincident with the star-forming models than AGN models. He /i.pc/i.pc-emitting star-forming galaxies at 𝑧 ∼ 2 . 5 -4 from Saxena et al. (2020) are shown for comparison as orange squares. Bottom right: GS-NDG-9422 lies at the tip of the parameter space covered by AGN models, while it is well within the bounds of that covered by star-forming models. \n<!-- image --> \nleft panel of Figure 2) and in a region that is difficult to reconcile with emission powered by an AGN. Non-detections of [N /i.pc/i.pc] 𝜆 6583 and [O /i.pc] 𝜆 6300 paint a similar picture in the [N /i.pc/i.pc]-BPT diagram (Baldwin et al. 1981) and the [O /i.pc]-VO87 diagram, although these are not shown. \nThe top right panel of Figure 2 shows that the He /i.pc/i.pc 𝜆 4686 / H 𝛽 ratio of GS-NDG-9422 exceeds that predicted by any of the star-forming models in Gutkin et al. (2016), especially those at 𝑍 = 0 . 001. However, it is well known that, at low metallicity, star-forming galaxies are often observed with He /i.pc/i.pc emission exceeding that which can be powered by standard stellar population models (Shirazi & \nBrinchmann 2012; Kehrig et al. 2015; Senchyna et al. 2017; Schaerer et al. 2019; Saxena et al. 2020). He /i.pc/i.pc-selected star-forming galaxies from Shirazi & Brinchmann (2012) and Senchyna et al. (2017) are shown as orange and green diamonds respectively. We find that while GS-NDG-9422isattheupperendoftheHe/i.pc/i.pc 𝜆 4686 / H 𝛽 distribution from these works, it does not exceed the very highest ratios, and also falls below the He /i.pc/i.pc maximum-starburst demarcation presented in Shirazi & Brinchmann (2012). The origin of strong He /i.pc/i.pc emission at low metallicity remains an unsolved problem with possible solutions including revised stellar wind properties, X-ray binaries, very hot binary stellar evolution products, rotating massive low-metallicity \nFigure 3. Observed Balmer decrements compared to theoretical values at different temperatures and densities. Green squares give the measured ratios from the Prism, while diamonds give ratios measured from gratings, where available. Solid black lines give the theoretical ratio for case B recombination at 𝑇 𝑒 = 10 4 K and 𝑛 𝑒 = 100 cm -3 . Coloured points show theoretical ratios for a range of temperatures (10 4 ≤ 𝑇 𝑒 ≤ 2 . 5 × 10 4 K; colour) and densities ( 𝑛 𝑒 = 10, 100, 1000 cm -3 ; marker size). The measured H 𝛼 /H 𝛽 is somewhat lower than the theoretical value, but we note that H 𝛼 may be underestimated in the prism spectrum (Table 2). \n<!-- image --> \nstars, and/or a top-heavy stellar initial mass function (Kehrig et al. 2015, 2018; Schaerer et al. 2019; Senchyna et al. 2020; Olivier et al. 2022). We return to this question in Section 5. We note that GS-NDG-9422 was included in a selection of narrow-line AGN in Scholtz et al. (2023) on the basis of the He /i.pc/i.pc 𝜆 4686 / H 𝛽 ratio. However, we consider the measured value of this line ratio to be inconclusive. \nIn the bottom left panel of Figure 2, we see that the O/i.pc/i.pc/i.pc] 𝜆𝜆 1660, 1666 / He /i.pc/i.pc 𝜆 1640 ratio measured for GS-NDG-9422 exceeds the AGN locus defined by Mingozzi et al. (2024) and has a value which cannot be reproduced by the Feltre et al. (2016) AGN models. The measuredratio is more in line with that measured in the 𝑧 ∼ 2 -4He/i.pc/i.pcselected star-forming galaxies from Saxena et al. (2020). Meanwhile the (C /i.pc/v.pc + C /i.pc/i.pc/i.pc]) / He /i.pc/i.pc ratio in the bottom right panel is only reproduced by the very tip of the AGN model parameter space, being more readily reproduced by the star-forming models. \nIn summary, AGN photoionisation models struggle to reproduce a number of the observed emission line ratios, especially those involving low-ionisation emission lines. In contrast, GS-NDG-9422 resides within regions of line-ratio space consistent with emission powered by stars. We therefore conclude that the ionising source in GS-NDG-9422 is most likely of stellar origin.", '3.2 Balmer decrements and dust extinction': 'Balmer decrements H 𝛿 /H 𝛽 and H 𝛾 /H 𝛽 from the Prism and H9/H 𝛽 from the grating are consistent with Case B values, indicating that there is no significant dust reddening in GS-NDG-9422 (Figure 3). Measured H 𝛼 /H 𝛽 ratios are lower than theoretically predicted for Case B at 𝑇 = 10 4 K. Note, this is not suggestive of dust reddening, which would act in the opposite direction. At higher temperatures, the theoretical ratio decreases, and our G395H/F290LP measurement is consistent with theoretical ratios with 𝑇 𝑒 ≳ 2 × 10 4 K, possibly indicative of a very hot nebula. As noted in Section 2.1, H 𝛼 may be underestimated in the Prism which could lead to the slightly \nlower observed ratio. H 𝛾 /H 𝛽 measured from the medium-resolution grating is marginally below the theoretical value. In isolation, this could suggest non-zero dust reddening; however, this evidence is outweighed by the other measured ratios. The high-resolution grating returns a much lower H 𝛾 /H 𝛽 = 0 . 3 ± 0 . 03. However, we note that the continuum is undetected in the high-resolution grating, which contributes uncertainty to the measured ratio. \nAdopting a luminosity distance of 𝐷 𝑙 = 58 . 5 Gpc, we derive 𝐿 H 𝛼 = 1 . 86 × 10 42 erg s -1 . Assuming no dust, a metallicity of 𝑍 = 0 . 1 𝑍 ⊙ , and a typical IMF, this would correspond to a star formation rate of ∼ 3 . 2 M ⊙ yr -1 (Eldridge et al. 2017). Under the assumption of no dust, Case B recombination, 𝑛 𝑒 = 100 cm -3 and 𝑇 𝑒 = 1 . 8 × 10 4 K, we derive a Ly 𝛼 escape fraction of 𝑓 esc , Ly 𝛼 = 0 . 29 ± 0 . 01 from the Ly 𝛼 /H 𝛼 ratio, or 𝑓 esc , Ly 𝛼 = 0 . 27 ± 0 . 01, measured from the Ly 𝛼 /H 𝛽 ratio, the latter of which may be more reliable owing to the H 𝛼 flux uncertainty discussed above.', '3.3 Electron temperature': 'The temperature-sensitive [O /i.pc/i.pc/i.pc] 𝜆 4363/ 𝜆 5007 ratio can be measured from each of the Prism, G395M and G395H observations, yielding three consistent, independent temperature measurements ( 𝑇 𝑒 = 1 . 83 ± 0 . 15, 1 . 99 ± 0 . 18, and 1 . 81 ± 0 . 18 × 10 4 K, respectively). The temperature derived from the medium-resolution O /i.pc/i.pc/i.pc] 𝜆 1666 / [O /i.pc/i.pc/i.pc] 𝜆 5007 ratio is somewhat lower ( 𝑇 𝑒 = 1 . 70 + 0 . 05 -0 . 06 × 10 4 K). However, the He /i.pc/i.pc+O /i.pc/i.pc/i.pc] flux measured from the medium-resolution G140M grating is significantly lower than that of the Prism. Instead, the measured temperature from above ( 𝑇 𝑒 = 1 . 83 × 10 4 K) implies 𝜆𝜆 1660,1666/ 𝜆 5007 = 0 . 08, which gives 𝑓 𝜆𝜆 1660 , 1666 = 8 . 2 ± 0 . 1 × 10 -19 erg s -1 cm -2 based on the [O /i.pc/i.pc/i.pc] 𝜆 5007 Prism flux, suggesting that O /i.pc/i.pc/i.pc] contributes ∼ 55 % of the measured Prism He /i.pc/i.pc+O /i.pc/i.pc/i.pc] blend. This is consistent with the O /i.pc/i.pc/i.pc] / He /i.pc/i.pc ratio measured in G140M. The implied He /i.pc/i.pc 𝜆 1640 flux (6 . 9 ± 1 . 1 × 10 -19 erg s -1 cm -2 ) gives a He /i.pc/i.pc 𝜆 1640/ 𝜆 4686 ratio of 6 . 3 ± 1 . 5 which is consistent with the theoretical value of 7.19 assuming Case B at 𝑇 𝑒 = 1 . 8 × 10 4 Kand 𝑛 𝑒 = 100 cm -3 . Note that this further supports the conclusion of a lack of dust in this system since the He /i.pc/i.pc 𝜆 1640 would be subject to extremely high extinction, relative to the 𝜆 4686 line. In systems with a strong nebular continuum component, 𝑇 𝑒 can also be constrained from the magnitude of the Balmer jump (e.g. Guseva et al. 2006; Pérez-Montero 2017). We return to this in Section 4 where we show that 𝑇 𝑒 (H + ) implied by the Balmer jump is consistent with 𝑇 𝑒 (O ++ ) measured from the [O /i.pc/i.pc/i.pc] auroral line ratio.', '3.4 Electron density': 'The density-sensitive doublets of C /i.pc/i.pc/i.pc] and [O /i.pc/i.pc] are not resolved in our observations, while N /i.pc/v.pc] and [S /i.pc/i.pc] are not detected. We report a marginal detection of the [Ar /i.pc/v.pc] 𝜆𝜆 4711, 4740 doublet in the Prism spectrum. These lines are partially blended with each other and with He /i.pc/i.pc 𝜆 4686, while [Ar /i.pc/v.pc] 𝜆 4711 is also completely blended with He /i.pc 𝜆 4713. Our three-component fit to this complex yields [Ar /i.pc/v.pc] 𝜆 4711/ 𝜆 4740 = 1 . 6 ± 0 . 8 after subtracting the predicted He /i.pc 𝜆 4713 contribution (assuming 𝜆 4713/ 𝜆 4471 = 0.15), consistent with the low-density limit ( 𝑛 𝑒 ≲ 10 3 cm -3 ; Kewley et al. 2019). A consistent density constraint arises if the UV continuum turnover in GS-NDG-9422is driven by two-photon nebular continuum, since the feature strongly suppressed by 𝑙 -changing collisions at higher densities. The presence of the two-photon continuum will be discussed in Section 4. \nTable 3. Physical properties and chemical abundances derived for GS-NDG9422.', '3.5 Chemical abundances': 'We derive chemical abundances for GS-NDG-9422 adopting 𝑇 𝑒 = 1 . 83 × 10 4 K and a density of 𝑛 𝑒 = 100 cm -3 following the procedure in Cameron et al. (2023a). Given the apparent agreement between 𝑇 𝑒 ( H + ) and 𝑇 𝑒 ( O ++ ) , we assume a constant temperature for all ionisation zones. We derive 12 + log ( O / H ) = 7 . 59 ± 0 . 01 from the [O /i.pc/i.pc], [O/i.pc/i.pc/i.pc], and H 𝛽 lines, obtaining a consistent result with both Prism and grating line fluxes. We derive the carbon abundance from the C /i.pc/i.pc/i.pc] 𝜆𝜆 1907, 1909 / [O /i.pc/i.pc/i.pc] 𝜆 5007 ratio measured from the Prism, assuming no dust, finding log ( C / O ) = -0 . 73 ± 0 . 03 after applying the ionisation correction factor (ICF) presented in Amayo et al. (2021). However, we note the significant emission from C /i.pc/v.pc in the spectrum. The ICF may not be representative of the extreme conditions in GS-NDG-9422 (e.g. Berg et al. 2019), and the quoted C/O may represent a lower limit. \nSince no nitrogen lines are detected in GS-NDG-9422, we can only place upper limits on the nitrogen abundances. The low O + abundance implies that the N + abundance is likely negligible. We instead consider the N ++ abundance using N /i.pc/i.pc/i.pc] 𝜆𝜆 1750 / [O /i.pc/i.pc/i.pc] 𝜆 5007 limit measured from the Prism, and the N /i.pc/i.pc/i.pc] 𝜆𝜆 1750 / O /i.pc/i.pc/i.pc] 𝜆 1666 limit measured from the grating. These yield 3𝜎 upper limits of log ( N ++ / O ++ ) < -0 . 85 and < -1 . 01 respectively. We adopt the former as our preferred limit. \nWe detect several helium recombination lines from both the singly and doubly ionised states. Prism measurements of He /i.pc 𝜆 4471 / H 𝛽 and He /i.pc 𝜆 5875 / H 𝛽 yield consistent measurements of He + / H + = 0 . 10 ± 0 . 005 and 0 . 10 ± 0 . 01 respectively. Deriving a He ++ / H + abundance using the He /i.pc/i.pc 𝜆 4686 line results in only a 6 ± 1 % contribution to the total helium abundance, giving He ++ / H + = 0 . 005 ± 0 . 001. Together, this implies a total helium abundance of He / H = 0 . 11 ± 0 . 01, higher than typical values observed in low-metallicity systems (Matsumoto et al. 2022), but consistent with some massive globular clusters (Piotto et al. 2007). However, we caution that our derived helium abundance does not account for collisional emission or self-absorption, which could be significant (e.g. Peimbert et al. 1992).', '3.6 Photoionization Modelling of the Emission Lines': 'The diagnostic diagrams, electron temperature, and line widths all point to a scenario where GS-NDG-9422 is powered by emission from young stars. We explore the feasibility of reproducing the emission lines with young stellar populations with photoionization models using CLOUDY v23 (Ferland et al. 2017). We assume that the intrinsic spectrum is powered by a standard young SSP model, adopting BPASS /v.pc2.2.1 SSP models including binary stellar evolution (Eldridge et al. 2017) with an IMF maximum mass of 300 M ⊙ and a high-mass slope of -2 . 35. We consider a population with an age of 3 Myr and a metallicity of 0.1 𝑍 ⊙ , consistent with that measured for the gas from the spectrum. We note that the abundance patterns assumed in these models are scaled solar, which may not be representative of the stellar populations forming at this redshift. We adopt a spherical geometry with an inner radius of 0 . 1 pc. The calculation is stopped at an electron fraction of 1% or when the neutral column density reaches 10 18 . 7 cm -2 , consistent with the minimal observed Ly 𝛼 emission offset ( ≲ 100 km s -1 ) (Verhamme et al. 2015). We then vary gas density, ionization parameter, metallicity (assuming solar abundance patterns from Grevesse et al. 2010), and carbon abundance, fixing the gas temperature to that measured from [O /i.pc/i.pc/i.pc] 𝜆 4363/ 𝜆 5007, until we reproduce the line strengths of [O /i.pc/i.pc/i.pc] 𝜆 5007, [O /i.pc/i.pc] 𝜆𝜆 3727, C /i.pc/i.pc/i.pc] 𝜆𝜆 1909, H 𝛼 , and H 𝛾 with respect to H 𝛽 . Fe is assumed to be heavily depleted or to have not yet been produced. We also assume resonant lines from low-ionization species have the same escape fraction as that measured for Ly 𝛼 . Note that by focusing on line ratios, the resulting continuum is a prediction of the model. The intrinsic spectrum of our best fit model ( 𝑛 = 10 3 cm -3 , log 10 ( 𝑈 ) = 1 . 2, log 10 ( 𝑍 O / 𝑍 ⊙ ) = -1 . 1, and log 10 ( 𝑍 C / 𝑍 ⊙ ) = -1 . 4) is shown as the magenta line in Figure 4. \nAs can be seen in Figure 4, the strengths of the emission lines are well reproduced by our best fit CLOUDY model, highlighting the fact that relatively metal-poor, young stellar populations are able to explain the emission lines seen in this galaxy. There remains a discrepancy between the observed He /i.pc/i.pc emission and that predicted by the photoionization models 1 , but this is consistent with numerous low-redshift metal-poor galaxies that show anomalous He /i.pc/i.pc emission with no signatures of a black hole or an AGN (Kehrig et al. 2015, 2018; Schaerer et al. 2019; Senchyna et al. 2020; Saxena et al. 2020).', '4 CONTINUUM SHAPE': 'We have shown in the previous section that the emission lines are consistent with ionisation by young metal-poor stellar populations. In this section, we explore the possible origin of the continuum emission in GS-NDG-9422.', '4.1 Balmer jump': 'Low-metallicity, young stellar populations can have ionizing photon production efficiencies that are high enough that the free-bound emission from recombining hydrogen can outshine the stellar continuum at optical wavelengths, leading to the observation of a Balmer jump (e.g. Byler et al. 2017). As seen in Figure 1, already from the broad- and medium-band photometry alone, there is evidence of a spectral discontinuity at the location of the Balmer limit. The Balmer jump is not typically observed as a sharp spectral feature since, at \nFigure 4. Prism spectrum of GS-NDG-9422 (black) compared with the best fit models using a standard SSP (magenta) accounting for both the 𝑧 = 6 IGM opacity (dashed yellow) and a DLA with column density of 10 23 . 1 cm -2 (dotted cyan). \n<!-- image --> \nrealistic spectral resolutions, blending of the tail of the Balmer series and strong emission lines like [O /i.pc/i.pc] 𝜆𝜆 3726, 3729 and [Ne /i.pc/i.pc/i.pc] 𝜆 3869 can occur (e.g. Schirmer 2016). We perform an empirical fit for this discontinuity by masking out regions contaminated by strong emission lines and fitting the spectrum (in units of 𝑓 𝜈 ) over the range 𝜆 rest > 1930 Å with a two-part linear function, broken at the Balmer limit ( 𝜆 rest = 3645 Å). This results in a clear detection of a spectral jump with 15 . 0 ± 0 . 9 nJy in the observed frame (second panel Figure 1; dotted line), demonstrating the strong contribution of nebular continuum to the spectrum of GS-NDG-9422. Such features have been observed in the spectra of metal-poor star-forming galaxies at low-redshift (Guseva et al. 2006, 2007) and at high-redshift (RobertsBorsani et al. 2024), while they have also been widely predicted at high-redshift in SED modelling of photometric data (Endsley et al. 2023; Topping et al. 2023) and in simulations (Katz et al. 2023a; Wilkins et al. 2023). \nAs shown in Figure 4 the spectral discontinuity at the location of the Balmer jump is well reproduced by our best fit photoionization model. It is important to emphasize here that the presence of spectral discontinuity in the continuum is a prediction of the photoionization model which was purely designed to reproduce the emission lines rather than the shape of the continuum.', '4.2 UV continuum turnover': 'Even more striking than the spectral discontinuity at the location of the Balmer jump is the presence of a steep turnover in the UV continuum of GS-NDG-9422 at 𝜆 obs ≈ 1 𝜇 m ( 𝜆 rest ≈ 1430 Å) (Figure 1). While our best-fit photoionization model is successful in reproducing the line emission and the Balmer jump, where the model fails is that it significantly over-predicts the emission at 𝜆 rest ≲ 1430 Å. Here we explore the origin of this discrepancy.', '4.2.1 Absorption from neutral hydrogen?': 'Similar UV turnovers are often observed as a result of absorption from foreground neutral hydrogen - either the neutral intergalactic medium(IGM)(Miralda-Escudé 1998) or Damped Lyman𝛼 absorption (DLA) systems (Heintz et al. 2023). \nBeginning with the IGM, we apply the 𝑧 = 6 IGM transmission curves from Garel et al. (2021), to our best-fit photoionization model and we find that IGM damping is unable to produce the observed UV turnover from this BPASS model (yellow dashed line in Figures 4 & 5). Indeed, the observed turnover in GS-NDG-9422 requires absorption far exceeding the maximal IGM damping wing, calculated following the formalism presented in Miralda-Escudé (1998) and Barkana & Loeb (2001). Similar results were found in Heintz et al. (2023) for targets at much higher redshift. This is not particularly surprising because if the IGM were responsible, many more galaxies at 𝑧 ≳ 6 would show very strong UV turnovers. \nWenextconsiderthepresence of a DLA. We model damping due to the presence of a DLA with CLOUDY by calculating the transmission of the best-fit BPASS model through slabs of neutral hydrogen with increasing column densities, finding that column densities of 𝑁 H ∼ 10 23 cm -2 are needed to reproduce the magnitude of the turnover. Such a value is higher than any previously reported galaxy-scale DLA system (e.g. Tanvir et al. 2019; Umeda et al. 2023; see Figure 5). \nThe primary issue with naively assuming a DLA is that such high column densities imply zero transmission at 1216 Å, conflicting with the strong observed Ly 𝛼 emission. The middle panel of Figure 5 shows that this can, in principle, be reconciled by invoking a DLA with 30 % leakage. However, the plausibility of such an extreme columndensity with a low covering fraction is unclear, given the fact that other known DLA systems with very high gas columns do not show Ly 𝛼 emission (Umeda et al. 2023; Heintz et al. 2023). In principle, one could shift the DLA to a lower redshift which would allow for \n<!-- image --> \n<!-- image --> \nFigure 5. Top: Zoom-in on UV turnover for models plotted in Figure 4. Middle: Fitting to GS-NDG-9422 using a DLA model with a 70 % covering fraction allows Ly 𝛼 escape, but implies an extremely high column density. Bottom: Comparison of implied column density with known DLAs. \n<!-- image --> \nsome Ly 𝛼 emission to escape as the emitted Ly 𝛼 will be redshifted out of resonance in the reference frame of the DLA; however, this would require much higher column densities than 10 23 cm -2 in order to reproduce the UV turnover. Furthermore, at such high column densities, the gas can self-shield from the local-radiation field, and the core would be expected to be fully molecular. It is not clear whether such high gas columns can be maintained without the gas collapsing and forming stars (Schaye 2001). Moreover, GS-NDG-9422 shows no signatures of dust extinction based on the Balmer decrement, the He II ratio, and the fact that the photoionization model can easily reproduce the observed UV slope without assuming dust (Section 3). If the DLA was present within GS-NDG-9422 at a metallicity of nearly 10% solar, one would need to explain why the dust has not formed or could not survive in a thick neutral cloud. Given the fine-tuned requirements, we consider this picket-fence scenario of a thick DLA with optically-thin channels highly unlikely. \nFigure 6. Schematic of the main nebular continuum components across the rest UV to optical range. Free-bound emission can give rise to a spectral discontinuity at the Balmer limit ( 𝜆 rest ≈ 3645 Å). Two-photon continuum has a fixed shape with a turnover at 𝜆 rest ≈ 1430 Å, but is usually subdominant compared to continuum emission of the ionising source. Free-free emission is usually sub-dominant at these wavelengths. The relative contribution of two-photon continuum is highly dependent on nebular conditions. \n<!-- image --> \nOther geometries may exist (apart from the picket fence or lowerredshift DLA) that could possibly explain GS-NDG-9422. For example, one could consider a foreground DLA and background or spatially offset clouds where emission could be reflected. These scenarios all must be reconciled with the very high Ly 𝛼 escape fraction of ∼ 27% which again seems unlikely. For this reason, we consider other alternatives for the origin of the UV turnover.', '4.2.2 Two-photon continuum emission': 'The nebular continuum consists of three components: free-bound, which gives rise to the spectral discontinuity at 𝜆 rest ∼ 3645 Å, freefree, which is typically subdominant compared to free-bound and only impacts 𝜆 rest ≳ 3000 Å, and two-photon emission (Figure 6). Two-photon emission arises from transitions from 2 𝑠 → 1 𝑠 in neutral hydrogen, resulting in the emission of two photons whose energies sum to that of Ly 𝛼 . The distribution of photons emitted via this process is symmetric around 2431 Å ( 𝜈 = 1 2 𝜈 Ly 𝛼 ) when expressed in terms of number of photons per second per frequency interval. However, when expressed in units 𝑓 𝜆 , it takes the form of a broad asymmetric peak which turns over at 𝜆 rest ≈ 1430 Å, remarkably close to the wavelength of the observed UV turnover in GS-NDG9422. The two-photon continuum is typically subdominant compared to the stellar continuum, but has been predicted to be observable in systems with extremely high ionising photon production efficiency (Fosbury et al. 2003; Raiter et al. 2010). Here we consider the possibility that the observed UV turnover is the two-photon continuum. \nTo test whether the continuum of GS-NDG-9422 is consistent with being primarily nebular, we consider a model where the spectrum consists of: \n- (i) Two-photon emission\n- (ii) Free-bound & Free-free emission\n- (iii) Young stars with ages < 10 7 . 5 yrs\n- (iv) Old stars with ages > 10 7 . 5 yrs \nThe shape of the spontaneous two-photon continuum is fixed 2 , so we only vary its overall normalization. The combination of freebound and free-free emission has a shape that is sensitive to gas temperature and thus we vary both gas temperature and the normalization of this component. Finally the shapes of the stellar spectra are sensitive to age, so we consider the normalizations and ages of each stellar population. In total, the model has seven free parameters. As above, we assume that the stellar component follows BPASS /v.pc2.2.1, while the nebular continuum components are computed with PYNEB (Luridiana et al. 2015). Posterior distributions for each parameter are computed using an MCMC (Foreman-Mackey et al. 2013). \nIn the top panel of Figure 7 we show the model corresponding to the 50th percentile distribution of each of the seven parameters (blue line) as well as the points used for the fit (red) and the observed spectrum (green). The MCMC prefers a model where the continuum at 𝜆 rest < 5800 Å is dominated by the nebular continuum. Indeed the two-photon continuum is able to reproduce of the shape of the UV downturn while free-bound emission peaks high enough to create the observed spectral discontinuity near the Balmer jump. Applying frequentest statistics, we find a reduced 𝜒 2 of 𝜒 2 𝜈 = 0 . 80 indicating that this 50th percentile model is a very good fit to the continuum of GS-NDG-9422. \nAlthough this nebular-dominated scenario provides a good fit to the continuum, it remains an open question of whether the model is consistent with the observed emission lines. This can be verified in twoways.Becausethefree-free and free-bound emission are sensitive to gas temperature, we can check whether the temperature predicted from the continuum is consistent with that measured from the oxygen emission lines. In the bottom panel of Figure 7, we show the posterior distribution on gas temperature predicted by the continuum fitting (black histogram) compared to that measured using the [O III] 𝜆 4363 auroral line (cyan), finding that the two are consistent within 1 𝜎 uncertainty. While formally the O /i.pc/i.pc/i.pc temperature does not have to be the same as that of the H II gas, empirical measurements from low-redshift galaxies suggest that the two temperatures are often very similar (e.g. Guseva et al. 2006, 2007). \nAn even stronger test is to compare the observed H 𝛽 equivalent width versus that predicted by the continuum-fitting model. The H 𝛽 emission should primarily arise from recombination, the same as the free-bound continuum. Thus, in a nebular-dominated scenario, EW(H 𝛽 ) should only depend on gas temperature and the relative contribution of free-bound and free-free to two-photon emission. The top panel of Figure 8 shows that the 1 𝜎 uncertainty on the observed EW(H 𝛽 ) significantly overlaps the 1 𝜎 spread in the posterior distribution of predicted EW(H 𝛽 ). This again demonstrates that the information in the continuum is sufficient to explain many of the properties of the emission lines. \nWe can apply the same test for H 𝛼 emission. Since the continuum was only fit up to rest-frame 5800 Å, predicting the H 𝛼 equivalent width requires extrapolating the model. In Figure 8 we compare the posterior distribution on predicted EW(H 𝛼 ) to that measured from the prism as well as that inferred from imaging. As discussed above, the EW 0 (H 𝛽 ) from the imaging data is somewhat higher than that measured from the prism (although the 1 𝜎 contours overlap). We \n2 A significant contribution of induced two-photon emission can alter the width of the probability distribution of emitted photons (e.g. Chluba & Sunyaev 2006), which can in turn shift the wavelength at which the two-photon continuum turns over when expressed in 𝑓 𝜆 . However, we found this effect to be completely negligible ( < 1 Å) in any of the modelling considered in this work. \n<!-- image --> \nRest \n- \nFrame Wavelength [ \n˚ A] \nFigure 7. Top: Best fit continuum model from fitting described in Section 4.2.2. Continuum points using in the fitting are shown in red, while the full spectrum is shown in green. Coloured lines show the continuum components arising from young stars (pink), old stars (light blue), free-bound + free-free nebular (brown), and two-photon (yellow) in the best fit model. The blue line shows the overall best fit. Bottom: Posterior distribution on gas temperature predicted by the continuum fitting. The median and 1 𝜎 values (red lines) are in good agreement with the nebular temperature measured from the [O/i.pc/i.pc/i.pc] 𝜆 4363/ 𝜆 5007 ratio (Table 3; blue). \n<!-- image --> \nfind that our predicted values fall high compared to the prism measurement, but the value from the imaging falls on top of our 1 𝜎 confidence interval. Hence our model is formally very consistent with the imaging data and consistent within 2 𝜎 of the prism. Because we have not fit the continuum near H 𝛼 the model could be missing a contribution from older stars, which can increase at these wavelengths without impacting our current fit. Since GS-NDG-9422 has metals, these must have originated somewhere. A metallicity of 0 . 1 𝑍 ⊙ is much higher than that predicted for the IGM at 𝑧 ∼ 6 (e.g. Madau & Dickinson 2014) and thus it is unlikely the system was enriched externally. While it is possible that we are witnessing the illumination of the immediate enrichment from the current population of massive stars, the abundance patterns are such that it is \nFigure 8. Top: Posterior distribution on H 𝛽 equivalent width predicted by the continuum fitting. The median and 1 𝜎 values (red lines) are consistent with that measured from the Prism spectrum (blue). Bottom: Same as top panel, but for H 𝛼 . Here, the median and 1 𝜎 values (red lines) are somewhat higher than that measured from the Prism spectrum (blue), but in good agreement with the equivalent width implied by the medium-band imaging. \n<!-- image --> \nlikely that there might be an underlying population of stars that is no longer UV-bright. Therefore our current model remains flexible enough to accommodate the scenario of an older population of stars that contributes to the continuum at wavelengths near H 𝛼 . \nNevertheless, given that our model prefers that the UV and optical part of the spectrum arises primarily from nebular emission, it poses the question: what drives this behaviour?', '5 WHAT COULD BE DRIVING TWO-PHOTON EMISSION IN GS-NDG-9422?': 'Under the assumption that the continuum of GS-NDG-9422 is nebular-dominated, we consider the numerous scenarios that may give rise to the observed spectrum. \nFigure 9. Top: Normalized spectrum of hydrogen-only gas with 𝑛 = 10 3 cm -3 irradiated by blackbodies of different temperatures (as given in the colour bar). The magenta line represents a blackbody temperature of 100,000 K. Bottom: Normalized spectrum of hydrogen-only gas irradiated by a blackbody with 𝑇 = 10 5 Kat different gas densities (as given in the colour bar). The magenta line represents a density of 10 3 cm -3 . \n<!-- image -->', '5.1 Is the ionising source present in the spectrum?': "Although Balmer jumps have commonly been observed in the spectra of young, low-metallicity star-forming galaxies (e.g. Guseva et al. 2006, 2007), the steep UV slopes associated with the spectra of these stellar populations means that the nebular contribution is completely sub-dominant in the FUV where the two-photon continuum peaks. However, one can envision a scenario where the nebular emission is offset from the location of ionising source. If the slit were to contain only nebular gas, the two-photon emission could dominate the observed spectrum. \nOne example of this is Hanny's Voorwerp, which has been suggested to be quasar light echo (Lintott et al. 2009). In this case, both the free-bound and two-photon emission are expected to be strong. However, there are three key differences between Hanny's Voorwerp and GS-NDG-9422: (1) He /i.pc/i.pc 𝜆 4686/H 𝛽 is > 6 × higher in the presumed QSO light echo compared to the galaxy studied here (Figure 2). He /i.pc/i.pc 𝜆 1640 is also much stronger in Hanny's Voorwerp (Keel et al. 2012), (2) while the two-photon continuum is likely important in Hanny's Voorwerp, another unidentified source has been postulated to explain the excess UV emission. In other words, the spectrum is not fully nebular dominated in the UV and may consist of additional scattered light, and (3) the spatial extent of Hanny's Voorwerp is tens of kpc. Conversely, GS-NDG-9422 is very compact, and it is well-centred within the NIRSpec micro-shutter (Figure 1). While other QSO light echoes have been detected (e.g. Schirmer et al. 2016), the spectra and morphologies do not seem to be consistent with GS-NDG-9422, leading us to disfavour this scenario. \nAn alternative explanation is that the ionization source has flickered on and off and we are capturing the system just after it has shut off. In this case, no continuum from the ionising source would be \nFigure 10. Top: Spectrum of GS-NDG-9422 (black) compared with the best fit blackbody model (magenta) with 𝑇 BB = 10 5 . 05 K and a He /i.pc/i.pc leakage fraction of 0.22 (Table 4; Section 5.2). Middle: Spectrum of GS-NDG-9422 (black) compared with photoionsiation models powered by various individual massive Pop. III stars with different effective temperatures. Bottom Spectrum of GS-NDG-9422 (black) compared with the photoionisation model powered by a model Wolf-Rayet SED with the most similar spectrum to the best-fit blackbody SED. All models can also reproduce the observed spectrum well (dotted lines). See Section 5.2 for details. \n<!-- image --> \nλ \nf \n1 \n1 \n. \n10 \n1 \n. \n15 \n1 \n. \n20 \nFigure 11. Zoominonthe region around He /i.pc/i.pc 𝜆 4686 demonstrating the need to reduce the He /i.pc/i.pc ionizing flux in the blackbody models. The spectrum of GSNDG-9422 is shown in black. The magenta line shows the best-fit model from Figure 10. Green to blue lines show decrease He /i.pc/i.pc leakage fraction applied to the blackbody, modelling the effect of a He-rich atmosphere. Models with a high leakage fraction clearly overpredict the He /i.pc/i.pc 𝜆 4686 line, although we note this feature is blended with nearby [Ar /i.pc/v.pc] lines. We note a slight flux excess blue-ward of He /i.pc/i.pc 𝜆 4686 which, from inspection of the 2D spectrum, appears to be driven by a noisy pixel. This excess flux is still below the level of the excess emission implied by the 100 % He /i.pc/i.pc leakage model shown in green. \n<!-- image --> \ndetected and we would be observing only the remnant nebular emission. However, the H + recombination timescale, assuming Case B recombination at a temperature of 18,300 K, is ∼ 5 , 000 yr for a density of 10 2 cm -3 or ∼ 500 yr for 10 3 cm -3 . This is much shorter than the main-sequence lifetimes of massive stars. Thus, in the case of flickering, it is highly unlikely that stars are powering the emission. Some QSO proximity zones show evidence for QSO lifetimes of < 10 4 yr (Eilers et al. 2018, 2021). While such QSOs are a rare subset of the general population, these lifetimes are broadly consistent with the recombination timescale. Because these QSOs are detected via their near zone, the QSOs are currently 'on' and thus the UV continuum is still dominated by the AGN and not the nebular emission. Evidence for AGN fading on such short time-scales has also been observed locally (e.g. French et al. 2023), but the spectra of these objects, in particular the low ionization state lines is inconsistent with GS-NDG-9422. Moreover, we have only considered the hydrogen recombination time scale. If we consider He ++ under the same assumptions, we arrive at a recombination timescale of ∼ 300 yr for a density of 10 2 cm -3 or ∼ 30 yr for 10 3 cm -3 . The fact that we observe He /i.pc/i.pc emission is thus difficult to reconcile with this 'flickering' scenario, and imply that we would be catching the source at a very specific moment time. Given the very special timing required, the identification of other objects with spectra like GS-NDG-9422 would seemingly disfavour this scenario. We discuss the identification of similar candidates in Section 6.3.", '5.2 Is GS-NDG-9422 powered by hot stars?': "Assuming that the ionizing source remains luminous and is present within the slit, in order for both the two-photon and free-bound continua to dominate over the stellar spectrum in the rest-frame UV and optical, the source population must have a much larger ion- \nising photon production efficiency ( 𝜉 ion ) than standard SSP models, necessitating hotter blackbody temperatures. We explore this by running CLOUDY simulations with input blackbody SEDs, with a setup closely based on that described in Section 4.2.1. To gain insight into the requirements for the two-photon continuum to dominate over the ionizing SED, we initialize hydrogen-only gas at constant gas temperature (measured from [O /i.pc/i.pc/i.pc] 𝜆 4363/ 𝜆 5007) and systematically vary the density and blackbody temperature (Figure 9). A weak UV turnover begins to appear at blackbody temperatures 𝑇 BB ≳ 65 , 000 K, which is hotter than a typical O star. A strong UV turnover requires at least 𝑇 BB ≳ 90 , 000 K, much hotter than massive O stars (up to ∼ 50,000 K; Walborn et al. 2004; Evans et al. 2011; Bressan et al. 2012). \nThe two-photon continuum is also sensitive to gas density because, at high densities, 𝑙 -changing collisions will suppress the two-photon emission relative to Ly 𝛼 . This is seen in the bottom panel of Figure 9 where we vary gas density for a fixed blackbody temperature of 100 , 000K.TheUVturnoverisstrongly suppressed at 𝑛 ≳ 10 3 cm -3 . We emphasize that the details of this calculation are sensitive to the chosen column density at which the model is truncated (which in our case is set by the velocity offset of Ly 𝛼 ). Furthermore, there exist significant differences in atomic data predictions for the strength of 𝑙 -changing collisions as a function of temperature (Guzmán et al. 2017). \nUnder the constraints determined above, we adopt an empirical approach to determine the possible underlying spectrum of GS-NDG9422. We assume that the ionizing spectrum, to first order, can be modelled as a blackbody. To optimize the blackbody model fit to GS-NDG-9422, we begin with the parameters of the best fit BPASS model (Section 4.2.1) and update ionization parameter, gas density, and blackbody temperature to simultaneously reproduce the emission line ratios and continuum shape. We allow for both density and ionization bounded nebulae by adding an additional stopping criterion to reproduce the measured [O /i.pc/i.pc/i.pc] 𝜆 5007/[O /i.pc/i.pc] 𝜆𝜆 3727 (O32) ratio. This stopping criterion often supersedes the neutral gas column stopping criteria used above. The O32 ratio and the gas temperature are allowed to vary within their observational uncertainties. Optimisation of this model (top panel of Figure 10) demonstrates that a blackbody model with 𝑇 = 10 5 . 05 K can provide a good fit to both the shape of the continuum and the majority of lower-ionization state emission line ratios in GS-NDG-9422. \nHowever, where our simple blackbody model fails is that it strongly overpredicts the flux of He /i.pc/i.pc lines compared to those observed in the spectrum (Figure 11). The weak He /i.pc/i.pc 𝜆 1640 and He /i.pc/i.pc 𝜆 4686 in GS-NDG-9422 places tight constraints on the ionizing spectrum at 𝐸 > 4 Rydberg. The opacity of helium in the stellar atmosphere is well known to play an important role in mediating the flux of stellar populations at these energies, suppressing the flux of He + -ionising photons ( 𝜆 < 228 Å; e.g. Smith et al. 2002). To explore this, we run models varying the leakage fraction of photons with 𝐸 > 4 Rydberg from 0% (i.e. no He + -ionizing photons) to 100% (unattenuated blackbody). We conclude that 70% -75% of the He + -ionizing photons emitted by the blackbody must be extinguished to match the observed He /i.pc/i.pc emission, with a leakage fraction of 0 . 25 in our best-fit model. The parameters for our optimized model are reported in Table 4. \nWenote that other sources, including active galactic nuclei (AGN) or high-mass X-ray binaries (HMXBs), can have significant highenergy photon outputs, and have been invoked to explain peculiar emission line ratios seen at high-redshift (Maiolino et al. 2023; Katz et al. 2023b). However, the weak He /i.pc/i.pc emission disfavours power-law SEDs that extend past the He + -ionizing edge. We exclude HMXBs \ndue to the strong over-prediction of He /i.pc/i.pc emission in these models (see Appendix B), while the presence of an AGN is discussed above. Finally, we note that spectral fitting of some low-redshift 'high-redshift analogues' has suggested a similar need for significant ionizing contribution from a hot ( 𝑇 > 80 , 000 K) blackbody (Olivier et al. 2022), however the key difference in GS-NDG-9422 presented here is the dominance of the two-photon continuum. This necessitates an extremely high 𝜉 ion that cannot be produced if a substantial fraction of the ionizing photons are emitted by typical OB stars. \nWe now turn to the question of what sort of stars have sufficiently high surface temperatures to reproduce our best-fit blackbody model with 𝑇 BB = 10 5 . 05 K. \nWolf-Rayet stars not only exhibit extremely high surface temperatures, but can also have helium atmospheres that provide the necessary opacity to reduce their He + ionizing output (Crowther 2007). We explore models from grid of the PoWR Wolf-Rayet models (Todt et al. 2015) 3 . Given the gas-phase metallicity measured for GS-NDG-9422, we consider the WNL-H40 grid of the PoWR Wolf-Rayet models (Todt et al. 2015) with 𝑍 = 0 . 07 𝑍 ⊙ , similar to that measured for GS-NDG-9422. We identify model 13-10 as most similar to our optimised blackbody, for which 𝑇 = 100 , 000 K and the luminosity is fixed at 𝐿 = 10 5 . 3 𝐿 ⊙ (Figure 12 top left). We note that this model assumes iron group abundances to be scaled solar, which may not be representative of stars forming at 𝑧 ∼ 6. Abundances of carbon, nitrogen, and oxygen are assumed to have undergone significant CNO burning (see Todt et al. 2015 for details). \nAdopting parameters from the optimised blackbody model but removing the He /i.pc/i.pc leakage parameter, we replaced the blackbody SED with this theoretical low-metallicity Wolf-Rayet star spectrum. No further optimisation is performed and the resulting spectrum is shown as the dashed orange line in Figure 10, nearly identical to the optimised blackbody model. \nAlthough Wolf-Rayet stars are typically associated with broad emission lines due to their high-velocity stellar winds, the absence of these features in GS-NDG-9422 could simply be the result of the extremely bright nebular component outshining these wind features, as predicted by our photoionization models. Furthermore, in the absence of iron, which is the dominant source of stellar atmospheric opacity, wind speeds can drop below 500 km s -1 , even at solar oxygen abundance (Gräfener & Hamann 2008). Hence, high-redshift Wolf-Rayet dominated galaxies may not show broad He /i.pc/i.pc 𝜆 1640 and He /i.pc/i.pc 𝜆 4686 lines (Gräfener & Vink 2015). \nStars stripped in binaries can also exhibit the required surface temperatures above 10 5 K(Götberg et al. 2018, 2023). Compared to our best-fit extinguished blackbody, we find that stripped star SEDs (Götberg et al. 2018) produce too few He + ionizing photons, and the He /i.pc/i.pc emission is underpredicted by these stars (Figure 12 top right). \nMassive metal-free Population III stars are predicted to have sufficiently high temperatures to power a two-photon dominated spectrum (Schaerer 2002; Trussler et al. 2023). While the IMF of Population III stars is highly uncertain (Klessen & Glover 2023), comparing extremely top-heavy (Pop. III.1) and moderately top-heavy (Pop. III.2) Yggdrasil models (Zackrisson et al. 2011) with our extinguished blackbody (Figure 12 bottom left) indicates these produce too many He + ionizing photons, while predictions for even less topheavy IMFs begin to fall short of the required 𝜉 ion . In contrast, some individual Pop. III star models from Larkin et al. (2023) with effective temperatures of ∼ 97 , 000 K and masses of 85 M ⊙ to 108 M ⊙ \nTable 4. Parameters for the optimized blackbody model. \nreasonably reproduce the required ionizing SED (Figure 12 bottom right). We strongly emphasize that the measured metallicity of 12 + log 10 ( O / H ) = 7 . 59 ± 0 . 01 suggests the presence of Pop. III stars is highly unlikely , unless we are witnessing the immediate enrichment and illumination of metals produced by primordial stars. Hence, we are not advocating that GS-NDG-9422 hosts a population of primordial stars. Nevertheless, stellar atmosphere models have large uncertainties at low metallicity, and few such models exist in the literature, so we consider these Pop. III star models in our analysis under the assumption that the atmospheres may be representative of hot massive stars at very low metallicity. Repeating the exercise described for our Wolf-Rayet models, this time replacing the blackbodySEDwithlow-metallicity massive star models from Larkin et al. (2023) with effective temperatures between 95,000 K and 99,000 K also results in a very good fit to the spectrum of GS-NDG-9422 (green dashed lines in Figure 10). \nIn summary, we identify three classes of model stellar SED that have the surface temperatures required to reproduce the observed two-photon continuum turnover in GS-NDG-9422 (Figure 10; see also schematic in Figure 13). Wolf-Rayet stars with 𝑍 = 0 . 07 𝑍 ⊙ , equal to the measured nebular metallicity, and hot massive stars with low-metallicity atmospheres provide a remarkably good fit, while existing model SEDs from stripped stars only fall short in having too strong suppression of He + -ionizing flux. While a perfect match to the observed spectrum of GS-NDG-9422 will require fine-tuning of the photoionization model, the fact that the continuum shape and the vast majority of the emission lines can be reproduced under very simple assumptions is encouraging that stars such as those considered in this section may be present in GS-NDG-9422.", '6.1 Implications for the stellar initial mass function if GS-NDG-9422 hosts a population of hot stars': 'We now consider what our modelling implies for the mass distribution of the stellar population in GS-NDG-9422. The progenitor masses of Wolf-Rayet stars at the metallicity of GS-NDG-9422 are not well constrained (Massey et al. 2001). Masses of ≥ 37 M ⊙ have been estimated for Wolf-Rayet stars in the Small Magellanic Cloud (SMC) with 𝑇 eff ≳ 100 , 000 K (Hainich et al. 2015), implying even higher progenitor masses. Meanwhile, the low-metallicity massive star models have somewhat higher progenitor masses of ∼ 100 M ⊙ . While our search may not be exhaustive, we conclude from Figure 12 that the nebular-dominated spectrum of GS-NDG-9422 is consistent with ionization powered by low-metallicity massive stars ( ≳ 50 M ⊙ ), perhaps in the Wolf-Rayet phase. \nIf we adopt a progenitor mass of 50 -100 M ⊙ , then, assuming a typical IMF, we expect to form one such star per every \nFigure 12. Best fit blackbody SED model with He-rich atmosphere (black solid) compared to hot star model SEDs from the literature. The dotted black line shows the blackbody spectrum prior to extinguishing the He /i.pc/i.pc ionizing radiation. The top left panel shows select 𝑍 = 0 . 07 𝑍 ⊙ Wolf-Rayet stars from Todt et al. (2015), selected to match the measured nebular metallicity. These show remarkable resemblance to our model blackbody. The top right panel shows stripped star models from Götberg et al. (2018). These attain sufficiently high surface temperatures, but the atmospheres in existing models exhibit too much suppression of the He /i.pc/i.pc ionising continuum. The bottom left panel shows Yggdrasil Pop. III models (Zackrisson et al. 2011) with two different assumed Pop. III IMFs (very top heavy, Pop. III.1; blue) and (moderately top heavy, Pop. III.2; magenta). These models strongly overpredict the He /i.pc/i.pc-ionising flux. The bottom right panel shows individual massive Pop. III stars from Larkin et al. (2023) with different effective temperatures as shown in the colour bar. Some of these models exhibit the required SED properties to reproduce the observed spectrum of GS-NDG-9422 (Figure 10), although we note the discrepancy between these zero-metallicity models and the measured nebular metallicity. Hence, we are not claiming the presence of Pop. III stars in GS-NDG-9422. \n<!-- image --> \nFigure 13. Schematic of how the hot star scenario gives rise to a nebular-dominated spectrum in a simple spherical nebula. The incident stellar spectrum ( 𝑇 eff = 10 5 . 05 K) is shown in blue. The portion of this transmitted through the nebula is shown in the darker shade, while the lighter shade shows the portion that is absorbed to photoionisation of the gas. The red shows the resulting nebular spectrum, with the black showing the sum of nebular and transmitted stellar, which is what an observer will see. For a fixed 1500 Å flux density, hot stars with 𝑇 eff ≳ 10 5 K emit significantly more ionising photons (light blue) than a typical stellar population, which in turn allows them to power much stronger nebular emission (red), which ultimately outshines their own spectrum at wavelengths longer than Ly 𝛼 (1216 Å). Note that a significant population of old stars can be accommodated within this scenario without affecting the nebular dominance at UV wavelengths, since these old stars primarly contribute flux at longer wavelengths. \n<!-- image --> \nRest-frame wavelength [ \n˚ \nA] \n∼ 1 , 300 -3 , 800 M ⊙ of stellar mass. To explore whether our model is consistent with this, we repeat our photoionization modelling with our best-fit massive star models while simultaneously adding a second component from a BPASS model. We assume the BPASS stellar population has the same metallicity as the gas. In the case of lowmetallicity massive stars, we assume the BPASS model has an age of 3 Myr (the approximate lifetime of massive stars), while for the Wolf-Rayet models, we assume the progenitor stars are lower mass (50 M ⊙ ) and live for slightly longer (5 Myr). We then progressively increase the ionization parameter of the second population until the UV turnover from the two-photon continuum begins to weaken. We calculate the stellar mass of a BPASS SSP that can be present per hot star as \n𝑀 SSP = 𝑈 SSP 𝑈 star 𝑄 star 𝑄 SSP , (1) \nwhere 𝑄 star = 10 49 . 36 𝛾 s -1 for the Wolf-Rayet star model or 10 49 . 98 𝛾 s -1 for the low-metallicity massive star model with 𝑇 eff = 97 , 352 K. 𝑄 SSP = 10 46 . 73 or 10 46 . 03 𝛾 s -1 M -1 ⊙ at 3 Myr and 5 Myr, respectively. In both cases, we find the maximal allowable contribution to be 𝑈 SSP 𝑈 star ≤ 6 . 3%. Therefore, we can place upper limits of the BPASS SSP mass of 𝑀 SSP ≤ 112 or 137 M ⊙ per one low-metallicity massive star or Wolf-Rayet star, respectively. This is clearly discrepeant from the values of 3 , 800 M ⊙ and 1 , 300 M ⊙ implied from a typical IMF. In other words, there is a 35 × excess in the number of massive stars in the case of our low-metallicity massive star model, or a 9 . 5 × excess for our Wolf-Rayet model. Either scenario implies that the IMF in GS-NDG-9422 is very top-heavy. \nWe note that the measurement presented here is sensitive to the IMF by way of constraining the ratio between the ionising photon production rate, dominated in our model by stars more massive than ≳ 50 M ⊙ , and the continuum flux at ∼ 1500 Å, which in standard young stellar population models is dominated by stars with M ≳ 5 M ⊙ (e.g. Byler et al. 2017). It is, therefore, only sensitive to the high-mass end (M ≳ 5 M ⊙ ) of the IMF, and is completely insensitive to the impact of stars with M ≲ 5 M ⊙ . \nTaken at face value, this calculation implies that the ratio of ≳ 50 M ⊙ to ∼ 5 -50 M ⊙ stars in GS-NDG-9422 is of order ∼ 10 × higher than that predicted by a typical IMF. However, the results of this calculation are very sensitive to the chosen underlying SSP, the assumed age, the progenitor masses of Wolf-Rayet stars, and the mass of the low-metallicity massive stars. For SSPs that have intrinsically lower 𝜉 ion , the excess drops as 𝑄 SSP appears in the denominator of Equation 1. If the ages at which the Wolf-Rayet stars evolve off the main sequence is longer than what we have assumed here, the excess will also decrease. More generally, we highlight that the origin of such hot stars in high-redshift environments and the modelling of their atmospheres is highly uncertain and motivates detailed characterisation of the effects of binary stripping and stellar wind mass loss, particularly at low metallicity (Götberg et al. 2019; Vink 2022). Thus, while we have demonstrated qualitatively that explaining the observed spectrum without variations to IMF is extremely difficult, a detailed characterisation of the high-mass end of the IMF in GS-NDG-9422 will require more advanced understanding of stellar evolutionary processes in these environments. \nUnderstanding the shape, upper-mass, and lower-mass cutoff of the IMF, and whether these evolve with initial conditions, is critical to the interpretation of nearly all extragalactic observables (Hopkins 2018). Theoretical works have predicted the IMF to get progressively more top-heavy for low-metallicity gas at high pressure (Chon et al. 2021, 2022; Sneppen et al. 2022), while others indicate that increased \nCMB temperature can cause the IMF to become more bottom-light at high-metallicity (Bate 2023). \nWhile many studies of local field stars and young clusters have found no strong evidence for variation in the IMF of these systems (e.g. Bastian et al. 2010 and references therein), a number of lines of evidence have emerged suggesting the IMF may vary in some environments. Top-heavy IMFs have been derived in some local massive young star clusters (Kalari et al. 2018; Schneider et al. 2018), while some models of globular clusters have invoked a topheavy IMF at early times to explain how gas expulsion proceeded from the young cluster (Marks et al. 2012). \nSpectral line indices in the spectra of early-type galaxies have been widely demonstrated to show that the IMF is bottom-heavy in these systems (van Dokkum & Conroy 2010; Spiniello et al. 2014; MartínNavarro et al. 2015; Conroy et al. 2017; Maksymowicz-Maciata et al. 2024). Similar conclusions have been drawn from gravitational lensing studies (Treu et al. 2010; Smith 2020) and dynamical modelling (Cappellari et al. 2012; Poci et al. 2022; Lu et al. 2023), both of which are senstive to the mass-to-light ratios. We note that these measurements are sensitive to the ratios of stars with 𝑀 ≲ 0 . 5 𝑀 ⊙ and 𝑀 ∼ 0 . 5 -1 𝑀 ⊙ , which our constraint is insensitive to. Furthermore, we note that chemical evolution modelling of massive ellipticals has shown that a time-invariant bottom-heavy IMF cannot explain the observed [Mg/Fe] and [Fe/H] abundance ratios in these systems with these studies instead suggesting that this bottom-heavy phase may have been preceeded by a short-lived top-heavy phase (Vazdekis et al. 1997; Weidner et al. 2013; Jeřábková et al. 2018; De Masi et al. 2019). \nAbundances of CNO elements and their isotopes are expected to be highly sensitive to variations in the high-mass end of the IMF (Romano 2022). Variations in the 13 C/ 18 Oratio observed in 𝑧 ∼ 2 -3 dusty starburst galaxies have been interpreted as arising from a topheavy IMF (Zhang et al. 2018), while a sub-solar C/O abundance ratios arising from a top-heavy IMF has been suggested to explain anomalous IR emission line ratios at high-redshift (Katz et al. 2022). Furthermore, a rapid starburst with a top-heavy IMF has been invoked as a possible explanation the enhanced N/O abundance ratio observed in GN-z11 (Bekki & Tsujimoto 2023). \nTop-heavy IMFs with low mass-to-light ratios have also been widely invoked as a possible cause of the surprising abundance of UV-bright galaxies observed at high redshift (e.g. Finkelstein et al. 2023; Harikane et al. 2024; Yung et al. 2024).', '6.2 Total cluster mass and number of hot stars': "The hydrogen ionising photon luminosity, 𝑄 , can be approximated from the H 𝛽 flux as: \n𝑄 ≈ 4 𝜋𝐷 2 𝑙 𝐼 H 𝛽 ( 1 -𝑓 esc ) ℎ𝜈 H 𝛽 𝛼 𝐵 𝛼 eff B , (2) \nwhere 𝐷 𝑙 is the luminosity distance, 𝐼 H 𝛽 is the measured H 𝛽 flux, ℎ is Planck's constant, 𝜈 H 𝛽 is the frequency of the H 𝛽 transition, 𝑓 esc is the escape fraction of ionising photons, 𝛼 𝐵 is the total case B recombination rate, and 𝛼 eff B is the effective H 𝛽 recombination rate. To calculate a luminosity distance, we adopt a cosmology with 𝐻 0 = 67 . 31 km s -1 Mpc -1 and Ω m = 0 . 315 (Planck Collaboration et al. 2016). This gives 𝐷 𝑙 = 58 . 5 Gpc, implying an H 𝛽 luminosity of 𝐿 H 𝛽 = 7 . 0 × 10 41 erg s -1 . Using the escape fraction derived from our best fitting blackbody photoionization model of 7 . 3% and recombination rates evaluated at 20,000 K (Osterbrock & Ferland \nRest frame wavelength [ ˚ A] \n<!-- image --> \nFigure 14. Zoom in on the rest-frame UV region for GS-NDG-9422 (black) compared with the spectra of the 𝑧 = 3 . 35 Lynx arc (top) and A2744-NDGZD4 at 𝑧 = 7 . 88 (bottom). We have normalised the spectra and shifted them all to the rest frame. \n2006), we estimate a hydrogen ionising photon luminosity of 𝑄 = 1 . 65 × 10 54 s -1 . \nThis value allows us to estimate the mass of the star clusters that have formed in GS-NDG-9422. The Wolf-Rayet star models presented in Section 6.1 have a fixed luminosity of 10 5 . 3 𝐿 ⊙ , which results in a hydrogen ionising photon luminosity of 𝑄 WR = 10 49 . 36 s -1 . However, known Wolf-Rayet stars in the most comparable local environments typically have significantly higher luminosities of ∼ 10 6 . 2 𝐿 ⊙ (Shenar et al. 2016), implying that 𝑄 WR ≈ 10 50 . 3 s -1 might be a more realistic value. Based on the H 𝛽 luminosity, this would imply ∼ 10 , 000 of the 𝑍 = 0 . 07 𝑍 ⊙ Wolf-Rayet stars would be needed to power the spectrum of GS-NDG-9422. In the alternative model, ∼ 17 , 000 very metal-poor stars of ∼ 100 M ⊙ with 𝑇 eff = 97 , 000 K would be required. Based on the calculations presented in Section 6.1, this implies a maximum star cluster mass of 10 6 . 22 M ⊙ for the Wolf-Rayet star model, or 10 6 . 55 M ⊙ for the lowmetallicity massive star model. We caution that these mass estimates should be treated only as a guide since they are very sensitive to the adopted hot star SED as well as the same assumptions outlined in Section 6.1. We also note that the quoted value includes only the mass of recently formed population contributing significant UV luminosity. A considerable mass of older stellar populations could also be accommodated within these models.", '6.3 Are there other galaxies like GS-NDG-9422?': 'While the physics driving the abnormal continuum shape of GSNDG-9422 remains uncertain, it is important to understand whether GS-NDG-9422 is unique, or whether other objects show similar spectral features. Identifying larger samples of objects that are similar to GS-NDG-9422willhelpruleoutscenarios that require fine-tuning. \nWhile we have not performed an exhaustive or systematic search, in Figure 14 we highlight two objects that both show Ly 𝛼 and a \nUV turnover, similar to GS-NDG-9422. The first is the Lynx arc (Fosbury et al. 2003), a gravitationally-lensed, extreme emission system identified at 𝑧 = 3 . 35. The emission lines of the Lynx arc differ considerably from GS-NDG-9422 in that He /i.pc/i.pc is weaker and there is strong N /i.pc/v.pc] emission which is not observed in GS-NDG9422. Nevertheless the UV downturn and Ly 𝛼 are remarkably similar. Indeed, analysis in Fosbury et al. (2003) came to similar conclusions as presented here for GS-NDG-9422 - namely, that hot stars ( 𝑇 eff ≳ 80 , 000 K) are powering the emission in this galaxy which leads to the visibility of the two-photon continuum. Hence, it is clear that the shape of the continuum in GS-NDG-9422 is not unique among the galaxy population. \nWe identify a second galaxy, A2744-NDG-ZD4 at 𝑧 = 7 . 88, observed as part of the Ultra-deep NIRCam and NIRSpec Observations Before the Epoch of Reionization (UNCOVER) program (ID: 2561, PI: Labbe; Bezanson et al. 2022) which also appears to show Ly 𝛼 escape and a UV turnover (Figure 14 bottom panel). The spectrum of this object, retrieved from Dawn JWST Archive (DJA) 4 which had been reduced using the custom-made pipeline MsaExp v.0.6.12 5 (see Heintz et al. 2023 for details), received only 2.3 hours integration. Thus, the continuum is not well detected in the rest-frame optical and we can not determine whether it shows a Balmer jump. Deeper data will be required to conclusively determine whether A2744-NDGZD4 is truly of the same nature as the Lynx arc and GS-NDG-9422.', '7 CONCLUSIONS': 'In this paper, we have presented a discussion of the possible physical origin of the observed continuum and line emission in GS-NDG9422, a galaxy at 𝑧 = 5 . 943. The identification of a spectral discontinuity of 15 ± 0 . 9 nJy at the location of the Balmer limit is clearly indicative of very strong nebular emission, as are the high equivalent widths observed for [O /i.pc/i.pc/i.pc] 𝜆 5007, H 𝛽 , and H 𝛼 . We measure high nebular temperatures ( ( 1 . 83 ± 0 . 15 ) × 10 4 K), low densities ( 𝑛 𝑒 ≲ 10 3 cm -3 ), and low metallicity (12 + log ( O / H ) = 7 . 59 ± 0 . 01; ∼ 8 % (O/H) ⊙ ) in this system. We show that, considering only the emission line ratios, this system is highly consistent with emission powered by a typical population of young, metal-poor stars, with no strong evidence for the presence of AGN activity. \nWhat is more difficult to explain is the observed shape of the continuum. In particular, the strong UV turnover observed at 𝜆 rest ≈ 1430 Å. The shape of this feature is a remarkably consistent with the hydrogen two-photon continuum, but could also be explained by a very high column density of neutral hydrogen. We consider several scenarios for how each of these effects could self-consistently give rise to the observed spectrum of GS-NDG-9422, all of which are summarized in Figure 15. \nTheprimarychallenge faced by models explaining this feature with a DLA is that the magnitude of the turnover implies unprecedented column densities of 𝑁 HI ≳ 10 23 cm -2 , higher than any known DLA. Furthermore, the observation of narrow, high EW Ly 𝛼 emission with a high escape fraction ( ∼ 27%) requires some mechanism by which this emission can reach the observer. This can be reconciled if Ly 𝛼 arises from a reflection spectrum (Scenario #1), although known examples of this are orders of magnitude larger in physical scale than GS-NDG-9422. Alternatively, the Ly 𝛼 might be allowed to reach the observer if the DLA has only partial coverage (Scenario #2) or \nFigure 15. Summary of the three classes of scenarios that can explain the spectrum on GS-NDG-9422. For each scenario, we show various cartoons for how the physics may manifest and give rise to a steep UV slope, a high Ly 𝛼 escape fraction, and the UV turnover redward of Ly 𝛼 . \n<!-- image --> \nis at lower redshift (Scenario #3); however, each of these scenarios imply even higher neutral gas column densities which means their plausibility is questionable. Why such high gas columns would not be fully molecular and why at this moderate metallicity there are no signatures of dust attenuation with so much neutral gas remain open questions in the DLA scenario. For these reasons, we do not favour the DLA solution. \nWedemonstrated in Figure 7 that modelling the continuum of GSNDG-9422 with strong two-photon, free-bound and free-free continuum plus a subdominant young stellar and old stellar component provides a very good fit to the spectrum, with the predicted nebular temperature and equivalent widths of Balmer emission highly consistent with what is measured from the emission lines. The main challenge for explaining the UV turnover with two-photon continuum emission is therefore simply: how can the system power such strong nebular emission without the rest-UV continuum becoming dominated by the spectrum of the ionising source? \nOne possibility is that no flux from the ionising source is observed, either due to a spatial offset whereby the ionising source falls outside the slit (Scenario #4) or because the ionising source has recently turned off and we are observing emission from its relic (Scenario #5). The former is disfavoured by the compact morphology of GSNDG-9422, which is well-centred in the slit (Figure 1). The predicted recombination timescales in Scenario #5 ( ∼ 500 - 5,000 yr for H + and ∼ 30-300 yr for He ++ ) are incredibly short meaning we would need to have caught this object at a very specific moment in time, although this could perhaps be explained with an AGN fading on a timescale of ≲ 10 4 yr (e.g. French et al. 2023). We consider this model to be possible, but the identification of larger samples of similar may rule this scenario out. \nThe final scenario we consider is that the ionising source is indeed present in the slit, but its ionising photon production efficiency is so large that it powers sufficiently strong nebular emission that the two-photon continuum provides the dominant contribution to the rest-UV continuum (Scenario #6 and Figure 13). This is a generic prediction of all photoionization models that include a significant population of hot stars with 𝑇 eff ≳ 10 5 K(e.g. Schaerer 2002; Raiter et al. 2010; Zackrisson et al. 2011; Inoue 2011; Trussler et al. 2023). We identify several model stellar SEDs that can attain these temperatures, showing that models with ionising spectra dominated by ∼ 7 % 𝑍 ⊙ Wolf-Rayet stars and/or very low-metallicity massive stars are able to reproduce the puculiar shape of the observed continuum. The critical aspect to this scenario is that it requires a top-heavy IMF, with the implied ratio of ≳ 50 M ⊙ to ∼ 5-50 M ⊙ stars significantly enhanced relative to typical stellar populations. We note that the precise magnitude of this excess is highly dependent on the assumptions around the evolution and atmospheric properties of these hot, massive, low-metallicity stars, and this work motivates a pursuing improved models of these stars. Nonetheless, we note that although such an IMF is unprecedented in nearby star clusters, it is broadly consistent with other suggestions of enhanced massive star formation in extreme systems at early times (see discussion in Section 6.1). \nAll three classes of solutions we have proposed to explain the spectrum of GS-NDG-9422 pose interesting physical questions about high-redshift galaxy formation. Either we have discovered the highest DLAcolumndensity for a star-forming galaxy, we have witnessed the flickering of a powerful ionising source on timescales much shorter than typical QSO lifetimes, or the ionizing source represents an exotic population of stars with high surface temperature. The similarities with the Lynx Arc at 𝑧 = 3 . 35 and A2744-NDG-ZD4 at 𝑧 = 7 . 88, indicate that GS-NDG-9422 may not be unique. This motivates a \nmore systematic search of such objects to unravel the origin of their unique spectra.', 'ACKNOWLEDGEMENTS': "We thank Bob Fosbury for useful discussions and providing us with the LRIS spectrum of the Lynx arc. We thank Paul Crowther for providing insightful and constructive comments. We thank Thibault Garel for providing us with the simulated IGM attenuated curves from the SPHINX simulation. We thank Anna Feltre and Stéphane Charlot for sharing their latest AGN photoionisation model grids. AJC, AS and AJB, acknowledge funding from the 'FirstGalaxies' Advanced Grant from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 789056). CW thanks the Science and Technology Facilities Council (STFC) for a PhD studentship, funded by UKRI grant 2602262. NL acknowledges support from the Kavli foundation.", 'DATA AVAILABILITY': 'Reduced spectra, imaging, and measured photometry of GS-NDG9422 were made publicly available as part of JADES Data Release 1 (Bunker et al. 2023; Rieke et al. 2023) and can be found at https://archive.stsci.edu/hlsp/jades . Photoionisation models presented in this work will be shared on reasonable request.', 'APPENDIX A: EMPIRICAL DLA COLUMN DENSITY MEASUREMENTS': 'In Section 3, we optimized a CLOUDY photoionization model in order to reproduce the emission lines of GS-NDG-9422, which then provided an estimate of the continuum. We then applied various DLA columns with leakage to this model in order to reproduce the UV turnover and the escape of Ly 𝛼 . However, the exact column density needed is degenerate with the underlying shape of the continuum. In this section, we repeat the experiment trying only to match the shape of the continuum. \nFollowing the approach in Section 4, we run an MCMC to fit the continuum of GS-NDG-9422. In this case, we do not allow the twophoton continuum to have a free normalization (and hence we remove this parameter), and we replace it with a variable DLA column density and leakage. The model thus has eight free parameters (compared to the seven used in Section 4). \nIn Figure A1 we show the fit corresponding to the 50th percentile model in the posterior distributions of each parameter as well as the marginalized posterior distributions on H I column density and leakage. Indeed we find values close to 10 23 . 3 cm -2 , which is slightly lower than reported in Section 3, but very consistent within 1 𝜎 . In either case, the DLA column required is exceptionally high compared to known DLAs. \nInterestingly, despite the fact that Ly 𝛼 was not included in the fitting, the MCMC tightly constrains the amount of leakage to be ∼ 35%. This is because the shape of the transmission curve for the DLAis inconsistent with the shape of the UV downturn in GS-NDG9422, while in contrast, it is much more consistent with that of the two-photon continuum. Nevertheless, it is encouraging to see that the MCMC requires the same leakage as discussed in Section 3. \nWhile this DLA model is able to reproduce the shape of the continuum, we find two inconsistencies. First, the 50th percentile temperature of 22,369 K is much higher than that measured by the [O /i.pc/i.pc/i.pc] 𝜆 4363 auroral line. As stated above, these two temperatures do not necessarily have to agree, but are typically observed to be close (Guseva et al. 2006, 2007). Moreover, in this model, the predicted H 𝛽 EWis 767 Å ± 40 Å which is very inconsistent with what is observed. This is in stark contrast to the model where the UV is dominated by two-photon emission. Hence the results in this section confirm and strengthen our conclusions that the spectrum of GS-NDG-9422 is more consistent with being primarily nebular in origin.', 'APPENDIX B: MODELS WITH XRBS': 'During the course of the modelling presented in Section 5, we also considered the possibility that the emission is powered by X-ray binaries. Following Senchyna et al. (2020) and Katz et al. (2023b), we model the X-ray sources using a modified blackbody spectrum (Mitsuda et al. 1984). We assume black hole masses in the range 6 -25 M ⊙ and disk radii between 10 3 -10 4 gravitational radii. In order for a UV turnover to appear, the ionizing photon output from the XRBs must dominate over the stellar population so that the nebular continuum can outshine the stars, implying a very high ratio of X-ray luminosity to star formation rate. In Figure B1, we show our XRB model for 25 M ⊙ black holes optimised to reproduce the continuum shape of GS-NDG-9422. This model significantly over-predicts the strength of the He /i.pc/i.pc 𝜆 1640 and 𝜆 4686 lines. Varying the black hole masses does not resolve this issue. We therefore conclude that XRBs are not the dominant ionizing source in GS-NDG-9422. \n6 \nFigure A1. Top: Best-fit continuum model from fitting described in Appendix A. Young stellar component + DLA is shown in pink, nebular component with a fixed low two-photon continuum contribution is shown in brown, while the old stellar component is shown in light blue. Middle: Resulting posterior distribution of neutral hydrogen column density. Bottom: Resulting posterior distribution of leakage fraction. \n<!-- image --> \nFigure B1. Spectrum of GS-NDG-9422 (black) compared with photoionization models that include high-mass X-ray binaries with a black hole mass of 25 M ⊙ (magenta). The solid and dashed magenta lines indicate models with different accretion disk radii. \n<!-- image --> \nThis paper has been typeset from a T E X/L A T E X file prepared by the author.'}
2024ApJ...970...31R	We characterize the earliest galaxy population in the JADES Origins Field the deepest imaging field observed with JWST. We make use of ancillary Hubble Space Telescope optical images five filters spanning 0.40.9 m and novel JWST images with 14 filters spanning 0.85 m including seven mediumband filters and reaching total exposure times of up to 46 hr per filter. We combine all our data at gt2.3 m to construct an ultradeep image reaching as deep as 31.4 AB mag in the stack and 30.331.0 AB mag 5 r 0.1 circular aperture in individual filters. We measure photometric redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts z 11.515. These objects show compact halflight radii of R SUB12SUB 50200 pc stellar masses of M SUBSUB 10SUP7SUP10SUP8 SUP M SUBSUB and star formation rates 0.11 M SUBSUB yrSUP1SUP. Our search finds no candidates at 15 lt z lt 20 placing upper limits at these redshifts. We develop a forwardmodeling approach to infer the properties of the evolving luminosity function without binning in redshift or luminosity that marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the impact of nondetections. We find a z 12 luminosity function in good agreement with prior results and that the luminosity function normalization and UV luminosity density decline by a factor of 2.5 from z 12 to z 14. We discuss the possible implications of our results in the context of theoretical models for evolution of the dark matter halo mass function.	2024-07-01T00:00:00Z	['10.3847/1538-4357/ad463d', '2024ApJ...970...31R', 'arXiv:2312.10033', '10.48550/arXiv.2312.10033', '2023arXiv231210033R']	['Early universe', 'Galaxy formation', 'Galaxy evolution', 'High-redshift galaxies', 'Reionization', '435', '595', '594', '734', '1383', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	Earliest Galaxies in the JADES Origins Field Luminosity Function and Cosmic Star Formation Rate Density 300 Myr after the Big Bang	2,024	191	0.69	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	58	https://arxiv.org/pdf/2312.10033.pdf	{'Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic Star-Formation Rate Density 300 Myr after the Big Bang': "Brant Robertson, 1 Benjamin D. Johnson, 2 Sandro Tacchella, 3, 4 Daniel J. Eisenstein, 2 Kevin Hainline, 5 Santiago Arribas , 6 William M. Baker, 3, 4 Andrew J. Bunker, 7 Stefano Carniani, 8 Courtney Carreira, 1 Phillip A. Cargile, 2 Stephane Charlot, 9 Jacopo Chevallard, 7 Mirko Curti, 10 Emma Curtis-Lake, 11 Francesco D'Eugenio, 3, 4 Eiichi Egami, 5 Ryan Hausen, 12 Jakob M. Helton, 5 Peter Jakobsen, 13, 14 Zhiyuan Ji, 5 Gareth C. Jones, 7 Roberto Maiolino, 3, 4, 15 Michael V. Maseda, 16 Erica Nelson, 17 Pablo G. P'erez-Gonz'alez, 6 D'avid Pusk'as, 3, 4 Marcia Rieke, 5 Renske Smit, 18 Fengwu Sun, 5 Hannah Ubler, 3, 4 Lily Whitler, 5 Christina C. Williams, 19 Christopher N. A. Willmer, 5 Chris Willott, 20 and Joris Witstok 3, 4 \n1 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 96054, USA 2 Center for Astrophysics \| Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138 USA \n- 3 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\n- 4 Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK \n5 \nSteward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA \nCentro de Astrobiolog'ıa (CAB), CSIC-INTA, Cra. de Ajalvir Km. 4, 28850- Torrej'on de Ardoz, Madrid, Spain \n6 \n- 7 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK 8 Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy\n- 9 Sorbonne Universit'e, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France 10 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany \n11 Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 9AB, UK \n12 Department of Physics and Astronomy, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218 13 Cosmic Dawn Center (DAWN), Copenhagen, Denmark \n- 14 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200, Copenhagen, Denmark\n- 15 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK\n- 16\n- NSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA 20 NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada \nDepartment of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706 USA 17 Department for Astrophysical and Planetary Science, University of Colorado, Boulder, CO 80309, USA 18 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK 19", 'ABSTRACT': "We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters spanning 0 . 4 -0 . 9 µ m) and novel JWST images with 14 filters spanning 0 . 8 -5 µ m, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data at > 2 . 3 µ m to construct an ultradeep image, reaching as deep as ≈ 31 . 4 AB mag in the stack and 30.3-31.0 AB mag (5 σ , r = 0 . 1' circular aperture) in individual filters. We measure photometric redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts z = 11 . 5 -15. These objects show compact half-light radii of R 1 / 2 ∼ 50 -200pc, stellar masses of M ⋆ ∼ 10 7 -10 8 M ⊙ , and star-formation rates of SFR ∼ 0 . 1 -1 M ⊙ yr -1 . Our search finds no candidates at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to infer the properties of the evolving luminosity function without binning in redshift or luminosity that marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results, and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2 . 5 from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical models for evolution of the dark matter halo mass function. \nKeywords: Early universe (435) - Galaxy formation (595) - Galaxy evolution (594) - High-redshift galaxies (734)", '1. INTRODUCTION': "JWST has pushed the forefront of our knowledge of galaxies in the distant universe to the first 350 million years of cosmic time. Within the first weeks of operations, surveys with JWST unveiled galaxy candidates beyond redshift z ∼ 12 in an epoch when only the most optimistic models of the cosmic star formation rate density predicted that galaxies would be easily discoverable (Naidu et al. 2022a; Castellano et al. 2022; Finkelstein et al. 2023a; Adams et al. 2023a; Atek et al. 2023; Donnan et al. 2023b; Harikane et al. 2023b; Morishita & Stiavelli 2023; Bouwens et al. 2023). The identification and spectroscopic confirmation by the JWST Advanced Deep Extragalactic Survey (JADES; PIs Rieke and Lutzgendorf; Eisenstein et al. 2023a) of the galaxies JADES-GS-z12-0 at z = 12 . 6 and JADES-GS-z13-0 at z = 13 . 2 affirmatively established for the first time the reality of galaxies at z > 12 (Curtis-Lake et al. 2023; Robertson et al. 2023; D'Eugenio et al. 2023). Subsequently, other galaxy candidates have been confirmed at z ∼ 12 -13 in other surveys (Fujimoto et al. 2023; Wang et al. 2023a) and many additional high-redshift candidates identified photometrically (e.g., Hainline et al. 2023a; P'erez-Gonz'alez et al. 2023a; Leung et al. 2023). \nThe discovery of these distant sources raises substantial questions about the nature of galaxy formation in the early universe (Ferrara et al. 2023; Mason et al. 2023; Dekel et al. 2023; Li et al. 2023; Lovell et al. 2023; Shen et al. 2023a; Yung et al. 2024). The earliest known galaxies appear relatively bright (e.g., Naidu et al. 2022a; Castellano et al. 2022; Treu et al. 2023; Finkelstein et al. 2023b), show a range of stellar masses M ⋆ ∼ 10 7 -10 9 M ⊙ , and have young stellar ages of t ⋆ ∼ 10 7 -10 8 yr (Robertson et al. 2023). Structurally, these galaxies show physical sizes of r ∼ 0 . 1 -1 kpc and star formation rate surface densities of ˙ Σ ⋆ ∼ 50 -100 M ⊙ yr -1 kpc -2 (Robertson et al. 2023; Arrabal Haro et al. 2023; Wang et al. 2023a). They are compact star forming galaxies undergoing rapid star formation on a timescale comparable to their local dynamical times. Individually, the properties of these objects are not extreme given the densities and dynamics of the early universe. Collectively, the apparent, albeit uncertain, abundance of such objects in the context of structure formation may be unexpectedly high. Resolving this essential quandary requires statistical constraints on the abundance of z > 12 galaxies and information on their possible origins through higher-redshift searches. \nTo answer these questions, this work presents first results on the search for distant galaxies in the JADES Origin Field (JOF; Program ID 3215, PIs Eisenstein and Maiolino; Eisenstein et al. 2023b). The JOF observations were designed to use JWST medium bands, including NIRCam F162M, to isolate the Lymanα break at z ≳ 12 and simultaneously control for contamination by lower-redshift line emitters that can mimic the broad-band spectral energy distributions (SEDs) of distant galaxies (Naidu et al. 2022b; Zavala et al. 2023; Arrabal Haro et al. 2023; P'erez-Gonz'alez et al. 2023b). In concert with ultra-deep broad-band observations from JADES, the 9 . 05 arcmin 2 JOF provides the best current dataset for finding and characterizing z ≳ 12 galaxies. We search the JOF for objects to an effective limiting depth of f ν ∼ 2 -3 nJy, performing SED fitting analyses to select the highest redshift candidates. We then use a forward-modeling approach to infer the character of the evolving luminosity function given the properties of our sample of high-redshift candidate galaxies. Our method accounts for the photometric redshift posterior constraints of our sample's galaxies without binning in redshift or luminosity. We employ our method to study the behavior of the evolving luminosity function beyond z ∼ 12 and the abundance of galaxies at earlier times. \nThis paper is organized as follows. In § 2 we review the JOF data, the observations, data reduction procedure, source detection, and photometry. In § 3, we describe our selection procedure based on SED template fitting. Forward modeling constraints on the galaxy candidate structural properties and inference of the distant stellar population properties are described in § 4. We characterize the galaxy luminosity functions at z ∼ 12 -15 and our constraints on the UV luminosity density at z ∼ 12 -20 in § 5, and report the inferred physical properties of the high-redshift candidates in § 6. We interpret the observational results in the context of galaxy formation theory in § 7. We summarize our conclusions and preview future work in § 8. Throughout this work, we use the AB magnitude system (Oke & Gunn 1983) and assume a flat Lambda cold dark matter (ΛCDM) cosmology with Ω m = 0 . 3 and H 0 = 70 km s -1 Mpc -1 .", '2. DATA': 'This work uses JWST observations in the JOF to discover and constrain the abundance and properties of z > 12 galaxies. In § 2.1 we review the JOF and accompanying JADES and Hubble Space Telescope (HST) \nobservations. In § 2.2, we present the data reduction methods used to process the imagery. The detection and photometric methods used to discover the objects are described in § 2.3.', '2.1. Observations': 'Eisenstein et al. (2023b) presents the JOF, a single JWST NIRCam pointing of exceptional depth, with about 7 days of exposure time spread between 14 filters covering an A ∼ 9 arcmin 2 area. The JOF began with the parallel imaging of deep JADES spectroscopy (Program ID 1210, presented in Bunker et al. 2023) that produced long F090W, F115W, F150W, F200W, F277W, F335M, F356W, F410M, and F444W exposures in a field adjacent to the Hubble Ultra Deep Field within the GOODS-S field. This campaign continued in Cycle 2 Program ID 3215, which observed in 6 JWST NIRCam medium bands-F162M, F182M, F210M, F250M, F300M, and F335M-again acquired in parallel to deep NIRSpec observations. We also include all JADES GOODS-S medium-depth imaging (Program ID 1180) that overlaps with the JOF. This area of GOODS-S partially overlaps with the FRESCO (Program ID 1895) F182M, F210M, and F444W data, which we incorporate. The field also has partial coverage of HST ACS F435W, F606W, F775W, F814W and F850LP images reduced and released through the Hubble Legacy Field program (Illingworth et al. 2016a) reductions of the Great Observatories Origins Deep Survey (GOODS; Giavalisco et al. 2004) and Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011) images. In total, these data provide nineteen JWST and HST photometric bands that we use to constrain the galaxy SEDs and particularly the Lymanα break.', '2.2. Data reduction': "Our image reduction methods were outlined in Rieke et al. (2023) and Eisenstein et al. (2023a), detailed in Tacchella et al., (in prep), and we provide a summary here. We process the images with the jwst Calibration Pipeline (version 1.11.4) and Calibration Reference Data System pipeline mapping (CRDS pmap) 1130, which includes in-flight NIRCam dark, distortion, bad pixel mask, read noise, superbias and flat reference files. \nWe use jwst Stage 1 to perform the detectorlevel corrections and ramp fitting. We run this stage with the default parameters, except for the correction of 'snowball' artifacts from cosmic rays. The identification and correction of snowballs represent a big challenge. Heuristically, we find that the following parameters provide reasonable snowball amelioration: max jump to flag neighbors = \n1, min jump to flag neighbors = 100000, min jump area = 5, min sat area = 1, expand factor = 2, min sat radius extend = 2 . 5, and max extended radius = 200. \nAs detailed in Rieke et al. (2023), we run jwst Stage 2 with the default parameters, but replace the STScI flats for all LW bands except F250M and F300M with custom super-sky flats. When we do not have sufficient images to produce a robust flat field, we interpolated the flatfield images from the bands adjacent in wavelength. Following Stage 2, we perform custom corrections for all additive effects including 1 /f noise, scattered light effects ('wisps' and 'claws'), and the large-scale background. Furthermore, we also updated the DQ data quality array to mask additional features imprinted visually onto the mosaics, including persistence, uncorrected wisp features, and unflagged hot pixels. \nBefore running jwst Stage 3, we perform astrometric registration to Gaia DR2 (G. Brammer priv. comm., Gaia Collaboration et al. 2018) with a modified jwst -pipeline tweakreg code. We apply both a rotation and offset to the individual level-2 images. For images taken in the A module with the medium bands F182M, F210M, and F335M, we replace the default distortion maps with the nearest (in effective wavelength) wideband distortion map for that detector. \nWe construct the mosaics using jwst Stage 3. We create single mosaics for each filter by combining exposures from all observations, and run jwst Stage 3 with the default parameter values while setting the pixel scale to 0.03 '/pixel and a drizzle parameter of pixfrac = 1 for the SW and LW images. Finally, we perform a custom background subtraction, following the procedure outlined in (Bagley et al. 2023a). For F090W, F115W, and F150W, hot pixels that pass median rejection are replaced with median filtered values from the local flux image.", '2.3. Detection and Photometry': "The detection and photometry methods are discussed in Rieke et al. (2023) and Eisenstein et al. (2023a) and will be detailed in Robertson et al. (in prep). \nTo perform source detection, an inverse varianceweighted stack of the long-wavelength NIRCam F250M, F277W, F300M, F335M, F356W, F410M, and F444W SCI and ERR channels are constructed. Small-scale noise residuals from incomplete masking in the jwst pipeline are median filtered from the ERR images. The signal-to-noise ratio (SNR) image created from the ratio of these images is used as the detection image. An initial source significance threshold of SNR > 1 . 5 is used to select regions of interest, and a series of custom compu- \ntational morphology algorithms inspired by NoiseChisel (Akhlaghi & Ichikawa 2015; Akhlaghi 2019) are applied to refine the segmentations. Stars and diffraction spikes are masked by constructing segmentations from stacks of all available filters and integrated into the detection segmentation map. The detection image segmentations are deblended using a logarithmic scaling of the F200W image. High-pass filtering is applied to the outer regions of large segmentations to isolate proximate satellite galaxies. After these refinements of the segmentation map, a final pass to detect potentially missed compact, faint sources is applied. The completeness as a function of flux and size for this detection algorithm has been calculated using source injection simulations and is presented in Section 4.1. \nAfter engineering the segmentation map, we perform a set of customized photometric measurements based on the photutils (Bradley et al. 2023) analysis package. Object centroids are computed using the 'windowed positions' used by Source Extractor (Bertin & Arnouts 1996). Apertures for measuring Kron (1980) fluxes are determined based on the stacked signal image (the numerator of the SNR detection image) using a Kron parameter of 2 . 5. We limit the area of the Kron aperture to be less than twice an object's segmentation area. In addition to Kron fluxes, we measure circular aperture photometry with aperture radii of r = { 0 . 1' , 0 . 15' , 0 . 25' , 0 . 3' , 0 . 35' , 0 . 5' } . To provide aperture corrections, we produce a model point spread function (mPSF) following the method of Ji et al. (2023), where we inject WebbPSF models into jwst level-2 images and mosaic them using the same exposure pattern as the JOF observations to provide a composite star field. An mPSF for each band (and observing program) is then constructed from these PSF-mosaics. The circular aperture corrections are measured and tabulated, and the Kron aperture corrections computed by integrating within the corresponding elliptical apertures placed on the mPSF. For HST, we measure empirical (e)PSFs using the photutils (Bradley et al. 2023) ePSF Builder with visually inspected stars in the field. \nWe perform a bevy of photometric validation tests. Cross validation against the CANDELS survey HST photometry using completely independent HST reductions from the Hubble Legacy Field program are presented in Rieke et al. (2023) for the broader JADES/GOODS-S field. We also compute median photometric offsets from SED templates using EAZY (Brammer et al. 2008), following the method described by Hainline et al. (2023a). We find these zeropoint offsets to be within 5 . 2%, and typically within 1%, for JWST filters. \nWe also forward model each galaxy's surface brightness profile using the Forcepho code (B. Johnson, in prep). We use Forcepho with custom model pointspread functions to model the surface brightness profile of each galaxy in our survey simultaneously with any nearby objects in each individual exposure where pixel covariance is minimized. We restrict the modeling to the F200W and F277W bands, to minimize the chance of any PSF mismatch or astrometric errors while maximizing S/N and resolution. The surface brightness profile is assumed to be a S'ersic (1968) model, with a fast Gaussian-based factorization of the model. Forcepho provides a Bayesian estimate of the surface brightness profile parameters, including the galaxy half-light radius. We have used Forcepho to study the structure of other extremely high-redshift galaxies (e.g., Robertson et al. 2023; Tacchella et al. 2023), and we refer the reader to Baker et al. (2023) for more details on our morphological analysis methods.", '2.4. Image Depths': "With the construction of our broad- and medium-band NIRCam mosaics and the long-wavelength ( λ > 2 . 3 µ m) detection image, we can use the photometry method described in § 2.3 to measure our image depths. In Table 1, we report the median aperture corrected 5 -σ point-source depth in each filter and the stack (using the F277W PSF to estimate the stack's aperture correction). When measuring the depth in each image, we use a dilated version of the segmentation map created by the detection algorithm to mask source pixels. We note that the single-band images depths listed in Table 1 are all within 10 -25% of the 5 -σ point-source depths we reported in Eisenstein et al. (2023b) that were computed from the JWST exposure time calculator, with the longest wavelength filters showing the most improved depth. Our single-band images reach 30.3 -31.0 AB, and the combined λ > 2 . 3 µ m stack reaches 31.4 AB depth. For comparison, we also list the 5 -σ pointsource depth the corresponding λ > 2 . 3 µ m stacks from available NIRCam long-wavelength images in NGDEEP (F277W+F356W+F444W; Bagley et al. 2023b), the MIRI-UDF NIRCam parallel (F277W+F356W; P'erezGonz'alez et al. 2023a), and the JADES GOODS-S Deep region that covers the Hubble Ultra Deep Field (Rieke et al. 2023). To measure their depths, we processed these fields using identical methods and used the same F277W PSF model to aperture correct them. We report depths for each program separately, and note that where the MIRI-UDF parallel and NGDEEP NIRCam \nTable 1. Depths of the JADES Origins Field \nNote -a Median r = 0 . 1 '' aperture corrected 5 σ point-source depth. b This depth reflects our independent processing of the NGDEEP data, and we refer the reader to Bagley et al. (2023b) for their depth measurements. c This depth reflects our independent processing of the MIRI-UDF data, and we refer the reader to P'erez-Gonz'alez et al. (2023a) for their depth measurements. \nimaging overlap the combined depths will be even more sensitive than listed in Table 1.", '3. SELECTION OF REDSHIFT Z ≳ 12 GALAXIES': "The photometric selection of high-redshift galaxies relies on identifying a strong Lymanα break in the restframe UV of a galaxy's SED (e.g., Guhathakurta et al. 1990; Steidel et al. 1995). Below, we detail our selection of z ≳ 12 galaxies based on this feature.", '3.1. Photometric Redshift Estimation': "To infer the photometric redshifts of galaxies in the JOF, we apply the techniques detailed in Hainline et al. (2023a) to fit templates of galaxy SEDs to our JWST and HST photometry, varying the redshift to assess the relative goodness of fit. To perform the SED fits, we use the EAZY code (Brammer et al. 2008) to compute rapidly the photometric redshift posterior distributions for each galaxy in the JOF survey. When fitting SED \ntemplates, we use the template suite described in Hainline et al. (2023a) that includes models with strong line emission and a range of UV continua. The photometric redshifts estimated from fits to these templates were shown to have an outlier fraction (defined as the fraction of sources with \| z phot -z spec \| / (1 + z spec ) > 0 . 15) of f out = 0 . 05 in Rieke et al. (2023), and f out = 0 for 42 sources at z > 8 in Hainline et al. (2023a). A range of potential redshifts z = 0 . 01 -22 in ∆ z = 0 . 01 increments were considered, and for selection, we adopt the use of the redshift corresponding to the minimum χ 2 from the fit, z a . For each nominal redshift, we use the Inoue et al. (2014) model for attenuation from the intergalactic medium (see also Madau 1995). We do not adopt any magnitude priors, we impose an error floor of 5% on the photometry, and allow for negative fluxes. When fitting the SED models to determine a photometric redshift, to maximize signal-to-noise ratios we use aperture-corrected r = 0 . 1' circular aperture fluxes on the native resolution JOF images without convolution to a common PSF, multiplied by the photometric offsets discussed in Section 2.3. We have checked that we obtain comparably high-redshift solutions when using common-PSF Kron aperture photometry with lower SNR, except where noted below. We note that for some objects, the best-fit SED model has Lymanα line emission. This feature arises as an artifact of the optimization process in EAZY that mixes templates with and without Lymanα emission. We do not claim this line emission to be real. The equivalent width of Lymanα is degenerate with the redshift of the break, which can contribute to a photometric redshift offset of ∆ z ≈ 0 . 2 -0 . 4 relative to a spectroscopic redshift. Local attenuation from the galaxy interstellar medium or circumgalactic medium can shift the photometric redshift by a similar amount (e.g., D'Eugenio et al. 2023; Heintz et al. 2023)", '3.2. Selection Criteria': "In the JOF, we apply the following criteria to identify our high-redshift sample. These criteria have been adapted from Hainline et al. (2023a) but further tailored to a 12 ≲ z ≲ 20 selection. We note that these criteria both select objects previously discovered, notably by Hainline et al. (2023a), and identify new objects. We provide the provenance of each object when discussing our samples below. Our selection criteria are: \n- 1. The redshift at the EAZY fit χ 2 minimum must be z a ≥ 11 . 5.\n- 2. Two of F277W, F356W, and F444W JWST NIRcam filters must show > 5 σ detections. \n- 3. All the long-wavelength NIRCam fluxes (F250M, F277W, F300M, F335M, F356W, F410M, F444W) must exceed 1 . 5 σ significance.\n- 4. The redshift posterior distribution must have an integral probability of P ( z a > 11) > 0 . 68, where we take P ( z ) ∝ exp( -χ 2 / 2).\n- 5. The goodness of fit difference between the best high redshift ( z > 11) and low redshift ( z < 7) solutions must satisfy ∆ χ 2 > 4, and for the best fit we require χ 2 < 100 summed over all 19 filters.\n- 6. The flux in F090W and F115W each must be below 2 . 5 σ significance, as we expect no robust detection of flux blueward of the Lymanα break.\n- 7. To avoid objects redder than the typically blue high-redshift objects (e.g., Topping et al. 2023), we require that sources cannot have both (F277WF356W) > 0 . 125 and (F356W-F444W) > 0 . 25.\n- 8. Each object must have F150W, F162M, F182M, F210M, and F277W coverage. This criterion limits our survey area to the F162M JOF footprint.\n- 9. The NIRCam short and long wavelength local exposure time must be within a factor of four, which avoids edge effects from the mosaic pattern.\n- 10. To avoid variable sources, the flux in the NIRCam Medium bands acquired in the second year of JWST operations must not exceed the broad band NIRCam fluxes acquired in the first year by more than 1 σ in all bands simultaneously. In practice, we treat overlapping medium and broad band filters as random samples of the same flux density, and then flag when the difference between such pairs of flux estimates exceeds the quadrature sum of each pair's errors when taken in different epochs.\n- 11. We require that the source not be covered by another galaxy as determined from the segmentation map, which lowers the available area by 22%. The final available area after accounting for foreground sources is approximately A ' = 7 . 06 square arcmin. \nWe note that without the data quality (criteria 910), minimum long-wavelength SNR threshold (criteria 2-3), or color criteria (criteria 6-7), fifteen objects would be selected. However, of these sources, one (JADES+53.05101-27.89787) sits in an oversubtracted area of a distant star diffraction spike and three more are covered by a stray light 'wisp' feature in F162M (JADES+53.08317-27.86572, \nJADES+53.07681-27.86286, and JADES+53.0496427.88605). For a discussion of wisp features in JWST, please see Rigby et al. (e.g., 2023). \nOf the remaining eleven, one fails the minimum SNR threshold (for JADES+53.07385-27.86072, all filters redward of F335M have f ν < 2nJy) and one fails the color criteria (JADES+53.08468-27.86666 is red). In total nine objects pass the selection; these comprise our Main Sample (see Table 2). While we will comment on the additional two interesting objects that nearly satisfy our selection, we do not consider them in our fiducial luminosity function analyses. We call this collection of two objects the 'Auxiliary Sample' at z > 11 . 5. There are also five sources in Hainline et al. (2023a) sample in the vicinity of the JOF with previously reported photometric redshifts z > 11 that are not in our sample. Of these, with the additional JOF data we find four objects to have revised photometric redshifts z < 7 or fail other selection criteria (JADES+53.02700-27.89808, JADES+53.03696-27.89422, JADES+53.0790127.87154, JADES+53.10469-27.86187). The fifth falls in the F162M gap (JADES+53.07076-27.86544; NIRCam ID 176151) and therefore does not reside in our effective area. \nWe note that the F250M SNR criterion fixes the upper redshift limit of our selection. If we remove this criterion and the z > 11 . 5 limit, we find one additional z ∼ 11 . 4 candidate (JADES+53.10131-27.85696) detected in all filters F150W and redder with f ν ≈ 2nJy, excepting F250M which is about 1 . 5 σ low. In other words, we would find no z ≳ 20 candidates by removing the weak F250M detection criterion. \nThe luminosity function analysis discussed below in § 5 enables the accounting of potential contributions to the inferred galaxy abundance from galaxies with maximum-likelihood photometric redshifts below the putative redshift of interest. We identified galaxies with maximum likelihood redshifts z > 8 and P ( z > 12) > 0 . 01 that otherwise satisfy the above selection criteria. There are three such galaxies, which fall in the photometric redshift z ≈ 10 . 5 -11 . 2 range, which will be referred to as the 'Contributing Sample' at z < 11 . 5. Two of these (NIRCam IDs 76035 and 172510) were previously found in Hainline et al. (2023a). A third galaxy, NIRCam ID 64312 with photometric redshift z ≈ 10 . 6 and P ( z > 12) ≈ 0 . 05 from photometry, was subsequently confirmed to lie at slightly lower redshift with P ( z < 12) < 0 . 01 and was not considered further. \nTable 2. High-redshift candidates in the JADES Origins Field \nNote -a The half-light size refers to the intrinsic, PSF-deconvolved size of each source, in milliarcseconds. b Best fit photometric redshift with 16- and 84-percentile uncertainties from the inferred photometric redshift distribution. c The posterior probability density for the photometric redshift of the candidate to lie at redshift z < 7, given the SED fitting method described in § 3.1. d Spectroscopically confirmed at z = 14 . 32 by Carniani et al. (submitted). e Fails red color limit. f Fails minimum SNR criterion. \nTable 3. Aperture-corrected HST/ACS photometry a in r = 0 . 1 '' circular apertures. \nNote -a These photometric measurements were made on the native-resolution Hubble Legacy Field images (Illingworth et al. 2016b). \nFigure 1. F444W/F200W/F090W false color red/green/blue image of the JADES Origin Field (background image; 27.5 arcmin 2 ), the JOF F162M footprint (jade outline) and F356W+F410M+F444W/F200W+F210M/F090W+F115W false color red/green/blue thumbnail images (each 0.86 arcsec 2 ) for z ≳ 12 high-redshift galaxy candidates. The RGB images of the galaxy candidates typically appear to have a green hue in this color space, as they are all detected in the filters used for both the green and red channels, but not the blue channel. Each inset thumbnail lists the best-fit EAZY photometric redshift and the JADES NIRCam ID, and we indicate the shared angular scale of the thumbnails with a scale bar showing 0.2'. Table 2 lists the designations of the objects based on [RA, Dec]. NIRCam ID 183348 was spectroscopically confirmed as JADES-GS-z14-0 by Carniani et al. (submitted) at z = 14 . 32. \n<!-- image --> \nTable 4. Aperture-corrected short-wavelength JWST/NIRCam photometry in r = 0 . 1 '' circular apertures. \ncandidates in our Main Sample, and the two auxiliary objects that have photometric redshifts z > 11 . 5 but that fail data quality or redness cuts.", '3.3. Sample': "Given the selection criteria presented in § 3.2, our entire 11 . 5 < z < 15 sample consists of eight galaxy candidates (our 'Main Sample'). Table 2 lists their designations based on [RA, Dec], the internal JADES NIRCam ID, and the best-fit redshift z a . Five of these objects (ID 16699, 33309, 13731, 11457, and 55733) were previously identified in Hainline et al. (2023a); the other three are new here. We also record galaxy sizes measured from our Forcepho modeling in Table 2. For each object, we provide r = 0 . 1' circular aperture photometry for HST ACS bands in Table 3, and JWST 0 . 1'-radius circular aperture photometry for the NIRCam short-wavelength and long-wavelength filters appear in Tables 4 and 5, respectively. Note the fluxes we report in Tables 3-5 are measured on the native resolution images and are not convolved to a common PSF. \nFigure 1 shows an F444W/F200W/F090W red/green/blue false color mosaic of the JOF region. Of the 27.5 arcmin 2 area shown, about 9.05 arcmin 2 has acceptable F162M coverage. The inset thumbnail images for each galaxy candidate show 0 . 86 arcsec 2 regions with red/green/blue colors provided by \nF356W+F410M+F444W/F200W+F210M/F090W+F115W, along with the best fit redshift z a and the JADES NIRCam ID referenced in Table 2. We plot the eight galaxy \nNext, in order of increasing photometric redshift, we introduce each galaxy candidate with some summary discussion and a figure of the SED fits to the 0 . 1'-radius circular aperture photometry. We show the photometric redshift posterior distribution and best-fit redshift for each and the redshift posterior distribution limited to z < 7, as well as the best-fit SED and most likely low-redshift SEDs. We also show the JWST filter transmission curves and the fourteen JWST filter cutouts for each galaxy.", '3.3.1. JADES+53.09731-27.84714; NIRCam ID 74977': 'Figure 2 shows the best-fit SED for object JADES+53.09731-27.84714 (NIRCam ID 74977). The object is remarkably faint with m AB ≈ 30 . 5 redward of Lymanα , and has a best-fit redshift of z a = 11 . 5. The best low-redshift solution has z low = 2 . 7, but exceeds the observed F115W constraint.', '3.3.2. JADES+53.02618-27.88716; NIRCam ID 16699': "Figure 3 shows the best-fit SED for object JADES+53.02618-27.88716 (NIRCam ID 16699, Hainline et al. 2023a). The best-fit redshift is z a = 11 . 6 for this faint source, which has m AB ≈ 30 . 2 -30 . 5 in the NIRCam long-wavelength channels. The best lowredshift solution has z low = 2 . 6. We note that using the BAGPIPES SED-fitting code (Carnall et al. 2018) to constrain the photometric redshift of this galaxy candi- \nTable 5. Aperture-corrected long-wavelength JWST/NIRCam photometry in r = 0 . 1 '' circular apertures. \nte provides a slightly lower redshift of z ≈ 11 . 3, without Lymanα emission and with still non-zero P ( z > 11).", '3.3.3. JADES+53.04017-27.87603; NIRCam ID 33309': 'Figure 4 shows the best-fit SED for object JADES+53.04017-27.87603 (NIRCam ID 33309, Hainline et al. 2023a). The best-fit SED model has z a = 12 . 1, while the best low-redshift solution has z low = 3 . 2. The source is also remarkably faint, with m AB ≈ 30 . 2 in the NIRCam long-wavelength filters.', '3.3.4. JADES+53.03547-27.90037; NIRCam ID 160071': 'Figure 5 shows the best-fit SED for object JADES+53.03547-27.90037 (NIRcam ID 160071). The flux densities of this object are f ν ≈ 3 . 5 nJy ( m AB ≈ 30). When fitting an SED model, the observed photometry, including the strong break in F150W, constrain the redshift to be z a = 12 . 4. The best solution at low redshift has z low = 3 . 4.', '3.3.5. JADES+53.06475-27.89024; NIRCam ID 13731': 'Figure 6 shows the best-fit SED for object JADES+53.06475-27.89024 (NIRCam ID 13731, Hainline et al. 2023a). The long-wavelength JWST/NIRCam photometry shows m AB ≈ 29 . 5, and constrains the posterior photometric redshift distribution to be peaked strongly near z a = 12 . 9. The best-fit low-redshift solution at z low = 3 . 5 would exceed the F090W, F115W, F150W, and F162M photometry by several standard deviations.', '3.3.6. JADES+53.02868-27.89301; NIRCam ID 11457': 'Figure 7 shows the best-fit SED for object JADES+53.02868-27.89301 (NIRCam ID 11457, Hainline et al. 2023a). The object has NIRCam flux densities redward of the break of f ν ≈ 4 -6 nJy ( m AB ≈ 29 . 5 -29 . 9), that constrain SED models to yield a bestfit redshift z a = 13 . 5. The best-fit low-redshift solution SED at z low = 3 . 6 exceeds the F090W and F150W constraints by several standard deviations. The second peak in the high-redshift p ( z ) at z ≈ 12 . 7 is driven by the marginal (2 . 3 σ ) detection in F162M, which if real would prefer a slightly lower redshift than the mode but still within our selection criteria. However, we caution that the F162M detection, the F182M-F210M color, and the rising SED shape longward of 3 . 5 µ m could indicate a potential low-redshift contaminant not well-modeled by our SED template set. We therefore proceed with caution while including this candidate in our sample.', '3.3.7. JADES+53.07557-27.87268; NIRCam ID 376946': "Figure 8 shows the best-fit SED for object JADES+53.07557-27.87268 (NIRcam ID 376946). This faint ( m AB = 30 . 5) object at redshift z a = 14 . 4 is slightly redder than most of the other candidates. JADES+53.07557-27.87268 displays an unusual SED in that either the F182M and F210M fluxes must be biased low by several sigma to be consistent with the F200W flux, and the high-redshift solution does not match well the observed F182M, F200W, and F210M data. The best solution at low redshift has z low = 3 . 8 with nearly 8% of the EAZY probability, although it overpredicts \nFigure 2. SED model, photometric redshift posterior distributions, and JWST NIRCam image thumbnails for galaxy candidate JADES+53.09731-27.84714 (NIRcam ID 74977). The upper left panel shows the aperture-corrected r = 0 . 1' flux density f ν in nJy of the NIRCam (purple points with 1 σ uncertainties) and HST/ACS (red points with 1 σ uncertainties) photometry for the object, with median photometric offset corrections applied. The best-fit SED is shown in blue, while the best fit low-redshift solution is shown in gray. The synthetic model photometry for both models are shown as open squares, and the JWST NIRCam filter transmission curves are shown as colored regions. The upper right panel shows the posterior distribution of photometric redshifts for the object (blue), the best-fit redshift (vertical dashed line), the photoz posterior if only redshifts z < 7 are considered (light gray), and the best-fit redshifts provided as an annotation, as is the posterior probability density at redshifts below z ∼ 7. The bottom panel shows inverted grayscale thumbnails of the fourteen NIRCam filters in a 0 . 93 × 0 . 93 arcsec 2 region around each object, the stretch applied to each filter scaled with the mean value in the thumbnail. The signal-to-noise ratio of the aperture-corrected r = 0 . 1' circular aperture photometry for each band is noted in the corresponding thumbnail. The JADES NIRCam ID is also provided on the left side of the image. \n<!-- image --> \nthe observed F150W flux. We note that when using the BAGPIPES SED-fitting code (Carnall et al. 2018) with a broad log-uniform prior ( M ⋆ ∈ [10 5 , 10 13 ] M ⊙ ) on stellar mass to constrain the photometric redshift of this galaxy candidate, we find a yet larger low-redshift probability density than with EAZY. The best fit redshift is still z > 14 and most of its photometric redshift posterior probability is at very high-redshift. We also note that this object has the largest increase in the lowredshift probability density when using common-PSF Kron aperture fluxes to fit a photometric redshift, but, given the loss in SNR for this exceedingly faint object, the photometric SED become much noisier.", '3.3.8. JADES+53.08294-27.8556; NIRCam ID 183348': "JADES+53.08294-27.8556 (NIRcam ID 183348) with redshift z = 14 . 4 is the most remarkable object in our sample, with a best-fit SED shown in Figure 10. The object appears relatively bright ( f ν ≈ 30nJy; r = 0 . 1' \nradius aperture) but shows strong break from F210M to F182M and no significant flux at shorter wavelengths. Before the JOF ultradeep JWST/NIRCam medium band data was acquired, based on JADES JWST/NIRCam broadband data Hainline et al. (2023a) first discussed this source with a photometric redshift of z phot = 14 . 51. Owing to the observed brightness of the source and its close proximity to another lower-redshift source, 183348 was rejected from their main sample. Subsequently, Williams et al. (2023) determined a lower photometric redshift z phot = 3 . 38, and found the source was detected by JWST/MIRI at 7 µ m from the SMILES program (PID 1207; PI Rieke). Given the addition of our ultradeep JOF JWST/NIRCam medium band data, we find the photometric redshift posterior of 183348 distribution is sharply peaked at z ∼ 14 . 4. This high redshift peak is now much more strongly favored than low redshift solutions as the new JOF medium band mea- \n20 \nFigure 3. Same as Figure 2, but for galaxy candidate JADES+53.02618-27.88716 (NIRCam ID 16699). \n<!-- image --> \nFigure 4. Same as Figure 2, but for galaxy candidate JADES+53.04017-27.87603 (NIRCam ID 33309). \n<!-- image --> \nFigure 5. Same as Figure 2, but for galaxy candidate JADES+53.03547-27.90037 (NIRCam ID 160071). \n<!-- image --> \nFigure 6. Same as Figure 2, but for galaxy candidate JADES+53.06475-27.89024 (NIRCam ID 13731). \n<!-- image --> \nFigure 7. Same as Figure 2, but for galaxy candidate JADES+53.02868-27.89301 (NIRCam ID 11457). \n<!-- image --> \nFigure 8. Same as Figure 2, but for galaxy candidate JADES+53.07557-27.87268 (NIRCam ID 376946). \n<!-- image --> \nFigure 9. Same as Figure 2, but for galaxy candidate JADES+53.10762-27.86013 (NIRCam ID 55733). \n<!-- image --> \nFigure 10. Same as Figure 2, but for galaxy JADES+53.08294-27.85563 (NIRCam ID 183348). We note this object has been discussed previously in Hainline et al. (2023a) and Williams et al. (2023), and spectroscopically confirmed by Carniari et al. (submitted). The F162M data for this object has been omitted because of data quality issues. \n<!-- image --> \nsurements better constrain the shape and depth of the break at ∼ 1 . 8 µ mwhile placing limits on strong emission lines redward of the break. While low-redshift solutions have low probability, the low-redshift photometric redshift posterior distribution is very sharply peaked near z low = 3 . 4 and requires a very red object with strong emission lines in F200W and F277W. A principal concern regarding 183348 is the close proximity of a neighboring galaxy (NIRCam ID 183349) that has a best-fit photometric redshift of z a ≈ 3 . 4. This alignment obviously supported the previous suspicion that 183348 was also at the lower redshift. However, our analysis of the initial JOF NIRCam medium-band photometry as well as JWST/MIRI photometry (Helton et al., submitted) further supported the higher redshift, and on that basis, the galaxy was selected for spectroscopic followup. Carniani et al. (submitted) present a spectroscopic redshift confirmation of z = 14 . 32, and we refer the reader to that work for a detailed analysis of the properties of this intriguing galaxy. Here, we do compare the properties inferred for this galaxy along with other objects in the Main Sample measured in the same manner. We note that the photometric and spectroscopic redshift distributions are very similar, and our choice to adopt its photometric redshift distribution during the luminosity function inference has little impact on our results. We also note that gravitational lensing by the neighbor is considered by Carniani et al. (submitted), but find the magnification to be small.", '3.3.9. JADES+53.10762-27.86013; NIRCam ID 55733': 'Figure 9 shows the best-fit SED for object JADES+53.10762-27.86013 (NIRCam ID 55733, Hainline et al. 2023a). The galaxy candidate has NIRCam long-wavelength fluxes of m AB ≈ 29 . 9 and a best-fit redshift of z a ≈ 14 . 6. The best low-redshift solution has z low = 3 . 9 with 2% of the EAZY probability, although the corresponding SED model would substantially exceed the observed F150W. We note that this object shows F162M flux at 1 . 1 σ significance, and confirmation of this hint of a signal would negate a possible high-redshift solution.', '3.3.10. Auxiliary Objects': 'We also provide SED fits for the Auxiliary candidates JADES+53.07385-27.86072 (Figure 11), and JADES+53.08468-27.86666. (Figure 12). \nJADES+53.07385-27.86072 (NIRcam ID 54586) is exceedingly faint and is relegated to our Auxiliary sample by failing the minimum SNR criteria, with some long-wavelength NIRCam filters showing m AB > 30 . 5 flux levels. The high-redshift posterior distribution for \nthis object is correspondingly broader, with a peak at z a = 13 . 1. The best low redshift solution has z low = 3 . 6. Finally, JADES+53.08468-27.86666 (NIRCam ID 44962, Hainline et al. 2023a) is in our Auxiliary sample owing to its red SED that increases from f ν ≈ 3 nJy in F182M to f ν ≈ 6nJy in F444W. The redshift posterior distribution is double-valued, with a peak at z a = 12 . 9. The best low-redshift solution has z low = 3 . 5.', '4. COMPLETENESS SIMULATIONS': 'The detection and selection of high-redshift galaxy candidates impose limitations that reduce the completeness of a sample. To convert the number of observed galaxies satisfying the selection criteria into a measurement of the galaxy number density, the completeness of the detection and selection process can be computed and incorporated. Below, in § 4.1 we use simulations to characterize our detection completeness and in § 4.2 we simulate our selection completeness. These calculations are used in § 5 to include completeness corrections in the rest-UV luminosity function. \nWe note that the requirement to compute the completeness suggests that the detection and selection process should be algorithmic and automatable. We therefore do not apply any cuts based on visual inspection or judgment beyond crafting the detection method described in § 2.3 or the selection criteria presented in § 3. This restriction allows us to simulate both the detection and selection completeness.', '4.1. Detection Completeness': "To compute the detection completeness of our photometric pipeline, we performed detailed source injection simulations using a wide range of input sources. First, we create a mock input galaxy catalog by drawing from randomized distributions of galaxy physical properties including redshift, star formation rate, stellar mass, size, S'ersic (1968) surface brightness profile index, position angle, and axis ratio. The objects are selected to have properties comparable to the z > 8 sources reported by Hainline et al. (2023a). We use the Prospector code (Johnson et al. 2021) to compute the object fluxes given their physical properties and redshift. With this mock catalog, we use the GalSim (Rowe et al. 2015) image simulation software to create simulated S'ersic (1968) profile objects distributed across a grid on the sky. We compute the overlap of the JOF mosaics in each filter with this grid of objects, and then add the randomized objects as injected sources in the JOF images. The result is a large set of synthetic JOF mosaics with injected sources. We can then process the images identically to the real data and attempt to discover sources. \nFigure 11. Same as Figure 2, but for Auxiliary galaxy candidate JADES+53.07385-27.86072 (NIRCam ID 54586). \n<!-- image --> \nFigure 12. Same as Figure 2, but for Auxiliary galaxy candidate JADES+53.08468-27.86666 (NIRCam ID 44962). \n<!-- image --> \nWith the injected images, we combine the longwavelength NIRCam images as for the real data, creating an ultradeep stack. Our pipeline detection algorithm is applied to the injected mosaic stack to create a new detection catalog with simulated sources. We can then characterize the completeness of our detection method as a function of the source properties. We repeat the simulations with ten separate realizations, such that a total of 115,000 injected sources with widely-ranging intrinsic properties are used. \nFigure 13 shows the detection completeness as a function of the two main factors affecting this completeness. The apparent brightness of the objects influence their signal-to-noise ratio in the stacked detection image. The size of the object affects the surface brightness, which in turn determines the per pixel SNR that governs the contrast an object of a given luminosity relative to the sky background. The detection algorithm reaches 90% completeness at around m AB ∼ 30 . 2 for small objects ( R 1 / 2 ≲ 0 . 1 arcsec). This completeness function can be integrated into an interpolator to allow for the object completeness as a function of apparent magnitude and size to be utilized in inferring the UV luminosity function. We note that through this simulation for the JOF we find that only about 78% of the pixels are not impacted by foreground objects, which we account for in computing our effective survey volume. Given that the objects of interest are small, only several pixels across, and our detection method reaches fairly low significance (SNR ∼ 1 . 5) per pixel such that the segmentations reach low surface brightnesses, we find this number to be representative of the impact of foreground sources on our detection completeness.", '4.2. Selection Completeness': 'To simulate the selection completeness, we can use the spectral energy distributions in our mock galaxy catalog and the photometric uncertainty measured for our galaxy sample to simulate the effects of photometric noise on our selection and consequently the inferred UV luminosity function. We create a sample of two million mock galaxies with model SEDs, induce photometric noise with a normal scatter in each HST and JWST filter of the magnitude of our measured sky background. Our measurement uncertainties are sky-dominated, so only include sky noise in our simulated fluxes. These two million noisy simulated SEDs are then provided to EAZY exactly in the same manner as our real catalog, and SED fitting is performed to each object. This enables us to estimate how the photometric noise can disrupt the mapping between true redshift and photometric redshift, and identify which redshift windows could provide non- \nFigure 13. Detection completeness in our JOF analysis as a function of intrinsic half-light radius and F277W apparent magnitude. The detection method is complete for small objects and bright magnitudes, and the differential completeness reaches about 90% at F277W ≈ 30 . 2AB for small objects. Shown is a two-dimensional normalized histogram of object size and flux indicating the fraction of sources with such properties detected by the pipeline. The method becomes highly incomplete fainter than m AB ∼ 31 or for halflight radii above about half an arcsec. Owing to pixels covered by foreground sources, the maximum detection completeness will be reduced to ∼ 78% of that shown here. \n<!-- image --> \nnegligible contamination for our selection criteria. For reference, we note that in our simulations, the fraction of objects with F200W SNR > 5 that are photometric redshift outliers with ( \| z a -z true \| / (1 + z true )) > 0 . 1 is 3.8%. \nFigure 14 shows the completeness of selection criteria as applied to our mock galaxy catalog, as a function of the true object redshift and absolute UV magnitude. The selection proves highly complete at M UV < -18 for redshifts z ≳ 12. At magnitudes fainter than M UV > -17 . 5, the photometric noise prevents the strict elimination of low-redshift solutions such that the objects fail the ∆ χ 2 selection described in 3. At the high-redshift end, the selection declines at z ≈ 20 when the Lymanα break affects F250M and our SNR requirement in that filter becomes limiting. As with the detection completeness, an interpolator can be constructed from the selection completeness and then used to correct the galaxy number counts for the lossy selection process. We note here that we define M UV as the rest-frame 1500 ˚ A UV luminosity density, computed by fitting a power-law to rest-frame UV photometry and marginalizing over any covariance with the spectral slope (for more details, see § 6.1). \nFigure 14. Completeness of our selection criteria as a function of galaxy redshift and absolute magnitude. For bright objects, the selection criteria described in § 3 produce a substantially complete sample. For fainter objects, the ∆ χ 2 criterion fails as the photometric noise prevents the SED fitting procedure from distinguishing robustly between high and low photometric redshifts. \n<!-- image -->', '5. REST-FRAME UV LUMINOSITY FUNCTION AT Z ≳ 12': 'To compute the rest-frame UV luminosity function from our sample of galaxy candidates and our completeness calculations, we can construct multiple measures of the galaxy abundance. We wish to account for several confounding effects. \nFirst, galaxies with a range of intrinsic redshifts will contribute to the the observed number counts of galaxies at a given photometric redshift. The degree of this contamination will depend on the abundance of galaxies at other proximate intrinsic redshifts whose photometric redshifts overlap with the epoch of our measurement. We must therefore account for the evolving luminosity function and mixing between populations at different redshifts. \nSecond, each individual galaxy has a posterior distribution for its photometric redshift. Rather than assign each galaxy to a specific redshift bin and absolute magnitude, we can allow for a posterior distribution on the photometric redshift to represent a track of inferred absolute magnitude and redshift. Each galaxy can make a fractional contribution to the UV luminosity function at redshifts where its posterior has support. \nGiven these considerations, we want to allow for flexibility in our representation of the UV luminosity function. We can either infer a parameterized luminosity function by computing the likelihood of observing each \ngalaxy, given the evolving distribution of galaxy counts with luminosity and redshift, fully without binning, or we could bin in magnitude and redshift but account for the photometric redshift posterior distributions of each object. In either case, with the known individual properties of each object, we want to treat the completeness of our detection and selection methods at the per-object level rather than through binning. Below, we present both methods, where we expand on the methods used by Leja et al. (2020) to infer the evolving stellar mass function at low redshift but now applied to the UV luminosity function evolution at high redshifts. We have tested both methods using mock galaxy samples constructed from specified luminosity functions and posterior photometric redshift distributions.', '5.1. Inferring Evolving Luminosity Function Parameters': 'The probability of observing an object with a given true luminosity and redshift is given by the product of the redshift dependent luminosity function Φ( L, z \| θ ), the selection function S ( L, z ), and the differential comoving volume element probed V ( z ). We can assume the luminosity function depends on some parameters θ . Unfortunately, we do not know the true luminosity and redshift of each galaxy i , but instead estimate it from photometric data D i , by using SED models to construct the likelihood function L ( D i \| L, z ). The likelihood of observing a galaxy with D i must then be marginalized over the unknown true parameters \nL ( D i \| θ ) ∝ ∫ dL ∫ dz L ( D i \| L, z ) λ ( L, z \| θ ) (1) \nλ ( L, z \| θ ) = Φ( L, z \| θ ) S ( L, z ) V ( z ) (2) \nΦ( L, z \| θ ) = ϕ ( z ) ( L/L ∗ ( z )) α ( z ) e -L L ∗ ( z ) (3) \nHere λ ( L, z ) is the differential number of objects expected to be selected from the survey, as a function of the true L and z . We have parameterized the luminosity function as a single Schechter function. The redshift evolution of the luminosity function can be treated with a dependence of the parameters on ( z -z ref ) where z ref is some reference redshift, e.g. the midpoint of the redshift range of interest. For our purposes, we will adopt either simple log-linear or log-exponential evolution with redshift. To compute the likelihood of each object marginalized over the true object redshift and luminosity we numerically integrate the marginalization integrals using samples from the probability distribution provided by EAZY. \nL ( D i \| θ ) ∼ ∑ j w i,j λ ( L i,j , z i,j \| θ ) / ∑ j w i,j (4) \nBy drawing fair samples from the probability distributions provided by EAZY, and noting that the effective \npriors on z and L were uniform, each sample has equal weight w i,j . With the ability to compute the likelihood of each object given the model, the likelihood for an ensemble of objects is then the product of the individual likelihoods. However, we must include the overall constraint given by the number of observed objects. The total expected number of selected objects is given by the integral of the product of the luminosity function and the effective volume, and the observational constraint is given by the Poisson likelihood of the actual number of observed objects 1 \nL ( D \| θ ) = e -N θ ∏ i L ( D i \| θ ) (5) \nN θ = ∫ dL ∫ dz λ ( L, z ) (6) \nHere N θ is the total number of observed objects. Note that for redshifts and luminosities for which our observations are complete, the method accounts for the likelihood of non-detections given the chosen luminosity function parameter values.', '5.2. Estimating a Step-Wise Luminosity Function': 'While the method in § 5.1 does not bin in redshift or luminosity, the observed candidate galaxies could be assigned to specific redshift and luminosity bins. If nothing else, binning allows for the measured galaxy abundance to be usefully plotted and compared with other measurements. The binned luminosity function summarizes the information retained by the unbinned parameterized LF for which representing constraints on the galaxy abundance requires access to samples of the posterior distribution of LF parameters. \nConsider the photometric redshift posterior distribution p i ( z ) of a candidate galaxy i with observed apparent magnitude m i . In the absence of photometric noise, the absolute magnitude of the object is M i = m i -DM ( z ), where DM ( z ) is the cosmological distance modulus including K -correction. Accounting for photometric noise, we will instead have some distribution of absolute magnitudes p ( M i \| m i , z ) for each object at a given photometric redshift. The distribution of inferred absolute magnitudes in some redshift bin z 1 to z 2 is \np ( M i \| z 1 , z 2 ) = ∫ z 2 z 1 dz ∫ dm i p ( M i \| m i , z ) p ( z ) . (7) \nThe contribution of a galaxy to an absolute magnitude bin would then be \nN i ( M 1 , M 2 , z 1 , z 2 ) = ∫ M 2 M 1 p ( M i \| z 1 , z 2 ) dM i . (8) \nThe total number density per magnitude n j in a magnitude bin M 1 < M j < M 2 would then be \nn j ( M 1 , M 2 , z 1 , z 2 ) = ∑ i N i ( M 1 , M 2 , z 1 , z 2 ) ( M 2 -M 1 ) V j (9) \nwhere V j is the average effective volume in the bin, allowing for the completeness to vary for each object i . In practice, evaluating these equations involves summing over samples from the photometric posterior distributions of the galaxies while accounting for samples that lie outside the redshift bin to enforce the posterior normalization constraint ∫ p ( z ) dz = 1. We note that when computing the samples in M UV and z , to compute M UV we use the 1500 ˚ A rest-frame flux computed in the appropriate JWST filter given a putative redshift z . When computing M UV , we use the total fluxes computed from the Forcepho morphological decompositions. \nProcedurally, for each redshift bin we take all ordered M UV samples and separate them into bins whose edges are set to maintain a comparable number of samples per bin. We sum the number of samples in each bin and divide by the total number of samples across all galaxies, which provides the (non-integer) number of galaxies per bin. The average completeness in the bin is computed from the per-object selection and detection completeness based on the object properties and the fraction of pixels in the image not covered by foreground sources. We then divide the number of galaxies in each bin by the bin width, the completeness, and the volume to get the number density. The uncertainties for each bin are estimated from number count statistics. \nWhile we report our step-wise estimate, which accounts for photometric scatter between magnitude bins and variable completeness, we consider these measurements estimated checks on the inferred LF constraints described in § 5.1 that do not bin in either redshift or magnitude and additionally account for potential contamination from proximate redshifts and the evolving shape of the luminosity function with redshift. We emphasize here that our formal derived constraints on the luminosity function are provided through our inference procedure in the form of the computed posterior distributions of the parameters of our model evolving luminosity functions.', '5.3. Luminosity Function Constraints': "Given the measured properties of our sample galaxies, their photometric redshift distributions p ( z ), and the method described in § 5.1, we can compute marginalized constraints of an evolving UV luminosity function once we adopt a parameterized form. \nTable 6. Step-wise Luminosity Function. \nNote -The ranges listed for each M UV reflect the widths of the magnitude bins, which are determined by the distribution of photometric redshift posterior samples for the objects contributing to each bin. \nTable 7. Luminosity Function Marginalized Parameter Constraints \nNote -a The lower limit on the LF normalization is not well constrained, but the 95% upper limit is log 10 ϕ ⋆, 0 < -3 . 84. b The 95% upper limit on the characteristic magnitude is M ⋆ < -19 . 9. c We constrain the evolution parameter to be η < -0 . 08 at 95%. \nFor the luminosity function, we adopt a redshiftdependent Schechter (1976) function \nϕ UV ( M UV , z ) = 0 . 4 log 10 ϕ ⋆ ( z )[10 0 . 4( M ⋆ -M UV ) ] α +1 × exp[ -10 0 . 4( M ⋆ -M UV ) ] (10) \nwhere the redshift-dependent normalization ϕ ⋆ ( z ) can be further parameterized. Our fiducial choice for the normalization evolution is \nlog 10 ϕ l ⋆ ( z ) = log 10 ϕ ⋆, 0 + η ( z -z 0 ) . (11) \nWe will refer to z 0 as the reference redshift, which we will take fixed at z 0 = 12 unless otherwise noted. The default parameters of the model then include ⃗ θ = [ M ⋆ , α, ϕ ⋆, 0 , η ], or the characteristic magnitude M ⋆ , the faint-end slope α , the normalization at the reference redshift ϕ ⋆, 0 , and the log-linear rate of change with redshift η . In practice, we fit in maggies l = -0 . 4 M UV and \nthen convert to absolute magnitudes after inference. We adopt log-uniform priors for ϕ ⋆, 0 and η , a uniform prior in magnitude, and a uniform prior in α . The priors are reported in Table 7, along with our inferred constraints on the parameters. We emphasize again that information from all redshifts where the selection function has non-negligible support is included by our inference procedure, which accounts both for regions of redshift and magnitude space with detections and those absent samples that could have been detected if present. The effective redshift range where our model is informative for the luminosity function is mostly set by the selection completeness (Figure 14), or roughly z ∼ 11 -20. We present the full posterior distributions on the parameters in Figure 15. We here emphasize that the clear covariance between ϕ ⋆ and M ⋆ mostly acts to keep the luminosity density ρ UV ∝ L ⋆ ϕ ⋆ roughly constant at a given redshift. This feature is reflected in our constraints on ρ UV shown in Figure 17. \nSince we constrain the abundance of galaxies at all selected and detectable redshifts and magnitudes simultaneously, evaluating the luminosity function at any one redshift requires computing the marginal distribution of the luminosity function equation 10 over the posterior distribution of parameters for a given redshift and range of absolute magnitudes. At each z and M UV , equation 10 is evaluated for all posterior samples, and the cumulative distribution of ρ UV weighted by the sample weights w k constructed. Figure 16 shows the marginal constraint on the UV luminosity function at redshift z = 12, with the 16-84% of ϕ UV shown as a shaded region and the median ϕ UV shown as a white line. We also show the median inferred ϕ UV at z = 14 as a light gray line. Note that none of these ϕ UV percentiles are guaranteed to follow equation 10 individually, but we do report the marginalized constraints on the luminosity function parameters in Table 7. We also show our stepwise luminosity function estimates computed in redshift bins of 11 . 5 < z < 13 . 5 and 13 . 5 < z < 15. These stepwise luminosity function measures are reported in Table 6. \nIn Figure 16, we also show z ∼ 12 -14 luminosity function determinations reported in the literature. These measurements include the z ∼ 12 data from Harikane et al. (2023b), Harikane et al. (2023a), P'erez-Gonz'alez et al. (2023a) and Willott et al. (2023), the Adams et al. (2023b) constraints at z ∼ 12 . 5, z ∼ 13 measurements from Donnan et al. (2023a) and McLeod et al. (2023), and the z ∼ 14 determinations from Finkelstein et al. (2023a). The median luminosity function constraints inferred from our sample and our forward modeling approach agree with the available observations to within \nFigure 15. Posterior distributions of the evolving luminosity function parameters. Shown are the posterior distributions for the luminosity function normalization log 10 ϕ ⋆ [Mpc -3 mag -1 ], the normalization evolution parameter η , the characteristic magnitude M ⋆ in absolute magnitude, and the faint-end slope α . Contours represent the 68% and 90% enclosed probabilities for each parameter. The marginalized posterior distributions for each parameter are shown at the top of each column, along with the 16%, 50% and 84% marginal constraints (see also Table 7). The lower limits on ϕ ⋆ and M ⋆ are not well constrained, but we constrain at 95% probability that log 10 ϕ ⋆ < -3 . 84 and M ⋆ < -19 . 9. We note that η < 0 with > 95% probability, indicating that we infer a declining luminosity density at z > 12. \n<!-- image --> \nabout 1 σ , excepting the z ∼ 14 constraints from Finkelstein et al. (2023a) that lie above our inference. We note here that the z ∼ 11 luminosity function constraints from Donnan et al. (2023a), McLeod et al. (2023), and Finkelstein et al. (2023a) lie above our 84% inference of the z = 12 luminosity function, and that our selection function (Figure 14) by design removes z ∼ 11 galaxies from our sample. We also emphasize that our results are completely independent of the other data shown in Figure 16.", '5.3.1. Luminosity Density Evolution': "Given the evolving luminosity function parameters inferred given the sample properties, the UV luminosity density evolution ρ UV ( z ) can be computed. Figure 17 presents the marginalized constraints on the UV luminosity density evolution. Shown are 16-84% (jade shaded region) and median ρ UV (white line) integrated to M UV < -17, along with measured (left panel) or extrapolated (right panel) constraints to M UV < -17 from the literature. Our measurements have sensitivity to objects at redshifts 11 ≲ z ≲ 20, and we indicate the luminosity density evolution inferred for the model represented by equations 10 and 11. As the Figure shows, we infer that the UV luminosity density declines at highredshift at a rate of η ≡ d log ϕ ⋆ /dz ≈ -0 . 2 per unit redshift. Between z = 12 and z = 14, we therefore infer that the luminosity density declines by a factor of 10 -0 . 2(14 -12) ≈ 2 . 5. Within our statistical uncertainties, this inference agrees with almost all the literature determinations including Ishigaki et al. (2018), Bouwens et al. (2022), McLeod et al. (2023), Donnan et al. (2023b), Harikane et al. (2023b), Harikane et al. (2023a), Adams et al. (2023b), P'erez-Gonz'alez et al. (2023a), Leung et al. (2023), and Willott et al. (2023). The constraints at z ∼ 11 from Finkelstein et al. (2023a) agree with our results, but their z ∼ 14 point lies above our constraints albeit with large uncertainties. If we extrapolate the UV luminosity evolution inferred by our model, we find good agreement with the literature measurements back to z ∼ 8 (e.g., Ishigaki et al. 2018; Bouwens et al. 2022; P'erez-Gonz'alez et al. 2023a; Willott et al. 2023; Adams et al. 2023b). Also shown in Figure 17 is the corresponding evolution in the cosmic star formation rate density ρ SFR , using the approximate conversion from ρ UV of κ UV = 1 . 15 × 10 -28 M ⊙ yr -1 erg -1 s Hz from Madau & Dickinson (2014). For comparison, we also show the Madau & Dickinson (2014) model for the evolving cosmic star formation rate density.", '5.4. Caveats': 'Of course, with only nine objects at these extreme distances and depths, there are important caveats to con- \nsider about the LF measurement. First, most of our objects are photometric candidates, and despite the closer spacing of the medium bands and our care in selection, we consider it possible that some might be lower redshift interlopers. A Lymanα break at z = 14 falls at the same wavelength as a Balmer break around z ≈ 4. We stress that false positives would likely have a redshift distribution that falls less slowly than the true Lymanα break population, so a population of false positives will typically cause the LF to appear to evolve more shallowly at extreme redshifts. However, the success of our selection method in providing a photometric redshift for 183348 of z = 14 . 32 that was confirmed by Carniani et al. (submitted) provides some evidence that our highest redshift candidates could bear out. \nSince the remaining candidates at z > 13 . 5 have some imperfection in their cases, as discussed in § 3.3, and to illustrate the relative impact of the highest-redshift objects on our inferences, we consider the impact on the LF estimate if we were to ignore the z > 14 objects. Removing these objects makes the inferred evolution of the LF notably steeper, which we show through the UV luminosity density evolution in Figure 17 where the light jade region and gray line report the marginalized 16-84% credibility interval and median ρ UV , respectively. Since the fiducial model assumes an evolution ϕ l ⋆ ( z ) that has a log-linear dependence on redshift, the ρ UV inferred by the model beyond the redshift of our observed sample can in principle be artificially inflated by the inferred trend at z ∼ 12 -14. Instead, when removing the z > 14 objects, we explore a more rapid decline given by \nlog ϕ e ⋆ ( z ) = log( ϕ ⋆, 0 ) × exp[( z -z 0 ) /h ϕ ] . (12) \nThis model enables a log-exponential drop in the galaxy abundance. Indeed, without the z > 14 objects the inferred ρ UV would drop much more rapidly than in the fiducial model based on the Main Sample. For reference, by z ∼ 16 the difference between the two inferences is more than an order of magnitude. Of course, given the small number statistics, we are also sensitive to the impact of a single false negative. If any of the remaining Auxiliary Sample objects in § 3.3 were to prove out, the LF would surely rise.', '5.5. Comparison with Halo Abundance and Large-Scale Structure': "The large-scale structure of the Universe is expected to present a substantial cosmic variance uncertainty given the small size of this field. High-redshift galaxies likely live in rare halos of high mass for their epoch, leading to a large clustering bias and substantial number density fluctuations. To investigate this, we utilize the halo \ncatalog from a cold dark matter simulation performed by the Abacus N-body code as part of the AbacusSummit suite (Garrison et al. 2021; Maksimova et al. 2021). This simulation used 6144 3 particles in a 300 h -1 Mpc box, resulting in a particle mass of 1 . 5 × 10 7 M ⊙ , and a force softening of 21 comoving kpc. Halos were found using the CompaSO algorithm (Hadzhiyska et al. 2021). While this simulation has high accuracy, we caution that the measurement of halo mass always depends on the halo-finding algorithm; we focus here on the relative trends across redshift and on the clustering. \nIn Figure 18, we compare our LF measurements to the cumulative halo mass function as a function of redshift. One sees that if the shallow LF is correct, then matching the abundance of these galaxies to the abundance of the most massive halos would require a strongly evolving halo mass. On the other hand, if one were to discard the objects at z > 14, then the result is more similar to the abundance of a constant mass, roughly of 10 10 M ⊙ . Of course, the galaxies may live in less massive halos, with a scatter between luminosity and mass (e.g., Shen et al. 2023b; Sun et al. 2023); indeed, some scatter is inevitable (Pan & Kravtsov 2023). In what follows, we therefore consider the properties of halos with virial masses of 10 9 . 7 M ⊙ , about 340 particles, which has comparable abundance to our galaxy sample at z ∼ 12 -14. \nWe then calculate the variation within the simulation of regions similar in size to the JOF. We use pencilshaped regions of 6 h -1 Mpc square, roughly 3 ' at z ∼ 12, with a depth appropriate to ∆ z = 1. We find that at z = 12 (11.5-12.5), there are an average of 8.3 halos above our mass threshold in a region, but with a standard deviation of 5.6. At z = 13, this abundance drops to 2 . 3 ± 2 . 3; at z = 14, the abundance drops further to 0 . 7 +1 -0 . 7 . The distribution of halo number counts becomes noticeably skewed, and by z = 14 we find that 6% of regions have ≥ 3 halos. Hence, we find that unless the host halos are much less massive (and their luminosity much more variable), the large-scale structure contributes an error at least as large as the Poisson error. We caution that this uncertainty could impact the observed rate of decline of the UV luminosity density, given our area, and motivates further studies over larger fields. However, to combat other systematics such studies should also leverage the depth and filter coverage comparable to that afforded by the JOF, which is challenging given the necessary exposure time. \nFinally, we note that we have neglected the effect of magnification by gravitational lensing in our inference of M UV . While none of our candidates show obvious lens morphology, the high-luminosity tail of the high-redshift \nFigure 16. UV luminosity at z ∼ 12 inferred from the JADES Origins Field (JOF). Using the method described in § 5.1, we compute the marginalized constraints on the UV luminosity function inferred from galaxies discovered in the JOF with photometric redshift distributions that overlap the redshift range 11 . 5 < z < 13 . 5. We account for photometric scatter, the photometric redshift distribution of each object, the selection completeness for each object, and potential contamination from proximate redshifts. The 16%-84% marginal constraints on the abundance ϕ UV as a function of absolute UV magnitude M UV are shown as a jade-shaded area and the median ϕ UV ( M UV ) is shown as a white line. For comparison, we also compute step-wise luminosity function constraints as described in § 5.2 at z ∼ 12 (solid black points) and at z ∼ 14 (open black circles). These step-wise estimates agree with the inferred ϕ UV , but the continuous constraints represent our results for the UV LF. We also show a variety of constraints from the literature at comparable redshifts (colored points), and note that none of these data were used to aid our inference of the UV LF. \n<!-- image --> \nluminosity function will likely be enhanced by lensing (e.g., Wyithe et al. 2011; Mason et al. 2015; Ferrami & Wyithe 2023), which might affect interpretations of the luminosity function in the context of theories of galaxy formation.", '6. PHYSICAL PROPERTIES OF THE HIGH-REDSHIFT POPULATION': 'Beyond the abundance and UV luminosity of these z ≳ 12 galaxies, the physical properties of the galaxies are of particular interest for understanding the process of galaxy formation at the earliest epochs. With the high-quality space-based optical-infrared photome- \nFigure 17. Evolution of the UV luminosity density ρ UV ( M UV < -17) with redshift derived from the JOF sample. Shown are literature values for ρ UV ( z ) measured (left panel) or extrapolated (right panel) to M UV < -17. In both panels, the shaded jade region shows the 16% and 84% marginal constraints on the luminosity density computed from the posterior samples of the evolving luminosity function inference, as well as the median luminosity density with redshift (white line). These constraints model a linear evolution in log 10 ϕ ⋆ and include a permissive prior on the faint-end slope α . Overall, our constraints agree well with prior literature results even as our inference is completely independent. The dark green lines extending to z ∼ 8 show the low-redshift extrapolation of the inferred ρ UV ( z ) evolution, while the shaded region indicates the redshift range where our detection and selection completeness is non-negligible. We also indicate an approximate cosmic star formation rate density (right axis; M ⊙ yr -1 Mpc -3 ) using the conversion κ UV = 1 . 15 × 10 -28 M ⊙ yr -1 erg -1 s Hz, and show the Madau & Dickinson (2014) model (left panel; dotted line). For comparison, inn the left panel, we show the corresponding constraint if the JOF high-redshift galaxies and candidates at z > 14 are excluded and log 10 ϕ ⋆ is fit with an exponential evolution. In this case, we would infer the light jade region (16%-84% marginal constraint) with gray line (median). \n<!-- image --> \ny available in the JOF, physical properties of the highredshift galaxy stellar populations can be inferred.', '6.1. Rest-frame UV Magnitude and Spectral Slope': 'Given the dramatic distances to these objects, the photometry obtained in the JOF primarily probes only their rest-frame UV spectra. Using common-PSF images and aperture-corrected Kron photometry as a proxy for the total fluxes, we can fit the rest-frame UV photometry with a power law f ν ∝ λ 2+ β and jointly constrain M UV and β given the object redshifts. Figure 19 shows the posterior distribution of M UV and β for the candidate galaxies in our Main Sample at z > 11 . 5. The posterior mean and standard deviation for each parameter are reported in Table 8, and for convenience we also report M UV in Table 2. The maximum likelihood values for the rest-frame spectral slope are -2 ≳ β ≳ -3. These values are comparable to the rest-frame spectral properties of high-redshift photometric samples (e.g., Cullen et al. 2023a; Topping et al. 2023), although not quite blue enough to suggest completely dust-free objects (e.g., Cullen et al. 2023b).', '6.2. Morphology and Size': "As expected, these galaxies show small angular sizes. As described in 2.3.1, we fit single S'ersic profiles to the individual exposures in the F200W and F277W filters, reporting the half-light radii in Table 2. The posterior distributions are often non-Gaussian and asymmetric. Unsurprisingly, most of the objects are small, with halflight radii below 50 mas, excepting the unusual z = 14 . 32 galaxy 183348. \nTo characterize the limiting angular resolution of our images, we have also fit S'ersic profiles to the exposures (separated by epoch of observation) in the same bands for known brown dwarfs of similar flux levels in the JOF and wider GOODS-S areas (Hainline et al. 2023b). As in our past work (Robertson et al. 2023), we find that brown dwarfs in the JADES Deep imaging are recovered with 95% upper limits on sizes of 20 mas in F200W, so we regard objects with 95% lower limits above 20mas as inconsistent with a point source. As such, candidates 16699, 160071, and 55733 are resolved, with half-light angular sizes up to 50 mas and half-light physical sizes of 132, 118, and 142 pc, respectively. The galaxy 183348 \nTable 8. Sample physical properties, assuming best-fit redshift. \nNote -The UV absolute magnitude M UV and rest-frame UV slope β are jointly fit to common-PSF Kron photometry for each object. We report here the mean and standard deviation of other posterior distributions for each parameter. The star formation rates are averaged over the last 10 Myr of the inferred star formation histories. \nFigure 18. Comparison of the inferred evolution of the JOF galaxy number density n ( z ) and the abundance of dark matter halos in cosmological simulations. Shown are the inferred number density constraints (dark jade region 16-84%, white line 50%) for model with a linear evolution in log 10 ϕ ⋆ with redshift z . The grid of gray lines show the abundance of dark matter halos with masses greater than log 10 M ∼ 9 . 4 -11 computed from the AbacusSummit simulation suite (Maksimova et al. 2021). In the inferred JOF n ( z ), if simply matched by abundance the halo mass of the typical galaxy would vary by roughly a factor of ∼ 10. If instead we were to discard the z > 14 objects and fit an exponential evolution to log 10 ϕ ⋆ , the typical galaxy would mostly track a halo mass = log 10 M ∼ 10 (light jade region). For reference, we indicate the extrapolation of the inferred number density constraints to lower redshifts with jade lines. \n<!-- image --> \nspectroscopically-confirmed at z = 14 . 32 by Carniani et \nFigure 19. Posterior distributions of rest-frame UV absolute magnitude M UV and spectral slope β for candidate galaxies in our Main Sample at z > 11 . 5. Shown as kerneldensity-estimated contours are the 68% and 95% credibility intervals on the joint posterior distributions for each object. The maximum likelihood values for the UV spectral slope are -2 ≳ β ≳ -3. \n<!-- image --> \nThe outlier at M UV ≈ -21 is 183348, spectroscopically confirmed at z = 14 . 32 (Carniani et al., submitted). \nal. (submitted) shows a size of 76 mas, or about 240 pc. The remaining sources are consistent with a point source, though many have non-negligible probability of having larger sizes. We note that objects 13731 and 376946 are both constrained to be very small. In addition to the multiband Forcepho fit reported in Table 2, \nindependent single-band Forcepho fits to the 13731 infer its size be less than 10 and 16 mas (95th percentile) in F200W and F277W respectively. While 376946 appears unresolved in F200W and F277W, it appears more extended in some medium band filters. Regardless, the sizes of these objects are small enough that we expect their extents do not impact their detection completeness (e.g., Figure 13). \nThese results are similar to those found in Robertson et al. (2023), where 2 of the 4 z > 10 galaxies were resolved. One consequence of being resolved is that the light from these galaxies cannot be purely from an accreting massive black hole (Tacchella et al. 2023). Other spectroscopically-confirmed galaxies at z > 12 have had size measurements inferred from scene modeling, and show sizes of R 1 / 2 ∼ 100 -300pc (e.g., Wang et al. 2023b). Collectively, these results indicate that compact sizes are a common property of many high-redshift galaxies and candidates.", '6.3. Star Formation Rate Histories': "To perform detailed modeling of the SEDs in terms of stellar populations, we use the Prospector code (Johnson et al. 2021), following the methods described in Tacchella et al. (2022, 2023). Briefly, we assume a variable star-formation history (SFH) with a bursty continuity prior, with 8 time bins spanning 0 -5 Myr, 5 -10 Myr and 6 logarithmically spaced up to z = 25. We allow the redshift to vary within the EAZY posterior. We adopt a single metallicity for both stars and gas, assuming a truncated log-normal centered on log( Z/Z ⊙ ) = -1 . 5 with width of 0.5, minimum of -2.0, and maximum of 0.0. We model dust attenuation using a two-component model with a flexible attenuation curve. For the stellar population synthesis, we adopt the MIST isochrones (Choi et al. 2016) that include effects of stellar rotation but not binaries, and assume a Chabrier (2003) initial mass function (IMF) between 0.08 and 120 M ⊙ . No Ly α emission line is added to the model to account for resonant absorption effects, while the IGM absorption model (Inoue et al. 2014; Madau 1995) is taken into account (normalization is a free parameter). We do not try to constrain independently the effects of possible additional Lymanα damping-wing absorption. For consistency with Figures 2-10, we use the r = 0 . 1' aperture fluxes, but we note that using r = 0 . 3' aperture fluxes provide quantitatively similar results for these compact objects. We put an error floor of 5% on the photometry. The rest of the nebular emission (emission lines and continuum) is self-consistently modeled (Byler et al. 2017) with two parameters, the gas-phase metallicity (tied to the stellar metallicity), and the ionization parameter \n(uniform prior in -4 < log( U ) < -1). By combining these inferred stellar population properties with the size measurements from ForcePho , we can additionally infer the stellar mass and star formation rate surface densities of the candidate galaxies. \nFigure 20 shows the resulting star formation rate histories (SFHs) of the eight galaxy candidates in our sample. The average SFR over the last 10 Myr is also reported for each candidate galaxy in Table 8. In each case, the continuity prior on the star formation history was used to inform the point-to-point star formation rate variations in the galaxies. For each object, the photometry listed in Tables 3-5 were used, except for the faintest object 74977 ( f ν ∼ 2 -3nJy) where the lower SNR Kron fluxes were used. We find that the typical star formation rate of these objects are SFR ≈ 0 . 1 -10 M ⊙ yr -1 over the last t ∼ 10 -30 Myr. The galaxies formed substantial fractions of their stars in the recent past, and have characteristic ages of just a few tens of millions of years. A few of the objects (NIRCam IDs 13731, 33309, 55733, 74977) show features in their SFHs roughly 10 -20 Myr before their observed epoch, with flat or even falling SFR thereafter. We speculate that these features may reflect 'mini-quenching' events where star formation shuts down briefly after exhausting or removing fuel (Looser et al. 2023). For the other objects, the SFHs appear to increase to the epoch of observation, suggesting some upswing in the star formation rate and luminosities of these objects. In two cases (NIRCam ID 74977 and 183348) the objects show evidence of comparable or higher star formation rates 100 Myr before the observed epoch. For 74977, this early star formation would correspond to z ∼ 14 . 2. For 183348, the early star formation would potentially start at z ∼ 20. The uncertainties on the SFH are large, and we cannot constrain well the star formation rate before z ∼ 15 for most objects. Given the physical sizes of the objects of R 1 / 2 ≈ 50 -200pc inferred from the ForcePho analysis, the star formation rate surface densities of these objects are Σ SFR ∼ 10 -100 M ⊙ yr -1 kpc 2 . Both the SFR and SFR surface densities are comparable to those found by Robertson et al. (2023) for spectroscopically-confirmed galaxies at z ∼ 12 -13, and consistent with being from the same galaxy population. \nThe above analysis assumes no luminous contribution from an active galactic nucleus. Of course, some of these galaxies may possibly host luminous AGN, as have been found or suspected in some other high-redshift galaxies (e.g., Goulding et al. 2023; Ubler et al. 2023; Kokorev et al. 2023; Maiolino et al. 2023a,b). AGN emission would decrease the inferred stellar emission and require a re-assessment of the star formation histories and stellar \nFigure 20. Star formation histories (SFHs) inferred using the Prospector code (Johnson et al. 2021), assuming a continuity prior and following the methods described in Tacchella et al. (2023). The galaxy candidates show star formation rates of SFR ≈ 0 . 1 -1 M ⊙ yr -1 over the last ∼ 10 Myr, measured backward from the epoch of observation. Roughly half of the objects show increasing star formation histories, while the others indicate a peak or burst in their star formation rates roughly 10 Myr before the observation epoch. This feature may indicate an episode of 'mini-quenching' (Looser et al. 2023) in these objects. Only one galaxy indicates a comparable or higher SFR t ∼ 100 Myr before the observation epoch, such that no object indicates evidence of substantial star formation before z ∼ 15. Each galaxy is labeled by both their [RA,Dec] designation, photometric redshift, and internal JADES NIRCam ID. \n<!-- image --> \nmasses, and possibly the photometric redshifts. We note that the fact that some of these galaxies are angularly resolved implies that some of the emission is stellar.", '6.4. Stellar Mass Distributions': 'Figure 21 presents the marginal stellar mass distributions inferred from Prospector fits to the observed photometry. The posterior samples of the galaxy properties were used to produce marginal distributions of the stellar mass, following the procedure described in Robertson et al. (2023). In agreement with Robertson et al. (2023), we find that the stellar masses of these z ∼ 12 -15 galaxies are M ⋆ ∼ 10 7 -10 9 M ⊙ . Given the sizes of R 1 / 2 ∼ 50 -200pc we measure from the surface brightness profiles, the stellar mass surface densities of the objects are then Σ ⋆ ∼ 10 3 -10 4 M ⊙ pc -2 . For a self-gravitating system, the dynamical timescale is then comparable to the star formation timescale inferred in § 6.3. Overall, in agreement with our previous findings in Robertson et al. (2023), these objects are consistent with rapidly star-forming, compact galaxies with formation timescales comparable to a few dynamical times. Using the simple abundance matching comparison with dark matter halos discussed in § 5.5, we note that matching to number densities would place these objects in M h ∼ 10 10 M ⊙ dark matter halos, with M ⋆ /M h ∼ 10 -1 -10 -3 , well above the present-day stellar mass to halo mass relations (e.g., Wechsler & Tinker 2018).', '7. DISCUSSION': "The luminosity function evolution remains the best current indicator of the connection between galaxies, dark matter halos, and cosmic reionization at the highest redshifts (for a review, see Robertson 2022). These results from the JADES Origins Field provide some new insight into the process of high-redshift galaxy formation. \nThe JOF provides the best currently available data for probing faint galaxies at redshifts z > 12, given its depth and filter array. Using an area twice the size of the Hubble Ultra Deep Field, the JOF area reaches a deeper limit (30 . 2 -30 . 5AB) and has fourteen JWST filters including the ultradeep JADES Program 1210. The inclusion of deep F162M provides an essential check on the reality of the highest-redshift candidates. \nOf our Main Sample, none of the galaxies are brighter than M UV = -18 . 6, and many have M UV > -18. The depth allows us to constrain the UV luminosity function to fainter limits at z ∼ 14 than previously possible, while retaining tighter control of systematics by having additional medium band filters to probe the Lyman \nFigure 21. Posterior distribution of stellar mass for candidate z > 11 . 5 galaxies. Shown are the stellar mass distributions constructed from posterior samples of the Prospector code (Johnson et al. 2021). The objects have inferred stellar masses of M ⋆ ∼ 10 7 -10 8 M ⊙ , comparable to that inferred for the spectroscopically-confirmed z ∼ 12 -13 analyzed by Robertson et al. (2023). Each galaxy candidate is labeled by its JADES NIRCam ID and photometric redshift, and color-coded the same in Figure 20. \n<!-- image --> \nbreak with more fidelity. Following the stellar population modeling procedure of Tacchella et al. (2023), we find that the star formation rate and stellar mass properties are comparable to galaxies spectroscopically confirmed at z ∼ 12 -13 (Robertson et al. 2023; Curtis-Lake et al. 2023; Wang et al. 2023b). Using the ForcePho forward model for the surface brightness distribution of these galaxies, we find that they have compact sizes of R 1 / 2 ∼ 50 -200pc, also in agreement with spectroscopically confirmed galaxies at these redshifts (Robertson et al. 2023; Wang et al. 2023b). \nIn agreement with previous determinations of UV luminosity function in extragalactic JWST fields (McLeod et al. 2023; Donnan et al. 2023b; Adams et al. 2023b; Harikane et al. 2023b,a; P'erez-Gonz'alez et al. 2023a; Willott et al. 2023; Finkelstein et al. 2023a), we find that the luminosity function of galaxies has smoothly declined from z ∼ 8, as first established by HST observations (e.g., McLure et al. 2013), to z ∼ 12. Our results for the abundance of galaxies at z ∼ 12 are in broad agreement with the literature values, as shown in Figures 16 and 17. We do note that our inferred UV luminosity density at z ∼ 14 is lower than that reported \nby Finkelstein et al. (2023a), but the uncertainties are large. \nHowever, our selection completeness using the JOF observations is sensitive to galaxies out to z ∼ 20 when the Lymanα break enters F250M. With a suitable revision to our selection, we would be sensitive to bright galaxies at even greater distances. Our work presents a new method for modeling the redshift-dependent UV luminosity function incorporating both detections and non-detections to constrain its evolution over the redshift range z ≈ 11 -20 where our completeness is high. From the lack of galaxy candidates at z > 15, we find that the decline to z > 14 continues at d log ϕ ⋆ /dz ∼ -0 . 2 with our nominal Main Sample presented in Tables 2-5. We note that uncertainties owing to cosmic variance are clearly non-negligible for the JOF, and a larger sample of galaxies at z > 11 . 5 is needed to confirm this decline. Nonetheless, we now know that the M UV ∼ -21 object NIRCam ID 183348 selected by our JOF Medium band photometry to be at a photometric redshift of z ≈ 14 . 4 has been spectroscopically confirmed at z = 14 . 32 by Carniani et al. (submitted). As Figure 18 shows, the evolving luminosity density at z > 14 we infer from 183348 and our photometric candidates, while declining, still requires a constant remapping between galaxy and halo abundance, with increasing efficiency in low-mass halos at higher redshifts. This evolution is in contrast to the possibility that z > 14 galaxies were not abundant, where a rapid drop in the UV luminosity density would track more closely the abundance of M vir ∼ 10 10 M ⊙ halos and the galaxy efficiency could stabilize at early times. Given the confirmation of 183348, we see no evidence for such a stabilization in the efficiency of galaxy formation out to z ∼ 14 or beyond. \nLastly, since our results are consistent with prior literature results at z ∼ 12, theoretical models that match those observations also match ours. For instance, the feedback-free models of Dekel et al. (2023) and Li et al. (2023) agree with our z ∼ 12 observations for an efficiency of ϵ max ≈ 0 . 2. Models for the evolving number counts of high-redshift galaxies based on dust-free populations (e.g., Ferrara et al. 2023) also predict a star formation rate density evolution to z ∼ 15 in agreement with our inferences, assuming all our candidates are really high-redshift sources (Ferrara 2023).", '8. SUMMARY AND CONCLUSIONS': "Using ultra-deep JWST observations of the JADES Origins Field (JOF), we search for the most distant galaxies in the universe. With fourteen JWST and up to nine Hubble Space Telescope filters covering the JOF, we can carefully select galaxies at z > 12 by identifying \ndropouts in NIRCam F162M and bluer filters using SED template-based photometric redshift fitting. Our findings include: \n- · We select nine galaxy candidates at z ∼ 12 -15 and no galaxy candidates at z ≳ 15. These objects include the most distant candidates detected in more than five filters and displaying a dropout in more than 10 filters. Our sample selection includes a galaxy at z = 14 . 32 since spectroscopically confirmed. Simulations of our detection and photometry methods and our prior spectroscopic confirmations of high-redshift JADES sources suggest that the other candidates without spectroscopic confirmation are robust. Several of our candidates have been identified in previous analyses, including Hainline et al. (2023a) and Williams et al. (2023).\n- · These objects show apparent total magnitudes of m AB ∼ 29 . 5 -30 . 5 in the rest-frame UV and blue rest-UV spectral slopes -2 ≳ β ≳ -3.\n- · Performing detailed structural modeling with ForcePho and stellar population inference using Prospector, we find that the galaxies have starformation rates of SFR ≈ 0 . 1 -10 M ⊙ yr -1 , stellar masses of M ⋆ ∼ 10 7 -10 9 M ⊙ , sizes of R ∼ 50 -200 pc, and stellar ages of t ⋆ ≈ 30 -50 Myr. The properties of our low-mass candidates are comparable to the properties of z ∼ 12 -13 galaxies with confirmed redshifts, as first identified by the JADES collaboration.\n- · We develop a new forward modeling method to infer constraints on the evolving UV luminosity function without binning in redshift or luminosity while marginalizing over the photometric redshift posterior distribution of candidates in our sample. This method allows for an accounting of potential contamination by adjacent redshifts and includes the impact of non-detections on the inferred galaxy luminosity function evolution.\n- · With the population of z > 12 galaxy candidates newly discovered in JOF, we provide an inference on the z ∼ 15 luminosity function and a refined measure of the luminosity function at z ∼ 12 in agreement with literature values. At z ∼ 15, we infer a continued decline from z ∼ 12. Over the redshift range z ∼ 12 -14, where we have detected galaxies, we infer a factor of 2 . 5 decline in the luminosity function normalization ϕ ⋆ and a corresponding decline in the luminosity density \nρ UV . We note that cosmic variance uncertainties for the high-redshift JOF sample are not negligible, and this decline should be confirmed with a larger sample over a wider area. \nThis demonstrates the immediate impact new JWST observations can have on our knowledge of the distant universe. With high-redshift galaxy populations now established fewer than 300 million years after the Big Bang, we have extended our reach into the cosmic past by 40% during the first eighteen months of JWST operations. \n- The JADES Collaboration thanks the Instrument De1\n- velopment Teams and the instrument teams at the Eu2\n- ropean Space Agency and the Space Telescope Science 3\n- Institute for the support that made this program pos4\n- sible. The authors acknowledge use of the lux super5\n- computer at UC Santa Cruz, funded by NSF MRI grant AST 1828315. 6 7\n- BER, BDJ, DJE, PAC, EE, MR, FS, & CNAW acknowl8\n- edge support from the JWST/NIRCam contract to the 9\n- University of Arizona, NAS5-02015. BER acknowledges 10\n- support from JWST Program 3215. DJE is supported 11 \n12 \nas a Simons Investigator. \nSA acknowledges support \n- from Grant PID2021-127718NB-I00 funded by the Span13\n- ish Ministry of Science and Innovation/State Agency of 14\n- Research (MICIN/AEI/ 10.13039/501100011033). WB, 15\n- FDE, RM, & JW acknowledge support by the Science 16\n- and Technology Facilities Council (STFC), ERC Ad17\n- vanced Grant 695671 'QUENCH'. AJB, JC, & GCJ ac18\n- knowledge funding from the 'FirstGalaxies' Advanced 19\n- Grant from the European Research Council (ERC) un20\n- der the European Union's Horizon 2020 research and 21\n- innovation programme (Grant agreement No. 789056). 22 \n- SC acknowledges support by European Union's HE 23\n- ERC Starting Grant No. 101040227 - WINGS. ECL 24\n- acknowledges support of an STFC Webb Fellowship (ST/W001438/1). FDE, RM, & JW acknowledge support by UKRI Frontier Research grant RISEandFALL. Funding for this research was provided by the Johns Hopkins University, Institute for Data Intensive Engineering and Science (IDIES). RM also acknowledges funding from a research professorship from the Royal Society. The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant DNRF140. PGP-G acknowledges support from grant PID2022-139567NB-I00 funded by Spanish Ministerio de Ciencia e Innovaci'on MCIN/AEI/10.13039/501100011033, FEDER, UE. DP acknowledges support by the Huo Family Foundation through a P.C. Ho PhD Studentship. RS acknowledges support from a STFC Ernest Rutherford Fellowship (ST/S004831/1). H U gratefully acknowledges support by the Isaac Newton Trust and by the Kavli Foundation through a Newton-Kavli Junior Fellowship. LW acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2137419. The research of CCW is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foun25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49\n- dation. 50", 'Facilities: HST(ACS,WFC3), JWST(NIRCam)': 'Software: astropy (Astropy Collaboration et al. 2018, 2022), EAZY (Brammer et al. 2008), Source Extractor (Bertin & Arnouts 1996), photutils (Bradley et al. 2023), nautilus (Lange 2023)', 'REFERENCES': 'Adams, N. J., Conselice, C. J., Ferreira, L., et al. 2023a, \nMNRAS, 518, 4755, doi: 10.1093/mnras/stac3347 \nAdams, N. J., Conselice, C. J., Austin, D., et al. 2023b, arXiv e-prints, arXiv:2304.13721, doi: 10.48550/arXiv.2304.13721 Akhlaghi, M. 2019, arXiv e-prints, arXiv:1909.11230, doi: 10.48550/arXiv.1909.11230 Akhlaghi, M., & Ichikawa, T. 2015, ApJS, 220, 1, doi: 10.1088/0067-0049/220/1/1 \nArrabal Haro, P., Dickinson, M., Finkelstein, S. L., et al. 2023, ApJL, 951, L22, doi: 10.3847/2041-8213/acdd54 Astropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f \nAstropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74 Atek, H., Shuntov, M., Furtak, L. J., et al. 2023, MNRAS, \n519, 1201, doi: 10.1093/mnras/stac3144 Bagley, M. B., Finkelstein, S. L., Koekemoer, A. M., et al. 2023a, ApJL, 946, L12, doi: 10.3847/2041-8213/acbb08 Bagley, M. B., Pirzkal, N., Finkelstein, S. L., et al. 2023b, arXiv e-prints, arXiv:2302.05466, doi: 10.48550/arXiv.2302.05466 Baker, W. M., Tacchella, S., Johnson, B. D., et al. 2023, arXiv e-prints, arXiv:2306.02472, doi: 10.48550/arXiv.2306.02472 \nBertin, E., & Arnouts, S. 1996, A&AS, 117, 393 \nWilliams, C. C., Alberts, S., Ji, Z., et al. 2023, arXiv e-prints, arXiv:2311.07483. \nhttps://arxiv.org/abs/2311.07483 \nWillott, C. J., Desprez, G., Asada, Y., et al. 2023, arXiv e-prints, arXiv:2311.12234, doi: 10.48550/arXiv.2311.12234 \nWyithe, J. S. B., Yan, H., Windhorst, R. A., & Mao, S. 2011, Nature, 469, 181, doi: 10.1038/nature09619 Wilkins, S. M., & Gardner, J. P. 2024, MNRAS, 527, \nYung, L. Y. A., Somerville, R. S., Finkelstein, S. L., 5929, doi: 10.1093/mnras/stad3484 \nZavala, J. A., Buat, V., Casey, C. M., et al. 2023, ApJL, 943, L9, doi: 10.3847/2041-8213/acacfe'}
2024A&A...690A.333S	We aim to constrain the chemodynamical properties of the Sagittarius Sgr dwarf galaxy using carbon abundances. At low metal licities in particular these properties reveal the early chemical evolution of a system tracing the contributing supernovae SNe and how much of their ejecta eventually made it into the next stellar generation. Our sample from the Pristine Inner Galaxy Survey PIGS includes 350 metalpoor FeH lt 1.5 stars in the main body of Sgr with good quality spectroscopic observations. Our metalpoor Sgr population has a larger velocity dispersion than metalrich Sgr from the literature which could be explained by outsidein star formation extreme Galactic tidal perturbations andor the presence of a metalrich disc and bar metalpoor halo. The average carbon abundance CFe in Sgr is similar to that of other classical dwarf galaxies DGs and consistently lower than in the Milky Way by 0.20.3 dex at low metallicities. The interstellar medium in DGs including Sgr may have retained yields from more energetic Population III and II supernovae SNe thereby reducing the average CFe. Additionally SNe Ia producing more Fe than C would start to contribute at lower metallicity in DGsSgr than in the Galaxy. The presence of a CFe gradient for Sgr stars with FeH 2.0 6.8 10SUP4SUP dex arcminSUP1SUP suggests that SNe la contributed to the system at those metallicities especially in its inner regions. There is a low frequency of carbonenhanced metalpoor CEMP stars in our Sgr sample. At higher metallicities and carbon abundances i.e. mostly CEMPs this may be due to photometric selection effects but those are less likely to affect nonCEMP stars. Given the lower average CFe in DGs we propose using the same CEMP definition CFe gt 0.7 as that applied to the Galaxy at large ends up underpredicting the number of CEMP stars in DGs. Burthermore for Sgr a cut at CFe 0.35 may be more appropriate which brings the frequency of CEMP stars in agreement with that of the whole Galaxy.	2024-10-01T00:00:00Z	['2024arXiv240618636S', '10.1051/0004-6361/202451258', '2024A&A...690A.333S', '10.48550/arXiv.2406.18636', 'arXiv:2406.18636']	['stars: abundances', 'stars: Population II', 'galaxies: abundances', 'galaxies: dwarf', 'galaxies: individual: Sagittarius', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics']	The Pristine Inner Galaxy Survey PIGS X. Probing the early chemical evolution of the Sagittarius dwarf galaxy with carbon abundances	2,024	191	0.55	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	7	https://arxiv.org/pdf/2406.18636.pdf	{'Probing the early chemical evolution of the Sagittarius dwarf galaxy with carbon abundances': 'Federico Sestito 1 , 2 , Anke Ardern-Arentsen 3 , Sara Vitali 4 , 5 , Martin Montelius 6 , Romain Lucchesi 7 , Kim A. Venn 1 , Nicolas F. Martin 8 , 9 , Julio F. Navarro 1 , and Else Starkenburg 10 \n- 1 Department of Physics and Astronomy, University of Victoria, PO Box 3055, STN CSC, Victoria BC V8W 3P6, Canada\n- 2 Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK e-mail: [email protected]\n- 3 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\n- 4 Instituto de Estudios Astrofísicos, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile\n- 5 Millenium Nucleus ERIS\n- 6 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands\n- 7 Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy\n- 8 Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France\n- 9 Max-Planck-Institut fur Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany\n- 10 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands \nReceived XX; accepted YY', 'ABSTRACT': 'We aim to constrain the chemo-dynamical properties of the Sagittarius (Sgr) dwarf galaxy using carbon abundances. Especially at low metallicity, these reveal the early chemical evolution of a system, tracing the supernovae (SNe) that contributed and how much of their ejecta made it into the next stellar generation. Our sample from the Pristine Inner Galaxy Survey (PIGS) includes ∼ 350 metal-poor ([Fe / H] < -1 . 5) stars in the main body of Sgr with good quality spectroscopic observations. Our metal-poor Sgr population has a larger velocity dispersion than metal-rich Sgr from the literature, which could be explained by outside-in star formation, extreme Galactic tidal perturbations and / or the presence of a metal-rich disc / bar + a metal-poor halo. The average carbon abundance [C / Fe] in Sgr is similar to that of other classical dwarf galaxies (DGs) and consistently lower than in the Milky Way by ∼ 0 . 2 -0 . 3 dex at low metallicity. The interstellar medium in DGs, including Sgr, may have retained yields from more energetic Population III and II supernovae (SNe), thereby reducing the average [C / Fe]. Additionally, SNe Ia, producing more Fe than C, would start to contribute at lower metallicity in DGs / Sgr than in the Galaxy. The presence of a [C / Fe] gradient for Sgr stars with [Fe / H] ≳ -2 . 0 ( ∼ 6 . 8 × 10 -4 dex arcmin -1 ) suggests that SNe Ia contributed in the system at those metallicities, especially in its inner regions. There is a low frequency of carbon-enhanced metal-poor (CEMP) stars in our Sgr sample. At higher metallicity / carbon abundance (mostly CEMP-s) this may be due to photometric selection e ff ects, but those are less likely to a ff ect CEMP-no stars. We propose that, given the lower average [C / Fe] in DGs, using the same CEMP definition ([C / Fe] > + 0 . 7) as in the Galaxy under-predicts the number of CEMP stars in DGs, and for Sgr a cut at [C / Fe] ∼ + 0 . 35 may be more appropriate, which brings the frequency of CEMP stars in agreement with that in the Galaxy. \nKey words. Galaxies: individual: Sagittarius dwarf galaxy - Galaxies: dwarf - Galaxies: abundances - Stars: abundances - Stars: Population II', '1. Introduction': 'The Sagittarius (Sgr) dwarf galaxy (Ibata et al. 1994), located approximately 26.5 kpc away from us towards the inner Galactic regions (Vasiliev & Belokurov 2020), experienced its first in-fall into the Milky Way (MW) about 5 Gyr ago (e.g. Ruiz-Lara et al. 2020). As it is being tidally stripped by the MW, its core and two stellar streams are now visible in the Sky (Ibata et al. 1994; Mateo 1998; Majewski et al. 2003; Law & Majewski 2010; Belokurov et al. 2014), as well as various associated globular clusters (Sbordone et al. 2007; Mucciarelli et al. 2017). Given its proximity, it is an ideal test-bed for galactic chemo-dynamical models. \nThe star formation history (SFH) of Sgr is characterised by multiple star formation episodes, investigated with both high- \nresolution spectroscopy (e.g. Bonifacio et al. 2000; Monaco et al. 2005; Chou et al. 2007; McWilliam et al. 2013; Hansen et al. 2018a; Hasselquist et al. 2017, 2021; Sestito et al. 2024b) and photometric techniques (e.g. Bellazzini et al. 1999; Layden & Sarajedini 2000; Siegel et al. 2007; Vitali et al. 2022). So far, studies have typically focussed on metal-rich and relatively young stars, given that they are the prevalent population. Further complicating the study of the oldest / metal-poor stars is the strong overlap in the colour-magnitude diagram between the Milky Way bulge population and stars in Sgr (Monaco et al. 2005; Mucciarelli et al. 2017), especially on the blue, metalpoor side of the red giant branch (RGB) of Sgr. However, the most metal-poor stars are key to understanding the early chemical evolution of Sgr. \nAn e ffi cient way to discover new members in dwarf galaxies is to use the exquisite Gaia (Gaia Collaboration et al. 2016, 2021, 2023) astrometry and photometry alone (e.g. Chiti et al. 2021; Filion & Wyse 2021; Yang et al. 2022; Waller et al. 2023; Sestito et al. 2023a,c; Hayes et al. 2023; Jensen et al. 2024) or to couple it with metal-poor dedicated photometric surveys, e.g. the Pristine survey (Starkenburg et al. 2017; Martin et al. 2023), as done in the Pristine dwarf galaxy survey (e.g. Longeard et al. 2022, 2023). \nAlong those lines, the Pristine Inner Galaxy Survey (PIGS) targets metal-poor stars towards the inner regions of the MW (Arentsen et al. 2020b), as well as the Sagittarius dwarf galaxy (Vitali et al. 2022). The latter work investigated the metallicity distribution of ∼ 50 , 000 Sgr candidate members as a function of their spatial location, and identified the largest sample of Sgr candidate members with [Fe / H] ≤ -2 . 0 ( ∼ 1200 stars). From PIGS, Sestito et al. (2024b) followed-up with MIKE highresolution spectroscopy 12 very metal-poor (VMP, [Fe / H] ≤ -2 . 0) Sgr members, the largest and most complete detailed chemical abundance analysis of the VMP Sgr component (vs 4 VMPsin Hansen et al. 2018a). The authors interpreted the chemical pattern of the most metal-poor stars as the result of a variety of type II supernovae and asymptotic giant branch stars. A wide range of energetic supernovae and hypernovae with intermediate mass (10 -70 M ⊙ ) are needed to account for the chemical abundances of the lighter elements up to the Fe-peak. The chemical trend of the heavier elements is interpreted as a mixture of yields from compact binary mergers and massive (up to ∼ 120 M ⊙ ) fastrotating stars (up to ∼ 300 km s -1 ). \nInvestigating the origin of carbon in a given stellar population is crucial to understand various astrophysical topics, for example the types of supernovae contributing in a given system, nucleosynthesis in massive stars and binary interaction mechanisms (e.g. Frebel et al. 2007; Vincenzo & Kobayashi 2018; Kobayashi et al. 2020). At low metallicity, many stars are found to be carbon-enhanced. Populations of these so-called carbonenhanced metal-poor (CEMP) stars, with [C / Fe] > + 0 . 7, are powerful probes of the underlying stellar population and the star formation history. Some CEMP stars are thought to carry the imprint of the first generations of supernovae, these are called CEMP-no stars and have sub-solar Ba, [Ba / Fe] < 0 . 0 (Beers & Christlieb 2005; Aoki et al. 2007). It has been suggested that classical DGs have a lower CEMP-no fraction than the MW halo and ultra-faint dwarfs (UFDs) (e.g. Starkenburg et al. 2013; Jablonka et al. 2015; Kirby et al. 2015b; Simon et al. 2015; Hansen et al. 2018b; Lucchesi et al. 2024; Skúladóttir et al. 2015, 2021, 2024b). \nOther types of CEMP stars are typically the products of mass transfer from binary interaction with a former asymptotic giant branch (AGB) star companion. These are Ba-rich ([Ba / Fe] > + 1 . 0) due to slow-process channels taking place in the AGB companion and are called CEMP-s stars (Beers & Christlieb 2005). The latter group is important to understand the properties of binary populations. In particular, their properties are instructive to understand the nucleosynthetic channels, convection and non-convective processes (e.g. Stancli ff e et al. 2007); the interaction mechanisms, such as the physics of Roche-lobe over-flow and wind accretion (e.g. Abate et al. 2013); and their influence on the measurement of the velocity dispersion in a system and its dynamical mass (e.g. Spencer et al. 2017; Arroyo-Polonio et al. 2023), such as its dark matter content. \nFrom medium-resolution spectroscopy, metallicities and carbon abundances have been measured in only 11 VMP stars in Sgr (Chiti & Frebel 2019; Chiti et al. 2020). In this work, we \nuse the data release of the PIGS low / medium-resolution spectroscopic campaign (Ardern-Arentsen et al. 2024) to select the largest sample of low-metallicity ([Fe / H] ≤ -1 . 5) Sgr members (356 stars) with measured metallicity, [C / Fe], and radial velocity to date. The dataset and a discussion on the photometric selection e ff ects due to the Pristine filter is reported in Section 2. The dynamical properties of the metal-rich and metal-poor populations in Sgr are outlined in Section 3. A comparison of the [C / Fe] abundances in Sgr with respect the other classical dwarf galaxies (DGs) and the MW halo and inner Galaxy is discussed in Section 4. We discuss the types and frequencies of CEMP stars in Sgr in Section 5, including a suggestion that the definition of CEMPmight need revision in DGs. Conclusions are summarised in Section 6.', '2. The Pristine Inner Galaxy Survey (PIGS)': 'PIGS targets the most metal-poor stars in the inner regions of the Milky Way (Arentsen et al. 2020b), using a metallicitysensitive narrow CaHK filter mounted at Canada-France-Hawaii Telescope (CFHT). Among the photometric metal-poor candidates, ∼ 13 235 stars have been observed with the Anglo Australian Telescope (AAT) using the AAOmega + 2dF spectrograph. We will refer to them as the PIGS / AAT sample, which is publicly available (Ardern-Arentsen et al. 2024). The AAT setup acquired spectra with low-resolution ( R ∼ 1800) in the blue and with medium-resolution ( R ∼ 11 000) around the calcium triplet. The analysis is described in detail in Arentsen et al. (2020a), but, briefly, the two arms were fit simultaneously with the FERRE code 1 (Allende Prieto et al. 2006) to obtain stellar parameters (e ff ective temperature and surface gravity), metallicities, and carbon abundances. The radial velocities (RVs) were derived by cross-correlation of the calcium triplet spectra with synthetic templates.', '2.1. PIGS target selection from photometry': 'Some of the PIGS / AAT fields overlap with the core of the Sagittarius dwarf galaxy and, in four fields, Sgr stars were specifically targeted. Two fields were observed in 2018 and served as a pilot program ( Field282.0-29.8\\_Sag , Field284.0-30.0\\_Sag ), two additional fields with more Sgr candidates were observed in 2020 ( Field282.9-32.1 , Field286.0-31.1 ). For the 2018 observations, Sgr stars were selected to be within a radius of 0.6 mas yr -1 around proper motion µα = -2 . 7 mas yr -1 and µδ = -1 . 35 mas yr -1 and parallax -parallax\\_error < 0.05 mas. This was relaxed a little in 2020, to a radius of 1 mas yr -1 around those proper motions and the parallax -parallax\\_error < 0.1 mas. In 2020, suspected variable stars were removed using the flux error and the number of observations (Fernández-Alvar et al. 2021). Both selections were done using Gaia DR2 (Gaia Collaboration et al. 2018). \nThe photometric calibration of the PIGS CaHK photometry was slightly di ff erent when the targets were selected compared to the current, final photometric catalogue, but changes are not expected to be major for the Sgr fields. For the fields from 2018, Sgr candidates were selected using a horizontal line in ( CaHK -G )0 -2 . 5( BP -RP )0 to select the best ∼ 100 Sgr targets per field (and the rest of the AAT fibres were filled with inner Galaxy targets). Observed targets can be seen as black / small coloured points in the Pristine colour-colour diagrams in the lefthand panels of Figure 1, compared to all Sgr candidates in the \nFig. 1. Colour-colour diagrams for the two fields observed in 2018 (left) and the two observed in 2020 (right). Grey dots are all stars with PIGS photometry in the targeted fields passing the respective Sgr selection criteria for those years (see Section 2.1) and G < 17. Black dots in the top row are all observed AAT stars in these fields, coloured small dots in the bottom panel are good quality AAT stars coloured by spectroscopic metallicity. Large dots denote Yoon et al. (2016) CEMP stars with logg < 2 . 5 and -3 < [Fe / H] < -2 . 0. Colour coding in the top row is [C / Fe], in the bottom is [Fe / H]. Vertical lines indicate colour cuts applied. The stars within the pink circles in the top left and right panels are discussed in Section 5.2 (in the right-hand panel it is at (x,y) ∼ (1.4,-1.3)). \n<!-- image --> \nfields in grey. A red cut at ( BP -RP )0 = 1 . 7 was also made. For the fields in 2020 a di ff erent strategy was used, the focus was completely on Sgr and inner Galaxy stars were mostly used as fillers if no fibres could be placed on Sgr stars. Sgr candidates were selected in two ways. The first group contained all stars brighter than G 0 = 15 . 5 and bluer than a [M / H] = -1 . 0 MIST isochrone (Choi et al. 2016; Dotter 2016), this was to get some red and bright targets that would have been missed in the 2018 selection. The next group contained the most promising metalpoor candidate stars according to CaHK , again using a horizontal selection in the colour-colour diagram, this time with factor of 3.0 instead of 2.5 in front of ( BP -RP )0. These selections can be seen as black / small coloured points in the right-hand panels of Figure 1. A colour cut of 1 . 0 < ( BP -RP )0 < 1 . 8 was also made.', '2.2. Selection effects with reference to CEMP stars': 'Photometric selections of metal-poor stars are plagued by selection e ff ects against carbon-rich stars, especially for cooler stars (e.g. Beers et al. 1999; Rossi et al. 2005; Goswami et al. 2006; Da Costa et al. 2019; Yoon et al. 2020; Arentsen et al. 2021; Martin et al. 2023). This is because carbon has many molecular features in the spectrum, a ff ecting both the narrow-band and broad-band photometry. \nWe empirically investigate possible selection e ff ects in our Sgr sample by comparing the location of our observed Sgr / AAT sample in the Pristine colour-colour diagram with known CEMP \nstars from Yoon et al. (2016, hereafter Y16). We select giant stars within the relevant Sgr range, making cuts on log g < 2 . 5 and -3 . 0 < [Fe / H] < -2 . 0. Almost all Y16 stars after this cut have T e ff > 4500 K. We use the synthetic CaHK catalogue from Martin et al. (2023), derived from Gaia XP spectra (Gaia Collaboration et al. 2023), and cross-match it with Y16 to obtain Pristine colour-colour diagram positions for these stars. All CaHK uncertainties for the Y16 stars are less than 0.075 mag, with more than 80% less than 0.05 mag. For the metal-poor regime in the Sgr / PIGS colour-colour diagrams, PIGS CaHK uncertainties are typically less than 0.025 mag. \nLarge symbols in Figure 1 are CEMP stars from Yoon et al. (2016) in the relevant Sgr range. Unfortunately, the Y16 catalogue does not contain many cool giants in this metallicity range, but a small sample of 48 stars remains that can be used. What is clear is that the CEMP stars are mostly not where they are expected to be, given their metallicity - they are further down in the colour-colour diagrams. A similar conclusion for the Pristine survey was reached by Martin et al. (2023), who reported that these stars have higher photometric metallicities than their spectroscopic metallicities (see also Ca ff au et al. 2020). Analogously, the SkyMapper survey, which is targeting metal-poor stars with the v filter also in the CaHK region, found a similar bias against CEMP stars, especially for those stars with very large carbon-enhancement (predominantly CEMP-s, Da Costa et al. 2019). \nFor the 2018 fields (left column of Figure 1), a large fraction of Y16 CEMP stars falls outside the selected region (y-axis \nµδ \n≲ -0 . 9). These are mostly stars with [Fe / H] > -2 . 5 and / or [C / Fe] > + 1 . 8 - the regime where CEMP-s stars dominate. Stars with [Fe / H] < -2 . 5 and [C / Fe] < + 1 . 8 fall within the selected range - this combination of [Fe / H] and [C / Fe] is in the regime of the Group II / CEMP-no stars. In the 2020 fields (right column of Figure 1), more Sgr stars were targeted and the selection boundary lies slightly lower in the colour-colour diagram. More Y16 CEMP stars now overlap with the selection range, although very much at the edge. The biases are similar to those of the 2018 selection, although a few more stars with [Fe / H] < -2 . 5 and [C / Fe] > + 2 . 0 are included now. From this analysis, we conclude that CEMP-no stars with moderate carbon-enhancement should likely be included in our selection (especially for the 2020 fields, where the majority of our sample comes from), but a large fraction of CEMP-s stars would likely have been excluded. \nFinally, we note that the Y16 sample does not have any stars cooler than 4500 K with [Fe / H] < -2 . 5 or with [Fe / H] > -2 . 5 and [C / Fe] < + 1 . 5. It is therefore di ffi cult to estimate the biases against these stars, although we expect them to be worse for such cool stars. Our analysis in this work is focused on slightly warmer stars so the details of these stars are not crucial.', '2.3. Sagittarius spectroscopic sample used in this work': "For this work, to remove the MW contamination from the Sgr candidates, a selection of the Sgr members is made based on the Gaia DR3 proper motions, position on the sky, and radial velocity. In particular, we use the reduced proper motions for Sgr 2 , as defined in Vasiliev & Belokurov (2020). This takes into account that the proper motion of the members changes as a function of the coordinates. We assume a star to be a Sgr member if it has a reduced proper motion of less than 0 . 6 mas yr -1 as in Vasiliev & Belokurov (2020) and Vitali et al. (2022). Additionally, Sgr members have RVs in the range from 100 to 200 km s -1 (e.g. Ibata et al. 1994; Bellazzini et al. 2008; Minelli et al. 2023). Finally, we limit our analysis to stars with RA > 280 · . This leads to a sample of 834 kinematically selected PIGS / AAT Sgr stars. \nNot all the AAT spectra have enough good quality to obtain reliable measurements of [Fe / H] and [C / Fe]. Therefore, bad measurements are removed from the kinematical selection using the flag good\\_ferre = True , as suggested in Ardern-Arentsen et al. (2024). This flag is based on the S / N of the blue spectra, the FERRE χ 2 and the CaT not being double-lined. This further cut leads to 631 Sgr members with available chemistry. The stars with bad S / Nin the AAT sample are partly due to issues with the 2dF fibre placement (see discussion in Arentsen et al. 2020a), which were particularly severe for the two fields observed in 2020 - this is why the upper / right parts of these fields in RA / Dec (see top-left panel of Fig. 2) do not have many stars in the final Sgr cut. \nThe stellar parameter grid used in FERRE is limited to 4500 ≤ Te ff (K) ≤ 7000 and 1 ≤ logg ≤ 5, implying that for stars at the edge of this grid, a wrong model atmosphere might have been adopted to derive the [Fe / H] and [C / Fe]. For the Sgr stars, this is particularly an issue at the cool end (see the bottom right panel of Figure 2); we, therefore, remove stars with Te ff < 4510 K to avoid stars close to the cool limit of the FERRE grid. For warm stars, the [C / Fe] abundances may not be reliable, we therefore remove stars with Te ff > 5700 K. Because we are interested in the \n- \nµ \n= \nµδ \n+ \n1 \n. \n35 \n+ \n0 \n. \n024 \n∆ \nα \n+ \n0 \n. \n019 \n∆ \nδ \n+ \n0 \n. \n00002 \n∆ \n3 \nα \n, \nchemistry, we only keep stars with reasonable uncertainties on [Fe / H] and [C / Fe] ( < 0 . 5 dex). After these cuts, the PIGS / AAT Sgr sample consists of 437 stars. However, in this work we are mainly interested in stars with [Fe / H] < -1 . 5, which results in a final selection of 356 metal-poor PIGS / AAT Sgr members with good measurements of [Fe / H], [C / Fe], and RV. A table of the Sgr members updated to Gaia DR3 will be available as online material. \nThe PIGS / AAT sample (13 235 stars, grey dots), the stars from the kinematical cut (834, coral circles), the final selection (356 stars, blue circles), and Sgr members from APOGEE DR17 (525 stars, brown crosses, Abdurro'uf et al. 2022) are shown in Figure 2. The figure displays the position on the sky zoomed in on the Sgr fields (top left panel), the reduced proper motion space (top right), the [Fe / H] -RV space (bottom left), and the Kiel diagram (bottom right). PIGS / AAT stars in grey dots that lie within the red circle in proper motion space (top right) do not have RV compatible with Sgr, and, similarly, PIGS / AAT stars in grey dots with similar RV as Sgr (bottom left) do not match its proper motion. The Kiel diagram clearly shows an overdensity of stars at the cool edge of the FERRE grid, which has been removed as outlined above. Most PIGS / AAT Sgr stars have 1 . 0 < log g < 2 . 5 and 4500 K < Te ff < 5300 K. \nPart of this work is focused on very carbon-rich objects (Section 5), so it is important to be certain that our spectroscopic quality cuts do not bias against such stars. The main quality cuts of relevance are the S / N and the FERRE χ 2 . The S / N is determined from the spectra independently of the FERRE fit, in two regions (4000 -4100 Å and 5000 -5100 Å), and is not expected to be strongly a ff ected by the carbon abundance, so cutting on it is unlikely to introduce a bias against CEMP stars. If FERRE cannot find a good fit or there are many bad regions in the spectrum, the χ 2 will be high. We inspect all fits of Sgr candidates with bad S / N or bad χ 2 by eye, and identify two clearly carbon-rich stars that are badly fitted, with a high χ 2 . Both of these are very cool, very carbon-enhanced and intermediate / very metal-poor, and they will be discussed in Section 5.2.", '3. On the RV distribution': "The RVs of the PIGS / AATSgr sample fall within the overall distribution of stars in Sgr's core, ranging between 100 -200 km s -1 (e.g. Ibata et al. 1994; Bellazzini et al. 2008; Minelli et al. 2023), see also Figure 2. Various studies have pointed out that the metalpoor population of Sgr, both in the core and in the stream, is more spatially extended and has a larger velocity dispersion σ RV and a larger systemic velocity < RV > than the more metal-rich population (e.g. Gibbons et al. 2017; Johnson et al. 2020; Peñarrubia &Petersen 2021; Vitali et al. 2022; Limberg et al. 2023; Minelli et al. 2023). With the PIGS / AAT Sgr sample, we update these quantities using a more metal-poor, and likely older, population than previous work. \nSgr stars have been divided into two populations, the metalpoor ([Fe / H] < -1 . 5) from PIGS / AAT and the metal-rich ([Fe / H] > -0 . 6) from APOGEE DR17. The number of stars in these two populations as a function of the projected elliptical distance from Sgr's centre is shown in Figure 3. The metal-rich population dominates over the metal-poor one in the very inner regions, until a projected elliptical distance of ∼ 0 . 25 half-light radii (rh). Then the two groups from the two surveys are similarly populated. \nThe metal-poor and the metal-rich populations are then divided into two sub-groups according to their projected elliptical distances: the inner group at < 0 . 25 rh vs the outer at ≥ 0 . 25 rh. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 2. Top left panel: On-sky position. Black plus symbol mark the position of the M54 star cluster. Contour lines represent the distribution of Gaia DR3 Sgr candidate members selected on their proper motions as in our kinematical cut. Contour lines are mark the position at which the number of Sgr candidate members decreases by a factor of 2, 4, and 6, respectively. Top right panel: reduced proper motion space. Bottom left: metallicity -radial velocity distribution. Bottom right: Kiel diagram of the PIGS / AAT data, including the Inner Galaxy and Sgr. Blue circles mark the final Sgr cut (356 stars), coral circles denote the Sgr stars from the kinematical selection (834 stars), brown crosses indicate the Sgr members selected from APOGEE DR17, and the grey dots correspond to the PIGS / AAT sample of the inner Galaxy. Sgr members from APOGEE DR17 have been selected imposing a similar RV and proper motion cut as our sample, a high signal-to-noise ratio in the spectra ( > 70) to ensure good quality in the RV and [Fe / H]. APOGEE DR17 stars are not displayed in the Kiel diagram to better highlight PIGS data. \n<!-- image --> \nTo derive the systemic RV and the RV dispersion, a Bayesian framework embedded in a Monte Carlo Markov chain, based on the Metropolis-Hastings algorithm, is employed. The prior probability distribution is a step function and it expects these quantities to be in the ranges 90 ≤ RV ≤ 220 km s -1 and σ RV ≤ 40 km s -1 . The likelihood is a Gaussian distribution centred on the systemic RV and with a dispersion that takes into account the intrinsic RV dispersion of the system and the uncertainties of the RV measurements. The systemic RV, < RV > , vs velocity dispersion, σ RV, are displayed in Figure 4 and reported in Table 1. As reference, Figure 4 also displays the values for the populations from Minelli et al. (2023), for metal-rich stars ([Fe / H] > -0 . 6, \nblue small circle) and metal-poor stars ([Fe / H] ≤ -0 . 6, but almost no stars with [Fe / H] < -1 . 0, black small circle). We checked for possible systematics between the APOGEE and PIGS radial velocities by comparing both surveys to Gaia radial velocities (not limited to Sgr to have many more stars). The di ff erence ∆ RV(PIGS -Gaia ) = + 0 . 5 km s -1 (Ardern-Arentsen et al. 2024) and ∆ RV(APOGEE -Gaia ) = + 0 . 2 km s -1 , implying there is only a ∼ 0 . 3 km s -1 systematic di ff erence between APOGEE and PIGS. \nThe overall metal-poor (large blue circle) and metal-rich (large black circle) populations have a systemic RV of 145 . 4 ± 0 . 9 km s -1 and of ∼ 142 . 6 ± 0 . 7 km s -1 , respectively. These val- \nFig. 3. Number of stars as a function of the projected elliptical distance. The metal-poor ([Fe / H] < -1 . 5) and metal-rich ([Fe / H] > -0 . 6) are denoted by the blue and black line, respectively. Right ascensions and declinations are converted to tangential plane coordinate assuming the centre of the system as α 0 = 283 . 764 deg and δ 0 = -30 . 480 deg. The half-light radius is assumed to be 2 . 6 kpc (Majewski et al. 2003; Mucciarelli et al. 2017) at a distance of 26 . 5 kpc (Vasiliev & Belokurov 2020), ellipticity ∼ 0 . 57 and position angle ∼ -104 deg as in Vitali et al. (2022). \n<!-- image --> \nare compatible with the ones inferred by Minelli et al. (2023, small circles) adopting a di ff erent cut in [Fe / H] and a di ff erent dataset. The di ff erence in the systemic RV between these populations is significant, given the uncertainties and the precision of the RVs. We did not take into account projection e ff ects. \nRecently, An et al. (2024) modelled the RV distributions of Sgr and M54, and inferred a di ff erence of 4 km s -1 between M54 (magenta cross marker) and the main body of Sgr (magenta small circle), with mean radial velocities of 139 . 6 ± 0 . 9 km s -1 for M54 and 143 . 7 ± 0 . 7 km s -1 for the main body, with a velocity gradient in the main body. While our sample does not include stars from M54, our estimate of the systemic velocity for the main body is 144 . 1 ± 0 . 5 km s -1 (magenta large circle), which is compatible with the value from An et al. (2024). Our results suggest that there is additionally a di ff erence between MP and MR Sgr field populations - there appears to be an increasing mean RV going from M54, to metal-rich field stars, to metal-poor field stars. \nIn agreement with previous work on the stream and core (e.g. Gibbons et al. 2017; Johnson et al. 2020; Peñarrubia & Petersen 2021; Vitali et al. 2022; Limberg et al. 2023; Minelli et al. 2023), we find that the overall metal-poor population has a velocity dispersion larger than the metal-rich counterpart, in our case σ RV ∼ 17 km s -1 vs σ RV ∼ 12 km s -1 , respectively. Also to be noted from Figure 4: the inner populations in our analysis (large squares), both MP and MR, have lower RV dispersion and lower systemic RV than their respective outer populations (large plus markers). For all the populations and subgroups, the velocity dispersion and the systemic velocity are found to be considerably higher than the values for M54 (magenta cross marker). The latter has been classified as a nuclear star cluster, which might explain its higher velocity dispersion compared to isolated globular clusters. Its velocity dispersion might have been inflated by the dark matter halo of Sgr, by tidal interactions, and by the multiple burst of star formation (e.g. Carlberg & Grillmair 2022; \n1 \nFig. 4. RV dispersion vs systemic RV. Large blue and black markers denote the metal-poor ([Fe / H] ≤ -1 . 5, MP) stars from Sgr / AAT and the metal-rich Sgr population ([Fe / H] > -0 . 6, MR) from APOGEE DR17, respectively. The large magenta circle marks the position of the main Sgr's body in this space, considering data from APOGEE and Sgr / AAT. Squares, plusses, and circles correspond to the inner (projected distance < 0 . 25rh), the outer ( ≥ 0 . 25rh), and the whole population, respectively. A systematic error of 2 km s -1 is added in quadrature to the RV uncertainties of Sgr / AAT data. The blue and black small circles mark the MP ([Fe / H] ≤ -0 . 6) and MR ([Fe / H] > -0 . 6) populations of Sgr from Minelli et al. (2023), respectively. Magenta cross and small circle denote M54 and the main body of Sgr as measured by An et al. (2024). Horizontal solid lines indicate the RV dispersion of the other classical DGs (McConnachie & Venn 2020). \n<!-- image --> \nKacharov et al. 2022; Herlan et al. 2023; Gray et al. 2024). The MP and MR populations should not be contaminated by many M54 members.", '3.1. Internal and external mechanisms in play': "Various internal and external mechanisms can a ff ect the chemodynamical properties of a system. For instance, the internal morphology can play a role. In this regard, a dynamically hotter MP and a colder MR population with weak rotation has been proposed to indicate the presence of a metal-rich thick and rotating disc or bar surrounded by a more dispersed and metal-poor stellar halo in Sgr (Mayer et al. 2001; Sánchez-Janssen et al. 2010; Kazantzidis et al. 2011; del Pino et al. 2021; Carlberg & Grillmair 2022; Minelli et al. 2023; Lokas 2024). Both observations \nTable 1. Systemic RVs and RVs dispersions. \nNotes. Systemic RVs and RVs dispersions for the metal-poor and metalrich populations and for the whole body. The values for the inner, outer, and whole groups are reported, together with the source of the datasets. \nand simulations suggest that the rotating bar should have a length of 2 -2 . 5 kpc (del Pino et al. 2021; Lokas 2024), which correspond to an elliptical radius of 0 . 8 -1 . 0rh. As shown in Figure 3, the majority of the stars from both APOGEE and PIGS lies within 1 half-light radius. The presence of such a rotating disc / bar would also explain some chemo-dynamical properties of the stellar streams associated with Sgr (Peñarrubia et al. 2010; Oria et al. 2022; Carlberg & Grillmair 2022). The fact that the MR population, either in the inner or in the outer regions, has a lower velocity dispersion and a lower systemic RV than the MP supports the idea that these two groups populate two di ff erent structures, such as a 'disc / bar' and a stellar 'halo' of Sgr. If so, projection e ff ects on the bar are another ingredient explaining the di ff erent systemic RV from the MP group. \nAdditionally, outside-in star formation has been proposed as one mechanism to explain the di ff erent spatial and kinematical properties between MP and MR populations in DGs, such as the gradient in the velocity dispersion (e.g. Tolstoy et al. 2004; Battaglia et al. 2006, 2008; Zhang et al. 2012; Hidalgo et al. 2013; Benítez-Llambay et al. 2016; Revaz & Jablonka 2018; Sestito et al. 2023a,c; Tolstoy et al. 2023). In this scenario, the oldest MP population would form spatially everywhere in the system, and their supernovae would enrich the ISM. Then some of the gas might have sunk to the inner region with time, forming younger and more metal-rich stars that are more gravitationally bound to the system. As a result of this, the MP population would be more spatially extended and kinematically hotter than the MR one, with the latter being confined mostly to the inner regions with a lower velocity dispersion. \nThe main external mechanisms that can a ff ect the dynamical properties of a DG are merging events and tidal stripping. In case of the former, stars will be heated up by the accreted system, and likely the less bound ones, such as in the outskirts, will be more a ff ected. Then, the additional gas from the accreted system (if it has any) can sink into the inner regions, triggering the formation of new stars that are more metal-rich (BenítezLlambay et al. 2016). In addition, tidal stripping also influences the distribution and kinematics of the outskirts, which are less \nbound, of a system. In fact, the ongoing stripping of Sgr resulted in the formation of the Sgr stellar streams, which are known to be more metal-poor on average than the core (e.g. Hayes et al. 2020; Limberg et al. 2023; Cunningham et al. 2024). It has been proposed that Sgr has interacted gravitationally with the MW for more than 8 Gyr, with its first pericentric passage likely to have happened around 5 -6 Gyr ago (Law & Majewski 2010; RuizLara et al. 2020). The MR population in Sgr has an estimated age spanning from 4 to 8 Gyr - their star formation, or part of it, might have have been triggered by Galactic perturbations at, or close to, the first pericentric passage. Investigations on simulated galaxies reveals that the extreme tidal e ff ects that Sgr is undergoing might have a ff ected the system's morphology, e.g. it could have reshaped its disc (if it had one) into a prolate rotating bar structure (Lokas 2024).", '3.2. Comparison to other DGs': "The values of the velocity dispersion for the other 6 classical DGs (horizontal lines, McConnachie & Venn 2020) are also reported in Figure 4 as a reference. The velocity dispersion for the MR population in Sgr is similar to Fornax's value, which is the highest among the DGs compilation. The σ RV for the MP population in Sgr is significantly higher than the averages for the other DGs. This could be due to an observational bias, such as the σ RV in the reference galaxies are calculated from the overall population, which is mostly more metal-rich than the MP population in Sgr. As an example, the velocity dispersion for the overall population in Sculptor is around 7 km s -1 , while restricting to the more dispersed metal-poor stars would provide a σ RV ∼ 10 -12 km s -1 (Tolstoy et al. 2004; Battaglia et al. 2008; Walker & Peñarrubia 2011; Tolstoy et al. 2023; Sestito et al. 2023a). In addition, Sgr has a total mass higher than the other DGs reported in Figure 4 and it is experiencing strong Galactic tidal stripping, which is far more extreme than in the other systems (e.g. Battaglia et al. 2022; Pace et al. 2022, and references therein), which both concur to inflate the σ RV of this system.", '4. Carbon trends in Sagittarius': 'We next focus our attention on the chemistry of Sgr, specifically the abundance of carbon. As discussed in the Introduction, carbon abundances can trace the early chemical evolution of a system (e.g. Frebel et al. 2007; Vincenzo & Kobayashi 2018; Kobayashi et al. 2020). Is the level of carbon in Sgr similar to that in the other classical DGs? What about in comparison with the inner Galaxy and the MW halo? To answer these questions, in Figure 5 we present the average [C / Fe] ratio as a function of the metallicity for Sgr (red circles) compared to the classical DGs (left panel) and compared to the inner Galaxy and the MW halo (right panel). All carbon abundances have been corrected for evolutionary e ff ects according to Placco et al. (2014), see Arentsen et al. (2021) for details. We find that the Sgr carbon abundance slightly rises with decreasing metallicity.', '4.1. Halo and inner Milky Way': "There are a number of studies that have explored the carbon abundance of low-metallicity stars in the MW halo, and to a lesser extent in the inner Galaxy. Arentsen et al. (2022) showed that trends involving carbon abundances are very sensitive to the assumptions made in the synthetic spectroscopic grids (e.g. the model atmospheres, the adopted atomic and molecular data) \nFig. 5. Average [C / Fe] vs [Fe / H] divided into 7 metallicity bins ( ∼ 0 . 25 dex). CEMP stars have been removed and carbon abundances have been corrected for evolutionary e ff ects according to Placco et al. (2014). Left panel: comparison with classical DGs (coloured markers), Car, Dra, Fnx, Scl, Sex, and Umi are from Lucchesi et al. (2024), while LMC data is from Chiti et al. (2024) and Oh et al. (2024). [C / Fe] in classical DGs for which there are less than 2 stars are not displayed. Right panel: Comparison with the MW. MW halo stars (black markers) are from Aguado et al. (2019), revised as in Arentsen et al. (2022). Inner Galaxy from PIGS / AAT (grey markers, Ardern-Arentsen et al. 2024) are divided into three groups, the whole sample (grey circles, solid line), the stars confined into the inner regions (grey crosses, dash-dot line, apocenter < 3 kpc) and the halo interlopers (grey plusses, dash-line, apocenter > 8 kpc). Bins populated by less than 5 stars are removed. MW stars from PIGS are selected to have log g < 2 . 3, while compilation from Aguado et al. (2019) is restricted to stars with log g < 3 . 0. In both cases, AGBs are removed. [C / Fe] ratios from all the datasets are corrected for the evolutionary e ff ects as in Placco et al. (2014). An o ff set of up to ± 0 . 05 is added to the metallicity bins of the MW and DGs compilations to better display the markers and the uncertainties on the average [C / Fe]. \n<!-- image --> \nand the employed pipeline, with large systematic o ff sets between di ff erent literature samples (see their Figure 4). To not bias our conclusions, in the comparison with the halo and the inner Galaxy, we restrict ourselves to [C / Fe] measured within PIGS and the Pristine survey, which have all been derived with the same methodology. \nThe inner Galaxy PIGS / AAT sample is selected from Ardern-Arentsen et al. (2024), restricted to those stars with good measurements of stellar parameters, metallicities and carbon abundances, as in our Sgr sample. An additional cut is imposed to select stars with similar surface gravity as the bulk of the Sgr sample (log g < 2 . 3) and to remove the region of early asymptotic giant branch stars (eAGBs) whose carbon abundances have been altered by stellar evolution (Arentsen et al. 2021). This selection is composed of 2318 stars with [Fe / H] < -1 . 5 (grey circles in the right-hand panel of Figure 5). Additionally, this sample is split into two sub-groups according to their Galactic apocentric distances, those that remain confined in the inner Galaxy (apocentre < 3 kpc, grey crosses) and the 'halo interlopers' (apocentre > 8 kpc, grey plus markers). The former and the latter are composed of 1032 and 276 stars, respectively. For the \nMWhalo, we include the Pristine medium-resolution follow-up sample from Aguado et al. (2019, 141 stars, black plus markers), with carbon abundances corrected for spurious log g determinations following Arentsen et al. (2022). This sample has a less restrictive cut on the surface gravity, namely log g < 3 . 0. \nAlthough the same trend is visible for the Milky Way and Sgr samples, namely a rise in carbon abundance with decreasing metallicity, the average carbon abundances are higher in the Milky Way samples compared to Sgr, and the rise appears to be less steep in Sgr. The [C / Fe] di ff erence between Sgr and the Milky Way starts at ∼ 0 . 1 dex for [Fe / H] = -1 . 6 and increases to 0 . 3 -0 . 4 dex for [Fe / H] < -2 . 5. \nThe average carbon abundance of the MW (inner regions and the halo) is also higher than most of the classical DGs, except for Fornax (Fnx, gold squares). The di ff erence in carbon abundances can be interpreted as a di ff erent population of SNe II and AGB stars that contributed to the chemical enrichment of the dwarf galaxies in comparison with the one of the Galaxy. In particular, a higher contribution of faint and core-collapse SNe could provide a higher [C / Fe] ratio (Umeda & Nomoto 2003; Limongi & \nChie ffi 2003; Iwamoto et al. 2005; Kobayashi et al. 2006, 2020; Vanni et al. 2023). \nThe physical and chemical properties of the building blocks that contributed to the formation of the proto-Galaxy are still under discussion (e.g. Schiavon et al. 2017; Helmi 2020; Santistevan et al. 2021; Sestito et al. 2021), as well as the importance of an ancient in-situ component (Belokurov & Kravtsov 2022, 2023). Did the early building blocks have a chemical evolution similar of the present UFDs? What about their masses and sizes, or, in other words, are the building blocks comparable to classical DGs or to smaller UFDs (see Deason et al. 2016)? \nFor the PIGS inner Galaxy sample, there is a slight di ff erence in the average level of carbon abundance between the 'confined' (plusses, lower [C / Fe]) and 'halo interloper' (crosses, higher [C / Fe]) samples, of the order of 0.05-0.10 dex. This could potentially be connected to di ff erent building blocks contributing to these populations, e.g. more chemically evolved ones to the confined population and more chemically pristine systems to the halo population (see also the discussion in Ardern-Arentsen et al. 2024). We further discuss the connection to dwarf galaxies and their chemical evolution in Section 4.3.", '4.2. Note on possible systematics': "As previously discussed, the PIGS / AAT inner Galaxy and the Sgr stars have been analysed with the same methodology applied to the same AAT spectra, and the Aguado et al. (2019) sample has been analysed with the same methodology as well, so systematic di ff erences should hopefully be minimal. One caveat here is that [ α / Fe] is fixed in the analysis, to + 0 . 4. However, various high-resolution spectroscopic works showed that the majority of the inner Galaxy VMP stars have similar [ α / Fe] compared to typical halo stars (Howes et al. 2014, 2015, 2016; Sestito et al. 2023b), and the α -abundances are also very similar between the MW and Sgr in the VMP regime (Hansen et al. 2018a; Sestito et al. 2024b). Therefore, we should not expect significant biases in the FERRE analyses due to [ α / Fe] di ff erences. \nWe note that the magnitude of the evolutionary carbon correction following Placco et al. (2014) also depends on the natal nitrogen abundances of stars, which may di ff er for each formation site, but are all assumed to be [N / Fe] = 0 . 0 in the calculations. However, the predicted e ff ect on the carbon corrections is much smaller than the di ff erence we find between Sgr and the Milky Way - Figure 1 of Placco et al. (2014) shows that for a [Fe / H] = -2 . 3 star, the di ff erence in the carbon correction between a [N / Fe] of -0 . 5 and + 0 . 5 at birth is at most ∼ 0 . 05 dex. Therefore, a di ff erent average level of [N / Fe] between the MW and Sgr would not impact our findings. The evolutionary corrections may also potentially be better or worse in some parts of the parameter space (e.g. depending on log g ), so it is crucial to compare stars in similar evolutionary phases. We attempted this by limiting the reference samples in log g , but the distributions of evolutionary phases are not exactly the same. \nWhat might be the e ff ect of photometric selection e ff ects on trends of carbon? As discussed previously, very carbon-rich stars are likely excluded from our selection because they look too metal-rich. Could our selection be biased even for 'carbonnormal' stars, selecting only those with relatively lower carbon abundances? This is unlikely to be the case, especially for [Fe / H] < -2, given that the carbon features are relatively weak for carbon-normal VMP stars and given that our selection was not only targeting VMP stars, but also probed the slightly more metal-rich population. \nFinally, we checked potential systematics on the mean [C / Fe] and its trend with metallicity as a function of the surface gravity. As a sanity check, we repeated the exercise of Figure 5, restricting the Sgr and MW compilations to stars with 1 . 8 < log g < 2 . 3 (for lower log g , the Placco et al. 2014 evolutionary carbon corrections become more important). We find no qualitative or quantitative di ff erences between this more strict cut and the one applied to produce Figure 5. However, we note that the MW halo sample from Aguado et al. (2019) would not have enough stars to populate all the metallicity bins for this limited log g selection.", '4.3. Dwarf galaxies': "To compare the average [C / Fe] of Sgr with classical DGs, stars with [Fe / H] ≤ -1 . 5 have been selected from the DG members summarised in Lucchesi et al. (2024), Chiti et al. (2024), and Oh et al. (2024). The compilation from Lucchesi et al. (2024) is composed of 442 stars (16 CEMP-no) and distributed in 7 classical DGs, namely Canes Venatici I (CVn I, 1 star, Yoon et al. 2020), Carina (Car, 8 stars, Venn et al. 2012; Susmitha et al. 2017; Lucchesi et al. 2024), Draco (Dra, 161 stars, Kirby et al. 2015a), Fornax (Fnx, 14 stars, Tafelmeyer et al. 2010; Kirby et al. 2015a; Lucchesi et al. 2024), Sculptor (Scl, 173 stars, Kirby & Cohen 2012; Kirby et al. 2015a; Skúladóttir et al. 2015, 2024b), Sextans (Sex, 4 stars, Tafelmeyer et al. 2010; Lucchesi et al. 2020), Ursa Minor (UMi, 81 stars, Kirby & Cohen 2012; Kirby et al. 2015a). The compilations from Chiti et al. (2024) and Oh et al. (2024) include members of the Large Magellanic Cloud (LMC) for a total of 21 stars (no CEMP). The systems from these compilations, excluding CVn I and CEMP-no stars, are displayed in Figure 5 with coloured circles, diamonds, squares, and plusses. \nThe average level of [C / Fe] in Sgr is within the wide range of the 7 classical DGs. In particular, the average carbon abundance in Sgr appears to be higher than in Scl for [Fe / H] > -2 . 4 by up to ∼ 0 . 3 dex. Compared to Car, Sgr's [C / Fe] level is also higher, for [Fe / H] ≲ -2 . 4 by at least ∼ 0 . 3 dex. As proposed by Skúladóttir et al. (2024b), the strikingly low amount of [C / Fe] in Scl and Car might be explained by a strong imprint of hypernovae from Pop III stars. Thus, classical DGs and stars with such low carbon level might be crucial for understanding the energy distribution of the primordial generation of stars (e.g. Koutsouridou et al. 2023). \nAnother nucleosynthetic channel that contributes to lower the [C / Fe] is from SNe Ia, in which the production of Fe exceeds that of C (Iwamoto et al. 1999). This event might be responsible for lowering the [C / Fe] in Dra and UMi for [Fe / H] ≳ -2 . 5, as also shown in Kirby et al. (2015b). Chemical abundance analysis from Cohen & Huang (2009) reveals that the level of [C / Fe] in Dra strongly decreases around [Fe / H] ∼ -2 . 5, such as the metallicity at which SNe Ia starts to kick in. Similarly, Sestito et al. (2023c) discovered that the contribution of SNe Ia in UMi starts at [Fe / H] ∼ -2 . 1. In Sgr, the contribution of SNe Ia is absent in the VMP regime. However, Sestito et al. (2024b) suggest that the trend of [Co / Fe] at [Fe / H] ≳ -2 . 0 might be an indication of a possible contribution of SNe Ia in Sgr. This can also explain the lower [C / Fe] at [Fe / H] ≳ -2 . 0 compared to the more metal-poor bins. A more thorough investigation of this metallicity regime in Sgr will be explored by PIGS in a coming paper (Vitali et al., in prep.). \nSestito et al. (2024b) discussed the early chemical enrichment phase of Sgr from the detailed chemical abundances of 11 VMP stars. The chemical pattern of Sgr stars has been interpreted as the result of a mixture of Pop III and II stars contribut- \ning (Sestito et al. 2024b). In particular, intermediate-mass highenergy and hypernovae are needed to explain the abundance patterns of the lighter elements up to the Fe-peak, while compact binary merger events and fast-rotating (up to ∼ 300 km s -1 ) intermediate-mass to massive metal-poor stars ( ∼ 25 -120 M ⊙ ) are needed to account for the level of the heavy elements. No evidence for contributions from pair-instability supernovae has been found in Sestito et al. (2024b). This mixture of various energetic SNe events appears to be common in classical DGs, and therefore explain the similarity in [C / Fe] between these systems and their lower level compared to the MW, see Section below for a further discussion on this topic.", '4.4. The different supernovae enrichment': "The di ff erent amount of [C / Fe] among the classical DGs and their lower level compared to the MW can be interpreted as the imprint of a di ff erent chemical evolution and a di ff erent e ffi -ciency in retaining the ejecta of SNe. For instance, the chemical evolution models from Vanni et al. (2023) suggest that DGs would have been polluted by a mixture of SNe II from Population III and II stars vs a more pristine population of SNe II in the building blocks of the MW halo (see also Skúladóttir et al. 2024b). The higher fraction of Pop II would have contributed to partially lower the average [C / Fe] (Vanni et al. 2023). \nIn addition, the ISM of classical DGs is considered to be homogeneously mixed, therefore able to have retained the ejected yields from the most energetic events (Skúladóttir et al. 2024b), such as high-energy SNe II, hypernovae, and potential pairinstability SNe II. The retention of the ejected yields from the most energetic events would lower the average amount of [C / Fe], given they would produce more Fe than C (e.g. Limongi & Chie ffi 2018; Kobayashi et al. 2020; Koutsouridou et al. 2023; Vanni et al. 2023). \nWhile there is a consensus that massive systems would contribute to the formation of the MW (e.g. Deason et al. 2016), it is still an open question whether the MW's building blocks resembled UFDs or DGs in terms of their ISM e ffi ciency in retaining SNe yields or regarding their star formation history or their initial mass function. We interpret the higher average [C / Fe] of the MW as an indication that the ISM e ffi ciency of the MW's building blocks is similar to UFDs, hence unable to retain the most energetic events (e.g. Ji et al. 2016; Roederer et al. 2016; Hansen et al. 2017; Kobayashi et al. 2020; Applebaum et al. 2021; Waller et al. 2023; Sestito et al. 2024a). Therefore, the ISM of the building blocks of the MW, should be the fossil of the lower energetic events only (Koutsouridou et al. 2023; Vanni et al. 2023; Skúladóttir et al. 2024b). Additionally, if inhomogeneous chemical enrichment is in place, asymptotic giant branch stars (AGBs) can also be an extra source for the level of carbon, even at lower metallicities (Kobayashi et al. 2014; Vincenzo & Kobayashi 2018; Kobayashi et al. 2020). \nFigure 5 also shows a di ff erence in the average [C / Fe] between the MW halo and the inner Galaxy, especially those stars confined within 3 kpc. Recently, Pagnini et al. (2023) suggested that a potential dearth of CEMP stars in the inner Galaxy could be due to the very high star formation rates at early times. The star formation would be so intense that stars massive enough to explode as pair-instability SNe would form, which would lower the average [C / Fe] compared to the halo. However, no star carrying the imprint of pair-instability SNe has been found so far in the Galaxy (e.g. Lucey et al. 2022; Sestito et al. 2023b; Skúladóttir et al. 2024a). \nFurthermore, SNe Ia can concur to lower the average [C / Fe] in a given system (Iwamoto et al. 1999). The contribution of SNe Ia might start at [Fe / H] ≳ -2 . 5 in some classical DGs (e.g. Cohen & Huang 2009; Venn et al. 2012; Kirby et al. 2015b; Sestito et al. 2023c), and likely between -2 . 0 ≲ [Fe / H] ≲ -1 . 5 for Sgr (Sestito et al. 2024b). This is not the case for the MW, where SNe Ia starts to kick in at higher metallicities, [Fe / H] ∼ -1 . 0 (e.g. McWilliam 1997; Matteucci 2003; Venn et al. 2004). Therefore, the lower average [C / Fe] at [Fe / H] ≳ -2 . 5 in DGs and at [Fe / H] ≳ -2 . 0 in Sgr can also be caused by the contribution of SNe Ia.", '4.5. The radial gradient of [C/Fe]': "Our sample is large enough and covers enough of Sgr to test whether there may be any radial gradients in [C / Fe]. To avoid potential systematic e ff ects in [C / Fe] between radial bins due to di ff erences in stellar parameter coverage, we limit the sample to 1 . 8 < log g < 2 . 3 for this analysis. We find that the general picture of our results does not change compared to using a more generous cut or the full sample, but the behaviour is cleaner for the limited sample. \nThe median [C / Fe] as a function of the projected elliptical distance is shown in Figure 6. The Sgr PIGS / AAT sample is divided into two sub-groups, the low-metallicity (blue circles, -2 . 5 ≤ [Fe / H] ≤ -2 . 0) and a slightly more metal-rich group (navy circles, -2 . 0 < [Fe / H] ≤ -1 . 5), and removing CEMP stars from the calculations. There is a net positive [C / Fe] gradient for the slightly more metal-rich sub-group, with a di ff erence of ∼ + 0 . 25 dex between the very inner region and the outskirts of Sgr. This leads to a positive gradient in [C / Fe] of about ∇ [C / Fe] ∼ 0 . 23 dex r -1 h or ∼ 8 . 8 × 10 -2 dex kpc -1 or ∼ 6 . 8 × 10 -4 dex arcmin -1 . \nRegarding the low-metallicity sub-group, a mild positive gradient is visible if the innermost bin is not considered. In this case, the di ff erence in [C / Fe] would be ∼ 0 . 2 dex between the inner to the outer Sgr's regions. To be taken into account, uncertainties on the average [C / Fe] are larger for the low-metallicity sub-group than the more metal-rich one. \nIs the more pronounced gradient at higher metallicities connected to a di ff erent chemical enrichment between the two populations? A couple of concurrent mechanisms might explain these gradients: outside-in star formation and the contribution of SNe Ia. \nThe former, as discussed in Section 3, implies that the oldest and most metal-poor stars should form everywhere in the system and would carry a similar imprint of nucleosynthetic events, if also homogeneous mixing applies to the system. In the case that the ISM is not completely homogeneously mixed between the inner regions and the outskirts, these two regions might carry di ff erent level of [C / Fe]. Likely, the outskirts would be less e ffi -cient in retaining the more energetic events as the inner regions, resulting in a higher average [C / Fe]. \nThe stellar feedback from the first supernovae would expel the gas outside the system, which then later would be reaccreted onto the inner regions, where slightly more metal-rich stars would form. These relatively more metal-rich inner stars might carry the imprint of SNe Ia as well. As discussed in Section 4.4, SNe Ia can lower the average [C / Fe] (Iwamoto et al. 1999), and the higher contribution of these events in the inner regions would explain the positive gradient in [C / Fe]. This result would be an indication, in addition to the trend of [Co / Fe] in Sestito et al. (2024b), that SNe Ia might have started to kick \nFig. 6. Median [C / Fe] as a function of the projected elliptical distance. Stars from the final selection of Sgr PIGS / AAT. The median is obtained removing the sample from CEMP stars and dividing it into distance bins and into two sub-groups, the more metal-poor (blue circles, -2 . 5 ≤ [Fe / H] ≤ -2 . 0) and the slightly more metal-rich (navy circles, [Fe / H] > -2 . 0). Stars have been selected to have 1 . 8 < log g < 2 . 3. \n<!-- image --> \nin in Sgr at metallicities between -2 . 0 < [Fe / H] < -1 . 5, well below what was previously inferred ([Fe / H] ∼ -1 . 27, e.g. de Boer et al. 2014).", '5. CEMP stars': 'As discussed in the Introduction, CEMP stars are of interest because they probe the properties of the First Stars and early chemical evolution (CEMP-no) and of binary populations (CEMP-s). Next, we investigate the properties of CEMP stars in Sgr with the PIGS / AAT Sgr data set, which is much larger than previous literature samples with [C / Fe] in Sgr. \nTo our sample of carbon measurements in Sgr, we add those of Chiti & Frebel (2019) and Chiti et al. (2020), who observed metal-poor Sgr stars with the Magellan Echellette (MagE) Spectrograph, measuring [C / Fe] for 4 and 18 targets, respectively. These stars have metallicities in the range -3 . 1 ≲ [Fe / H] ≲ -1 . 5, similarly to the PIGS / AAT range. None of these stars are CEMP according to the standard definition ([C / Fe] > + 0 . 7). Other Sgr members with measured [C / Fe] that are not included are the targets analysed in Hansen et al. (2018a) and from APOGEE DR17. Hansen et al. (2018a) measured [C / Fe] in 12 stars with metallicity -2 . 95 ≲ [Fe / H] ≲ -1 . 40. These targets were observed with UVES high-resolution spectrograph at VLT. However, as shown in Sestito et al. (2024b), the [C / Fe] ratios from Hansen et al. (2018a) are systematically lower than the ones from Chiti & Frebel (2019), Chiti et al. (2020), and this work (see Figure 5 in Sestito et al. 2024b). APOGEE stars are not included, since the C-measurements are in non-local thermodynamic equilibrium (non-LTE) and in the infra-red, which have o ff sets compared to LTE measurements in the optical (Jönsson et al. 2020).', '5.1. New CEMP stars in Sgr': "The distribution of [Fe / H] vs A(C) for Sgr stars is shown in Figure 7 (blue circles). According to the classical definition of \n- PIGS/AAT\n- DGs\n- Sagittarius \nP185855-301522 \nP190122-304744 \n<!-- image --> \n<!-- image --> \nFig. 7. Abundance of C, A(C), as as function of [Fe / H]. The Sagittarius sample (blue circles) includes stars from the final selection made in Section 2.3, from Chiti & Frebel (2019), and from Chiti et al. (2020). The DGs compilation (orange circles) is from Lucchesi et al. (2024). Inner Galaxy stars (grey circles) are selected from Ardern-Arentsen et al. (2024) to have good quality of the AAT spectra and good FERRE measurements as in our sample. Star P185855 -301522 (red pentagon) is analysed in Sestito et al. (2024b) and confirmed to be CEMP-s from high-resolution spectroscopy. Star P190122-304744 (purple square) is one of the two cool CEMP candidates discussed in Section 5.2. Horizontal green dashed line tentatively separates CEMP-s from CEMP-no as in Yoon et al. (2016). Stars on the left of the dashed black line have [C / Fe] > + 0 . 7 as defined in Aoki et al. (2007). The dashed red line denotes the tentative new limit for CEMP in Sgr ([C / Fe] = + 0 . 35). Sgr stars with [C / Fe] > + 0 . 35 are displayed with their errorbars to highlight that they are significantly distant from the bulk of the system's distribution. \n<!-- image --> \nCEMP stars ([C / Fe] > + 0 . 7), only 3-4 stars in the PIGS / AAT Sgr sample are classified as CEMP. One of them (red pentagon) has previously been studied in Sestito et al. (2024b), and was confirmed to be a CEMP-s star based on the over-abundance of s-process elements ([Ba / Fe] ∼ + 1 . 2). For the other two CEMP candidates, Ba measurements are not available. We compare the distribution of metallicities and carbon abundances with those for the inner Galaxy (grey circles) and DGs (Lucchesi et al. 2024, orange circles). Note that the DG sample only includes carbon-normal and spectroscopically confirmed CEMP-no stars. \nWithout measurements of Ba or Sr, it is not possible to classify CEMP stars with certainty, although a rough classification can be made based on [Fe / H] and A(C) alone (e.g. Yoon et al. 2016). CEMP-s stars typically have higher A(C) than CEMPno stars and are more common at higher metallicities, and a tentative separation between the two groups has been placed at A(C) = 7 . 1 (Yoon et al. 2016) and [Fe / H] ≳ -3 . 3. It is not entirely clean - there is some known contamination when using such a simple division without detailed chemistry, for example the Sestito et al. (2024b) CEMP-s star lies in the CEMP-no region based \non [Fe / H] and A(C) alone, and some DG CEMP-no stars lie in the CEMP-s region. Similarly, a contamination of CEMP-no in the CEMP-s region is also found for MW halo stars (e.g. Norris & Yong 2019). However, without better data, we may propose that the two new Sgr CEMP stars are likely of the CEMP-s kind given their metallicity and high carbon abundances.", '5.2. Two cool candidate CEMP stars': "We noticed that there are two stars in the AAT / Sgr sample (not passing our FERRE quality cuts, based on χ 2 ) that by eye appear to be very carbon-rich from their spectrum. These stars, Pristine\\_185524.38-291422.5 (Gaia DR3 source\\_id = 6761678859361894912) and Pristine\\_190122.55-304744.3 (6760545743905626496) are highlighted with pink circles in the Pristine colour-colour diagram in the top left and right panels of Figure 1. It is curious that one of them is located above the primary Sgr sequence in the Pristine colour-colour diagram. They are also shown on the CMD with large red symbols in the top panel of Figure 8. The same star that is an outlier in the colourcolour diagram is located beyond the metal-rich side of the RGB, which is also curious. If the star is truly a Sgr star (and there is no reason to suspect it is not given its radial velocity and proper motions), it cannot be an intrinsic carbon star, because it is not evolved enough. \nBoth stars have FERRE T e ff ∼ 4500 K, which is at the cool boundary of the FERRE grid, therefore they could be even cooler. Inspection of the spectroscopic fit shows that the FERRE fit is bad in both the blue and the CaT regions: there is a strong discrepancy between the carbon features in the star and those in the FERRE grid, although it is clear that the star is very carbonrich. This is potentially due to the assumptions on nitrogen in the FERRE grid (see below). \nTo further constrain the stellar parameters for these stars, we employ a di ff erent grid of synthetic spectra originally created for use in the Segue Stellar Parameter Pipeline ( SSPP , Lee et al. 2008a,b, grid from Y.S. Lee, private communication). An important di ff erence between the FERRE and SSPP grids is that the former assumes [N / Fe] = 0, while the latter assumes [C / N] = 0 - this is potentially particularly important for fitting the CN features in the CaT. We use a cool subset of the grid with the following stellar parameters: Te ff = [4000, 4250, 4500, 4750] K, log g = 1 . 0 (we checked that varying log g does not make a di ff erence), [Fe / H] from -3 . 0 to -1 . 0 in steps of 0.25 dex and [C / Fe] from 0 . 0 to + 3 . 0 in steps of 0.25 dex. After normalising both the observed and synthetic spectra with a running median of 200 pixels (50 Å), we search for the best matching spectrum by minimising the residuals. We do this separately for the CaT and the blue and combine the χ 2 values afterwards, giving more weight to the CaT because of its high resolution and because it is less sensitive to the shape of the molecular bands. \nFor both of the stars there is no clear best-fit stellar parameter combination, because there are strong degeneracies between Te ff , [Fe / H] and [C / Fe]. For Pristine\\_185524.38-291422.5, the outlier in photometry, the main constraint is placed on the absolute carbon abundance: for the 5% best fits, A(C) = 8 . 7 ± 0 . 4 (mean and standard deviation). The mean metallicity is -1 . 5 ± 0 . 4 and the temperature is not well-constrained within the limit of our small grid. The other star, Pristine\\_190122.55-304744.3, is more metal-poor and slightly less carbon-rich - the mean A(C) = 8 . 0 ± 0 . 5 and [Fe / H] = -2 . 2 ± 0 . 5 for the 5% best fits, and the temperature is again not well-constrained. For each of these stars, we present one of the best matching synthetic spectra in \nFig. 8. Top: colour-magnitude diagram of stars in Sagittarius (same samples as grey dots in Figure 1), with the CEMP candidates Pristine\\_185524.38-291422.5 (circle) and Pristine\\_190122.55304744.3 (square) highlighted with large red symbols. Middle: one of the best matching spectra from the SSPP synthetic grid for Pristine\\_185524.38-291422.5 on top of its AAT spectrum. Bottom: same but for Pristine\\_190122.55-304744.3. \n<!-- image --> \nFigure 8, with the observed spectrum in black and the synthetic one in red. We applied a by-eye linear normalisation to the blue arm synthetic spectrum to roughly match the shape of the observed spectrum rather than showing the normalised version, so the match is not perfect. \nWe conclude that these stars are likely CH- or CEMP-s stars. The location of the more metal-rich star in the Pristine colour- \ncolour diagram and the CMD is likely strongly a ff ected by the very large carbon bands, causing the star to look fainter and redder compared to where a 'normal' metal-poor star would be. This e ff ect appears to be less strong for the more metal-poor star, although it is on the border of having been included in our selection according to Figure 1. Such extreme stars have likely been missed in other selections of metal-poor stars as well, in DGs and the Milky Way, possibly leading to an underestimate of the number of binary mass-transfer type stars at intermediate metalpoor metallicities.", '5.3. Fraction of CEMP stars': 'In the Galactic halo, the cumulative fraction of CEMP stars for [Fe / H] < -2 . 0 has been found to be of the order of 20 -30%, rising to 30 -40% for [Fe / H] < -3 . 0 (Lee et al. 2013; Placco et al. 2014; Arentsen et al. 2021). There are various caveats complicating the exact determination of the overall CEMP and separate CEMP-no and CEMP-s fractions in the Galactic halo (Arentsen et al. 2021), but the consensus is that there is a significant fraction of these stars at low metallicity. As shown in Figure 7, only three out of 356 PIGS / AAT Sgr stars is classified as CEMP and none from Chiti & Frebel (2019) and Chiti et al. (2020), giving a total percentage of ∼ 3% for [Fe / H] < -2 . 0 and ∼ 5% for [Fe / H] < -2 . 5 - much lower than that claimed in Galactic halo samples. This could partially be the result of our photometric metal-poor candidate selection being biased against carbon-rich stars, especially those at slightly higher metallicity ([Fe / H] > -2 . 5) and / or higher carbon abundance ([C / Fe] > + 1 . 5) - the realm of the CEMP-s stars. \nThe CEMP fraction in Sgr is also low for [Fe / H] < -2 . 5, and we find that none of the 8 Sgr stars with [Fe / H] < -2 . 7 are CEMP. This is interesting given that in our test of the selection function in Section 2.2, we found that CEMP-no stars in this metallicity range should typically not have been excluded from our selection. This finding is consistent with previous observations suggesting that classical DGs are poor in CEMP-no stars in comparison to the MW and UFDs (e.g. Starkenburg et al. 2013; Jablonka et al. 2015; Kirby et al. 2015b; Simon et al. 2015; Hansen et al. 2018b; Lucchesi et al. 2024; Skúladóttir et al. 2015, 2021, 2024b; Chiti et al. 2024).', '5.4. Redefining CEMP stars in DGs': "Given that the average carbon abundance is ∼ 0 . 3 dex lower in Sgr compared to the Milky Way (Figure 5), is it fair to use the same definition of carbon-enhancement as in the Milky Way? This seems to be a generic question for classical DGs, as most of them have lower average [C / Fe] than the Milky Way, as discussed in the previous section, and they would therefore need a larger carbon 'boost' to be classified as CEMP. The LMC also has lower carbon abundances compared to the MW halo (although similar to the inner Galaxy), with a dearth of CEMP stars (Jönsson et al. 2020; Chiti et al. 2024; Oh et al. 2024). The first definition of CEMP stars was [C / Fe] > + 1 . 0 (Beers &Christlieb 2005), which was refined empirically by Aoki et al. (2007) to [C / Fe] > + 0 . 7 based on a sample of observations of MW stars, using the gap between carbon-normal stars and outliers with high carbon abundances. This definition is therefore a relative one, specifically for the Milky Way 'field' population, raising the question whether it should it be redefined for dwarf galaxies. \nInspecting Figure 7, there are a significant number of Sgr stars that appear to be outliers in A(C) from the main Sgr trend, although they do not make it to above the classical CEMP definition of [C / Fe] > + 0 . 7. For [Fe / H] ≲ -2 . 5 in the PIGS / AAT inner Galaxy sample, the average [C / Fe] ≈ + 0 . 3 with a dispersion of 0.2 dex (conservative estimate) - meaning that the [C / Fe] = + 0 . 7 CEMP definition selects stars that are ∼ 2 σ outliers, roughly 0.4 dex higher than the mean trend. The average [C / Fe] in Sgr in the lowest metallicity bins ([Fe / H] ≲ -2 . 5) is ∼ -0 . 05, therefore, adopting a similar conservative dispersion, stars with [C / Fe] > -0 . 05 + 0 . 4 > + 0 . 35 could be considered outliers in Sgr, and therefore CEMP. This working definition of CEMP stars in Sgr is shown in Figure 7 with a dashed red line. Using this new definition, ∼ 20 Sgr members would be classified as CEMP stars (vs 3 -4 from the classical definition). This would lead to a carbon-enhanced percentage of ∼ 15% for [Fe / H] < -2 . 0, which is much less in tension with the results in the MW (20 -30%, Lee et al. 2013; Placco et al. 2014; Arentsen et al. 2021). The percentage would be ∼ 12% for -2 . 5 < [Fe / H] < -2 . 0 and ∼ 30% for [Fe / H] < -2 . 5 (or ∼ 35% if only Sgr / AAT data are considered), compatible with the frequency of CEMP stars in the MW. \nSimilarly, for Dra, UMi, and Scl (selecting stars between -2 . 4 < [Fe / H] < -1 . 9), the new [C / Fe] threshold for a member star to be a CEMP would be ∼ + 0 . 3 , + 0 . 3 , + 0 . 1, respectively. This new limit would suggest that the percentage of CEMP in Dra, UMi, and Scl would be ∼ 16% , 27% , 19%, respectively. However, the latter values refer only to the CEMP-no population, given that the compilation from Lucchesi et al. (2024) does not contain CEMP-s stars. \nWe want to highlight that our new definition of CEMP is strictly empirical and based on the position of outliers in the [C / Fe] or A(C) distribution - they could be enhanced in carbon for a number of reasons. A more physically driven definition should take into account the IMF and the energy ranges of the SNe II exploded in a given system, the contribution of SN Ia and AGBs, the binary fraction, and the e ffi ciency of the system's ISM in recycling the ejected yields. Additionally, investigations of the chemical properties of CEMP candidates based on our relative CEMP definition will be necessary to test whether they show di ff erences in their abundance patterns compared to stars in the bulk of the carbon-metallicity distribution, and whether they are truly a di ff erent population of stars.", '6. Summary': "The chemo-dynamical properties of the low-metallicity regime of the Sagittarius dwarf galaxy are explored using the low / medium-resolution AAT spectra observed by the Pristine Inner Galaxy Survey (PIGS). The PIGS dataset contains measurements of RVs, stellar parameters, [Fe / H], and [C / Fe] for stars towards the inner Galaxy and Sgr. We summarise below the main conclusions of this work: \n- I) We provide a clean list of low-metallicity ([Fe / H] ≤ -1 . 5) members stars selected according to their RV from AAT and proper motion and on-sky position from Gaia, as in Vitali et al. (2022), and updated to DR3 (Figure 2). A table updated to Gaia DR3 will be available as online material.\n- II) The metal-poor ([Fe / H] ≤ -1 . 5) population (PIGS / AAT) of Sgr has a larger velocity dispersion and systemic RV than the metal-rich ([Fe / H] ≥ -0 . 6, APOGEE) as shown in Figures 3 and 4. Additionally, the velocity dispersion and the systemic RV increase in the outer regions for both \npopulations. This e ff ect might be caused by the contribution of various mechanisms, such as the complex structure in Sgr (MR / disc + MP / halo), the outside-in star formation, and the extreme Galactic tidal perturbations acting in the system. \n- III) The average [C / Fe] of Sgr is similar to the range displayed by the other classical DGs (Figure 5). However, the level of [C / Fe] is higher in Sgr than in Car and Scl. This can be explained by di ff erences in the IMF and in the energy distribution of the SNe II among these systems, with a predominance of more energetic events in Car and Scl.\n- IV) The average [C / Fe] of Sgr, and of the other classical DGs, is lower than in the MW at fixed [Fe / H], either when compared to inner Galactic or halo-like stars (Figure 5). The ISM of classical DGs might have been able to retain the ejecta of energetic events, such as hypernovae, while this would not have been the case for the building blocks of the Galaxy, where stochasticity might have played an important role. In this scenario, classical DGs should display the imprint of Population III and II high energy SNe II, which would act to lower the average [C / Fe]. Instead, less energetic events, faint- and core-collapse SNe II from a more pristine population should be imprinted in the stars of the MWbuilding blocks, hence the higher [C / Fe]. On the other hand, some studies (e.g. Deason et al. 2016) suggest that the majority of the MW building blocks should be similar in size to present DGs. However, their chemical evolution still remain an open question. Our results indicate a di ff erent supernovae imprint between Sgr (and classical DGs) vs the MW building blocks.\n- V) SNe Ia can also lower the average [C / Fe]. This kind of event would be already present at [Fe / H] ∼ -2 . 0 in classical DGs and absent in the MW stars at the same metallicities. Indications of the SNe Ia contributions in Sgr starting at -2 . 0 < [Fe / H] < -1 . 5 are the lower median [C / Fe] at these metallicities vs the higher [C / Fe] at lower metallicities (see Figure 5) and also the lower [C / Fe] in the inner regions (see Figure 6), inhabited by a slightly more metalrich population. The presence of SNe Ia at the aforementioned metallicities would also be confirmed by the trend of [Co / Fe] found by Sestito et al. (2024b).\n- VI) We find a positive [C / Fe] gradient of ∇ [C / Fe] ∼ 0 . 23 dex r -1 h or ∼ 8 . 8 × 10 -2 dex kpc -1 or ∼ 6 . 8 × 10 -4 dex arcmin -1 for stars with -2 . 0 < [Fe / H] < -1 . 5 (Figure 6), which we interpret as the e ff ect of contributions by SNe Ia. \nVII) We identify 4 new CEMP stars in Sgr. Figure 7 suggests that the empirical distinction between CEMP-s and CEMPno solely based on A(C) does not work well for Sgr and the classical DGs. We therefore cannot reach definitive conclusions on the nature of the new CEMP stars, however, we may propose that they are likely of the CEMP-s kind given their [Fe / H] and high A(C). \nVIII) The AAT spectra of two carbon-rich candidates, Pristine\\_185524.38-291422.5 and Pristine\\_190122.55304744.3, are re-analysed with the SSPP grid of synthetic spectra (Figure 8) because they had high χ 2 in the FERRE fit and were at the edge of the FERRE grid. They have [Fe / H] ∼ -1 . 5 and -2 . 2 with very high carbon abundances (A(C) ∼ 8 . 8 and 8 . 0, respectively), making them CHor CEMP-s candidates. The C-bands of the former star strongly a ff ect its colour, magnitude and its position in the Pristine colour-colour diagram (Figures 1 and 8). Similar stars could have been missed in other metal-poor (DG) selections as well. \n- IX) The photometric selection e ff ects in the various PIGS fields that include Sgr targets are discussed, showing there is a bias against CEMP stars in the sample (Figure 1), specifically those of the CEMP-s (binary interaction) type. CEMP-no stars (connected to early chemical evolution), however, are less likely to have been excluded from the selection and their frequency in our sample should be largely unbiased.\n- X) Following the classical definition of CEMP stars ([C / Fe] > + 0 . 7), the fraction of CEMP stars in our sample is very low: ∼ 3% for [Fe / H] < -2 . 0 and ∼ 6% for [Fe / H] < -2 . 5. However, the low mean abundance of [C / Fe] in Sgr (and other classical DGs) as well as the clear presence of outliers of the distribution at 'intermediate' carbon abundances, lead us to propose a new definition for CEMP stars. Rather than a fixed threshold, the limit should depend on the average [C / Fe] of a given system. For Sgr, stars with [C / Fe] ≳ + 0 . 35 can be considered CEMP in this case, as they are outliers from the bulk of the system's distribution (see Figure 7). The new frequency of CEMP in Sgr according to this definition would be ∼ 12% for -2 . 5 < [Fe / H] < -2 . 0 and ∼ 30 -35% for [Fe / H] < -2 . 5, much more in agreement with frequencies in the MW. \nThis work, which complements the high-resolution investigation by Sestito et al. (2024b), provides a novel glimpse into the early chemical evolution of Sgr by exploring its level of carbon. These works will be beneficial for upcoming spectroscopic surveys, for example 4DWARFS (Skúladóttir et al. 2023), which will observe a larger number of stars in the Sgr core and in its streams. \nAcknowledgements. We acknowledge and respect the l @ Ĳ k w @ ŋ @ n peoples on whose traditional territory the University of Victoria stands and the Songhees, Esquimalt and WSÁNE ' C peoples whose historical relationships with the land continue to this day. \nWe thank the Australian Astronomical Observatory, which have made the PIGS spectroscopic follow-up observations used in this work possible. We acknowledge the traditional owners of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past and present. \nWe want to thank the anonymous referee for their helpful and insightful comments. \nWe thank Vini Placco for calculating the carbon evolutionary corrections. We thank Young Sun Lee for providing the SSPP synthetic spectra. \n- FS and KAV thank the National Sciences and Engineering Research Council of Canada for funding through the Discovery Grants and CREATE programs. AAA acknowledges support from the Herchel Smith Fellowship at the University of Cambridge and a Fitzwilliam College research fellowship supported by the Isaac Newton Trust. SV thanks ANID (Beca Doctorado Nacional, folio 21220489) and Universidad Diego Portales for the financial support provided. SV acknowledges the Millennium Nucleus ERIS (ERIS NCN2021017) and FONDECYT (Regular number 1231057) for the funding. NFM gratefully acknowledges support from the French National Research Agency (ANR) funded project 'Pristine' (ANR-18-CE31-0017) along with funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 834148). ES acknowledges funding through VIDI grant 'Pushing Galactic Archaeology to its limits' (with project number VI.Vidi.193.093) which is funded by the Dutch Research Council (NWO). This research has been partially funded from a Spinoza award by NWO (SPI 78-411). \nThe spectroscopic follow-up used in this work was based on selection from observations obtained with MegaPrime / MegaCam, a joint project of CFHT and CEA / DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. \n- This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. \nThis research has made use of the SIMBAD database, operated at CDS, Strasbourg, France (Wenger et al. 2000). This work made extensive use of TOPCAT (Taylor 2005). \nAuthor contribution statement. FS led the analysis and the various discussions in this work, contributed to write most of this draft, and created most of the Figures. AAA led the PIGS / AAT target selection and observations, co-led the PIGS / AAT spectroscopic analysis with David Aguado (not a co-author here), analysed the cool candidate CEMP stars discussed in Section 5.2, created the respective section with figures, and was closely involved in shaping the manuscript and contributed to the scientific discussion. SV contributed to the discussion and revision of the paper. MM identified one of the cool candidate CEMP stars using photometry and contributed to the discussion. RL provided the [C / Fe] dataset of the various dwarf galaxies. KAV, NFM, JFN, and ES provided insightful scientific and editorial comments on the manuscript.", 'References': "- Abate, C., Pols, O. R., Izzard, R. G., Mohamed, S. S., & de Mink, S. E. 2013, A&A, 552, A26\n- Abdurro'uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35\n- Aguado, D. S., Youakim, K., González Hernández, J. I., et al. 2019, MNRAS, 490, 2241\n- Allende Prieto, C., Beers, T. C., Wilhelm, R., et al. 2006, ApJ, 636, 804\n- An, Z., Walker, M. G., & Pace, A. B. 2024, arXiv e-prints, arXiv:2404.16184 Aoki, W., Beers, T. C., Christlieb, N., et al. 2007, ApJ, 655, 492\n- Applebaum, E., Brooks, A. M., Christensen, C. R., et al. 2021, ApJ, 906, 96 Ardern-Arentsen, A., Monari, G., Queiroz, A. B. A., et al. 2024, MNRAS, 530, 3391\n- Arentsen, A., Placco, V. M., Lee, Y. S., et al. 2022, MNRAS, 515, 4082\n- Arentsen, A., Starkenburg, E., Aguado, D. 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2024PSJ.....5..153M	Ganymedes auroras are the product of complex interactions between its intrinsic magnetosphere and the surrounding Jovian plasma environment and can be used to derive both atmospheric composition and density. In this study we analyzed a time series of Ganymedes optical auroras taken with Keck IHIRES during eclipse by Jupiter on 2021 June 8 UTC one day after the Juno flyby of Ganymede. The data had sufficient signaltonoise in individual 5 minute observations to allow for the first highcadence analysis of the spatial distribution of the optical aurora brightness and the ratio between the O I 630.0 and 557.7 nm diskintegrated auroral brightnessesa quantity diagnostic of the relative abundances of O OSUB2SUB and HSUB2SUBO in Ganymedes atmosphere. We found that the hemisphere closer to the centrifugal equator of Jupiters magnetosphere where electron number density is highest was up to twice as bright as the opposing hemisphere. The dusk trailing hemisphere subjected to the highest flux of charged particles from Jupiters magnetosphere was also consistently almost twice as bright as the dawn leading hemisphere. We modeled emission from simulated OSUB2SUB and HSUB2SUBO atmospheres during eclipse and found that if Ganymede hosts an HSUB2SUBO sublimation atmosphere in sunlight it must collapse on a faster timescale than expected to explain its absence in our data given our current understanding of Ganymedes surface properties.	2024-07-01T00:00:00Z	['arXiv:2409.06055', '10.48550/arXiv.2409.06055', '2024arXiv240906055M', '2024PSJ.....5..153M', '10.3847/PSJ/ad49a2']	['Aurorae', 'Ganymede', 'Natural satellite atmospheres', 'Optical astronomy', '2192', '2188', '2214', '1776', 'Astrophysics - Earth and Planetary Astrophysics', 'Physics - Space Physics']	Shorttimescale Spatial Variability of Ganymedes Optical Aurora	2,024	191	0.45	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.06055.pdf	{"Short-Timescale Spatial Variability of Ganymede's Optical Aurora": '<!-- image --> \nZACHARIAH MILBY \n, \nKATHERINE DE KLEER \n, \nCARL SCHMIDT \n, \nAND FRANÇOIS LEBLANC \n1 Division of Geological and Planetary Sciences, California Institute of Technology \n3 Laboratoire Atmosphères, Milieux, Observations Spatiales, Centre National de la Recherche Scientifique, Sorbonne Université \n2 Center for Space Physics, Boston University', 'ABSTRACT': "Ganymede's aurora are the product of complex interactions between its intrinsic magnetosphere and the surrounding Jovian plasma environment and can be used to derive both atmospheric composition and density. In this study, we analyzed a time-series of Ganymede's optical aurora taken with Keck I/HIRES during eclipse by Jupiter on 2021-06-08 UTC, one day after the Juno flyby of Ganymede. The data had sufficient signal-to-noise in individual 5-minute observations to allow for the first high cadence analysis of the spatial distribution of the aurora brightness and the ratio between the 630 . 0 and 557 . 7 nm disk-integrated auroral brightnesses-a quantity diagnostic of the relative abundances of O, O 2 and H 2 O in Ganymede's atmosphere. We found that the hemisphere closer to the centrifugal equator of Jupiter's magnetosphere (where electron number density is highest) was up to twice as bright as the opposing hemisphere. The dusk (trailing) hemisphere, subjected to the highest flux of charged particles from Jupiter's magnetosphere, was also consistently almost twice as bright as the dawn (leading) hemisphere. We modeled emission from simulated O 2 and H 2 Oatmospheres during eclipse and found that if Ganymede hosts an H 2 O sublimation atmosphere in sunlight, it must collapse on a faster timescale than expected to explain its absence in our data given our current understanding of Ganymede's surface properties. \nKeywords: Aurorae (2192); Ganymede (2188); Natural satellite atmospheres (2214); Optical astronomy (1776)", '1. INTRODUCTION': "Ganymede's atmosphere is influenced by the presence of an internally-generated dipolar magnetic field offset by about 10 · from its rotation axis (Kivelson et al. 1996). This field redirects charged particles from Jupiter's magnetosphere towards Ganymede's planetographic poles, producing auroral \nThe first detection of the presence of a tenuous atmosphere around Ganymede was made using stellar occultation measurements over half a century ago (Carlson et al. 1973). The initial occultation measurements suggested a surface pressure of 0 . 1 Pa , and follow-up photo-chemical modeling showed that photolysis of H 2 O and preferential escape of H would produce an O 2 -dominated atmosphere (Yung & McElroy 1977). Kumar & Hunten (1982) showed there was an additional stable equilibrium in the Yung & McElroy (1977) model at a much lower surface pressure of 10 -7 Pa , consistent with the upper limit of 10 -6 Pa found using data taken during the Voyager 1 flyby (Broadfoot et al. 1979). \nCorresponding author: Zachariah Milby \[email protected] \novals similar to those on Earth. The interaction between Ganymede's magnetic field and the Jovian magnetosphere in which it resides modifies its structure, in particular the planetographic latitudes on Ganymede at which the boundaries between open and closed field lines occur. Because Jupiter's magnetic field rotates with a sidereal period of around 10 hours, much faster than Ganymede's 172 hour orbital period, pressure from Jupiter's magnetosphere forces the boundary to higher latitudes on Ganymede's ram-facing trailing hemisphere (Kivelson et al. 2004; Jia et al. 2009; Duling et al. 2022). \nFeldman et al. (2000) observed the far-ultraviolet aurora on Ganymede with the Space Telescope Imaging Spectrograph (STIS) on the Hubble Space Telescope. They used a slit wider than Ganymede's angular diameter, giving them the ability to \nHall et al. (1998) reported the first detection of emission from Ganymede's atmosphere. They observed two atomic oxygen lines in the far-ultraviolet at 130 . 4 and 135 . 6 nm , and concluded the brightness ratio of the lines meant they were produced from dissociative electron-impact on O 2 . They also found the emission was spatially-confined near the satellite's poles, matching the magnetic field model of Kivelson et al. (1996). \nimage the spatial distribution of any monochromatic emission. They detected emission at both 130 . 4 and 135 . 6 nm , narrowly confined at latitudes above ±40 · which was consistent with the expected location of the boundary between the open and closed field lines. They found the brightness both spatially and temporally variable, which led them to conclude the emission was auroral and driven by interactions between Ganymede's magnetosphere and plasma trapped in Jupiter's rotating magnetic field. \nMcGrath et al. (2013) observed differences in the morphology of the ultraviolet aurora for the leading, trailing and subJovian hemispheres. These observations showed the emission at higher latitudes on the trailing/upstream hemisphere, consistent with the magneto-hydrodynamic simulations of Jia et al. (2009). Later observations exhibit this same hemispheric morphology, demonstrating the intrinsic shape of Ganymede's auroral ovals and their spatial correlation with the open/closed field line boundary of Ganymede's magnetosphere (Musacchio et al. 2017; Molyneux et al. 2018; Roth et al. 2021; Greathouse et al. 2022; Marzok et al. 2022; Saur et al. 2022). \nGanymede's aurora are produced from the dissociation of atmospheric molecules by electrons trapped in Jupiter's magnetosphere, where rotational forces dominate the distribution of plasma. The best constraints on magnetospheric plasma properties currently available come from Galileo and Voyager 1 data. Bagenal & Delamere (2011) used these data to model the space environment around Jupiter and calculate densities, energies and scale heights of electrons in the plasma sheet as a function of distance from Jupiter. Eviatar et al. (2001) analyzed the intensity of the ultraviolet observations of Feldman et al. (2000) and found that direct impact from electrons in Jupiter's rotating magnetosphere could not excite the observed aurora brightnesses given the Galileo measurements of electron number density. They concluded that the electrons must be accelerated to higher energies by magnetospheric interactions at the open/closed field line boundary where Ganymede's magnetic field reconnects with Jupiter's magnetic field. \nA major outstanding question is the composition of Ganymede's atmosphere. Most previous far-ultraviolet aurora observations (Hall et al. 1998; Feldman et al. 2000; McGrath et al. 2013) concluded that the ratio of the 130 . 4 and 135 . 6 nm aurora brightnesses were indicative of an O 2 atmosphere with a column density between 10 18 and 10 19 m -2 . However, Roth et al. (2021) found a difference in the ratio of the two emissions between the disk center and limb using high spatial-resolution spectra from Hubble/STIS. They attributed this variability to the presence of a localized H 2 O atmosphere around the sub-solar point (near the disk center) with a peak column density of around 6 × 10 19 m -2 in sunlight, which exists in addition to the global O 2 atmosphere. \nThough ground-based optical observations have lower spatial resolution, there are four independent optical oxygen emissions (the 557 . 7 nm emission line, the 630 . 0/636 . 4 nm doublet, the 777 . 4 nm triplet and the 844 . 6 nm triplet) compared to just two detected in the ultraviolet (the 130 . 4 nm triplet and the 135 . 6 nm doublet). Because optical wavelengths can be observed from the ground, large telescopes provide the ability for observing cadences with shorter integration times and better signal-to-noise. de Kleer et al. (2023) published the first optical wavelength observations of Ganymede's aurora, where they found evidence for an O 2 atmosphere with a column density of (4 . 7 ± 0 . 1) × 10 18 m -2 . They found an upper-limit on the H 2 O column density of 3 × 10 17 m -2 , giving a maximum hemisphere-averaged H 2 O/O 2 column density ratio of just 0.06 in eclipse. The H 2 O distribution and density in the sunlit atmosphere proposed by Roth et al. (2021) would produce 46 R of Hα emission, whereas de Kleer et al. (2023) did not detect any Hα (in eclipse) and placed a 2 𝜎 upper limit of 1.8 R. This major difference in H 2 O abundance between observations is suggestive of day-night difference, which could be investigated by observing any atmospheric changes on short timescales during eclipse ingress as the satellite passes into the shadow. \nTo evaluate potential variability in Ganymede's atmosphere, we conducted a time-series analysis of its optical aurora using data taken on 2021-06-08 UTC. The spatially- and temporally-averaged aurora brightnesses from this observation were published as a part of the broader data set in de Kleer et al. (2023). For this study, we examined the spatial and temporal variability between the individual observations. We analyzed changes in the hemispheric spatial distribution and brightness of the 557 . 7 and 630 . 0 nm atomic oxygen aurora lines to evaluate evidence for any short-timescale changes in atmospheric composition. We also quantitatively evaluated whether the H 2 O atmosphere modeled by Leblanc et al. (2023) to explain the Roth et al. (2021) observations would be detectable under our optical observational constraints in order to provide additional confidence in conclusions about H 2 O column densities from the optical aurora. We took our optical data one day after the Juno flyby of Ganymede on 2021-06-07 UTC which included ultraviolet observations from the onboard Ultraviolet Spectrograph (UVS) instrument (Greathouse et al. 2022) and complementary HST/STIS observations just before and just after the flyby (Saur et al. 2022).", '2. OBSERVATIONS AND DATA REDUCTION': "We analyzed 17 spectra of Ganymede in eclipse (see table 1), taken on 2021-06-08 UTC using the High Resolution Echelle Spectrometer (HIRES, Vogt et al. 1994) on the Keck I telescope at the summit of Maunakea. Average seeing over the course of the eclipse observations was about \nFigure 1. Line-of-sight observing geometry of Ganymede during eclipse by Jupiter on 2021-06-08 UTC. The inset axis shows a zoom-in on the Jovian system with the positions and orbits of the other Galilean satellites. Earth's position relative to Jupiter allowed us to observe Ganymede as it passed into Jupiter's umbra. We retrieved the positions of planets and their orbital elements using JPL Horizons, and we have exaggerated the physical sizes of the planets and Jupiter's umbra (but not the relative spacing of the orbits) for illustrative purposes. \n<!-- image --> \n0 . '' 55 . 1 Ganymede's average velocity relative to Earth was about -24kms -1 (the negative sign indicating motion toward Earth), a velocity sufficient to Doppler-shift Ganymede's monochromatic auroral emission from telluric emission line counterparts. Between 12:48 and 16:15 UTC, Ganymede passed through Jupiter's umbra, allowing observation of the faint auroral emissions from its atmosphere without the overwhelming presence of reflected solar continuum. Though the full set of eclipse observations includes 17 spectra, we analyzed only 15; we eliminated the last two spectra (taken during nautical twilight) due to large systematic contamination of scattered light from Earth's atmosphere as the Sun rose. The partial umbral eclipse lasted for about 8 minutes at the beginning and end of the umbral eclipse. The positions of Earth and Jupiter relative to the Sun allowed the telescope's line-of-sight to see Ganymede enter Jupiter's umbra just beyond Jupiter's eastern limb (see figure 1 for a graphical depiction of this viewing geometry). We used the JPL Horizons Ephemeris Service 2 (hereafter called JPL Horizons) to \nFigure 2. Hemispheric observing geometry of Ganymede on 202106-08 between 12:58 and 15:08 UTC. Due to the geometry of eclipse observations, the line-of-sight intersected near the center of the sub-Jovian hemisphere. The sub-observer longitude changed over the duration of the observations by ±2 . 3 · from the central longitude in this projection. Labels mark latitude and planetocentric east longitude; the thicker black line shows the prime meridian. The dashed black lines show the locations of the auroral ovals (Duling et al. 2022), marking the boundary between the open and closed field lines of Ganymede's magnetic field. In this view the leading hemisphere is on the left and the trailing hemisphere is on the right. Ganymede surface map courtesy of the United States Geological Survey (USGS 2020). \n<!-- image --> \n-90 \ndetermine when Ganymede would be eclipsed by Jupiter and observable from Maunakea. \nTwilight on the summit of Maunakea began at 14:18 UTC and astronomical twilight ended at 14:49 UTC. Consequently, the last two observations exhibit greater uncertainty due to higher background from scattered sunlight. Sunrise occurred at 15:46 UTC, preventing any observations of the end of the eclipse. \nObservations of Ganymede as it passes through Jupiter's shadow exclusively measure the sub-Jovian hemisphere. The observations in this data set had a sub-observer east longitude between 8 . 7 · and 13 . 3 · . Figure 2 shows an average view of Ganymede as observed on 2021-06-08 UTC. (The subobserver longitude only varied from this projection by ±2 . 3 · over the duration of the observations.) \nThe HIRES instrument allows an observer to change the angles of the echelle and cross-disperser gratings in order to optimize wavelength coverage on the detectors. The setup chosen for these observations includes emission for atomic oxygen at 557 . 7 , 630 . 0 , 636 . 4 , 777 . 4 and 844 . 6 nm along with \nHα at 656 . 3 nm . de Kleer et al. (2023) reported the first analysis of these data, evaluating disk-integrated auroral brightnesses retrieved from spectra averaged over all individual observations. The brighter oxygen aurora emissions at 557 . 7 and 630 . 0 nm have sufficient signal-to-noise in the individual five-minute exposures to allow for an analysis of both spatial and temporal variability, which we will present in this study. \nIn addition to the eclipse spectra of Ganymede, our data included 10 bias exposures, 4 flat lamp exposures and 5 thorium-argon (ThAr) arc lamp exposures taken a few hours prior to the eclipse. For flux calibration, we took a spectrum of Jupiter's central meridian with the slit oriented north-south in the center of the disk. Because Ganymede was eclipsed by Jupiter and not visible for guiding, we used JPL Horizons to calculate offsets from another nearby Galilean satellite (the 'guide' satellite) and tracking rates in right ascension and declination. We then slewed the telescope manually from the guide satellite to the expected position of Ganymede and took five-minute exposures with manual tracking rates. After each exposure, we offset back to the expected position of another nearby Galilean satellite and took a fiducial exposure to record the expected position of Ganymede within the slit. The guide satellite was initially Io, but after two exposures we switched to Europa because Io began transiting Jupiter's disk.", '2.1.1. Reduction': "We reduced the data using an improved version of the pipeline described in de Kleer et al. (2023). The pipeline exists in two parts: the data reduction pipeline (Milby 2024a) is a generic HIRES pipeline, while the data calibration/brightness retrieval pipeline (Milby 2024b) is specific to the aurora observations. For each calculation the data reduction and flux calibration pipelines propagate errors as described below. We used version 2.1.0 of the data reduction pipeline and version 2.15.0 of the data calibration/brightness retrieval pipeline. \nWe designed our data reduction pipeline to work with data taken both before the 2004 detector upgrade (the single 2048 × 2048 pixel detector, hereafter called 'legacy' data) and after the detector upgrade (the current three-detector mosaic setup, hereafter called 'mosaic' data). The data reduction pipeline first combines the mosaic images into a single image with proper physical separation so that orders crossing between the detectors can still be partially used (this step is skipped for the legacy data). It uses a standard star observation (or similar bright point source) to find traces along the spectral dimension for each echelle order. Using the pixel positions of the traces, it constructs an order mask image using the slit length and pixel dimensions, both in units of arcseconds. This mask is an array of zeros everywhere except half of \nthe length of the slit above and below the traces, which are set to ones. To find the edges of the orders, it cross-correlates the mask image with the master flat-field along the spatial axis, which is able to account for the effect of overlapping orders, orders crossing between detectors and a reference trace which isn't precisely centered along the spatial axis. It extracts and rectifies individual orders by taking each pixel along the spatial dimension which fell within the boundaries of the mask order at the maximum offset of the cross correlation.", '2.1.2. Flux Calibration': "To produce a wavelength solution, the pipeline takes the rectified ThAr arc lamp spectra, averages them along the spatial axis (to produce a one-dimensional spectrum) and normalizes them. It then produces a two-dimensional 'image' of each one-dimensional spectrum stacked together. The HIRES data reduction pipeline MAKEE 3 includes templates taken at a variety of echelle and cross-disperser angles which identify order numbers, wavelengths and central pixel positions of lines in the thorium-argon arc lamp spectra. Our pipeline finds the template closest to the echelle and cross-disperser angles used in the observations, then constructs a similar twodimensional template of stacked one-dimensional spectra. It simulates individual lines within the one-dimensional spectra by using a Gaussian line profile at the central pixel position with a full-width at half-maximum (FWHM) equal to the slit width. It then cross-correlates the template reference spectrum with the observed arc lamp spectrum along both the spatial and spectral axes. It uses the maximum cross-correlation to construct an initial wavelength solution. It then fits a Gaussian function to the observed spectrum at the initial pixel position and assigns the center of the bestfit as the fractional pixel position corresponding to the wavelength guess. After assigning refined pixel positions to every identified wavelength, the pipeline fits a two-dimensional polynomial surface to construct a complete wavelength solution for each spectral-dimension pixel in each order. The use of lines identified in adjacent orders allows for a better wavelength solution in orders with fewer identified lines. Finally, it reduces the data by subtracting a median bias and flat-fielding using a normalized median flat-field, then corrects for wavelength-dependent airmass-extinction based on the median curve from Buton et al. (2013), removing the diluting effects of Earth's atmosphere from all science images, including flux calibration images of Jupiter's central meridian (described in the following section). \nThe calibration pipeline uses a spectrum of Jupiter's central meridian taken on the same night as the eclipse observations, a solar radiance reference spectrum 𝐹 E at 1 au (Coddington et al. 2023) and Jupiter's spectral reflectivity \n( 𝐼 / 𝐹 ) from Woodman et al. (1979) to calibrate the data from [ counts s -1 ] to rayleighs [R], a unit of photon column emission commonly used for aurora and airglow, defined as 1 R ≡ (10 10 /4π) ph s -1 m -2 sr -1 . \nFor Ganymede (and the other icy satellites Europa and Callisto), the pipeline retrieves aurora brightnesses for atomic oxygen emission at 557 . 7 , 630 . 0 , 636 . 4 , 777 . 4 and 844 . 6 nm and for atomic hydrogen emission at 656 . 3 nm . To calculate the expected spectral brightness 𝐵 J of Jupiter, it scales the solar reference spectrum with units of [R nm -1 ] by the square of the distance between the Sun and Jupiter 𝑎 J at the time of the observation and applies Jupiter's wavelength-dependent spectral reflectivity: \n𝐵 J = 𝐹 E ( 1 au 𝑎 J ) 2 ( 𝐼 𝐹 ) . (1) \nJupiter fills the slit, so to determine the observed flux rate ̇ 𝑁 J from Jupiter at a particular wavelength 𝜆 , the pipeline calculates the median value for the column in the twodimensional Jupiter meridian spectrum containing the desired wavelength, then multiplies that median value by the number of pixels subtended by the slit on the detector, thereby estimating the total count rate at a given wavelength 𝜆 from the slit. \nFor each individual observation of Ganymede, the pipeline takes the section of the two-dimensional spectrum within ±0 . 25 nm for targeted auroral wavelength (Doppler-shifted by Ganymede's velocity relative to Earth), then produces calibrated two-dimensional images by multiplying the ratio between the observed count rate per bin from Ganymede to the observed count rate from Jupiter by the physical unit conversion factor 𝐵 J described above, \nIndividual HIRES mosaic detector pixels have angular dimensions of 0 . '' 119 × 0 . '' 179 along the spatial and spectral dimensions. The data in this study have 3 × 1 spatial/spectral binning, so each bin has a projected size of 0 . '' 358 × 0 . '' 179 on the sky, or an angular area of about 0 . 0641 arcsec 2 ( 1 . 51 × 10 -12 sr ). HIRES users can choose from a series of slit length and width combinations by means of a series of deckers (movable metal plates containing the slits). The D3 decker used for these observations has a projected angular size of 1 . '' 722×7 '' , so the entrance area corresponds to 188 bins on the detector. \n𝐵 G = 𝐵 J ( ̇ 𝑁 G Ω bin )( Ω slit ̇ 𝑁 J ) 𝑤 slit Δ 𝜆, (2) \nwhere ̇ 𝑁 G is the observed flux rate from Ganymede in [ counts s -1 bin -1 ], Ω bin is the solid angular size of one detector bin in [ sr bin -1 ], ̇ 𝑁 J is the observed flux rate from Jupiter in [ counts s -1 ] calculated as described above and Ω slit is the solid angular size of the slit in [sr]. To account for the spectral \nresolution of the slit, it multiplies by the width of the slit 𝑤 slit in [bins] and the wavelength dispersion Δ 𝜆 in [ nmbin -1 ] at the targeted wavelength. \nTo calculate disk-integrated brightnesses for the science target satellite, it averages the emission over a user-defined circular aperture Ω 𝑎 which is larger than the apparent angular size of the target. Because it assumes emission from a disk with the angular area of Ganymede Ω G , it scales the average by the ratio (Ω 𝑎 /Ω G ) 2 .", '3. ANALYSIS': "Our analysis makes use of the right-handed equivalent of the System III coordinate system (with longitudes measured positively to the east rather than to the west). In this frame, the Joviographic rotation axis 𝛀 defines latitude 𝜆 III and the rotation of Jupiter's magnetic field defines longitude 𝜙 III,RH such that the magnetic field's rotation axis is offset from the Joviographic rotation axis by 9 . 5 · toward 159 · longitude (Connerney et al. 1998). We also calculated magnetic latitudes 𝜆 m defined by the magnetic rotation axis 𝛀 m by converting System III Joviographic latitude 𝜆 III to the magnetospheric reference frame using 𝜆 m = 9 . 5 · cos( 𝜙 III,RH -159 · ) -𝜆 III. Longitudes in the magnetospheric coordinate system are the same as the right-handed System III Joviographic coordinate system ( 𝜙 m = 𝜙 III,RH).", '3.1. Retrieval of Disk-Integrated Brightnesses': "We used the standard deviation of the spectrum near the emission to estimate the random error from instrumental effects and photon counting. To estimate systematic error, we compared the ratio of the 630 . 0 nm brightness to the 636 . 4 nm brightness to see how many of the observed ratios deviated from the expected value. The lifetime for O( 1 D 2 → 3 P 2 ) 630 . 0 nm emission is 178 s while the lifetime for O( 1 D 2 → 3 P 1 ) 636 . 4 nm emission is 549 s (Wiese et al. 1996), so if there is collisional quenching, the observed 636 . 4 nm emission should be suppressed more than the 630 . 0 nm emission and the 630 . 0 nm/636 . 4 nm emission ratio should be larger. However, the value cannot be any lower than the expected ratio of 3.09 for a collisionless atmosphere (Wiese et al. 1996). \nTable 1 lists the observation parameters and retrieved brightnesses for each of the 17 eclipse spectra in the time series. The apparent emission covers an area on the detector larger than the size of Ganymede's apparent disk because of the blurring effect of atmospheric seeing and telescope pointing variability between observations. To determine an appropriate aperture size, we used the two-dimensional spectra containing the bright 630 . 0 nm emission. We found an aperture with a radius of 1 . '' 75 fully enclosed the apparent flux, so we used this aperture size for all of the different spectral lines. This was about 2 . 25× the size of Ganymede's apparent angular radius. \nTable 1. Overview of the Keck/HIRES observations of Ganymede in eclipse on 2021-06-08 UTC. \nNotes: The average values and analyses in this paper do not include the last two observations at 15:01:18 and 15:08:29 which were taken during nautical twilight and exhibited significant scattered light contribution from Earth's atmosphere, affecting retrieved brightnesses and background subtraction. \n- b UTC time on Earth corrected for light-travel time between Ganymede and Maunakea.\n- d Sub-observer east longitude on Ganymede as observed from Maunakea. \n- f Magnetospheric longitude of Ganymede (the same as the System III west longitude, converted here to east longitude). \nWefound that including 9% systematic error in addition to the random error derived from the standard deviation of the spectrum resulted in approximately two-thirds of the observed ratios to be within 1 𝜎 of the expected value. All uncertainties listed in this paper are the quadrature sum of these two errors.", '3.2. Calculation of Incident Electron Densities': "The distribution of plasma in Jupiter's magnetosphere is affected by both rotational forces and particle energies. Phipps &Bagenal (2021) showed the plasma density reaches its maximum along a centrifugal equator which exhibits a variable latitudinal offset from the magnetic field equator depending on distance from Jupiter. Bagenal & Delamere (2011) mod- \neled the plasma distribution and derived how the vertical scale height of the electrons varies with radial distance from Jupiter due to competition between centrifugal forces and thermal pressure. The combination of these two effects with Jupiter's rotating magnetosphere affect the incident plasma density exciting the aurora. \nTo evaluate this effect quantitatively, we calculated the location of the plasma sheet mid-plane using the methodology outlined in Phipps & Bagenal (2021). Assuming symmetry along the azimuth axis, they derived an empirical fit to a dipole magnetic field which gives the Joviographic latitude 𝜆 ceq of the plasma density maxima as a function of distance \nTable 2. Retrieved optical aurora brightnesses for emission from atomic oxygen and atomic hydrogen. \na Calculated using weighted averages as ⟨ 𝐵 ⟩ = ∑ 15 𝑖 =1 ( 𝐵𝑖𝑤𝑖 )/ ∑ 15 𝑖 =1 𝑤𝑖 with average uncertainty ⟨ 𝜎 ⟩ = 1/ √ ∑ 15 𝑖 =1 𝑤𝑖 , where 𝑤𝑖 = 1/ 𝜎 2 𝑖 is the inverse variance. \nNotes: We retrieved each brightness from a circular aperture with a radius of 1 . '' 75 . Assuming emission from a disk with the solid-angular size of Ganymede (it had an apparent angular radius of 0 . '' 775 on 2021-06-08 UTC), we scaled the brightness by the ratio (1 . '' 75/0 . '' 775) 2 . Listed errors include 9% systematic uncertainty. \nJoviographic equator \nFigure 3. Geometric relationships relevant to the retrieval of electron properties. Ganymede orbits Jupiter at a distance of 𝑟 G . However, because of the tilt of the magnetic field rotation axis 𝛀 m relative to the Joviographic rotation axis 𝛀 , Ganymede's distance 𝑑 from the centrifugal equator varies over the approximately 10 hour magnetospheric rotation period of Jupiter. See Phipps & Bagenal (2021) for a detailed description of this geometry. Image credits: Jupiter (JPL 2001), Ganymede (USGS 2020). \n<!-- image --> \nfrom Jupiter 𝑟 and sub-Jovian east longitude 𝜙 \n𝜆 ceq ( 𝑟, 𝜙 ) = [ 𝑎 tanh ( 𝑏 𝑟 𝑅 J -𝑐 ) + 𝑑 ] sin( 𝜙 -𝑒 ) , (3) \nwhere the empirically-derived best-fit constants are 𝑎 = 1 . 66 · , 𝑏 = 0 . 131 rad , 𝑐 = 1 . 62 rad , 𝑑 = 7 . 76 · and 𝑒 = 249 · . \nWe retrieved the distance between Jupiter and Ganymede 𝑟 G and the sub-Jovian west longitude 𝜙 III,RH and latitude 𝜙 III,RH in Jupiter's System III Joviographic reference frame using the JPL Horizons. We first made an initial query targeting Ganymedeasobserved from Maunakea and retrieved the light travel time between Ganymede and Earth. We then subtracted that travel time from the observation start time and made a second query targeting Jupiter as observed from Ganymede at the new time, retrieving ephemeris information in the Jupiter system at the time of the observation. We converted all System III coordinates to the right-handed Joviographic coordinate system. \nFigure 3 shows the geometric relationships relevant to the calculation of the plasma density. Ganymede orbits Jupiter at a distance 𝑟 G , but because of the relative 9 . 5 · tilt of the magnetic field axis toward 159 · east longitude (Connerney et al. 1998), the height of Ganymede above the highest-density mid-plane of the plasma sheet 𝑑 , which falls along the centrifugal equator, varies with the approximately 10-hour rotation period of Jupiter's magnetosphere. To calculate the effective plasma density 𝑛 incident at Ganymede, we used a mid- \n[R] \nightness \nBr \nFigure 4. Disk-integrated brightness of Ganymede's 630 . 0 nm aurora as a function of distance from the plasma sheet mid-plane. The black points are the observations with error bars showing the combined random and systematic uncertainties. The gray line is a fit of equation (5) with 𝐵 0 = (130 ± 5) R and the scale height 𝐻 fixed to 2 . 78 R J (Bagenal & Delamere 2011). The shaded gray region shows the uncertainty in the peak brightness ( 𝐵 0 ) of the fit. The fit shows that the brightness exhibits a moderate correlation with distance from the plasma sheet mid-plane, and has an expected peak brightness of about 130 R if the oscillation is due to localized variations in plasma density. In this figure, the timing of the observations moves from right to left: Ganymede was above the plasma sheet mid-plane at the start of the observations, and moved through the center to below the sheet by the end of the night. \n<!-- image --> \nJ \nplane density of 𝑛 0 = 20 cm -3 measured during the Voyager flyby (Scudder et al. 1981) scaled with a Gaussian vertical distribution of the form \n𝑛 = 𝑛 0 e -( 𝑑 / 𝐻 ) 2 , (4) \nwhere 𝐻 is the scale height of the plasma at Ganymede's orbital distance from Jupiter 𝑟 G and 𝑑 is the minimum distance between Ganymede and the centrifugal equator, equivalent to the distance to the tangent to the centrifugal equator from which a normal line intersects Ganymede's orbit (Gledhill 1967). We used a fixed scale height value of 2 . 78 R J and orbital distance of 14 . 936 R J (Bagenal & Delamere 2011). Because the auroral brightness is directly proportional to the density of the exciting electrons in the case of a thin, noncollisional atmosphere (de Kleer & Brown 2018), this equation also describes the expected change in brightness 𝐵 with distance from the centrifugal equator (see figure 4), such that it can be rewritten as \n𝐵 = 𝐵 0 e -( 𝑑 / 𝐻 ) 2 (5) \nwhere 𝐵 0 is the peak brightness when Ganymede is at the high-density plasma sheet mid-plane.", '4. RESULTS AND DISCUSSION': "4.1. Disk-Integrated Brightness Variability \nThe disk-integrated brightnesses of Ganymede's aurora do not correlate solely with its location relative to the plasma sheet (see figure 4). Instead, the brightness appears to exhibit a bimodal distribution, reaching local maxima when Ganymede is both above and below the mid-plane of the plasma sheet (the peaks near 0 . 7 R J and -0 . 3 R J , respectively). There aren't enough high-cadence observations of Ganymede's aurora as it passes through the plasma sheet midplane to determine whether the dimming of the brightness between 0 . 5 R J and the mid-plane is physically meaningful rather than either stochastic variability in local electron number density or an observational effect such as Ganymede drifting in and out of the slit. Therefore, we cannot know if this apparent bi-modality about the plasma sheet centrifugal equator is a persistent feature. However, the brightness varies smoothly in time, suggesting it isn't due to random noise. Additionally, the variability exceeds the statistical error in the individual observations. Note that time moves from rightto-left in this image; the first observations were taken when Ganymede was above the plasma sheet, and the last were taken when it was below. \nIo's 135 . 6 nm auroral limb glow exhibits a comparable decrease in brightness with distance from the plasma sheet centrifugal equator (Retherford et al. 2003). Oliversen et al. (2001) and Schmidt et al. (2023) observed the same effect in Io's 630 . 0 nm auroral emission, but the higher cadence of the optical observations revealed additional variability which Schmidt et al. (2023) attributed to heterogeneity in the local plasma density that sweeps past the satellite. Analysis of Cassini data taken during it's flyby of Jupiter showed variations in electron number density with magnetic longitude \nThese results are less clear than similar analyses of Europa, which showed a direct correlation between distance from the plasma sheet and disk-integrated aurora brightness (Roth et al. 2016; de Kleer et al. 2023). Europa's lack of a magnetic field simplifies the excitation process in comparison to Ganymede, since incident electrons at Europa are neither locally accelerated nor restricted to particular geographic locations like they are for Ganymede. However, de Kleer et al. (2023) did find a potentially similar decrease in brightness when Europa was near the plasma sheet mid-plane in two sets of observations taken on 2021-06-21 UTC and 2021-07-16 UTC, though neither data sub-set observed Europa both above and below the mid-plane. Musacchio et al. (2017) analyzed UV observations taken by Hubble/STIS and found an increase in brightness on the leading hemisphere and a decrease in brightness on the trailing hemisphere when Ganymede was near the plasma sheet mid-plane. In contrast, the optical observations view the sub-Jovian hemisphere, and we didn't observe a change in the brightness ratio between the dusk-dawn (leading-trailing) hemispheres as Ganymede crossed through the mid-plane (see section 4.2 and figure 7). \n(Steffl et al. 2006, 2008). This suggests the bi-modal brightness modulation apparent in figure 4 may have simply reflected local upstream plasma conditions changing over the course of the observations. Simulations of interactions between Ganymede's magnetosphere and Jupiter's also suggest that magnetic reconnection rates vary on the order of tens of seconds, affecting the supply of electrons into Ganymede's atmosphere and subsequently the number of electron-impact excitations leading to auroral emission (Jia et al. 2009). \nFigure 4 shows a fit of equation (5) with the scale height 𝐻 fixed to a value of 2 . 78 R J as calculated by Bagenal & Delamere (2011) for Ganymede's orbital distance from Jupiter. The resulting fit has a Pearson correlation coefficient of 0.415 and a p -value of 0.124 (not achieving statistical significance for a 95% confidence threshold). Though we expect there to be a correlation with plasma sheet distance, the poor correlation suggests two simultaneous phenomena may be affecting Ganymede's auroral brightness: first-order brightness variation from scale-height-induced density variation as its position changes relative to the plasma sheet mid-plane and second-order brightness bi-modality from localized density variations in the plasma as it sweeps past. These observations suggest the longitudinal density heterogeneity overwhelms the scale-height dependence, but further observations will be needed to confirm this conclusion statistically.", '4.2.1. North-South Asymmetry': "The 630 . 0 nm emission data show both north-south and dusk-dawn (equivalent to trailing-leading in optical observations of the sub-Jovian hemisphere) hemispheric asymmetries (figure 5). \nWhen Ganymede is above the plasma sheet mid-plane (the first two rows in figure 5), the southern mid-plane-facing hemisphere is brighter. As the high-density center of the plasma sheet moves past Ganymede, the brightness is more evenly spread across the disk. Once Ganymede is below the mid-plane, the northern hemisphere is brighter. In all of the images, the peak brightness appears shifted toward dusk (trailing) hemisphere longitudes. \nRetherford et al. (2003) showed that electrons in the Jovian magnetosphere impact Io's atmosphere along two different pathways: the bulk rotation of the plasma sheet which produces a flux directed at the trailing hemisphere, and bounce motion along flux tubes constrained by the morphology of Jupiter's magnetic field lines and how they connect with Io's \nIn order to quantify these asymmetries, we calculated the hemisphere-integrated brightnesses from semi-circular apertures with radii of 2 . 25 R G . For pixels that fell in both hemispheres we allocated the brightness based on the relative pixel area in each hemisphere. Table 3 lists the brightnesses of each relevant hemisphere. \nTable 3. [OI] 630 . 0 nm brightnesses calculated for the northern, southern, dawn (leading) and dusk (trailing) hemispheres. \nmagnetic field. They analyzed a similar north-south brightness asymmetry observed in Io's UV aurora and showed that modeled field-aligned electron motion along a flux tube accurately reproduced the brightness ratio they observed. Assuming the symmetric Gaussian profile around the centrifugal equator with a scale height of 𝐻 = 2 . 78 R J (equation 4), we numerically integrated the ratio 𝑅 N/S of flux tube electron column densities for a distance 𝑑 from the plasma sheet centrifugal equator \n𝑅 N/S = ∫ ∞ 𝑑 e -( 𝑥 / 𝐻 ) 2 d 𝑥 ∫ 𝑑 -∞ e -( 𝑥 / 𝐻 ) 2 d 𝑥 . (6) \nFigure 6 shows the ratio of the northern hemisphere brightness to the southern hemisphere brightness with a logarithmic vertical axis so that the spacing of the ratios is meaningful. The diagonal gray line shows the result of the integral ratio in equation (6) evaluated for the range of plasma sheet distances across the Ganymede observations. This line is not a fit to the data; the correlation coefficient between the expected ratio and the observations is 0.952 with 𝑝 ≪ 0 . 001 , clearly demonstrating that the asymmetric column densities along the flux tubes intersecting each hemisphere quantitatively matches the observed north-south brightness asymmetry of Ganymede's aurora. \ne \nv \nbo \nA \nw \nBelo \nFigure 5. Calibrated images of Ganymede's 630 . 0 nm auroral emission displayed in 20 R contours. To better reveal the spatial variability we smoothed the data using a Gaussian kernel with a FWHM of 0 . '' 5 approximating typical seeing conditions for the morning of 2021-06-08 UTC. The time in the upper left of each image is the UTC time at the start of the observation, the annotation in the lower right is the distance to the plasma sheet centrifugal equator; positive when Ganymede is above the mid-plane and negative when it is below. As Ganymede moved vertically through the plasma sheet, the hemisphere closest to the mid-plane exhibited the brightest aurora. From approximately 14:00 to 14:45 UTC, Ganymede is within ±0 . 5 R J and the brightness is more evenly-distributed across the disk. Ganymede passed through the highestdensity plasma mid-plane at 14:20 UTC, so the top two rows display data when Ganymede was above the mid-plane (with enhanced brightness at southern latitudes) and the bottom row displays data when Ganymede was below the mid-plane (with enhanced brightness at northern latitudes). Because of the blurring effect of atmospheric seeing, the calibrated brightness distribution of the individual pixels is lower than the disk-averaged value reported in de Kleer et al. (2023) and table 1, which assume all emission originates from a disk with Ganymede's solid-angular size. The white grid shows the physical size and orientation of Ganymede, with north pointing upward (for more details on this observing geometry, see figure 2). \n<!-- image --> \nRatio \nightness \nBr \nFigure 7 shows the ratio of the dusk hemisphere brightness to the dawn hemisphere brightness. We found the dusk (trailing) hemisphere was almost always twice as bright as the dawn (leading) hemisphere. McGrath et al. (2013) ob- \n<!-- image --> \nJ \nFigure 6. Ratio of Ganymede's 630 . 0 nm auroral emission between its northern and southern hemispheres out to an angular distance of 2 . 25 R G. The horizontal dashed gray line indicates a ratio of 1 (equal average brightness between hemispheres), and the vertical dashed gray line shows where Ganymede passed through the center of the plasma sheet. The diagonal line shows the expected brightness ratio due to asymmetric flux tube electron column densities (equation 6). \nSaur et al. (2022) looked at this same hemispheric brightness ratio in 135 . 6 nm UV data taken on 2021-06-07 UTC (the day before these HIRES observations), but the low signal-to-noise at ultraviolet wavelengths required them to integrate for longer, reducing the cadence of their time series observations. Regardless, they found a similarly-variable hemispheric brightness ratio in the ultraviolet data, and their north-south ratio varied between 1.6 and 0.4 (see Saur et al. 2022, figure 5) which matches the extremes in the 630 . 0 nm HIRES observations. At an orbital distance of 14 . 936 R J we calculated Ganymede reaches a maximum height above or below the plasma sheet centrifugal equator of 2 . 44 R J , so from equation (6) the peak north-south hemispheric brightness ratio should be about 4.35 at -2 . 44 R J and 0.230 at 2 . 44 R J .", '4.2.2. Dusk-Dawn Asymmetry': "2 \nRatio \nightness \nBr \nFigure 7. Ratio of Ganymede's 630 . 0 nm auroral emission between its dusk and dawn hemispheres out to an angular distance of 2 . 25 R G. The horizontal dashed gray line indicates a ratio of 1 (equal average brightness between hemispheres), and the vertical dashed gray line shows where Ganymede passed through the center of the plasma sheet. The horizontal line is a best-fit constant and the shaded gray area shows the uncertainty in the fit. \n<!-- image --> \nserved a similar brightness asymmetry in HST observations of Ganymede's 135 . 6 nm emission. Musacchio et al. (2017) and Molyneux et al. (2018) did not observe the same relative enhancement of the leading hemisphere, but they observed just the leading or trailing hemisphere (rather than the simultaneous leading-trailing geometry of the sub-Jovian optical observations), so they wouldn't be able to see the same kind of inter-hemisphere enhancement that we did. \nThough we assumed an increased electron flux on the trailing hemisphere contributes to the dusk-dawn asymmetry, the physical structure of Ganymede's magnetosphere and the way \nLeblanc et al. (2017) simulated Ganymede's atmosphere and showed an enhancement in O 2 column density toward the dusk hemisphere in eclipse (see their figure 5). Because O 2 does not readily condense on the Ganymede's surface, they suggested the asymmetry is due to a combination of the thermal inertia of the surface ice, the morphology of the magnetic field and the incident sputtering particles originating from the direction of the trailing hemisphere. In particular, thermal lag in the surface ice causes the surface to be warmer toward dusk compared to dawn. A higher temperature on the hemisphere subjected to sputtering particle flux allows for a larger sputtering rate (Cassidy et al. 2013). Oza et al. (2018) estimated latitude-averaged O 2 column densities on several tidally-locked Solar System moons, including Ganymede and Europa, and estimated a hemisphericallyaveraged dusk-dawn ratio of 1.22, far below our minimum estimated ratio of 1 . 84 ± 0 . 11 . The higher asymmetry found in the aurora brightnesses is likely the product of both the larger column density and a larger electron flux on the trailing hemisphere, however the optical aurora data cannot decouple their relative contributions. \n4 \nTable 4. Modeled emission ratios relative to 557 . 7 nm [O I] for Ganymede's optical aurora, assuming an electron population with a Maxwell-Boltzmann distribution centered at 100 eV and a number density of 20 cm -3 . \nit connects with Jupiter's almost certainly complicates the electron flux path. Kivelson et al. (2004) showed how the interaction between Jupiter's magnetosphere and a magnetized moon like Ganymede results in field line reconnection which restricts plasma flow and redirects it toward a narrow range of magnetic latitudes near the magnetic poles, allowing for electron flux on both the leading and trailing hemispheres (rather than concentrating the bulk of the flux on the trailing hemisphere). Our analysis of the north-south hemispheric brightness ratio provides evidence for flow along the field lines connecting to Jupiter and therefore access to the full vertical extent of the plasma split between the hemispheres. In contrast, simple impact on the trailing hemisphere from the rotation of Jupiter's magnetosphere would produce a much smaller ratio between the northern and southern hemispheres since the relative difference in electron flux could only come from the vertical extent of Ganymede's physical cross section. Eviatar et al. (2001) assumed an increased electron energy to account for the effects of Ganymede's magnetic field, while Saur et al. (2022) increased both the number density and the peak of the electron energy distribution (to double that of Eviatar et al. 2001). Though we used the electron densities derived from Voyager data (Scudder et al. 1981) and the energy distribution given by Eviatar et al. (2001), we evaluated the effect of the electron properties given by Saur et al. (2022) on observed optical aurora brightnesses as detailed in section 4.5.", '4.3. Constraints on Atmospheric Composition Variability over Eclipse': "The ratio of the 630 . 0 nm brightness to the 557 . 7 nm brightness is particularly sensitive to the presence of H 2 O as a parent molecule (table 4, see also de Kleer et al. 2023, ta- \nRatio \nightness \nBr \nFigure 9 shows the modeled H 2 O/O 2 column density ratio for each observation. The weighted average (shown in \n<!-- image --> \nSolar Altitude Observed from Maunakea [deg] \n12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 16:30 \nStart Time of Observation [UTC] \n2021-06-08 \nFigure 8. Time series of the ratio of the aurora brightness at 630 . 0 nm to the brightness at 557 . 7 nm . We have selected only data points for which the ratio signal-to-noise was greater than 2, which eliminated the observation at 14:15:28. Dashed horizontal lines show the modeled ratios for atmospheres of pure O, O2, H2O and CO2 \n. The ratios remain relatively constant over the duration of the observed eclipse and do not show evidence for the presence of significant H2O or CO2. The dark gray background region spanning most of the time range shows the 3-hour, 47-minute duration of the umbral eclipse. The light gray regions on either side are the approximately 8-minute duration of the partial umbral eclipse. The top axis shows the altitude of the Sun as observer from the summit of Maunakea. Sunrise on Maunakea occurred at 15:46 UTC, preventing observations of the end of the eclipse. \nble 5). Since the publication of de Kleer et al. (2023) we have expanded the aurora model to include cross-sections for electron impact on CO 2 producing emission at 630 . 0 nm and 636 . 4 nm (Strickland & Green 1969) and 777 . 4 and 844 . 6 nm (Zipf 1984). Using the same cross sections for O, O 2 and H 2 O listed in de Kleer et al. (2023), we've calculated a 630 . 0 nm to 557 . 7 nm emission ratio of 13 . 5 for electron impact on O 2 , 2 . 72 for for electron impact on O, 0 . 943 for for electron impact on H 2 Oand 0.892 for electron impact on CO 2 . The H 2 O ratio is nearly 1, so we should observe approximately equal brightnesses at 557.7 and 630 . 0 nm if the primary source of the auroral emission was electron impact on water molecules. \nFigure 8 shows a time-series of the 630 . 0 nm to 557 . 7 nm emission ratio retrieved from the individual Ganymede observations. The ratio appears relatively constant over the duration of the eclipse, though on average lower than expected for a pure-O 2 atmosphere, suggesting the presence of an atmospheric species beyond O 2 . If Ganymede's atmosphere is collisional, quenching of the longer-lived O( 1 D 2 ) atoms (Wiese et al. 1996) could also lower the ratio of 630 . 0 nm to 557 . 7 nm emission. (Transitions from O( 1 D 2 ) emit both the 630 . 0 and 636 . 4 nm photons when relaxing to the O( 3 P 2 ) and \nRatio \nDensity \nColumn \n2 \n<!-- image --> \nStart Time of Observation [UTC] \nFigure 9. H 2 O/O 2 column density ratio derived from MCMC fits to retrieved aurora brightnesses. The dashed gray line shows the weighted average of the individual points and the shaded gray region shows the asymmetric uncertainty of the weighted average. This average column density ratio of 0 . 10 +0 . 05 -0 . 04 matches the upper limit found by de Kleer et al. (2023) for the same date. \nO( 3 P 1 ) ground states, respectively.) However, we did not find any evidence of collisional quenching in our analysis of the 630 . 0 nm/636 . 4 nm emission ratio when estimating systematic error (see section 3.1). \nWe used the same Markov chain Monte Carlo (MCMC) method described in de Kleer et al. (2023) to estimate the relative column densities of a four-species atmosphere consisting of O, O 2 , H 2 O and CO 2 using an electron number density of 20 cm -3 (Scudder et al. 1981) and a Maxwell-Boltzmann velocity distribution centered at 100 eV (Eviatar et al. 2001). Table 5 lists the median atmospheres for each observation along with the median atmosphere to the average brightnesses listed in table 1. \nThe relatively constant ratio suggests atmospheric composition was not changing over the course of the observations; if there was an H 2 O atmosphere, it either collapsed within the first ten minutes of umbral eclipse (an effect we analyzed in section 4.4) or the emissions we observed included only a minor contribution from H 2 O. \nFor each individual observation, the median atmospheres tended toward a primarily-O 2 composition, but included minor contributions from both O and H 2 O to account for the brightness ratio of less than 13.5 expected for a pure O 2 atmosphere. CO 2 is a trace component of Ganymede's surface ice (McCord et al. 1998), and our model found a correspondingly minor CO 2 component about one order of magnitude lower than H 2 O for each observation. The brightness contribution from a CO 2 column of this magnitude is less than the measurement uncertainties of the auroral brightnesses, so we treat this result as an upper limit on a potential CO 2 component in Ganymede's atmospheric composition. \nTable 5. MCMC median atmospheres using retrieved brightnesses listed in table 1. The uncertainties listed are the 16th and 84thpercentile quantiles. \ngray) is 0 . 10 +0 . 05 -0 . 04 . This ratio matches the 2 𝜎 upper limit of 0.10 found by de Kleer et al. (2023) for the 2021-06-08 UTC observations using averages of the spectra and still suggests an O 2 -dominated atmosphere. (Though de Kleer et al. (2023) report an upper limit of 0.06 for the H 2 O/O 2 column ratio, they calculated this value using averages over multiple nights of Ganymede observations; their best-fit atmosphere for the 2021-06-08 UTC data had an upper limit ratio of 0.10.) \nde Kleer et al. (2023) found that the column density of H 2 O derived by Roth et al. (2021) would have produced more emission at 557 . 7 and 656 . 3 nm than they detected in the average disk-integrated optical observations. However, because O 2 and H 2 O may have different spatial distributions (Roth et al. 2021; Leblanc et al. 2017, 2023), the observing geometry during eclipse could reduce observed emission from electron impact on H 2 O due to the physical separation between the open/closed field line boundary and the higher-density H 2 O column near the disk center. In order to explore this spatial effect on the aurora emission, we simulated optical observations assuming the modeled atmosphere from Leblanc et al. (2023), which was tuned to reproduce the Roth et al. (2021) UV observations.", '4.4. Detectability of an H 2 O Atmosphere with Keck/HIRES': "We evaluated simulations of both O 2 and H 2 O atmospheres (figure 10). Each time step was averaged over 2 · of Ganymede's orbit around Jupiter (a duration of 57 minutes). \nTo simulate eclipse conditions, the effects of solar photon ionization and dissociation were turned off between orbital angles of 176 . 26 · and 183 . 74 · , measured in a right-handed coordinate system from a reference position of 0 · pointing toward the Sun. These angles correspond approximately to the midpoint of the partial umbral eclipses. In our analysis of the simulations (figures 11 and 12), we converted this angular coordinate system to time relative to the eclipse midpoint. \nWeused the same Maxwell-Boltzmann electron population centered at 100 eV as described in Leblanc et al. (2023), with the density modified spatially to account for acceleration near the open/closed field line boundaries. We set the background number density to 20 cm -3 , increasing to a peak of 70 cm -3 at the latitudes of the ovals from Duling et al. (2022) following a Gaussian shape with a FWHM of 20 · , chosen by Leblanc et al. (2023) to match simulations of Ganymede's magnetosphere (Jia et al. 2009). \nThe simulated O 2 atmospheres have a disk-integrated column density between 4 . 45 × 10 18 and 4 . 5 × 10 18 m -2 , which are between 5 and 13% lower than the average column density from our best-fit model atmosphere and that of de Kleer et al. (2023). The simulated H 2 O disk-integrated column densities vary between about 10 19 m -2 before eclipse ingress to a minimum of about 7 × 10 16 m -2 just before eclipse egress. The peak simulated H 2 O column density decreases from 6 × 10 19 m -2 to 2 × 10 16 m -2 . \nWe calculated the auroral emission at the native resolution of the simulations, then rebinned the results to the detector resolution of HIRES. Finally, we smoothed the data using a two-dimensional Gaussian kernel with a FWHM of 0 . '' 5 , representative of the typical seeing conditions of the night of 2021-06-08 UTC. We then calculated the disk-integrated brightnesses with the same aperture size we used for the HIRES observations. \nFigure 11 shows a comparison between the simulated and observed brightnesses for 557 . 7 nm [O I ] and 656 . 3 nm H I auroral emission. The six time steps in the simulated brightnesses shown in each plot in red correspond to the total emission from both the H 2 O and O 2 simulations. The simulated column density of O 2 doesn't change substantially over the course of the eclipse, maintaining a steady simulated diskintegrated 557 . 7 nm brightness of about 13 R . The simulated steady-state H 2 Oatmosphere in full sunlight has a diskintegrated column density ratio with the O 2 atmosphere of about 3, and our aurora model produces a simulated 20 R of emission at 557 . 7 nm and 45 R of emission at 656 . 3 nm from the model H 2 O atmosphere alone. Combining the model H 2 O and O 2 atmospheres, the simulated 557 . 7 nm brightness reaches 33 R , almost triple the average observed value of (11 . 5 ± 0 . 6) R . de Kleer et al. (2023) showed Ganymede's disk-integrated aurora were extremely consistent across three different nights of observation spanning more than 20 years, \nO \nO \nH \nFigure 12 shows the simulated surface temperature of Ganymede's sub-solar point before, during and after eclipse (Leblanc et al. 2017) along with the number of H 2 O molecules in the Monte Carlo simulation. The model only tracks molecules in gas phase, so as the H 2 O molecules freeze onto the surface, the number of simulated particles proportionally decreases. The simulated surface temperature \n<!-- image --> \n10 \nFigure 10. Simulation of Ganymede's O2 (top row) and H2O (bottom row) atmospheric column densities before, during and after eclipse by Jupiter as observed from Earth. Each time step was averaged over 2 · (57 minutes) of Ganymede's orbit. The coordinate system is right-handed, measured counter-clockwise along Ganymede's orbit with 0 · pointing toward the Sun and 180 · at the mid-point of the eclipse. Small-scale structures prominent in the H2O column densities are artifacts from the Monte Carlo simulation method. These simulations suggest the H2O atmosphere, if present, collapses rapidly after the onset of the eclipse, and recovers quickly after Ganymede emerges back into sunlight. \nso random variability in the incident electron densities alone likely cannot account for the difference between the modeled sunlit H 2 O and O 2 brightness and the observed brightness. \nA possible explanation for the apparent lack of emission from electron impact on H 2 O is the rapid condensation of the H 2 O atmosphere onto the surface at the onset of eclipse, an effect seen in the simulated H 2 O atmosphere. The simulated column density of H 2 Odecreases substantially between the second simulation time step (which includes the onset of eclipse) and the third simulation time step (fully in eclipse). The modeled sunlit 557 . 7 and 656 . 3 nm emissions from electron impact on H 2 O are well above our detection threshold, whereas the modeled in-eclipse emissions are near or below our detection threshold. (The plotted detection thresholds are the average of the combined random and systematic uncertainties and represent the typical noise level at a given wavelength.) This suggests we would be able to detect the presence of an actively-condensing H 2 O atmosphere as modeled by Leblanc et al. (2023) with observations taken sufficiently early during the eclipse. Instead, the observed emissions show no temporal changes in either the 557.7 or 656 . 3 nm brightnesses over the first tens of minutes of eclipse (figure 11). If there was a localized sublimation H 2 O atmosphere present in sunlight that froze out in eclipse, the deposition process must occur within the first ten to fifteen minutes of Ganymede entering the full umbral eclipse. \nof Ganymede decreased from 143 K to 113 K at a rate of -1Kmin -1 during the first 30 minutes of the eclipse. By the end of the eclipse the surface temperature had decreased by a further 12 K to a minimum of 101 K . Within the first hour, the number of simulated H 2 Omolecules (a quantity proportional to H 2 O density) decreased by an order of magnitude. \nBy the start of the time interval covered by the second observation (integrated from 17 to 22 minutes after the start of the full umbral eclipse), the simulated temperature decrease has slowed and the surface temperature only changed by 1 K from 114 K to 113 K . The column density decreased from 63%to45%ofthepre-eclipse maximum, so from electron impact on H 2 Owewould expect 10 . 5 R of emission at 557 . 7 nm for a total of 24 R ( 5 𝜎 above the observed brightness) and 21 . 6 R of emission at 656 . 3 nm ( 2 . 4 𝜎 above the noise threshold). \nWe evaluated the change in H 2 O column density over the duration of the first three observations to see what emission we could expect to detect from the collapsing H 2 O atmosphere in the simulation. During the time interval that corresponds to the first HIRES observation (integrated from 10 to 15 minutes after the start of the full umbral eclipse), the model temperature fell from 122 K to 117 K and the column density decreased from 73% to 66% of the pre-eclipse maximum. Auroral brightness is directly proportional to column density, so at 557 . 7 nm we would expect to measure the 13 . 5 R of constant emission from electron impact on O 2 and an equivalent brightness from electron impact on H 2 O, for a total of about 27 R , almost 8 𝜎 above the observed brightness of (12 ± 2) R . At 656 . 3 nm we would expect to measure 27 . 6 R of emission from electron impact on H 2 O, about 2 𝜎 above the observed brightness of (9 ± 9) R . \n[R] \nightness \nBr \nFigure 11. Comparison between 557 . 7 nm [O I] and 656 . 3 nm HI (Hα ) disk-integrated brightness observations and predicted brightnesses from modeled column densities. Black circles with vertical error bars show observations and their associated uncertainties. Red squares show predicted brightnesses from column densities averaged over 57 minutes ( 2 · ) of Ganymede's orbit (the horizontal bars show the extent of the averaging window). The dashed gray horizontal line in each plot shows the detection threshold (the typical standard deviation of the observations). The 557 . 7 nm [O I] modeled brightness from the O2 atmosphere does not substantially change over the course of the eclipse, so we have shown its value as a solid horizontal dark gray line. The shaded gray regions in the background are the same eclipse boundaries shown in figure 8. For both the 557 . 7 nm [O I] and 656 . 3 nm HI observed brightnesses, the H2O contribution quickly drops below the detection threshold. Because the first two observations should exhibit some contribution from H2O above the detection threshold but do not appear significantly brighter than the rest of the eclipse observations, we concluded there was no emission from a localized H2O atmosphere present in this data set. \n<!-- image --> \nAt the start of the time interval covered by the third observation (integrated from 34 minutes to 39 minutes after the start of the full umbral eclipse), the surface temperature has dropped to about 111 K and the column density has decreased to 9% of the pre-eclipse maximum. The contribution to the brightnesses at both 557 . 7 and 656 . 3 nm from electron impact on H 2 O are below the noise level of the individual observations. \nThese simulations suggest we should detect a contribution from electron impact on H 2 O to the observed brightnesses at 557 . 7 and 656 . 3 nm in the first two observations. However, \n[K] \nemperature \nT \nCount \nFigure 12. Simulated sub-solar surface temperature (Leblanc et al. 2017) and number of simulated H2O molecules before, during and after eclipse. Ganymede's surface temperature dropped rapidly at the onset of eclipse, decreasing by 30 K in the first half hour and a further 5 K by the end of the first hour. The number of simulated H2O molecules in the simulation (proportional to the H2O number density) decreased by an order of magnitude within the first hour, and remained relatively constant for the rest of the eclipse even though the temperature continued to decrease by an additional 7 K before the end of the eclipse. \n<!-- image --> \nfor both observations the observed brightnesses are consistent with the presence of at most a minor H 2 O atmospheric column.", '4.5. Potential Impact of New Juno-Derived Electron Properties on Aurora Interpretation': "Saur et al. (2022) provide an important caveat to the interpretation of modeled aurora brightnesses for Ganymede. Because of local electron acceleration within Ganymede's magnetic field, the upstream electron distribution used in most models (including ours) does not accurately represent the electron distribution that produces the aurora, and the 70 cm -3 density enhancement near the auroral ovals may not be physically accurate. Greathouse et al. (2022) reported Juno observations of the 130 . 4 and 135 . 6 nm UV aurora at very high spatial resolution. They showed the emission was narrowly-confined in latitude near the open/closed field line boundary, with combined brightnesses peaking around 1000 R . Saur et al. (2022) evaluated the electron distribution necessary to excite aurora of these combined brightnesses and found the electrons must have a Maxwell-Boltzmann distribution centered at 200 eV (twice the assumed energy of the upstream electrons) and a density of 950 cm -3 (about 50times higher than the assumed upstream electron number density). The significantly smaller brightnesses found by our work and previous UV studies are due to coarse spatial resolution spreading the brightness over resolution elements much larger than the emitting region. \nTo explore the manifestation of this effect in the HIRES observations, we evaluated the 630 . 0 nm emission using these \nFigure 13 shows the simulated physical emission from Ganymede with these parameters at full-resolution in the top image and at HIRES detector resolution in the bottom image. Both of these simulations use the modeled O 2 column densities shown in figure 10. The total simulated disk-integrated brightness is 149 R , which we retrieved from the simulation using the same aperture size as the data. This result is extremely close to the best-fit value of (130 ± 5) R at 630 . 0 nm retrieved from the HIRES observations for the center of the plasma sheet (figure 4), especially considering the simplistic assumptions of this simulation; small changes in the electron gradient could easily produce a result matching the observed brightnesses. As a result, we concluded that the observations are therefore consistent with the electron properties derived by Saur et al. (2022) when coupled with the emission spatial distribution and hemispheric brightness asymmetry from Greathouse et al. (2022) and the pixel scale of the HIRES detectors. \n<!-- image --> \n0 \n200 \nFigure 13. Simulation of 630 . 0 nm auroral emission from electron impact on O2 incorporating an updated electron energy distribution and emitting region. We used a Maxwell-Boltzmann electron energy distribution centered at 200 eV with a number density of 950 cm -3 (Saur et al. 2022), a typical O2 column density during eclipse and we restricted the aurora emission to a Gaussian shape along the open/closed field line boundaries with a FWHM of 5 · (Greathouse et al. 2022). The top high-resolution image shows the spatiallyconfined emission from discrete elements 0 . '' 01 wide, with brightnesses peaking above 10 kR at the limb. The bottom image shows this same image scaled down to the detector resolution and binning used in the Ganymede eclipse observations on 2021-06-08 UTC. We convolved this image with a two-dimensional Gaussian kernel with a FWHM of 0 . '' 5 , approximating typical seeing conditions on the summit of Maunakea. The white circle shows the size of the apparent disk of Ganymede. This simulation has equal electron flux for the northern and southern hemispheres, and is therefore representative of the plasma conditions encountered by Ganymede at the plasma sheet mid-plane. \n<!-- image --> \nelectron properties and the simulated O 2 column density from 177 · to 179 · (see figure 10). We restricted the emission to the latitudes of the open/closed field line boundaries (Duling et al. 2022) using a Gaussian profile with a FWHM of 5 · (Greathouse et al. 2022). To account for the leading-trailing brightness asymmetries apparent in the HIRES observations, we applied a sinusoidal scaling to the electron densities with longitude, decreasing from 950 cm -3 at the trailing hemisphere limb longitude to 26% of this value ( 244 cm -3 ) at the leading hemisphere limb longitude. This produces an electron flux with a dusk-dawn asymmetry matching the value of 1.84 we found in our analysis of the HIRES observations. Though we assume this brightness gradient is due exclusively to spatially-varying electron number density, it could also incorporate a higher O 2 column density on the trailing hemisphere (Oza et al. 2018). \nThis further demonstrates that either choice of electron energy distribution, number density and spatial distribution is consistent with the derived column densities of this and other works. Table 6 lists the expected emission ratios for this higher energy and higher number density Juno-derived electron distribution. The 200 eV electron energy distribution from Saur et al. (2022) is consistent with the in situ measurements of the low-energy electron distribution reported by Allegrini et al. (2022), which were made from within Ganymede's magnetosphere by the Jovian Auroral Distributions Experiment (JADE) during the Juno flyby of Ganymede on 2021-06-07 UTC. They did not report any corresponding measurements of electron number density from within Ganymede's magnetosphere, but outside of it they found densities that were typically between 5 and 20 cm -3 , more consistent with the Voyager flyby results (Scudder et al. 1981). Similarly, Kurth et al. (2022) used measurements of total electron densities made by the Juno/Waves instrument during the same \nTable 6. Modeled emission ratios relative to 557 . 7 nm [O I] for Ganymede's optical aurora, assuming the Juno-derived electron population with a Maxwell-Boltzmann distribution centered at 200 eV and a number density of 950 cm -3 . \nGanymede flyby and found they varied between 15 cm -3 to 30 cm -3 . However, when we simulated auroral emission like that shown in figure 13 assuming the same column densities but using with a Maxwell-Boltzmann electron energy distribution centered at 200 eV and a number density of 20 cm -3 , we found a predicted emission of only 3 R of 630 . 0 nm , suggesting more complex acceleration processes must be occurring within Ganymede's magnetic field. \nIn this study we presented a time-series analysis of 17 highresolution spectra of Ganymede's auroral emission at optical wavelengths taken with Keck I/HIRES on 2021-06-08 UTC. This study was the first to resolve and analyze the spatial variability of Ganymede's aurora at optical wavelengths and the first at any wavelength to evaluate these changes on a cadence of just a few minutes. We observed Ganymede during eclipse by Jupiter, which allowed us to capture the aurora without the overwhelming presence of reflected solar continuum. The timing of the eclipse also allowed us to observe Ganymede as it passed through the mid-plane of the Jovian plasma sheet where it was subjected to the highest incident electron number density. The high cadence of the observations also let us evaluate the potential for the rapid collapse of a localized H 2 O atmosphere near the sub-solar point as Ganymede passed into Jupiter's shadow.", '5. CONCLUSIONS': "We observed Ganymede's plasma sheet mid-plane-facing hemisphere brightening relative to the opposite hemisphere, reaching a peak hemispheric brightness ratio of nearly 2 when it was about 1 R J from the centrifugal equator of the plasma sheet. We did not observe the same exponential drop-off of brightness with distance from the plasma sheet mid-plane \nthat has been seen at Europa (Roth et al. 2016; de Kleer et al. 2023), though there was additional variability in the brightness which we attributed to longitudinal density heterogeneities in the Jovian plasma sheet. Additional observations could help to provide a better understanding of the nature of the variability seen in the brightness. \nOur MCMC model found an median atmospheric H 2 O/O 2 column density ratio of 0 . 10 +0 . 05 -0 . 04 , matching the upper limit of 0.10 found by de Kleer et al. (2023) for 2021-06-08 UTC but well below the H 2 O abundance found by Roth et al. (2021) from sunlit UV observations. The 630 . 0 nm/557 . 7 nm brightness ratio is well suited to differentiating between electron impact on O 2 and H 2 O as the source of the excited O atoms producing the auroral emission. We quantitatively tested our ability to detect an H 2 O atmosphere localized to near the sub-solar point by using the Monte Carlo simulations of Leblanc et al. (2023) for Ganymede's O 2 and H 2 O atmospheres in eclipse. We combined these simulations with our aurora model to simulate emission components proposed by Roth et al. (2021) and evaluate their detectability with HIRES. \nIn addition to evaluating the north-south hemispheric brightness ratio, we also compared the dusk-dawn (trailingleading) hemispheric brightness ratio. This ratio did not change with plasma sheet distance like the north-south ratio, but we did find the dusk hemisphere was almost always nearly two-times brighter than the dawn hemisphere. This effect is likely a combination of both higher incident electron number density on the ram-facing trailing hemisphere and a higher O 2 column density toward dusk due to higher afternoon surface temperatures predicted by models (e.g., Leblanc et al. 2017; Oza et al. 2018). Unfortunately we cannot decouple these effects with eclipse observations because of the viewing geometry limitations. \nThe modeled emission from the simulated sunlit H 2 O and O 2 atmospheres produced brightnesses of 33 R at 557 . 7 nm and 45 R at 656 . 3 nm , well above our typical observed brightnesses of (11 . 5 ± 0 . 6) and (0 ± 2) R , respectively. Our results are therefore inconsistent with the presence of a localized high-density H 2 O atmosphere near the disk center proposed by Roth et al. (2021) based on sunlit observations. \nEven though the simulations of the eclipse atmospheres suggest the H 2 O column density decreases rapidly during eclipse ingress, emission from electron impact on the actively-condensing H 2 Oatmosphere should still be a significant component of the observed 557 . 7 and 656 . 3 nm brightnesses for the first two observations in our data set. We did not detect any variability in the brightness which would suggest an actively-condensing H 2 O atmosphere during eclipse ingress, implying that if a sublimation H 2 Oatmosphere exists in sunlight and freezes back onto the surface during eclipse, the timescale for condensation must be around 10 minutes or less. \nKdK acknowledges support from NASA through a grant to program HST-GO-15425 from the Space Telescope Science Institute, which is operated by the Associations of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. FL acknowledges support by l'Agence Nationale de la Recherche (ANR) under projects ANR-22CE49-005-002 and ANR-21-CE49-0019. CS gratefully acknowledges the auspices of NASA's Solar System Observations program under contract 80NSSC22K0954. \nThe data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University \nThis work benefited from scientific exchanges that took place within International Space Sciences Institute (ISSI) international Team #559 and as part of an ISSI workshop, Team #515. \nof California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. \nThe authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. \nThis research has made use of the Keck Observatory Archive (KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronautics and Space Administration. \nFacilities: W. M. Keck Observatory (Keck I/HIRES)", 'REFERENCES': 'Greathouse, T. K., Gladstone, G. R., Molyneux, P. M., et al. 2022, Geophysical Research Letters, 49, e2022GL099794, doi: 10.1029/2022GL099794 \nJupiter, https://www.jpl.nasa.gov/images/ \nHall, D. T., Feldman, P. D., McGrath, M. A., & Strobel, D. F. 1998, The Astrophysical Journal, 499, 475, doi: 10.1086/305604 Jet Propulstion Laboratory (JPL). 2001, High Resolution Globe of \npia02873-high-resolution-globe-of-jupiter \nKivelson, M. G., Bagenal, F., Kurth, W. S., et al. 2004, in Jupiter: The Planet, Satellites and Magnetosphere, ed. F. Bagenal, T. E. Dowling, & W. B. McKinnon, Cambridge Planetary Science (Cambridge University Press), 513-536 \nJia, X., Walker, R. J., Kivelson, M. G., Khurana, K. K., & Linker, J. A. 2009, Journal of Geophysical Research: Space Physics, 114, A09209, doi: 10.1029/2009JA014375 \nKivelson, M. G., Khurana, K. K., Russell, C. T., et al. 1996, Nature, 384, 537, doi: 10.1038/384537a0 \nKurth, W. S., Sulaiman, A. H., Hospodarsky, G. B., et al. 2022, Geophysical Research Letters, 49, doi: 10.1029/2022GL098591 \nKumar, S., & Hunten, D. M. 1982, in Satellites of Jupiter, ed. D. Morrison, Space Science Series (University of Arizona Press), 782-806 \nLeblanc, F., Oza, A. V., Leclercq, L., et al. 2017, Icarus, 293, 185, doi: 10.1016/j.icarus.2017.04.025 \n115557, doi: 10.1016/j.icarus.2023.115557 \nLeblanc, F., Roth, L., Chaufray, J. Y., et al. 2023, Icarus, 399, \nMarzok, A., Schlegel, S., Saur, J., et al. 2022, Journal of \ndoi: 10.1029/2022JE007256 \nGeophysical Research: Planets, 127, e07256, \nMcCord, T. B., Hansen, G. B., Clark, R. N., et al. 1998, Journal of Geophysical Research, 103, 8603, doi: 10.1029/98JE00788 \n- McGrath, M. A., Jia, X., Retherford, K., et al. 2013, Journal of Geophysical Research: Space Physics, 118, 2043, doi: 10.1002/jgra.50122\n- -. 2024b, hiresaurora : (Somewhat) Automated Galilean Satellite Eclipse Aurora Brightness Retrievals, 2.15.0, Zenodo, doi: 10.5281/zenodo.10946628\n- Milby, Z. 2024a, hirespipeline : A Keck/HIRES Data Reduction Pipeline, 2.1.0, Zenodo, doi: 10.5281/zenodo.10946624\n- Molyneux, P. M., Nichols, J. D., Bannister, N. P., et al. 2018, Journal of Geophysical Research: Space Physics, 123, 3777, doi: 10.1029/2018JA025243\n- Oliversen, R. J., Scherb, F., Smyth, W. H., et al. 2001, Journal of Geophysical Research, 106, 26183, doi: 10.1029/2000JA002507 Oza, A. V., Johnson, R. E., & Leblanc, F. 2018, Icarus, 305, 50, doi: 10.1016/j.icarus.2017.12.032\n- Musacchio, F., Saur, J., Roth, L., et al. 2017, Journal of Geophysical Research: Space Physics, 122, 2855, doi: 10.1002/2016JA023220 \nPhipps, P., & Bagenal, F. 2021, Journal of Geophysical Research: Space Physics, 126, e28713, doi: 10.1029/2020JA028713 Retherford, K. D., Moos, H. W., & Strobel, D. F. 2003, Journal of Geophysical Research (Space Physics), 108, 1333, doi: 10.1029/2002JA009710 \n- Roth, L., Saur, J., Retherford, K. D., et al. 2016, Journal of Geophysical Research: Space Physics, 121, 2143, doi: 10.1002/2015JA022073\n- Roth, L., Ivchenko, N., Gladstone, G. R., et al. 2021, Nature Astronomy, 5, 1043, doi: 10.1038/s41550-021-01426-9\n- Saur, J., Duling, S., Wennmacher, A., et al. 2022, Geophysical Research Letters, 49, e2022GL098600', 'APPENDIX': 'Table 7 . Data files used in this study and their corresponding observation type and target. All Ganymede observations were taken during eclipse. \nTable 7 lists the file names and corresponding observation type for each FITS file used in this study. All data are available from the Keck Observatory Archive (KOA) 4 . These data were taken as a part of program ID C294 with principal investigator Katherine de Kleer. \nScience \nEuropa'}
2024arXiv240912204A	We revisit the dynamics of the postNewtonian PN twobody problem for two inspiraling compact bodies. Starting from a matteronly reduced Hamiltonian we present an adapted framework based on the Lie series approach enabling the derivation of complete perturbative solutions within the conservative sector. Our framework supports both circular and eccentric orbits and is applicable to any perturbation respecting rotational invariance and timeindependence. In the context of the ArnowittDeserMisner ADM canonical formalism this includes up to at least 3PN order and local terms beyond. We provide an example application at 2PN recovering classical periapsis advance and orbital period corrections alongside the full orbital evolution in time coordinates. We discuss eventual extension to spinning and timedependent systems.	2024-09-01T00:00:00Z	['arXiv:2409.12204', '10.48550/arXiv.2409.12204', '2024arXiv240912204A']	['General Relativity and Quantum Cosmology']	Hamiltonian normal forms for the postNewtonian binary problem	2,024	191	0.34	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.12204.pdf	{'Hamiltonian normal forms for the post-Newtonian binary problem': "C. Aykroyd, 1, ∗ A. Bourgoin, 1 and C. Le Poncin-Lafitte 1 1 SYRTE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universit´es, UPMC Univ. Paris 06, LNE, 61 avenue de l'Observatoire, 75014 Paris, France (Dated: September 20, 2024) \nWe revisit the dynamics of the post-Newtonian (PN) two-body problem for two inspiraling compact bodies. Starting from a matter-only reduced Hamiltonian, we present an adapted framework based on the Lie series approach, enabling the derivation of complete perturbative solutions within the conservative sector. Our framework supports both circular and eccentric orbits, and is applicable to any perturbation respecting rotational invariance and time-independence. In the context of the Arnowitt-Deser-Misner (ADM) canonical formalism this includes up to at least 3PN order and local terms beyond. We provide an example application at 2PN, recovering classical periapsis advance and orbital period corrections, alongside the full orbital evolution in time coordinates. We discuss eventual extensions to spinning and time-dependent systems.", 'I. INTRODUCTION': "The Einstein field equations form the cornerstone of the general theory of relativity (GR), offering an elegantly profound description of interacting matter, energy, and geometry. Yet these equations are notoriously complex due to their nonlinear, tensorial and interdependent nature. As a consequence, the discovery of exact solutions is chiefly limited to highly idealized symmetrical scenarios. Real astrophysical phenomena, however, are rarely encapsulated by such pristine conditions. Gravitationally radiating systems are a prime example of inherently asymmetric systems, particularly evident in the dynamics of compact binaries. Their study typically segments the process into distinct regimes, each requiring its own set of numerical or analytical approximation tools. The PN framework, popularized from the earliest days of GR [1-4], excels in addressing the inspiralling two-body problem. Initially, PN approximations assume wide, bound orbits with weakly or moderately relativistic speeds. Yet in practice, these solutions have been 'unreasonably effective' [5] in describing a range of compact systems, achieving remarkable accuracy up to the innermost stable circular orbit. This success is partly attributed to GR's 'effacing' property, which permits the omission of self-gravitation and internal structure at lower orders. \nThe evolution of the PN framework has closely followed technological advancements, notably with the discovery of the Hulse-Taylor pulsar [6], which underscored the necessity for a formalism compatible with high-order calculations. The era of gravitational wave (GW) astronomy, inaugurated by the detection of GW150914 [7], has reinforced this need, aligning analytical methods with the capabilities of second-generation detectors such as Advanced LIGO [8], VIRGO [9], KAGRA [10] (LVK), and the International Pulsar Timing Array [11]. As we approach the advent of third-generation detectors, includ- \ning LISA [12], the Einstein Telescope [13], and the Cosmic Explorer [14], further refinement of the PN framework remains imperative. \nPN schemes typically rely on decoupling the field and matter degrees-of-freedom. This can be performed by 'integrating out' the metric either at the level of the field equations or directly at the action-via a combination of perturbative techniques such as multipolar expansions [15], asymptotic matching [16, 17], and Effective Field Theory [18-20]. Eventually, the aim is to establish a matter-only Fokker Lagrangian [21], or to directly compute the particles' equations of motion [16]. Canonical formulations like ADM instead search for a reduced matter-only Hamiltonian or Routhian functional [22, 23]. Accordingly, different harmonic and ADM-TT gauge formulations have been found to be in agreement [16, 24]. But while there has been a notable focus in this fieldmatter decoupling step of PN schemes, there remains a substantial gap in furnishing comprehensive analytical solutions of the reduced functionals. The quasi-Keplerian parametrisation [25-27] has been sucessfully determined up to 4PN to describe the orbits of non-spinning binaries. However, a systematic extension of this approach to broader perturbative situations including spinning or dissipative systems remains a challenge. In many other instances, orbital evolutions are only determined approximately, restricted to secular contributions or assumptions of (quasi-)circular orbits. A common approach employs the Lagrange Planetary Equations [28, 29], which, fundamentally, entails solving by hand the equations of motion for the Keplerian elements, and can hardly be achieved exactly. In the context of canonical formalisms, Hamilton-Jacobi methods [23, 30, 31]-typically represented in action-angle coordinates-provide a straighter path to determining long-term dynamical invariants, but often lack explicit time-coordinate solutions. \nDeparting from the reduced matter-only canonical formalism, the Lie series approach can instead allow us to achieve an analytic portrayal of the dynamics that captures both long- and the short-term phenomena. Notably, the Lie framework provides a very structural path \nfor extending 'secular averaging' procedures to high orders and to a broad range of perturbative scenarios. In [32], Lie methods have been employed for determining symplectic ADM centre-of-mass coordinate transformations. They have also been applied to solve the binary problem at 1PN order, in the spinless case [33] and in the spinning case [34, 35]. To our knowledge, a general two-body Lie framework that is directly applicable to post-Newtonian systems (at higher than first order) is not yet established. \nIn this work, we aim to describe a canonical framework for orbital determination that can be readily applied to a wide range of perturbed two-body systems. We accordingly furnish parametric solutions for time-independent and rotational-invariant perturbations. These assumptions are valid in the ADM (spinless) conservative sector, up to at least 3PN order and to local terms beyond. Our provided solutions shall include the complete analytical orbital behaviour at some PN truncation order, including secular and oscillatory terms, and valid for fully eccentric systems. We shall prescribe a treatment which does not require explicit usage of action-angle coordinates, but which is instead established in terms of Keplerian functions or other parametrisations that can be easily adapted to the problem at hand. This freedom facilitates eventual applications of the framework to spinning systems and perturbations of non-gravitational nature. In Sec. III, we shall apply our framework to the 2PN ADMTT gauge Hamiltonian, not only re-deriving standard periapsis advance and orbital period corrections, but also thoroughly exploring the full long- and short-term orbital evolution. We then verify our results in a numerical example. \nNotation and conventions. \nWe presently introduce the notation used throughout the paper. Bold symbols are reserved for 3-vectors, decomposed into norm u = \| u \| (light-typeface) and direction ˆ u = u /u (overhat). Complex conjugation shall be denoted by the dagger operator ( · ) † . We shall adopt the following Poisson bracket convention: \n{ f, g } := ∂f ∂ r · ∂g ∂ p -∂f ∂ p · ∂g ∂ r , (1) \nfor phase space coordinates ( r , p ).", 'II. TWO-BODY PERTURBATIVE FRAMEWORK': "We dedicate this section to recalling the general Lie perturbative method and formulating it in the context of the post-Newtonian two-body problem. Consider a matter-only, local, autonomous Hamiltonian for two point-particles of the form: \nH ( r , p ) = K -1 ∑ ℓ =0 c -2 ℓ H ℓ ( r , p ) + O ( c -2 K ), (2) \nexpressed in a center-of-mass coordinate system C . The zeroth-order Newtonian term H 0 is assumed integrable, which is not the case for the higher-order ℓ -th PN perturbations c -2 ℓ H ℓ ( r , p ). Fortunately, not all is lost, and a complete time-integrated solution can often be obtained via the appropriate choice of coordinates. \nThe essence of the Lie series method is to establish a coordinate system C ∗ in which the Hamiltonian becomes integrable up to an optimal order Q ≤ K -1: \nH ∗ ( r ∗ , p ∗ ) = Q ∑ ℓ =0 c -2 ℓ H ∗ ℓ ( r ∗ , p ∗ ) + K -1 ∑ ℓ =0 c -2 ℓ R ℓ ( r ∗ , p ∗ ) + O ( c -2 K ), (3) \nwhere the starred expressions refer to the new set of canonical variables in C ∗ . The new integrable Hamiltonians H ∗ ℓ are in mutual involution. The terms R ℓ are nonintegrable remainders which include eventual resonances from the low orders ℓ < Q ; in general, such resonances must be carefully handled (see e.g. [36] for details). The above expression is known as a Birkhoff normal form. \nFor the class of systems we shall be considering in this paper, each Hamiltonian term shall be a rotationalinvariant scalar, depending only on r = ∥ r ∥ , p r := p · r /r and p = ∥ p ∥ . We shall demonstrate a posteriori that, remarkably, resonances are absent in our systems-namely, R ℓ = 0 , ∀ ℓ ≤ Q . Furthermore, we shall henceforth restrict H to be of PN order sufficiently low that Q = K -1. In this case, Eq. (3) is reduced to the integrable terms H ∗ ℓ only. \nThe general strategy to obtain (3) is to consider a family of 'near-identity' coordinate transformations which are canonical by construction. Namely, we consider the group of Hamiltonian symplectomorphisms, coordinate transformations T g generated by the action of smooth Hamiltonian functions g . Equipped with the Poisson bracket these functions form a Lie algebra with elements L g = {· , g } ; each vector element may act infinitesimally on phase-space functions f : \nδf = L g ( f ) = { f, g } . (4) \nThe near-identity condition requires the generator to be of PN order O ( c -2 ), with potentially higher-order terms: \ng ( r , p ) = K -1 ∑ ℓ =1 c -2 ℓ g ℓ ( r , p ) + O ( c -2 K ), (5) \nwhere each g ℓ is explicitly independent of c . To recover the global coordinate transformation one then exponentiates L g , giving rise to the Lie series : \nT g ( f ) = e L g ( f ) := ∞ ∑ ℓ =0 1 ℓ ! L ℓ g ( f ) = f + { f, g } + 1 2 { { f, g } , g } + . . . , (6) \nwhere L ℓ g denotes the ℓ -th repeated composition of L g . By construction, these transformations preserve Poisson brackets. Note that the converse is not true, not all symplectic transformations can be expressed as the application of a Hamiltonian flow. Under this formalism, the phase-space coordinates transform as: \nr = T g ( r ∗ ), p = T g ( p ∗ ), (7) \nwith the inverse given by negating the sign of g .", 'A. Homologic equations': 'Time-independent canonical transformations preserve the value of the Hamiltonian, in which case it holds that H ∗ ( r ∗ , p ∗ ) = H ( r , p ) = T g ( H )( r ∗ , p ∗ ). Expanding the Lie operator on H and equating similar-order terms yields the homologic functional relations: \nH 0 = H ∗ 0 (8a) \n{ g 1 , H 0 } = H 1 -H ∗ 1 , (8b) \n{ g 2 , H 0 } = H 2 + 1 2 { H 1 + H ∗ 1 , g 1 } -H ∗ 2 , (8c) \n. \n. \n. \nwhere the unknowns are the generator sequence ( g k ) and the Hamiltonian sequence ( H ∗ k ). The homologic equations can be solved sequentially from the top, plugging the solutions from the previous iterations; generically, they have the form: \n{ g ℓ , H 0 } = P ℓ -H ∗ ℓ , (9) \nwhere the perturbation P ℓ is some function of the solutions up to order ℓ -1. Taking the average along the Newtonian flow Φ H 0 t directly yields a solution for H ∗ ℓ : \nH ∗ ℓ = ⟨P ℓ ⟩ , (10) \nwhere \n⟨ f ⟩ = lim T →∞ 1 T ∫ T/ 2 -T/ 2 f · Φ H 0 t d t . (11) \nNext, we determine the generator by leveraging the integrability of the Keplerian system. Namely, H 0 is a maximal superintegrable system of 6-dimensional phasespace with 5 independent integrals of motion. These integrals can be specified in many forms; we shall supply some of them in the following section. In this manner, the system can be expressed in terms of five freely-chosen (independent) constants I = ( I 1 , . . . I 5 ), and a remaining Keplerian periodic degree-of-freedom, which we shall call θ . We stress that although these six parameters must be one-to-one with ( r , p ), they are not required to form a \ncanonical set, allowing more flexibility in the specification of the results 1 . We may subsequently develop the left-side of the homologic equation via chain rule: \n{ g ℓ , H 0 } = ∂g ℓ ∂θ { θ, H 0 } + ∂g ℓ ∂ I · { I , H 0 } , (12) \nwhere the five brackets { I , H 0 } identically vanish. A formal solution is thus obtained from Eq. (9) by integrating over θ : \ng ℓ = ∫ ( P ℓ -H ∗ ℓ ) ( d θ d t ) -1 kep d θ , (13) \nwhere we have defined the Keplerian time-derivative \n( d d t ) kep ≡ {· , H 0 } . (14) \nThe primitive is taken to be average-free, which avoids the emergence of Poisson terms. \nThe construction of an explicit parametric expansion to these integral solutions requires a convenient description of the system coordinates. For this we propose the Keplerian parametrisation of the following section.', 'B. The Keplerian parametrisation': "We shall now recall the Keplerian osculating parameters, which are typically employed in celestial mechanics (see e.g. [37]) and are convenient for the Lie formalism. We stress that this differs from the post-Keplerian formalism [25, 26], where Newtonian relationships are adjusted with PN corrections. Instead, one adopts phasespace relations which are exact, independently of the system perturbation. Namely, every given point ( r , p ) in phase-space C (or likewise in C ∗ ) is uniquely characterised by six orbital elements; for instance: the semimajor axis a , the eccentricity e , the true anomaly v , the argument of the periapsis ω , the inclination ι , and the longitude of the node Ω. These do not form a canonical set of variables, yet their Poisson brackets do not require PN corrections, facilitating the integration of the Newtonian-level derivatives in the homologic equation. We shall also introduce the eccentric and mean anomalies E and M , the longitude of the periapsis ϖ = ω +Ω, and the total orbital action L = √ a , all of which can be related to the aforementioned six elements. To specify all these parameters, we henceforth assume that the coordinates are appropriately rescaled, so that the Keplerian potential adopts the form: \nH 0 ( p , r ) ≡ p 2 2 -1 r = -1 2 L 2 . (15) \nAt each point in C , we introduce an ellipse tangent to the motion, lying within a coordinate plane perpendicular to the orbital angular momentum J ( r , p ) = r × p . Accordingly, we decompose the motion onto a right-handed orbital basis ( ˆ A , ˆ B , ˆ J ): \nr = r (cos v ˆ A +sin v ˆ B ), (16a) \np = 1 J ( -sin v ˆ A +( e +cos v ) ˆ B ) , (16b) \nwith r = J 2 / (1 + e cos v ) and e = √ 1 -J 2 /L 2 . The vectors A = e ˆ A and B = e ˆ B are the periapsis and binormal vectors. Together with J = J ˆ J they form nine (dependent) first integrals of the Newtonian motion, of which A and B are only conserved for perfect inverse-square central forces. The last two can be readily expressed in functional form: \nA = p × J -r r , B = J p -J × r Jr . (17) \nThe orbital basis can be further related to a non-rotating reference orthonormal basis ( ˆ X , ˆ Y , ˆ Z ) via a rotation operator R : \nˆ A = R ( ˆ X ), ˆ B = R ( ˆ Y ), ˆ J = R ( ˆ Z ). (18) \nIn celestial mechanics, R is typically specified via z -x -z Euler angles (Ω , ι, ω ). Finally, the three anomalies v , E and M are defined to satisfy the following relationships: \nM = E -e sin E (19) \nand \ncos v = 1 e ( J 2 r -1 ) , sin v = Jp r e , (20a) \ncos E = 1 e ( 1 -r L 2 ) , sin E = rp r eL . (20b) \nThese angles are chosen on the same Riemann sheet, ensuring that the difference between any two of them does not exceed π . \nWe emphasize that the Keplerian parameters are to be seen as functions of phase-space and applicable to both the C and C ∗ coordinate systems. Accordingly, for each parameter q , we introduce their time-evolutions under Hamiltonian H and H ∗ : \n˜ q ( t ) = q ( r ( t ) , p ( t )), (21a) \n¯ q ( t ) = q ( r ∗ ( t ) , p ∗ ( t )). (21b) \nWe shall refer to ¯ q as the secular evolution of q and to ˜ q as the complete time evolution of q . We can immediately deduce the following relationship: \n˜ q = T g ( q ). (22) \nThe evolution equation of each parameter can be extracted from the generalized Hamilton's equations, \n˙ ˜ q = ˜ { q, H} , (23a) \n˙ ¯ q = { q, H ∗ } , (23b) \nvalid for non-canonical variables. In these equations, the application of tilde (resp. bar ) denote that expressions are evaluated in C (resp. C ∗ ) coordinates [cf. Eq. (21)], after computation of the brackets. The advantage of the Lie approach lies in that solving Eq. (23b) and then applying (22) is in general much easier than directly solving (23a). The Poisson bracket structure of each parameter q can be deduced from the provided definitions, and is already known from celestial mechanics 2 . We remark that these brackets remain unchanged under the symplectic transformation T g .", 'C. Solutions to the homologic equation': 'The integral solutions to the homologic equation can be explicitly solved for a large class of rotationally-invariant problems. Namely, we consider the asymptoticallyvanishing polynomial expansion \nP ℓ = A ℓ, 0 + A ℓ, 1 r -1 + . . . + A ℓ,s r -s , (24) \nexpressed such that the coefficients A ℓ, 0 , . . . , A ℓ,s depend solely on the Keplerian integrals of motion I (via use of the relationships from Sec. II B). In this case, we develop Eqs. (10, 13) with: \ng ℓ = s ∑ k =1 A ℓ,k ( ξ k -M n 0 h k ) , H ∗ ℓ = s ∑ k =0 A ℓ,k h k , (25) \nwith n 0 = ( -2 H 0 ) 3 / 2 .The basis elements turn out to be: \nξ k = ∫ 1 r k ( d θ d t ) -1 kep d θ , h k = 〈 1 r k 〉 , (26) \nwhere the primitive is chosen ensuring that ⟨ ξ k ⟩ = ( ⟨ M ⟩ /n 0 ) h k = 0. Explicit integrated expressions for ξ k and h k are provided in Appendix A. \nWith this the Hamiltonian H ∗ = ∑ ℓ c -2 ℓ H ∗ ℓ and the generator g = ∑ ℓ c -2 ℓ g ℓ are now fully determined. The explicit coordinate-time evolutions r ∗ ( t ) and p ∗ ( t ) can hence be obtained from the integrable equations of motion; for instance, via the Kepler elements ¯ q ( t ) outlined in the previous section. Initial conditions in C ∗ -space are determined through application of the inverse Lie operator T -g . Eventually, the phase-space evolution in the original coordinate system C can be recovered via application of T g . \nAs an immediate example, we shall apply the framework to the conservative sector of the ADM matter Hamiltonian at 2PN order.', 'III. APPLICATION TO THE CANONICAL ADM HAMILTONIAN AT 2PN': "Asymptotically flat coordinates in GR admit the Poincar'e group as a global symmetry [24, 31]. The generators of this group are realised as 10 surface integrals, conserved on-shell as dictated by Noether's theorem: the total energy E and total linear momentum Π , generators of space-time translations; the angular momentum ϑ , generator of rotations; and the generator of boosts Κ = Γ -t Π associated to the centre-of-mass constant Γ . In canonical formalisms, each of these quantities is seen as a function of phase-space. The ADM-TT gauge Hamiltonian is not manifestly Poincar'e invariant since most quantities are realised in nonlinear fashion. Instead, this invariance must be explicitly shown as done in [23, 25, 38-40]. However, for non-spinning particles, the ADM-TT gauge does manifestly respect the Euclidean group, and thus the momenta are simply Π = ∑ a p a and ϑ = ∑ a r a × p a , valid at all post-Newtonian orders (where ( r a , p a ) are the coordinates of particle a ). The energy, as usual, refers to the conserved Hamiltonian, namely E = Hmc 2 , where m = ∑ a m a represents the combined mass parameter of the system. When including the radiative sector to account for losses via the gravitational radiative reaction, the Noether quantities are no longer conserved; at orders 2 . 5PN and above, one must consider flux-balance-type equations (see e.g. [21, 24, 41]) or time-dependent Hamiltonians [23, 42]. In our case we are concerned in studying the conservative sector of the two-body problem. \nFrom the freedom of performing a Poincar'e transformation, we adopt coordinates of the centre-of-mass (CM) where Π = Γ = Κ = 0. Rothe and Schafer [32] explicitly determine a symplectic transformation to a canonical CM frame which is valid for any Π and Γ . In practice, under CM coordinate conditions ( Π = Γ = Κ = 0), the frame transformation reduces to the Newtonian-like relations p = p 1 = -p 2 , r = r 2 -r 1 (see also [22, 30, 43]). As our starting point, we shall adopt the rescaled CM coordinates: \np ' = p µ , r ' = r Gm , H ' = Hmc 2 µ , t ' = t Gm , (27) \nwhere µ = m 1 m 2 /m and ν = µ/m . We also rescale E ' = E /µ and ϑ ' = ϑ / ( Gmµ ). For notational simplicity, we shall henceforth omit the primes. In rescaled coordinates, the reduced matter-only ADM-TT Hamilto- \n[23, 30]: \nH 1 ( r , p ) = 1 2 r 2 -1 8 (1 -3 ν ) p 4 -1 2 r ( (3 + ν ) p 2 + ν ( n · p ) 2 ) , (28a) H 2 ( r , p ) = 1 16 ( 1 -5 ν +5 ν 2 ) p 6 + 1 8 r ( ( 5 -20 ν -3 ν 2 ) p 4 -2 ν 2 p 2 ( n · p ) 2 -3 ν 2 ( n · p ) 4 ) + 1 2 r 2 ( (5 + 8 ν ) p 2 +3 ν ( n · p ) 2 ) -1 4 r 3 (1 + 3 ν ), (28b) \nwith n = r /r and leading zeroth-order term H 0 as specified in Eq. (15).", 'A. Solution and observables': 'We employ the methods outlined in Sec. II to transform the rescaled CM Hamiltonian. Specifically, Eq. (25) parametrically furnishes the normal-form and generator as a function of the perturbation at each PN order. We consider iteratively the two perturbations: \nP 1 = H 1 , P 2 = H 2 + 1 2 {H 1 + H ∗ 1 , g 1 } . (29) \nEach perturbation is then expanded into the form (24) by appropriately expressing momenta in terms of the first integrals and the separation, with the aid of the following two expressions: \np 2 = 2 r -1 L 2 , p 2 r = p 2 -J 2 r 2 . (30) \nThe procedure leads to the following Hamiltonian: \nH ∗ ( J, L ) = -1 2 L 2 + 1 c 2 { 15 -ν 8 L 4 -3 JL 3 } + 1 c 4 { 5(2 ν -7) 4 J 3 L 3 -27 2 J 2 L 4 + 3(35 -4 ν ) 4 JL 5 -ν 2 -15 ν +145 16 L 6 } . (31) \nWe recall that the parameters L = L ( r ∗ , p ∗ ) and J = J ( r ∗ , p ∗ ) are treated as pure Newtonian-order functions of phase-space coordinates. This enables calculations to be treated agnostically in terms of the parameter choice and more readily adapted to a new problem. Simultaneously, Hamiltonian (31) can also be regarded as if it were expressed in Delaunay variables, since the involved transformation is canonical. Unsurprisingly, we find that our expression precisely matches the Delaunay form that can be derived via Hamilton-Jacobi methods (cf. [23, 30]). Nevertheless, Lie methods have the advantage of also providing the generator: \ng = 1 c 2 { p r r ( J (4 -ν ) -6 L ) 2 JL 2 -6 J arctan ( Lrp r JL + r ) + νp r 2 } + 1 c 4 { p r r 2 ( J 2 ( ν 2 -2 ν ) 8 -3 J 3 ν 4( J + L ) ) + p r r ( 9 J 2( J + L ) + 6 -ν 4 ) + p r ( 5(2 ν -7) 4 J 2 + 3( ν +2) 4 J ( J + L ) + 3 JL + -ν 2 +10 ν -32 8 L 2 ) + p r r ( 5(2 ν -7) 4 J 3 L + 9(3 -ν ) 4 JL 3 -ν +16 8 L 4 ) + ( 10 ν -35 2 J 3 + 15 -6 ν 2 JL 2 ) arctan ( Lrp r JL + r ) } . (32) \nWe have verified the consistency of the obtained generator, checking that our expressions for H ∗ and g identically satisfy the homologic equations-to order O ( c -6 )-with T g correctly transforming H into H ∗ . In the next steps, we shall derive the integrable dynamics from Hamiltonian (31), and eventually incorporate the short-term oscillatory behaviour to obtain the complete dynamics via application of the Lie transform generated by (32).', '1. Integrable dynamics': "Effectively, the parameter space C ∗ encodes the secular or 'average' behaviour of the system. The dynamics can be easily determined from the transformed Hamiltonian H ∗ via the generalised Hamilton's equations (23b), without explicitly resorting to the variable transformations alluded to in the previous section. We presently provide the expression of six independent osculating Keplerian parameters, following the bar-tilde conventions from Eq. (21). The remaining parameters, such as ¯ v ( t ) and ¯ E ( t ), are obtained via the Keplerian relationships (19)-(20), applied at each moment in time. The full set of parameters can then be used to reconstruct the evolution of the phase-space coordinates ( r ∗ , p ∗ ) under the integrable Hamiltonian H ∗ . As expected, the secular PN perturbations act only on the orbital phase and the periapsis angle: \n¯ a ( t ) = ¯ a 0 , ¯ e ( t ) = ¯ e 0 , ¯ M ( t ) = ¯ M 0 + ∂ H ∗ ∂L t , (33a) \n¯ ι ( t ) = ¯ ι 0 , ¯ Ω( t ) = ¯ Ω 0 , ¯ ϖ ( t ) = ¯ ϖ 0 + ∂ H ∗ ∂J t . (33b) \nThe subscripted zeros refer to initial values obtained from the osculating functions applied to ( r ∗ 0 , p ∗ 0 ). Two of the constants, ¯ a 0 and ¯ e 0 , are directly linked to the Noether quantities E and ϑ , which are also the dynamical invariants of the transformation T g . Namely, this symplectomorphism inherits both the time and the rotation symmetries of the system, as is clear from Eq. (32). Rotational invariance in particular ensures the preservation of ϑ , so that J = ϑ = ¯ J . Moreover, the energy relation E = H ∗ ( ¯ L, J ) can be inverted order-by-order, yielding \na useful expression: \n¯ L = 1 √ -2 E + 1 c 2 { 3 J + ν -15 8 √ -2 E } + 1 c 4 { 35 -10 ν 4 J 3 -6 ν -15 2 J E + 3 ν 2 +30 ν +35 128 ( -2 E ) 3 / 2 } . (34) \nThe values for ¯ a 0 and ¯ e 0 can be deduced from these expressions by inputting ¯ a 0 = ¯ L 2 and ¯ e 0 = (1 -¯ J 2 / ¯ L 2 ) 1 / 2 . Next, we extract the precession and mean orbital fre- \nquencies from the normal-form Hamiltonian: \n˙ ¯ M ( t ) = 1 ¯ L 3 + 1 c 2 { 9 ¯ J ¯ L 4 + ν -15 2 ¯ L 5 } + 1 c 4 { 15(7 -2 ν ) 4 ¯ J 3 ¯ L 4 + 54 ¯ J 2 ¯ L 5 + 60 ν -525 4 ¯ J ¯ L 6 + 3 ( ν 2 -15 ν +145 ) 8 ¯ L 7 } , (35a) \n˙ ¯ ϖ ( t ) = 1 c 2 3 ¯ J 2 ¯ L 3 + 1 c 4 { 15(7 -2 ν ) 4 ¯ J 4 ¯ L 3 + 27 ¯ J 3 ¯ L 4 + 12 ν -105 4 ¯ J 2 ¯ L 5 } . (35b) \nThese frequencies are associated to well-known observables of the integrable PN dynamics. These are the periapsis-to-periapsis period T and the periapsis advance per orbit ∆ ϕ : \n˙ ¯ M = Gm ( 2 π T ) , ˙ ¯ ϖ = ∆ ϕ T . (36) \nFor convenience, we provide the observables T and ∆ ϕ in physical units. Explicit expressions for P and ∆ ϕ have also been derived by Damour, Jaranowski & Schafer [30] in the framework of the Hamilton-Jacobi formalism. It is worth mentioning that their expressions perfectly align with our results when we incorporate Eq. (34).", '2. Complete dynamics of the original system': "We may now return to the original coordinate system C . Applying T g to each Keplerian parameter, we extract their full temporal evolution as a function of the secular variables derived in the previous section [cf. Eq. (22)]. \nAfter some lengthy bracket computations, we obtain the expressions that follow, strictly valid at O ( c -6 ): \n˜ a = ¯ a + 6 ∑ k =0 A ( a ) k cos( k ¯ v ), (37a) \n˜ e = ¯ e + 6 ∑ k =0 A ( e ) k cos( k ¯ v ), (37b) \n˜ v = ¯ v + 6 ∑ k =1 A ( v ) k sin( k ¯ v ) + (¯ v -¯ E ) ( 2 ∑ k =0 B ( v ) k cos( k ¯ v ) ) , (37c) \n˜ ϖ = ¯ ϖ + 6 ∑ k =1 A ( ϖ ) k sin( k ¯ v ) + B ( ϖ ) (¯ v -¯ M ). (37d) \nThe Kepler parameters ˜ ι = ¯ ι and ˜ Ω = ¯ Ω are identically unchanged by the T g . Each of the above coefficients is time-independent; they are reported in Appendix B. \nWe remark that certain coefficients [Eqs. (37b-37d)] contain coordinate singularities in the quasi-circular regime (¯ e → 0 or equivalently ¯ J → ¯ L ). Specifically, their Laurent expansions contain single or double poles in ¯ e = 0, and for small enough eccentricity (¯ e ≲ 1 /c 2 ) the residuals are unbounded. This is an artefact of expanding the post-Newtonian osculating parametrisation and can be bypassed by adopting 'Poincar'e-like' elements: \nz = e exp(i ϖ ), λ = v + ϖ , ζ = sin( ι/ 2) exp(iΩ), (38) \nWe have determined the full temporal evolution of the aforementioned 'regularised' elements: \n˜ z = ¯ z +e i ¯ ϖ { 6 ∑ k = -6 ( A ( z ) k +i B ( z ) k ( ¯ v -¯ E ) + C ( z ) k (¯ v -¯ E ) 2 ) e i k ( ¯ ϖ -¯ λ ) + 2 ∑ k = -2 ( D ( z ) k +i E ( z ) k ( ¯ v -¯ E ) ) e i k ( ¯ ϖ -¯ λ E ) } , (39a) \n˜ λ = ¯ λ + 4 ∑ k =1 i A ( λ ) k ( ¯ z k -(¯ z † ) k ) cos( k ¯ λ ) + B ( λ ) k ( ¯ z k +(¯ z † ) k ) sin( k ¯ λ ) \n+(¯ v -¯ E ) ( 2 ∑ k =0 C ( λ ) k ( ¯ z k +(¯ z † ) k ) cos( k ¯ λ ) + i D ( λ ) k ( ¯ z k -(¯ z † ) k ) sin( k ¯ λ ) ) + B ( ϖ ) (¯ v -¯ M ), (39b) \n˜ ζ = ¯ ζ , (39c) \nwith λ E = E + ϖ . \nFor convenience, the eccentricity can also be extracted via the exact expression instead of the truncated series: \n˜ e = √ 1 -¯ J 2 / ˜ a , (40) \nwith ˜ a given by Eq. (37a). 3", '3. Numerical example': "The analytical expressions provided in the previous section solve the ADM Hamiltonian (28) at the O ( c -6 ) level. We shall now illustrate their validity numerically, for an unequal-mass moderately-relativistic binary in both quasi-circular and eccentric orbit configurations. For each scenario, we integrate H by employing an 8thorder implicit Runge-Kutta scheme. Integration is performed in polar coordinates ( r, λ, p r , J ), where Hamilton's equations reduce to (see for instance [31]; rotational invariance is required): \n˙ r = ∂ H ∂p r , ˙ p r = -∂ H ∂r , (41a) \n˙ λ = ∂ H ∂J , ˙ J = -∂ H ∂λ = 0, (41b) \nwhere an implicit change of variables is performed with the aid of Eqs. (30). Figure 1 depicts the phase error by which the perturbative solution (in C coordinate space) deviates from the numerical integrator in each scenario. As we might anticipate, the analytical residue manifests close to 2PN marks below that of a standard Keplerian solution. This residue arises from the truncation of higher-order terms in the Lie expansion of T g , as portrayed in Fig. 2. Specfically, the secular Kepler parameters ¯ q ( t ) are obtained from a trunction of H ∗ to its integrable terms, while a second truncation occurs in the transformation ˜ q ( t ) = T g (¯ q )( t ). Consequently, this residue manifests as terms of order O ( c -6 ) that are linear in time for ˜ ϖ ( t ) and ˜ M ( t ) and periodic for the other variables, which is consistent with a 2PN solution. \nFIG. 1. Evolution of the phase residue ( λ ) of the truncated analytical solution (in solid red ) when compared to numerical integration evolving under the 2PN ADM Hamiltonian [Eq. (28)], for a moderately-relativistic binary of velocity p/c ∼ 10 -3 and mass-ratio ν = 2 / 9. Two eccentricity configurations are considered: on the left , a quasi-circular orbit ( e = 0 . 01); on the right , a highly eccentric orbit ( e = 0 . 8). The solid blue benchmark denotes the error of a purely Keplerian analytical solution. The numerical accuracy baseline at 10 -15 is represented in dashed black , estimated from the residue of the Keplerian solution when c →∞ . We have also verified that the conservation of the ADM constants E and J respect this accuracy level. \n<!-- image --> \nFIG. 2. Schematics of the Lie canonical framework. The leftmost column conceptualises a complete relativistic Hamiltonian system with an infinite series of PN terms. The middle column represents a truncation at O ( c -2 K ). The rightmost column corresponds to the secular system with Hamiltonian H ∗ in normal form, obtained by applying the Lie transformation T g to H . Indeed, the red pathway illustrates the key transformation steps of the Lie approach; meanwhile, the blue arrow represents direct integration of H , a nonlinear process with a potentially infinite amount of terms. In practical computations, each application of the Lie transform T g is truncated to order O ( c -2 K ), leading to a computation of ˜ q ( t ) that is formally accurate to the same order. \n<!-- image --> \nℓ \n=0", 'IV. CONCLUSIONS': 'We have provided a canonical Lie framework for analyzing the perturbed two-body problem under generic conditions of time-independence and rotational invariance. Our framework allows for the computation of the complete coordinate-time evolution in the PN conservative sector, using centre-of-mass coordinates C . It is also applicable to more general perturbative two-body systems under similar conditions. The formalism can theoretically be applied to perturbations of arbitrary order, although the residue is expected to grow quickly at very high orders (see e.g. [36]). We provide a generic family of solutions [Eq. (25)] for systems within the aforementioned conditions. Accordingly, we specify an equivalent Hamiltonian in a new set of phase-space coordinates C ∗ , which is integrable at given order; as well as the corresponding Lie transformation T g : C ∗ → C , enabling the conversion between both coordinate systems. From the boundedness of the obtained Lie generator, it is clear that resonances are absent from the class of systems which \nhave been studied. We have also applied our framework to the local, conservative ADM sector of the non-spinning point-mass binary problem, recovering classical results. \nGoing forward, our formalism may be broadened to other contexts, such as spinning pole-dipole models [23, 40, 44, 45] or time-dependent systems, including the ADM dissipative sector and non-local terms at 4PN [23]. Notably, the radiation-reaction term at 2 . 5PN may be treated by either doubling the phase-space variables [23, 46-48] or incorporating explicit time-dependence. In general, the presence of non-autonomous perturbative terms no longer leads to the preservation of the value of the conservative Hamiltonian by the Lie transform T g ; regardlessly, similar homologic relations to Eq. (8) can be derived, but in extended phase-space. Spinning systems may benefit from the phase-space description of the Keplerian variables specified in Sec. II B; when properly normalised, the angular momentum and eccentricity vectors form the algebra of SO (3) × SO (3) ∼ SO (4), potentially expressible in spinor form (see [49] for an example). We conclude by remarking that in instances where \nresonances are present, order-by-order execution of the normal-form process is recommended (see e.g. [36] for details).', 'SUPPLEMENTARY INFORMATION': "We provide source files in Mathematica language, organised as follows: \n- · A Wolfram Language package file 'KeplerPert.wl' containing methods for computing the canonical Lie transform and for solving autonomous rotationally-invariant two-body homologic equations;\n- · A Wolfram Notebook file 'ADM 2PN.nb' exemplifying the package's use applied to the 2PN ADM Hamiltonian;\n- · An auxiliary package 'PoissonBracket.wl' implementing some basic properties of Poisson bracket algebra;\n- · An auxiliary package 'Utils.wl' with a collection of generic helper functions.", 'ACKNOWLEDGMENTS': "C.A. acknowledges the joint finantial support of Centre National d' ' Etudes Spatiales (CNES) and ' Ecole Doctorale Astronomie et Astrophysique d'Ile de France (ED127 AAIF). This work was also supported by the Programme National GRAM, by PNPS (CNRS/INSU), by INP and IN2P3 co-funded by CNES, and by CNES LISA grants at CEA/IRFU. We are very grateful to G. Faye for the constructive comments and discussions.", 'Appendix A: Flow integrals': 'In this section we search for explicit expressions to the basis elements ξ k and h k , which parametrise the solutions to the homologic equations. We recall their definitions: \nξ k = ∫ 1 r k ( d θ d t ) -1 kep d θ , h k = 〈 1 r k 〉 , \nwith the integration constant chosen such that ⟨ ξ k ⟩ = 0. It is immediately clear from the definition that h 0 = 1. Constraining θ to a Kepler angle further leads to h k = ( n 0 / 2 π ) ( ξ k ( π ) -ξ k ( -π ) ) , where n 0 = 1 /L 3 . For higher orders of ξ k and h k we shall split the computation into two distinct cases. Namely, using the definitions in Sec. II B, we work out the relationships that follow: \n- · For k = 1, identifying θ = E , we have: \n( d E d t ) kep = 1 Lr , E = M + rp r L . (A1) \n- · For k ≥ 2, identifying θ = v , we have instead: \n( d v d t ) kep = J r 2 , v = M + rp r L +2arctan ( Lrp r JL + r ) . (A2) \nThe integrals for k = 1 , 2 are trivial and yield: \nξ 1 = LE , h 1 = Ln 0 . (A3) \nξ 2 = v J , h 2 = n 0 J . (A4) \nThe evaluation of higher values of k involves a little more work. Namely, we consider, for integer k ≥ 3: \nξ k = 1 J 2 k -3 k -2 ∑ j =0 ( k -2 j ) e j ∫ cos j v d v . (A5) \nFortunately, the integrals of powers of cosine on the right K α = ∫ cos α v d v can be determined analytically. They follow from the recurrence relationship: \nK α = ( α -1 α ) K α -2 + 1 α sin v cos α -1 v , K 0 = v . (A6) \nWe verify via induction that the recurrence is satisfied by: \nK α = sin v ∑ β ∈ A β !! α !! ( α -1)!! ( β -1)!! 1 β cos β -1 v + α e ( α -1)!! α !! v , \n(A7) \nwhere ( · )!! is the double factorial, α e := 1 -( α mod 2), and the sum is performed over the set A = { α e +1 , α e + 3 , . . . , α -2 , α } . The last term is only present when α is even, as evidenced by the presence of the α e factor. Putting everything together and rearranging: \nξ k = p r J 2 k -4 k -2 ∑ α =0 ( k -2 α ) × ∑ β ∈ A β !! α !! ( α -1)!! ( β -1)!! e α -β β ( J 2 r -1 ) β -1 + v J 2 k -3 ⌊ k 2 ⌋-1 ∑ α =0 ( k -2 2 α )( 2 α α ) ( e 2 ) 2 α . \nWe can now extract h k : \nh k = n 0 J 2 k -3 ⌊ k 2 ⌋-1 ∑ α =0 ( k -2 2 α )( 2 α α ) ( e 2 ) 2 α , (A8) \nvalid for integer k ≥ 2, and where ⌊ x ⌋ is the largest integer smaller or equal to x . The angular relationships provided [Eqs. (A1, A2)] allow us to express g ℓ = ∑ k A ℓ,k ( ξ k -( M/n 0 ) h k ) solely as a function of r , p r and p . Finally, we can confirm via parity arguments that the average of each term does not contribute to g ℓ : \n〈 ξ k -M n 0 h k 〉 = 0. (A9)', 'Appendix B: Coefficients of the Keplerian coordinate transformation': 'We provide the coefficients of the 2PN ADM transformations for the Keplerian parameters from C ∗ to C as detailed in Sec. III A 2:', '· coefficients for a:': 'A ( a ) 0 = 1 c 2 { 4 -ν -6 ¯ L ¯ J + ¯ L 2 (2 ν -7) ¯ J 2 + ¯ L 4 (9 -ν ) ¯ J 4 + 1 c 2 ( 7 ( 7 ν 2 -96 ν +360 ) ¯ L 6 16 ¯ J 8 -( 207 ν 2 -1876 ν +4360 ) ¯ L 4 32 ¯ J 6 + ( 17 ν 2 -98 ν +208 ) ¯ L 2 4 ¯ J 4 + 3(7 ν -90) ¯ L 3 4 ¯ J 5 -( ν -32) ¯ L 4 ¯ J 3 -3(2 ν -5) 2 ¯ J ¯ L -27 ν 2 -196 ν +40 32 ¯ J 2 + -ν -16 4 ¯ L 2 ) } (B1) \nA ( a ) 1 = ¯ e c 2 { ¯ L 2 (7 ν -16) 4 ¯ J 2 + ¯ L 4 (48 -7 ν ) 4 ¯ J 4 + 1 c 2 ( -24 ¯ J ( ¯ J + ¯ L ) + 640 + 124 ν -13 ν 2 32 ¯ J 2 -3(9 ν +40) ¯ L 8 ¯ J 3 (B2) + ( 75 ν 2 -426 ν +1306 ) ¯ L 2 16 ¯ J 4 + 3(35 ν -312) ¯ L 3 8 ¯ J 5 -( 313 ν 2 -2296 ν +3684 ) ¯ L 4 32 ¯ J 6 + ( 11 ν 2 -147 ν +504 ) ¯ L 6 2 ¯ J 8 ) } , \nA ( a ) 2 = ¯ e 2 c 2 { ¯ L 4 (3 -ν ) ¯ J 4 + 1 c 2 ( 6( ν -7) ¯ J ( ¯ J + ¯ L ) -6( ν -7) ¯ J 2 + 6( ν -7) ¯ L ¯ J 3 + ( 67 ν 2 -632 ν +2120 ) ¯ L 2 32 ¯ J 4 + 3(5 ν -24) ¯ L 3 ¯ J 5 -( 105 ν 2 -532 ν +468 ) ¯ L 4 16 ¯ J 6 + ( 127 ν 2 -1536 ν +4032 ) ¯ L 6 32 ¯ J 8 ) } (B3) \nA ( a ) 3 = ¯ e c 2 { ¯ L 2 ν 4 ¯ J 2 -¯ L 4 ν 4 ¯ J 4 + 1 c 2 ( -29 ν 2 +200 ν -256 64 ¯ J 2 -3(11 ν -16) ¯ L 16 ¯ J 3 + ( 121 ν 2 -710 ν +1116 ) ¯ L 2 32 ¯ J 4 + 3(55 ν -144) ¯ L 3 16 ¯ J 5 -( 357 ν 2 -2084 ν +2744 ) ¯ L 4 64 ¯ J 6 + 3 ( 3 ν 2 -29 ν +48 ) ¯ L 6 4 ¯ J 8 ) } (B4) \nA ( a ) 4 = ¯ e 2 c 4 { (3 ν -4)(11 ν -48) ¯ L 2 32 ¯ J 4 + 3(5 ν -6) ¯ L 3 4 ¯ J 5 -( 63 ν 2 -260 ν +192 ) ¯ L 4 32 ¯ J 6 + 3 ( 5 ν 2 -32 ν +24 ) ¯ L 6 16 ¯ J 8 } (B5) \nA ( a ) 5 = ¯ e 2 c 4 { 3 ν (16 -3 ν ) 64 ¯ J 2 -9 ν ¯ L 16 ¯ J 3 + ν (17 ν -54) ¯ L 2 32 ¯ J 4 + 9 ν ¯ L 3 16 ¯ J 5 -ν (41 ν -108) ¯ L 4 64 ¯ J 6 + ( ν -3) ν ¯ L 6 4 ¯ J 8 } (B6) \nA ( a ) 6 = ¯ e 2 c 4 { ν 2 ¯ L 2 32 ¯ J 4 -ν 2 ¯ L 4 16 ¯ J 6 + ν 2 ¯ L 6 32 ¯ J 8 } , (B7)', '· Coefficients for e :': 'A ( e ) 0 = ¯ e c 2 { 3 ¯ J ¯ L -3 ¯ J ( ¯ J + ¯ L ) -ν -9 2 ¯ J 2 + ν -4 2 ¯ L 2 + 1 c 2 ( -181 ν 2 +1144 ν -496 128 ¯ J 2 ¯ L 2 + 2 ν +13 2 ¯ J 3 ( ¯ J + ¯ L ) + 94 -35 ν 8 ¯ J 3 ¯ L + 3(8 ν -29) 4 ¯ J ¯ L 3 + 2 ¯ J 3 ( ¯ L -¯ J ) -9 2 ¯ J 2 ( ¯ J + ¯ L ) 2 + 157 ν 2 -2296 ν +5080 128 ¯ J 4 + 3 ν 2 -23 ν +64 8 ¯ L 4 ) } , (B8) \nA ( e ) 1 = 1 c 2 { 7 ν -16 8 ¯ L 2 -7 ν -48 8 ¯ J 2 + 1 c 2 ( -89 ν 2 +724 ν -1174 32 ¯ J 2 ¯ L 2 -9(7 ν -24) 16 ¯ J 3 ¯ L + 9(11 ν -24) 16 ¯ J ¯ L 3 + 107 ν 2 -1596 ν +3516 64 ¯ J 4 + 71 ν 2 -404 ν +640 64 ¯ L 4 ) } , (B9) \nA ( e ) 2 = ¯ e c 2 { -( ν -3) 2 ¯ J 2 + 1 c 2 ( -113 ν 2 +352 ν +32 256 ¯ J 2 ¯ L 2 + 3 ν -29 2 ¯ J 3 ( ¯ J + ¯ L ) -3 2 ¯ J 3 ¯ L -2 ¯ J 3 ( ¯ L -¯ J ) + 49 ν 2 -1632 ν +3744 256 ¯ J 4 ) } , (B10) \nA ( e ) 3 = ¯ e 2 c 2 { ν 8 ¯ L 2 -ν 8 ¯ J 2 + 1 c 2 ( 39 ν 2 -340 ν +828 64 ¯ J 2 ¯ L 2 + 3( ν +16) 32 ¯ J ¯ L 3 + 9(13 ν -48) 32 ¯ J 3 ¯ L -73 ν 2 -224 ν +248 128 ¯ J 4 + -5 ν 2 +104 ν -256 128 ¯ L 4 ) } , (B11) \nA ( e ) 4 = ¯ e c 4 { 45 ν 2 -328 ν +384 128 ¯ J 2 ¯ L 2 + 3(5 ν -6) 8 ¯ J 3 ¯ L -53 ν 2 -168 ν +168 128 ¯ J 4 } , (B12) \nA ( e ) 5 = 1 c 4 { 9 ν 32 ¯ J 3 ¯ L + (11 ν -36) ν 64 ¯ J 2 ¯ L 2 -9 ν 32 ¯ J ¯ L 3 -(13 ν -24) ν 128 ¯ J 4 -3(3 ν -16) ν 128 ¯ L 4 } , (B13) \nA ( e ) 6 = ¯ e 3 c 4 { -ν 2 256 ¯ J 4 } , (B14)', '· Coefficients for v :': "A ( v ) 1 = 1 c 2 { -2 ¯ J ( ¯ L -¯ J ) -4 ¯ J ( ¯ J + ¯ L ) + 13 ν -32 8 ¯ J 2 + 1 c 2 ( 43 ν 2 -260 ν +1152 64 ¯ J 2 ¯ L 2 + 9(23 ν -72) 16 ¯ J 3 ¯ L + 43 ν -74 16 ¯ J 3 ( ¯ L -¯ J ) + 178 -41 ν 16 ¯ J 3 ( ¯ J + ¯ L ) -3 ¯ J 2 ( ¯ J + ¯ L ) 2 + ν 2 +288 ν -68 64 ¯ J 4 ) } , (B15) \nA ( v ) 2 = 1 c 2 { -3 2 ¯ J ¯ L + ν -2 ¯ J 2 + 4 -ν 2 ¯ L 2 + 1 c 2 ( -53 ν 2 +688 ν -1568 128 ¯ J 2 ¯ L 2 -3(14 ν -47) 8 ¯ J ¯ L 3 + 79 -8 ν 8 ¯ J 3 ¯ L + 4 ¯ J 3 ( ¯ L -¯ J ) -10 ¯ J 3 ( ¯ J + ¯ L ) + 117 ν 2 -80 ν +608 128 ¯ J 4 + -ν 2 +7 ν -32 4 ¯ L 4 ) } , (B16) \nA ( v ) 3 = ¯ e c 2 { ν 8 ¯ J 2 + 1 c 2 ( -67 ν 2 +600 ν -1280 128 ¯ J 2 ¯ L 2 + 3 -ν ¯ J 3 ( ¯ L -¯ J ) -3(41 ν -176) 32 ¯ J 3 ¯ L + -19 ν -36 8 ¯ J 3 ( ¯ J + ¯ L ) + 163 ν 2 -580 ν +792 128 ¯ J 4 ) } , (B17) \nA ( v ) 4 = 1 c 4 { -21 ν 2 +126 ν -92 32 ¯ J 2 ¯ L 2 + 3( ν -4) 4 ¯ J ¯ L 3 -3(7 ν -8) 8 ¯ J 3 ¯ L + ( ν -2)(21 ν -32) 32 ¯ J 4 + ( ν -4) 2 8 ¯ L 4 } , (B18) \nA ( v ) 5 = ¯ e c 4 { -9 ν 32 ¯ J 3 ¯ L -3(3 ν -16) ν 128 ¯ J 2 ¯ L 2 + (17 ν -36) ν 128 ¯ J 4 } , (B19) \nA ( v ) 6 = ¯ e 2 c 4 { ν 2 128 ¯ J 4 } , (B20) \nB ( v ) 0 = 1 c 4 { 9(2 ν -5) 4 ¯ J 4 -3(2 ν -5) 4 ¯ J 2 ¯ L 2 } , (B21) \nB ( v ) 1 = ¯ e c 4 { 3(2 ν -5) ¯ J 4 } , (B22) \nB ( v ) 2 = ¯ e 2 c 4 { 3(2 ν -5) 4 ¯ J 4 } . (B23) \n- · Coefficients for ϖ : \nA ( ϖ ) 1 = ¯ e c 2 { 2 ¯ J ( ¯ L -¯ J ) -2 ¯ J ( ¯ J + ¯ L ) -5 ν -48 8 ¯ J 2 + 1 c 2 ( -7 ν 2 +28 ν -128 64 ¯ J 2 ¯ L 2 -9(3 ν -8) 16 ¯ J 3 ¯ L + 74 -43 ν 16 ¯ J 3 ( ¯ L -¯ J ) + 25 ν -362 16 ¯ J 3 ( ¯ J + ¯ L ) -6 ¯ J 2 ( ¯ J + ¯ L ) 2 + 11 ν 2 -712 ν +2100 64 ¯ J 4 ) } , (B24) \nA ( ϖ ) 2 = 1 c 2 { -ν -3 2 ¯ J 2 + 1 c 2 ( 21 ν 2 -304 ν +1296 128 ¯ J 2 ¯ L 2 + 3( ν -14) 4 ¯ J 3 ¯ L -4 ¯ J 3 ( ¯ L -¯ J ) -8 ¯ J 3 ( ¯ J + ¯ L ) -53 ν 2 -176 ν +336 128 ¯ J 4 ) } , (B25) \nA ( ϖ ) 3 = ¯ e c 2 { -ν 8 ¯ J 2 + 1 c 2 ( 11 ν 2 -104 ν +256 128 ¯ J 2 ¯ L 2 + 3( ν -16) 32 ¯ J 3 ¯ L + ν -3 ¯ J 3 ( ¯ L -¯ J ) + 19 ν -72 8 ¯ J 3 ( ¯ J + ¯ L ) -107 ν 2 -564 ν +824 128 ¯ J 4 ) } , (B26) \nA ( ϖ ) 4 = 1 c 4 { 13 ν 2 -86 ν +96 32 ¯ J 2 ¯ L 2 + 3(5 ν -6) 8 ¯ J 3 ¯ L -17 ν 2 -66 ν +60 32 ¯ J 4 } , (B27) \nA ( ϖ ) 5 = ¯ e c 4 { 9 ν 32 ¯ J 3 ¯ L + 3(3 ν -16) ν 128 ¯ J 2 ¯ L 2 -(17 ν -36) ν 128 ¯ J 4 } , (B28) \nA ( ϖ ) 6 = ¯ e 2 c 4 { -ν 2 128 ¯ J 4 } , (B29) \nB ( ϖ ) = 1 c 2 { 3 ¯ J 2 + 1 c 2 ( 3(2 ν -5) 4 ¯ J 2 ¯ L 2 -15(2 ν -7) 4 ¯ J 4 ) } . (B30) \n- · The non-zero coefficients for z are: \nA ( z ) 0 = ¯ e c 2 { 3 ¯ J ¯ L -3 ¯ J ( ¯ J + ¯ L ) -ν -9 2 ¯ J 2 + ν -4 2 ¯ L 2 + 1 c 2 ( -21 ν 2 +133 ν -10 16 ¯ J 2 ¯ L 2 -29( ν -2) 8 ¯ J 3 ¯ L + 3(8 ν -29) 4 ¯ J ¯ L 3 + 2 ν +17 2 ¯ J 3 ( ¯ J + ¯ L ) -9 2 ¯ J 2 ( ¯ J + ¯ L ) 2 + 17 ν 2 -251 ν +446 16 ¯ J 4 + 3 ν 2 -23 ν +64 8 ¯ L 4 ) } (B31) \nA ( z ) 1 = 1 c 2 { -ν ¯ e 2 8 ¯ J 2 + 1 c 2 ( -13 ν 2 +63 ν -149 16 ¯ J 2 ¯ L 2 -3( ν -72) 16 ¯ J 3 ¯ L + 3(5 ν -8) 16 ¯ J ¯ L 3 + 15 ν 2 -92 ν -150 32 ¯ J 4 + 11 ν 2 -58 ν +64 32 ¯ L 4 ) } (B32) \nA ( z ) -1 = 1 c 2 { 3(8 -ν )¯ e 2 4 ¯ J 2 + 1 c 2 ( -3 ( 19 ν 2 -188 ν +276 ) 32 ¯ J 2 ¯ L 2 -9(3 ν -8) 8 ¯ J 3 ¯ L + 3(17 ν -48) 8 ¯ J ¯ L 3 + 65 ν 2 -1280 ν +3528 64 ¯ J 4 + 49 ν 2 -288 ν +512 64 ¯ L 4 ) } , (B33) \nA ( z ) 2 = ¯ e c 4 { -ν 2 +8 ν -48 16 ¯ J 2 ¯ L 2 + 3( ν +6) 8 ¯ J 3 ¯ L + ν 2 -14 ν +12 16 ¯ J 4 } (B34) \nA ( z ) -2 = ¯ e c 2 { -ν -3 2 ¯ J 2 + 1 c 2 ( -7 ν 2 +22 ν +34 16 ¯ J 2 ¯ L 2 + 3( ν -11) 2 ¯ J 3 ( ¯ J + ¯ L ) -3(3 ν -2) 8 ¯ J 3 ¯ L + 3 ν 2 -112 ν +366 16 ¯ J 4 ) } (B35) \nA ( z ) 3 = 1 c 4 { 3 ν 32 ¯ J 3 ¯ L + ( ν -10) ν 64 ¯ J 2 ¯ L 2 -3 ν 32 ¯ J ¯ L 3 -( ν -4) ν 128 ¯ J 4 -( ν -16) ν 128 ¯ L 4 } (B36) \nA ( z ) -3 = 1 c 2 { ν 8 ¯ L 2 -ν 8 ¯ J 2 + 1 c 2 ( 7 ν 2 -67 ν +183 16 ¯ J 2 ¯ L 2 + 3(3 ν +8) 16 ¯ J ¯ L 3 + 3(17 ν -72) 16 ¯ J 3 ¯ L -13 ν 2 +8 ν -82 32 ¯ J 4 + -ν 2 +22 ν -64 32 ¯ L 4 ) } (B37) \nA ( z ) -4 = ¯ e c 2 { 5 ν 2 -39 ν +48 16 ¯ J 2 ¯ L 2 + 3(5 ν -6) 8 ¯ J 3 ¯ L -5 ν 2 -9 ν +12 16 ¯ J 4 } (B38) \nA ( z ) -5 = 1 c 4 { 9 ν 32 ¯ J 3 ¯ L + 3(3 ν -10) ν 64 ¯ J 2 ¯ L 2 -9 ν 32 ¯ J ¯ L 3 -3(3 ν -4) ν 128 ¯ J 4 -3(3 ν -16) ν 128 ¯ L 4 } (B39) \nB ( z ) 0 = ¯ e c 2 { 3 ¯ J 2 + 1 c 2 ( 3(4 ν -13) 4 ¯ J 2 ¯ L 2 + 9 ¯ J 3 ¯ L -9 ¯ J 3 ( ¯ J + ¯ L ) -3(12 ν -53) 4 ¯ J 4 ) } , (B40) \nB ( z ) 1 = ¯ e 2 c 4 { -3 ν 8 ¯ J 4 } , (B41) \nB ( z ) -1 = 1 c 4 { 3(3 ν -8) 4 ¯ J 2 ¯ L 2 -9( ν -8) 4 ¯ J 4 } (B42) \nB ( z ) -2 = ¯ e c 4 { -3( ν -3) 2 ¯ J 4 } (B43) \nB ( z ) -3 = ¯ e 2 c 4 { -3 ν 8 ¯ J 4 } (B44) \nE \n( \nz \n) \n1 \n= \nC ( z ) 0 = 1 c 4 { -9 2 ¯ J 4 } (B45) \nD ( z ) 1 = 1 c 2 { -3¯ e 2 2 ¯ J 2 + 1 c 2 ( -3( ν -10) 4 ¯ J ¯ L 3 -3(3 ν -10) 2 ¯ J 2 ¯ L 2 -9 2 ¯ J 3 ¯ L + 15(2 ν -7) 8 ¯ J 4 + 3(4 ν -13) 8 ¯ L 4 ) } , (B46) \nD ( z ) -1 = 1 c 2 { 3¯ e 2 2 ¯ J 2 + 1 c 2 ( -3( ν +2) 4 ¯ J ¯ L 3 + 3(3 ν -10) 2 ¯ J 2 ¯ L 2 -9 2 ¯ J 3 ¯ L -15(2 ν -7) 8 ¯ J 4 -3(4 ν -13) 8 ¯ L 4 ) } , (B47) \nD ( z ) 2 = ¯ e 3 c 4 { 9 8 ¯ J 4 } , (B48) \nD ( z ) -2 = ¯ e 3 c 4 { 9 8 ¯ J 4 } , (B49) \n1 \n2 \nc \n- \n2 \n9 \n¯ \nJ \n4 \n} \n, \n(B50) \nE ( z ) -1 = 1 c 2 { 9 2 ¯ J 4 } (B51) \nThe remaining coefficients are identically zero. \n- · The non-zero coefficients for λ are: \nA ( λ ) 1 = 1 c 2 { ( ν +2) 2 J 2 -3 J ( J + L ) + 1 c 2 ( ( 3 ν 2 -106 ν +508 ) 32 J 4 -(2 ν +23) 4 J 3 ( J + L ) + 9(5 ν -16) 8 J 3 L + ( 9 ν 2 -58 ν +256 ) 32 J 2 L 2 -9 2 J 2 ( J + L ) 2 ) } (B52) \nA ( λ ) 2 = 1 c 2 { ( ν -1) 4 J 2 -3 4 J ( J + L ) + 1 c 2 ( ( 4 ν 2 +6 ν +17 ) 16 J 4 -(11 ν +2) 4 J 3 ( J + L ) + 3(14 ν -47) 16 J 3 L + ( ν 2 -7 ν +32 ) 8 J 2 L 2 + 9 2 J 2 ( J + L ) 2 ) } (B53) \nA ( λ ) 3 = 1 c 4 { ( 7 ν 2 -2 ν -4 ) 32 J 4 -3(5 ν -2) 8 J 3 ( J + L ) + 27 8 J 2 ( J + L ) 2 } (B54) \nA ( λ ) 4 = 1 c 4 { ( ν -1) 2 16 J 4 -3( ν -1) 8 J 3 ( J + L ) + 9 16 J 2 ( J + L ) 2 } (B55) \nB ( λ ) 1 = 1 c 2 { ν +2 2 J 2 -3 J ( J + L ) + 1 c 2 ( 3 ν 2 -106 ν +508 32 J 4 + 9(5 ν -16) 8 J 3 L + -2 ν -23 4 J 3 ( J + L ) + 9 ν 2 -58 ν +256 32 J 2 L 2 -9 2 J 2 ( J + L ) 2 ) } (B56) \n{ \nB ( λ ) 2 = 1 c 2 { ν -1 4 J 2 -3 4 J ( J + L ) + 1 c 2 ( 4 ν 2 +6 ν +17 16 J 4 + 3(14 ν -47) 16 J 3 L + -11 ν -2 4 J 3 ( J + L ) + ν 2 -7 ν +32 8 J 2 L 2 + 9 2 J 2 ( J + L ) 2 ) } (B57) \nB ( λ ) 3 = 1 c 4 { 7 ν 2 -2 ν -4 32 J 4 -3(5 ν -2) 8 J 3 ( J + L ) + 27 8 J 2 ( J + L ) 2 } (B58) \nB ( λ ) 4 = 1 c 4 { ( ν -1) 2 16 J 4 -3( ν -1) 8 J 3 ( J + L ) + 9 16 J 2 ( J + L ) 2 } (B59) \nC ( λ ) 0 = 1 c 4 { 15(2 ν -7) 4 J 4 -3( ν -5) J 4 -3(2 ν -5) 4 J 2 L 2 } (B60) \nC ( λ ) 1 = C ( λ ) 2 = 1 c 4 { 3(2 ν -5) 2 J 4 } (B61) \nD ( λ ) 1 = D ( λ ) 2 = 1 c 4 { -3(2 ν -5) 2 J 4 } (B62) \nThe remaining coefficients are identically zero. \n- [1] A. 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2024arXiv240911558B	Pulsars are rotating neutron stars that are observed to be slowing down implying a loss of their kinetic energy. There can be several different physical mechanisms involved in their spindown process. The properties of fastrotating pulsars depend on the nature of the neutron star matter which can also affect the spindown mechanisms. In this work we examine three different physical phenomena contributing to the spindown magnetic dipole radiation gravitational mass quadrupole radiation due to the mountain formation gravitational mass current quadrupole radiation or the rmodes and calculate the expressions for the braking indices due to all of them. We have also considered jointly the implications of the uncertainties of the equation of the state of neutron star matter and rapid rotation on the braking indices corresponding to the aforementioned processes and their combinations. In all cases the rapid rotation results in a departure from the standard values in the literature for the braking index when the rotational effects are ignored. If generated with a saturation amplitude within the range of 104 101 the rmode oscillations dominate the spindown of millisecond pulsars. We also explore the braking index in the context of millisecond magnetars. This study examines two braking index measurements in the context of newly born millisecond magnetars from two observed short gammaray bursts. The measured braking indices for these objects are consistent with our estimation which allows us to conclude that the spin frequency of the remnants is within the range of sim 550850 Hz.	2024-09-01T00:00:00Z	['arXiv:2409.11558', '2024arXiv240911558B', '10.48550/arXiv.2409.11558']	['Astrophysics - High Energy Astrophysical Phenomena']	Structural response of neutron stars to rapid rotation and its impact on the braking index	2,024	191	0.49	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.11558.pdf	{'Structural response of neutron stars to rapid rotation and its impact on the braking index': "Avishek Basu, 1, ∗ Prasanta Char, 2, 3, † and Rana Nandi 4, ‡ \n1 Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK 2 Departamento de F'ısica Fundamental, Universidad de Salamanca, Plaza de la Merced S/N, 37008 Salamanca, Spain 3 Space Sciences, Technologies and Astrophysics Research (STAR) Institute, Universit'e de Li'ege, Bˆat. B5a, 4000 Li'ege, Belgium 4 Department of Physics, School of Natural Sciences, Shiv Nadar \nInstitution of Eminence, Greater Noida 201314, Uttar Pradesh, India \nPulsars are rotating neutron stars that are observed to be slowing down, implying a loss of their kinetic energy. There can be several different physical mechanisms involved in their spin-down process. The properties of fast-rotating pulsars depend on the nature of the neutron star matter, which can also affect the spin-down mechanisms. In this work, we examine three different physical phenomena contributing to the spin-down: magnetic dipole radiation, gravitational mass quadrupole radiation due to the 'mountain' formation, gravitational mass current quadrupole radiation or the r-modes, and calculate the expressions for the braking indices due to all of them. We have also considered jointly the implications of the uncertainties of the equation of the state of neutron star matter and rapid rotation on the braking indices corresponding to the aforementioned processes and their combinations. In all cases, the rapid rotation results in a departure from the standard values in the literature for the braking index when the rotational effects are ignored. If generated with a saturation amplitude within the range of 10 -4 -10 -1 , the r-mode oscillations dominate the spin-down of millisecond pulsars. We also explore the braking index in the context of millisecond magnetars. This study examines two braking index measurements in the context of newly born millisecond magnetars from two observed short γ -ray bursts. The measured braking indices for these objects are consistent with our estimation, which allows us to conclude that the spin frequency of the remnants is within the range of ∼ 550-850 Hz.", 'I. INTRODUCTION': "The majority of neutron stars (NS) are observed as pulsars, which are rapidly rotating and observed to spin down over time, indicating a loss of rotational kinetic energy. The spin-down rate is given as ˙ Ω = -k Ω n [1], where n is the braking index and k is a positive constant. The braking index captures information about the mechanism of energy loss. Depending on the emission loss mechanism k can be a function of radius, moment of inertia, magnetic field strength etc (discussed later in the text). In most cases, the pulsar's rotational energy loss is attributed to the magnetic dipolar radiation (MDR) from the pulsar magnetosphere for which the braking index is 3 [2]. However, pulsars can lose energy via gravitational wave (GW) radiation if there is non-axisymmetric deformation or r-mode instability, for which the braking index is 5 and 7 respectively [3]. Therefore, it is natural to expect a rapidly rotating neutron star to lose energy via all three channels if the conditions are favourable. Apart from these three channels of energy loss, pulsars also lose energy due to the particle wind [4, 5]. \nThe spin-period of pulsars spans a large range. The slowest pulsar has a rotation period of 76 s [6] to the most rapidly rotating being 1.39 ms [7]. The total population is typically categorized as normal (or 'slow') pulsars \nand millisecond pulsars, with the demarcation around 16 ms [8]. The braking index has been measured for a few younger pulsars and found be less than 3 [9-15], which is often attributed to the evolution of the magnetic field, modified magnetosphere or even the interaction of supernova fallback disk [4, 16, 17]. A similar sub-3 braking index can be achieved in rapidly rotating pulsars, which is purely due to structural change in the star due to rapid rotation. In such a scenario, the k cannot be assumed independent of the spin frequency [18], which has been discussed in detail in Sec. IV. \nThe effect of structural evolution is prominent over a spin frequency of 200 Hz [18]. Currently, the ATNF Pulsar Catalogue 1 [19] records 392 pulsars with spinfrequency more than 200 Hz, which corresponds to ∼ 11% of the total pulsar population discovered till date. These rapidly rotating pulsars are important in probing the physics of dense nuclear matter. The physics of the nuclear interaction and the constituents of the matter determines the dense matter equation of state (EOS), which in turn governs the stellar structure and its response to rapid rotation. To study the frequency evolution of the stellar structure and the braking index, we have adopted a semi-agnostic approach to construct the dense matter equation of state following the formalism of Gandolfi et al. [20], which is summarized in Sec. II. \nThere are various types of constraints that can be applied to the EOS model. Astrophysical observations \nprovide several constraints relevant to the high density part of the EOS. In particular, the radio observations of the most massive pulsar, PSR J0740+6620 [21, 22] provide the most stringent constraint on the NS EOS. The tidal deformability measurements from the multimessenger observations of the binary neutron star merger event GW170817 put additional constraints on the EOS [23-25]. In recent years, the Neutron star Interior Composition Explorer (NICER) collaboration has reported a few simultaneous mass-radius measurements of PSR J0030+0451, PSR J0740+6620, and PSR J0437-4715, respectively, that also help us to constraint the Mass-radius plane and consequently the EOS parameter space [2632]. On the other hand, the latest constraint on the lowdensity part of the EOS comes from the advancements of the chiral effective field theory ( χ -EFT) calculations [33-37]. \nWe have employed a Bayesian analysis framework, supplemented by the measurements from the astrophysical observations to constrain the EOS model parameters and present the results of the analysis in Sec. II. In Sec. III, we used the constrained EOS to investigate their effects on stellar structure under rapid rotation. In Sec. IV, we have presented an ab initio derivation of the braking index under the assumption of energy loss through MDR and GWs due to finite deformation and r-mode oscillation. The analytical expressions have been later combined to study the frequency-dependent evolution of the braking index in context to millisecond pulsars and nascent millisecond magnetars born from NS-NS mergers or other formation scenarios. Finally, we summarise our findings of this work in Sec. V.", 'II. DESCRIPTION OF NEUTRON STAR MATTER': 'In this section, we describe the EOS of dense nuclear matter. We follow the work by Gandolfi et al. [20] to construct a double polytrope EOS for our purpose. With this particular form of the function, Gandolfi et al. [20] could parametrize the results of the ab-initio microscopic calculations using Quantum Monte Carlo methods. We use this parametrization to create the EOS directly from nuclear empirical parameters (NMPs). We summarize the necessary equations in the following. The energy per particle for the pure neutron matter (PNM) is expressed as a function of density, \nE PNM ( ρ ) = a ( ρ ρ 0 ) α + b ( ρ ρ 0 ) β , (1) \nwhere, ρ 0 is the nuclear saturation density. \nThe symmetry energy is defined as, E sym ( ρ ) = E PNM ( ρ ) -E SNM ( ρ ), where E SNM ( ρ ) is the energy per particle of the symmetric nuclear matter (SNM). Then, the slope of symmetry energy becomes, L ( ρ ) = \n3 ρ 0 ∂E sym ( ρ ) /∂ρ . At saturation, they can be expressed as, \nJ = E sym ( ρ 0 ) = a + b -E SNM , L ( ρ 0 ) = 3( aα + bβ ) . (2) \nThe density dependence of the symmetry energy can be parametrized by, \nE sym ( ρ ) = C ( ρ ρ 0 ) γ . (3) \nThus, we get, C = a + b -E SNM from equation 2 at ρ = ρ 0 . Finally, from the condition, P = ρ 2 ∂E SNM ∂ρ \| ρ = ρ 0 , we get, \nγ = aα + bβ a + b -E SNM . (4) \nUsing the relations above, the general expression of the EOS as a function of baryon number density ( ρ ) and proton fraction ( x = ρ p /ρ , where ρ p is the proton density) becomes, \nE ( ρ, x ) = E PNM ( ρ ) + C ( ρ ρ 0 ) γ [ (1 -2 x ) 2 -1 ] . (5) \nTo describe the NS matter, we also need to include the contribution from the lepton. In a cold NS, the lepton density is regulated by the β -equilibrium condition, µ n = µ p + µ e . Beyond 2 ρ 0 , we construct the EOS using the piecewise constant speed of sound EOS model [38] up to the density of 12 ρ 0 . We randomly sample the density points within this interval of the c s -ρ plane and the corresponding speed of sound points complying with the constraints 0 ≤ c s < 1. For the low-density part of the EOS, we have used the standard Baym-PethickSutherland (BPS) EOS ( < ∼ 0 . 001 fm -3 ) for the outer crust and Negele-Vautherin (NV) EOS (up to 0 . 08 fm -3 ) for the inner crust[39, 40]. The crust EOS is smoothly joined with the core EOS given by equation 5. \nAfter setting up our EOS model, we continue to explore the parameter space of our model using various theoretical, experimental, and observational constraints within a Bayesian analysis. We describe briefly the constraints and their implementation in the following. \nFirst, we consider constraints around saturation density. The EOS generated within our model must satisfy the constraints coming from the chiral effective field theory ( χ -EFT) calculations. We used the results from Drischler et al. [36] to implement the ( χ -EFT) constraints for the EOSs in β -equilibrium in the range (0 . 5 , 1 . 1) ρ 0 . The priors of our model parameters are given in table I. The ranges of ρ 0 and E SNM are chosen according to the experimental knowledge. Another condition is on the calculated values of E sym and L at saturation. We have imposed the E sym to be in the range of 25 to 40 MeV and L to be within 30 and 80 MeV. \nFIG. 1. Shows the posterior distribution of the model parameters along with the symmetry energy at the saturation density ( J ) and the slope of symmetry energy at the saturation density ( L ). \n<!-- image --> \nWe have used the mass-measurements of massive pulsars, and combined tidal deformability ( ˜ Λ) from GW170817 as our observational constraints on the EOS. In particular, we have used the mass-measurement of J0740+6620 as 2 . 08 ± 0 . 07 M ⊙ reported by [22] as a Gaus- \nlikelihood, \nP (data M max \| X ) = 1 2 [ 1 + erf ( M max ( X ) /M ⊙ -2 . 08 0 . 07 √ 2 )] , (6) \nwhere, ( X = a, α, b, β, E SNM , ρ 0 ) represents our model parameters. Then, we applied the improved constraints on ˜ Λ from Abbott et al. [25]. The value of ˜ Λ de- \nFIG. 2. The brown lines indicate the constrained EOSs ( ∼ 6K) corresponding to the posterior distributions of figure 1 and the blue lines indicate the EOS which falls within the 68% confidence interval of the constrained posterior distribution of the EOS parameters. \n<!-- image --> \nn the mass ratio q = m 1 /m 2 and the chirp mass ( M chirp = ( m 1 m 2 ) 3 / 5 ( m 1 + m 2 ) 1 / 5 ), where m 1 , m 2 are the masses of primary and secondary objects of the binary system, respectively. The chirp mass has been determined rather precisely to be M chirp = 1 . 186 ± 0 . 001 M ⊙ for GW170817. We use the publicly available data from LIGO-VirgoKagra (LVK) collaboration 2 assuming the low-spin prior and construct the likelihood as, \nP (data LVK \| X ) = ∫ dm 1 ∫ dm 2 P ( m 1 , m 2 \| X ) × P (data LVK \| m 1 , m 2 , Λ 1 ( m 1 , X ) , Λ 2 ( m 2 , X )) , (7) \nwhere, P ( m 1 , m 2 \| X ) is the prior distribution for the component masses of the binary. For simplicity, we have chosen a uniform prior for m 1 and m 2 . We use a Gaussian kernel density estimator to construct the GW likelihood from the discreet data. We have fixed the chirp mass to its median value because of the high precision of the measurement. Then, we construct binaries corresponding to that chirp mass by varying m 1 and determine the corresponding m 2 . For each EOS, we compute the tidal deformabilities (Λ 1 , Λ 2 ) for the pairs of ( m 1 , m 2 ), and find the probability using equation 7. \nThe final likelihood function with all the astrophysical \nFIG. 3. The M-R diagrams for all ∼ 6K EOSs from figure2 have been shown in the brown coloured lines and the blue lines indicate the M-R relations corresponding to the 68% confidence interval of the posterior distribution of the constrained EOS parameters. \n<!-- image --> \nTABLE I. Summary of the prior range used for our EOS model parameter. \nconstraints is simply the product, \nP (data astro \| X ) = P (data M max \| X ) × P (data LVK \| X ) . (8) \nFinally, we sample the posterior using a nested sampling algorithm [41], in the dynesty software package [42]. The posterior distributions of the EOS parameters for nuclear matter are shown in Fig. 1. We have also included the derived values for E sym and L at ρ 0 . We have found E sym = 31 . 49 +3 . 43 -3 . 11 MeV and L = 58 . 82 +10 . 24 -8 . 44 MeV at 68% CI. These values are well within the agreed ranges found in literature [43, 44]. This affirms the robustness of our EOS samples used to study the rotational properties of pulsars in this work. We have shown the corresponding EOSs and their mass-radius curves in Fig. 2 and 3, respectively. We can see from Fig. 3 that the EOSs generated in our exercise span a large range of radii for a 1 . 4 M ⊙ star. Therefore, these curves are able to represent the parameter space of neutron star matter. Note \nthat, we have not used the data from NICER measurements in our analysis as we see that our 68% mass-radius curves in Fig. 3 are consistent with 68% of those sources.', 'III. ROTATIONAL EFFECTS ON THE STRUCTURE OF NEUTRON STARS': 'To study the effect of rotation on the macroscopic quantities of the star we have used the RNS code 3 [45, 46]. For every EOS computed in the Sec II, we calculate the gravitational mass ( M ), the equatorial radius ( R e ), the ratio of the polar to the equatorial radius ( R p /R e ) and the moment of inertia ( I ) as a function of rotation frequency ( f = Ω / 2 π , where Ω is the angular velocity of the star), keeping the baryon mass ( M b ) fixed at 2 M ⊙ . \nA rapidly rotating star can sustain more mass than a non-rotating star with the same central density. To obtain the sequence of the above-mentioned macroscopic quantities with rotation frequency, we treat central density as the parameter and search for the frequency which a star of baryon mass of 2 M ⊙ can sustain. However, since a low central density leads to a low-mass star, the frequency needed to meet the M b = 2 M ⊙ condition may be more than the Kepler frequency (defined as the maximum frequency of rotation that a star can support before mass-sheading) if the central density is too low. Hence, the RNS code does not converge. As we increase the central density, the M b = 2 M ⊙ criteria is fulfilled at increasingly lower frequencies. When the central density is very high, the condition of M b = 2 M ⊙ is achieved at a frequency ( < ∼ 200 Hz) that is too low for the RNS code to handle. Therefore, we identify the densities corresponding to the highest and the sequence between those two densities for all ∼ 6000 EOSs in an automated fashion in our pipeline built around the RNS code. \nThe rapid rotation generates a large amount of centrifugal force, which leads to the departure from the spherical shape of the star that is characterised by the ratio of the polar to the equatorial radius R p /R e . The dependence of this ratio on frequency is shown in the bottom left panel of Fig. 4 for all EOS.lowest frequencies for which the RNS code converges. Subsequently, we generate the Due to the deformation in the shape of the star, the definition of the radius becomes ambiguous in such a case, yet, in many scenarios, for estimating various physical quantities, the radius of the star is required as discussed in Sec. IV. Hence, in this work, we define a frequency-dependent radius by averaging over the circumferential radius of a rotating star given as \nR ≡ ⟨ R (Ω) ⟩ = ∫ 2 π 0 R ( θ, Ω) dθ ∫ 2 π 0 dθ . (9) \nThe expression of circumferential radius R ( θ, Ω) = R e [1 -¯ Ω(0 . 788 -1 . 03 x ) cos 2 θ ] is taken from the previous work by AlGendy and Morsink [47] and Suleimanov et al. [48], where θ is the polar angle, ¯ Ω = Ω( R 3 e /GM ) 1 / 2 and x = GM/c 2 R e . The EOS and rotation frequency-dependent average radius of the star is shown in the lower right panel of Fig. 4, which can be seen to decrease strongly at higher frequencies ( > ∼ 850Hz). Also at the higher frequency, the R p can be ∼ 25% smaller than the R e , therefore our method of computing a unique value of a radius and attributing a spherical shape to the star may have some shortcomings even if it mathematically renders a valid solution. In the upper left and right panels of Fig. 4 we show the variation of the gravitational mass and moment of inertia as a function of the rotation frequency for all EOS. It is evident from Fig. 4 that all these physical quantities have significant frequency dependence and will contribute to the braking index of the NS which we will discuss in the next section. We have confined our studies to the rotational frequency above > ∼ 200 Hz following Hamil et al. [18], where the frequency dependence of the macroscopic quantities has been shown to be prominent above > ∼ 200 Hz only. \nThe high-frequency behaviour of the macroscopic quantities observed in Fig. 4 is controlled by the stiffness of the EOSs. At a given density, a stiffer EOS produces more pressure than a softer one and, therefore, can defy gravity more easily. Hence, if there is no rotation, the stiffer EOS will make a star of a larger radius and a lower central density if a certain number of baryons (equivalent to a fixed baryon mass) are put together. Consequently, the star with the stiffer EOS will be less bound, resulting in a higher gravitational mass. Note the gravitational binding energy is defined as M b -M . This behaviour remains even if the stars rotate. Therefore, the upper curves in M -vsf , R -vsf , and I -vsf plots in Fig. 4 correspond to stiffer EOSs. With increasing frequency, the equatorial radii of all the stars (irrespective of the underlying EOS) grow, making them less bound and having higher gravitational mass. Furthermore, since at a given rotational frequency, the star with the stiffer EOS is less bound than the softer EOS, it becomes unstable at a lower frequency. In other words, the Kepler frequency is smaller in the case of stiffer EOSs. It explains the behaviour seen in the M -f plot. The R p /R e -vsf curve captures the evolution of the stellar deformation as the function of rotation frequency. It is clear from the above discussion that for the stiffer EOSs, the equatorial radius increases much faster with the small change to the rotation frequency compared to the softer EOSs. Consequently, the ratio R p /R e falls faster in the case of stiffer EOSs. Therefore, the left curves in the R p /R e -vsf plot belong to stiffer EOSs. \n45 \nFIG. 4. This figure shows the variation of four structural quantities of neutron stars as a function of the rotation frequency for all EOS obtained in Sec II. The blue lines indicate the relation for all ∼ 6K EOS and the purple lines corresponds to the solutions of EOS constructed from the 68% C.I. of the posterior distribution of parameters shown in Fig. 1. The upper left panel shows the variation of the gravitational mass as the function of the rotation frequncy, the upper right panel shows the variation of the moment of inertia, the lower left panel shows the variation of the ratio of the polar to equitorial radius and the lower right panel shows the variation of average radius of the star as the function of rotation frequency. For every EOS the baryon mass of the star is kept fixed at 2 M ⊙ , further details can be found in the Sec. III. \n<!-- image -->', 'IV. BRAKING INDEX': 'In this section, we present the derivation of the braking index and its dependence on the spin frequency. Later in this section, we discuss the results in the context of millisecond pulsars and newly born millisecond magnetars in detail.', 'A. Calculation of braking index': "Pulsars spin down at the cost of their rotational kinetic energy. If the spin-down energy ˙ E through a mechanism is proportional to -Ω n +1 , the spin-down relation is obtained as \nd dt ( 1 2 I Ω 2 ) = I Ω ˙ Ω ∝ -Ω n +1 ⇒ ˙ Ω = -k Ω n , (10) \nwhere k is a constant which depends on various physical parameters of the neutron stars depending on the mechanism of the emission loss discussed later in this section. The braking index n depends both on the first and the second derivative of the spin frequencies as \nn = Ω ¨ Ω ˙ Ω 2 . (11) \nThe numeric value of the braking index is the signature of the mechanism which dominates the spin-down of a pulsar. \nThe energy loss by the MDR is given as \n\| ˙ E EM \| = 2 3 µ 2 Ω 4 sin 2 α ≡ E EM R 6 Ω 4 , (12) \nwhere the magnetic moment \| µ \| 2 = B 2 p R 6 / 4, with the surface magnetic field strength B p , stellar radius R , α is the angle of inclination between the rotation and the magnetic axis and E EM = B 2 p sin 2 α/ 6. The corresponding spin-down rate using Eq. 10 leads to \n˙ Ω MDR = -2 B 2 p R 6 sin 2 α 3 I Ω 3 ⇒-k MDR Ω 3 , (13) \n̸ \nindicating the MDR leads to braking index 3. However, in Sec. III we have noticed at higher spin-frequency the R and depends on f making k MDR dependent on f and hence the braking index. On the other hand, the f dependence of I , will not allow us to arrive at the expression 13, as dI/d Ω = 0. Hence, for a rapidly rotating star, the modification to the spin-down expression is essential. \nThe expression given in Eq. 12 is valid if the external magnetic field is purely dipole. However, studies show the presence of complex structures of higher order magnetic multipole near the stellar surface [49]. But, at a larger distance from the surface, the approximation of the dipolar field line is still valid. Therefore in this work, we have assumed that the pulsar magnetosphere is purely dipolar. Apart, from the geometrical consideration of the magnetic field, we also ignore any possible evolution of the dipolar field with time. Otherwise, it would contribute by an amount 2Ω ˙ B/ ˙ Ω B to the braking index. Similarly, the evolution of α would allow the departure of the braking index by an amount 4Ω ˙ α/ ˙ Ωtan α , which has been also ignored for the current study. As we do not consider any evolution in B and α we use them as a parameter in our model. \nA long-lived, non-axisymmetric deformation of neutron stars would result in energy loss via the continuous GW (details can be found in a recent review by Gittins [50]). These deformations are referred to as mountains and generate a certain amount of ellipticity ( ϵ ), whose magnitude is still uncertain, depending on the process leading to the formation of the mountains and the crustal EOS. However, a typical value of maximum ellipticity that a neutron star crust can sustain is ∼ 10 -6 [51-53]. The strong magnetic field can lead to the deformation of a star generating ϵ ∼ 10 -12 and the misalignment between the magnetic axis and the rotational \naxis leads to the GW emission [54-56]. Similarly, mountains are generated from the accretion process, they can be either thermal mountains [57] or magnetic mountains. The thermal mountains produced from the temperaturesensitive nuclear reactions on the surface of the neutron stars survive for a much shorter duration ∼ 0 . 2 yrs (this a typical value for more details refer to Gittins [50] and the references therein) between the accretion-mediated outburst phase of neutron stars. However, the magnetic field-supported mountains of the accreted material may survive over much longer time scales > ∼ 10 8 yr [58] giving rise to ϵ ∼ 10 -7 to 10 -8 . Implies, that the mountain formed at the end of the accretion phase (even after the companion's disappearance) can survive for a decent fraction of a neutron star's lifetime. The amount of energy loss via the GW emission is proportional to the sixth power of the spin frequency and the ellipticity of the star, and is given as \n\| ˙ E Q GW \| = 32 5 G c 5 I 2 ϵ 2 Ω 6 ≡ E Q I 2 ϵ 2 Ω 6 , (14) \nwhere E Q = 32 G/ 5 c 5 . The corresponding spin-down rate, following the Eq. 10 is \n˙ Ω Q GW = -E Q Iϵ 2 Ω 5 ⇒ = -k Q GW Ω 5 , (15) \nindicating the braking index in this case is 5. However, a similar argument like in the case of MDR spin-down rate given in Eq.13, is valid here, where rotational effects on I would not lead to the expression we have arrived at Eq. 15. \nThe all-sky search for the continuous gravitational waves did not lead to the discovery of a signal but has enabled to establish an upper limit of ϵ < ∼ 10 -6 (depending on the distance and the frequency) [59]. Hence, in our case, we use ϵ = 10 -7 which is an order of magnitude lower than the established upper limit. \nThe r-modes are quasi-toroidal oscillations in neutron stars where the Coriolis force acts as the restoring force [60]. Counterrotating r-modes can become unstable to gravitational radiation reaction via the ChandrasekharFriedman-Schutz (CFS) mechanism [61, 62]. It has been shown that in the absence of any fluid dissipation, CFS instability can arise at any rotational frequency of the star [63-65]. In realistic neutron stars, however, several damping mechanisms are present, such as due to the bulk and shear viscosities, which lead to the r-modes amplitude achieving a saturation value ( α S ), as shown in Owen et al. [60]. The microscopic origin of the damping mechanisms relates to the nature and the temperature of the neutron star matter. In the angular velocitytemperature (Ω -T ) plane, the interplay between the gravitational wave emission timescale and viscous dissipation timescales curve out an instability region. When the star resides within the unstable region, the gravitational wave emission contributes significantly to the rotational energy loss becoming another mechanism for the spindown of the star. The frequency of the emitted GW \nis 4/3 the rotation frequency of the star [3]. Considering previous studies, the r-mode saturation amplitude can attain a value between 10 -4 to 10 -1 . For these values of α S , the energy loss due to r-mode is given by [60, 66], \n\| ˙ E R GW \| = ( 4 3 ) 8 4 πG 25 c 7 α 2 S M 2 R 6 Ω 8 J 2 ≡ E R α 2 S M 2 R 6 Ω 8 J 2 \n(16) \nwhere, E R = ( 4 3 ) 8 4 πG 25 c 7 , and J is given by, \nJ = 1 MR 4 ∫ R 0 ρ ( r ) r 6 dr. (17) \nThe expression 10 gives the spin-down rate as \n˙ Ω R GW = -E R α 2 S M 2 R 6 J 2 I Ω 7 , (18) \nindicating the spin-down due to r-mode leads to a braking index of 7. The above expression 18 is under the approximation of the no rotational effect on the stellar structure. However, we have shown that for rapid rotation there exists significant structural evolution, hence a revision to the above expressions is important. \nWhen all the above three mechanisms of energy loss are active, the rotational energy loss is given by \nd dt ( 1 2 I Ω 2 ) = -˙ E EM -˙ E Q GW -˙ E R GW , (19) \nwhere ˙ E EM , ˙ E Q GW and ˙ E R GW are the rate of loss of energy via the electromagnetic radiation, gravitational quadrupolar radiation and the r-mode. The relation between the spin-down rate and the other macroscopic properties of the neutron star's interior and magnetosphere can be obtained by expanding the left-hand side derivative of the Equation 19, where we consider a nonzero time evolution of the moment of inertia. \n˙ I +2 ˙ Ω I = -2 E EM R 6 Ω 3 -2 E Q I 2 ϵ 2 Ω 5 -2 E R α 2 S M 2 R 6 Ω 7 J 2 (20) \nIn Equation 20, we use the chain rule to write ˙ I = dI d Ω d Ω dt = I ' ˙ Ω and further simply the equation to obtain ˙ Ω as \n˙ Ω = -2 E EM R 6 Ω 3 +2 E Q I 2 ϵ 2 Ω 5 +2 E R α 2 S M 2 R 6 Ω 7 J 2 I ' Ω+2 I (21) \nSimilarly by taking the derivative of Equation 20 and applying the chain rule to write I = I '' ˙ Ω 2 + I ' ¨ Ω, we obtain \n¨ Ω = [ -6 E EM Ω 2 ˙ Ω R 5 (2 R ' Ω+ R ) -2 E Q Iϵ ˙ ΩΩ 4 (2 I ' ϵ Ω+5 Iϵ ) -2 E R α 2 S MR 5 Ω 6 ˙ Ω J × (2 M ' R Ω+6 MR ' Ω+2 MR Ω J ' +7 MR ) -3 I ' ˙ Ω 2 -I '' ˙ Ω 2 Ω] / [2 I + I ' Ω] . (22) \nAll the quantities with ( ' ) denote the derivative with respect to the angular velocity Ω. Combining the ˙ Ω and ¨ Ω from the expressions in the given in the Equation 21 and 22, we obtain the braking index using the relation 11 as a function of the spin angular velocity Ω. \nn (Ω) = n EM + n Q GW + n R GW + n I , where n EM = 6 E EM R 5 Ω 3 (2 R ' Ω+ R ) 2 E EM R 6 Ω 3 +2 E Q I 2 ϵ 2 Ω 5 +2 E R α 2 S M 2 R 6 Ω 7 J 2 , n Q GW = 2 E Q I Ω 5 ϵ 2 (2 I ' Ω+5 I ) 2 E EM R 6 Ω 3 +2 E Q I 2 ϵ 2 Ω 5 +2 E R α 2 S M 2 R 6 Ω 7 J 2 , n R GW = 2 E R α 2 S R 5 M Ω 7 J × (2 M ' R Ω J +6 MR ' Ω J +2 MR J ' Ω+7 MR J ) 2 E EM R 6 Ω 3 +2 E Q I 2 ϵ 2 Ω 5 +2 E R α 2 S M 2 R 6 Ω 7 J 2 , n I = 3 I ' Ω+ I '' Ω 2 . (23) \n-2 I + I ' Ω \nThe expression of n (Ω) above shows the evolution of the braking index with the spin-frequency, which is dependent on the EOS through the frequency derivative of the macroscopic quantities like the M , R and I . The n EM term is the contribution of MDR loss, n Q GW is the contribution from the GW radiation due to long-lived deformations, n R GW is the contribution from the emission loss via the r-mode oscillations. The term n I depends \npurely on the moment of the inertia and its higher order derivatives, which has been previously derived in Glendenning et al. [67]. The quantities E EM , ϵ and α S can be considered as the switches which by setting to zero help in analysing the braking index in the absence of a particular channel of energy loss. For example, setting α S = 0 will help in understanding the evolution of the braking index purely due to the electric dipolar emission \nloss and the GW emission loss due to finite ϵ . In the limit of slow rotation where all the derivatives w.r.t Ω tend to zero, by setting ϵ and α S to zero we obtain n = 3, by setting E EM and α S = 0, we obtain braking index 5 and finally by setting E EM = ϵ = 0, we obtain n = 7, which are in agreement with the limiting cases known in the literature.", 'B. Effect of rapid rotation on n EM , n Q GW and n R GW': "In Sec IV A, we have shown how the expressions in Eq. 23 reduce to the known values of the braking index in the literature when the rotational effects are ignored by setting all derivatives w.r.t Ω to zero. However, when the effect of rotation is taken into account the departure of n EM , n Q GW and n R GW from 3, 5 and 7 occurs, which depends on the EOS through the variation of the macroscopic quantities and their derivatives as the function of the Ω. \nIf the energy loss is considered through the magnetic dipolar radiation only, then \nn EM = 3 ( 1 + 2 R ' Ω R ) + n I , (24) \nwhich can be obtained by setting ϵ and α S = 0. Similarly by assuming the loss of energy is via the GW emission due to the non-axisymmetric long-lived deformation (that is by setting B p = α S = 0) the braking index is given as \nn Q GW = 5 ( 1 + 2 I ' Ω 5 I ) + n I . (25) \nLikewise, for the emission to happen purely from the rmode oscillation in the absence of the electromagnetic dipolar radiation and the GW emission from deformations the braking index takes the form \nn R GW = 7 ( 1 + 2 M ' Ω 7 M + 6 R ' Ω 7 R + 2 J ' Ω 7 J ) + n I , (26) \nwhich is obtained by setting B p = ϵ = 0. In Fig. 5 we have shown the dependence of these three braking indices n EM , n Q GW and n R GW separately as a function of f , for three different EOS. The solid yellow line corresponds to one of the EOS described in Sec. II with the highest weight (the parameters of the EOS are: α = 0.699, β = 2.151, a = 9 . 423 MeV, b = 5 . 662 MeV, ρ 0 = 0 . 152 fm -3 and E SNM = -17 . 934 MeV), the blue dashed line is for the density-dependent relativistic mean field (RMF) EOS, DDME2 [68] and another RMF EOS with nonlinear meson field couplings, S271v6 [69] is shown by the black solid line. The departure of n EM , n Q GW and n R GW from the known constant value of 3, 5 and 7, respectively is due to the rotational effect. We have used B p = 10 8 G and α = 45 · to obtain the dependence of n EM as a function of Ω, for n Q GW and n R GW we use ϵ = 10 -7 and α S = 10 -4 respectively. \nThe response to the rapid rotation depends on the stiffness of the EOS, which is different for different EOSs (refer to Sec. III for detailed discussion), giving rise to different braking index curves. However, the difference in response to the n EM and n R GW at frequencies < ∼ 550 Hz is negligible for at least these three representative EOS shown in the Fig. 5, unlike in the case of n Q GW . Both n EM and n R GW systematically decrease with the increase in the spin frequency, which indicates at frequencies > ∼ 200 Hz the contribution of n I is significant. The importance of n I can be argued from Fig. 4, where the R is almost constant at f ∼ 200 -750 Hz, indicating R ' ∼ 0 which would lead to n EM ≃ 3 of Eq. 24 at this frequency range, hence, the deviation seen in n EM from 3 in Fig. 5 can be attributed to n I . A similar trend in the variation of n R GW can be seen in Fig. 5, which is also attributed to the n I . The finite (but small positive) value of M ' /M and finite (but small negative) value of J ' / J leads to a different slope than the n EM in the initial fall of n R GW from the value 7. \nThe spin frequency evolution of n Q GW depends only on I and its derivative which changes appreciably even at f > ∼ 200 Hz, making n Q GW always > 5 (the first term of Eq. 25) and an increasing function of f in the absence of n I . Whereas n I is always < 0, and the nature of its fall depends on the EOS. Therefore, there may exist a range of frequencies for which the n Q GW > 5 in the presence of the n I term as shown in Fig. 5. However, unlike n EM and n R GW (at least with the example for these EOS) we find the EOS-dependent variation in n Q GW is distinguishable even at the frequencies ∼ 250 -300 Hz. But as the spin frequency increases the variation of all three braking indices for different EOS can be distinguished at the f ∼ 716 Hz, the spin frequency of PSR J1748 -2446ad (maximum spin frequency of a pulsar known till date), an eclipsing binary millisecond pulsar in the globular cluster Terzan 5 [7]. However, at much larger frequencies ( f > ∼ 850 Hz), we find the large change in radius (see Fig. 5) indicating the size decreases with frequency, which could be an artefact because of forceful fitting of a sphere to a largely deformed star. Hence, the results beyond f > ∼ 850 Hz must be interpreted with care.", 'C. Effect of B p , ϵ and α S on the braking index.': "In the previous Sec. IV B, we have discussed the rotational effect on all three separate components of the braking index individually in the absence of the other two channels of energy loss. Here we present the result of Ω dependent braking index taking all three energy loss channels into account for the same set of three representative EOSs used in Sec. IV B. We further investigate the dependence of the braking index for various physically motivated combinations of parameters of B p , ϵ and α S . The results are shown in Fig. 6, where the green colour indicates the results for the EOS obtained from our agnostic EOS model (see Sec. II for details). The same \nFIG. 5. This figure shows the effect of rotation on the braking index of pulsars (see Sec. III for detailed discussions and references). The left panel indicates the braking index purely due to magnetic dipolar radiation of magnetic field strength B p = 10 8 G and α = 45 · . The middle panel shows the variation of the braking index if the energy loss happens only through the GW emission in the presence of long-lived deformations generating ϵ = 10 -7 (r-mode and magnetic dipole radiation ignored). The right panel shows the variation of the braking index as the function of rotation frequency when the energy loss happens through the r-mode only with the amplitude α S = 10 -4 . The solid yellow line represents one of the EOS from our sample (see Sec. IV B for the details), the blue dashed line indicates the DDME2 EOS and the black solid line indicates S271v6 EOS. The vertical magenta line indicates f = 716 Hz line which is the maximum spin-frequency of a pulsar discovered to date. \n<!-- image --> \nsample EOS has been used in Sec. IV B and has been shown with the solid yellow line in Fig. 5. The blue and red colour corresponds to S271v6 and DDME2 EOS, and different line style indicates the different combinations of B p , ϵ and α S . \nThe upper left panel of the Fig. 6 shows the variation of n (Ω) for neutron stars with a typical value B p = 10 8 G, ϵ = 10 -7 and for two values of α S = 10 -1 and 10 -4 shown using the solid and dashed line respectively. The justification for using such value of ϵ and α S has been presented in Sec. IV A. To arrive at the typical value of B p , we find the median value of B p of pulsars with f > 200 Hz from the ATNF pulsar catalogue [19]. The overlap of the dashed and solid lines for all the EOSs indicates that the effect of two different parameter sets can not be distinguished. The braking index of ∼ 7 around f ∼ 200 -400 Hz indicates that the evolution is mostly dominated by the GW emission due to r-mode instability. The MDR loss and GW radiation from the mass quadropolar moment do not have any significant contribution towards the braking index. However, the departure from the value of 7 has a contribution from n I and other frequency-dependent structural quantities (as discussed in Sec. IV B) which makes n < 7. Therefore, this \nfigure indicates if the r-mode instability is generated in a typical millisecond pulsar (which is dependent on the spin frequency and the redshifted stellar temperature, for details see Kraav et al. [70]), with α S ranging between 10 -4 -10 -1 , then the spin-down would be dominated by the GW radiation through the r-mode instability. \nAt higher frequency f ∼ 716 Hz the contribution of n I and rotational effects contributing to n R GW leads to 6 < n < 7. At a much higher frequency of f > ∼ 850 Hz, the rapid decline in braking index is due to a sharp fall in the radius as discussed previously in Sec. IV B and III. \nIn the upper right-hand side panel of Fig. 6, we explore the evolution of the braking index in millisecond pulsars in the absence of radiation in GW through r-mode instability. We use B p = 10 8 G, and two separate values of ϵ = 10 -7 and 10 -10 shown in the solid and dashed line respectively. Both lines overlap with braking index ∼ 3, the deviation from n = 3 is due to the rotational effect. This also indicates that the emission loss through GWis insignificant even with a deformation of ϵ = 10 -7 . Our findings are consistent with the current observation limit [59]. We find that a large deformation which can generate ϵ > ∼ 10 -3 can result in the emission of GW large enough to impact the rotational evolution of the neutron \nFIG. 6. This figure captures the dependence of braking index as the function of the rotation frequency on the magnetic field strength B p , the ellipticity due to long-lived deformation ϵ and the r-mode saturation amplitude α S . However, we use a constant inclination angle α = 45 · . The variation has been studied using three representative EOSs, the red colour lines indicate DDME2 EOS, the blue colour line indicates S271v6 EOS and the green line indicates an EOS from our sample (see Sec. IV C) for details. The results for different combinations of B p , ϵ and α S have been shown with different line styles. Refer to Sec. IV C for detailed discussion and implications. \n<!-- image --> \nstars. However, such a high magnitude of ellipticity is unlikely for normal millisecond pulsars [71]. \nThe lower left panel of Fig. 6 shows the evolution of the braking index if a neutron star has an ultra-strong magnetic field B p = 10 15 G, making them fall in the class of magnetars. The main motivation for using such a strong magnetic field is to study the frequency-dependence on the braking index of a newly born millisecond magnetar [72], which may form due to the accretion-induced collapse of white dwarfs [73] or a merger of two neutron stars [74, 75]. The formation of these objects is believed to manifest as the Gamma Ray bursts (GRBs) [76]. In the lower-left panel of Fig. 6, we find if α S = 10 -4 and ϵ = 10 -7 at lower frequency f ∼ 200 -600 Hz the braking index is ∼ 3, indicating the magnetic dipolar radiation dominates the spin-down rate of the star. At higher frequencies, f > ∼ 600 Hz the departure from n ∼ 3 is due to \nthe rotational effects and is sensitive to the EOS. \nWhen the saturation amplitude of the r-mode is large, that is, α S = 10 -1 (shown in the lower left panel of Fig. 6), we find at low rotation frequency ( f ∼ 200Hz) the braking index is ∼ 3, but finally rises to ∼ 6 close to f = 716 Hz. The increase in the braking index from ∼ 3.5 to ∼ 6 is the manifestation of Ω 8 dependence on the energy loss via the r-mode. The braking index cannot reach the value of 7 because of the rotational effects where n I is also dominant along with the contribution from R ' /R and J ' / J , where all of them are negative quantities. The behavior of the braking index found here illustrates the importance of incorporating the rotational effects to interpret the spin-down evolution of the remnants produced from the astrophysical transient events discussed above. For example, if the rotational effect is not considered, one can misinterpret a measurement of \nFIG. 7. Evolution of the braking index as the function of rotation frequency. The blue band and the red band show the 90 percentile and 68 percentile of the braking index respectively, which originates from the uncertainties in the EOS computed in Sec. II. The over plot of the yellow and grey band is the braking index of GRB 140903A and GRB 130603B with their 1 σ uncertainty measured from their light curves. \n<!-- image --> \nn = 5 for a magnetar as the spin-down dominated by the GW emission due to a finite mass quadrupolar moment. \nWe have further investigated the evolution of the braking index in the parlance of a newborn millisecond magnetar with a much higher value of ellipticity ( ϵ = 10 -3 ), based on the measurement of Xie et al. [77]. The results are shown in the lower right panel of Fig. 6, we use the same colour convention of EOSs mentioned earlier in the section. The dashed line style indicates the braking index evolution with B p = 10 15 G, ϵ = 10 -3 and α S = 10 -1 , this is the combination with the highest magnetic field, ellipticity and r-mode saturation amplitude. We find the evolution of the braking index is exactly identical to the parameter set B p = 10 15 G, ϵ = 10 -7 and α S = 10 -1 (shown by the solid lines in lower left panel of Fig. 6), which implies that even with 4 orders of magnitude larger ellipticity the evolution is still dominated by the r-mode instability if the saturation amplitude is 10 -1 . Whereas, if the α S ≤ 10 -4 , we find the evolution of the braking index is dominated by the magnetic dipolar radiation, which is shown by the solid line with the B p = 10 15 G, ϵ = 10 -3 and α S = 10 -4 . The dashed-dotted line indicates the evolution of braking index in absence of r-mode but finite deformation of ϵ = 10 -3 and B p = 10 15 G, which indicates again the magnetic dipole radiation is the dominant mode of the energy loss from the neutron star.", 'D. EOS uncertainties and the braking index of millisecond magnetar': 'The long and short GRBs are observed to show an X-ray plateau phase after the prompt emission, which lasts for ∼ 10 -1000 s indicating an ongoing energy injection from the central engine [74, 75, 77-80]. There exists debate on the nature of the central engine but millisecond magnetar is one of the plausible models often used in literature. The spin-down of the magnetar is believed to power the X-ray plateau region of the light curve [81, 82], which is modelled to obtain the braking index [77, 83]. We have shown above that the braking index depends on the structure of the neutron stars which is pronounced at higher frequencies. Given the remnant is a newly born millisecond magnetar with spin period P ∼ 1 -3 ms [77] it is expected to have the dependence from EOS on its braking index. Therefore, we compute the range of possible braking index at every given rotational frequency, which gets translated from the uncertainties in the EOS obtained in Sec. II. We have used Eq. 23 to compute the braking index for all the EOS constructed in Sec. II, which has been supplemented with all the frequency-dependent profiles of M,I,R and J (as discussed in the Sec. III and shown in Fig. 4) and their derivatives. We have used B p = 10 15 G, ϵ = 10 -7 and α S = 10 -4 , where the values of ϵ and α S is conservative estimates based on our results of Sec. IV C. The result of our analysis is shown in Fig. 7. The 90 percentile and the 68 percentile in the braking index as the function of frequency are shown by the blue and red band respectively which have been computed from the weights on the EOS parameter obtained from the Bayesian analysis in Sec. II. The range of the braking index gets wider at higher frequencies due to the contribution of the rotational effects from different EOS of various stiffness. The yellow and the grey horizontal bands are the braking index of GRB 140903A and GRB 130603B with the 1 σ uncertainty measured from their lightcurve by Lasky et al. [83]. The measured braking index passes through our estimated band of braking index indicating if the millisecond magnetar is formed with a typical value of B p = 10 15 G, ϵ = 10 -7 and α S = 10 -4 or less, the spin frequency of the central engine can be between ∼ 550 -850 Hz for GRB 140903A and f < ∼ 650 Hz for GRB 130603B.', 'V. CONCLUSION': "In this paper, we have investigated the effect of stellar rotation on the braking index and its dependence on the EOS. We have generated a large number of EOS by employing a semi-agnostic model and constrained them using the latest astrophysical observations from the gravitational and radio wave bands within a Bayesian framework. We find our constrained parameter space is also consistent with the observational findings of neutron star's mass and radius from X-ray bands. The con- \nstrained nuclear EOSs have been used to study the effect on the macroscopic quantities of rapidly rotating neutron stars like mass, radius, moment of inertia and the ratio of polar to equatorial radius. Our analysis shows neutron stars with stiffer EOS experience larger deformations in the shape at a given frequency, in comparison to the softer EOS. Consequently, neutron stars with stiffer EOS have smaller Kepler frequencies. \nNext, we study the effect of rotation on the braking index of neutron stars. We present the first calculation of the braking index accounting for the rotational effects in the presence of energy loss by magnetic dipole radiation and GWs due to non-axisymmetric deformations and r-mode oscillations. We have shown that the rotation of the star can impact its braking index significantly. The deviation of the braking index from 3 to smaller values in the case of magnetic dipole radiation is due to the rotational effects. This rotational effect is prominent only when the spin frequency > ∼ 200 Hz. However, the low braking index in normal pulsars has been explained through various other mechanisms, such as, the evolution of magnetic inclination angle, which has not been considered here. Similarly, we have shown if the neutron stars spin-down is purely due to GW radiation from non-asymmetric deformations or r-mode oscillations, the effect of rapid rotation leads to the significant departure of braking index from their classical value of 5 and 7 respectively. \nWe further explore the dependence of the braking index on the rotation frequency for various physically motivated combinations of surface magnetic field strength, ellipticity of the star and the amplitude of r-mode oscillations. We find if there is r-mode instability generated in a typical millisecond pulsar with α S between 10 -4 -10 -1 , then angular momentum loss through GW r-mode instability will dominate the spin-down of the pulsar. Whereas, extremely large deformation of ϵ > ∼ 10 -3 is required for the spin-down of a typical millisecond pulsar to be dominated by GW emission due to time-dependent mass quadrupolar moment, which has been ruled out under various physical conditions. Our analysis shows that if the ϵ < ∼ 10 -7 the spin-down is dominated by the magnetic dipole radiation in the absence of r-mode instability, which is consistent with the current limit of the GW observations. \nWe extended our study in the light of millisecond magnetars, which are believed to be born from NS-NS mergers, core-collapse of a massive star or accretion-induced \ncollapse of a white dwarf. The spin-down energy is believed to power the X-ray plateau phase of GRBs. These millisecond magnetars can have large magnetic field strength ∼ 10 15 G. We find at large spin-frequencies > ∼ 600 Hz the presence of large amplitude r-mode oscillations dominates the spin-down rate. However, the effect of rapid rotation suppresses the braking index closer to 6 instead of 7. Whereas, the impact of non-axisymmetric deformation on braking index is less pronounced unless ϵ > 10 -3 . However, if the birth spin frequency is low ∼ 200 Hz or r-mode amplitude is less than 10 -1 , spindown is always dominated by the magnetic dipole radiation. \nWe have used the constrained EOS to compute the permissible range of the braking index as a function of the rotational frequency. The width of the range of the braking index increases with rotation frequency, indicating a wide range of EOS-dependent structural change of neutron stars in response to large centrifugal force. We find a millisecond magnetar with the magnetic field strength 10 15 G, deformation ϵ = 10 -7 and r-mode amplitude α S = 10 -4 has braking index ∼ 3 at low frequency and n < 3 at higher frequency 4 as shown in Fig. 7. Our band of computed braking is in good agreement with the measured braking index of GRB 140903A and 130603B, from where we find the spin frequency of the central engine of GRB 140903A could be between ∼ 550 -850 Hz, which is subjected to systematics at higher spin-frequency end. Our model of spin-frequency dependent braking index hints towards a potential method to constrain the dense mater EOS by precisely measuring the braking index and the spin frequency from the high-energy GRB light curves.", 'ACKNOWLEDGEMENTS': "Pulsar research at Jodrell Bank Centre for Astrophysics and Jodrell Bank Observatory is supported by a consolidated grant from the UK Science and Technology Facilities Council (STFC). P.C. is supported by European Union's HORIZON MSCA-2022-PF-01-01 Programme under Grant Agreement No. 101109652, project ProMatEx-NS. P.C. also acknowledges past support of the Fonds de la Recherche Scientifique-FNRS, Belgium, under Grant No. 4.4501.19. Part of the computation has been performed in the high-performance computing facility 'Magus,' which is available at the Shiv Nadar Institution of Eminence. \nof the rigid co-rotating magnetosphere), which naturally lead to n < 3 scenario. However, this modification in the magnetosphere is beyond the scope of this paper and has been confined to the simple dipolar approximation of the magnetosphere. \n- [1] R. N. Manchester, J. M. Durdin, and L. M. Newton, Nature 313 , 374 (1985).\n- [2] J. E. Gunn and J. P. Ostriker, Nature 221 , 454 (1969).\n- [3] K. Riles, Living Reviews in Relativity 26 , 3 (2023), arXiv:2206.06447 [astro-ph.HE].\n- [4] A. K. Harding, I. Contopoulos, and D. 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2024arXiv240807053S	We introduce a new software package DarkMatters which has been designed to facilitate the calculation of all aspects of indirect dark matter detection of WIMPs in astrophysical settings. Two primary features of this code are the improvement in performance compared to existing tools and higher levels of accuracy when determining radio synchrotron emission associated with WIMP annihilations both of which are enabled by the employment of a set of modern and novel numerical techniques. The code also includes functionality for a multiwavelength set of output products including gammaray radio and neutrino fluxes which can be saved in common formats used by the astronomical community such as the FITS data file format. The calculations may be tailored to work with a wide range of astrophysical target structures from dwarf galaxies to galaxy clusters and the configuration of the underlying calculations is managed by a set of keyvalue dictionary entries that are easy to understand and use. The code base is publicly accessible through an online repository with a permissive MIT source code licence.	2024-08-01T00:00:00Z	['10.48550/arXiv.2408.07053', 'arXiv:2408.07053', '2024arXiv240807053S']	['High Energy Physics - Phenomenology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena', 'Physics - Computational Physics']	DarkMatters A powerful tool for WIMPy analysis	2,024	191	0.41	['EPRINT_HTML', 'EPRINT_PDF']	1	https://arxiv.org/pdf/2408.07053.pdf	{'No Header': '- b', 'DarkMatters: A powerful tool for WIMPy analysis': 'Michael Sarkis a,b , Geoff Beck a \na Centre for Astrophysics and School of Physics, University of the Witwatersrand, 1 Jan Smuts Ave, Johannesburg, 2101, South Africa Department of Physics, Stellenbosch University, 111 Merriman Ave, Stellenbosch, 7602, South Africa', 'Abstract': 'We introduce a new software package, DarkMatters , which has been designed to facilitate the calculation of all aspects of indirect dark matter detection of WIMPs in astrophysical settings. Two primary features of this code are the improvement in performance compared to existing tools, and higher levels of accuracy when determining radio synchrotron emission associated with WIMP annihilations, both of which are enable by the employment of a set of modern and novel numerical techniques. The code also includes functionality for a multi-wavelength set of output products including gamma-ray, radio and neutrino fluxes which can be saved in common formats used by the astronomical community, such as the FITS data file format. The calculations may be tailored to work with a wide range of astrophysical target structures, from dwarf galaxies to galaxy clusters, and the configuration of the underlying calculations is managed by a set of key-value dictionary entries that are easy to understand and use. The code base is publicly accessible through an online repository with a permissive MIT source code licence. \nKeywords: dark matter theory, dark matter simulations, absorption and radiation processes', '1. Introduction': 'The nature of the Dark Matter (DM) in our universe remains a significant unsolved problem in modern physics. The accumulation of evidence over the past several decades, including observations of anisotropies in the Cosmic Microwave Background (CMB) [1], galaxy cluster and merger dynamics [2], galactic rotation curves [3] and gravitational lensing [4], as well as implications from the current view of the large-scale structure of the universe [5], all point toward a likely particle nature of this currently unidentified substance (see [6] for a recent review). There is thus a global effort underway to try and detect DM, either directly through terrestrial observatories and particle colliders, or through characteristic signatures of astrophysical DM observed indirectly with telescopes. These methods are often complementary in nature, probing different aspects of candidate particle properties, and have grown in scale such that there are now multiple dedicated \nEmail addresses: [email protected] (Michael Sarkis), [email protected] (Geoff \nBeck) \nSeptember 10, 2024 \nexperiments that generate huge datasets to search through. Of particular interest in this work, the field of indirect detection (ID) presently contains many studies that span multi-wavelength and multi-messenger disciplines, producing constraints on DM model parameters from radio to gamma-ray, comsic ray and neutrino observations. \nEven with the given observational constraints, there are a multitude of viable candidate particle models that have been proposed to fit the role of DM (see [7]). Weakly Interacting Massive Particles (WIMPs) are a generic class of models that contain some neutral, weaklyinteracting species with properties that not only provide a clear candidate for the cosmic abundance of DM, but also appear naturally in extensions to the Standard Model (SM) of particle physics. These particles are often considered to be collisionless and cold, due to constraints determined by the layout of large-scale structure [5, 8], which allow them to fit as a component into the standard ΛCDM model of cosmology. The presence of WIMPs within astrophysical structures can be probed through ID, which is predicated on the expectation that they may self-annihilate and produce a set of SM products, which may in turn be observable by existing telescopes and astronomical observatories. \nGiven the non-detection of any (confirmed) WIMPs to date, a set of limits on viable particle parameters that generate more indirect emissions than expected have been found (for reviews of the extensive amount of study in this area, see of [7, 9, 10]). Prompt gamma-ray fluxes have typically been the most popular search channel in ID studies, with significant interest in the literature propelled by data from the Fermi-LAT space telescope [11] and a tantalising excess signal found at the galactic centre [12, 13], which is still generating debate and further investigation [14, 15]. There has also been a significant effort to search the dwarf spheroidal (dSph) satellite galaxies of the Milky Way (MW), as these targets are highly DM-dominated structures with relatively low baryonic foreground emissions, with gamma-ray observations of dozens of dSphs used in recent studies [16, 17, 18, 19]. Radio wavelength ID studies have also enjoyed recent interest, driven by the excellent observing capabilities of modern radio interferometer telescopes like the LOFAR [20], ATCA [21], JVLA [22], and most recently the SKA precursor instruments like MeerKAT [23] and ASKAP [24]. Since the reference work of [25], the number of radio wavelength ID studies has grown substantially, and now includes notable works regarding dSphs [26, 27, 28, 29], galaxies like M31 [30] and the Large Magellanic Cloud [31], and galaxy clusters [32, 33, 34, 35, 36]. Finally, the use of multi-messenger species in ID studies has also advanced substantially in the past several years, with notable studies including [37, 38, 39, 40] for neutrino searches and [41, 42, 43, 44, 45, 46] for cosmic ray searches. \nThe studies highlighted above display the abundance of both data availability and interest in the literature, and new generations of observatories are likely to only amplify this abundance. To keep up with this incoming data, it is vital for ID studies to have access to a modelling framework that provides the accuracy needed for high-resolution data and the computational efficiency needed for high data volumes. There are existing open-source tools that are able to compute the multi-wavelength emissions from WIMPs, the most notable of which is the RX-DMFIT [47] package which was developed to provide the earlier DM-FIT [48] package with X-ray and radio flux capabilities, and the recent DM-related patches which extend the functionality of the GALPROP [49] package by the author of [30] 1 . The GALPROP [49] and DRAGON(2) [50] code packages provide solutions \nto the related problem of cosmic-ray transport in the galaxy, and some of the numerical techniques used in these packages have inspired development of segments of the code presented in this work. However, each of these open-source tools lack some functionality in what we would consider as a complete solution for a highly accurate, efficient and generalised ID DM tool. We have thus developed the DarkMatters package to overcome these limitations and we provide the tool in a permissive and open-source format to the community. \nThe structure of this paper is as follows. Section 2 contains a description of the theoretical framework for all physical models, including DM halo and particle models, gas density and magnetic field profiles, and resulting multi-wavelength emission. Section 3 includes details of how to interface with the code and general outlines for obtaining and using the software. In Section 4, a set of comparison calculations with an existing software package, RX-DMFIT , are described and any differences apparent in the results are analysed. Section 5 contains a discussion on the use of this package in the wider context of astrophysical DM indirect detection, as well as a set of concluding remarks. A detailed technical description of the numerical method used to solve the electron propagation equation is additionally provided in Appendix A.', '2.1. DM halos': "There are several physical characteristics that play a role in the modelling of DM halos. Although there is evidence to suggest that DM halos are likely to have tri-axial shapes (see [51, 52]), we make the common assumption that the halos are spherically symmetric. This assumption correlates with the spherically symmetric magnetic field and gas density profiles used in the code, and should provide a good approximation for many astrophysical scenarios. The full halo parameters can be divided into two categories. The first are the characteristic values of density and radius: ρ s and r s respectively. The second set contains the virial mass, radius, and concentration: M vir , R vir , and c vir . The minimum information required to specify a halo is simply a virial mass/radius. This is combined with numerical fitting functions for the concentration to fully specify the halo parameters. Otherwise, the following pairs are accepted as minimum information: ( r s , any), ( R vir , c vir ), and ( M vir , c vir ). We will now proceed to specify the definitions of each variable. \nFirst, we note that the virial radius is determined according to \n1 4 3 πR 3 vir ∫ R vir 0 4 πr 2 ρ ( r ) dr = ∆ c ( z ) ρ c ( z ) , (1) \nwith the critical density being given by \nρ c ( z ) = 3 H ( z ) 2 8 πG , (2) \nwhile the virial contrast ∆ c is \n∆ 18 π 2 82 x 39 x 2 , (3) \nc ≈ -- \nwhere x = 1 -Ω m ( z ). Here Ω( z ) is the usual cosmological density parameter, for matter ( m ) or the cosmological constant (Λ), and Ω m ( z ) is given by \nΩ m ( z ) = 1 1 + Ω Λ (0) Ω m (0) (1 + z ) -3 . (4) \nWe can then define the virial mass via \nM vir = 4 3 π ∆ c ρ c R 3 vir , (5) \nand the virial concentration by \nc vir = R vir r -2 . (6) \nNote that r -2 is the radius where the logarithmic slope of the density profile ρ ( r ) is equal to -2. This is not always equal to r s but the code accounts for this discrepancy in all of the predefined density profiles. The virial concentration can also be determined by one of a suite of numerical fitting functions from Prada 2012 [53], Mu˜noz-Cuartas 2011 [54], Colafrancesco 2006 [25], or Bullock 2001 [55]. \nThe first halo density profile available in the code is the Navarro-Frenk-White (NFW) profile [56] \nρ nfw ( r ) = ρ s ( r r s )( 1 + r r s ) 2 . (7) \nThere is also a generalised form of this profile \nρ gnfw ( r ) = ρ s ( r r s ) α ( 1 + r r s ) 3 -α , (8) \nwith free parameter α . We also have the Einasto profile [57] \nρ ein ( r ) = ρ s exp { -2 α [( r r s ) α -1 ]} , (9) \nagain with free parameter α . Then there is Burkert's profile [58] \nρ bur ( r ) = ρ s ( 1 + r r s ) ( 1 + ( r r s ) 2 ) , (10) \nand the pseudo-isothermal profile \nρ iso ( r ) = ρ s 1 + ( r r s ) 2 . (11)", '2.2. Particle spectra': "The observable radiation from the halos described above is also dependent on the nature of the constituent DM particles. The generic WIMP models considered here are parameterised by two quantities: the WIMP mass ( m χ ), and either cross-section ( ⟨ σv ⟩ ) or decay rate (Γ) in the case of annihilation or decay, respectively. The energy spectrum of the SM particles produced by each WIMP annihilation or decay is then represented by dN i /dE , where N i is the produced multiplicity, i refers to the species of the final state stable SM particle, and E is the corresponding particle energy. The final state is reached through various intermediate particle interactions - known as channels and denoted by f - which can be represented generally through the reaction pathway of χχ → f → i ( χ → f → i ) in the case of annihilation (decay). A general form of the produced energy spectrum is given by \nd N i d E = ∑ f B f d N f → i d E , (12) \nwhere the considered spectrum is a combination of all relevant intermediate channels, and the contribution of each channel is determined by the branching ratio B f . As is standard in the indirect-detection literature, we consider results for individual channels (corresponding to a branching ratio of B f = 1 for the channel of interest) that can be specified explicitly for each computation within the code (See Section 3.2). The final state species are likewise chosen according to the desired observable to be computed ( i would be gamma-ray photons for the case of calculating prompt gamma-ray emissions, for example), and these will be discussed in further detail in Section 2.5. \nThe produced particle spectra, as calculated from any source, can be specified as an input to the code. By default, DarkMatters makes use of the latest version of the pre-computed and model-independent numerical tables provided by the Poor Particle Physicist Cookbook for Dark Matter Indirect Detection (PPPC4DMID) [59], which are also hosted in an online repository 2 . The tables ('ingredients') provided by the PPPC4DMID have also been computed with corrections for electroweak radiative effects, following the prescription given in [60]. These corrections can have an impact on the final energy spectrum, especially when particle energies are larger than the electroweak scale, which is relevant for DM models with a large WIMP mass. The accessible and model-independent nature of the results provided by the PPPC4DMID allows for a convenient utilisation of these particle spectra, which has made this package a popular resource in recent indirect-detection literature. \nTo then determine the spatial and energy distribution of the particles that are injected into the halo from WIMP annihilation and decay, the particle spectrum defined above is multiplied by the number density of WIMPs to yield the source function Q ( r, E ), which can be written as follows: \nQ ( r, E ) = 1 2 ( ρ χ ( r ) m χ ) 2 ⟨ σv ⟩ d N i d E , (annihilation) ( ρ χ ( r ) m χ ) Γ d N i d E . (decay) (13) \nIn these equations, ρ χ represents the radial density of WIMPs in the halo as described by the profiles given in the previous section. The number density in the case of annihilation assumes Majorana WIMP pairs, and one should include an additional factor of 1/2 for the case of Dirac WIMPs.", '2.3. Electron propagation': "Following the annihilation or decay of WIMPs, the indirect detection of the observable products will depend on the final state of the SM particles. In the case of gamma-ray photons or neutrinos, their propagation from the DM halo to the Earth is relatively straightforward, as they do not interact with intermediate environments like magnetic fields (see Sections 2.5.2 and 2.5.3). However, for radio-frequency synchrotron emissions (resulting from χχ → f → e ± ), the evolution of the charged final state electrons is goverened by several physical interactions between the electrons and magnetic fields and thermal gas populations that are ubiquitous in large astrophysical structures. To model these effects, one typically employs a cosmic-ray transport equation, such as the one used in the GALPROP package [61], and presented in the Appendix of [49]. Although this form encapsulates all possible physical interactions, the two dominant effects in most astrophysical scenarios that pertain to the indirect detection of WIMPs in large DM halos are those of spatial diffusion and energy losses. By neglecting all sub-dominant terms, the full transport equation reduces to the standard diffusion-loss equation, \n∂ψ ( x , E ) ∂t = ∇· ( D ( x , E ) ∇ ψ ( x , E )) + ∂ ∂E ( b ( x , E ) ψ ( x , E )) + Q ( x , E ) . (14) \nIn this equation, we have represented the spatial and energy distribution of electrons by ψ , as in Galprop . We further simplify this equation by assuming spherical symmetry, so that x → r , which should be valid for typical cool-core galaxy clusters and dSphs (for an example of the detailed solution to this equation without this assumption in the M31 galaxy, see [30, 62]). The solution to this partial differential equation is the equilibrium electron distribution, which can be found through one of several mathematical techniques (this is discussed in further detail in Appendix A). \nThe effects of diffusion and energy loss are determined by the corresponding coefficients in Equation (14), D ( r, E ) and b ( r, E ), respectively. In DarkMatters , we utilise the following general form of the diffusion coefficient, \nD ( r, E ) = D 0 ( E 1 GeV ) δ ( B ( r ) B (0) ) -δ , (15) \nwhere D 0 is the diffusion normalisation constant, δ is the diffusion index and B ( r ) , B 0 represent the magnetic field profile and normalisation, respectively. When modelling the effects of spatial diffusion with Equation (15), it is necessary to choose a description of the turbulence in the magnetic field that gives rise to the diffusion of the charged particles in that field. This choice determines the value of the index δ , and common descriptions include the cases of Kolmogorov ( δ = 1 / 3), Kraichnan ( δ = 1 / 2), and Bohm-like ( δ = 1) diffusion. The factor of ( B ( r ) /B 0 ) -δ in Equation (15) traces the radial profile of the magnetic field, which results in an equilibrium electron distribution that accurately reflects the geometry of the environment when using numerical solution methods, and a form of this factor has been used before in the works by [27, 63]. \nThe energy-loss coefficient, defined as the energy-loss rate b ( r, E ) ≡ d E/ d t , encapsulates all of the physical process that reduce the total energy of the electron distribution. The dominant effects at high energies (relevant for typical WIMP models) are synchrotron emission and Inverse Compton (IC) scattering, though we also include the effects of Coulomb scattering and bremsstrahlung, which can be relevant at lower energies. The full form of the energy-loss coefficient that is used in DarkMatters when solving Equation (15) is given below, \nb ( r, E ) = b IC ( U ph eV / cm 3 )( E 1 GeV ) 2 + b synch ( E 1 GeV ) 2 ( B ( r ) 1 µ G ) 2 + b Coul ( n e ( r ) 1 cm -3 )( log ( γ n e ( r ) / 1 cm -3 ) +73 ) + b brem ( n e ( r ) 1 cm -3 ) γ (log ( γ ) + 0 . 36) , (16) \nwhere n e ( r ) is the thermal gas number density, U ph is the target photon energy density ( U ph = 0 . 26 eV / cm 3 in the case of CMB photons) and γ is the usual Lorentz factor. Each of the effect's loss rate coefficients are taken from the values presented in [30], which have recently been corrected from errors in previous works. The coefficients, in units of GeVs -1 , are thus b IC = 1 . 0 × 10 -16 , b synch = 0 . 025 × 10 -16 , b Coul = 7 . 6 × 10 -18 , b brem = 7 . 1 × 10 -20 . \nWith the explicit forms of the functions Q,D and b given by Equations (13),(15) and (16) above, the diffusion-loss equation can be solved for an equilibrium distribution of electrons. An example solution is provided in Figure 1, which shows the full 2-dimensional electron equilibrium distribution that has been computed for the Coma galaxy cluster environment and a set of typical WIMP parameters. Note the larger concentration of low-energy electrons within the inner regions of the halo, which is a common feature of the equilibrium distributions.", '2.4. Gas density and magnetic field': 'A variety of gas density and magnetic field strength radial profiles are available by default within DarkMatters . These consist of the constant (flat), power law (pl), β , double β , and exponential profiles. Every profile has a normalisation constant n 0 or B 0 for gas density and magnetic field respectively. We then use a notation system that a physical scale associated with the profile is either r e or r b while an exponent is β e or β b . In the case of the double β profile there exist two scales, normalisations, and exponents. \nBelow we list the gas density and magnetic field strength profiles in their functional \nFigure 1: The equilibrium distribution of electrons, computed numerically as the solution to the diffusionloss equation (Equation (14)). This solution is for a set of sample DM halo and particle parameters, namely for the Coma galaxy cluster with an NFW halo profile and 100 GeV WIMPs annihilating solely through the χχ → bb → e ± channel. \n<!-- image -->', 'forms': 'n e, flat ( r ) = n 0 , (17) \nn e, pl ( r ) = n 0 ( r r e ) β e , (18) \nn e, beta ( r ) = n 0 ( 1 + ( r r e ) 2 ) 3 βe 2 , (19) \nn e, d-beta ( r ) = n 0 ( 1 + ( r r e ) 2 ) 3 βe 2 + n 0 , 2 ( 1 + ( r r e, 2 ) 2 ) 3 β e, 2 2 , (20) \nn e, exp ( r ) = n 0 exp ( -r r e ) , (21) \nand \nB flat ( r ) = B 0 , (22) \nB pl ( r ) = B 0 ( r r b ) β b , (23) \nB beta ( r ) = B 0 ( 1 + ( r r b ) 2 ) 3 β b 2 , (24) \nB d-beta ( r ) = B 0 ( 1 + ( r r b ) 2 ) 3 β b 2 + B 0 , 2 ( 1 + ( r r b, 2 ) 2 ) 3 β b, 2 2 , (25) \nB exp ( r ) = B 0 exp ( -r r b ) . (26)', '2.5. Observables from DM': 'Once the physical environment of the DM halo and the particle model is fully specified, various multi-messenger observables can be calculated in DarkMatters . There are three broad categories that we focus on, which include radio emissions from synchrotron radiation, gamma rays produced from both prompt annihilation and secondary mechanisms, and prompt neutrino emmission. The details of the implementation of each of these emissions are described below.', '2.5.1. Radio': "The method used to calculate the radio emissivity from relativistic electrons follows a standard framework, as laid out in [25]. Firstly, the isotropically-averaged power of the synchrotron emission from a population of electrons is expressed as in [25], \nP ( ν, E, r ) = ∫ π 0 d θ sin( θ ) 2 2 π √ 3 r e m e cν g sin( θ ) F ( x/ sin( θ )) , (27) \nwhere r e , m e and ν g are the electron's classical radius, mass and non-relativistic gyrofrequency, respectively. The function F and the variable x are defined as follows: \nF ( t ) ≡ t ∫ ∞ t d z K 5 / 3 ( z ) (28) \nand \nx ≡ 2 ν (1 + z ) 3 ν 0 γ 2 [ 1 + ( γν p ν (1 + z ) ) 2 ] 3 / 2 , (29) \nwhere K 5 / 3 is a modified Bessel function of order 5/3 and ν p = 8980( n e ( r ) / 1 cm -3 ) 1 / 2 Hz is the plasma frequency of the gas. We have used the fitting formulae provided by [64] to calculate the kernel function in Equation 28. The synchrotron power is then used to calculate the emissivity, by \nj synch ( ν, r ) = ∫ m χ m e d E ( ψ -( r, E ) + ψ + ( r, E )) P ( ν, E, r ) , (30) \nwhere ψ ± are the equilibrium electron and positron distributions found by solving Equation (14). Various common observable quantities, such as the radio surface brightness or the flux density spectrum, can now be calculated from the emissivity. Firstly, the radio surface brightness is defined as \nI synch ( ν, s ) = ∫ d l j ( ν, √ s 2 + l 2 ) 4 π , (31) \nwhere l is the line-of-sight coordinate along a cylinderical axis and s is the corresponding cylindrical radius to a point within the halo. Sample results calculated using Equation (31) can be seen in Figures 2 and 3. As modern radio datasets are often accessible as 2dimensional images, DarkMatters provides a mapping of the above surface brightness profile onto a 2-dimensional spatial grid, which allows for convenient comparisons between model and data. The sky coordinates and desired pixel size or image resolution of the produced image are all adjustable parameters when interfacing with the code, and DarkMatters also provides an option to save these images conformant to the FITS data file format 3 , which is a standard multi-dimensional data format used extensively in the radio astronomy community. A sample 2-dimensional image of radio surface brightness emissions (calculated with Equation (31)) is shown in Figure 4. \n0 \n0 \nFigure 2: Sample calculations for the radio surface brightness emissions from the Coma galaxy cluster for a wide range of frequencies and two annihilation channels; (a): bottom quarks and (b): muons. Also shown is the characteristic scale radius r s of the DM halo, in equivalent angular units, as a vertical line. \n<!-- image --> \nThe final radio-frequency observable that we consider is the integrated radio flux density, defined as \nS synch ( ν, R ) = ∫ R 0 d 3 r ' j ( ν, r ' ) 4 π ( d 2 L +( r ' ) 2 ) , (32) \nwhere R represents the maximum radius of the halo within which the emission is integrated, and d L is the luminosity distance to the halo. A set of sample calculations of the integrated flux, as calculated with Equation (32), is shown in Figure 5. \nFigure 3: Equivalent surface brightness calculations as shown in Figure 2, except for the Reticulum II dSph target. \n<!-- image --> \nFigure 4: Sample 2-dimensional radio surface brightness image, computed by mapping the results of the radial surface brightness calculation in Equation (31) onto a 2-dimensional grid. Calculation parameters follow the sample for the Coma galaxy cluster with an NFW halo profile and 100 GeV WIMPs annihilating through the bottom quark channel. Results shown here are for synchrotron emission frequency of 1 GHz. Note that this image can be saved in the FITS data format by DarkMatters . \n<!-- image --> \nFigure 5: Sample calculations for the integrated radio flux density from the (a): Coma galaxy cluster and (b): Reticulum II dSph. Results shown vary the WIMP mass and annihilation channel for comparative purposes. Note the change in flux scale between the two figures. \n<!-- image -->", '2.5.2. Gamma-ray': "Primary (prompt) γ 's The differential flux of gamma rays that are produced directly from WIMP annihilations and decay, referred to as the 'prompt' emission, is calculated as follows. \nJ (∆Ω , l ) = ∫ ∆Ω ∫ l ρ 2 ( r ' )d l ' dΩ ' , (33) \nD (∆Ω , l ) = ∫ ∆Ω ∫ l ρ ( r ' )d l ' dΩ ' , (34) \nS γ ( E ) ≡ dΦ γ d E = ⟨ σv ⟩ 8 πm 2 χ J d N i d E , (annihilation) Γ 4 πm χ D d N i d E , (decay) (35) \nwhere the energy spectra of the produced gamma rays are found from Equation (12). \nSecondary (ICS) γ 's This is composed of high-energy photons produced via the equilibrium electron population within the DM halo. The mechanisms used are Inverse Compton Scattering (ICS) and bremsstrahlung. \nThe ICS power at observed frequency ν from an electron with energy E at redshift z is determined via [65, 66] \nP IC ( ν, E, z ) = cE γ ( z ) ∫ dϵ n ( ϵ ) σ ( E,ϵ, E γ ( z )) , (36) \nwhere E γ ( z ) = hν (1 + z ), ϵ is the seed photon energy distributed according to n ( ϵ ). The Klein-Nishina cross-section is defined according to \nσ ( E,ϵ, E γ ) = 3 σ T 4 ϵγ 2 G ( q, Γ e ) , (37) \nwhere σ T is the Thompson cross-section and \nG ( q, Γ e ) = 2 q ln q +(1 + 2 q )(1 -q ) + (Γ e q ) 2 (1 -q ) 2(1 + Γ e q ) , (38) \nwith \nq = E γ Γ e ( γm e c 2 + E γ ) , \nΓ e = 4 ϵγ m e c 2 , (39) \nwhere γ is the electron Lorentz factor. \nFor bremsstrahlung of an electron colliding with target species j we use [65, 66] \nP B ( ν, E, r ) = cE γ ( z ) ∑ j n j ( r ) σ B ( E γ , E ) , (40) \nwhere the notation remains the same as for ICS. Note that n j is the target species number density and the cross-section is given by \nσ B ( E γ , E ) = 3 ασ T 8 πE γ [( 1 + ( 1 -E γ E ) 2 ) ϕ 1 -2 3 ( 1 -E γ E ) ϕ 2 ] , (41) \nwhere ϕ 1 and ϕ 2 are species dependent factors [65, 66]. In DarkMatters we use only protons as the target species and equate n j to the gas density within the halo environment. \nThe fluxes from the mechanisms is determined following the same process as for radio emissions, but with the appropriate choice of power function in Eq. 30.", '2.5.3. Neutrino': 'Neutrino fluxes are computed in the same manner as prompt gamma-rays \nS ν ( E ) ≡ dΦ ν d E = ⟨ σv ⟩ 8 πm 2 χ J d N ν d E , (annihilation) Γ 4 πm χ D d N ν d E , (decay) (42) \nwhere J and D have been previously defined in Eqs. (33) and (34). A set of sample calculations using the above equation can be seen in Figure 7 for the Coma galaxy cluster. \nFigure 6: Spectral Energy Distribution (SED) for the multi-wavelength WIMP-related emissions from the (a): Coma galaxy cluster and (b): Reticulum II dSph. The dominant contributions from each major emission type (corresponding to the radio synchrotron, prompt gamma-ray and secondary gamma-ray emissions described above) are labelled at the relevant locations on the curves. \n<!-- image --> \nFigure 7: Sample neutrino flux output from the Coma galaxy cluster, calculated using Equation (42). \n<!-- image -->', '3.1. Installation': 'The package is installed merely by cloning the GitHub repository and ensuring all the dependencies are installed (a file requirements.txt is provided for this purpose).', '3.2. Specifying input files': 'DarkMatters uses either yaml or json files for input and output, the default option is yaml. Below we specify a yaml input for an example calculation \n```\nhalo\\_data: name: "segue1" profile: "einasto" index: 0.3333 rho\\_norm: value: 1.738e+8 unit: "Msun/kpc^3" scale: value: 0.15 unit: "kpc" distance: value: 23.0 unit: "kpc" mag\\_data: profile: "flat" mag\\_norm: value: 2.e-6 unit: "gauss" gas\\_data: profile: "exp" gas\\_norm: value: 1.e-4 unit: "1/cm^3" scale: value: 38.0 unit: "pc" diff\\_data: diffindex: 0.7 diff\\_rmax: value: 1.6 unit: "kpc" diff\\_constant: value: 3.0e+26 unit: "cm^2/s"\n``` \n```\npart\\_data: part\\_model: "bb" em\\_model: "annihilation" calc\\_data: calc\\_mode: "flux" freq\\_mode: "radio" m\\_wimp: value: - 10 - 100 - 1000 unit: "GeV" f\\_sample\\_limits: value: - 5.e+1 - 2.e+4 unit: "MHz" f\\_sample\\_num: 40 f\\_sample\\_spacing: "log" calc\\_angmax\\_integrate: value: 4 unit: "degree" os\\_delta\\_t\\_min: value: 0.1 unit: "yr"\n``` \nThe above example input also serves to illustrate the data structures used in the package, these being a set of python dictionaries halo\\_data , mag\\_data , gas\\_data , diff\\_data , part\\_data , and calc\\_data . These store the properties associated with the halo, magnetic field, gas, diffusion environment, particle physics, and calculation to be performed. Another feature to remark on is the specification of unitful quantities: they are given by a dictionary with a \'value\' and \'unit\' entry. Any unit of same type (mass,distance,density) can be given and astropy is used to convert it into the internal unit system of DarkMatters . This requires that the unit entry be specified in an astropy friendly format. \nOne other dictionary exists: cosmo\\_data . This contains the cosmological parameters to be used in calculations. If left unspecified, it defaults to the Planck 2018 results [1].', '3.2.1. halo data': "Here we specify the information on the DM halo. A summary of how modelling parameters link to the dictionary is presented in Table 1. Note that the density profiles can be accessed by specifying the halo\\_profile parameter with the allowed values 'nfw', 'gnfw', 'einasto', 'burkert', and 'isothermal'. Additional profiles can be added by modifying the halo density profiles.yaml file, in the dark\\_matters/config folder, as well as adding a function recipe to the halo\\_density\\_builder function in the dark\\_matters/astro\\_cosmo/astrophysics.py file. \nTable 1: Examples of DarkMatters halo parameter correspondence to modelling.", '3.2.2. mag data': "The magnetic field data is linked to the modelling parameters in Table 2. The common options for mag\\_profile are 'flat', 'powerlaw', 'beta', 'doublebeta', and 'exp'. \nTable 2: Examples of DarkMatters magnetic field parameter correspondence to modelling from Eq. (22).", '3.2.3. gas data': "The gas density data is linked to the modelling parameters in Table 3. The common options for gas\\_profile are 'flat', 'powerlaw', 'beta', 'doublebeta', and 'exp'. \nTable 3: Examples of DarkMatters gas density parameter correspondence to modelling from Eq. (17).", '3.2.4. diff data': 'This dictionary is quite simple, containing the radius at which the boundary conditions are applied diff\\_rmax as well as D 0 , the diffusion constant, in diff\\_constant . The parameter diffindex specifies the index δ in Eq. (15). \nIt is also possible to choose the target photon field for ICS by specifying the photon temperature (at present the code only accommodates black-body target fields) through the parameter photon\\_temp . Additionally, the energy density of ICS seed photons (for \ncalculating energy loss in Eq. (16)) can be specified via photon\\_density which has units of energy density.', '3.2.5. part data': "Here we specify the particle physics data. The annihilation/decay channel is given by part\\_model and we choose between 'annihilation' or 'decay' options using em\\_model . Custom models can also be included by setting up an input file with appropriate structure. See the Wiki 4 for more detail.", '3.2.6. cosmo data': "This is not displayed as it defaults to Planck 2018 [1]. Note the numerical fitting function for c vir can be chosen here using the parameter cvir\\_mode . The options are 'p12' [53], 'munoz 2011' [54], 'cpu 2006' [25] and 'bullock 2001' [55]. Note that the default value is 'p12'.", '3.2.7. calc data': "In this dictionary we actually specify what we want to calculate with all the other input data. This is principally chosen via two parameters: calc\\_mode and freq\\_mode . The first can take options 'flux', 'sb', or 'jflux'. These refer to integrated flux, surface brightness, and flux from a J or D factor respectively. The second parameter decides on the mechanisms to be considered. Its options are 'all', 'gamma', 'radio', 'pgamma', 'sgamma' or 'neutrinos x' (where x can be mu, tau, or e). These refer to: all electromagnetic emissions, primary plus secondary high-energy emissions, radio only, primary high-energy only, secondary high-energy only, and neutrinos respectively. \nFurther parameters specify more detail. For example m\\_wimp provides a list of WIMP masses (in energy units) to perform the calculation for. Then f\\_sample\\_limits , f\\_sample\\_num , and f\\_sample\\_spacing decide what frequencies to perform the calculation at. These can be manually specified using f\\_sample\\_values , which provides a list of the values, instead. The parameter calc\\_angmax\\_integrate gives the radius within which to integrate the flux (a physical radius can be specified instead with calc\\_rmax\\_integrate ). Finally, os\\_delta\\_t\\_min tells the algorithm the smallest time spacing value to use. This tends to need to be adjusted from the default only for small structures like dwarf galaxies (see the Wiki for further detail).", '3.3. Running the code': 'The following simple script will run DarkMatters \nimport sys \nsys.path.append(dark\\_matters\\_path) \nfrom dark\\_matters.input import read\\_input\\_file \nfrom dark\\_matters.calculations import run\\_calculation \nfrom dark\\_matters.output import make\\_output \ndata\\_sets = read\\_input\\_file("example.yaml") \noutput\\_data = run\\_calculation(data\\_sets) make\\_output(output\\_data,out\\_mode="yaml") \nThis assumes the path to the folder containing the dark\\_matters folder is specified in place of dark\\_matters\\_path and that the input file is called example.yaml . The code will produce an output yaml file with a name that reflects some of the associated calculation parameters. This file will contain all of the input dictionaries and include the results in calc\\_data["results"] , which is itself a dictionary. The file contents is identical to the contents of the output\\_data variable above. \nWhen the code is executed it conducts a set of consistency checks on the input dictionaries and calculates all the supplementary parameters needed to perform the requested tasks. It then proceeds to execute the instructions in the calc\\_data dictionary.', '4. Comparisons with existing tools': "Aside from an internal test suite, a set of simple calculations with the Radio and X-ray DMFIT ( RX-DMFIT ) package [47] have been performed to compare and validate the results from DarkMatters . The RX-DMFIT package 5 was originally written to extend the DMFIT [48] package with radio and X-ray fluxes. At the time of writing, surface brightness outputs (of the form in Equation (31)) are not supported in RX-DMFIT , so the comparisons are focused on emissivity and integrated flux outputs. For this, we have considered two typical and common targets of study in the literature - the Coma galaxy cluster and the Reticulum II dSph. These targets exist on opposite ends of size and mass ranges for astrophysical structures, and provide edge cases for the physical environments that our package is designed to process. For these tests, we have kept the input configurations between the two codes as similar as possible, to highlight differences in the underlying method rather than any differences in parameter choices. The input configurations used in these tests have been provided as online material alongside this manuscript through the Zenodo doi 6 \nThe comparisons between both the radio emissivity and integrated flux are displayed in Figures 8 and 9. We have also calculated simple relative differences between the methods to quantify the changes, where each result has been found relative to the larger value, i.e. we take ( \| y 1 -y 2 \| /y 1 ) × 100%, where y 1 > y 2 . In the Coma galaxy cluster, we see DarkMatters produce ∼ 46% higher emissivity values throughout the halo. The integrated flux values are likewise higher, by ∼ 30% at 10 MHz, down to ∼ 15% at 10 GHz with an average of ∼ 23% difference over all frequencies in this range. In Reticulum II, we observe larger overall differences, with the emissivities from DarkMatters higher by ∼ 76% at the center of the halo, dropping to ∼ 60% when averaged over all radii. Notably, the emissivities converge at a point within the halo, with values from RX-DMFIT larger at large radii - this trend will be discussed further below. The integrated flux outputs from Reticulum II have a maximum difference of ∼ 80% at 10 MHz, which drops to ∼ 50% averaged over all frequencies. \nThe trends between each package's results shown above can be attributed to the treatment of the spatially dependent variables in the solution to the diffusion-loss equation. This is one of the primary differences between the two packages, with RX-DMFIT using the semi-analytical technique first described in [25], and DarkMatters using the numerical \nFigure 8: Direct comparison of outputs calculated using the DarkMatters (black curve) and RX-DMFIT (orange curve) software packages. Results shown here are the emissivity (Equation (30)) in the left panel and the integrated flux (Equation (32)) in the right panel, for the Coma galaxy cluster modelled with parameters shown on the figure. For a full list of parameters used in these calculations, see the supplementary online material provided with this manuscript. \n<!-- image --> \nFigure 9: Direct comparison of outputs calculated using the DarkMatters (black curve) and RX-DMFIT (orange curve) software packages. Results shown here are the emissivity (Equation (30)) in the left panel and the integrated flux (Equation (32)) in the right panel, for the Reticulum II dSph modelled with parameters shown on the figure. For a full list of parameters used in these calculations, see the supplementary online material provided with this manuscript. \n<!-- image --> \ntechnique outlined in Appendix A. One consequence of using the semi-analytical method of [25] is that all parameters involved in the diffusion-loss equation cannot depend on spatial variables (in this case the radius variable r ). To illustrate how this point affects the observable emissions, consider first that a factor of the magnetic field strength appears in both the diffusion and energy-loss functions (Equations (15) and (16) respectively). To satisfy the above condition, an average value for the halo is calculated in RX-DMFIT according to \nB avg = 1 r h ∫ r h 0 d r B ( r ) , (43) \nwhere r h defines the maximum radius of the diffusive environment [47]. As the relevant astrophysical magnetic field profiles B ( r ) monotonically decrease from their central strength B 0 , in general we have B avg < B 0 , especially when r h is larger than the scale radius of the magnetic field. Since synchrotron radiation (which depends on B ( r )) is the dominant energy-loss mechanism at higher electron energies, the overall energy-loss function b ( E ) would be reduced at the center of the halo when an average value of the magnetic field is used over the full profile. Now, the semi-analytical solution method of [25, 47] makes use of a set of Green's functions that are defined with a variable v ( E ) as follows: \nv ( E ) = ∫ M χ E d ˜ E D ( ˜ E ) b ( ˜ E ) . (44) \nThe quantity √ v , which represents the mean distance that an electron would travel while losing a specified amount of energy, depends inversely on b ( E ). Lower values of b ( E ) that are caused by averaging the magnetic field will thus result in larger values of v and electrons that diffuse further in the halo while losing the same energy. \nThis interpretation can be used to explain the differences in the emissivity results seen between the two packages. With a central magnetic field strength that is relatively large in the DarkMatters case, the electrons are confined to the central regions of the halo where they continue to radiate. However, with the larger diffusion scale length determined by the lower overall magnetic field strength, electrons in the RX-DMFIT case travel further through the halo to the outer regions, while losing an equivalent amount of energy. This effect can be seen in Figure 9, which shows relatively less emissivity from RX-DMFIT at the center of the halo, but slightly more as r → r h . This is likely a stronger effect in Reticulum II than in Coma (Figure 8) because of the steeper exponential magnetic field profile found and comparatively smaller diffusion region in Reticulum II, both of which more effectively reduce the average magnetic field strength from the central value of B 0 . Then, we note that the integrated flux calculations show larger fluxes from DarkMatters for all frequencies. This is expected behaviour given the larger emissivity values for each target, especially at the center of the halo where the emission is strongest. \nFinally, we note the presence of some minor numerical artefacts in the emissivity calculations from RX-DMFIT at large radii in the Coma cluster, which are also present in the results displayed in [47]. These artefacts have been observed to worsen significantly with different values of the diffusion radius r h , especially if this value becomes comparable to or larger than the virial radius of the DM halo. The artefacts seem to originate from the numerical integration scheme used in the solution method of [25], and should be accounted for in the RX-DMFIT package through the use of alternative integration schemes provided by the GNU Scientific Library (GSL) [67]. We note that the results shown here \nwere found with the default method, a quadrature non-adaptive Gauss-Kronod technique (QNG), though the quadrature adaptive integration with singularities (QAGS) method produced similar results. The use of the recommended scheme options for computational speed in RX-DMFIT , i.e. the use of QNG integration, with a lookup table for the Green's function calculations, resulted in a average computation time of 295 seconds (taken over all performed calculations on the author's personal desktop computer). In comparison, the average computation time for the results from DarkMatters , performed on the same hardware, was 28 seconds. Of this time, an average of 24 seconds was used for finding the equilibrium distribution of electrons as the solution to the diffusion-loss equation, with the remaining time taken for emissivity or flux calculations. Profiling the code reveals that the vast majority of the compute time is made up of calls to the linear algebra method used to solve the matrix equations described in Appendix A, and the specifics of these methods will be discussed further therein. The code profiling results shown here correspond to the claim made in [26] that the numerical solution method is faster than the semi-analytical one. The use of advanced numerical techniques - like completely vectorised, sparse, block-matrix algebra with dedicated SciPy [68] libraries that are designed for fast numerical calculations and an accelerated timestep-switching technique inspired by GALPROP , have further improved computational performance of the DarkMatters package.", '5. Discussion and conclusions': 'In this paper we have introduced DarkMatters , the codebase which was developed to calculate all of the relevant physical properties of WIMPy DM halos and simulate their observable multiwavelength emissions. The functionality and configurability of the software should provide a complete solution - from basic input parameters of the system to final observable quantities - for a wide range of astrophysical environments. To fulfil this purpose, we have included and pre-configured a comprehensive set of parameters for possible modelling scenarios. At the current release, there are 6 separate profiles for the DM halo density and magnetic field strength and 5 separate profiles for gas density, with a straightforward method for adding further custom profile functions. There is also support for both WIMP annihilation and decay modes, with a default interface with the excellent PPPC4DMID [59] resource which provides particle spectra for all relevant SM products from these annihilations and decays. Outside of the typical input the user is expected to provide, the data structures used in the code are written with the intention of being simple to understand and comprise mostly of python dictionaries with explicit key-value pairs which makes the identification and manipulation of relevant variables as convenient as possible. \nWhen compared to existing open-source tools, DarkMatters provides several scientific and practical benefits, particularly when calculating radio emissions. In the RX-DMFIT package, the calculation of the equilibrium electron distribution follows a semi-analytical prescription. This method results in an effective reduction of the magnetic field strength at the centre of the halo, while allowing diffusing electrons to travel further throughout the halo. This effect can thus underestimate the radio emissivity in these regions, particularly in compact objects with steep magnetic field profiles and small diffusion regions. In regards to practical use cases, the solution method adopted by DarkMatters also not only provides quicker computation times, but is more robust to artefacts associated with \nintegration over the diffusion zone. We note that the GALPROP package is able to calculate the equilibrium electron distribution using a similar numerical solver to the one used here, thus also avoiding the issues mentioned above. However, it does not provide the host of auxiliary functions related to determining radio emissions from WIMPs that are included in DarkMatters , and has been designed primarily for use with galaxies that have a similar morphology to the MW, which limits its usefulness when considering a wide range of astrophysical targets and the observations associated with them. \nThe aspects of the code mentioned above thus allow the indirect emissions from any DM halo target, from small dSph satellite galaxies to large galaxy clusters, to be used in the ongoing effort by the astronomy sector to complement terrestrial DM experiments. This is a vital aspect to indirect detection experiments, as the current generation of astronomical observatories are already producing huge, high-resolution and high-sensitivity datasets from a large number of targets, through both directed observations and survey programmes. The use of this data in DM searches thus requires not only accurate modelling of the emission process, but also the quick and practical computation provided by DarkMatters to keep up with the available data. The DarkMatters package has already been utilised successfully in a recent indirect detection study [36] using MeerKAT Galaxy Cluster Legacy Survey [69] observations of several radio-faint galaxy clusters, setting stringent constraint on the WIMP annihilation cross-section. The open-source nature of this code should also allow for extra features to be added as they are developed by the community, so that this tool may come to play an increasingly helpful role in the ongoing hunt for DM in our universe.', '6. Acknowledgments': 'We thank Andrei Egorov for helpful discussions and suggestions regarding some calculations. GB acknowledges funding from the National Research Foundation of South Africa under the Thuthuka grant no. 117969. The work of MS was supported by the National Research Foundation of South Africa (Bursary No. 112332). This work makes use of the following software packages and code libraries: AstroPy [70], SciPy [68], NumPy [71], SymPy [72] and Matplotlib [73].', 'Data availability statement': 'Supplementary data used in this work is hosted in the Zenodo repository located at the doi: https://doi.org/10.5281/zenodo.13312389 . The source code of the software package presented in this work is available through the GitHub repository https:// github.com/Hyperthetical/DarkMatters/ .', 'Appendix A. Solving the diffusion-loss equation': "The standard, spherically symmetric diffusion-loss equation used to describe the evolution of cosmic rays in a magnetic field is given by \n∂ψ ( r, E ) ∂t = ∇· ( D ( r, E ) ∇ ψ ( r, E )) + ∂ ∂E ( b ( r, E ) ψ ( r, E )) + Q ( r, E ) . (A.1) \nSeveral techniques to solve this second-order partial differential equation for the equilibrium distribution of electrons exist in the literature. These include the semi-analytical methods of [25] (used in a multitude of WIMP indirect detection studies and the method of choice in the RX-DMFIT software package [47]), and the method developed in [74] (also recently used in indirect WIMP searches [28, 29]). The DarkMatters package employs a set of numerical techniques to solve this equation. Some of these techniques were inspired by existing codebases, like the GALPROP [49] and DRAGON(2) [50] packages, while others have been developed in this work. We also note that the authors of [26] have used a similar numerical technique in a series of WIMP indirect detection studies [75, 27, 31]. \nThe numerical solution used here is based on the Crank-Nicolson (CN) finite-differencing of Equation (A.1). The discretised operators of each dimension, namely the spatial and energy (momentum) dimensions, are then solved individually and alternatively in a technique known as Operator Splitting (OS), which leads to a drastic reduction in the computational requirements of the solution. Further, the full spatial dependence of magnetic field and gas density profiles that are relevant to the evolution of the system can in this case be considered at each step of the modelling process, unlike in semi-analytical methods. \nFirstly, we take Equation (A.1) and make the variable transformations ˜ r = log 10 ( r/r 0 ) and ˜ E = log 10 ( E/E 0 ), where r 0 and E 0 are characteristic scaling values. In DarkMatters , the defaults for these values are r s , the scale radius of the DM halo, and 1 GeV, respectively. With the aforementioned assumption of spherical symmetry, the diffusion-loss equation then becomes \n∂ψ ∂t = 10 -2˜ r (ln(10) r 0 ) 2 ([ ln(10)2 D + ∂D ∂ ˜ r ] ∂ψ ∂ ˜ r + D ∂ 2 ψ ∂ ˜ r 2 ) + 10 -˜ E ln(10) E 0 ( ∂b ∂E ψ + b ∂ψ ∂E ) + Q, (A.2) \nwhere ψ = ψ (˜ r, ˜ E ). This transformation of variables results in function evaluations and discrete grid points that are more convenient to handle, given that the relevant physical scales can encompass many orders of magnitude. To discretise Equation (A.2), we make use of the CN scheme, which is described in [76]. This scheme uses the average of explicit and implicit differencing terms, which has the benefit of having the unconditional stability of implicit methods as well as the second-order accuracy for small-scale effects. A representation of this scheme applied to a one-dimensional diffusion-loss equation can be written as in [49], \nψ n +1 i -ψ n i ∆ t = α 1 ψ n +1 i -1 -α 2 ψ n +1 i + α 3 ψ n +1 i +1 2∆ t + α 1 ψ n i -1 -α 2 ψ n i + α 3 ψ n i +1 2∆ t + Q i . (A.3) \nHere the functions have been discretised over the grid with spatial and temporal indices given by i and n , and the grid spacing is ∆ t = t n +1 -t n . The α coefficients are generalised coefficients containing some combination of the diffusion and energy loss functions and grid spacings which will be defined below. The form of Equation (A.3) shows the use of both explicit ( n +1 terms) and implicit ( n terms) differencing techniques. This equation is also commonly written in the form \n-α 1 2 ψ n +1 i -1 + ( 1 + α 2 2 ) ψ n +1 i -α 3 2 ψ n +1 i +1 = Q i ∆ t + α 1 2 ψ n i -1 + ( 1 -α 2 2 ) ψ n i + α 3 2 ψ n i +1 , (A.4) \nwhere implicit and explicit terms have been grouped to yield the overall updating equation \nAψ n +1 = Bψ n + Q. (A.5) \nThe matrices in this updating equation have tri-diagonal forms which contain the α -coefficients. The tri-diagonal nature of these matrices is an important factor in the computation of solutions, as specialised algorithms have been developed that perform the calculation more efficiently than fully populated matrices. However, when generalising this scheme to multiple dimensions, the tri-diagonality of the matrices is lost. The OS method is thus used to retain the benefit of the tri-diagonal matrix structure. Representing the diffusion-loss equation as \n∂ψ ∂t = L ψ, (A.6) \nwhere L ψ is a linear combination of independent operators, then each linearly independent operator can be applied in turn during an iterative solution algorithm, as long as each operator has a valid finite differencing scheme in the absence of all others. In this case, the operators for the spatial and energy dimensions in Equation (A.2) can be split and written as \nand \nL r = 10 -2˜ r (10 ln(10) r 0 ) 2 ( 2 D ∂ ∂ ˜ r + ∂D ∂ ˜ r ∂ ∂ ˜ r + D ∂ 2 ∂ ˜ r 2 ) (A.7) \nL E = ∂b ∂E + b ∂ ∂E . (A.8) \nRepresenting the finite differencing schemes of these operators symbolically as Ψ r and Ψ E , we obtain the following \nL r → Ψ r = 10 -2˜ r i (10 ln(10) r 0 ) 2 [ ψ i +1 -ψ i -1 2∆˜ r ( ln(10)2 D + ∂D ∂ ˜ r ) + ψ i +1 -2 ψ i + ψ i -1 ∆˜ r 2 D ] (A.9) \nand \nL E → Ψ E = 10 -˜ E j ( E 0 ln(10)) [ bψ j +1 -bψ j ∆ ˜ E ] . (A.10) \nWe have only considered the upstream case for the energy dimension, as in GALPROP and [26], under the assumption that electrons only lose energy. In the above schemes the ∆ quantities have their usual meanings and the subscripts i and j have been reserved for the spatial and energy grids, respectively. The overall updating scheme for the points n to n +1, using the OS method, can then be expressed symbolically as \nψ n +1 / 2 = Ψ r ( ψ n , ∆ t/ 2) ψ n +1 = Ψ E ( ψ n +1 / 2 , ∆ t/ 2) . (A.11) \nThe computation of Equation (A.11) requires the solution of large matrix equations at each iteration, which can draw significant compute power, especially in the case of high-resolution grids. To mitigate these computational requirements, we describe below a technique to reduce the number of iterations and method calls necessary for the algorithm to run, significantly speeding up the runtime of the code. Now, the discretised 2-dimensional electron distribution ψ i,j is represented by a 2-dimensional array in the code \nof form ψ ∈ R I × J , where the 2 spatial indices i, j run from 0 to I, J , respectively. This function is flattened to a single column vector in the operation ψ ∈ R I × J → ψ ∈ R IJ × 1 . There are two ways to perform this operation, and in each dimension (˜ r or ˜ E ) the flattened vector will have blocks of sub-matrices that are composed solely of function values that are independent of the alternate dimension. For example, when performing the ˜ r step in the algorithm, the first I elements in the column vector will compose one block of values that are independent of ˜ E . Consequently, each of the matrices A and B in the general updating system (Equation (A.5)) will then also be composed of an equal number of sub-matrix blocks that act on the independent blocks in ψ . We represent these tridiagonal block matrices as M , and they have the following form (derived from Equation A.4): \nM (˜ r, E j ) = ( 1 + α 2 (˜ r 0 , ˜ E j ) 2 ) -α 3 (˜ r 0 , ˜ E j ) 2 -α 1 (˜ r 1 , ˜ E j ) 2 ( 1 + α 2 (˜ r 1 , ˜ E j ) 2 ) . . . ( 1 + α 2 (˜ r I -1 , ˜ E j ) 2 ) -α 3 (˜ r I -1 , ˜ E j ) 2 -α 1 (˜ r I , ˜ E j ) 2 ( 1 + α 2 (˜ r I , ˜ E j ) 2 ) (A.12) \nand similarly for M (˜ r i , ˜ E ). With this definition, the full A matrices in each step of the algorithm then resemble: \nA ˜ r = M (˜ r, ˜ E 0 ) 0 M (˜ r, ˜ E 1 ) . . . M (˜ r, ˜ E J -1 ) 0 M (˜ r, ˜ E J ) (A.13) \nfor the spatial dimension and \nA ˜ E = M (˜ r 0 , ˜ E ) 0 M (˜ r 1 , ˜ E ) . . . M (˜ r I -1 , ˜ E ) 0 M (˜ r I , ˜ E ) (A.14) \nfor the energy dimension (the B matrices are constructed similarly). The total size of each matrix is then the total size of each block matrix times the number of block matrices, for example M (˜ r, E j ) ∈ R I × I and A ˜ r ∈ R IJ × IJ . Each half-step of Equation (A.5) can now be solved with a single matrix equation solution function call, instead of I or J times. \nThe forms of the matrices in Equation (A.13) and (A.14) are clearly extremely sparse. In most practical scenarios, the sparsity index (calculated by S = ( N -N nz ) /N × 100% , where N and N nz are the total number and number of non-zero elements in the matrix) is > 99 . 9%. Thus, when working with these matrices in the code we have made use of the scipy.sparse Python module, which stores the above matrices in various memory-efficient storage configurations. In particular, the matrices are stored in the 'csr' (compressedsparse-row) storage format, which allows for fast array vector operations. Since the \nsolution of the matrix equation is required at every iteration of the algorithm, the efficiency of the overall numerical solution is directly linked to the efficiency of this library's sparse matrix implementation. \nThe forms of the discretised operators that appear in the OS scheme in Equation (A.11) are now accessible through the method of equating coefficients, and we find: \nΨ ˜ r : α 1 ∆ t = C ˜ r ( -ln(10) D (˜ r i , E j ) + ∂D ∂ ˜ r ∣ ∣ ˜ r i ,E j 2∆˜ r + D (˜ r i , E j ) ∆˜ r 2 ) α 2 ∆ t = C ˜ r ( 2 D (˜ r i , E j ) ∆˜ r 2 ) α 3 ∆ t = C ˜ r ( ln(10) D (˜ r i , E j ) + ∂D ∂ ˜ r ∣ ∣ ˜ r i ,E j 2∆˜ r + D (˜ r i , E j ) ∆˜ r 2 ) (A.15) \nfor the spatial dimension and \nΨ E : α 1 ∆ t = 0 α 2 ∆ t = C ˜ E b j ∆ E α 3 ∆ t = C ˜ E b j +1 ∆ E (A.16) \nfor the energy, where the coefficients are defined by C ˜ r = 10 -2˜ r i (ln(10) r 0 ) 2 and C ˜ E = 10 -˜ E j ln(10) E 0 . ˜ ) as the initial condition, and we apply \nWe use the DM electron source function Q (˜ r, E the following Dirichlet and Neumann boundary conditions: \nψ (˜ r = ˜ r max ) = 0 , ∂ψ ∂ ˜ r (˜ r = ˜ r min ) = 0 . (A.17) \nThe Dirichlet condition corresponds to the free-streaming of the electrons out of the halo at the maximum diffusion radius, and is enforced simply by setting all values of ψ at ˜ r = ˜ r max to 0 at each iteration of the OS method. The Neumann condition is satisfied by the calculation of new α -coefficients for the points ˜ r = ˜ r min , which represents the smallest sampled radius in the spatial grid. Analogously to Equation (A.15), we find \nΨ ˜ r =˜ r min : α 1 ∆ t = 0 α 2 ∆ t = C ˜ r ( 4 D (˜ r i , ˜ E j ) ∆˜ r 2 ) α 3 ∆ t = C ˜ r ( 4 D (˜ r i , ˜ E j ) ∆˜ r 2 ) (A.18) \nAs described in Section 3, the values of ˜ r max and ˜ r min are configurable in the input parameter file. \nFinally, we discuss the conditions which determine the continuation and eventual termination of the algorithm to find the equilibrium electron distribution. These conditions are closely related to the various physical timescales that are involved in the problem, which we define as follows. The timescales of energy-loss and diffusion are τ E = ˜ E/b ( ˜ E ) and τ D = ∆˜ r 2 /D (˜ r, ˜ E ) respectively, and the timescale of changes to the ψ distribution are τ ψ = ψ/ ∣ ∣ ∣ ( ∂ψ ∂t )∣ ∣ ∣ which we estimate using a simple first-order Forward-Time difference. We have implemented two separate methods to advance each step of the updating algorithm given in Equation (A.11), each inspired by the techniques developed in the GALPROP package (see [49] or the Explanatory Supplement 7 ). \nIn the first method, called the 'accelerated' (ACC) method, the initial timestep ∆ t i is chosen to be larger than the timescales of both energy-loss and diffusion. Then, after a set number of iterations, ∆ t is reduced by a factor of 2, after which the entire process repeats until ∆ t reaches a minimum value. This timestep-changing technique allows for all relevant physical scales to be included in the final solution, while reducing the total number of iterations required to find an accurate solution. The motivation for this method arises from considering a variety of astrophysical structures and scenarios, which could have a wide range of timescales related to energy-loss or diffusion impacting the evolution of the electron distribution. If the timestep ∆ t is small enough to capture the short timescale effects, converging on the final equilibrium distribution could require a huge number of iterations. Conversely, while needing only relatively few iterations, large timesteps can lead to inaccurate final solutions as the impact from short-timescale processes are lost during each iteration. If the variables governing the ACC method (through the parameters os\\_delta\\_ti , os\\_max\\_steps and os\\_delta\\_t\\_min ) are chosen appropriately, the dynamic nature of the timestep value solves both of the above issues during a typical run, and it is thus the recommended method when using DarkMatters . \nThe other possible method has no cap on the number of iterations, and has a constant step size (CSS) for the timestep ∆ t . In this method a tolerance T can be set for the relative difference between the electron distribution at subsequent iterations, and if the relative difference is below this tolerance, i.e. ∣ ∣ 1 -ψ n -1 /ψ n ∣ ∣ < T , then the algorithm is considered to be stable. This condition ensures that the changes in the distribution over all the grid points (which are initially large) approach zero instead of diverging or oscillating due to potential numerical effects. Because of the range of timescales discussed above, this method typically takes much longer to find the solution as the ACC method. However, we have also utilised this method to perform a set of benchmarking tests that have strict criteria for the final convergence of the algorithm to the equilibrium distribution. In these tests, we use a very small timestep (∆ t < min { τ E , τ D } ) with no cap on the iteration number. Then, similarly to the tests performed in GALPROP , we signal convergence with the following condition: \n∂ψ ∂t = 0 ⇒ ψ n -1 -ψ n ∆ t = 0 , (A.19) \nwhere Equation (A.19) must hold for all points in the electron distribution. While ensuring the maximum level of accuracy in the final solution, these runs are most useful as a comparative tool, as the number of iterations required to satisfy the above condition is usually prohibitive for practical purposes. As the relative difference in the electron distributions between iterations also tends to rapidly diminish at large iteration numbers, these final steps can in general be avoided without significant risk of a loss of accuracy. \nOutside of benchmarking and accuracy tests, the general condition for convergence to the equilibrium distribution is set by \nτ ψ > max { τ E , τ D } , (A.20) \nwhich ensures that the timescale of changes to the electron distribution is larger than the timescale of any physical effects involved.", 'References': "- [1] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, R. Battye, K. Benabed, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, J.-F. Cardoso, J. Carron, A. Challinor, H. C. Chiang, J. Chluba, L. P. L. Colombo, C. Combet, D. Contreras, B. P. 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2024ApJ...975L..14C	Astrophysical explosions that contain dense and rampressuredominated ejecta evolve through an interaction phase during which a forward shock FS contact discontinuity CD and reverse shock RS form and expand with time. We describe new selfsimilar solutions that apply to this phase and are most accurate in the limit that the ejecta density is large compared to the ambient density. These solutions predict that the FS CD and RS expand at different rates in time and not as single temporal power laws are valid for explosions driven by steady winds and homologously expanding ejecta and exist when the ambient density profile is a power law with a powerlaw index shallower than 3 specifically when the FS does not accelerate. We find excellent agreement between the predictions of these solutions and hydrodynamical simulations both for the temporal behavior of the discontinuities and for the variation of the fluid quantities. The selfsimilar solutions are applicable to a wide range of astrophysical phenomena andalthough the details are described in future workcan be generalized to incorporate relativistic speeds with arbitrary Lorentz factors. We suggest that these solutions accurately interpolate between the initial coasting phase of the explosion and the later energyconserving phase or if the ejecta is homologous and the density profile is sufficiently steep the selfsimilar phase described in R. A. Chevalier.	2024-11-01T00:00:00Z	['arXiv:2409.10600', '10.3847/2041-8213/ad87cc', '2024arXiv240910600C', '2024ApJ...975L..14C', '10.48550/arXiv.2409.10600']	['Analytical mathematics', 'Core-collapse supernovae', 'Transient sources', 'Hydrodynamics', 'Shocks', '38', '304', '1851', '1963', '2086', 'Astrophysics - High Energy Astrophysical Phenomena']	From Coasting to Energyconserving New Selfsimilar Solutions to the Interaction Phase of Strong Explosions	2,024	191	0.56	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.10600.pdf	{'From coasting to energy-conserving: new self-similar solutions to the interaction phase of strong explosions': 'Eric R. Coughlin 1 \n1 Department of Physics, Syracuse University, Syracuse, NY 13210, USA', 'ABSTRACT': "Astrophysical explosions that contain dense and ram-pressure-dominated ejecta evolve through an interaction phase, during which a forward shock (FS), contact discontinuity (CD), and reverse shock (RS) form and expand with time. We describe new self-similar solutions that apply to this phase and are most accurate in the limit that the ejecta density is large compared to the ambient density. These solutions predict that the FS, CD, and RS expand at different rates in time and not as single temporal power-laws, are valid for explosions driven by steady winds and homologously expanding ejecta, and exist when the ambient density profile is a power-law with power-law index shallower than ∼ 3 (specifically when the FS does not accelerate). We find excellent agreement between the predictions of these solutions and hydrodynamical simulations, both for the temporal behavior of the discontinuities and for the variation of the fluid quantities. The self-similar solutions are applicable to a wide range of astrophysical phenomena and - although the details are described in future work - can be generalized to incorporate relativistic speeds with arbitrary Lorentz factors. We suggest that these solutions accurately interpolate between the initial 'coasting' phase of the explosion and the later, energy-conserving phase (or, if the ejecta is homologous and the density profile is sufficiently steep, the self-similar phase described in Chevalier 1982). \nKeywords: Analytical mathematics (38); Core-collapse supernovae (304); Hydrodynamics (1963); Shocks (2086); Transient sources (1851)", '1. INTRODUCTION': "The formation of a shockwave is an inevitable consequence of the injection of energy into a cold medium, making them ubiquitous in astrophysical settings. The dissipation of bulk kinetic energy at a shockwave occurs at the expense of the production of internal energy and radiation, making them not just ubiquitous but signposts of high-energy phenomena, including gammaray bursts (e.g., Rees & Meszaros 1992; M'esz'aros & Rees 1997; Sari et al. 1998), core-collapse (e.g., Woosley et al. 2002; Khatami & Kasen 2023) and type-Ia (e.g., Chevalier 1988; Hillebrandt & Niemeyer 2000) supernovae, colliding-wind binaries (e.g., Stevens et al. 1992; Abaroa et al. 2023), galactic (e.g., Holzer & Axford 1970; Thompson & Heckman 2024), stellar (e.g., Castor et al. 1975; Weaver et al. 1977), and AGN outflows (e.g., Begelman et al. 1984; Begelman & Cioffi 1989), and tidal \[email protected] \ndisruption events (TDEs; e.g., Rees 1988; Gezari 2021; Alexander et al. 2016). \nWhile an explosion and the outward motion of gas must initially be powered by some other (i.e., nonkinetic) energy reservoir, the thermal content of the expanding gas generally declines adiabatically and more rapidly than the kinetic energy, such that the 'ejecta' of an explosion becomes cold and kinetically dominated soon after it is initiated. If, however, there is a medium surrounding the object at the time of the explosion, then the interaction between the expanding ejecta and that medium provides an additional source of dissipation and corresponding emission. This type of interaction is thought to power the late-time rebrightenings of some supernovae (i.e., type-IIn; e.g., Chevalier & Fransson 1994; Sollerman et al. 2020), is the most widely accepted mechanism for generating the afterglow associated with GRBs (e.g., Kumar & Zhang 2015), and can generally supplement - and conceivably dominate - the overall energetics of highly energetic, cosmic explosions (e.g., \nDong et al. 2016; Arcavi et al. 2017; Andrews & Smith 2018; Sollerman et al. 2022). \nDuring this 'interaction phase,' the collision between the ambient gas and the ejecta creates a forward shock (FS) with radius R s that advances into the ambient medium, a reverse shock (RS) with radius R r that propagates into the ejecta, and a contact discontinuity (CD) with radius R c that separates the two shocked fluids (see Figure 1 of Khatami & Kasen 2023 for an illustration). Once the ejecta density becomes comparable to the ambient density (equivalent, in order of magnitude, to the condition that the swept-up mass be comparable to the initial explosion mass; e.g., Ostriker & McKee 1988), we expect the FS to be set by the (potentially time-dependent, if a steady wind drives the outflow; e.g., Weaver et al. 1977) explosion energy, at which point the blastwave enters the self-similar Sedov-Taylor phase (Sedov 1959; Taylor 1950). Chevalier (1982) showed that, when the ejecta is homologously expanding, there are also self-similar solutions that describe the initial interaction phase, and these solutions have time-independent ratios of R r /R c and R s /R c . They also require a constant and specific ratio of the ejecta density, ρ ej , and the ambient density, ρ a , and are only attained if the (initial) ejecta density falls off as a power-law steeper than ∝ r -5 . The origin of the latter constraint is that it is only for these steep power-law profiles that the kinetic energy of the ejecta is infinite 1 ; flatter profiles have a finite energy and must be governed by the conservation of that energy at sufficiently late times. \nHere we show that there are distinct self-similar solutions that describe the interaction phase, which are valid when the density of the ejecta, ρ ej , exceeds the density of the ambient medium, ρ a , and are most accurate when ρ ej ≫ ρ a . During this phase the forward shock 'coasts' at a fixed value of the ejecta velocity, meaning that the self-similar FS shell thickness, ∆ s = R s /R c -1, is constant in time. The thickness of the reverse shock, ∆ r = 1 -R r /R c , scales as √ ρ a /ρ ej and is therefore generally time-dependent, implying that the self-similar solutions possess different rates of expansion of the FS, CD, and RS. Our solutions are valid while ∆ r ≲ ∆ s , after which we expect the solution to transition either to Chevalier (1982)'s if the ejecta density profile is sufficiently steep, or to the Sedov-Taylor solution when it is less so (or that of Weaver et al. 1977 for wind-powered explosions), and therefore interpolate between the coasting and energy-conserving phases of a strong explosion. \nIn Section 2 we provide motivation and physical arguments for the existence and properties of the self-similar solutions, and in Section 3 we derive the solutions and provide comparisons to numerical simulations. We discuss various aspects of the solutions and analyze applications in Section 4 before briefly summarizing in Section 5.", '2. BASIC CONSIDERATIONS': "Let a spherically symmetric and kinetic-energydominated outflow (the 'ejecta') impinge upon a static and cold ambient medium. At the moment that the impact occurs, we let the ejecta have density ρ ej and velocity V ej at the radius of impact, and the ambient medium have density ρ a (also at that radius). Then the initial velocities of the FS, RS, and CD from the jump conditions are, respectively, \nV s = γ +1 2 1 1 + √ ρ a /ρ ej V ej , V r = 1 -γ -1 2 √ ρ a /ρ ej 1 + √ ρ a /ρ ej V ej , V c = 1 1 + √ ρ a /ρ ej V ej . (1) \nHere γ is the adiabatic index of the gas (we are assuming that both the shocked ejecta and ambient fluids are characterized by the same adiabatic index, though this assumption could be trivially relaxed). \nFollowing the initial interaction, the RS, CD, and FS spatially separate from one another owing to their different velocities, and the shocked fluid variables at the RS and the FS will depend on the ejecta and ambient properties at those respective radii, i.e., the spatial variation of the ejecta and ambient fluids is relevant in addition to their values characterizing the initial impact. The fluid variables at the FS and RS that arise from the jump conditions are then: \nv ( R s ) = 2 γ +1 V s , v ( R r ) = 2 γ +1 V r + γ -1 γ +1 v ej ( R r ) , ρ ( R s ) = γ +1 γ -1 ρ a ( R s ) , ρ ( R r ) = γ +1 γ -1 ρ ej ( R r ) p ( R s ) = 2 γ +1 ρ a ( R s ) V 2 s , p ( R r ) = 2 γ +1 ρ ej ( R r ) ( v ej ( R r ) -V r ) 2 . (2) \nHere v is the fluid velocity and p is the pressure, and we have incorporated the spatial dependence on the ejecta and ambient densities by explicitly writing the radii at which they are evaluated. We have also written the ejecta velocity as v ej , i.e., v ej ( r, t ) refers to the \nspatially and temporally varying velocity profile of the inner ejecta ( V ej , being the velocity that the leading edge of the ejecta would have if the ratio ρ a /ρ ej ≡ 0, has distinct physical significance and will be relevant, and we therefore do not re-use this variable). \nWhile Equation (1) is only strictly valid for the initial interaction between cold ejecta impinging on cold ambient gas, it shows that the RS-CD-FS structure expands (or, if the ambient density profile is sufficiently steep, contracts) as a function of the ratio ρ a /ρ ej . If R s , R c , and R r vary in such a way that ρ a ( R s ) /ρ ej ( R r ) is timeindependent, Chevalier (1982) showed that self-similar solutions can be found that satisfy R r /R c ∝ R s /R c ∼ const. and R c ∝ t α with α a constant. But in general - and for ejecta density profiles that do not initially conform to steep power-laws in radius - such solutions cannot be found, and the RS, CD, and FS propagate at different speeds. \nThere is, however, another scenario in which the interaction between the ejecta and ambient gas simplifies and for which approximate self-similar solutions could plausibly exist: if ρ a /ρ ej ≪ 1, Equation (1) shows V r = V ej ( 1 + O [ √ ρ a /ρ ej ]) and V c = V ej ( 1 + O [ √ ρ a /ρ ej ]) , such that as ρ a /ρ ej → 0, the RS 'piles up' into the CD and the velocity is continuous to the CD; this behavior is reasonable, because when ρ ej /ρ a ≫ 1, we would expect the ejecta to expand uninhibited by the ambient medium. In this same limit, the velocity of the ejecta is unaltered by the interaction, meaning that the velocity at the RS satisfies v ej ( R r ) = V ej and is given by its initial value, i.e., V r = V c = V ej for all t > 0. In this 'coasting' regime (e.g., Ostriker & McKee 1988) during which the CD has a constant speed, the FS must have V s ∝ V ej = const. to not be overtaken by the CD (i.e., it cannot decelerate, and we are assuming it does not accelerate; see Section 4 for a description of where the latter breaks down) and ∆ s ≡ R s /R c -1 = const. Self-similar solutions would then yield the values of ∆ s and V s /V ej . \nAdditionally, the difference between the velocity of the CD, V c , and the velocity at the leading edge of the ejecta in the limit that ρ a /ρ ej ≡ 0, V ej , constitutes a smallness parameter in this regime, and from Equation (1) scales as ∝ √ ρ a /ρ ej ≡ δ . As the ejecta expands, this ratio can be evaluated at the CD by extrapolating the inner and outer density profiles, and δ will therefore depend on time. The same smallness parameter characterizes the difference between V r and V c in this limit, such that the relative shell thickness between the CD and the RS, ∆ r = 1 -R r /R c , scales as δ . To satisfy the jump conditions across the RS and be continuous across the \nCD, the pressure within the RS shell must then satisfy p ∝ ρ ej V 2 c ∆ 2 r . Because of the time dependence contained in the shell thickness, exact self-similar solutions for the RS during this phase likely do not exist 2 , but these observations suggest that approximate self-similar solutions could be found for which the fluid variables are written as power series in the small quantity δ . \nWe derive these self-similar solutions in the next section. The zeroth-order (in δ ) FS solutions (Section 3.1) are contained in the Sedov space and simply have zero acceleration, and depend on neither the density nor the velocity profile of the inner ejecta. Despite being fairly trivial, we have not seen these solutions discussed in this context or this generally 3 . The RS solutions (Section 3.2) are constrained simultaneously by the (known, ∝ δ ) expansion rate of the shell ∆ r and the continuity of the velocity and pressure at the CD. The latter are satisfied for special deceleration parameters that act as eigenvalues, but instead of permitting the smooth passage of the fluid variables through a sonic point (as is typically the case for type-II similarity solutions; Zel'dovich & Raizer 1967), they satisfy boundary conditions at the CD. The solutions apply to both wind-driven and homologouslyexpanding-ejecta-driven explosions, both of which we analyze. Finally, we construct first-order 'corrections' to the FS and shocked fluid (Section 3.3), which describe the response of the FS to finite δ ; analogous to the RS, these require a precise growth rate for ∆ s (alongside the constant and self-similar width) that serves to satisfy continuity conditions at the CD. \nBefore constructing and analyzing the self-similar solutions, we establish and define the relevant radii and the physical setup that will be used throughout the remainder of the paper. Three time-dependent radii that were already introduced are the RS radius R r ( t ), the CD radius R c ( t ), and the FS radius R s ( t ). From these we construct the relative FS shell thickness, ∆ s = R s /R c -1, and the relative RS shell thickness, ∆ r = 1 -R r /R c , \nboth of which are time-dependent. There is a fourth radius, R ej ( t ), that is the radius of the leading edge of the ejecta in the limit that the ratio of the ambient to ejecta density is identically zero (which would coincide with the radius of the CD and the RS in the same limit), which we write as \nR ej ( t ) = R i ( 1 + V ej t R i ) . (3) \nThe radius R i is the radius of R ej at t = 0 and V ej is the (constant) velocity of the leading edge of the ejecta, which was also introduced above. R i is distinguished from (and is not equal to) the radius of the CD at t = 0, because all three discontinuities and the ejecta radius are coincident at the origin at time t = -R i /V ej . The inner ejecta (i.e., interior to the RS) is cold and expanding ballistically, such that the velocity profile is conserved in a Lagrangian sense, i.e., each fluid element preserves its initial velocity. The two types of outflow we consider are homologously expanding ejecta and a constant-velocity wind, such that the Eulerian velocity at the RS is \nv ej ( R r ) = V ej R r ( t ) R ej ( t ) , homologous expansion V ej , time-steady wind . (4) \nIn the case of a wind, time steadiness demands that the ejecta density fall off with Eulerian radius as ∝ r -2 . The spatial dependence of homologously expanding ejecta can be more general, e.g., Chevalier (1982) considered the case where the ejecta has a power-law density profile. However, in the limit that we are analyzing where the ejecta is nearly unaltered by the ambient gas, it is only the time dependence near the RS (which, in turn, is nearly equal to R c and R ej ) that enters, such that ρ ej ∝ r -3 to leading order in δ . We therefore have \nρ ej ( R r ) = ρ ej ( R r ( t ) R i ) -3 , homologous expansion ρ ej ( R r ( t ) R i ) -2 , time-steady wind ≡ ρ ej ( R r ( t ) R i ) -m . (5) \nFinally, we will let the ambient density profile conform to a power-law in radius, such that the ambient density evaluated at the FS is \nρ a ( R s ) = ρ a ( R s ( t ) R i ) -n . (6) \nIn Equations (5) and (6) we re-used ρ ej and ρ a to refer to the ejecta and ambient densities measured at R i for \nsimplicity of notation; in all of what follows, any appearance of ρ ej and ρ a will refer to these normalizations, and any explicit time dependence will be included. \nFigure 1 illustrates the four radii in the problem as well as the various regions, including the unshocked ejecta, shocked ejecta, shocked ambient gas, and ambient gas. The inset on the top-right illustrates the variation of the density throughout the entire shell, as well as the temporal decline of the ejecta density with time (as ∝ R -m r ), and the spatial decline of the ambient density (as ∝ r -n ).", '3.1. Forward shock': 'We assume that the ratio of the ambient to ejecta density is sufficiently small that V c ≃ V ej . We define the fluid and self-similar variables as \nv = V c f s ( η ) , ρ = ρ a ( R c R i ) -n g s ( η ) , p = ρ a V 2 c ( R c R i ) -n h s ( η ) , η = r/R c -1 R s /R c -1 ≡ r/R c -1 ∆ s , (7) \nwhere 0 ≤ η ≤ 1 and ∆ s a constant. Then the fluid equations in spherical symmetry, \n∂ρ ∂t + 1 r 2 ∂ ∂r [ r 2 ρv ] = 0 , ∂v ∂t + v ∂v ∂r + 1 ρ ∂p ∂r = 0 , ∂ ∂t ln ( p ρ γ ) + v ∂ ∂r ln ( p ρ γ ) = 0 , (8) \nrespectively become \n-ng s -1 ∆ s (1 + ∆ s η ) dg s dη + 1 ∆ s (1 + ∆ s η ) 2 d dη [ (1 + ∆ s η ) 2 f s g s ] = 0 , (9) \n-(1 + ∆ s η -f s ) df s dη + 1 g s dh s dη = 0 , (10) \nn ( γ -1) -1 ∆ s (1 + ∆ s η -f s ) d dη ln ( h s g γ s ) = 0 , (11) \n0. 0.2 Figure 1. A diagram of the relevant radii, being the reverse shock (RS), the contact discontinuity (CD), and the forward shock (FS), as well as the ejecta radius R ej ; the latter is the radius that the CD and RS would have in the limit that the ratio of the ambient to ejecta density were identically zero. The top-right inset shows how the ejecta density declines with time as R -m r , where m = 2 for a steady wind and m = 3 for homologously expanding ejecta, as well as the power-law decline of the ambient density as ∝ r -n , and the overall variation of the density throughout the shocked FS and RS shells. The inset on the bottom-right shows a zoom-in on the RS-CD-FS structure, as well as the locations of the three discontinuities in the top-right inset. \n<!-- image --> \nwhile the boundary conditions at the shock, from Equation (2), are \nf s (1) = 2 γ +1 (1 + ∆ s ) , g s (1) = γ +1 γ -1 (1 + ∆ s ) -n , h s (1) = 2 γ +1 (1 + ∆ s ) 2 -n , (12) \nand that at the CD is \nf s (0) = 1 . (13) \nEquations (9) - (11) are solved by integrating from the shock ( η = 1) and iterating on ∆ s until Equation (13) is satisfied 4 .', '3.2. Reverse shock': 'From our considerations in Section 2, we expect the radius of the CD to vary as \nR c = R ej ( t ) { 1 -κ c δ ( R c ) } , (14) \nwhere κ c is an unknown constant and \nδ ( R c ) = √ ρ a ρ ej ( R c ( t ) R i ) m -n 2 . (15) \nWe define the RS radius R r and self-similar variable as \nR r = R c (1 -∆ r ) ≡ R c { 1 -κ r δ ( τ ) } , η = r/R c -1 1 -R r /R c = r/R c -1 ∆ r , (16) \nwhere κ r is another unknown parameter and 5 -1 ≤ η ≤ 0. Because ∆ r is time-dependent, the usual self-similar \nparameterization of the fluid variables (e.g., that used for the FS) will not lead to self-consistent equations for f r , g r , and h r . However, and again from the physical considerations in Section 2, we expect the following to hold to leading order in ∆ r : \nv = V c { 1 + ∆ r f r ( η ) } , ρ = ρ ej ( R c R i ) -m g r ( η ) p = ρ ej ( R c R i ) -m V 2 c ∆ 2 r h r ( η ) . (17) \nThe self-similar equations are derived by keeping lowestorder terms in ∆ r in the fluid equations and using the definitions of R c , R r , and ∆ r in terms of δ , and are \n-m ) g r -( 1 + m -n 2 ) η dg r dη + d dη [ f r g r ] = 0 , \nmγ -n + ( f r -η -m -n 2 η ) d dη ln ( h r g γ r ) = 0 . \n(2 (18) -κ c κ r ( m -n 2 )( 1 + m -n 2 ) + m -n 2 f r + ( f r -η -m -n 2 η ) df r dη + 1 g r dh r dη = 0 , (19) (20) \nThe boundary conditions at the shock can be determined from the jump conditions (2) and are \nf r ( -1) = 2 γ +1 ( 1 + m -n 2 )( γ -1 2 κ c κ r -1 ) -γ -1 γ +1 ( 1 + κ c κ r ) ( m -2) , (21) \nh r ( -1) = 2 γ +1 ( 1 + κ c κ r ) 2 ( 1 + m -n 2 -( m -2) ) 2 , \ng r ( -1) = γ +1 . (23) \n(22) γ -1 \nThe factors proportional to ( m -2) account for the additional terms that enter in the case of homologously expanding ejecta ( m = 3) vs. a steady wind ( m = 2), i.e., Equations (21) - (23) are correct for both m = 2 and m = 3. The continuity of the velocity at the CD also requires \nf r (0) = 0 . (24) \nEquations (18) - (24) depend only on the unknown ratio κ c /κ r ; as for the FS shell thickness ∆ s , κ c /κ r acts as an eigenvalue that can be determined iteratively by shooting from the RS until the condition f r (0) = 0 is met. The individual values of κ r and κ c are recovered \nby requiring pressure continuity at the CD: since the pressure at the CD from the forward-shocked material is given by the self-similar parameterization in Section 3.1, we have \nρ ej R -m c V 2 c κ 2 r δ 2 h r (0) = ρ a R -n c V 2 c h s (0) ⇒ κ r = √ h s (0) h r (0) , (25) \nthus closing the system. \nIn imposing Equation (25) we assumed that h r is finite at the CD. In Appendix B we analyze the asymptotic behavior of the solutions near the CD and show that the pressure does indeed converge to a finite value, but the convergence is slow - especially as n approaches 3 for m = 3.', '3.3. Corrections to the forward shock': 'The self-similar solutions for the FS in Section 3.1 set ρ a /ρ ej = 0 and are independent of this parameter. However, the FS will be modified analogously to the CD (i.e., Equation 14) owing to the finite value of ρ a /ρ ej , and there will be modifications to ∆ s that scale as δ . We account for these effects by letting f s ( η ) → f s ( η ) + δ ( τ ) f 1 ( η ), and similarly for the functions g s and h s , in Equation (7), and letting \n∆ s ( τ ) → ∆ s + κ s δ ( τ ) , (26) \nwith κ s an unknown parameter. \nWith this parameterization, the boundary conditions at the FS (read off from Equation 2) are \nf 1 (1) = 2 γ +1 ( 1 + m -n 2 ) κ s , g 1 (1) = -γ +1 γ -1 (1 + ∆ s ) -n -1 nκ s , h 1 (1) = 2 γ +1 (1 + ∆ s ) 1 -n ( m +2 -2 n ) κ s , (27) \nwhile the continuity of the velocity at the CD demands \nf 1 (0) = 0 . (28) \nLinearizing the fluid equations in δ yields the equations for f 1 , g 1 , and h 1 ; they are lengthy, and to preserve readability, we put them in Appendix A. The solution for κ s is found, as for κ c /κ r , by shooting from the FS until f 1 (0) = 0.', '3.4. Total solutions': "Figure 2 shows the self-similar solutions for m = 2 (ejecta in the form of a time-steady wind; left panels) \nFigure 2. The self-similar density (top), velocity (middle), and pressure (bottom) for a wind-driven explosion ( m = 2, left) and that driven by homologous ejecta ( m = 3, right) for the values of the ambient density power-law index n shown in the legend. In all cases we set γ = 5 / 3. \n<!-- image --> \nand m = 3 (homologously expanding ejecta; right panels) for the n in the legend of the top panels and γ = 5 / 3; the solutions are qualitatively similar for γ = 4 / 3. The density (top panels) of the shocked ejecta increases and diverges at the CD for all values of n , such that most of the mass is concentrated near the CD. The velocity profile is roughly linear throughout the shell, while the pressure displays nontrivial variation throughout the RS shell and is approximately uniform for the FS. Figure 3 \nis analogous to Figure 2, but illustrates the corrections to the FS fluid variables. Table 1 provides values of ∆ s and the various κ 's for both values of m and a range of n . \nThe solid, orange curves in the top-left, top-right, and bottom-left panels of Figure 4 give the density, velocity, and pressure, respectively, from a hydrodynamical simulation run with flash (Fryxell et al. 2000), while the bottom-right panel shows the solution for the RS-shell \nFigure 3. Analogous to Figure 2, but here we are plotting the corrections to the FS density g 1 , velocity f 1 , and pressure h 1 . \n<!-- image --> \nwidth (i.e., 1 -R r /R c ) as a function of the position of the CD. We initialized the simulation with a constantvelocity, cold wind, such that v = V ej = 1 and ρ = r -2 for r < 1, and v = 0 and ρ = 10 -4 ( n = 0) for r > 1, i.e., δ ( r = 1) = 10 -2 . The parameters controlling numerical accuracy (e.g., interpolation order) are identical to those described in Paradiso et al. (2024), and we used a resolution of ∆ r = 7 . 64 × 10 -5 . The dashed curves are the predictions from the self-similar solutions. In all cases the agreement is extremely good. Figure 5 is the analogous set of plots for a homologously expanding shell ( m = 3) \nimpacting a wind-fed medium ( n = 2), where here the initial density ratio was set to ρ a /ρ ej = 4 × 10 -4 and the resolution ∆ r = 10 -3 . For this setup the ejecta density profile was flat (independent of radius) and the initial velocity profile was v = V ej r/R ej , with V ej = R ej = 1.", '4. DISCUSSION': 'Here we discuss various aspects of the self-similar solutions. \nTable 1. ∆ s , κ r , κ c , and κ s for a time-steady wind ( m = 2, left column) and homologous ejecta ( m = 3, right column) for various ambient density power-law indices n . \nFigure 4. The density (top-left), velocity (top-right), and pressure (bottom-left) profiles for a wind-driven explosion ( m = 2, left) in a constant-density ( n = 0) ambient medium and γ = 5 / 3. The black-dashed curves give the predictions from the self-similar solution, while the orange curves are from a flash hydrodynamical simulation. The bottom-right panel shows the RS-shell width, ∆ r , as a function of the radius of the CD, both from flash (orange) and as predicted by the self-similar solution (black-dashed). \n<!-- image --> \nFigure 5. Same as Figure 4 but for homologously expanding ejecta ( m = 3) impacting a wind-like medium ( n = 2). \n<!-- image -->', '4.1. Duration of the self-similar phase and limiting values of n': 'As for any self-similar solution, those described here do not account for initial conditions. In the idealized scenario of cold ejecta impinging on a cold medium, the initial velocities of the discontinuities are given in Equation (1), which allows us to estimate the time taken for the self-similar stage to be reached: since the initial growth rate of the FS shell is d ∆ /dt = ( γ -1) / ( γ +1) × V c /R c , the self-similar solutions will be approximately valid once ∆ ≃ ∆ s , or in a time of \nt ss , begin ≃ R i V ej e ∆ s γ +1 γ -1 . (29) \nFor all values of n , ∆ s ≲ few × 0 . 1 (see Table 1), meaning that the self-similar phase should be reached within only a few dynamical times of the ejecta; this is consistent with the numerical solutions in Figures 4 and 5. We expect the self-similar phase to last until the RS shell ∆ r satisfies ∆ r ≃ ∆ s . With ∆ r = κ r δ , this gives \nt ss , end ≃ R i V ej ( ∆ s κ r ) 2 m -n ( ρ ej ρ a ) 1 m -n . (30) \nThe scaling with the ejecta and ambient density is expected on simple arguments related to the ejecta vs. swept up mass, but since ∆ s /κ r ≳ few , this timescale can be considerably longer for relatively steep ambient density profiles. \nWe expect the FS self-similar solutions to be valid provided that the FS does not accelerate. There are second-type self-similar and accelerating solutions described in Waxman & Shvarts (1993) for n ≥ n acc , where n acc ≃ 3 . 25 for γ = 5 / 3 and n acc ≃ 3 . 12 for γ = 4 / 3. The RS solutions exist when n < 3; as n → 3 the RS shell thickness goes to zero and the velocity is continuous to the CD. For 3 ≤ n ≤ n acc , we expect the type-III solutions given in Gruzinov (2003) to describe the flow.', '4.2. Stability': 'For m < n , the influence of the RS on the FS is decaying with time, meaning that the FS solution asymptotically approaches the self-similar solution. The rate at which it does so is proportional to t ( m -n ) / 2 , provided that other perturbations affecting the solution are both stable and decay at a faster rate than t ( m -n ) / 2 . These additional perturbations are characterized by an eigenvalue that controls the rate at which they decay (for stable solutions) or grow (for unstable solutions), and exist even in the limit that ρ a /ρ ej → 0 and denote, e.g., the corrections to the shock propagation that arise from initial conditions. The equations describing the perturbations can be derived in the same way as those that \narise from finiteδ , except the contribution from the deceleration of the CD is ignored, and the eigenvalue σ is determined by requiring that f 1 (0) = 0. We have investigated these eigenvalues: all of them are stable and generally smaller than ( m -n ) / 2, until the density profile of the ambient medium approaches n acc , at which point the eigenvalue approaches zero. We therefore would expect the solutions in steep density profiles ( n ∼ 3) to be most heavily modified by initial conditions, rather than the RS. \nThe shocked-ejecta density diverges at the CD for all solutions (see Appendix B), seemingly implying that the RS solutions are Rayleigh-Taylor unstable. However, the deceleration of the CD and RS are themselves small in this limit, and if the Rayleigh-Taylor instability is present we expect it to grow as a power-law in R c , i.e., much more weakly than the exponential growth that accompanies a static interface. We defer a detailed investigation of the stability of the RS solutions to future work.', '4.3. Relativistic generalizations': "When the fluid becomes relativistic, the finite speed of light typically implies that exact self-similar solutions cannot be found for arbitrary Lorentz factors. This is because of the inherent time dependence contained in V c /c , and only when the fluid is ultra-relativistic can self-similar solutions be found (Blandford & McKee 1976; Best & Sari 2000; but see Coughlin 2019). However, this is not the case if both the shock speed and the shell thickness are constant, as the Lorentz factor is still purely a function of the self-similar variable if we adopt the same parameterization as in Section 3.1. There are thus self-similar solutions for the FS and FS shell during the coasting phase for arbitrary shock speeds, spanning the non-, mildly, and ultra-relativistic regimes. \nThere are also analogous and relativistic solutions for the RS, but only if the condition Γ 2 c ∆ r ≲ 1 is satisfied, where Γ c = (1 -V 2 c /c 2 ) -1 / 2 . This condition effectively states that the 'natural' shell thickness recognized by Blandford & McKee (1976), R c / Γ 2 c , must be larger than R c ∆ r for the system to be primarily determined by ∆ r . As for the Newtonian solutions, the FS ultra-relativistic generalizations are contained within the BMK space and are obtained trivially by setting d Γ c /dt = 0 in their selfsimilar equations. The RS solutions, however, are not, and are in fact dominated by the rest-mass inertia of the fluid to maintain pressure balance across the CD (see Equation 17). We analyze the relativistic generalizations in future work.", '4.4. Exact and higher-order solutions': 'The self-similar solutions described here are accurate to first order in the reverse-shock shell thickness ∆ r . In general, the time-dependent nature of the shell thickness implies that exact solutions that hold for arbitrarily late times are not possible. The exception to this is if m = n = 2, in which case ∆ r is a constant, and one does not need to maintain only leading-order terms in ∆ r in the fluid equations and maintain the self-similar approximation (this is also why, in Table 1, the corrections to the FS are zero for m = n = 2). While these solutions are interesting from an academic standpoint, the requirement of a very specific ambient power-law index makes them less practical (although a wind being driven into a wind-fed medium seems a likely astrophysical scenario), and we do not analyze them further here. \nIt is also possible to extend the solutions presented here to higher-order in the shell thickness δ simply by adding more terms to the various functions, e.g., let f r → f r + δf 1 in Equation (17). Presumably the resulting expressions would possess a finite radius of convergence in δ , which would signify the transition to the Sedov-Taylor, Weaver et al., or Chevalier self-similar solution as a function of m and n (and γ ), analogous to the results presented in Paradiso et al. (2024) for the case of a shock stalling in a gravitational field.', '4.5. Time-explicit representations and examples': "The self-similar solutions are written in terms of the CD radius R c , as this is the most obvious parameter to use, it makes the self-similar variable η manifestly range from -1 to 1, and it yields simple boundary conditions at the RS, FS, and CD. Nonetheless, from an observational standpoint it is more useful to write the expressions for the FS and RS explicitly in terms of time, which is straightforward because R c is, to leading selfconsistent order in δ , \nR c = R ej ( t ) ( 1 -κ c √ ρ a ρ ej ( R ej ( t ) R i ) m -n 2 ) , (31) \nwhere, again, \nR ej ( t ) = R i ( 1 + V ej t R i ) . (32) \nThen the FS and RS radii are \nR r = R ej ( t ) ( 1 -( κ c + κ r ) √ ρ a ρ ej ( R ej ( t ) R i ) m -n 2 ) , R s = R ej ( t ) (1 + ∆ s ) × ( 1 + ( κ s 1 + ∆ s -κ c )√ ρ a ρ ej ( R ej ( t ) R i ) m -n 2 ) . (33) \nAs an example of the application and implications of these solutions, many TDEs have recently been found to exhibit late-time radio emission (Horesh et al. 2021; Cendes et al. 2024), which can be plausibly attributed to winds and outflows launched from the TDE and interacting with the circumnuclear medium (e.g., Matsumoto & Piran 2021; Hayasaki & Yamazaki 2023; Matsumoto &Piran 2024). If the mass outflow rate is even 1% of the mass supply rate to the SMBH and the wind is launched at ∼ 0 . 1 c from tens of gravitational radii, the outflow density should be far in excess of the CNM density, i.e., ρ ej ≫ ρ a . With an ambient density profile as steep as ∝ r -2 . 5 (Alexander et al. 2020), the RS shell thickness decreases with time (assuming the velocity profile of the ejecta is ∼ const.) as ∆ r ∝ t -1 / 4 , while the FS shell thickness would be ≃ 0 . 25 (assuming γ = 5 / 3, or ∼ 0 . 17 for γ = 4 / 3; see Table 1). As described by Matsumoto & Piran (2024), the density of the CNM could flatten considerably (with n ≃ 0) near the sphere of influence of the SMBH, implying that at this point the RS shell thickness would grow approximately linearly with time. \nAs another example, the fast blue optical transient (FBOT; Drout et al. 2014) CSS-161010 (Coppejans et al. 2020) was interpreted as a 'dirty fireball,' i.e., a highmass and baryon-rich explosion powering a blastwave through a dense circumstellar medium. Because of the large ejecta density, it is possible that the system was even hundreds of days post-explosion - still in the phase described here. If the ambient medium was wind-fed ( n = 2) and the ejecta was homologously expanding ( m = 3), we would expect the CD (where most of the mass is contained) to evolve with time as \nR c = R ej ( t ) ( 1 -0 . 895 √ ρ a ρ ej ( R ej ( t ) R i ) 1 / 2 ) . (34) \nThese solutions should also be applicable to other FBOTs, including AT2018cow (Margutti et al. 2019; Perley et al. 2019) and ZTF18abvkwla (Ho et al. 2020).", '5. SUMMARY': 'We analyzed new self-similar solutions to the interaction phase of both wind-driven and homologously expanding ejecta-driven explosions, for which the forward shock, reverse shock, and contact discontinuity propagate at distinct temporal rates (see Equation 33 and Table 1). While these solutions are most applicable when the density of the ejecta is larger than that of the ambient medium, we expect them to provide reasonable estimates of the blastwave evolution until the Sedov-Taylor stage is reached (or the Weaver et al. 1977 solution if energy is continuously injected in the form of a wind) or, if the ejecta has a steep power-law density profile, \nthe Chevalier 1982 stage where all three discontinuities propagate with the same power-law dependence. These solutions therefore interpolate between the coasting and energy-conserving phases of strong explosions and apply to a wide range of astrophysical phenomena (see those described in Section 1 and 4.5). \n- I thank the anonymous referee for useful comments and suggestions. I acknowledge support from NASA through the Astrophysics Theory Program, grant 80NSSC24K0897.', 'A. EQUATIONS FOR THE CORRECTIONS TO THE FORWARD SHOCK': 'The equations describing the corrections to the FS that arise from finite values of δ can be derived by letting f s ( η ) → f s ( η ) + δf 1 ( η ) etc. in Equation (7), inserting the result into the fluid equations, and keeping first-order terms in δ . The result is \n∆ s m -n 2 g 1 g s -nκ s +( f s -1 -∆ s η ) d dη [ g 1 g s ] + ( f 1 -( 1 + m -n 2 ) κ s η ) d dη ln g s + df 1 dη + 2∆ s 1 + ∆ s η f 1 + 2 κ s (1 + ∆ s η ) 2 f s = 0 , (A1) \n∆ s m -n 2 ( f 1 -κ c ( 1 + m -n 2 )) +( f s -1 -∆ s η ) df 1 dη + ( f 1 -( 1 + m -n 2 ) κ s η ) df s dη + 1 g s dh 1 dη -g 1 g 2 s dh s dη = 0 , (A2) \n∆ s m -n 2 ( h 1 h s -γg 1 g s -2 κ c ( 1 + m -n 2 )) + n ( γ -1) κ s +( f s -1 -∆ s η ) d dη [ h 1 h s -γg 1 g s ] + ( f 1 -( 1 + m -n 2 ) κ s η ) d dη ln ( h s g γ s ) = 0 . (A3) \nThese are solved alongside the boundary conditions (27) - (28).', 'B. BEHAVIOR OF THE REVERSE-SHOCK SOLUTIONS NEAR THE CONTACT DISCONTINUITY': 'The boundary condition needed to determine the deceleration rate of the RS and the CD, Equation (25), necessitates that the pressure remain finite at the CD. To demonstrate that this is the case, we write the leading-order (in η ) expansions of the functions f r , g r , and h r about the CD as \nf r = Fη, g r = Gη α , h r = H 0 + H 1 η β (B4) \nwith F , G , H 0 , H 1 , α , and β constants; we demand that H 0 > 0 and β > 0 to ensure physical and self-consistent solutions. Inserting these relationships into the three self-similar Equations, ( ?? ) - ( ?? ), yields the following four conditions that must be satisfied: \n(2 -m ) -( 1 + m -n 2 ) α + F ( α +1) = 0 , β = 1 + α, H 1 G β = κ c κ r ( m -n 2 )( 1 + m -n 2 ) , mγ -n -γα ( F -1 -m -n 2 ) = 0 . (B5) \nThe third of these is independent of the remaining three and is not needed to address the self-consistency of the solutions, but serves to demonstrate that the pressure increases (decreases) near the CD if m > n ( m < n ), which is apparent from Figure 2. Combining the remaining three relations yields \nα = mγ -n n + γ ( n -m 2 -3 ) , β = γ (6 -m -n ) γ ( m +6) -n ( γ +2) , F = n -2 γ γ . (B6) \nThe left panel of Figure 6 shows β for γ = 5 / 3 (blue) and γ = 4 / 3 (orange/yellow) for m = 3 as a function of n , which shows that the power-law index of the pressure near the CD is always positive, but tends toward zero as n → 3. This shows that the pressure does converge to a finite value at the CD, but the very weak dependence on η , coupled \nFigure 6. Left: β (the leading non-zero power-law index of the pressure) as a function of n for γ = 5 / 3 (blue) and γ = 4 / 3 (orange/yellow) and m = 3. Since β > 0, the pressure converges near the CD, but the rate of convergence is extremely slow as n → 3. Right: the self-similar pressure in the RS shell, h r , as a function of -η on a logarithmic scale. The numerical accuracy needs to be extremely high, with the functions resolved at \| η \| ≤ 10 -11 for n = 2 and \| η \| ≤ 10 -16 for n = 2 . 5, to correctly determine the pressure at the CD and thus the eigenvalues controlling the expansion of the RS shell and the deceleration of the CD. \n<!-- image --> \nto the fact that H 1 diverges in the same limit (see the third of Equation (B5) and the fact that H 1 ∝ 1 /β ), implies that extremely high numerical accuracy is required to correctly deduce that value. To demonstrate this directly, the right panel shows the self-similar pressure h r throughout the RS shell, with -η on the horizontal axis on a logarithmic scale. 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2024ApJS..274...45B	We present a catalog of 71364 pointlike UV sources with Sloan Digital Sky Survey SDSS photometry and GALEX farUV FUVnearUV NUV 0.1 mag. The limit corresponds to stellar T SUBeffSUB 1500020000 K slightly depending on gravity but nearly reddening independent for Milky Waytype dust. Most sources are hot white dwarfs WDs and subdwarfs. Comparing the spectral energy distribution SED GALEX FUV NUV SDSS ugriz of 35294 sources having good photometry with colors of stellar models and known objects we identify 12404 inlineformula inlineformula binary hotcompact stars with a cooler lessevolved companion with a possible 815 contamination by lowredshift QSOs and 22848 inlineformula inlineformula singlestar candidates. Singlestar counts are an upper limit because pairs of similar stars have singlestarlike SEDs and hot WDs with mainsequence companions of certain types depending on the WDs radius are missed or counted as single in the available wavelength range and selection. The catalog offers unique leverage for identifying hot WDs elusive at longer wavelengths when a cooler larger companion dominates opticalIR fluxes 51 of the binarystar and 20 of the singlestar candidates are previously unknown objects. Gaia DR3 provides a parallax with error 20 for 34 of the binarystar candidates and 45 of singlestar candidates allowing T SUBeffSUB E SUB BV SUB radius and L SUBbolSUB to be derived from SED analysis. The binarycandidate sample usefully expands the overall current binaryWD census to subpopulations elusive to Gaia and to other searches. The binary fraction among this specific sample of hot compact objects albeit with the mentioned biases b SUB f SUB 46 compared with that of their progenitors gt8050 for mass range 81M SUBSUB according to M. Moe implies a lower merging rate than found for massive stars by H. Sana et al.	2024-10-01T00:00:00Z	['2024ApJS..274...45B', 'arXiv:2409.04626', '10.48550/arXiv.2409.04626', '2024arXiv240904626B', '10.3847/1538-4365/ad6e7c']	['Celestial objects catalogs', 'White dwarf stars', 'Ultraviolet sources', 'Ultraviolet photometry', 'Evolved stars', 'Binary stars', 'Late stellar evolution', 'Stellar properties', 'Stellar colors', 'Stellar effective temperatures', '212', '1799', '1741', '1740', '481', '154', '911', '1624', '1590', '1597', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies']	Hot Stars in the GALEX Ultraviolet Sky Surveys GUVcatAISxSDSSHS and the Binary Fraction of Hot Evolved Stars	2,024	191	0.54	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	1	https://arxiv.org/pdf/2409.04626.pdf	{'Stars': 'Luciana Bianchi 1 \[email protected] \nReceived \n; \naccepted \nReference: Bianchi, L. 2024, ApJS, in press, DOI: 10.3847/1538-4365/ad6e7c', 'ABSTRACT': "We present a catalog of 71,364 point-like UV sources with SDSS photometry and GALEX FUV-NUV ≤ 0.1mag. The limit corresponds to stellar T eff ≳ 15,000 -20,000K, slightly depending on gravity but nearly reddeningindependent for Milky-Way-type dust. Most sources are hot white-dwarfs (WDs) and sub-dwarfs (SDs). Comparing the SED (GALEX FUV, NUV, SDSS u,g,r,i,z ) of 35,294 sources having good photometry with colors of stellar models and known objects, we identify 12,404 ± 1871 1267 binary hot-compact stars with a cooler, less-evolved companion (with a possible 8% -15% contamination by low-redshift QSOs), and 22,848 ± 1267 3853 single-star candidates. Single-star counts are an upper limit because pairs of similar stars have single-star-like SED, and hot-WDs with main-sequence companions of certain types (depending on WD's radius) are missed or counted as single in the available wavelength range and selection. The catalog offers unique leverage for identifying hot WDs, elusive at longer wavelengths when a cooler, larger companion dominates optical-IR fluxes: 51% of the binary- and 20% of the single-star candidates are previously un-known objects. Gaia DR3 provides a parallax with error ≤ 20% for 34% of the binaries- and 45% of single-star candidates, allowing T eff , E B -V , radius and L bol to be derived from SED analysis. The binary-candidate sample usefully expands the overall current binary-WD census to subpopulations elusive to Gaia and to other searches. The binary fraction among this specific sample of hot-compact objects, albeit with the mentioned biases, b f ≳ 46%, compared with that of their progenitors ( > 80% -50% for mass range 8 -1 M ⊙ , Moe (2019)), implies a lower merging rate than found for massive stars by Sana et al (2017). \nSubject headings: Catalogs: Celestial object catalogs, white dwarf stars, ultraviolet: \nultraviolet photomety; evolved stars ; binary stars; stellar evolution: star counts, late stellar evolution ; stellar prperties: stellar colors, stellar effective temperature", '1. Introduction. Leveraging the GALEX UV Surveys to Identify Optically-Elusive Hot Stellar Sources.': "Hot stars are difficult to identify and to precisely characterize at optical wavelengths because their optical colors are saturated at the T eff of the earlier spectral types. Hot white dwarfs (WDs) are especially elusive at all wavelengths except the UV, due to their small radii and low optical luminosity. These hottest stars stand out, and their physical parameters are easily characterized, if UV measurements are available, in particular at far-UV (FUV) wavelengths and shortwards. UV colors not only are more sensitive to the hottest T eff 's than optical-IR colors are, but - combined with data at longer wavelengths they make it possible to remove the known degeneracy between T eff and E B -V for hot stars from SED analysis, and to derive an unbiased T eff value concurrently with extinction (e.g., Bianchi et al. (2014b) - their Figure 1, Bianchi et al. (2011a,b, 2018b)). \nThe Galaxy Evolution Explorer (GALEX, Martin et al. (2005); Morrissey et al. (2007); Bianchi (2009)) has surveyed the sky at ultraviolet (UV) wavelengths for almost a decade. GALEX has imaged most of the sky in two Ultraviolet bands, FUV ( λ eff ∼ 1528 ˚ A, 1344-1786 ˚ A) and NUV ( λ eff ∼ 2310 ˚ A, 1771-2831 ˚ A) simultaneously, until the FUV detector stopped working in May 2009, and observations continued with the NUV detector only. With a field of view of ≈ 1.2 · diameter and a spatial resolution of ≈ 4.2 '' (FUV) and 5.3 '' (NUV) (Morrissey et al. 2007), GALEX performed nested surveys with different area coverage and depth, see e.g., Bianchi (2009); Bianchi et al. (2011a, 2014a, 2017); these works show also maps of the sky coverage of the surveys. The resulting photometric database contains about 600 million UV measurements of almost 300 million UV sources, and > 120,000 UV spectra (Bianchi et al. 2018b). The GALEX database is a unique resource for studies of hot stars and other classes of objects, including extra-Galactic objects such as star-forming galaxies and redshift ≲ 2 QSOs. Given that there was essentially no precursor \nall-sky or wide-area coverage UV imaging survey (Bianchi 2016a,b) and that a better one is still a few years away at best, the GALEX UV sky surveys remain for now a unique resource for statistical studies of hot stellar sources. \nOn average, only about 10% of the NUV sources are detected also in FUV, owing to the stellar IFM being skewed towards cooler (less massive) stars, and UV-bright galaxies being more rare than red ones. For detailed maps of source content of the sky as a function of UV magnitudes and colors, see Bianchi et al. (2014a) and Bianchi et al. (2017). \nThe GALEX database 1 contains repeated measurements of some sources, when fields have been re-observed or partly overlap. Science-enhanced catalogs of unique UV sources, i.e. removing duplicate measurements, have also been produced (Bianchi et al. (2011a,b, 2014a) ( BSCcat ), Bianchi et al. (2017) ( GUVcat , revised in 2020). These catalogs support a variety of statistical studies and especially the matching with other databases, which requires a unique source list without duplicate entries. To create a list of unique UV sources, in GUVcat AIS Bianchi et al. (2017) associated all existing measurements of the same source that may appear with distinct GALEX IDs in the database for the All-sky Imaging Survey (AIS), the survey with the largest area coverage (tags PRIMGID and GROUPGID). When constructing GUVcat , Bianchi et al. (2017) also identified and corrected a number of bad 'co-adds' in the GALEX database, that had resulted in wrong exposure time and FOV RADIUS values in tables such as photoobjall in the GALEX database. GUVcat also includes tags to facilitate science analysis, notably INLARGEOBJ and LARGOBJSIZE, \n1 The GALEX data, including integrated images and source catalogs at single- and multi-visit (coadd) depths, are archived in the Mikulski Archive for Space Telescopes (MAST) archive at the Space Telescope Science Institute (STScI), that also hosts the afore-mentioned GUVcat , GUVmatch , and other extracted catalogs in the Casjobs context 'GALEX catalogs' \nthat flag sources in the footprint of extended objects larger than 1 ' , such as large galaxies or stellar clusters, where crowding and underlying unresolved galaxy light may compromise source photometry. Bianchi et al. (2017) noted that 1 ' is a very conservative limit, for the purpose of eliminating crowded regions, but a user can choose to worry only about larger objects by using a combination of these two tags. 2 \nIn this work we present a science-enhanced catalog of (mostly) hot stellar sources (Section 2.1), with corollary data from SDSS, Gaia (Section 2.2) and Simbad (Section 2.3). In Section 3 we perform bulk analysis of the sources, after culling the sample for quality (Section 3.1). By comparing the sources seven-band SED (GALEX FUV , NUV, SDSS u, g, r, i, z ) with model colors and with a subsample of known objects with classification, we identify binaries consisting of a hot compact object and a cooler, less-evoled star (Section 3.2) and characterize some examples (Section 3.3). The results are discussed in Section 4 and summarized, along with the features of the released catalogs of relevance for future users, in Section 5. Appendix A contains details about the match of the UV sources with their individual GALEX observations, and Appendix B describes the released catalogs and where to find them.", '2.1. The Sample. Selection of Hot Stars from the GALEX FUV-NUV Color': "The GALEX broad-band FUV-NUV is essentially a reddening-free color, for typical Milky-Way type extinction with R V =3.1 (Bianchi et al. (2017) - their Table 1), because the very wide NUV passband includes the 220nm extinction feature, which compensates the steeper extinction at FUV wavelengths by small dust grains. Therefore, the GALEX color alone yields a robust selection of hot stellar sources, and a good indication of stellar T eff , nearly independent of reddening, as illustrated in Figure 1. \nHowever, to also characterize the type of the hot stellar objects, and especially to identify those in binary systems with a cooler companion, we use as a starting point the GUVcat AIS catalog matched to the SDSS data release 14, GUVmatch AISxSDSSdr 14 (Bianchi & Shiao 2020). The match of the whole GUVcat AIS ( ≈ 83 million UV sources) with SDSS data release 14 (DR14) yielded 23,310,532 counterparts to 22,207,563 unique GUVcat AIS sources, 10,167,460 of which are point-like 3 , over a total overlap area of about 11,100 square degrees (area calculated with AREAcat , Bianchi et al. (2019); sky-coverage maps are shown in Bianchi et al. (2014a)). SDSS adds five optical magnitudes, u, g, r, i \n3 given the higher spatial resolution of SDSS, ≈ 1 '' , compared to the 4.2/5.3 '' (FUV/NUV) of GALEX, we use the SDSS definition of STAR/GALAXY in the SDSS photometric database, where 'STAR' designates point-like sources, i.e. mostly stars and QSOs (Bianchi et al. 2011a) and 'GALAXY' generically refers to extended sources \nand z , to the UV photometry. Bianchi & Shiao (2020) had also estimated the statistical incidence of spurious matches, i.e. positional coincidences of non physically associated sources. They matched the same SDSS database to a 'fake GUVcat', i.e. a clone of the GUVcat catalog where the position of the sources was offset by 5 ' , so to create a catalog of non-existing sources with the same magnitude and color distributions as the real UV-source catalog. The fraction of spurious matches turned out to be negligible, less than half percent in our pointlike sample; in more detail, as shown in their Figure 1 where the number of real and fake matches is compared, the fraction of spurious matches increases with match separation, as expected, because when including sources within a progressively increasing radius, the area increases with R 2 , and the probability of finding random sources is higher in a larger area. On the contrary, the number of real matches is higher at short separations. This comparison was also used by Bianchi & Shiao (2020) to optimize the choice of match radius, as the best compromise to minimize both inclusion of accidental matches and loss of real matches, for an essentially complete matched catalog. \nWeinitially selected matched sources from Bianchi & Shiao (2020)'s GUVmatch AISxSDSSdr 14 with FUV-NUV ≤ 0.1mag and photometric error ≤ 0.3 mag in both GALEX filters, regardless of SDSS photometric errors or quality. The FUV-NUV ≤ 0.1mag color cut corresponds to T eff hotter than ∼ 15,000-20,000K, slightly depending on gravity and stellar type but nearly independent of reddening unless the extinction is extremely high, as shown in Figure 1; E FUV -NUV / E B -V =0.11 for Milky Way dust with R V =3.1, see Table 1 of Bianchi et al. (2017). The GUVmatch AISxSDSSdr 14 catalog by Bianchi & Shiao (2020) includes all SDSS matches to a GALEX source within 3 '' . We use the flag DISTANCERANK to extract a unique source list, selecting the GALEX sources that have only one SDSS match (DISTANCERANK=0) and, for those with multiple SDSS matches, retaining the closest match (DISTANCERANK=1). These tags are propagated in the resulting catalog, because for UV sources that have additional SDSS matches within the match radius, \ni.e. that are resolved into multiple optical sources within 3 '' , the UV flux might be composite of the optically-resolved matches, and the UV-optical color of the closest match may be biased. Therefore, sources that have additional matches (DISTANCERANK=1) must be treated with caution. For sources of interest, the additional matches can be examined in the GUVmatch AISxSDSSdr 14 catalog by Bianchi & Shiao (2020). The FUV-NUV ≤ 0.1mag selection yields 278,433 sources (we refer to the initial selection as GUVcat AISxSDSS HSall ). We eliminate 36 sources with FUV ARTIFACT=32 and 22 sources with NUV ARTIFACT=32 (rim artifact, Bianchi et al. (2017)), resulting in a sample 278,375 GALEXxSDSS matched sources before further culling, which is discussed later. \nFUV MAGs range from 11.72 to 22.52 ABmag in the sample, and NUV MAGs are between 12.37 and 22.80 ABmag; therefore, the sample includes some sources in the non-linear or saturated count-rate regime. These are not eliminated upfront, because they might be interesting sources for follow-up with other instruments. Non-linearity roll-off at 10% level sets in at 13.73 ABmag for FUV and 13.85 ABmag for NUV, see Morrissey et al. (2007) and in particular their Figure 8. \nAs mentioned in Section 1, GUVcat AIS 's tag INLARGEOBJ flags sources that are in the footprint of extended objects such as galaxies or stellar clusters larger than 1 ' . For a foreground stellar source in the line of sight of a bright galaxy disk or in a crowded cluster, the source identification and photometry can be extremely problematic, as illustrated in Figure 5 of Bianchi et al. (2017). The initial GUVcat AISxSDSS HSall sample includes 915 sources with tag INLARGEOBJ not 'N'; these are in 133 distinct extended objects, of sizes ranging from 301 ' to 1 ' (see Bianchi et al. (2017) for definition of sizes) 4 . Several of the \nextended objects that contain sample sources are stellar clusters (INLARGEOBJ=OC:or GC:- or SC:- ; see column 94 in Table 1); we do not discard the hot sources in clusters from the initial catalog, although the standard pipeline photometry is sometimes unreliable (see later). Out of the 133 unique extended objects including GUVcat AISxSDSS HSall sources, 59/68/74/78 have sizes ≤ 2 ' /3 ' /4 ' /5 ' , and all of these are galaxies. Some GUVcat AISxSDSS HSall sources are also found in larger galaxies: GA:NGC0224, \nOC:MWSC4301 SC:DOL-DZIM6 OC:ALESSI62 GA:NGC2403 OC:MWSC5154 SC:COLLINDER21 GA:IC1613 GC:NGC7089 GA:NGC4565 GA:NGC4244 OC:MWSC5033 OC:MWSC5901 OC:MWSC4602 OC:NGC2420 OC:ALESSI10 OC:MWSC4572 OC:MWSC4383 OC:FSR1064 OC:MWSC5804 OC:MWSC5828 OC:ASCC41 OC:MWSC5016 OC:FSR1102 GA:NGC4472 OC:FSR0080 GA:NGC0628 GA:NGC5033 SC:MWSC5044 OC:MWSC5018 SC:FSR0876 OC:FSR0765 OC:MWSC5735 SC:MWSC5038 OC:MWSC5076 SC:NGC6481 GA:UGC07698 OC:FSR0161 GA:NGC3945 GC:KOPOSOV2 GA:UGC02275 GA:NGC4579 GA:UGC02302 GA:UGC05829 GA:UGC01176 GA:UGC09242 GA:NGC4651 GA:UGC01133 GA:UGC07608 GA:IC3687 GA:SDSSJ113637.41+112327.0 GA:NGC6255 GA:NGC3104 GA:NGC4151 GA:NGC1199 GA:NGC4340 GA:UGC08331 GA:NGC5866B GA:UGC08651 GA:UGC09537 GA:UGC01547 GA:IC2329 GA:UGC05340 GA:UGC04499 GA:UGC08683 GA:UGC04837 GA:UGC00501 GA:UGC03966 GA:NGC0807 GA:UGC07307 GA:UGC07719 GA:PGC045321 GA:UGC08572 GA:PGC032620 GA:UGC09828 GA:IC3105 GA:PGC007998 GA:UGC05907 GA:IC2421 GA:UGC00035 GA:UGC08426 GA:NGC7469 GA:UGC01519 GA:UGC00052 GA:IC0949 GA:PGC022928 GA:UGC12388 GA:PGC3130476 GA:NGC0317A GA:NGC7603 GA:NGC5790 GA:PGC071675 GA:UGC07906 GA:UGC01697 GA:UGC01318 GA:PGC071916 GA:2MASXJ11052812+2500595 GA:PGC057881 GA:UGC06086 GA:UGC05999 GA:UGC04249 GA:PGC037722 GA:IC2568 GA:SDSSJ161622.19+041407.7 GA:UGC10025 GA:UGC00370 GA:PGC053682 GA:PGC2806871 GA:UGC04940 GA:SDSSJ114553.33+635411.6 GA:UGC06980 GA:2MASXJ09520676+2638256 GA:UGC06027 GA:UGC09458 GA:UGC09698 GA:PGC061304 GA:PGC023515 GA:PGC135848 GA:PGC060039 GA:PGC023508; \nsizes in arcmin: 301.2,217.2,177.8,158.4, 66.0, 62.1, 56.4, 36.8, 32.4, 30.0, 28.8, 26.4, 24.0, 24.0, 22.8, 20.4, 20.4, 20.0, 19.2, 19.2, 18.3, 16.8, 16.7, 16.2, 15.6, 15.6, 15.6, 15.0, 15.0, 15.0, 13.8, 13.2, 13.2, 12.6, 11.4, 10.8, 10.8, 10.2, 10.0, 9.9, 9.8, 9.6, 9.6, 9.6, 9.6, 9.0, 8.4, 8.4, 7.8, 6.3, 6.0, 5.5, 5.2, 5.0, 5.0, 4.7, 4.5, 4.4, 4.2, 3.9, 3.5, 3.4, 3.3, 3.2, 3.1, 3.0, 2.9, 2.8, 2.8, 2.6, 2.3, 2.3, 2.2, 2.0, 2.0, 2.0, 1.9, 1.9, 1.9, 1.8, 1.7, 1.7, 1.7, 1.6, 1.6, 1.6, 1.6, 1.6, 1.5, 1.5, 1.4, 1.4, 1.4, 1.4, 1.4, 1.3, 1.3, 1.3, 1.3, 1.3, 1.3, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0 \nGA:NGC0598, GA:UGC10822, GA:PGC088608, GA:NGC5457, GA:NGC2403, GA:IC1613, GA:NGC4565, GA:NGC4244, GA:NGC4472, GA:NGC0628, GA:NGC5033, GA:UGC07698, GA:NGC3945, GA:UGC02275, GA:NGC4579. GALEX images of all the extended objects larger than 5 ' , with GUVcat sources overlaid, can be viewed in the GUVcat web site. 5 Sources in large galaxies are likely integrated fluxes of UV-bright regions (see Figure 5 of Bianchi et al. (2017)); therefore, they are meaningless for our purpose and they will be discarded from the analysis of stellar hot sources. There might be bright stellar sources, such as novae, in nearby galaxies images, but these cases would require performing custom photometry rather than using the automated pipeline extraction. \nOf the 278,375 GALEX AISxSDSS matched sources with FUV-NUV ≤ 0.1mag, 71,364 are defined point-like in the SDSS database. SDSS TYPE ='STAR' indicates 'point-like' source morphology; therefore, the source could be a star or a QSO. Of the remaining 207,011, ten have SDSS type ='UNKNOWN' and the other 207,001 have type ='GALAXY', indicating that they are treated as extended sources by the SDSS pipeline. \nIn this work, we focus our analysis on the 71,364 sources deemed point-like by the SDSS pipeline, because SDSS has ≈ 3 × higher spatial resolution than GALEX images. We refer to this subset as the master catalog (released as GUVcat AISxSDSS HSpoint ), the starting point for the analysis. Of the 71,364 point-like sources, 360 are flagged to be in extended objects, mostly open clusters (INLARGEOBJ= OC:MWSC5735, OC:ALESSI10, OC:MWSC5828, OC:FSR0161, OC:FSR0765, OC:ALESSI62, OC:FSR0080, OC:NGC2548, OC:FSR1064, OC:NGC2420, OC:MWSC4383, OC:MWSC4288, OC:NGC2632, \nOC:MWSC5154, OC:MELOTTE111) or globular clusters (GC:NGC7089, GC:NGC6341, \nGC:NGC5272) or other stellar clusters (SC:NGC7772, SC:COLLINDER21, SC:PLATAIS2, \nSC:NGC6481, SC:FSR0876, SC:DOL-DZIM6, SC:MWSC5038, SC:MWSC5044), but some sources measured as point-like by SDSS are also in the footprint of galaxies. 6 Hot stars in stellar clusters are of course of great interest, but require custom photometry, because the standard GALEX pipeline is not robust in crowded fields: see, e.g., de Martino et al. (2008) and Figure 2 of Bianchi (2014). Therefore, all sources flagged in the footprint of extended objects are kept in the GUVcat AISxSDSS HSpoint catalog for completeness. Although most of these sources can probably be recovered by using the aperture photometry (included in the master catalog) or custom photometry, they are not used in the present analysis in favour of a clean, homogeneous sample with consistent processing, in particular the pipeline defined 'best' magnitude, recorded as FUV MAG and NUV MAG. This initial point-like sample will be further culled for the intended analysis (Section 3.1) but it is provided publicly in its entirety for possible other uses.", '2.2. Match with Gaia DR3': "Wematched the input catalog of 71,364 point-like sources GUVcat AISxSDSS HSpoint with Gaia DR3 release (Gaia Collaboration 2022) (before culling the sample for analysis, as will be discussed later in Section 3.1). A 3 '' match radius was used; this value was found by Bianchi & Shiao (2020) to be a good compromise to minimize both embarking spurious matches and missing real matches. For point sources, the difference between the GALEX position and the position of the Gaia match is usually much smaller than 3 '' (Bianchi & \nShiao (2020), see their Figure 1). Using a conservatively large match radius, however, is needed so that possible close-by sources that may be unresolved by GALEX (making the UV -optical colors biased) are also recorded, as well as possible matches with high proper motion sources, at the risk of embarking some spurious additional matches. In the final match output, therefore, GALEX sources with multiple matches or with one match close to 3 '' must be examined with caution. \nThe epoch of Gaia (J2016) matched source and the date of the GALEX observation (2003-2013) may differ by up to 13 years. Therefore, a simple cross-match between coordinates from the two databases could include Gaia sources within 3 '' of the GALEX source position that were slightly farther than 3 '' at the epoch of the GALEX observation, or exclude sources that were closer than 3 '' at the time of the GALEX observations, but beyond 3 '' when measured by Gaia. Such cases are very rare, because they require a large proper motion. However, because we are dealing with a stellar sample, rather than simply cross-matching the input list with the Gaia DR3 J2016 positions we followed a more precise procedure, accounting for position shifts due to proper motion. We performed an initial match of the GALEX sources catalog to Gaia DR3 with a very large match radius (15 '' ) which is the maximum possible displacement in 13 years among Gaia sources, considering the highest proper motion values in the entire Gaia DR3. We performed this initial cross-maching at the Casjobs interface at MAST. Then, for all potential Gaia matches within 15 '' that have a significant proper motion measurement (a fraction of the matches), we used the proper motion (DR3 tags: PMRA, PMDEC) to 'regress' the Gaia J2016 position to where it was at the epoch of the GALEX observation. The 15 '' -radius cross-matching returned 95,400 Gaia sources, including some multiple matches of the same GALEX source; of these, 4,370 input sources have a Gaia match within 15 '' but not within 3 '' . And 10,399 GALEX sources do not have a Gaia match even within 15 '' . \nRegressing the Gaia position of each potential match to the time of the GALEX observation requires an additional step back from the GALEX extracted catalog. The sources were selected from the GALEX database that combines all GALEX observations of the same source (when multiple visits exist); therefore, the sources' RA,DEC values were measured on a combination ('coadd') of all existing images of that source in most cases (see Bianchi et al. (2017) for more details). By consequence, the GALEX source position in the original database does not have a corresponding observation epoch (the misleading tag 'EPOCH' in the Casjobs GALEX database is actually the equinox of the coordinate system, J2000). In order to obtain a reference position consistent with an observing date for the GALEX sources, we matched the sample with all GALEX measurements from individual observations (GALEX 'visits'), by cross-matching our GUVcat AISxSDSS HSpoint catalog with table visitphotoobjall in the Casjobs GALEX database, that contains measurements from all GALEX data at individual-visit level (about 600 million source measurements). A 3 '' match radius was used. When multiple visits were found for a source, we selected the 'best' visit, chosen as the observation with the longest exposure time, excluding those visits where the source was near the edge of the field, which suffers by some distortion. We then used the Ra,DEC,Epoch of the GALEX best visit for a refined Gaia matching. The match of the input source list with the information at individual visit level is described in Appendix A and the full results are also made available, given that a number of sources have multiple observations, and these can be useful for different purposes, including a serendipitous variability search between repeated observations, or for examining different images of specific sources of interest. Relevant parameters of the GALEX 'best-visit' are included in the master catalog (Table 1). For 25 input sources a visit-level match was not found within 3 '' (see Section 6); these are eventually excluded from the analysis by other culling criteria, as explained in Section 3.1. \nHaving chosen for each source a GALEX 'best visit' from the individual observations, \nwe recomputed the spherical distance between the best-visit's GALEX RA,DEC and all potential Gaia matches found within 15 '' , with their DR3 position 'rewinded' to the date of GALEX's 'best-visit' (if a significant proper motion value exists). The results from the Gaia match include, in the master catalog, two tags for the separation between the GALEX source position and the Gaia counterpart: SEPBACK ARCSEC (the separation from the GALEX best-visit position and the Gaia match accounting for proper motion correction, i.e. the Gaia source position at the epoch of the GALEX best visit) and DISTARCMIN (the separation as returned from the initial DR3 J2016 positional match). The 'rewinded' Gaia position at the epoch of GALEX's best visit is also given, tags: Gaia RAback, Gaia DECback (while GAIA RA and GAIA DEC are the epoch=J2016 position from DR3). See Table 1 for description of all tags. When there is no significant proper motion, the two separations and the two Gaia positions coincide, and for the 25 input sources without a visit-level detection within 3 '' , the proper-motion correction cannot be applied either. Of all Gaia sources within 15 '' , 58,980 have SEPBACK ARCSEC ≤ 3 '' , i.e. are within a 3 '' match radius using the position corrected for proper motion (58,269 of these have also DISTARCMIN60. ≤ 3 '' ), and 58,538 have DISTARCMIN60. ≤ 3 '' , regardless of SEPBACK ARCSEC, i.e. before applying proper motion corrections. Because the two cuts (i.e., accounting for proper motion or not) yield some sources that are not in common, we retain in the final Gaia matched catalog all matched sources that have separation ≤ 3 '' either before or after accounting for proper motion, a total of 59,249 entries. The matches can be trimmed to a smaller match radius using the tags DISTARCMIN and SEPBACK ARCSEC. There are 427 sources that have a primary match within 3 '' after registering the Gaia positions to the GALEX best-visit epoch using the proper motion, but were beyond 3 '' using the DR3 J2016 position. Their proper motion values range from 0.259288 to 965.74628 mas and the GALEX epochs between years 2003.4742 and 2011.7510 (year.decimal). Also, 45 sources that are within 3 '' using the Gaia J2016 position are \nfarther than 3 '' after applying the proper motion (PM range: 0.15602224 to 55.018314 mas, GALEX epoch: 2003.7570 to 2009.4041). The sources that are within 3 '' after correcting for proper motion but are not according to their J2016 position, or viceversa, include cases with high proper motion values or close to the 3 '' match-radius cut off limit. For example, the highest discrepancy between the Gaia -GALEX separation before and after proper-motion correction is for source GALEX ID=6373800939499817688; the Gaia match (Gaia ID = 943770757800160384) has a proper motion of 965.746 mas; using the DR3 (epoch J2016) coordinates, the Gaia source is 10.5335 '' away from the GALEX input position, but after correcting for proper motion it was 1.6888 '' away from the GALEX best-visit RA,DEC (epoch = 2006.9764, about 10 years earlier than the Gaia observations). Its parallax of 58.5336 mas with a small error ( < 1%), places the source at a distance of only 17.08 pc. This case and other similar ones are retained because, in favour of completeness, we did not limit proper motion corrections based on proper motion uncertainties. The uncertainties are propagated in the master catalog, and can be examined to refine the sample according to specific needs and criteria. The above numbers do not include the 15,106 GALEX input sources that do not have a Gaia match (10,399 without a match even within 15 '' and 4,707 with a match within 15 '' but beyond 3 '' ). \nFollowing Bianchi & Shiao (2020), we built tags to track input sources with multiple Gaia matches within 3 '' , as described in Table 2. These tags are also propagated in the master catalog (Table 1). In particular, MMRANK GAIA is defined as: \n- · MMRANK GAIA = 0 if the Gaia match is the only one for the GALEX source (53,331 cases)\n- · MMRANK GAIA = 1 if the Gaia match is the closest of more than one match (2,927 cases) within the match radius\n- · MMRANK GAIA > 1 in order of increasing distance from the GALEX position, for \nadditional Gaia matches (2,991 sources)); there are up to 4 matches for one source 7 \n- · MMRANK GAIA = -888 (as all other Gaia related tags) if there is no Gaia match within 3 '' (15,106 sources, about 21% of the input master catalog) \nWhen a GALEX input source has multiple Gaia matches within the match radius, all the matches are listed in the full output (Section 7, i.e. Appendix B), in subsequent rows where the GALEX input source is repeated. To retrieve the list of unique input sources, one can select MMRANK GAIA ≤ 1 from the full match output tables. In the master catalog, only the closest match is retained, but the 'MMRANK GAIA' and 'NMMRANK GAIA' tags carry the information of whether multiple matches exist. \nOf the 59,249 input sources that have at least one Gaia match, i.e. the primary matches that are not 'null' (null matches have GAIA ID = '-888' in the master catalog), 53,331 have only one match within 3 '' , and only 2,927 have additional matches. Such figures imply that there is little ambiguity on the actual Gaia counterpart for most GALEX sources. Of these Gaia matches, 6,757 have no parallax measurement, and of the 52,492 matches with a parallax value in DR3, 46,443 have a parallax > 0. mas, while 6,049 have a negative parallax (a value resulting from DR3 pipeline, not null). Negative parallax values, when not due to large uncertainties, may indicate a failure of the solution parallax -proper-motion, which in our sample could be related to binarity, either owing to orbital motion not yet disentangled from proper motion in DR3, or to semi-resolved binaries causing a shift of the source centroid among the repeated Gaia observations. Therefore, although a DR3 negative parallax prevents absolute estimates of the stellar parameters, possible failed-solution cases \n(negative parallax) are potentially interesting targets for the purpose of investigating very close binaries, augmenting the impressive results produced by Gaia on binaries that can be measured by Gaia (e.g., El-Badry (2024)). As we will see, the present sample can especially stretch the census of binaries containing a hot compact object, that may be lacking in the Gaia binary-WDs population. \nIn sum, out of the 71,364 GALEX sources in the input catalog (before further culling for science analysis) 15,106 GALEX sources have no Gaia match; they are retained in the master catalog with values of Gaia tags SOURCE ID as well as all other Gaia-related parameters set to -888. \nIf we now consider the 56,258 sources with either a unique Gaia match, or the closest match in case of multiple matches (i.e., we do not include additional matches beyond the closest match in the statistics), 50,098 primary matches have a parallax value in Gaia DR3, but only 44,258 have parallax > 0., and of these, 19,258 have a parallax error ≤ 20%. We will give particular emphasis in the analysis to the subsample for which a good direct distance estimate is available, so that absolute stellar parameters can be derived (Section 3), after further sample cleaning in Section 3.1. \nThe Gaia DR3 matches to the GALEX sources have also been cross-linked with DR3 table vari summary ; the resulting full match output catalog (Section 7 (Appendix B) ) includes all tags from both the main Gaia source table and the variability information when available, i.e. when variability has been detected by Gaia. \nThe most relevant tags of the matched sources are included in the master catalog of point-like sources, as well as some distilled information on Gaia (serendipitous) variability from vari summary . The full matching results, with all tags from both Gaia source and vari summary are also available online as a separate file (Section 7).", '2.3. Match with the SIMBAD database': "The input list of point-like hot GALEXxSDSS sources was also matched with the CDS Simbad database, using the Vizier interface at http://cdsxmatch.u-strasbg.fr , specifically http://cdsxmatch.u-strasbg.fr/#tab=xmatch& . \nWe used two values of match radius, 5 '' and 10 '' , larger than the value adequate for Gaia matching, because bright stellar objects with entries in Simbad from old catalogs may have rounded or less precise coordinates, and we aimed at collecting a complete set of possible matches, to be culled later in the analysis if necessary. \nThe Simbad cross-match to GUVcat AISxSDSS HSpoint returns 34,831 and 35,665 matches within match radius = 5 '' and 10 '' respectively; some of these are multiple matches of the same source. That is, less than half of the input point-like GALEXxSDSS hot sources are known objects. The full results from the Simbad matches are given in separate tables (see Section 7 i.e. Appendix B). Among all the Simbad matches within match radius =5 '' (=10 '' ), 2008 (2152) have Simbad MAIN TYPE = Star, 8 and 14,830 (14,842) have MAIN TYPE = WD* Candidate, to cite the more numerous classes. These numbers highlight the power of UV data to identify hot WDs, by confirming the high percentage of WD candidates among the known objects on the one hand, and - on the other hand - by showing the limitation of current known samples, and the capability of UV-data to significantly extend the census of such objects, which are elusive at other wavelengths. Only 34 (37) do not have Simbad tag OTHER TYPES set; 13,099 (13,137) have a reported spectral type (Simbad tag SP TYPE, propagated in the master catalog as SIMBAD SP TYPE NEAREST), while about two thirds, 21,732 (22,528), do not have \na reported SP TYPE. These statistics refer to the full Simbad match results, including multiple matches. These tags, and other information distilled from the Simbad match, are included in the master catalog (see Table 1 for field description, columns 223-231); if there are multiple matches to a GALEX source, in the master catalog the values are given for the nearest match, and additional matches can be found in the full output. The tags of the closest match propagated in the master catalog are extracted from the 5 '' match-radius results; if there is no match within 5 '' , the values are set to '==', even if there is a match further away than 5 '' and within 10 '' , because the chance that such distant matches are spurious (positional coincidence) is high. The number of Simbad matches within 5 '' and 10 '' is also reported in the master catalog, in tags N SIMBAD MATCHES 5AS and N SIMBAD MATCHES 10AS respectively. Therefore, if there is no match within 5 '' but there are matches within 10 '' , or in case of multiple matches, these are not included in the master catalog, but the tag N SIMBAD MATCHES 10AS > 0 indicates that they exist and can be retrieved in the complete Simbad cross-match output files for the sources of interest (see Section 7). There are up to five (eleven) Simbad matches to one GALEX source within 5 (10) '' . Of the 71,364 GUVcat AISxSDSS HSpoint sources, 37,273 have no Simbad match within 5 '' and 36,985 do not have any match even within 10 '' . Such small difference indicates that we can probably ignore matches beyond 5 '' . In some cases, multiple Simbad matches of a source could be the same object, that exists in the CDS database with different identifiers. Of the known objects, 21,263 do not have a spectral type classification.", '3.1. Culling the Sample for Analysis': "For the analysis that follows, we cull the GUVcat AISxSDSS HSpoint catalog by eliminating sources that may have bad or inconsistent (between GALEX and SDSS) \nphotometric measurements, resulting in a biased UV -optical SED, and other warnings. First, we exclude all sources with flag INLARGEOBJ not equal 'N' (Section 2.1), at the expense of eliminating interesting sources in stellar clusters; this criterion reduces the sample from 71,364 to 71,004 sources. \nThen, we examine the tag 'SEP', the separation between the FUV and NUV source positions from the pipeline detection, that were merged into one GALEX source. For most GALEX visits, FUV and NUV were exposed simultaneously, yielding one FUV image and one NUV image of the same field; the source extraction was performed by the GALEX pipeline in the NUV and FUV images separately, then measurements in the two bands were merged to create the source catalog. SEP ranges between 0. and 6.986 '' in the sample 9 . We eliminate 1,259 sources with SEP > 3.0 '' , reducing the analysis sample to 69,745 sources 10 , with most values ≤ 2 '' (Figure 2, left panel). Based on the histogram (right panel) in Figure 2, we further restrict the analysis sample to sources with NUV FWHM IMAGE and FUV FWHM IMAGE ≤ 8 pxl (i.e., 12 '' ; a GALEX virtual pxl is 1.5 '' ): 48,560 sources. If we retained sources with NUV FWHM IMAGE and FUV FWHM IMAGE ≤ 10.pxl (15 '' ), the analysis sample would contain 59,374 sources. Most of the sources eliminated with these criteria can be recovered with custom photometry in the individual GALEX observations, but for the present bulk analysis we prefer a reliable UV photometry at the expense of reducing the sample numerically. \n9 However, in the individual visits where one of the two detectors was not exposed (mostly FUV), SEP=-999.0 and only the magnitude from the recorded image is recorded in the GALEX Casjobs database, the other magnitude has a value of -999.0 \n10 eliminating sources with SEP > 3 '' is a very conservative limit; a cut at 'SEP' > 5 '' will only eliminate 260 sources. For this type of analysis, we choose a robust, albeit reduced, sample rather than a numerically larger sample possibly contaminated by lower quality data \nThe reason why we find some GALEX sources measured by the pipeline as more extended than the instrumental PSF among sources that the SDSS has treated as point-like (taking the pipeline-choosen 'best'-mag measurements FUV MAG and NUV MAG), could arise from the very different morphology that some sources may have in UV versus optical wavelengths (e.g., Bianchi (2014)) and from the lower GALEX spatial resolution, whereby two or more very close sources may not be resolved by the GALEX pipeline but are resolved at the SDSS almost 3 × better resolution. In fact, if we examine separately NUV FWHM IMAGE and FUV FWHM IMAGE, there are more NUV sources with NUV FWHM IMAGE larger than 10pxl than there are FUV ones (8,300 vs 3,529). The difference can be explained by the NUV sources being more numerous (typically ∼ 10 times), and therefore more crowded than FUV-detected sources, that are more rare and sparse, causing the pipeline to merge close NUV sources in crowded fields while resolving the rarer sources in the FUV image. The point-spread-function (psf) is also slightly broader in NUV than in FUV, especially for very bright sources. Among the sources with NUV FWHM IMAGE ≥ 10 '' , 16.3% have multiple SDSS matches to the GALEX GUVcat sources, while in the whole point-like sample only 7.7% have multiple SDSS matches. Most of these sources can be recovered by custom photometry (e.g., de Martino et al. (2008); Bianchi (2014)) or in some cases by using small aperture photometry (also included in the catalog) and applying the aperture correction from Morrissey et al. (2007); de la Vega & Bianchi (2018), instead of using the 'best' pipeline measurements that treated the source as extended. We keep in the master catalog all sources, and all the existing magnitude measurements, so that interested users know that GALEX imaging is available, and quality photometry can be recovered, for sources of particular interest; the appropriate method may vary, according to object type and purpose. In this work we perform a statistical analysis of a large sample; therefore, we must use a consistent set of measurements, and restrict the analysis to the subsample where these standardized measurements are expected \nto be reliable. \nAll the 25 sources that have no visit-level detection within 3 '' of the merged-catalog position in the full point-like sample, are eliminated by the above cuts in this culled sample (48,560 sources). \nOne source, GALEX ID=6378832251021428138, has SDSS measurements only in the r-band, although it is rather bright at UV wavelengths: FUV MAG=18.14 ± 0.076 and NUV MAG=18.62 ± 0.045 ABmag. The FUV-NUV= -0.476 ± 0.089 color makes it a candidate for a hot isolated star. The Simbad database yields only one match with a known source, SDSS J134459.14-011038.2, classified as a Blue straggler. We do not eliminate this source for now, although its SDSS SED is incomplete 11 . \nFinally, for a number of sources the SDSS magnitude is saturated in one or more filters, which significantly alters the SED shape, hampering the SED model-analysis. After discarding sources with saturation flag in g, r, i , the analysis sample is reduced to 46,818 sources. Excluding also sources saturated in the z-band, there are 46,744 sources left. As many as 1234 sources are saturated in SDSS g -band, and these might be very interesting sources for our objectives. However, we eliminate from the analysis in this work all sources with saturation in any of the SDSS bands g, r, i, z ; they might be recovered in future works with additional complementary optical data. The analysis sample is thus reduced to 46,744 sources with no SDSS saturation flags. \nOf the 46,744 sources in the culled analysis sample, 5,891 ( ∼ 12.6%) have no Gaia \nmatch, 40,853 (87.4%) have a match in Gaia DR3, as defined in Section 2.2, of which 38,728 have a unique Gaia counterpart (MMRANK GAIA =0) and only 2,125 have additional Gaia sources within the match radius (MMRANK GAIA =1) 12 . Among the 40,853 primary Gaia matches, 37,411 (80%) have a parallax value in DR3; but as many as 3,885 (9.5%) have a negative parallax value; of the 33,526 (82% of the Gaia matches) Gaia counterparts with a parallax measurement > 0mas (and PARALLAX ERROR is also positive), 15,586 (38% of the Gaia matches) have a parallax error better than 20%. This subsample is particularly valuable because, thanks to the availability of a good direct distance measurement, absolute stellar parameters such as Radius and L bol can also be derived from the stellar SEDs (Section 3.3). Not surprisingly, the fraction of Gaia matches with good measurements is higher in the culled sample than in the overall sample (Sections 2.1 and 2.2). \nAbout 58% (27,440 sources) of the analysis sample have a Simbad match, all of which have SIMBAD MAIN TYPE NEAREST not null. Among the Simbad matches, SIMBAD MAIN TYPE NEAREST is 'Star' for 1,116 sources and 'WD* Candidate' for 12,178 sources (26% of the sample). Only a few cases have SIMBAD MAIN TYPE NEAREST indicating an extra-Galactic object: 65 have type = 'Galaxy', and from zero to a few sources have other extra-Galactic types, with the exception of the SIMBAD MAIN TYPE NEAREST = QSO (1053) and = Seyfert 1 (581); such contamination will be discussed in the following sections (see also Bianchi et al. (2009, 2011a)). \nWe have strived in the selection so far to increase the purity of the sample without giving weight to the quality of the SDSS photometry, lest interesting UV-identified hot WDs might be eliminated because they have poor SDSS measurements. We examine now \nthe error distribution of the SDSS magnitudes, that is relevant for the analysis that follows and in the interpretation of the results. \nWhile the GALEX FUV and NUV magnitudes have a maximum error of 0.3 mag by construction of the database, we have not applied so far any error cut to the SDSS counterparts. SDSS photometric errors are > 0.3mag in g / r / i / z for 354 / 810 / 1833 / 10542 SDSS counterparts (the distribution reflecting probably our selection of hot stellar sources), with a few sources in each band having error up to a few hundred mags. But most sources that have a large error in one band do not necessarily have large errors in other filters. As far as there are a few good colors, the SED can still be usefully analyzed, because each photometric band is given a weight inversely proportional to its error, for example in the SED fitting procedure with model colors. Therefore, we do not eliminate these sources from the online master catalog, as it can be used by others in other ways, and augmented by other data. But we note here that 233 SDSS counterparts have error > 0.3mag in all g, r, i, z bands, 268 SDSS counterparts have error > 0.3mag in g, r, i , and 296 have error > 0.3mag in SDSS g and r . \nFor the following analysis, which involves both GALEX and SDSS measurements, we further restrict the sample by imposing error cuts of 0.2mag in all SDSS bands except for SDSS z , resulting in a total of 35,294 sources. We used the SDSS pipeline 'PSFMAG' measurements. Of the 35,294 sources, 10,598 do not have a Simbad match within 5 '' (both SIMBAD MATCH NEAREST and SIMBAD MAIN TYPE NEAREST are '=='), while for the 13,846 known objects that do not have a spectral type classification the field SIMBAD SP TYPE NEAREST is set to '==' if there is no counterpart, and to '-888.' if there is a known counterpart but no spectral type.", '3.2. Identification of Single and Binary Candidates from color-color Diagrams': "In view of the growing evidence that the majority of massive stars are formed in binaries and that ∼ 70% of these binaries interact and 20-30% possibly merge before reaching their final evolutionary stage (e.g., Sana et al (2014, 2017); Moe & di Stefano (2017); Abadie et al (2010); Patrick et al (2019, 2020)) and - on the other hand - of the unique sensitivity of our UV catalog to identify hot WDs, the evolved descendant of intermediate-mass stars, it is interesting to search in the present source catalog, which consists essentially of hot WDs (mostly) or SDs, for candidate binaries including a hot evolved star and a cooler, less evolved companion. The purpose is to increase the statistical samples of such binaries, that are elusive in databases at longer wavelengths. A large, unbiased sample can support a variety of follow-up studies to clarify binary evolution (such as, for example, the intial-final mass relation). Here we perfom bulk color analysis to estimate the binary fraction of these evolved stellar objects, and to infer - by comparison with the fraction of unevolved binaries, measured in several studies by others - the probability that binaries merge before the post-AGB stage. \nIn color-color diagrams combining the GALEX and SDSS photometric bands, the single hot WDs populate a sequence well separated by other classes of astrophysical objects, as shown in Figure 4 (see also Bianchi & Shiao (2020) - their Figures 4 and 5, Bianchi (2009); Bianchi et al. (2011a)). Also, binary hot WDs with companions cooler than spectral type ≈ A or ≈ F (main sequence or giants) can be separated from single hot WDs and from other astrophysical objects, with the possible exception of some rare QSOs that have typical optical colors but anomalous FUV-NUV colors, due to enhanced Ly α emission in the low-redshift sample ( ∼ 0.1 to 1.47) and to a dust phase for a smaller sample at redshift ∼ 2 (Bianchi et al. 2009). These QSOs, discovered by Bianchi et al. (2009), represent a small fraction of the QSO population ( ≲ 5%, according to the estimate by Bianchi et al. (2009)). \nMost QSOs have typically fainter UV magnitudes than the present GALEX stellar sample, which is extracted from the AIS 13 . \nAs mentioned in the previous section, for this analysis, which involves both GALEX and SDSS measurements, we restrict the sample as culled in Section 3.1 by imposing error cuts of 0.2mag in all SDSS bands except for SDSS z , resulting in a total of 35,294 sources. We used the SDSS pipeline 'PSFMAG' measurements. \nFigure 4 shows a combination of one GALEX band, NUV, with SDSS g and i ; the data-points (blue dots) are shown as well as model-colors constructed from model atmospheres for main sequence and supergiants, and for compact stars with log g = 7 and 9 (models described in Bianchi & Shiao (2020); Bianchi (2024)). As previously noted by Bianchi & Shiao (2020) and Bianchi et al. (2011a), Figure 4 shows that the hot-star sequences with log g = 7 and 9 enclose the majority of data-points for the current sample. Because of the FUV-NUV ≤ 0.1mag cut, all single stars cooler than spectral type ∼ A are eliminated in this sample, while they can be seen to populate well the main-sequence and supergiant T eff color-sequences in the full sample of Bianchi & Shiao (2020). Only the locus of the hottest massive early-type stars is adjacent to the hot WD locus at the hottest T eff , and these would be hard to distinguish from the hot WDs given the photometric errors on the colors, except for the sources with a Gaia distance, for which a radius can be derived from SED analysis, see Section 3.3. However, massive stars (O and B types) are statistically very rare, because the IMF is skewed towards low masses and because massive stars evolve very fast, and for the purpose of source counts their numbers hardly affect the hot-WD population. \nFigure 4 shows also composite model colors for binaries composed of a hot WD (examples for three WD T eff values are shown: T eff =100,000K, 30,000K and 20,000K, each for two extreme values of radius, R WD =0.02 and 0.1 R ⊙ ) and cooler, less evolved companions of representative spectral types, both main sequence and giants. Although only a few examples are shown to avoid crowding the plot, these binary model-color sequences span all the color-color space occupied by the data-points outside the single-WD color locus. The model colors are shown for unreddened stars, but extinction arrows are drawn on a few sample model colors. The two-colors combination in Figure 4 also shows that binaries with companions hotter than ∼ A (depending on the ratio of radii within the stellar pair) are either excluded by our sample's UV color restriction, or would hardly be distinguishable from a single A-type (and earlier) star. Comparing this figure with Figure 4 of Bianchi & Shiao (2020) we also note that a large number of sources in their sample occupy the color-color locus covered by galaxy models: those data-points are all extended sources (black points in Bianchi & Shiao (2020) figures), which are excluded from this sample of point-like sources. The only extra-Galactic objects that may intrude the point-like sources are QSOs or AGNs, with a central light peak dominating the faint underlying galaxy, at low red-shift because of the FUV-NUV ≤ 0.1mag cut. \nIn Figures 4 to 6 sequences of QSO models colors are also shown from the 'average QSO' template (cyan symbols), and in Figures 5 and 6 also from the enhanced-Ly α template (dark blue) of Bianchi et al. (2009). QSO model symbols (diamonds) mark redshift = 0, 0.2, 0.4, 0.6, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 3.0 and 4.0; some values are labelled. The redshift=0 model locus is shown with a large diamond. As expected from the GALEX pass-bands, and from our UV-color selection, only low-redshift QSOs are included. In Figure 5 we mark with large cyan dots the 954 sources classified as extra-Galactic (QSO or AGN or Seyfert) in Simbad. \nBased on the loci defined by the model colors, we define a separation between single and binary stellar candidates, shown with a green line in Figure 5. The boundary follows the model sequence of the WDs with log g =9 (the single-WD sequence closest to the binary locus) with minimal extinction ( E B -V =0.1mag) and extends in NUVg to redder colors, because single stars cooler than T eff ∼ 15,000K are excluded from the sample by the FUV-NUV color limit, as is also evident from the color-color figures, while binaries owing to the presence of the hot WD- can occupy redder colors in the diagram. In Figure 5 the binary candidates are marked with orange dots, and the single hot-star candidates with green dots. We track these candidates with the same color-coding in Figure 6, showing the FUV-NUV color in the Y-axis. Note in this plot that the extinction arrows are nearly horizontal, because the broad FUV-NUV color is essentially reddening-free for Milky-Way type extinction (Section 2.1). The single-hot-star candidates (green dots) still occupy a defined locus, although with a wider spread in NUVg (X-axis) compared to the previous color combination, due also to the larger reddening effect because the color spans a wider wavelength range (note also the different X-axis range in the figures). For this reason, the initial selection was made in the color combination of Figure 5. Instead, the binary candidates, the orange points, occupy the whole width of the diagram, including the single-star locus, consistently with the binary model predictions. In FUV-NUV, the Ly α enhanced QSOs at low redshift (dark blue model colors) stand out well separated from the standard QSO template (cyan). The overplotted known QSOs + AGNs + Seyfert (large cyan dots) occupy a well defined locus in both Figures 5 and 6, overlapping with the locus of certain binary combinations. Figure 6 also shows that the hottest sources, with FUV-NUV ≤ -0.3mag, are rather free of QSO contamination. \nFrom the two-color separation illustrated in Figure 5, we count 22,848 ± 1267 3853 sources in the 'single-hot-star' locus (green dots) and 12,404 ± 1871 1267 binary candidates (orange dots). The uncertainty in the counts is estimated by applying to each color the combined \nphotometric errors of the two band measurements, and recounting the numbers in the single and binary loci adding and subtracting the error to the nominal colors of each source. As discussed above, the stellar-binaries locus contains a small number of intruding QSOs; therefore, the binary candidates count of 12,404 is an upper limit. We estimate a correction based on the known objects, and on the Gaia-match statistics. Excluding from the binary candidates the 954 sources with an extra-Galactic classification in Simbad, the binary fraction would be b f = 0.50%. The known QSOs or AGN or Seyfert are 7.7% of the sources in the binary-defined locus. But only a fraction of the sample has a Simbad match to a known object: only 48.6% (6,030) of the binary candidates have a Simbad identification (6,274, about 51%, have SIMBAD MAIN TYPE = '==', i.e. are unknown objects). If the ratio of extra-Galactic over Galactic sources were the same in the un-classified half of the sample, the contamination by extra-Galactic objects would be twice the 7.7% fraction of the known ones. With this correction, the binary fraction would be b f =0.46%. On the other hand, because the most elusive astrophysical objects are the hot WDs, in surveys at optical wavelengths, assuming a contamination by extra-Galactic objects in the culled sample of binary candidates of 15% is probably an overestimate. \nWe also recall that, with the presently available colors (FUV to near-IR), companions of types ∼ A and hotter (and in some cases M0V and cooler, depending on the ratio of the stellar radii) may escape the identification as binaries, as shown also in Figure 7, where we plot the composite SED of GALEX+SDSS bands of a number of cool star types with hot WDs of representative T eff and radii. Single stars of these spectral types and later are excluded from the sample because of the FUV-NUV color cut, as shown in Figures 4 to 6. Therefore, they would be missed from both the binary- and the single-star counts. The count of sources in the 'single hot star' locus must be regarded as an upper limit to the number of single WDs or SDs, because the SED of a stellar pair with very similar T eff will not be distinguishable from a single-star SED with the same T eff . Although such \n'identical twins' may be rare cases among WDs, and the number of massive OB stars is also negligible, the counts of single hot WDs is in principle an upper limit. With all these caveats, the fraction of binaries detectable as composite SED (i.e. not merged) is then b f ≳ 50% -46%, less than the ∼ 75% reported by most works for massive stars (Patrick et al 2019; Sana et al 2014, 2017; Moe & di Stefano 2017), but similar to the fraction reported for the lower mass range of the WD progenitors, from > 80% to 50% in the mass range from 8 to 1 M ⊙ according to Moe (2019). \nIf we consider the hottest stars among the sample, for example the sources with FUV-NUV ≤ -0.3mag ( T eff ≳ 20,000-30,000K), the binary fraction is 57% (10,122 single and 5,894 binary candidates including only 74 known objects classified as QSO or AGN or Seyfert) or 56.7% if we assume twice the known extra-Galactic objects contamination. \nIn sum, considering that the single-candidate counts are an upper limit, because they may contain identical-twin binaries, and that hot WDs with companions earlier than ∼ A types are missed in binaries counts, the binary fraction for evolved stars is estimated as ≳ 46% for the whole sample (subtracting an assumed 15% contamination by QSOs and AGNs), and 56.7% for the hottest sample.", '3.3. SED analysis to Characterize Binary and Single WD Candidates': "Using the color separation between single and binary hot stars from Section 3.2, we performed preliminary SED-fitting of the UV -optical SED with the model grids of Bianchi (2024), from which the model colors shown in Figures 4 to 6 are constructed. A full analysis will be the subject of another work; here we only show some examples in Figures 9, 10 and 11 to illustrate the potential use of this catalog. When the 7-bands SED can be fitted with a single-star model, both T eff and E B -V can be derived concurrently, because there are more \nindependent colors available than free parameters; we performed the fitting with model grids of different log g values (from 3.0 to 5 for Kurucz models, solar metallicity, and 6 to 8 for Tlusty pure-H models for the compact stellar objects). We fitted the SEDs with T eff and E B -V as free parameters (interpolating in the model grids), then choose the log g value that yielded the lowest best-fit χ 2 from all existing log g values. Once T eff , E B -V and log g are derived from the SED shape, scaling the best-fit model to the observed SED, accounting for the derived extinction, yields the stellar radius modulo distance, R 2 /D 2 . For the Gaia sources with a good parallax measurement (we choose a limit of parallax error ≤ 20%), R can be scaled to an actual value, and L bol is then derived from R and T eff . \nFor the binary candidates, that have a composite SED, only the FUV-NUV color can be used to derive T eff of the hot component; therefore, only one parameter, T eff , can be derived independently. We initially assumed log g =7 (which is an average value for the sample, from Figure 4), and a minimal reddening E B -V =0.05mag; this value turned out to be likely underestimated except at the high Galactic latitudes, because it yields many very small WD radii. Therefore, scaling the model of the appropriate T eff to the model grid yields a lower limit to the WD radius, and - when a Gaia distance is available - to L bol . We use the redder SDSS magnitudes to derive T eff and E B -V for the cool companion, and constrain its log g value, although these parameters are more uncertain than for the cases where all bands can be fitted as a single-star SED. With further iterations, the hot component of the stellar pair could be assumed to have the same E B -V as derived from the cool companion, when the result is robust, and T eff be derived from the FUV-NUV color with Tlusty models appropriately reddened. Such more detailed analysis will be the subject of a follow-up work, where additional data will be combined. In this work the scope was limited to a bulk exploratory analysis, in the interest of releasing the catalog for possible exploitation by others. \nIn Figures 9 to 11 we show examples of preliminary SED model analysis, for binary and single hot-star candidates. Figure 9 shows some of the binary WDs with the hottest derived T eff : most of these do not have a Simbad identification. The reason is quite evident from the examples in Figure 9: without the UV measurements, the optical-IR SED would be fitted with a single cooler star, and the presence of a hot secondary component would not be detectable, as it contributes negligible flux to the optical bands, unless the cool companion is also a subdwarf. From the preliminary analysis we also noted that, for the binaries, the SED fitting results are very uncertain when the cooler companion's T eff approaches 10,000K or hotter; reliable fits are usually achived with companions cooler than 8,000K, as could also be guessed from the color-color plots in Figures 4 to 6 and from the model SED examples in Figure 7.", '4. Discussion. The Binary Fraction of Hot Evolved Stars': "An interesting result from this work is the estimate of the fraction of binary vs single candidates hot-compact stars in the range of parameters where reliable counts can be derived from the present data. A wealth of studies has been recently devoted to estimate the binary fraction of massive stars and their immediate descendants, the Wolf-Rayet (WR). For example, Kobulnicky et al (2014) infer a fraction of ∼ 55% for binaries with orbital periods P < 5000days, studying 48 massive systems in Cyg OB2; Sana et al (2012) found b f (observed)=0.71 or b f (intrinsic)=0.69 from a sample of 71 O-type stars in six open clusters; b f ∼ 0.4 was found from a compilation of 227 Galactic WR (van der Hucht 2001)), b f (observed) ∼ 0.36 from RV monitoring of 11 WN (Dsilva et al 2023), b f (observed)=0.44 or b f (intrinsic)=0.56 for 16 WNE (Dsilva et al 2022), b f (observed)=0.58 for 12 WC stars (Dsilva et al 2020) but b f (intrinsic) ≥ 0.72 according to Dsilva et al (2022), and b f =0.70 for LBVs (Mahy et al. 2022). The ensemble of such studies (and the list is not complete) \nshowed that the binary fraction in massive stars is very high, with a significant number of main-sequence O-type stars also in hierarchical triple systems (Patrick et al 2019; Sana et al 2014, 2017; Moe & di Stefano 2017; Abadie et al 2010), and that many of these binaries exchange mass during their lifetime (71% of all massive stars), 20% to 30% of which will merge (Sana et al 2012), based on the distribution of the orbital periods. The cited studies are based on detailed observations of individual objects, including radial velocity (RV) monitoring of the binary systems to assess period distribution in order to correct for sample-selection biases; their only limitation is essentially the necessarily small numerical sample. The estimate of the number of very massive stars that undergo mass exchange, stripping, or merging with a close companion is relevant to explain for example the different classes of supernovae and their relative numbers (e.g., Sana et al (2012) and references therein). The progenitors of hot WDs are intermediate-mass stars (masses up to ∼ 8-9 M ⊙ ), i.e. late-O -early B type and later, although the frequent occurrence of mass exchanges within binary pairs at early stages could stretch the zero-age mass range both ways, in case of significant mass transfer to a companion or mass accretion from a companion. The estimated binary fraction for stars with initial masses between 8 and 1 M ⊙ varies from > 0.8 to 0.5 (Moe 2019). The binary candidate overall fraction derived in this work, b f ≳ 50% or ≳ 46% (assuming a 7.7% or 15% QSO contamination), refers to our sample of hot compact objects (mostly WDs) with a detectable cooler (less evolved) companion vs hot compact stars with a single-star-like SED, i.e. it includes mostly pairs in which one object has evolved through the post-AGB phase. The single stars of type ≳ A are excluded from the present sample by the FUV-NUV ≤ 0.1mag limit. \nOur identification of binary candidates is based on broad-band photometry, and, as discussed in previous sections, and as Figures 4 and 7 illustrate, the probability of distinguishing a binary from a single star largely varies depending on the ratio of T eff and radii within the stellar pair. Because the very hot evolved stars in binaries with an optically \nbrighter companion are elusive at all wavelengths except the UV (Figure 9), this work offers a unique opportunity to estimate the binary fraction of hot compact stars from an unprecedentedly large sample, two orders of magnitude larger than the ensemble of the detailed studies on which our knowledge of stellar multiplicity for early-type stars is based. The binary-candidate sample identified in this work is ∼ 4 × larger numerically than the astrometric WD-binaries detected by Gaia (about 3,200 WD-binaries, of which about 110 with a known UV excess) and comparable to the number of Gaia wide (resolved) WD binaries, about 16,000 (Shahaf et al (2024); El-Badry (2024); Garbutt et al (2024) and references therein). Given the paramount progress enabled by Gaia in this field, it is important to point out that our sample extends the Gaia WD-binary census not simply numerically, but with a higher sensitivity to the sub-population that may be elusive to Gaia. \nIt is interesting to look at the Gaia data among binary- and single-star candidates. Out of the 35,294 sources in the culled analysis sample, 95% (33,531 sources) have a Gaia match, and 82% of these (28,791 sources) have also a parallax measurement, of which only 14,490 sources (41%) with parallax error better than 20%. In more detail, the fraction of sources with a good parallax measurement is significantly higher for the single-star candidates (45%) than for the binary candidates (34%). The higher fraction of good parallax measurements among single WDs seems counter-intuitive, because hot compact objects have low optical luminosity (owing to their small radii), and the Gaia limit for detecting nearby WDs is stretched to larger distances when there is a main-sequence or giant late-type companion, which is much brighter than a hot WD at optical wavelengths, hence it is measureable by Gaia farther than the WD. In fact, the distribution of parallax-derived distances (Figure 12) shows that, among the sources with a good Gaia parallax (error ≤ 20%) binary candidates are identified at larger distances than single hot-star candidates. The relatively fewer good parallax measurements available among the binaries could then be explained if most systems \nwere unresolved or semi-resolved, making it difficult to solve for parallax, proper motion and orbital motion, and possibly also making the position measurements more uncertain than for a single, perfectly point-like source. We recall that DR3 can resolve nearby sources with angular separation down to 0.18 '' in general; however, the hot WD can be much fainter than the close companion at optical wavelengths (Figure 9). Such possibility is supported also by comparing the fraction of sources with a parallax measurement, regardless of error: 10,751 or 87% of the 12,404 sources in the binary locus, 20,789 or 91% of the 22,848 single-star candidates. Among the Gaia matches with a reported parallax value, as many as 1,676 (16%) have a negative parallax value in the binaries sample, versus 5% (1,102) in the single-star sample. Finally, of the 9,075 parallaxes with a positive value for binaries (19,687 for singles), 4213 (34%), and 10,261 (46%) have parallax error ≤ 20% for binary- and single candidates respectively. \nIn sum, among the Gaia counterparts there are relatively fewer good parallax measurements for binary candidates than for sources with single-star-like SED. As mentioned, binarity can offer an explanation for the difficulty of measuring parallaxes, especially if most systems are very close (projected on the sky), as is expected given the distribution of distances and the generally lower quality of parallax measurements among binary candidates. An alternative explanation for the relatively higher number of Gaia sources with no parallax measurements could be the presence of point-like QSOs or AGNs, whose colors intrude part of the binaries color locus (Figures 4 to 6), as discussed in Section 3.2. If all the binary candidates lacking a parallax measurement were extra-Galactic objects, the contamination would be 13%, close to the conservative limit we estimated based on the sources with Simbad identification. If all the Gaia sources with a negative parallax values were also extra-Galactic objects, the correction to the binary-candidate counts would be 27%. \nWe also recall that half of the binary candidates identified by this study are unknown objects. Only 6,030 out of the 12,404 binary candidates have a known object match in Simbad (including the 941 QSOs or AGNs or Seyfert1), and 2,651 of these have a spectral type; also, in several of the known stars the WD companion is previously undetected, whereas only ∼ 20% of the single-star candidates are un-known objects: 18,634 (7,148 with a spectral type) out of 22,848 sources have a Simbad match. \nIn Section 3.2 we have counted the number of candidate single hot stars and of binaries (hot star plus a cooler companion) using their multi-band color separation, as informed by stellar model colors and by the subsample with a known classification, and have discussed the limits and biases of the relative counts. Some 'single' sources may be the result of binary merging in prior phases, which is what ultimately we would like to estimate, by comparison with binary fractions of their progenitors at earlier phases. The merging ratios inform theory of binary evolution and stellar population models. But the sources in the 'single-star' color locus could also be pairs consisting of two identical (or almost identical) stars, whose T eff s do not differ enough to make the broad-band colors appear like a two-component SED in the wavelength range available, within typical photometric uncertainties of the present sample (see Figure 7). For the sub-sample with good Gaia distances, R ∗ and L bol can be derived from SED-fitting, once T eff and E B -V are derived, as shown in Section 3.3; however, for a hot WD of a given T eff , the radius R ∗ may easily vary by a factor of two or more, depending on its mass and log g , making it impossible to identify 'identical twins' with apparently single-star SED based on their luminosity. Therefore, the counts of sources in the 'singles' locus is an upper limit to the actual number of WDs evolved from initially single stars. By consequence, the estimated binary fraction would be a lower limit, if we have reasonably corrected the sources in the binary locus the from QSOs contamination. As shown in Figures 4 -6, a small region of the colors locus of binaries consisting of a hot WD and a cooler, less evolved star, is also shared by low-redshift QSOs \nor AGN. Because half of the binary-candidate sources are unknown objects, the estimate of the contamination by extra-Galactic objects is uncertain. Among the known objects (with classification in Simbad), the number of QSOs+AGN+Seyfert are 7.7% of the binary candidates; if the relative proportions were the same among the unclassified sources, which are about half of the binary candidates, the contamination would be ∼ 15%. Assuming this correction to the binary counts (0.85% of 12,404), the derived binary fraction is b f ≥ 46% in the range examined. The derived binary fraction for our specific sample is much higher than earlier estimates, which reported 18% to 26% of WDs to be in binary systems (Holberg 2009; Toonen et al 2017), and is higher for example than the wide-binaries fraction derived for polluted WDs and field WDs by Noor et al. (2024), of about 10% or less. Future work should characterize the stellar parameters of individual pairs, when possible (examples of preliminary results were given in Figure 9), to also explain the differences between the binary fraction derived in our specific sample versus that found in other samples selected in different ways and with different criteria. \nIn the count of binaries, companions with spectral type earlier than ∼ A cannot be always distinguished from single stars (depending on the ratio of radii, etc.); therefore, some types of binaries would be missed. However, because the IMF is skewed towards lower masses, this bias should not be significant because the later-types, lower-mass companions should be much more numerous. Single stars of intermediate and late types are eliminated by the FUV-NUV color cut of the sample, that includes sources hotter than T eff ∼ 15,000-20,000K, depending on gravity (Figure 1). \nThe measured binary fraction is b f ≳ 50 or ≳ 46%, after correcting the binary-candidate counts for a 7.7 or 15% contamination by extra-Galactic objects. Among the hottest sources in the sample, with FUV-NUV ≤ -0.3mag (corresponding to T eff ≳ 30,000K, depending on gravity), the binary fraction is higher, b f ∼ 57%. In this UV color range, the contamination \nby extra-Galactic objects is negligible, based on the known samples. Because the single-star counts could include binaries consisting of two similar stars, this fraction is a lower limit. Compared with the binary fraction of intermediate-mass unevolved stars reported by Moe (2019) (b f from > 80% to 50% in the mass range 8 -1 M ⊙ ), this result implies that ≲ 20% of the initial binaries merge, or much less if most of the current WDs' progenitors were closer to the lower mass range of Moe (2019), as is likely. Such implied merging-rate is lower than the result obtained by Sana et al (2012) based on the orbital period distribution of their O-stars sample: they estimate that ∼ 23% of the very massive stars merge during their lifetime. However, if the contamination by extra-Galactic objects in our binary counts were underestimated, b f would be lower, and the implied merging rate higher.", '5.1. Summary of Results, of Catalog Features and Unique Advantages': "We have constructed a catalog of hot 71,364 UV GALEX sources with FUVNUV ≤ 0.1mag and optical SDSS counterparts, deemed point-like at the SDSS resolution, with additional data to facilitate science analysis. The sources are mostly stellar, with some small contamination by low redshift QSOs or AGNs, and are mostly hot WDs and hot SDs. We reduced the sample to 35,294 sources with good SDSS photometry for analysis (Section 3.1). From the UV-optical colors in the seven bands FUV, NUV, u, g, r, i, z , analyzed with model-color grids, we identified in the sample 12,404 ± 1871 1267 candidate binaries, consisting of a hot WD or SD and a less evolved (in most cases) companion of spectral type later than ∼ A (the exact limit depending on gravity, and ratio of stellar radii within the pair, e.g. Figure 7), and 22,848 ± 1267 3853 sources with an apparently single-star SED, that could be largely bona fide single hot stars but could also contain binaries with very similar stellar parameters, or with an unevolved companion not detectable in the wavelength range of the UV -optical \nSED. \nAfter correcting for an estimated ∼ 7.7 -15% contamination of the binary candidates by extra-Galactic QSOs and AGNs, the resulting binary fraction is ∼ 50 -46% for the sample ( T eff ≳ 15,000K), and ∼ 57% for the sources with FUV-NUV ≤ -0.3mag, i.e. with T eff hotter than ∼ 20,000-30,000K (the T eff limit depends on gravity, see Figure 1). The estimate of T eff is rather robust, given that the GALEX FUV-NUV color is essentially reddening-free for Milky-Way type dust with R V =3.1 (Section 1). However, the reddening E B -V , and its uncertainty, affects the estimate of stellar Radius and L bol , which is obtained by scaling the best-fit model SED to the observed magnitudes, accounting for reddening. The derivation of individual stellar parameters will be the subject of a future work. \nOnly a fraction of the sources has a Gaia DR3 identification, and only 34% of the binary candidates and 45% of the single-star candidates have a parallax with error better than 20%, in spite the binaries span larger distances (as estimated from the subsample with good parallax measurements, see Figure 12). The subsample with a good distance estimate is particularly valuable because, besides T eff and extinction, stellar radius and L bol can also be derived from the observed SED (Section 3.3). Another interesting result is that about half of the entire sample has a known-object counterpart in Simbad; but, in the culled sample that was analyzed here, the fraction of previously known objects is ∼ 81% for the single-star candidates, and only 48% for the candidate binaries; in addition, among these known objects the presence of a hot companion was often unreported. This large discrepancy is not surprising, because the presence of a hot WD in a close binary system with a cooler unevolved companion would be mostly undetected unless FUV data are available, as is evident from the examples in Figure 9. \nIn sum, in our census of binary candidates, only ∼ 48% are previously known objects according to the Simbad database, while ∼ 80% of the [apparently] single evolved hot-star \ncandidates are known objects. But among the known objects in our binary sample, the hot-WD companion was often undetected prior to this work. Among the Gaia DR3 matches to our culled analysis sample of 35,294 UV sources, only forty are flagged as 'NON SINGLE STAR', of which thirty-five are selected as binary candidates in this work, and five as 'singles'. In the less restricted sample of 46,744 sources we only find five more Gaia matches tagged as 'NON SINGLE STAR', fortyfive in total. In the analysis sample, sixtytwo (seventytwo in the larger sample) have DR3's tag IN VARI ECLIPSING BINARY set to 'True'; fifty-one of the sixty-two are binary candidates according to our selection and eleven are single-WD candidates, confirming that - as previously explained - the 'singles' locus may contain some types of binaries, while the selection of binary candidates is very robust at least for hot WDs with companions cooler than ∼ 8-10,000K. The Gaia census of binaries, including those with a hot WD component, is expected to increase significantly by the end of the mission (El-Badry 2024). However, the comparison of our results with the current identification of binaries from the DR3 Gaia pipeline highlights the unique leverage offered by our sample with UV -optical wavelength coverage. In addition to the DR3 classification, several recent works used various combinations of criteria to identify different types of binaries from the Gaia data. Shahaf et al (2024) identified ∼ 3,200 astrometric WD binares, about one fourth of our binary-candidate sample size, and about 16,000 wide WD binaries have been reported (El-Badry (2024) and references therein). More relevant than the numerical differences, which are difficult to interpret because each method uses specific criteria that select subsamples with different characteristics, is the complementarity of the selections, in particular the leverage offered by the UV catalog to identify WD-binaries in a parameter regime (very hot, or very tight orbit, or low-mass) that may elude Gaia's selections, contributing a different piece of the puzzle towards a complete picture (see also El-Badry (2024) for a discussion on Gaia results). \nThe manyfold increase of the census of binaries with a hot compact object is relevant \nbecause a comparison between the binary fraction of hot evolved objects, estimated for the first time from a large statistical sample, with the binary fraction of their progenitors, the massive and intermediate-mass main sequence stars, translates into the percentage of the initial binaries that merge before one component reaches the post-AGB phase. Intermediate-mass stars are the major providers of C, N and other abundant elements, including elements supporting life as we know it. Statistically significant, unbiased samples of their evolved descendants could help clarify their evolutionary path (in particular the yield of chemical elements from the so called 'third dredge-up'), ultimately relevant for understanding the chemical evolution of the universe, as well as to inform star formation and stellar evolution theories, which underpin our interpretation of the integrated colors of galaxies at cosmological distances, where stellar populations cannot be resolved into their individual stellar constituents. Studies of stellar binarity are being extended to the Magellanic clouds to explore effects of metallicity (e.g., Dorda & Patrick (2021); Patrick et al (2019, 2020, 2022) with samples of cool supergiants having a hotter ( ∼ B-type) companion, where a binary fraction of ∼ 15 ± 4% (SMC) and 14 ± 5% (LMC) was found by Dorda & Patrick (2021). The catalog presented here allows the investigation to be extended to the hot post-AGB population, so far only accessible in the Milky Way, and in particular to extend the census of hot-WD in binary systems in the parameters ranges that make them elusive to Gaia detection methods, potentially filling gaps in poorly explored regimes. \nOnly a fraction of our UV-optical source catalog has a Gaia DR3 match, in spite it was extracted from the GALEX shallowest sky survey (AIS) with rather bright UV magnitude limits (see e.g., Bianchi (2009); Bianchi et al. (2014a, 2017)). In the wider catalog of 71,364 sources, of the 59,249 input UV sources that have at least one Gaia match, i.e. the primary matches that are not 'null' (null matches have GAIA ID = '-888' in the master catalog), 53,331 have only one match within 3 '' , and only 2,927 have additional matches. Such figures imply that in the vast majority of cases there is no ambiguity on the actual \nGaia counterpart of the GALEX source. Out of the 35,294 sources in the culled analysis sample, 95% (33,531 sources) have a Gaia match, and 82% of these (28,791 sources) have also a parallax measurement, 14,490 of which (41%) with error better than 20%. In more detail, the fraction of sources with a good parallax measurement is higher for the single-star candidates (45%) than for the binary candidates (34%) (Section 4). Because most of the sources are hot WDs, these numbers show the power of the UV source catalog to identify such very hot, optically-faint stellar objects, and how elusive they are even in the best available optical surveys todate. \nThe unique leverage of the UV catalog to identify hot WDs is also evident from the statistics of known objects, resulting from the match with the Simbad database (Section 2.3). The majority of the known objects in our catalog are classified as WD* Candidate (14,830 matches) and similar stellar classes, according to the Simbad database, highlighting the advantage of the UV-color selection with respect to other wavelengths. On the other hand, only about half of our catalog sources are known objects (27,440, or 58.7% in the conservatively culled analysis sample), and only 11,315 have a reported spectral type, highlighting again the power of the presented UV catalog to extend the known samples of elusive hot compact stars, with significant purity. As reported in Section 3.2, there is a remarkable difference between the fraction of known objects in the single- (80%) or binary(49%) candidates. The UV colors are particularly relevant to identify hot WDs in binary systems with a cooler, larger, optically brighter companion, whose flux makes the hot WD contribution negligible, hence its presence undetectable, at optical and IR wavelengths (examples in Figure 9). \nThe initial selection in this work was made for sources with FUV-NUV ≤ 0.1mag, and with a SDSS match. In the restricted 46,744 analysis sample (regardless of SDSS data quality), 20,782 (44%) have FUV-NUV ≤ -0.2 mag, and 28,279 (60%) have FUV-NUV \n≤ -0.3 mag. Therefore, the present sample contains an unprecedented high number of hot WDs (see Figure 1) with respect to known samples.", '5.2. Possible Applications and Follow-up Work': "Because we do not know the orbital separation of the binary candidates (all pairs are unresolved in GALEX and SDSS imaging), we cannot identify which of the binaries found in this work have likely exchanged mass, and which ones are separated enough that they can be reasonably assumed to both have evolved as a single star. While recent efforts to understand binary evolution concentrate on the effects of mass exchange (e.g., de Mink et al (2013, 2014); Sen et al (2022)), which affects the majority of massive stars, the binaries that have not exchanged mass can offer insight into the initial-final mass relation, that maps the WD mass to the initial mass of its progenitor. If no mass transfer has taken place, the mass of the cooler, unevolved companion is a lower limit to the mass of the initially more massive WD progenitor; and when the evolutionary age can be inferred from the cooler companion, the mass of the WD progenitor can be further constrained, assuming the pair was born coeval (physical binary, not the result of a capture). Comparison of the evolutionary time on the cooling track (from the current WD location in post-AGB HR-diagrams) with the total age anchored to the less evolved companion, yields the presumed pre-AGB lifetime of the WD progenitor, and constrains its mass. We are pursuing the identification of binaries that evolved with no mass exchange in two HST programs (Bianchi's HST-GO-14119 and HST-GO-15832), with HST high resolution imaging of over 100 candidates, selected from the broad sample presented here. We have identified about two dozens such cases, currently being analyzed (Bianchi et al (2024)). \nBesides the ongoing follow-up with HST of the selected subsamples, possible future work will explore a machine learning (ML) approach to better separate single and binary \ncandidates, and to better estimate the limits and biases, based on the model colors and the SED of known objects in the catalog. Other follow-up work such as radial-velocity monitoring may be of interest to others, to characterize our UV-selected sample with respect to samples identified by other methods. \nFinally, the presented sample, and the derived binary fraction among post-AGB objects, can contribute to clarifying the formation of planetary nebulae (PNe). In particular, two aspects are long debated and still need firm observational constraints. The first concerns the formation of PNe from low-mass stars (mass ≤ 2.3 ± 0.3 M ⊙ ): the expected stellar mass loss is insufficient to shed enough mass to reach the PN central star (CSPN) stage, unless it is enhanced by mass transfer or common envelope, related to binarity of the CSPN (Moe & de Marco 2011). The second addresses the statistical distribution of axy-simmetric PNe: to produce bi-polar PN morphology, the formation of an equatorial disk is postulated, when mass loss in the previous phases is enhanced by rotation. The equatorial overdensity would allow the PN shell to expand more in the polar directions when the circumstellar material is plowed outwards by the CSPN radiation-pressure-driven supersonic wind in the post-AGB phase (e.g., Herald & Bianchi (2011); Keller et al (2011); Bianchi (2012) and references therein). Again, the equatorial overdensity of the AGB-expelled circumstellar material is believed to be insufficient to account for the observed distribution of PNe geometries; but the preferentially-equatorial mass loss can be easily enhanced in very close binary CSPNe, possibly contributing to the formation of bipolar PN shells (e.g., G'omez-Munoz et al (2023); Ali et al. (2023) and references therein).", '5.3.1. There Are Many More GALEX Sources!': 'The sample presented in this work represents a homogeneous, but not complete, subset of the available GALEX UV sources. The sample was extracted from the GALEX AIS survey, which has the shallowest depth but the widest area coverage, as shown in Figure 1 of Bianchi et al. (2014a), and in particular only from the AIS fields that have both detectors (FUV and NUV) exposed. We used as a starting point GUVcat AIS (Bianchi et al. 2017) because it is a homogeneous catalog of unique GALEX sources in which some sources that have incorrect measurements in the official GALEX GR6+7 online release have been corrected, and repeated observations of the same source have been distilled to a unique-source list, suitable for matching with databases at other wavelengths and for statistical studies. The sample was further restricted to the availablity of SDSS coverage, about one fourth of the GUVcat AIS sources. We have used the GUVcat AIS matched with SDSS ( GUVmatch AISxSDSSdr 14, Bianchi & Shiao (2020)), because our goal of distinguishing single hot WDs vs binaries requires SED wavelength coverage spanning from FUV to near-IR wavelengths. The SDSS match also restricts the survey area covered by the extracted sample to the overlap of the two surveys, given the limited SDSS coverage (shown by Bianchi et al. (2014a): their Figure1 bottom; the overlap area is ≈ 11,000 square degrees, from Bianchi et al. (2019)). GUVmatch AISxSDSSdr 14 contains tags to identify multiple matches, useful to clean the extracted sample, and the five optical bands provided by SDSS allow us to select binaries in interesting, as well as poorly explored, regimes, and to estimate their parameters. 278,375 sources with FUV-NUV ≤ 0.1mag are extracted from the matched GUVmatch AISxSDSSdr 14 database ( ∼ 22.2 million) and further restricted to 71,394 point-like sources. Many more UV sources exist in the GALEX database, with FUV-NUV > 0.1mag, and many more (at least 10 × ) with NUV-only measurements: \n≈ 83 million sources just from the AIS survey (Bianchi et al. 2017).', '5.3.2. Variable Sources': 'An obvious warning concerns variable sources. The GALEX, SDSS, and Gaia data have been taken at different epochs, up to two decades apart. For stellar sources, such as, e.g., cataclismic variables (CVs) and other interacting or eclipsing binaries, and for any type of variable sources (including QSOs), colors from combined GALEX+SDSS SED or GALEX+Gaia photometry would be misleading if the source varied between the times of the respective observations. As described in Section 2.2, we have also linked the Gaia DR3 main catalog with the Gaia variability information, which is included in the master catalog. Out of the 35,294 sources in the culled sample analyzed in Section 3, only 1,763 sources do not have a Gaia match (tag GAIA ID = -888). Among the 33,531 sources with a Gaia counterpart, 2,104 have the tag PHOT VARIABLE FLAG set to \'VARIABLE\'. These sources are interesting for other purposes, but their SED could have biased UV -optical colors in the matched catalog, where only average magnitudes from all existing observations are recorded. Gaia DR3 tag IN VARI CLASSIFICATION RESULT is set to \'True\' for 1,971 of them; these include 62 variables flagged as IN VARI ECLIPSING BINARY, 24 as IN VARI SHORT TIMESCALE, 39 as IN VARI ROTATION MODULATION, 3 as IN VARI RRLYRAE, 3 as IN VARI MS OSCILLATOR. Interestingly, the highest number of Gaia counterparts with reported variability, 1,028, are flagged as IN VARI AGN. This number is very close to the 954 known extra-Galactic objects found from the Simbad classification in the binaries color-locus. Indeed, most of the sources with Gaia counterpart with flag IN VARI AGN set are those classified in Simbad as \'QSO\' or \'Seyfert 1\' (very few as \'Blazar\' or \'BLLAC\' or \'AGN\'), and only 228 do not have a Simbad object classification. This comparison also suggests that the contamination of the binary \ncandidates by extra-Galactic objects may be closer to the 8% estimated from the known objects in Simbad than to twice as much, which we have conservatively used to correct the sample of candidate variables. \nFig. 1.- Broad-band GALEX color FUV-NUV as a function of T eff from stellar models: Kurucz models with solar metallicity, with log g = 5.0 and 3.5, and Tlusty pure-H models with log g = 7.0. The effect of reddening is negligible for small E B -V values (for E B -V =0 and =0.25mag, shown in black and green, the lines are almost indistinguishable) and becomes appreciable at values higher than E B -V =0.40mag. Reddening is applied for a Milky-Waytype extinction curve with R V =3.1. The reddening effect is larger for UV-steep extinction curves such as those found in the Magellanic Clouds (see e.g., Table 1 of Bianchi et al. (2017) \n<!-- image --> \n). \n<!-- image --> \nFig. 2.Left: Distribution of the separation between FUV and NUV position of the extracted sources (\'SEP\' tag from the GALEX pipeline); Right: Histogram of [F/N]UV FWHM IMAGE (in pxl, 1pxl=1.5 \'\' ) measured by the GALEX pipeline. There is no correlation between these two extraction parameters over the sample. \n<!-- image --> \n<!-- image --> \nFig. 3.Left: Distribution of combined errors in the culled analysis sample for the FUV-NUV color (purple), for FUV,NUV and SDSS u, g, r, i (black), and for the five SDSS bands u, g, r, i, z combined (red). The dashed histograms are the errors combined and divided by the number of colors, i.e. 1 for FUV-NUV, 5 and 4 for the other two combinations. Right: Histogram of photometric errors in individual filters. For the majority of sources, the SDSS photometric errors are smaller than the GALEX photometric errors; therefore, the SDSS magnitudes have more relative weight for example in driving the SED fitting results. However, a number of sources have SDSS photometric errors up to ∼ 10 2 mag. \n<!-- image --> \nFig. 4.- The sources (blue dots) are shown in color-color space, along with model colors for hot compact stars (log g = 7 and 9, T eff = 100,000K to 15,000K, purple lines), main sequence stars and supergiants (red and yellow sequences, respectively), sample binaries consisting of a WD of T eff =100,000K, 30,000K, and 20,000K (purple, black, purple respectively), radius=0.02 R ⊙ and 0.1 R ⊙ , and a main sequence or giant star companion of representative spectral types. The QSO locus is also shown (cyan; the large diamond marks redshift=0). Low-redshift QSOs occupy a locus approximately in the center of this plot, which overlaps with some types of stellar binaries. Model colors are unreddened; thin arrows shown on some models indicate E B -V =0.2mag reddening. \n<!-- image --> \nFig. 5.- Same as in Figure 4; a green line shows the adopted separation between candidate single (green dots) and binary (orange dots) hot evolved stars. Sources with Simbad type = QSO or AGN or Seyfert are overplotted with cyan dots, of larger size than the other sources for visibility, but they are only 941 sources out of the 35,294 total analysis sample; they occupy exactly the locus of the QSO average-template colors for redshift ∼ 0.5 -1.5 (cyan for the standard QSO template, dark blue for the Ly α -enhanced template of Bianchi et al. (2009)). Triangles on the purple single WD sequences (log g =7 and 9) mark T eff =200kK, 30kK, 20kK, 15kK. \n<!-- image --> \nFig. 6.FUV-NUV versus NUV-r. Data and model colors as in the previous figures. The sample is restricted to FUV-NUV ≤ 0.1mag, but a wider Y-axis range is plotted to show also types of single stars and binaries excluded by the color cut. NUV-r separates main-sequence and giant stars from the WDs (log g = 6, 7 and 9 shown) more than in previous plots. In FUV-NUV, standard-template QSO colors (cyan) differ significantly from Ly α -enhanced QSOs (dark blue, Bianchi et al. (2009)). Known QSOs+AGN+Seyfert, marked with large cyan dots, mostly cluster around FUV-NUV ≳ 0, where the template colors predicts them to be. Reddening arrows for E B -V =0.2, shown on some models, are nearly horizontal, because FUV-NUV is essentially reddening free. The spread of the observed QSOs around the un-reddened template colors is due to photometric uncertainties and variations around an average template, the larger spread in the extinction direction is due to reddening. At FUV-NUV ≤ -0.3mag, the contamination from QSOs and B supergiants is negligible. Single-star and binary candidates selected in Figure 5 are marked with green and orange dots, respectively. \n<!-- image --> \nFig. 7.Left: model SEDs in GALEX+SDSS filters, covering a range of stellar pairs targeted by our study. Model mags for cool stars (Main Sequence (MS) case shown) in black, WD models colorcoded (the stellar parameters of the purple model are appropriate for a central star of planetary nebula). Right: black dots are the single-star MS models as in the left panel, dashed/dotted lines are composite WD+MS model magnitudes for sample pairs. At the ∼ 4.5 \'\' and 1.4 \'\' resolution of GALEX and SDSS imaging, all pairs are unresolved and the SED looks like the dashed or dotted model-mag lines. The figure illustrates the power of FUV -optical catalogs to identify candidate binaries in a regime different from what is accessible from optical data alone. Model magnitudes are scaled to a distance of 1kpc, and to the stellar radii given in the legend. \n<!-- image --> \n<!-- image --> \nFig. 8.- Histogram of FUV and NUV magnitudes of the total clean sample (SDSS errors ≤ 0.2mag), and of the single (green) and binary (orange) hot-star candidates, and of the known sources classified as QSOs or AGN or Seyfert (cyan). The contamination by the known extra-Galactic objects is minimal ( ≤ 8%, but only about half of the sample consists of known objects with classification). At the depth of the GALEX AIS, from which this sample is extracted, QSOs do not stand out for increasing at faint magnitudes, as they do in the deeper MIS survey (Bianchi et al. 2011a), where the number of extra-Galactic sources with respect to Milky Way stellar sources increases rapidly at fainter magnitudes. Therefore, in this sample, the QSO contamination in the binary locus cannot be eliminated by their magnitude and colors. \n<!-- image --> \nFig. 9.- Examples of two-component SED-fitting with stellar model magnitudes (Kurucz for the cooler companion, Tlusty for the hot companion). The Simbad identification, and spectral type (in parentesis), are given when available. Resulting T eff values are printed for both components, and of E B -V (derived for the cool star, assumed =0.05mag for the hot star). Radii derived from scaling the best-fit model are given first for a reference distance of 1kpc, then rescaled for the distance derived from the Gaia parallax. Dots are observed magnitudes, the black thick line connects the best-fit model magnitudes, thin black lines show the acceptable range of solutions. The dashed line is the composite model SED from both components. \n<!-- image --> \n9 \n12 \n22 \n24 \n26 \n12 \n12 \n20 \n24 \n26 \n26 \n12 \n12 \n22 \n24 \n12 \n12 \n20 \n24 \n24 \n26 \n14 \n16 \n18 \n20 \n22 \n14 \n18 \n20 \n22 \n24 \nE(B-V)-0.13 \n200 \n400 \n600 \n800 \n1000 \nWavelength [nm] \n6371302860892671324 \n6377847130478874908 \nSimbad: WD* (DA) \nE(B-V)-0.09 \nDkpc)= 1.000 -> R= 0.017 Rsun \n200 \n400 \n600 \n800 \n1000 \n200 \n400 \n600 \n800 \n1000 \nWavelength [nm] \nWavelength [nm] \n6374926871618914557 \n6378410065399909864 \nE(B-V)-0.04 \nTeff-55191K \nTeft-46079K \nDkpc) = 1.000 -> R= 0.022 Rsun \n200 \n400 \n600 \n800 \n1000 \n200 \n600 \n800 \n1000 \nWavelength [nm] \nWavelength [nm] \n6374399037318107361 \n6374574946293646256 \nSimbad: WD" (DA) \nE(B-V)-0.01 \nTeff-67293K \nTeff-38624K \n200 \n400 \n600 \n800 \n1000 \n200 \n400 \n600 \n800 \n1000 \nWavelength [nm] \nWavelength [nm] \nSimbad: WD\' (DA) \nE(B-V)-0.0o \nlogg-6.00 \nlogg-8.00 \nSimbad: WD\' (DA) \nE(B-V)-0.02 \nlogg-7.00\\_ \nlogg-6.00 \nSimbad: WD\' (DA) \nE(B-V)-0.04 \nTeff-47350K \nD(kpc)= 1.000 -> R*= 0.040 Rsun \nlogg-6.00 \n14 \n16 \n18 \n20 \n22 \n24 \nFig. 10.- Examples of SED-fitting with Tlusty high gravity stellar models for single-star candidates with derived high T eff \'s. The Simbad identification, and spectral type in parentesis if available, are given. The values of T eff and E B -V (derived as free parameters, and the bestfit log g ) are printed; the radius, derived from the best-fit-model scaling, is given for a reference distance of 1kpc as well as rescaled for the distance derived from the Gaia parallax. Symbols as in Figure 9. \n<!-- image --> \n9 \n8 \n9 \n9 \nlogg-7.00 \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 11.- Examples of SED-fitting for single hot-star candidates with derived T eff between 30kK and 20kK, from both Tlusty high gravity stellar models (log g range 6 -8) and Kurucz stellar models (log g range 3 -5). Resulting values, printed on the legend for both solutions, are generally consistent; the two best-fit model mag lines, both plotted with a black line, are indistinguishable on this scale. Most sources have a small radius, compatible with a compact evolved object; the solution from Kurucz model magnitudes finds the highest gravity available in the grid (log g =5) as the best-fit case, but the actual gravity is likely higher in these examples, which is reflected in the discrepancy among the derived radii. \n<!-- image --> \n<!-- image --> \nFig. 12.- Distances derived from Gaia DR3 parallax for the binary- and single-star candidates that have a Gaia match and a parallax measurement with error ≤ 20%, shown with a thin and thick line respectively. Actual values in the sample stretch to ∼ 6kpc, with very few data-points beyond the range shown. \n<!-- image -->', '6. Appendix A. Match of the point-source sample with individual GALEX observations.': "As mentioned in Section 2.2, we matched the selected catalog of point-sources with all GALEX data at visit level (i.e., measurements from individual observations), cross-matching the GUVcat AISxSDSS HSpoint catalog with table visitphotoobjall in the Casjobs database 14 . A 3 '' match radius was used. When multiple visits were found for the same source, we selected the 'best' visit, defined as the observation with the longest exposure time with both detectors exposed, excluding - when possible - visits where the source was on the edge of the field, which suffers from some distortion, and then used the RA,DEC,Epoch of the GALEX best visit for the Gaia cross-matching. The full match output of the individual GALEX visits is made available, given that a number of sources have multiple observations, and these can be useful for different purposes, including a limited serendipitous variability search between repeated observations, or for examining different images of specific sources of interest. \nThe match returned 352,642 GALEX sources measured in individual visits, within 3 '' of the 71,364 input point sources, indicating that multiple observations exist for a number of sources; these repeated visits allow a serendipitous variability search. For 25 input sources no counterpart was found within 3 '' in the visit-level source catalog. These sources are kept in the output for completeness. The input catalog was extracted from sources measured mostly in coadded images of available observations (but see Bianchi et al. (2017) about issues with some coadded images), which may result in a detection in the coadded \ndata even for sources undetected in individual images; also, for a source in a complex field or for an extended object, the pipeline may center the source in the coadded image at a position RA,DEC which differs by more than 3 '' (our match radius) from potential counterparts identified in the individual visits. All such cases probably are not bona-fide point-like sources, or well-resolved sources; therefore, we eventually eliminate them from the analysis sample (Section 3.1) although they are kept in the general point-like catalog for completeness, to indicate that GALEX images in those positions exist and can be used for custom-photometry if the source is of interest. \nFor most of the GALEX visits both FUV and NUV detectors were on, but there are cases in which one of the two was not exposed (mostly FUV, that stopped working in 2009 but was also off occasionally in the earlier part of the mission). We recall, as pointed out by Bianchi et al. (2011a,b, 2017) that a non-detection of a source in one of the bands may mean that that detector was not exposed, or that both detectors were exposed but the source was too faint to be detected in one of the two. Such important distinction cannot be found from the GALEX standard database catalogs (both non-exposure and non-detection have magnitude = -999) unless the exposure times are extracted; we provide exposure times in all our catalogs. On average, only about 10% of the NUV sources are also detected in FUV, for comparable exposure times, because the stellar and galaxy populations are skewed towards cooler sources, and 'hot' (i.e., FUV-bright) sources are more rare (Bianchi et al. 2014a, 2017). However, there are also cases where the pipeline detected a source in the FUV image and not a counterpart in NUV; such cases mostly occur because NUV sources, being more numerous, are also more crowded, and in dense fields such as, e.g., in stellar clusters, they can be so crowded that the pipeline merges several nearby sources into one extended source in the NUV image, while it resolves them in the FUV image. This is illustrated in Figure 5 of Bianchi et al. (2017). Of the 352,617 non-null visit matches, 117,878 have no FUV measurements and 462 have no NUV measurement. \nRelevant parameters of the GALEX 'best-visit' are also included in the master catalog for point sources, with tags: GALEX BESTVIS OBSDATE, GALEX BESTVIS NUVTIME, \nGALEX BESTVIS FUVTIME, GALEX BESTVIS EPOCHDECIMAL, GALEX BESTVIS FUV WEIGHT, GALEX BESTVIS NUV WEIGHT, \nGALEX BESTVIS FOV RADIUS, GALEX BESTVIS RA, GALEX BESTVIS DEC, GALEX BESTVIS FUVMAG, GALEX BESTVIS FUVMAGERR, GALEX BESTVIS NUVMAG, GALEX BESTVIS NUVMAGERR, GALEX NVIS FUV, GALEX NVIS NUV, GALEX MIN FUVMAG, \nGALEX MIN FUVMAGERR, GALEX MAX FUVMAG, GALEX MAX FUVMAGERR, GALEX MIN NUVMAG, GALEX MIN NUVMAGERR, \nGALEX MAX NUVMAG, and GALEX MAX NUVMAGERR . The tags are described in Table 1. Tags starting with 'GALEX BESTVIS' are values imported from visitphotoobjall and visitphotoextract . We also constructed tags GALEX NVIS FUV and GALEX NVIS NUV that give the number of images in which the source is found that have the FUV and NUV detector exposed, respectively. Note that in some cases the source is not detected, because the exposure is too short and the source may fall below detection threshold. In this case, the minimum value among all measured magnitudes (GALEX MIN FUVMAG or GALEX MIN NUVMAG) is -999. We included in the report of minimum and maximum values also the non-detections, because they can be informative of variability. Instead, to select the 'best visit', we examined only the visits where both the exposure time and the measured magnitude are above zero. If such cases exist, the 'best visit' is chosen as the one with the longest (NUV) exposure time; if no visit exists with both detectors exposed and the source detected in both bands, then the 'best visit' is chosen among all NUV exposures regardless of FUV. The measurements from this visit are propagated in the master catalog in the tags listed above. \nOne extreme example of a faint source can illustrate the -sometimes complicatedprocedure to distill the visit-level information into the master catalog. Source GALEX ID = 6376686083780907764 has five visits with both FUV and NUV detectors exposed. The FUV effective exposure times are FUV WEIGHT= 59.8594, 60.2578, 78.9844, 56.8984, and 65.2344 sec, and the respective FUV MAG are = -999.000, 21.9650, 21.4533, -999.000, and \n-999.000 ABmag. That is, in three of the five FUV images this source does not have a significant detection. For NUV, the exposure times are NUV WEIGHT=78.1406, 78.1719, 68.7969, 71.5938, and 67.3125 sec, and in the second visit the source is not detected: NUV MAG= 21.4124, -999.000, 21.7302, 21.3130, 21.5072 ABmag. So, there is only one visit with both detectors exposed and the source detected in both; therefore, this is chosen as the 'best visit', resulting in tags: GALEX BESTVIS FUVMAG=21.4533 ABmag and GALEX BESTVIS NUVMAG=21.7302 ABmag, with all the other 'best visit' related parameters (error, position, epoch, exposure time). However, the resulting minimum and maximum value among all existing visits with non null exposure time are GALEX MIN FUVMAG=-999. and GALEX MAX FUVMAG=21.9650 ABmag, GALEX MIN NUVMAG=-999. and GALEX MAX NUVMAG=21.7302 ABmag. Among the five repeated visits, SEP=0.49210 '' for the only visit where the source was detected in both FUV and NUV, and SEP=-999. '' for the others. We expounded here on the methodology and the criteria for clarity, for a corrrect interpretation of the TAGS provided in our catalog, and for future use also by others who may want to serendipitously search for variability among the repeated observations of sources in the GALEX databases. \nWe point out here another caveat, not previoulsy reported, because it may help users of the standard GALEX database, in particular when using misleading information from Casjobs table visitphotoobjall . In this table, the tag 'BAND' indicates which detectors were exposed during the visit; namely, BAND=1 indicates that NUV was exposed, BAND=2 that FUV was exposed, BAND=3 that both detectors were exposed. However, this information is not consistent with the exposure times FUV weight and NUV weight. For example, of the 71,364 input point sources, after excluding the 25 that do not have a visit match within 3 '' as mentioned, 8,075 have only one match within 3 '' in visitphotoobjall . Of these 8,075 individual visits, 7,989 have BAND=3 in the visit-level catalog (and all have FUV and NUV exposures > 0.sec, as expected), 75 have BAND=1, and all have NUV \nexposure > 0.sec but 61 of these also have FUV exposure > 0.sec, i.e. both detectors were exposed but the observation was not recorded with BAND=3. Finally, 11 of these visits are catalogued with BAND=2, but both detectors were exposed in all 11 cases. This example only indicates that the tag 'BAND' (in table visitphotoobjall from the GALEX pipeline) cannot be used to select individual observations according to FUV-NUV color availability. We recall that our specific sample was selected by a cut in FUV-NUV color; therefore, all sources that have only one visit should have had both detectors exposed in that visit, logically. In the case of the 14 sources with BAND=1 and no FUV exposure within 3 '' , probably the source was extracted from a coadd of different visits and merged with a separation larger than 3 '' between FUV and NUV detections (we recall that source detection is performed by the pipeline separately in the FUV and in the NUV image, then the FUV and NUV detections are merged; see discussion about the 'SEP' tag in Section 3.1). Here we have been referring to the BAND tag as recorded in the individual visits; the tag 'BAND' in the master catalog, however, is propagated from the merged catalog GUVmatch , which was built on the coadded database ( GUVcat (Bianchi et al. 2017), mostly from photoObjall ), and because we selected sources with a FUV-NUV cut, i.e. that have necessarily a measurement in both detectors, in the master catalog BAND is always =3. \nIn the point-source master catalog 25 sources have GALEX NVIS NUV=0, 8,166 have GALEX NVIS NUV=1, 15,102 have GALEX NVIS FUV=1 (there are repeated observations in NUV-only more than there are with both detectors), 8,061 sources have both GALEX NVIS NUV and GALEX NVIS FUV=1, and 14 have GALEX NVIS NUV=1 and GALEX NVIS FUV=0 (8 of which with SEP ≤ 3.0 '' , where SEP is the separation between the NUV-detection and FUV-detection position of the merged sources, not in the individual visits). \nThe full match with the GALEX observations at visit-level can be found at: http://dolomiti.pha.jhu.edu/hotwd/GUVHScat/GUVcat\\_AISxSDSS\\_HSpointXvisits\\_ 3arcsec.fits http://dolomiti.pha.jhu.edu/hotwd/GUVHScat/GUVcat\\_AISxSDSS\\_HSpointXvisits\\_ 3arcsec.csv \nThe catalog contains 386 tags (columns), the first six: GALEX ID, GALEX RA, GALEX DEC, GALEXID, GALEXRA, and GALEXDEC are the input catalog source ID and coordinates (repeated twice for database convenience), the seventh, OBJID, is the matched-source ID from visitphotobjall (the visit-level matched source), followed by tags taken from Casjobs table visitphotobjall , except for the last tags GALEX ID VIS, DISTARCMIN, GALEX RA VIS, GALEX DEC VIS, where DISTARCMIN is the separation between the input and the match, VPE NOBS DAT, VPE NTIME OB, VPE FOBS DAT, VPE FTIME OB, VPE ECLIPSE, and VPE LEG are useful parameters (date and time of the observation, 'ECLIPSE' = a unique GALEX identifier for each visit) from Casjobs table visitphotoextract , that we linked to the visitphotoobjall matches in order to obtain the epoch of observation, and finally EPOCHNUV DECIMAL and EPOCHFUV DECIMAL were constructed from the date and time of each observation, to give the epoch in year.decimal, which is convenient for applying the proper motion correction to Gaia (and other) matches in order to register them to the epoch of the GALEX visit.", '7. Appendix B. Available Data Products.': "Here we list the catalogs resulting from this project and publicly released. They are available from the author's web site at http://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/ . \nAll our data products are also available at MAST as a High Level Science Product via https://doi.org/10.17909/w9k5-tm92 (url: https://archive.stsci.edu/hlsp/ guvcat-hotstars/ ), and CDS Vizier.", 'GUVcat AISxSDSS HSpoint (71,364 sources)': "Point-like sources, selected from SDSS tag type ='STAR' from 278,375 sources in GUVmatch AISxSDSS (Bianchi & Shiao 2020) with FUV-NUV ≤ 0.1mag; no other culling. This is the 'master catalog' used as a starting point in the present work; it contains, in addition to the tags from the initial GUVmatch AISxSDSS , additional information to facilitate the science analysis. Specifically, columns 223-231 contain distilled information from the Simbad match (Section 2.3), columns 233-257 contain some parameters from the GALEX best visit (individual observation, see Appendix A) among all the visits from which the source was extracted, and columns 258-314 contain information distilled from the Gaia DR3 match (Section 2.2) including serendipitous detection of variability in the Gaia DR3 database, and tags constructed to track multiple matches. The columns are described in Table 1. See section 2.1 for other useful tags, such as 'INLARGEOBJ'; sources in the footprint of stellar clusters are excluded from the analysis in this work, but retained in the master catalog, and may be of interest for a variety of purposes but require careful checking of photometric quality. The catalog can be downloaded at: \nhttp://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/GUVcat\\_AISxSDSS\\_HSpoint.fits \nhttp://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/GUVcat\\_AISxSDSS\\_HSpoint.csv", 'GUVcat AISxSDSS HSpointXGaiaDR3': "The complete Gaia DR3 match results to the 71,364 GUVcat AISxSDSS HSpoint \npoint-like sources. See Section 2.2. It contains 59,249 Gaia counterparts to 56,258 GALEX input sources; the 10,399 input sources that had no match within 15 '' , and the 4,707 that have a match within 15 '' but not within 3 '' , are excluded. Columns 1-222 are from the GUVcat AISxSDSS HSpoint catalog. The subsequent columns contain all the tags from the Gaia main catalog ( source table in DR3), followed by the tags from the Gaia DR3 vari summary table. These files are the full match output, where all Gaia multiple matches within the match radius are retained, as defined in Section 2.2; in the master catalog we distilled only the Gaia tags of the 'primary' match, i.e. the only match if MMRANK GAIA = 0, or the closest among multiple matches (MMRANK GAIA =1), in order to keep a list of unique sources. Therefore, if multiple matches are found and should be examined, they can be found in this catalog. For sources with significant proper motion in Gaia DR3, the catalog also contains the Gaia source position 'rewinded' to the time of the GALEX visit (Section 2.2). Table 2 explains the columns. The full Gaia-match output is available at : http://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/GUVcat\\_AISxSDSS\\_HSpointXGaia3arcsecREWINDED\\_noNULLS.fits \nhttp://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/GUVcat\\_AISxSDSS\\_HSpointXGaia3arcsecREWINDED\\_noNULLS.csv", 'GUVcat AISxSDSS HSpointXsimbad [5 and 10]arcsec': "Full results of the match of GUVcat AISxSDSS HSpoint with the Simbad database (Section 2.3), with match radius of 5 '' and 10 '' . All the Simbad columns of the resulting matched sources are appended after columns 1-222 of Table 1. The relevant parameters of the nearest Simbad match, and the number of Simbad matches within 5 '' and 10 '' , are also included in the master catalog of point-like sources, see Table 1 - columns 265-271. Values are set to '==' if there is no match, and to '-888.' if no value of a given tag is given for the match. \nThe catalogs (original outputs from Vizier) 15 can be downloaded at: \nhttp://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/catHSpointXsimbad\\_5arcsec\\_1681832401503A.csv \nhttp://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/catHSpointXsimbad\\_10arcsec\\_1681832559998A.csv", 'GUVcat AISxSDSS HSpointXvisits': "Match of the master catalog with GALEX database table visitphotoobjall , to link each source to all its existing GALEX observations (when a match at visit-level is found within 3 '' of the master catalog position), as described in Appendix A. The full results are given in this catalog. Some relevant parameters, such as the number of visits found in FUV and NUV, and the best-visit parameters, are also distilled in the master catalog. The file below is the full output of all visits returned with sources within 3 '' of the input list: 352,641 rows. Description for all tags from the visitphotoobjall table are found in the MAST Casjobs Context 'GALEX GR6plus7' 16 . Values returned as 'null' are set to -888. The catalogs can be downloaded at: \nhttp://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/GUVcat\\_AISxSDSS\\_HSpointXvisits\\_3arcsec.fits \n15 we report that during this work, the first match obtained with Vizier X-match tool gave incorrect results, i.e, it returned matches with a reported distance smaller than the chosen match radius, but the actual separation between the coordinates of the input source and of the returned Simbad match were very different than the separation returned by Vizier, and in many case much larger than the match radius; we had incurred in the same problem before, and reported it to the CDS team, who fixed the synchronization with Simbad. We have repeated the match after a few months and checked that these files give correct distances between input source and match \nhttp://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/GUVcat\\_AISxSDSS\\_HSpointXvisits\\_3arcsec.csv", 'GUVcat AISxSDSS HSCulled': "The analysis sample of 35,294 sources after removing sources with SDSS saturation, possible inconsistencies between the source measurement in the GALEX FUV and NUV images, and trimming to SDSS photometric errors ≤ 0.2mag in u, g, r, i (Section 3.1). This subset of GUVcat AISxSDSS HSpoint , as culled for quality in Section 3.1, is used for SED analysis in Section 3. It includes all the same tags as the master catalog, described in Table 1, plus an additional tag 'COLOR LOCUS' that flags whether the source is in the binary-candidate (COLOR LOCUS=B) or single-candidate (COLOR LOCUS=S) color locus defined in Figure 5 and described in Section 3.2. COLOR LOCUS=[B] and [S] indicate sources that are not included in the respective color contours according to their nominal photometry values, but are included when color errors are applied. There are 11,123 sources with COLOR LOCUS=B and 1,281 with COLOR LOCUS='B [S]' (for a total of 12,404 binary candidates, 1,281 of which overlap with the single-star locus when 1σ color errors are added); 20966 sources have COLOR LOCUS=S and 1882 have COLOR LOCUS=[B] S (for the total 22,848 single-star candidates from their nominal colors, of which 1,882 falling in the binary-locus when color errors are applied); there are no sources classified as '[B] [S]', and 38 are unclassified. The catalog can be downloaded at: http://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/GUVcat\\_AISxSDSS\\_HSculled.csv http://dolomiti.pha.jhu.edu/uvsky/GUVcatHS/GUVcat\\_AISxSDSS\\_HSculled.fits", 'GUVcat AISxSDSS HSpoint ∗': 'Table 1. Description of Columns of the Master Catalog of Point Sources \nTable 1-Continued \nTable 1-Continued \nTable 1-Continued \nTable 1-Continued \na SDSS ID NCHILD SDSS RA SDSS RA ERROR SDSS DEC SDSS DEC ERROR SDSS TYPE PSFMAG U PSFMAG G PSFMAG R PSFMAG I PSFMAG Z PSFMAGERR U PSFMAGERR G PSFMAGERR R PSFMAGERR I PSFMAGERR Z EXPMAG U EXPMAG G EXPMAG R EXPMAG I EXPMAG Z EXPMAGERR U EXPMAGERR G EXPMAGERR R EXPMAGERR I EXPMAGERR Z EXPAB U EXPAB G EXPAB R EXPAB I EXPAB Z EXPABERR U EXPABERR G EXPABERR R EXPABERR I EXPABERR Z DEVMAG U DEVMAG G DEVMAG R DEVMAG I DEVMAG Z DEVMAGERR U DEVMAGERR G DEVMAGERR R DEVMAGERR I DEVMAGERR Z DEVAB U DEVAB G DEVAB R DEVAB I DEVAB Z DEVABERR U DEVABERR G DEVABERR R DEVABERR I DEVABERR Z PETROMAG U PETROMAG G PETROMAG R PETROMAG I PETROMAG Z PETROMAGERR U PETROMAGERR G PETROMAGERR R PETROMAGERR I PETROMAGERR Z FLAGS1 FLAGS2 FLAGS U FLAGS G FLAGS R FLAGS I FLAGS Z EDGE U SAT G SAT R SAT I SAT Z SAT U CR G CR R CR I CR Z CR PROBPSF U PROBPSF G PROBPSF R PROBPSF I PROBPSF Z PSFFWHM U PSFFWHM G PSFFWHM R PSFFWHM I PSFFWHM Z SPECOBJ ID PLATE MJD FIBER ID REDSHIFT REDSHIFTERR SPECTYPE CLASS SUBCLASS CLASS PERSON ELODIESPTYPE B V COLOR TEFF LOGG METALLICITY ELODIE REDSHIFT ELODIE REDSHIFTERR PROPERMOTION USNO RED1 USNO RED2 \nUSNO BLUE1 USNO BLUE2 RUN RERUN CAMCOL FIELD \nc value is =-888 if no match is found \nTable 2. 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Zaggia, SPRINGER, Astrophysics and Space Science Library N. 435 ( ISBN 978-3-319-31004-6 (print), ISBN 978-3-319-31006-0 (eBook)) DOI 10.1007/978-3-319-31006-0, p. 713 \nBianchi, L. 2016 in: 'From the Realm of the Nebulae to the Populations of Galaxies', editors M. D'Onofrio, R. Rampazzo, and S. Zaggia, SPRINGER, Astrophysics and Space Science Library N. 435 ( ISBN 978-3-319-31004-6 (print), ISBN 978-3-319-31006-0 (eBook)) DOI 10.1007/978-3-319-31006-0, p. 436 \nBianchi, L. 2014, Ap&SS, 354, 103; DOI: 10.1007/s10509-014-1935-6 \nBianchi, L., Conti, A. and Shiao, B. 2014a, J. Adv. Space Res., 53, 900 ; DOI:10.1016/j.asr.2013.07.045 ; astro-ph 1312.3281 (BCScat) Bianchi, L. et al. 2014b, J. Adv. Space Res., 53, 928; DOI: 10.1016/j.asr.2013.08.024 Bianchi, L. , 2012, in Proceedings of the International Astronomical Union Symp. 283, eds. A. Manchado, L. Stanghellini, and D. 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Rev. 45, 135 \nWenger, M., Ochsenbein, F., Egret, D., Dubois, P., Bonnarel, F. et al., Ap&SS, 2000, 143, p.9 ; HAL Id: insu-02889432 https://insu.hal.science/insu-02889432 \nSen, K., Langer, N., Marchant, P., et al. 2022, A&A, 659, A98 \nToonen S., Hollands M., Gansicke B. T., Boekholt T., 2017, A&A, 602, A16 \nLB acknowledges support from NASA ADAP grants 80NSSC19K0527 and NNX17AF35G. I am grateful to the referee whose comments prompted useful clarifications of the manuscript, to Bernie Shiao for helpful discussions about issues in the GALEX database, to Kareem El-Badry for insightful suggestions concerning Gaia data and the science results, and to Stavros Akras and Anastasios Karagiannis for beta-testing the catalog for possible applications. \nThis work includes results from the European Space Agency (ESA) space mission Gaia, Data Release 3. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. Gaia data were accessed through the MAST interface. \nFacilities: \nGALEX, MAST \nORCID iDs Luciana Bianchi https://orcid.org/0000-0001-7746-5461"}
2024arXiv240905156D	FullStokes polarimetric datasets originating from slitspectrograph or narrowband filtergrams are routinely acquired nowadays. The data rate is increasing with the advent of bidimensional spectropolarimeters and observing techniques that allow longtime sequences of highquality observations. There is a clear need to go beyond the traditional pixelbypixel strategy in spectropolarimetric inversions by exploiting the spatiotemporal coherence of the inferred physical quantities. We explore the potential of neural networks as a continuous representation of the physical quantities over time and space also known as neural fields for spectropolarimetric inversions. We have implemented and tested a neural field to perform the inference of the magnetic field vector approach also known as physicsinformed neural networks under the weakfield approximation WFA. By using a neural field to describe the magnetic field vector we can regularize the solution in the spatial and temporal domain by assuming that the physical quantities are continuous functions of the coordinates. We investigated the results in synthetic and real observations of the Ca II 8542 A line. We also explored the impact of other explicit regularizations such as using the information of an extrapolated magnetic field or the orientation of the chromospheric fibrils. Compared to the traditional pixelbypixel inversion the neural field approach improves the fidelity of the reconstruction of the magnetic field vector especially the transverse component. This implicit regularization is a way of increasing the effective signaltonoise of the observations. Although it is slower than the pixelwise WFA estimation this approach shows a promising potential for depthstratified inversions by reducing the number of free parameters and inducing spatiotemporal constraints in the solution.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.05156', 'arXiv:2409.05156', '2024arXiv240905156D']	['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics']	Exploring spectropolarimetric inversions using neural fields. Solar chromospheric magnetic field under the weakfield approximation	2,024	191	0.52	['EPRINT_HTML', 'EPRINT_PDF']	1	https://arxiv.org/pdf/2409.05156.pdf	{'Solar chromospheric magnetic field under the weak-field approximation': 'C. J. Díaz Baso 1 , 2 , A. Asensio Ramos 3 , 4 , J. de la Cruz Rodríguez 5 , J. M. da Silva Santos 6 , and L. Rouppe van der Voort 1 , 2 \n- 1 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway\n- 2 Rosseland Centre for Solar Physics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway\n- 3 Instituto de Astrofísica de Canarias, C / Vía Láctea s / n, E-38205 La Laguna, Tenerife, Spain\n- 4 Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain\n- 5 Institute for Solar Physics, Dept. of Astronomy, Stockholm University, AlbaNova University Centre, SE-10691 Stockholm, Sweden\n- 6 National Solar Observatory, 3665 Discovery Drive, Boulder, CO 80303, USA \ne-mail: \[email protected] \nDraft: compiled on September 10, 2024 at 12:51am UT', 'ABSTRACT': 'Context. Full-Stokes polarimetric datasets, originating from slit-spectrograph or narrow-band filtergrams, are routinely acquired nowadays. The data rate is increasing with the advent of bi-dimensional spectropolarimeters and observing techniques that allow long-time sequences of high-quality observations. There is a clear need to go beyond the traditional pixel-by-pixel strategy in spectropolarimetric inversions by exploiting the spatiotemporal coherence of the inferred physical quantities that contain valuable information about the conditions of the solar atmosphere. \nAims. We explore the potential of neural networks as a continuous representation of the physical quantities over time and space (also known as neural fields), for spectropolarimetric inversions. \nMethods. We have implemented and tested a neural field to perform one of the simplest forms of spectropolarimetric inversions, the inference of the magnetic field vector under the weak-field approximation (WFA). By using a neural field to describe the magnetic field vector, we can regularize the solution in the spatial and temporal domain by assuming that the physical quantities are continuous functions of the coordinates. This technique can be trivially generalized to account for more complex inversion methods. \nResults. We have tested the performance of the neural field to describe the magnetic field of a realistic 3D magnetohydrodynamic (MHD) simulation. We have also tested the neural field as a magnetic field inference tool (approach also known as physics-informed neural networks) using the WFA as our radiative transfer model. We investigated the results in synthetic and real observations of the Ca i i 8542Å line. We also explored the impact of other explicit regularizations, such as using the information of an extrapolated magnetic field, or the orientation of the chromospheric fibrils. \nConclusions. Compared to the traditional pixel-by-pixel inversion, the neural field approach improves the fidelity of the reconstruction of the magnetic field vector, especially the transverse component. This implicit regularization is a way of increasing the e ff ective signal-to-noise of the observations. Although it is slower than the pixel-wise WFA estimation, this approach shows a promising potential for depth-stratified inversions, by reducing the number of free parameters and inducing spatio-temporal constraints in the solution. \nKey words. Sun: atmosphere - Sun: chromosphere - Sun: magnetic fields - Methods: observational - Sun: activity - Radiative transfer', '1. Introduction': "During the past thirty years, inversion methods have proven to be one of the most robust ways of establishing a quantitative relation between the observed intensities and the underlying physical state of the atmospheric plasma. With the use of fast slit spectropolarimeters and 2D filterpolarimeters, maps of di ff erent regions in the solar atmosphere are routinely taken in which the four Stokes parameters are observed at several points along one or several spectral lines. The rate of new high-quality 2D observations is increasing with the advent of bi-dimensional spectropolarimeters and observing techniques that allow long time sequences of high-quality observations (Dominguez-Tagle et al. 2022; van Noort et al. 2022). With some exceptions (discussed below), in the overwhelming majority of studies available in the literature, such observations have been interpreted by assuming that all pixels are completely unrelated and by applying the inference \ntechniques (commonly known as inversion codes) on a pixel-bypixel basis. However, the spatial complexity of the observations is not as chaotic as one would expect from the high dimensionality of the data (spatial, temporal, spectral), but is rather coherent as a consequence of the physical processes that dominate the solar dynamics. \nFor example, the magnetic field in the chromosphere, where the magnetic pressure is larger than the gas pressure, tends to be rather smooth and slow-varying over space. However, polarimetric signals induced by those chromospheric magnetic fields are particularly weak, and in most cases very close to the detection limit of current instrumentation (e.g., Díaz Baso et al. 2019b; Yadav et al. 2021). Given these inherent properties, incorporating these characteristics in the inference would significantly help to constrain better the solar atmosphere. \nThis spectral and spatial coherence has been exploited to e ff ectively reduce the noise in solar and stellar spectropolarimetric observations (Martínez González et al. 2008; Díaz Baso et al. 2019a) and avoid averaging in time or space, which could lead to a loss of important information. The new generation of inversion codes is also starting to make use of this coherency to improve the fidelity of their algorithms. A study by van Noort (2012) proposed to couple the solution of neighboring pixels using the telescope point spread function. This inspired the recent development of a non-linear spatially-regularized and multi-resolution inversion technique by de la Cruz Rodríguez (2019). \nA di ff erent approach to take into account spatial coherency was presented by Asensio Ramos & de la Cruz Rodríguez (2015), using the concept of sparsity and compressibility, by linearly transforming the physical parameters to a di ff erent space in which their representation is compact. They used proximal algorithms (Parikh & Boyd 2014) to impose sparsity in the wavelet domain, decreasing the number of free parameters to reproduce the observables while simultaneously favoring spatial coherency. Recent studies (de la Cruz Rodríguez 2019, Morosin et al. 2020, de la Cruz Rodríguez & Leenaarts 2024) proposed adding spatiotemporal constraints by explicitly imposing a Tikhonov regularization on the physical parameters, thereby improving the fidelity of the reconstruction. At the same time, advances in deep learning have introduced the potential for data-driven regularizations. These methods encode complex priors from simulations into neural networks (Asensio Ramos & Pallé 2021; Liaudat et al. 2023). The ideas developed in the context of deep learning have also inspired other works, such as (Štˇepán et al. 2022, 2024), which propose to solve the 3D inversion problem by including the 3D non-local thermodynamic equilibrium consistency as a regularization, together with additional physical constraints and stochastic gradient descent techniques to mitigate issues of local minima. Lastly, automatic di ff erentiation frameworks, such as PyTorch (Paszke et al. 2019), facilitate the implementation of these ideas by e ffi ciently computing gradients and optimizing models to reproduce observations (De Ceuster et al. 2024). \nMotivated by the development of new instruments with an increasing field of view, we believe that implicit methods that describe the physical parameters in the whole domain with a compact representation can be of great help to reduce the dimensionality of the problem and, consequently, the computational load of the inference process. Here we investigate a di ff erent way of parametrizing the physical quantities by using a neural network as a continuous approximation to introduce spatio-temporal constraints. Recent works have demonstrated the potential of this idea in coronal tomography (Asensio Ramos 2023), source reconstruction under strong gravitational lenses (Mishra-Sharma & Yang 2022), magnetic field extrapolations (Jarolim et al. 2024) and interstellar chemistry (Asensio Ramos et al. 2024). These neural networks, usually termed implicit neural representations, neural fields (NF), or coordinate-based representations, are used to map coordinates on the space (or space-time) to coordinate-dependent field quantities. They have many desirable properties: they are e ffi cient in terms of the number of free parameters, they have controllable implicit bias, they produce di ff erentiable quantities that can be part of more elaborate optimizations, and they generate continuous signals that are ideal for imposing spatio-temporal constraints in noisy scenarios. In this work, we will study the case where the magnetic field is inferred under the weak-field approximation (WFA; Landi Degl'Innocenti & Landi Degl'Innocenti 1973), a powerful method to estimate the magnetic field from plage (da Silva Santos et al. 2023) to flare scenarios (Vissers et al. 2021) and simple enough to focus our attention on this particular \nnew implementation. The formalism also remains identical for LTE or non-LTE inversions of chromospheric lines. We plan to extend the use of NFs to more complex radiative transfer models in the near future. \nThe paper is organized as follows. We start with a brief introduction to their basic principles, and how we implement the new approach to perform spectropolarimetric inversions. Later we show the application of the NF on some examples and introduce some additional explicit regularizations. Finally, we provide a brief discussion about the implications of this work and outline potential extensions and improvements.", '2.1. Weak-field approximation': "The weak-field approximation (WFA; Landi Degl'Innocenti & Landi Degl'Innocenti 1973) is an analytical solution of the radiative transfer equation. This allows us to derive the emerging Stokes Q , U , and V parameters describing the polarization of the light as a function of Stokes I and its derivatives as a function of wavelength. The fundamental assumptions are that the magnetic field vector is constant with depth and that the splitting induced by the Zeeman e ff ect ( ∆ λ B ) is significantly smaller than the Doppler width of the line ( ∆ λ D ). This weak field regime occurs at di ff erent field strengths for di ff erent spectral lines (depending on the sensitivity to the magnetic field, the local temperature, etc). \nAt first order in the magnetic field strength, the relation between Stokes V and Stokes I is given by the following expression: \nV ( λ ) = -∆ λ B ¯ g cos Θ B dI d λ (1) \nwhere ∆ λ B = 4 . 6686 · 10 -13 λ 2 0 B , Θ B is the inclination of the magnetic field (angle between the observer's line-of-sight and the normal to the solar surface) and ¯ g is the Landé factor (Landi Degl'Innocenti & Landolfi 2004). The central wavelength of the line, λ 0, is given in Å while the magnetic field strength, B , is given in G. The same perturbation analysis allows obtaining Stokes Q and U , which only appear at second-order. In particular, we will use the equations that describe the dependence of Stokes Q and U in the wings ( λ ≫ ∆ λ D ) of the line: \nQ ( λ w ) = 3 4 ∆ λ 2 B ¯ G sin 2 Θ B cos 2 Φ B 1 λ w -λ 0 GLYPH<18> dI d λ GLYPH<19> U ( λ w ) = 3 4 ∆ λ 2 B ¯ G sin 2 Θ B sin 2 Φ B 1 λ w -λ 0 GLYPH<18> dI d λ GLYPH<19> (2) \nwhere ¯ G is a parameter that gives the magnetic sensitivity of linear polarization to B ⊥ , which depends on the quantum numbers of the transition (Landi Degl'Innocenti & Landolfi 2004). In both equations, there is a term depending on Φ B , which is the azimuth angle of the magnetic field with respect to a reference direction.", '2.2. Magnetic field inference': 'Once the model is set, we aim to infer the magnetic field vector from the interpretation of the observations of a set of spectral lines. Assuming that the weak-field approximation can be applied to the observed spectral lines and that observations are corrupted with uncorrelated Gaussian noise, we can use a least-squares estimator (maximum likelihood) to retrieve the magnetic field vector. The merit function L , the well-known χ 2 , can be defined for a particular pixel as the mean squared di ff erence of the observed \nFig. 1: Sketch of the NF approach. The physical quantities are described by a neural network that takes the coordinates as input and outputs the physical quantities at that point ( x , y ). The output of the network is then used to compute the observables from the model using a radiative transfer (RT) module (WFA in this case). The synthetic observables are compared with the observations and the error is back-propagated through the network to obtain the optimal solution. \n<!-- image --> \npolarization signals and the synthetic ones predicted by the model, normalized by the variance of the noise: \nL = L V + L Q + L U , (3) \nwhere \nL V = X i ( V obs i -V syn i ) σ 2 V , i , L Q = X i ( Q obs i -Q syn i ) σ 2 Q , i , L U = X i ( U obs i -U syn i ) σ 2 U , i . (4) \nThe sub-index i is used as a label for the spectral wavelength points. The previous merit function considers the general case in which the standard deviation is di ff erent for Stokes Q , U , and V . The formal simplicity of Eqs. (1) and (2) is one of the most important reasons why WFA has been useful. This linear dependence on the model parameters allows for an analytical optimization of the χ 2 (see, e.g., Martínez González et al. 2012). Since our approach here is more general (taking spatial correlation into account and considering more complex radiative transfer models in the near future), we do not use the analytical optimization of the χ 2 . Rather, we consider the numerical optimization of the χ 2 using gradient-based methods. \nFrom a practical point of view, using B , Θ B , and Φ B as free variables often leads to problems in the optimization. They are subject to several physical restrictions like the positive definiteness of B or are plagued with discontinuities (the azimuth is periodic in the interval [0 , π ], leading to problems during the optimization). For this reason, we opted to use the following variables, obtained as combinations of B , Θ B , and Φ B : \nB ∥ = B cos Θ B BQ = ( B sin Θ B ) 2 cos 2 Φ B BU = ( B sin Θ B ) 2 sin 2 Φ B . (5) \nThese three quantities are defined in ( -∞ , + ∞ ) and are continuous functions of the magnetic field vector. They also have the additional advantage of decoupling the problem into three unrelated sub-problems: Stokes V is only dependent on B ∥ , Stokes Q is only dependent on BQ and Stokes U is only dependent on BU . Therefore, one could solve each sub-problem independently. This is useful, for instance, if one Stokes parameter has much stronger signals than the rest (typically the case of Stokes V ), which could dominate the optimization process. In that case, one could use \ndi ff erent weights for each term in the merit function. Note that this last point is only valid for the WFA and not for the general case.', '2.3. Neural fields': "We propose here a general and powerful technique for the inference of magnetic field (and potentially thermodynamical parameters when using models more complex than the WFA) from observations using neural networks. The general idea is depicted in Fig. 1 and described in the following. We use a neural network, f θ ( x , y ), to describe the components of the magnetic field vector as a function of the coordinates. In this case, we describe the magnetic field components in a 2D plane in Cartesian coordinates ( x , y ). These coordinates represent the plane of 'e ff ective' formation height at which the polarization is generated. In the general case in which stratification and time evolution are taken into account, the input coordinates will be the ( x , y , z , t ). For convenience, we normalize all coordinates so that they are mapped to the interval [ -1 , 1]. The magnetic field components are then given by the following simple, but flexible, fully-connected neural network f θ : \nB = ( B ∥ , BQ , BU ) = f θ ( x ) , (6) \nwith θ the internal parameters of the neural network and x = ( x , y ). \nThis approach of describing the magnetic field using NFs has several advantages. First, the number of tunable parameters of the neural network can be potentially fewer than the number of unknowns in all pixels. This might not be especially relevant for the WFA model since there are only 3 unknowns per pixel. However, this will be crucial when applied to stratified inversions where we have many more physical quantities per pixel (temperature, velocity, magnetic field, microturbulence, etc.) in a very dense grid, not only spatial but also in the optical depth domain. Secondly, a NF is a global function in the space. This means that the information introduced by the observation of a single pixel informs the whole solution, leading to a very pronounced regularization e ff ect. This e ff ect is similar to the global character induced by the wavelet decomposition used by Asensio Ramos & de la Cruz Rodríguez (2015). Thirdly, a NF is a continuous and di ff erentiable function of the input coordinates. Therefore, the result of the inversion process is a continuous function that can be evaluated at any arbitrary point. Having a continuous and di ff erentiable magnetic field map can be greatly beneficial for the computation of current sheets though spatial derivatives of the magnetic field (see the application to inversion methods of Pastor Yabar et al. 2021), which otherwise will be greatly a ff ected by \nthe (potential) presence of noise and inversion artifacts. Finally, it is straightforward to include any explicit additional regularization term in the optimization that depends on the output or any derivative of the output with respect to the input coordinates.", '2.4.1. Parametric representation': 'Before showing the use of neural networks for the WFA problem, we will test first the performance of the neural network to describe the 2D spatial properties of the physical quantities of interest. To that end, we parameterize the magnetic field of a realistic 3D radiative magneto-hydrodynamics (rMHD) simulation. We have used one snapshot from a publicly available enhanced network simulation (Carlsson et al. 2016) performed with the Bifrost code (Gudiksen et al. 2011). Snapshots from this simulation have been extensively used in previous studies to test di ff erent diagnostic strategies (e.g., Leenaarts et al. 2012, 2013; Sukhorukov & Leenaarts 2017; Jurˇcák et al. 2018). The longitudinal magnetic field at 1500 km from the mean continuum formation layer is shown in the left panel of Fig. 2. This panel shows a chromospheric landscape with elongated magnetic features connecting two opposite-polarity patches. This image can be represented with the aid of a NF by minimizing the following loss: \nL B ∥ = X x , y GLYPH<0> f θ ( x , y ) -B ∥ ( x , y ) GLYPH<1> 2 , (7) \nwhere we utilize a fully-connected NN with ReLU (Rectified Linear Unit; Fukushima 1969) activation functions. Note that we do not train the neural network in the traditional manner for generalization across multiple datasets; instead, each network is uniquely optimized for a specific dataset to function as a tailored parametric tool, not as a generalized predictive model. The converged solution is found in the second panel of Fig. 2. The solution only approximates the general behavior of the original magnetic field distribution but is too smooth. This smoothness is a direct consequence of the implicit bias of NN, also known as spectral bias (Rahaman et al. 2018), which prevents standard networks from learning high-frequency functions 1 . This is an active area of research, and several strategies have been proposed to alleviate this problem and introduce an implicit bias towards high-frequency signals. One of the most successful strategies is to pass the input coordinates through a Fourier feature mapping γ ( x ), which allows the NF to correctly generate high spatial frequencies (Tancik et al. 2020). This mapping projects the input coordinates onto a high-dimensional space with a set of trigonometric functions: \nγ ( x ) = [cos(2 π Gx ) , sin(2 π Gx )] T (8) \nwhere G ∼ N (0 , ω ), i.e., each entry of the vector G is a frequency sampled from a Gaussian distribution with zero mean and standard deviation ω . The parameter ω controls the range of spatial frequencies that the network can reproduce. This will be helpful as a regularization, as we show later. After computing the Fourier features, we pass them through the neural network to obtain the magnetic field. By using a Fourier mapping, we can e ffi ciently reproduce both low and high spatial frequencies, being able to represent the magnetic field with a high fidelity. The third and fourth panels of Fig. 2 show the results of using the Fourier \nfeatures with di ff erent values of ω . Using ω = 10 (third panel), allows us to capture the lower spatial frequencies, but by increasing it to ω = 60 (fourth panel) we can reliably reproduce all the high-frequency spatial details of the magnetic field map. In the bottom right of each panel, we also quantify the mean absolute error (MAE) between the original simulation and the di ff erent results, decreasing from an average error of 5 G with the ReLU NNto 0.09 G using the NF ( ω = 60). We note that other strategies to alleviate the spectral bias exist. It is worth mentioning that the use of suitable activation functions, such as sinusoids, has also been shown to be very e ff ective in generating high-frequency functions (SIREN; Sitzmann et al. 2020). All these strategies can be seen as a nonlinear extension of Fourier series. \nIt is important to stress the fact that the number of learnable parameters of the NF is not dependent on the number of pixels in the observations, but rather on the properties of the spatial distribution of the physical quantities. In the example shown in Fig. 2, although the number of pixels is 500 × 500, the number of parameters of the network is ten times smaller, amounting to just 25k. This compact representation is arguably associated with the fact that the magnetic field in the chromosphere, where the magnetic pressure is larger than the gas pressure, tends to be rather smooth and slow-varying over space. This is not to be expected for other quantities, such as the temperature or the velocity, which vary at much smaller spatial scales. \nThe representations shown in Fig. 2 have been trained by optimizing the loss function of Eq. (7) computed for all pixels in the field-of-view (FoV). However, since the NF is a global function in x , the loss in a given pixel contains some information about the properties of the NF in the surroundings. For this reason, one can train the NF using mini-batches of pixels randomly chosen in the FoV, instead of summing over the whole FoV. To show this, Fig. 3 shows the mean absolute error as a function of iteration when di ff erent mini-batches are selected for each iteration. We can see that the convergence properties when using 10% of the total pixels are already as good as those obtained when using the whole FoV, but 10 times faster in terms of computing time. The sudden decrease in the merit function during the optimization is produced by the scheduler, a module that decreases the learning rate (step size) if there is no improvement after some iterations.', '2.4.2. Magnetic field reconstruction': 'The previous experiment demonstrates that a NF can correctly recover the details of the quantities of interest by their direct observation. But in spectropolarimetry, we only have access to the Stokes profiles and one has to pass through the radiative transfer model to infer the physical quantities. Here we demonstrate that this can be done by interpreting synthetic observations of the Ca i i 8542Å line. We have chosen this line because the WFA approximation on this line is reliable for field strengths up to ∼ 1200 G (Centeno 2018). For this test, we have created synthetic observables from the Bifrost simulation and we run our WFA NF to reproduce the polarization signals that emerge from the simulation. Our goal is to assess the improvement delivered by our new method when compared to the traditional pixel-by-pixel WFA. To this end, and to discard any influence of the line formation details, we follow Morosin et al. (2020) and set the magnetic field in each pixel equal to its vertical average from z = 1000 km to z = 1500 km. Heights are measured assuming that z = 0 km corresponds to the mean continuum formation layer. We compute the synthetic observables using the non-local thermodynamic \nFigure 4 shows the reconstruction of the magnetic field vector in the synthetic case used before in Fig. 2. The first column shows the original magnetic field vector, given in terms of B ∥ , B ⊥ , and Φ B . The second column shows the inferred magnetic field using \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n- \n60 \nFig. 2: Comparison of the parametric representation power of each method. From left to right: the original longitudinal magnetic field at z = 1500 km in the Bifrost simulation, represented using a ReLU neural network, and using a NF with ω = 10 and ω = 60.Fig. 3: Performance of the NF when evaluating di ff erent batches of pixels randomly chosen in the field of view in every iteration. \n<!-- image --> \nequilibrium (NLTE) radiative transfer STiC code 2 (de la Cruz Rodríguez et al. 2016, 2019) and add uncorrelated Gaussian noise to the Stokes Q , U , and V profiles. It is important to note that the average magnetic field of this particular snapshot is about 50-100 G, a value much lower than the typical magnetic field found in plage ( ∼ 400 G, Pietrow et al. 2020; Morosin et al. 2022). Consequently, to avoid the signal being buried in the noise, we use a standard deviation of the noise lower than the typical one found in solar observations. \nThe architecture chosen for the NF is a residual network (ResNet, He et al. 2015) with 2 residual blocks, 64 nodes per layer, and the ELU activation function (Exponential Linear Unit; Clevert et al. 2015). This architecture has proven its e ff ectiveness by showing e ffi cient training and good performance in capturing complex patterns. For the Fourier feature mapping, we have used ω = 60 for the longitudinal magnetic field and ω = 30 for the transverse components. The reason is that the latter is typically more a ff ected by the noise. Using a smaller value of ω encourages the NF not to fit the noise, which is of very high spatial frequency. The magnetic field generated by the NF is used to compute the synthetic observables via the WFA approximation, which are then compared with the observations. In particular, for the longitudinal \nmagnetic field, and using Eq. (1), we have used the following loss function: \nL V = X x , y X λ -CVB ∥ ( x , y \| θ ) dI obs x , y d λ -V obs x , y 2 (9) \nwhere CV = 4 . 6686 × 10 -13 ¯ g λ 2 0 is a constant for the line of interest, B ∥ ( x , y \| θ ) is the NF used to represent the longitudinal field, dI obs x , y / d λ is the derivative of the observed Stokes I profile at pixel ( x , y ), and V obs x , y is the observed Stokes V signal at the same pixel. The derivative of the intensity does not change during the optimization, so it can be pre-calculated and stored before the optimization process occurs. \nFor the transverse components, using Eq. (2), the loss functions are given by: \nL Q = X x , y X λ CQU · BQ ( x , y \| θ ) · 1 λ -λ 0 dI obs x , y d λ -Q obs x , y 2 (10) \nL U = X x , y X λ CQU · BU ( x , y \| θ ) · 1 λ -λ 0 dI obs x , y d λ -U obs x , y 2 (11) \nwhere CQU = 1 . 6347 × 10 -25 λ 4 0 ¯ G . Thanks to the definition of the BQ and BU variables, the losses can be optimized separately. This helps in imposing di ff erent implicit or explicit regularization methods for every variable. \nThe optimization of every loss function with respect to θ is carried out using the Adam optimizer with a learning rate of 3 · 10 -4 for 100 -500 epochs (depending on the dataset). The derivatives for the backpropagation through the network and the physical model (WFA) are carried out with the automatic di ff erentiation package PyTorch. Finally, after convergence of the NFs, one can recover the full magnetic field vector from B ∥ , BQ , and BU by a final transformation: \nB ∥ ( x , y ) = B ∥ ( x , y \| θ ) B ⊥ ( x , y ) = h BQ ( x , y \| θ ) 2 + BU ( x , y \| θ ) 2 i 1 / 4 Φ B ( x , y ) = arctan BU ( x , y \| θ ) BQ ( x , y \| θ ) ! . (12) \n[pixels] \ny \nx [pixels] \n[G] \nFig. 4: Comparison of the reconstruction of the magnetic field vector from the synthetic Ca i i 8542Å spectra calculated from the simulation (in rows: line-of-sight component, transverse component, and azimuth angle). From left to right: original magnetic field from the simulation, magnetic field inferred by the pixel-wise WFA, and results using the WFA NF. \n<!-- image --> \nthe traditional pixel-based WFA and the third row shows the inferred magnetic field using the NFs. The quality of the results is a ff ected, fundamentally, by the noise level and the regularization properties of NF. We have tested the performance of the NF for di ff erent noise levels and values of ω , finding the same consistent behavior. The results summarized in Fig. 4 are for the particular configuration with a noise level of 10 -3 for Stokes V and 10 -4 for Stokes Q and U , both given in units of the continuum intensity. The longitudinal field inferred with the NF is very similar to that of the traditional WFA in general terms. The NF strongly damps the high-frequency components of the magnetic field associated with the presence of noise, but the overall structure is very similar. The key reason is that the longitudinal magnetic field is a quantity that is well constrained by the observations even in the presence \nof noise because the Stokes V signals are typically above the noise and the estimated value is statistically unbiased and coincides with the original value (see Martínez González et al. 2012). \nThe NF produces a much better result for the transverse component and the azimuth of the magnetic field, which are much closer to the real ones than those obtained with the traditional WFA. In this case, the implicit spatial regularization of the NF is able to provide a much better solution. The pixel-by-pixel WFA tends to overestimate the transversal component of the magnetic field, producing a background component over the FoV which is proportional to the noise level. This e ff ect is well-known and produced because the maximum-likelihood solution is biased Martínez González et al. (2012). The standard approach to deal with this bias is to avoid the regions where this e ff ect has a strong \nimpact on any subsequent analysis. Here we show that the NF is able to provide a better estimate of the magnetic field by finding a solution that is more coherent with the surroundings. This produces a mitigation of the bias of the transversal component. At the bottom right of each panel, we have quantified both approaches, retrieving an average error 2.5 -4.5 times lower with the NF WFA compared with the pixel-wise approach. Our results confirm the advantages of the spatial regularization found by Morosin et al. (2020). Note that these small errors do not fully encapsulate the complexities found in actual observations, which are influenced by various systematic factors such as noise or spectral and spatial degradation. Nonetheless, within our idealized setting, the inference using the NF WFA is indicative of the potential improvements. Lastly, our calculations provide two more conclusions. The first one is that, if we decrease the value of ω , the NF tends to produce an excessively smooth solution. Although this is bad in general, it can be an advantage in very noisy observations. The second one is the finding that the larger the noise level, the better the reconstruction of the magnetic field using the NF WFA is compared to the traditional WFA (see Fig. A.1 in the Appendix for a comparison using an increased noise level).', '3.1. Example with real observations': 'After confirming with simulations that the WFA NF is able to recover the magnetic field with a higher fidelity than the standard pixel-by-pixel WFA in noisy cases, we apply it to real observations. We have used observations (Leenaarts et al. 2018; Yadav et al. 2023) from the active region NOAA 12593 observed on 2016-09-19 between 09:31:29 UT and 09:57:03 UT with the CRISP (Scharmer et al. 2008) instrument at the Swedish 1-m Solar Telescope (SST; Scharmer et al. 2003). The selection of the regularization frequencies ω was guided by empirical testing: we experimented with di ff erent values to find an optimal balance that minimizes noise while preserving critical features of the magnetic field, particularly the transverse component. The results from the analysis of the first time frame are shown in Fig. 5. The first row shows the results with the pixel-wise WFA and the second row displays those obtained when applying the WFA NF. As expected, since the Stokes V signals are well above the noise, the inference with both methods is very similar. In contrast, the pixel-wise WFA tends to estimate the transverse magnetic field larger than 1000 G in many locations of the field of view. This is the mentioned bias e ff ect, with very strong values because in the presence of a hot magnetic canopy, the line source function is very shallow in the chromosphere and the resulting line profile has a flat line core (see Appendix A of Morosin et al. 2020). In other words, the pixel-wise WFA compensates the almost-zero Stokes I derivative with a very large magnetic field value. This e ff ect is more noticeable in Fabry-Pérot observations due to the poor sampling of Stokes I , as shown in Díaz Baso et al. (2023). The NF, on the other hand, is able to provide a more coherent solution in the spatial domain without using unrealistic values. The maximum values are now below 800 G, where most of them are in the edge of the lower-right polarity (penumbra of the sunspot) and in between the small polarities in the FoV.', '3.2. Challenging case and additional spatial regularization': 'We have used observations of the active region NOAA AR 12723 recorded on 2018-09-30 in the Ca i i 8542 Å line by Vissers et al. (2022) with the CRISP instrument at the SST consisting of a \nmosaic of four overlapping pointings. This target was selected to explore the behavior of the NF when the magnetic field is particularly weak, but with a simple enough connectivity to investigate what additional information we could add to the inference problem. In this case, the NF WFA does a job similar to the pixel-wise WFA because the magnetic field is mainly concentrated at the sunspots and decreases rapidly for increasing distances from the sunspots. However, the formulation of the inversion process in terms of a NF allows us to include additional regularizations. \nThe results of the inversions are shown in Fig. 6. The longitudinal magnetic field, the transverse magnetic field, and the azimuth of the magnetic field are shown in the first, second, and third columns, respectively. The results clearly show that a NF is marginally regularizing the inference. The quality of the inferred magnetic field far from the sunspots is similar between both the NF WFA and the pixel-wise WFA. The estimation of the magnetic field is slightly more spatially coherent, as expected, but the overall structure is very similar. This is a clear example of a limiting case in which there is not enough information to constrain the solution. \nIn order to improve the magnetic field estimation, we have implemented an explicit regularization. It is implemented by guiding the NF to simultaneously obey another physical constraint, apart from fitting the spectropolarimetric observations. This constraint is implemented as a regularization term in the loss function parameterized with a hyperparameter λ , which penalizes the distance between the magnetic field configuration and the magnetic field configuration from the external source of information. Given that the longitudinal magnetic field is well constrained, we will focus here on adding a regularization to the transverse component, i.e.: \nL = L Q + L U + λ L reg . (13) \nAs a first example, the third row of Fig. 6 shows the result when L reg = L pot, where L pot is the mean squared di ff erence between the NF and a pre-calculated potential field extrapolation B pot from the well-constrained longitudinal magnetic field component. One would want to calculate the di ff erence between these two magnetic fields defining L pot following: \nL pot = X x , y GLYPH<16> B ⊥ -B ⊥ , pot GLYPH<17> 2 + X x , y GLYPH<16> Φ B -Φ B , pot GLYPH<17> 2 . (14) \nHowever, as we have not disambiguated the magnetic field azimuth, we have to calculate the distance between the corresponding analog quantities BU , pot, and BQ , pot : \nL pot = X x , y GLYPH<16> BQ -BQ , pot GLYPH<17> 2 + X x , y GLYPH<16> BU -BU , pot GLYPH<17> 2 . (15) \nWhen the value of λ is increased during the optimization, the NF starts to incorporate the information of the potential field extrapolation. However, even with a small regularization value, we can already see from Fig. 6 that the azimuth of the magnetic field becomes more spatially coherent, but the new transverse magnetic field becomes much stronger than the one estimated only from reproducing the polarization signals. This experiment shows that this particular region is far from being potential and incorporating this information will worsen the fits as soon as we increase the value of λ . More complex extrapolations can be performed but the idea here is to show how to incorporate external information into the inference problem. \nFinally, we also explore the idea of incorporating the orientation of the chromospheric fibrils as if they were aligned with \nFig. 5: Magnetic field reconstruction from a Ca i i 8542Å observation using the pixel-wise WFA (top row) and using the NF WFA with ω = 120 for B ∥ and ω = 80 for B ⊥ & Φ B (bottom row). Columns show the magnetic field in terms of the longitudinal component, the transverse component, and the azimuth angle. \n<!-- image --> \n‖ \n⊥ \nthe magnetic field. In fact, this idea has been used to improve nonlinear force-free modeling of coronal fields (Wiegelmann et al. 2008), given the limitations of the force-free assumption of the photospheric boundary (DeRosa et al. 2015). Extrapolations performed starting from a chromospheric vector boundary condition (Fleishman et al. 2017) or starting from the photosphere and adding even an incomplete set of chromospheric magnetic field data (Fleishman et al. 2019) can measurably improve the reconstruction of the coronal magnetic field, connectivity, and electric currents. This information can potentially also improve the inference of the magnetic field, especially in areas away from strong magnetic field concentrations. \nTo calculate the distance between the magnetic field azimuth Φ B and the direction of the fibrils Φ fib, we need to convert all angles to corresponding points on the unit circle to avoid the ambiguity problem, and then we can compute the distance of \nthese points. The fourth row of Fig. 6 shows the result when L reg = L fib, with \nL fib = X x , y [sin(2 Φ B ) -sin(2 Φ fib)] 2 + X x , y [cos(2 Φ B ) -cos(2 Φ fib)] 2 , (16) \ni.e., the mean squared di ff erence between the NF azimuth and the direction of the fibrils, as calculated from the intensity image. In order to infer the orientation of the fibrils from the observations we have used the following procedure: i ) we have used the core of the Ca i i 8542 Å line to detect the fibrils in the chromosphere 3 , \n‖ \n⊥ \n[pixels] \ny \nFig. 6: Magnetic field inference from a Ca i i 8542Å observation using the pixel-wise WFA, the NF WFA, and two additional approaches introducing an explicit regularization term: using the information of a potential extrapolation of the magnetic field (third row) and using the orientation of the fibrils (fourth row). The first column shows the longitudinal magnetic field, the second column shows the transverse magnetic field and the third column shows the azimuth of the magnetic field. \n<!-- image --> \ni i ) we apply a Sobel operator in each axis and take the arctangent to retrieve the orientation of the fibrils, which is collapsed to the range (0,180) degrees to avoid the 180 degrees ambiguity and i i i ) a Gaussian filter is applied to remove small artifacts at the edges of the fibrils (see Fig. A.2 in the Appendix for more information). As a result of the inference, Fig. 6 shows a magnetic field that is aligned in general with the fibrils while still reproducing the polarization signals. The middle panel shows that after incorporating the orientation of the fibrils, the transverse component remains almost the same. In fact, the estimation of the orientation of the fibrils fails in the umbra where fibrils are not visible (see Fig. A.2) \nbut the strong polarization signals are enough to compensate for that. This guided inferred magnetic field can be a much better boundary condition for coronal field extrapolations. \nOther potential regularizations that we should explore in the future are forcing the divergence-free condition or the suppression of strong electric currents. Both constraints can be computed from derivatives of the output of the neural network with respect to the input coordinates, which can be computed e ffi ciently with techniques from automatic di ff erentiation. This could allow us to resolve the Zeeman-180 degree azimuth ambiguity at the same time we are reproducing the spectra. \npixel-wise \n[G] \nneural-field \n[G] \nFig. 7: Longitudinal magnetic field (upper row) and transverse component (lower row) inferred using the pixel-wise WFA and the temporal WFA NF for a time series of the active region NOAA 12593 observed on 2016-09-19 with the SST / CRISP. The rightmost column shows the temporal evolution of the magnetic field for a particular pixel indicated as the intersection of the dashed lines in the middle panel. \n<!-- image -->', '3.3. Temporal regularization': 'The NF can be easily extended to the case where the magnetic field is not static but evolves with time. By imposing temporal coherence in the solution, one can obtain a better estimate of the magnetic field, as shown by de la Cruz Rodríguez & Leenaarts (2024). From a technical point of view, adding the time dependence can be done by adding t as an additional input parameter of the NF: \nB = ( B ∥ , BQ , BU ) = f θ ( x , t ) . (17) \nApart from this change, inferring the magnetic field proceeds exactly as before, with the only change that the optimization process requires minimizing the loss function over the spatio-temporal domain. The number of unknowns in the explicit temporal regularization of de la Cruz Rodríguez & Leenaarts (2024) scales linearly with the number of observed time steps ( nt ) since it infers the value of the magnetic field at all nxny pixels of the field-ofview. This requires solving a very large linear system of equations of size nxnynt × nxnynt . On the contrary, the NF produces a much more compact representation of the magnetic field because it only \nrequires adding a few weights from the input layer to the Fourier feature mapping layer. Since the magnetic field contains higher frequencies in the spatial directions than in the temporal direction, we implement the Fourier feature mapping with ω t for the time and ω xy for the spatial coordinates, with ω t < ω xy . \nTo showcase this approach, we have used a 26-minute time series of the active region NOAA 12593 described in Sec. 3.1. We optimize the temporal WFA NF on the Ca i i 8542 Å observations of this time series. For this particular example, we have used a temporal regularization of ω t = 3 and a spatial regularization of ω xy = 80. Higher values prevented the neural network from converging correctly. The results of the inference are shown in Fig. 7 for the longitudinal magnetic field and the transverse component. The rightmost column shows the temporal evolution of the magnetic field component for the particular pixel shown in the middle panel. \nIn the case of the longitudinal magnetic field (first row of Fig. 7), the NF is able to capture the details of the spatial distribution of the emerging flux region. In the extracted pixel, the NF is also able to capture the temporal evolution of the magnetic \nfield. The fluctuations of pixel-wise inversion are of the order of ∼ 100 G, which are compatible with the noise amplitude of the observations. This makes us confident that the implicit bias introduced in the NF with the small value of ω t correctly captures the time variation of the longitudinal component of the magnetic field. However, since this particular region is very dynamic, one could argue that a less restrictive temporal regularization could be needed to properly capture the details of the small-scale changes and a higher ω t would be necessary in this type of scenario. \nIn the case of the transverse magnetic field (second row of Fig. 7), there is a much larger di ff erence between the output of the NF and that of the traditional pixel-wise WFA. The traditional WFA tends to estimate a transverse magnetic field larger than 1000 G in the center of the FoV far from the sunspot, where we have weak polarization signals. This is again the bias produced by the noise described before. The NF, on the other hand, is able to provide a more coherent solution both spatially and temporally. Both approaches retrieve very similar magnetic field values where the signals are stronger. When compared with the previous section where only spatial regularization was employed, temporal regularization is particularly e ff ective in reducing the background noise (de la Cruz Rodríguez & Leenaarts 2024).', '4. Summary and conclusions': "In this study, we investigated the use of neural fields to parameterize the magnetic field for magnetic field inference. We have shown the capabilities of NFs to solve static and time-dependent magnetic field inferences. This implementation 4 comes with several important advantages. \nFirst, by reformulating the problem as a global problem (instead of pixel-wise independent problems), the NF can produce a much more compact representation, decreasing the number of parameters to optimize. This reduction generally depends on both the spatial complexity of the inferred physical quantities and the e ffi ciency of the chosen neural architecture to represent such complexity. We believe that, for three-dimensional cases like the inversion of Stokes profiles with stratified atmospheres, NFs will represent an exceptionally compact representation of the physical conditions (see, for example, Asensio Ramos 2023). So, we anticipate that NFs will become crucial as new modern instruments and telescopes are providing bigger FoV of more complex data (e.g. at the SST, the recently installed CRISP cameras o ff er a FoV diameter of 87 '' and the forthcoming CRISP2 will o ff er a FoV of 2 arcminutes). \nSecond, using a NF to describe the magnetic field as a continuous di ff erentiable function allows us to obtain a better estimation of the magnetic field in places where the signals are buried in the noise, without smearing out the details of locations where the signals are strong. Choosing adequate regularization frequencies ω is crucial to obtain results that are not overly smooth. These frequencies should be chosen so that the quality of the fit is not degraded but still maintains sharp gradients in locations where the signals significantly impact the quality of the fit. In fact, NFs seem to be key in the inference of the transverse magnetic field, which the pixel-wise WFA tends to overestimate. This new implementation can help us to better understand and quantify chromospheric heating and its relation with the strength of the horizontal magnetic field in the low chromosphere (Leenaarts et al. 2018; Díaz Baso et al. 2021; da Silva Santos et al. 2022). \nThird, the present work is based on a simple forward model for the Stokes profiles. A NF version of the WFA is not realistically competitive in terms of speed with other WFA implementations. For instance, the explicitly regularized WFA implementation of Morosin et al. (2020) can estimate the magnetic field of the FoV in less than a second, while the NF version requires ∼ 30 seconds. The temporal regularization takes also more time, almost ∼ 3 minutes in an o ff -the-shelf GPU, compared with ∼ 10 minutes in a CPU. The main reason for this large processing time is the relatively slow convergence of the NF given that it is optimized using first-order gradient-based techniques. On the other hand, the loss function in spectropolarimetric inversions is often optimized using (quasi-) second-order methods such as the Levenberg-Marquardt algorithm. Second-order methods require the construction of a very large approximate Hessian matrix that can impact memory consumption. For instance, for an observation with 1000 × 1000 pixels and 50 time steps and using a Milne Eddington model ( ∼ 10 free parameters), the Hessian matrix is of size 5 × 10 8 × 5 × 10 8 . In any case, the global Hessian matrix is relatively sparse. Depending on the sparsity degree of the Hessian matrix, the resolution of the coupled linear system of equations can pose a severe computational challenge, even when e ffi cient iterative methods are used (e.g., GMRES or BICGSTAB). Imposing nearest neighbor regularization yields a very compact and sparse matrix that can be e ffi ciently inverted. More dense cases, originating for example from the inclusion of an extended spatial point-spread-function or horizontal coupling by 3D radiative transfer e ff ects, could be trivially included in this framework, whereas it is not trivial to model the inverse problem. \nFourth, modern automatic di ff erentiation frameworks, like PyTorch, allow us to seamlessly use more complex forward models, such as a Milne-Eddington model, or solve the non-LTE inversion problem in a stratified atmosphere. In Asensio Ramos & Díaz Baso 2019, we already anticipated that traditional predictive neural networks, despite their remarkable speed, are not explicitly fitting Stokes profiles and the introduction of a di ff erentiable forward synthesis would notably increase the optimization time. Considering that the forward models are the bottleneck in complex spectropolarimetric inversions, emulators of the radiative transfer model (e.g., Asensio Ramos et al. 2024) emerges as a crucial strategy to speed up the calculations. Additionally, these frameworks allow us to seamlessly speed up the calculations using GPUs. Adding extra regularization terms to guide the solution is straightforward in these frameworks. As mentioned, by improving the estimation of the magnetic field at the chromospheric level, we can perform better coronal magnetic field extrapolations, and the chromospheric fibrils contain valuable information about the non-potentiality of the magnetic field which should be integrated into the inference (Jing et al. 2011). Apart from the ones used in this work (alignment of the magnetic field with the chromospheric fibrils or similarity to a pre-computed magnetic field extrapolation), one can think of adding Tikhonov regularization on the physical quantities or their spatial derivatives, sparsity constraints using linear transformation like wavelets (e.g., Asensio Ramos &de la Cruz Rodríguez 2015) or a divergence-free constrain. By extending this model to include additional constraints we could potentially resolve the Zeeman-180 degree azimuth ambiguity directly within the inversion process (Štˇepán et al. 2022, 2024). This would enhance the accuracy and reliability of the inferred magnetic field vectors. \nFinally, a natural extension would be a probabilistic reconstruction (i.e., providing the uncertainty of the magnetic field) by using gradient-based variational inference, as shown by MishraSharma & Yang (2022) in the context of gravitational lens recon- \nstruction. This could be accomplished, for example, by modeling the magnetic field distribution using a normalizing flow (Díaz Baso et al. 2022). Another point to explore is the accuracy in reproducing very high-frequency features in the observations because our Fourier Feature layer becomes less stable when we increase the complexity of the problem. A way of mitigating this problem can be using a multi-scale representation (Dolean et al. 2023; Saragadam et al. 2022) or add higher frequencies progressively as the training progresses (Asensio Ramos et al. 2024). Having demonstrated the potential of the method in a proof-of-principle setting, we leave these extensions to future work. \nIn summary, NFs allow us to improve the reconstruction of the magnetic field properties of the solar atmosphere, especially suitable for imaging spectropolarimeters with large fields of view and a scarce wavelength sampling but also for integral-field spectropolarimeters (van Noort et al. 2022; Rouppe van der Voort et al. 2023) where, although the FoV is smaller, the temporal cadence is very high and the temporal regularization will hold much better. These properties make NFs valuable for the data taken with the next generation of telescopes such as the existing Daniel K. Inouye Solar Telescope (DKIST; Rimmele et al. 2020) and the upcoming European Solar Telescope (EST; Quintero Noda et al. 2022). \nAcknowledgements. CJDB acknowledges M. L. DeRosa for valuable discussions on future application of NF inversion methods. This research is supported by the Research Council of Norway, project number 325491, and through its Centres of Excellence scheme, project number 262622. AAR acknowledges financial support from the Agencia Estatal de Investigación del Ministerio de Ciencia, Innovación y Universidades (MCIU / AEI) and the European Regional Development Fund (ERDF) through project PID2022-136563NB-I00. This project has been funded by the European Union through the European Research Council (ERC) under the Horizon Europe program (MAGHEAT, grant agreement 101088184). The NSO is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. The Swedish 1-m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. The Institute for Solar Physics is supported by a grant for research infrastructures of national importance from the Swedish Research Council (registration number 2021-00169). We acknowledge the community e ff ort devoted to the development of the following open-source packages that were used in this work: numpy ( numpy.org ), matplotlib ( matplotlib.org ), scipy ( scipy.org ), astropy ( astropy.org ) and sunpy ( sunpy.org ). This research has made use of NASA's Astrophysics Data System Bibliographic Services.", 'References': "- Asensio Ramos, A. 2023, Sol. Phys., 298, 135\n- Asensio Ramos, A. & de la Cruz Rodríguez, J. 2015, A&A, 577, A140\n- Asensio Ramos, A. & Díaz Baso, C. J. 2019, A&A, 626, A102\n- Asensio Ramos, A. & Pallé, E. 2021, A&A, 646, A4\n- Asensio Ramos, A., Westendorp Plaza, C., Navarro-Almaida, D., et al. 2024, MNRAS, 531, 4930\n- Carlsson, M., Hansteen, V. H., Gudiksen, B. 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From left to right: original magnetic field from the simulation, magnetic field inferred by the pixel-wise WFA and results using the WFA NF. \n<!-- image --> \nFig. A.2: Steps to retrieve the orientation of the fibrils using the intensity of the Ca i i line: the orientation (middle panel) is obtained by applying Sobel filters to the monochromatic image at the core (top panel), which is later filtered (bottom panel). \n<!-- image -->'}
2024arXiv240113742T	Our knowledge of relations between supermassive black holes and their host galaxies at zgtrsim1 is still limited even though being actively sought out to zsim6. Here we use the high resolution and sensitivity of JWST to measure the host galaxy properties for 107 Xrayselected typeI AGNs at 0.68ltzlt2.5 with restframe opticalnearinfrared imaging from COSMOSWeb and PRIMER. Black hole masses logleftMrm BHModotrightsim6.99.6 are available from previous spectroscopic campaigns. We extract the host galaxy components from four NIRCam broadband images and the HSTACS F814W image by applying a 2D image decomposition technique. We detect the host galaxy for sim90 of the sample after subtracting the unresolved AGN emission. With host photometry free of AGN emission we determine the stellar mass of the host galaxies to be logleftMModotrightsim9.511.6 through SED fitting and measure the evolution of the mass relation between SMBHs and their host galaxies. Considering selection biases and measurement uncertainties we find that the Mmathrm BHM ratio evolves as left1zright0.480.620.31 thus remains essentially constant or exhibits mild evolution up to zsim2.5. We also see an amount of scatter sigmamu0.300.140.13 similar to the local relation and consistent with lowz studies and a noncausal cosmic assembly history where mergers contribute to the statistical averaging towards the local relation is still feasible. We highlight improvements to come with larger samples from JWST and particularly Euclid which will exceed the statistical power of current wide and deep surveys.	2024-01-01T00:00:00Z	['arXiv:2401.13742', '2024arXiv240113742T', '10.48550/arXiv.2401.13742']	['Astrophysics - Astrophysics of Galaxies']	The Mrm BHM relation up to zsim2 through decomposition of COSMOSWeb NIRCam images	2,024	191	0.61	['EPRINT_HTML', 'EPRINT_PDF']	15	https://arxiv.org/pdf/2401.13742.pdf	{'No Header': '9', 'The M BH -M ∗ relation up to z ∼ 2 through decomposition of COSMOS-Web NIRCam images': "<!-- image --> \n- 1 Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan\n- 2 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan\n- 3 Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan \n4 \nCenter for Astrophysical Sciences, Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA \n5 \nMax-Planck-Institut fur Astronomie, Konigstuhl 17, D-69117 Heidelberg, Germany \n6 School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel \n7 NASA-Goddard Space Flight Center, Code 662, Greenbelt, MD, 20771, USA \n8 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China \nTechnical University of Munich, TUM School of Natural Sciences, Department of Physics, James-Franck-Str. 1, D-85748 Garching, \nGermany \n10 Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany \n11 \nINAF-Osservatorio Astronomico di Roma, Via Frascati 33, I-00078, Monte Porzio Catone, Italy \n12 Caltech/IPAC, 1200 E. California Blvd. Pasadena, CA 91125, USA \n13 Cosmic Dawn Center (DAWN), Denmark \n14 DTU Space, Technical University of Denmark, Elektrovej, Building 328, 2800, Kgs. Lyngby, Denmark \n- 15 Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA\n- 16 Institute for Physics, Laboratory for Galaxy Evolution and Spectral Modelling, EPFL, Observatoire de Sauverny, Chemin Pegasi 51, 1290 Versoix, Switzerland \n17 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA \n18 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, NL-9700AV Groningen, the Netherlands \n19 DTU-Space, Technical University of Denmark, Elektrovej 327, DK2800 Kgs. Lyngby, Denmark \n20 \nNiels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200 Copenhagen N, Denmark \n21 Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, Padova, I-35122, Italy \n22 The University of Texas at Austin, 2515 Speedway Boulevard Stop C1400, Austin, TX 78712, USA \nDepartment of Physics and Astronomy, University of Hawaii, Hilo, 200 W Kawili St, Hilo, HI 96720, USA \n24 \nDepartment of Computer Science, Aalto University, PO Box 15400, Espoo, FI-00 076, Finland \n25 \nDepartment of Physics, Faculty of Science, University of Helsinki, 00014-Helsinki, Finland \n26 Laboratory for Multiwavelength Astrophysics, School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623, USA \n27 Max-Planck-Institut fur extraterrestrische Physik, Gießenbachstraße 1, 85748 Garching b. Munchen, Germany 28 Institut d'Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Universit /acute.ts1 e, 98 bis boulevard Arago, F-75014 Paris, France 29 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91001, USA 30 \n- Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA", 'ABSTRACT': 'Our knowledge of relations between supermassive black holes and their host galaxies at z ≳ 1 is still limited, even though being actively sought out to z ∼ 6. Here, we use the high resolution and sensitivity of JWST to measure the host galaxy properties for 107 X-ray-selected type-I AGNs at \[email protected] \n23 \n0 . 68 < z < 2 . 5 with rest-frame optical/near-infrared imaging from COSMOS-Web and PRIMER. Black hole masses (log ( M BH /M ⊙ ) ∼ 6 . 9 -9 . 6) are available from previous spectroscopic campaigns. We extract the host galaxy components from four NIRCam broadband images and the HST/ACS F814W image by applying a 2D image decomposition technique. We detect the host galaxy for ∼ 90% of the sample after subtracting the unresolved AGN emission. With host photometry free of AGN emission, we determine the stellar mass of the host galaxies to be log ( M ∗ /M ⊙ ) ∼ 9 . 5 -11 . 6 through SED fitting and measure the evolution of the mass relation between SMBHs and their host galaxies. Considering selection biases and measurement uncertainties, we find that the M BH /M ∗ ratio evolves as (1 + z ) 0 . 48 +0 . 31 -0 . 62 thus remains essentially constant or exhibits mild evolution up to z ∼ 2 . 5. We also see an amount of scatter ( σ µ = 0 . 30 +0 . 14 -0 . 13 ), similar to the local relation and consistent with lowz studies, and a non-causal cosmic assembly history where mergers contribute to the statistical averaging towards the local relation is still feasible. We highlight improvements to come with larger samples from JWST and, particularly, Euclid, which will exceed the statistical power of current wide and deep surveys. \nKeywords: AGN host galaxies (2017) - Active galactic nuclei (16) - Active galaxies (17) - Galaxy evolution (594)', '1. INTRODUCTION': "With our understanding that galaxies grow by increasing their stellar mass through mergers and in situ star formation from gas accretion, there are still many unresolved questions in galaxy evolution. One of the most important challenges in galaxy formation is understanding the physical processes that relate the growth of supermassive black holes (SMBHs) alongside the growth of the galaxies that harbor them. Observational studies, mainly in the local universe, have unveiled tight correlations between the mass of SMBHs ( M BH ) and the physical properties of their host galaxies, such as stellar velocity dispersion σ ∗ and stellar mass M ∗ (Magorrian et al. 1998; Ferrarese & Merritt 2000; Marconi & Hunt 2003; Haring & Rix 2004; Gultekin et al. 2009; Graham et al. 2011; Beifiori et al. 2012; Kormendy & Ho 2013; Reines & Volonteri 2015). How and through what physical processes such a tight relation is formed is still unclear, and the origin of the mass relation can shed light on the evolution of not only SMBHs but also galaxies. \nA widely considered scenario, in answer to this question, is a co-evolution scheme, where galaxies and black holes mutually increase their mass at a correlated pace. As a potential physical cause for a co-evolution scenario, some studies implement active galactic nuclei (AGN) feedback, where the energy released from AGNs heats the gas and controls star formation or gas accretion through radio jets or an AGN winds (e.g. Springel et al. 2005; Di Matteo et al. 2008; Hopkins et al. 2008; Fabian 2012; DeGraf et al. 2015; Harrison 2017). Additionally, studies support a common gas supply simultane- \nously fueling both SMBHs and their host galaxies by increasing the BH accretion rate and star formation rate (SFR, Cen 2015; Menci et al. 2016). On the other hand, others have shown that, even in the absence of a close physical connection between SMBHs and host galaxies, the mass relation can be achieved through a non-casual connection; major mergers have averaged the mass relation statistically (cosmic averaging scenario; Peng 2007; Hirschmann et al. 2010; Jahnke & Macci'o 2011). \nTo unravel the cause of the mass relation, an effective approach is observing the relation between M BH and M ∗ throughout cosmic history. With such observations, we can directly determine whether the relation in the local universe, both its ratio and dispersion, evolves with redshift. Then, comparisons of the observational results with simulations (e.g., Ding et al. 2020; Habouzit et al. 2021; Ding et al. 2022b) based on various physical models can allow us to discuss the physical processes that establish the galaxy-BH relations and further constrain the physics of black hole formation and galaxy evolution. \nBefore the advent of James Webb Space Telescope (JWST), statistical studies using 2D image decomposition analyses were conducted using images obtained by Hubble Space Telescope (HST) (e.g., Peng et al. 2006; Jahnke et al. 2009; Bennert et al. 2011a; Cisternas et al. 2011; Simmons et al. 2011, 2012; Schramm & Silverman 2013; Mechtley et al. 2016; Ding et al. 2020; Bennert et al. 2021; Li et al. 2023b) and ground-based telescopes including Subaru's Hyper Suprime-Cam (HSC) (Ishino et al. 2020; Li et al. 2021), and Pan-STARRS1 (PS1) (Zhuang & Ho 2023). In summary, these studies have concluded that the relation between M BH and M ∗ does not evolve with redshift at z ≲ 2. However, studies using a large statistical and universal sample have yet to be \nachieved at z > 1. At these redshifts, we can obtain information longer than the 4000 ˚ A break to constrain M ∗ from observations at near-infrared wavelengths. However, there are a limited number of previous studies using near-infrared data at z ≳ 1 (e.g., Jahnke et al. 2009; Simmons et al. 2012; Mechtley et al. 2016; Ding et al. 2020, utilizing HST/NICMOS or HST/WFC3). Other studies carry out statistical AGN samples using spectral energy distribution (SED) fitting based 1D decomposition method (e.g., Merloni et al. 2010; Sun et al. 2015; Suh et al. 2020). \nJWST is now revolutionizing the field of AGN - host galaxy relations up to z ∼ 6 and beyond based on its high spatial resolution and unprecedented sensitivity. For instance, Ding et al. (2022a) applied a 2D decomposition analysis on early JWST NIRCam data from CEERS that successfully detected the host galaxies of five quasars at z ∼ 1 . 6 -3 . 5 from the SDSS DR17Q catalog (Lyke et al. 2020). They also succeeded in detecting clear substructure and performed pixel-by-pixel SED fitting for one of the five targets, SDSSJ1420+5300A, at z ∼ 1 . 6. Other studies have also performed 2D decompositions of AGN host galaxies using JWST imaging; for instance, Li et al. (2023a) have analyzed a galaxy that is one of the most promising candidates for having a recoiling SMBH ( z ∼ 0 . 36) while Kocevski et al. (2023) present the host properties of five X-ray-luminous AGNs (3 < z < 5) in CEERS. Zhang et al. (2023) also utilized JWST NIRCam data to assess the validity of M ∗ estimated from 1D decomposition (spectrum-based) method for the HETDEX type-I AGNs (2 < z < 2 . 5). \nAt z > 6, Ding et al. (2023) conducted decomposition analysis of two low-luminosity quasars, thus representing the highest-redshift record for detection of host stellar emission. They suggest that z ∼ 6 low luminosity quasars have a mass relation consistent with the local relation after considering selection biases and measurement uncertainties, albeit with a small sample. Equally remarkable, JWST studies of highz AGNs are revealing a higher abundance of lower mass black holes that are actively accreting within very dusty and compact galaxies (e.g., Onoue et al. 2023; Kocevski et al. 2023; Matthee et al. 2023; Harikane et al. 2023; Maiolino et al. 2023; Greene et al. 2023; Kocevski et al. 2024; Wang et al. 2024; Akins et al. 2024). However, in general, the sample sizes of these highz AGNs and the accuracy of M ∗ estimation are still limited. Therefore, the redshift evolution of the mass relation remains highly uncertain. \nTherefore, as an important next step to investigate the evolution of the mass relation, we perform a 2D decomposition analysis with JWST/NIRCam data for a sample whose redshift range significantly improves upon \nwhat was statistically analyzed before the JWST era, further bridging the gap between lowz statistical and limited highz studies. We use a sample larger than previous studies using JWST that reaches up to z ∼ 2 -3 when AGN and star formation activities peaked in cosmic history. With NIRCam images of N = 107 AGNs in COSMOS-Web (Casey et al. 2023) and PRIMERCOSMOS, we conduct 2D decomposition analyses and then statistically discuss the evolution of the M BH -M ∗ relation with consideration of the selection bias and measurement uncertainty (Lauer et al. 2007; Shen & Kelly 2010; Schulze & Wisotzki 2011, 2014). In addition, we report on the ability to accurately model the JWST PSF in each band and the impact on the derived host galaxy properties. \nThis paper is organized as follows. Section 2 describes the JWST/NIRCam data and the sample selection. Section 3 describes the detailed analysis method including 2D image decomposition and careful PSF modeling, SED fitting, and constrcution of mock data. In Section 4, we present our fitting results and discuss the PSF effect on the results. Then, we show the evolution of M BH -M ∗ relation in Section 5 with considering the selection bias. Also, we discuss the possibility of scatter evolution in the mass relation and summarize the challenges with 2D decomposition methods in Section 6. We present the conclusion and prospects for future studies in Section 7. In this paper, all magnitude are AB magnitude (Oke 1974), and we assume a standard cosmology with H 0 = 70 km s -1 Mpc -1 , Ω m = 0 . 30, and Ω Λ = 0 . 70.", '2.1. COSMOS-Web': "COSMOS-Web (PI: Jeyhan Kartaltepe and Caitlin Casey, GO1727, see Casey et al. 2023, for the overview) is a 270-hour treasury survey program in JWST Cycle 1, covering 0.54 deg 2 with NIRCam (Rieke et al. 2023) in four filters (F115W, F150W, F277W, F444W) and 0.19 deg 2 with MIRI (Bouchet et al. 2015) using F770W. Due to the large field, the COSMOS-Web field was split into twenty tiles. \nThe data are reduced with the JWST Calibration Pipeline 1 (Bushouse et al. 2023) version 1.14.0 and the calibration Reference Data System version 1223. The 5 σ depth in an aperture with a radius of 0 . '' 15 ranges from 26.7 to 27.5 mag in F115W and 27.5 to 28.2 mag in F444W, depending on the number of integrations (also see Section 2.1 of Casey et al. 2023). The mosaic images \nFigure 1. Location of the sample compared to the footprints of COSMOS-Web and PRIMER-COSMOS. The background is the mosaic image of HST/ACS F814W (Koekemoer et al. 2007) for the COSMOS field. Red stars show the position of each AGN. The region enclosed by the solid red line shows the PRIMER-COSMOS field, and gray shaded region corresponds to the COSMOS-Web field. \n<!-- image --> \nhave a resolution of 0 . '' 030/pixel. Details of the image reduction process will be described in Franco et al.,(in preparation). In addition to NIRCam four-band data, we use HST/ACS F814W data (Koekemoer et al. 2007). In this study, we do not use MIRI data because it is challenging to apply the 2D decomposition method (Section. 3.2) due to its lower spatial resolution and larger PSF.", '2.2. PRIMER': "Public Release IMaging for Extragalactic Research (PRIMER, PI: James Dunlop, GO1837) is a 195-hour treasury program of JWST Cycle 1, targeting two equatorial HST CANDELS Legacy Fields: COSMOS and UDS. PRIMER-COSMOS covers 144 arcmin 2 with eight NIRCam (Rieke et al. 2023) filters (F090W, F115W, F150W, F200W, F277W, F356W, F410M, F444W) and 112 arcmin 2 with two MIRI (Bouchet et al. 2015) filters (F770W and F1800W) in the COSMOS field. The processed COSMOS-PRIMER data consists of one mosaic image. In this analysis, we use NIRCam eight-band data and HST/ACS F814W data. The PRIMER-COSMOS data are reduced with the JWST Calibration Pipeline (Bushouse et al. 2023) version 1.8.3 and the calibration Reference Data System version 1017. The 5 σ depth in an aperture with a radius of 0 . '' 15 has a wide range \nfrom 27.9 to 28.3 mag in F090W and 28.4 to 28.9 mag in F444W, depending on the number of integrations, ∼ 1 mag deeper than the COSMOS-Web data. The mosaic images have a resolution of 0 . '' 030/pixel.", '2.3. Broad-line AGN sample': "To evaluate the relation between M ∗ and M BH , we use the type-I AGN sample with M BH estimates available in Schulze et al. (2015, 2018). Schulze et al. (2015) presents the redshift evolution of AGN population based on spectroscopically observed type-I AGNs from zCOSMOS (Lilly et al. 2007, 2009), VVDS (Le F'evre et al. 2005, 2013; Garilli et al. 2008), and SDSS (Schneider et al. 2010; Shen & Kelly 2012). Schulze et al. (2018) provides the properties of X-ray selected and spectroscopically-confirmed type-I AGNs in the FMOSCOSMOS survey (Kashino et al. 2013; Silverman et al. 2015). Here, we select the targets in Schulze et al. (2015) and Schulze et al. (2018) that are also detected by Chandra (Chandra-COSMOS Survey; Elvis et al. 2009; Civano et al. 2012) or XMM-Newton (XMM-COSMOS; Cappelluti et al. 2009; Brusa et al. 2010). The 2-10 keV flux sensitivity is 7 . 3 × 10 -16 erg cm -2 s -1 for Chandra and 3 × 10 -15 erg cm -2 s -1 for XMM-Newton. \nWe have M BH estimates from spectra acquired by the FMOS-COSMOS and zCOSMOS surveys with some AGNs having measurements from both. Considering the quality of the spectroscopy, the error on M BH estimation, and the fact that H β line is used for calibrating virial mass estimators, we use the M BH from H β (FMOS-COSMOS), H α (FMOS-COSMOS), and Mg ii (zCOSMOS-Bright, Deep) in order of preference; e.g., FMOS H β is used for an object with both FMOS H β and zCOSMOS Mg ii estimation. As shown in Figure 10 of Schulze et al. (2018), there is a very good agreement between the FMOS H α - and H β -based M BH compared to those using MgII. Schulze et al. (2018) also compared FWHM measurements taken at different times for each object (see Fig.7 of Schulze et al. 2018) and confirmed that the FWHM values are consistent with each other. Note that 19 AGNs in our sample are listed in the SDSS DR16 quasar catalog (Lyke et al. 2020); we do not use these SDSS Mg ii -based M BH measurements given the benefits of the FMOS and deeper zCOSMOS spectroscopy. The number and redshift range of each measurement are summarized in Table 1. \nFor H β , Schulze et al. (2018) used the virial mass estimation relation by Vestergaard & Peterson (2006), \nM BH (H β ) = 10 6 . 91 ( L 5100 10 44 erg s -1 ) 0 . 5 ( FWHM H β 1000 km s -1 ) 2 M ⊙ , (1) \nFigure 2 shows the distribution of 2-10 keV X-ray luminosity L [2 -10 keV] (panel a) and M BH (panel b) as a function of redshift, where L [2 -10 keV] is calculated with an X-ray spectral index Γ = 1 . 8 (e.g., Brightman et al. 2013). Since the sample consists of X-ray-selected objects and is flux-limited, there is a tendency for higherz objects to have larger L [2 -10 keV] over the sensitivity limit. We can also see the trend of M BH increasing with redshift. The sample biases likely influence this trend from the flux sensitivity and the availability of broad-line FWHM measurements from spectroscopic data. Figure 2 (c) displays the relation between the Eddington ratio L bol /L edd and M BH with Eddington ratio decreasing as M BH increases. This trend is due to the observational flux limitation and that L edd is propor- \n<!-- image --> \n<!-- image --> \n<!-- image --> \n/circledot \nFigure 2. Characteristics of the type-I AGN sample with JWST imaging from COSMOS-Web and PRIMER: (a) Distribution of 2-10 keV X-ray luminosity L [2 -10 keV] as a function of z . Black dashed lines indicate the F [2 -10 keV] sensitivity in the sample, (b) Black hole mass M BH as a function of z , and (c) relation between the Eddington ratio L bol /L edd and M BH . The color and shape of the points indicate the JWST survey field and the source of M BH estimation, respectively, as shown. \nTable 1. Sample size for each single-epoch M BH estimation \nwhere L 5100 is continuum luminosity at 5100 ˚ A and FWHM H β is the full-width at half-maximum (FWHM) of H β broad line. Then, the H α -based masses are calculated as given in Schulze et al. (2017); Schulze et al. (2018), \nM BH (H α ) = 10 6 . 71 ( L H α 10 42 erg s -1 ) 0 . 48 ( FWHM H α 1000 km s -1 ) 2 . 12 M ⊙ , (2) \nL H α is the H α luminosity and FWHM H α is the FWHM of the broad H α emission line. For Mg ii -based M BH estimation (zCOSMOS-bright Mg ii , and zCOSMOSdeep Mg ii ), the calibration by Shen et al. (2011) is used, \nM BH (MgII) = 10 6 . 74 ( L 3000 10 44 erg s -1 ) 0 . 62 ( FWHM MgII 1000 km s -1 ) 2 M ⊙ , (3) \nwhere L 3000 is continuum luminosity at rest-frame 3000 ˚ A and FWHM MgII is FWHM of Mg ii broad emission line. These single-epoch virial mass estimations have an uncertainty due to possible variability and uncertainties in the modeling of broad-line regions (c.f., Shen 2013). In this paper, we use the M BH uncertainties that also consider uncertainties from the single-epoch virial mass estimation, typically ∼ 0 . 4 dex. We also consider this uncertainty in generating mock data (Section 3.6). \nFrom the parent catalog, we select broad-line AGNs that reside in the COSMOS-Web and PRIMER fields. The final sample size has 107 AGNs with black hole mass estimation summarized in Table 1. Figure 1 shows the spatial location of the AGN sample within the COSMOS-Web footprint. We use five broad-band images from HST/ACS (F814W) and JWST/NIRCam (F115W, F150W, F277W, and F444W) for the sample residing in COSMOS-Web. Four more broad-band (F090W, F200W, F356W) and medium-band (F410M) images are available for five galaxies in the PRIMER field. \nWe also compare the optical color index g -i calculated based on the COSMOS2022 photometry (Weaver et al. 2022) with SDSS quasars and hard-X-ray-detected AGNs at the same redshift range (Figure 6 in Silverman et al. 2005). Our sample has a wide g -i distribution of g -i = 0 -2 . 4. Thus, our sample includes both unobscured and dust-obscured AGNs and is not significantly biased to either sample. \ntional to M BH . While there is a correlation between M BH and L bol , dividing L bol by L edd to calculate the Eddington ratio cancels out this weaker correlation between L bol and M BH . \nFigure 3 shows the original F277W images, i.e., before the decomposition analysis, of representative AGNs in our sample. In some targets, we can recognize the extended components of the host galaxies far from the central PSF-like feature. However, a central AGN component, especially those with spiky diffraction features in the outer part, dominates the system and buries the host galaxy component. These dominant PSF components prevent us from obtaining host galaxy information directly and make the 2D decomposition analysis necessary. \nFigure 3. Original images in F277W of some targets in the order of redshift. The target IDs and the redshifts are shown in the left corner of each image. The white bars indicate a 1 '' scale. Depending on the host-to-total flux ratio ( H/T ) that varies with the sample, a central AGN component can dominate an entire system, thus burying a host galactic component. \n<!-- image -->", '3. METHOD': "To extract host galaxy components from the original AGN + host galaxy NIRCam images, we apply a 2D image analysis tool galight (Ding et al. 2020). With galight , we perform forward modeling of each image as a superposition of a PSF component and PSF-convolved S'ersic components corresponding to the light from an AGN and its host galaxy, respectively. We then obtain images of the host galaxy, free of the AGN, by subtracting the fitted PSF component from the original image.", '3.1. Comparative analysis of model PSF construction': 'Considering that the AGN can account for up to ∼ 95% of the total flux (e.g., Ding et al. 2020, 2022a), \nthe results significantly depend on the accuracy of reconstructing the PSF. There are different strategies to reconstruct PSF images based on either using theoretical PSFs (e.g., Suess et al. 2022) or stellar images (e.g., Nardiello et al. 2022; Ding et al. 2022a; Ding et al. 2023; Zhuang & Shen 2023; Baker et al. 2023). The former uses the theoretical PSF model such as WebbPSF (Perrin et al. 2012, 2014), and the latter uses natural stellar images as the PSF directly or modeled PSF with tools such as PSFEx (Bertin 2011). Many previous studies concluded that the synthetic PSF simulated by WebbPSF is narrower than the PSFs reconstructed with the natural stars (Ono et al. 2022; Ding et al. 2022a; Onoue et al. 2023). Note that Ito et al. (2023) used the intermediate method between the former and the latter; they used theoretical PSFs with WebbPSF and smoothed it by comparing the surface brightness profile of natural stellar images. In this paper, we reconstruct the PSF using three methods based on natural star images for each region and filter. Then, we compare the results with the different PSF reconstruction methods and discuss the host galaxy characteristics with the method dependence.', '3.1.1. χ 2 ν -based methods': "First, we follow the strategy of Ding et al. (2020, 2022a). They first construct an empirical PSF library for which each PSF is represented by the image of a single star. Then, the 2D decomposition analysis is run with all single PSFs in the library, separately. Then, they sort the results in the order of reduced chi-square χ 2 ν and stack the PSFs with the top 3, 5, and 8 χ 2 ν values. Using the single PSFs and the stacked PSFs, they select the PSF with the smallest χ 2 ν as the final PSF. This method is based on the χ 2 ν ; i.e., they assumed that the lower χ 2 ν is indicative of a better (the closer to the more accurate) PSF. \nFollowing their strategy, we apply the find PSF function in galight to list PSF candidates, then select PSF candidates manually for each mosaic image and filter. In this manual selection process, only obvious PSF candidates with the PSF-like complex hexagonal and spiky diffraction features and without a galaxy-like broad component are selected. We cropped the images of the selected PSF candidates for the short-wavelengthchannel filters (F090W, F115W, F150W, and F200W) and long-wavelength-channel filters (F277W, F356W, F410M, and F444W) as squares with 150 and 240 pixels per side, corresponding to 4 . '' 5 and 7 . '' 2, respectively. After removing neighbor objects using clean PSF function in galight , the PSF libraries contain ∼ 30 -50 PSF candidates depending on the filter and region. Then, \nwe fit each AGN target with a superposition of a PSFconvolved S'ersic profile and each single PSF candidate in the PSF library. With the fitting results of each single PSF, we select PSFs with the top-5 χ 2 ν and top-75% χ 2 ν and stack them to generate an averaged PSF image. These top-5 stacked and top-75% stacked PSFs are finally used to estimate the parameters in this method. Note that each target has its own top-5 and top-75% PSFs generated from single PSFs with the lowest χ 2 ν selected for each target.", '3.1.2. Modeling method': "Zhuang & Shen (2023) compare JWST/NIRCam PSFs modeled with different methods ( Swarm , photutils , and PSFEx ), and concluded that PSFEx reconstructed PSFs provide the best performance. From the 2D decomposition of simulated broad-line AGNs, Zhuang & Shen (2023) also suggested that smaller χ 2 ν values do not necessarily provide a means to distinguish which PSFs are more likely to characterize the AGN with higher accuracy. Following the conclusion by Zhuang & Shen (2023), we use PSFEx and compare the results with χ 2 ν -based selected PSFs (Section 3.1.1). \nPSFEx constructs an empirical PSF model based on the output catalog of SExtractor (Bertin & Arnouts 1996). We first run SExtractor for source detection, and then run PSFEx for modeling the PSF for each mosaic image and filter. PSFEx can also reconstruct local PSFs as a function of positions on the detector. We do not use local PSFs because Zhuang & Shen (2023) also concluded that a universal or global PSF usually shows 'satisfactory' fitting results, and the sample region (COSMOS-Web and PRIMER-COSMOS) has a much smaller number of stars than the south continuous viewing zone, which Zhuang & Shen (2023) tested local PSF reconstruction.", '3.1.3. Comparing the final PSFs': 'Now, we have three final PSFs for comparison; the top-5 χ 2 ν stacked PSF, the top-75% χ 2 ν stacked PSF, and the PSFEx PSF (Sections 3.1.1 and 3.1.2). To compare PSFs, we perform, a 2D Gaussian fitting for each PSF image and measure the FWHMs along the semi-major axis. Figure 4 compares the FWHMs of PSFs obtained with each method, target, and filter. The x and y-axis of Figure 4 show the ratio FWHM Top -5 /FWHM PSFEx and FWHM Top -75% /FWHM PSFEx , where FWHM is the value along the semi major axis. \nFirstly, regardless of the filters, we can see that the distribution extends further in the x-axis direction than the y-axis. This can be attributed to greater variation in the FWHMs for the top-5 stacked PSFs. We use the same PSFEx PSF for each target in the same field, and \nFigure 4. Comparison of FWHM (semi-major axis) for PSFs from different PSF reconstruction methods for all individual AGNs. The x-axis represents the ratio of the FWHMs for the top-5 PSFs to those for the PSFEx PSFs, and the yaxis represents the ratio of the FWHMs for the top-75% PSFs to those for the PSFEx PSFs. Blue, green, orange, and red colors indicate the different filters: F115W, F150W, F277W, and F444W. Median values and 1 σ confidence range are denoted by star symbols and error bars. The gray dashed line indicates y = x , i.e., the same FWHMs for the top-5 and top75% PSFs. Notably, F277W and F444W exhibit a FWHM bias among different PSF reconstruction methods. \n<!-- image --> \nthe top-75% stacking in the same field uses mostly the same single PSFs in the field. In contrast, top-5 stacking employs only the best-fit single PSFs with the lowest χ 2 ν . As a result, the FWHM variation for each galaxy is largest for the top-5 PSF followed by the top-75% PSF and PSFEx PSF, and FWHM Top -5 /FWHM PSFEx have a larger scatter than FWHM Top -75% /FWHM PSFEx . \nSecondly, we focus on the FWHM bias between the methods for each filter. As suggested by Zhuang & Shen (2023), for short-wavelength filters (F115W and F150W), there is a significant scatter in the FWHM ratio. On the other hand, for long-wavelength filters (F277W and F444W), the scatter is smaller than the short-wavelength side. These results imply that, in the long-wavelength filters, PSFEx PSFs are sharper than the top-5 and Top-75% PSFs. The impact of these trends on the 2D decomposition analysis is discussed in Section 4.2. \nNote that these trends can depend on the visual inspection performed when constructing the PSF library (Section 3.1.1) and the settings used for Sextractor and PSFEx (Section 3.1.2). For example, visual inspection can be biased by the hexagonal diffraction features of the JWST PSF, an appropriate FWHM range, and the absence of extended structures originating from host galaxies. If this selection process is strongly biased by the hexagonal features, it might lead to a selective choice of brighter PSFs. As a result, the parameter distributions presented here may not necessarily match the distribu- \ntion of the actual PSF. Nonetheless, even different PSF reconstruction methods can result in different FWHMs. Therefore, when performing a 2D decomposition analysis with only one PSF reconstruction method and not considering the possibilities of other PSFs, 2D decomposition results can be biased by a specific PSF. Because determining the PSF shape perfectly is challenging, it is also important to discuss uncertainties by considering the results obtained with possible different PSFs. We discuss how different PSF reconstruction methods affect the results of the 2D decomposition and the final M ∗ estimation in Section 4.2, and we perform a detailed comparison of the obtained final PSFs in appendix B.', '3.2. Decomposition': "Using galight , we fit the AGN + host galaxy images with the composite model of a PSF component and a single S'ersic profile (S'ersic 1968) convolved by a PSF. Note that we do not assume the same morphology in every band, i.e., the fitting is performed in each filter independently. In cases where there are nearby galaxies that can affect the fitting, these galaxies are also modeled as S'ersic components and fitted simultaneously. Our S'ersic model has seven free parameters: amplitude, S'ersic index n , effective radius r e , coordinates of the center x c , y c , and ellipticity e 1 , e 2 . The PSF model corresponding to the AGN component has three free parameters: amplitude and coordinates of the center x c , y c . Thus, the number of free parameters for PSF + single S'ersic component is ten in total. Note that the actual number of free parameters in the fitting changes depending on the number of nearby objects also fitted with a S'ersic profile. \nTo avoid unphysical results, the range of n and r e is constrained to [0 . 3 , 7] and [0 . '' 06 , 2 . '' 0], respectively. Note that some galaxies show clear substructures that do not suit a S'ersic profile, such as bars and spiral arms. Thus, using a S'ersic profile is a first-order approximation to model the global component of AGN host galaxies. \nIn the fitting process, we cut the image into square regions centered on the target with a radius seven times the standard deviation along the semimajor axis of the 2D Gaussian fitting with photutils (Bradley et al. 2023). Then, the above model is optimized with Particle Swarm Optimizer (PSO; Kennedy & Eberhart 1995). galight also supports Markov Chain Monte Carlo (MCMC) to estimate the posterior parameter distributions. As suggested by Ding et al. (2023), we also confirm that the uncertainty estimated with MCMC is much smaller than the uncertainties from different PSFs. Thus, we do not use MCMC in the decomposition analysis, and we estimate the uncertainty from the results \nwith different single PSFs (Section 3.4). We set the supersampling factor relative to pixel resolution to 3, which controls interpolation within a pixel to perform a subpixel shift of the PSF (c.f., Ding et al. 2023). \nAs described above, using the PSF library constructed with galight , we fit with every single PSF in the library and sort them in order of χ 2 ν . Then we fit with the three final PSFs; top-5 χ 2 ν stacked, top 75% χ 2 ν stacked, and PSFEx . Figure 5 compares the fitting results with the final PSFs for the example galaxy CID-62 in the F444W. We can find that the host component is more prominent than the central PSF component at larger radii, and 2D decomposition makes it possible to detect the host galaxy initially buried under the PSF component with all three model PSFs. \nFor targets that have n > 6 . 5 in any band other than F277W, we rerun the fit while fixing n to the value found for the F277W band. F277W has the lowest number of values hitting the upper limit on n , falls above the rest-frame 4000 ˚ A break, and is somewhat central to JWST wavelength coverage. This pertains to 42, 30, and 26 sources detected in F115W, F150W, and F444W, respectively. From mock tests, we confirm that galight can return n ∼ 7 even if the actual value is much smaller (Appendix C). This is likely due to fitting where some of the AGN emission is attributed to a central stellar concentration of the host thus overestimating the host galaxy flux as well.", '3.3. Detection of host galaxy': "Figure 5 shows an example where the host galaxy is clearly detected. However, for some galaxies, the strong PSF component dominates the total flux maybe due to not only a low host-to-total flux ratio H/T but also compact morphology, making it challenging to distinguish the host signal from their PSF component. To gauge, in a quantitative manner, which AGNs have accurate host galaxy information, we exclude cases where the host galaxy is undetectable, following three strategies below. \n(1) Bayesian Information Criteria: In addition to the PSF + S'ersic model (PS+SE model) described above, we also fit with a model containing only a PSF component (PS model). Then we calculate the Bayesian Information Criteria (BIC; Schwarz 1978) for the two models, PS+SE and PS model, as, \nBIC = χ 2 + k ln ( n ) , (4) \nwhere k is the number of free parameters, and n is the number of data points. We regard that the PS+SE model provides a better description of the data than the PS model when BIC PS+SE is much smaller than BIC PS , \nas \nBIC PS+SE < BIC PS -10 , (5) \nFigure 5. Example fits for CID-62 ( z ∼ 1 . 92) using F444W. Each row shows the result for a different PSF (top-5 stacked, top-75% stacked, and PSFEx , from top to bottom). In each row, the images are as follows: original, model, data - model point source (host galaxy only), and normalized residuals, from left to right. χ 2 ν values are shown in the panels of data - model point source. The right panel shows the 1D surface brightness profile where dashed lines indicate half-width at half-maximum (HWHM) of each PSF. In the data-point source image, we reveal a disk-like host galaxy which is buried under the PSF before subtraction, regardless of the PSF reconstruction method. \n<!-- image --> \nWe decide the threshold value of the BIC difference of 10 based on Kass & Raftery (1995). \n(2) S/N of the host galaxy: To estimate the significance of the detection of the host galaxy, we calculate the S/N of the host galaxy as done in Ding et al. (2023). We construct an error map of the PSF-subtracted images considering two sources of error: noise from the observed images and the uncertainty propagated from different PSF reconstructions. Due to the intrinsic variations of the PSF image even in the same FoV (Zhuang & Shen 2023; Yue et al. 2023) and errors in the observed image used in PSF reconstruction, the reconstructed PSFs contain uncertainties. Thus, considering the uncertainty from PSF reconstruction is indispensable to calculate the S/N of detected host galaxies. For the PSF uncertainty, we use the pixel-by-pixel standard deviation of the fitted PSF components when fitting the PS+SE model with each single PSF in the PSF library. Then, we calculate the final noise map as a composite of the observed noise map and the PSF uncertainty map. With the final noise map, we calculate the signal- \nto-noise ratio of host galaxy S/N host within a radius of 2 r e . Then, we define the detection as cases with high S/N host , as \nS/N host > 5 . (6) \n(3) Manual inspection and removal: We find that some objects have invalid central values in their F444W image and shallower surface brightness profiles than any PSFs. The fitting of these galaxies fails even with applying a mask in the central region. We find four cases (CID-50, CID-208, CID-668, and CID-112) with such features and label them as non-detections. \nCID-142 is located near the edge of the image, and the host galaxy is partially cut off. It is also the possible that CID-142 has a mismathced PSF from PSFs in other fields. Therefore, obtaining accurate photometry of CID-142 is challenging, thus we manually exclude CID-142 in the following discussion. \nWe also find some obvious false detections in F814W, where the image shows a dominant PSF feature and no extended host-like feature, and galight fit the PSF-like \nfeatures as a host galaxy with H/T ∼ 1. We confirm that the decomposition of the JWST images clearly detects a host-like extended feature. Thus, in addition to the above strategies, we recognize 31 obvious false detections in F814W as non-detection cases. \nWith the above three strategies, we confidently report the detection of an AGN host galaxy for those that fulfill both the conditions (5) and (6) for each of the final PSFs; i.e., we determine whether the host is detected or not for each top-5, top-75%, and PSFEx PSF, separately. Also note that this decision is made for each sample and each filter, i.e., a galaxy detected in one filter may not be detected in another filter. The number of objects detected over two filters of NIRCam is 102, occupying ∼ 95% out of the entire N = 107 sample with the top5 PSF (see Appendix A for the number of detection in each filter).", '3.4. Photometry of host galaxies': "We calculate the flux of the host galaxies, using the S'ersic fits, considering Galactic dust extinction (Schlegel et al. 1998) for the detected cases. For the photometric accuracy, we set an error of 0.2 mag, which represents likely systematic uncertainties (e.g., Ding et al. 2022a; Zhuang & Shen 2023; Zhuang et al. 2023) and errors discussed in Section 3.3, considering both observational errors and PSF uncertainty in a radius of 2 r e . We use the 3 σ value for the undetected filters as the upper limit. \nHere, we also fit with a model containing only a S'ersic component (SE model). For targets with BIC PS+SE > BIC SE -10, the SE model is better or flexible enough to describe the data than the PS+SE model, and we use the S'ersic photometry calculated with the SE model instead of the PS+SE model. Such cases are observed only in F444W and are very limited (two or three objects depending on the PSF).", '3.5. SED fitting': 'We fit the photometry of the host galaxies with CIGALE ( v2022.1 , Boquien et al. 2019; Yang et al. 2022) SED fitting library. For a stellar population, we use the single stellar population model by Bruzual & Charlot (2003) ( bc03 module) and the Chabrier initial mass function (IMF; Chabrier 2003) with the M ∗ cutoff of 0 . 1 M ⊙ and 100 M ⊙ (Bruzual & Charlot 2003). We assume a delayedτ model for a star-formation history (SFH), where SFR at each look-back time t is modeled as \nSFR ( t ) ∝ ( t -t age ) exp ( -t -t age τ ) ( t > t age ) , 0 ( t < t age ) , (7) \nwhere t age and τ indicate the starting time of star-formation activity and the declining timescale of SFR. We also consider a nebular emission with nebular module and a dust attenuation with dustatt modified starburst module that assumes the modified Calzetti et al. (2000) law. We set E ( B -V ), M ∗ , t age , and τ as free parameters; their grid values are decided basically following Zhuang et al. (2023) CIGALE run and summarized in Table 2. To avoid unphysical solutions, we set the upper limit of t age to 0 . 95 t H , where t H indicates the cosmic age at each redshift. Otherwise, stellar metallicity and the ionization parameter U are fixed at Solar metallicity and log U = -2. \nWe also apply the Bayesian-based spectral energy density (SED) fitting code, Prospector (Leja et al. 2017; Johnson et al. 2021), to assess the uncertainty M ∗ with different SED fitting codes. Prospector is based on the Flexible Stellar Population Synthesis ( FSPS , Conroy et al. (2009); Conroy & Gunn (2010)) to generate model SEDs of galaxies. For comparison, we use almost the same settings with CIGALE : a Chabrier IMF, a delayedτ SFH model, Solar metallicity, Calzetti et al. (2000) dust attenuation law, and nebular emission with log U = -2. Prospector can fit with the non-parametric SFH, which separates galaxy formation history into several age bins and assumes a constant SFR in each bin (e.g., Leja et al. 2019). Lower et al. (2020) input cosmological hydrodynamic simulation data into Prospector and concluded that non-parametric SFH tends to reconstruct M ∗ more accurately than parametric SFHs. Lower et al. (2020) also suggested that parametric SFH tends to underestimate M ∗ . However, in this study, the available photometry is limited in the number and wavelength range (only five/nine bands in the near-infrared wavelength range for the COSMOS-Web/PRIMER-COSMOS field). Furthermore, the photometry derived in Section 3.4 contain uncertainties from the 2D decomposition analysis. Therefore, we choose to use the parametric SFH (delayedτ model) instead of the non-parametric assessment. The parameter prior settings in the MCMC run are summarized in Table 2. We compare the results with CIGALE and Prospector in Section 4.4.', '3.6. Generating mock data to consider selection effects': 'As mentioned, our sample is X-ray-flux limited (Section 2.3), raising the possibility of bias toward larger M BH or higher Eddington ratio (Lauer et al. 2007; Schulze & Wisotzki 2011, 2014). Due to this selection effect, a direct comparison of the observational results with the local relation is not appropriate. Thus, in this study, we generate a mock AGN-galaxy catalog based on the procedure in Li et al. (2021) and apply the mock \nTable 2. Major parameter settings in SED fitting \nFigure 6. Procedure for generating mock galaxies at z ∼ 1 . 5 with an example parameter set of α = α local , β = β local , γ = 0, and σ µ = 0 . 3. Orange and blue contours show the distribution of the entire sample (1 σ -5 σ ) and that above the detection limit of the observed sample (1 σ -3 σ , see Sections 5.1 for the definition of the detection). The individual panels are as follows: (a) true black hole mass M BH , mock vs. true stellar mass M BH , mock , before adding observational uncertainties, (b) 2 -10 keV flux ( F [2 -10 keV]) , mock ), (c) H α FWHM (FWHM H α, mock ), (d) pseudo-observed virial mass M vir , mock vs. true black hole mass M BH , mock where M vir , mock is the mock black hole mass after considering the observational bias. (e) pseudo-observed stellar mass ˜ M ∗ , mock vs. stellar mass M ∗ , mock where ˜ M ∗ , mock is the stellar mass after considering the observational bias, and (f) pseudoobserved virial mass M vir , mock vs. pseudo-observed stellar mass ˜ M ∗ , mock . The latter is to be compared with the observed M BH -M ∗ plane shown in the left panel of Figure 12. Black dashed lines in panels (a) and (f) indicate M BH -M ∗ local relation obtained from the fitting of the local galaxies (Haring & Rix 2004; Bennert et al. 2011b, , see Section 5), and gray dashed lines in panels (d) and (e) indicate y = x line. \n<!-- image --> \n∗ \n∗ \nobservation (adding selection biases and observational effects) to discuss the intrinsic evolution of the mass relation. The procedure for generating the mock catalogs is described below. \nFirst, we generate the mock redshift z mock and mock true stellar mass M ∗ , mock based on the COSMOS2020 stellar mass function (SMF) by Weaver et al. (2023). \nNext, we use the M ∗ , mock to generate the mock true BH mass M BH , mock for the mock sample. Here, we assumed the local relation as \nlog ( M BH M ⊙ ) = α log ( M ∗ M ⊙ ) + β, (8) \nwhere α and β indicate the slope and the intercept of the local M BH -M ∗ plane. Then, assuming a normal \ndistribution, we calculate M BH , mock as, \nlog M BH , mock = N ( α log ( M ∗ M ⊙ ) + β + γ log (1 + z ) , σ 2 µ ) . (9) \nHere, the parameter γ indicates the strength of redshift evolution of the mass relation, and σ µ is the intrinsic scatter of the mass relation. Figure 6 (a) shows M BH , mock vs. M ∗ , mock distribution with the example parameter set of γ = 0 and σ µ = 0 . 3. \nThen, based on z mock and M BH , mock , we assign mock Eddington ratio λ Edd , mock by sampling the Eddington ratio distribution function by Schulze et al. (2015). Using the λ Edd , mock and M BH , mock , we calculate the bolometric luminosity L bol , mock . \nFrom L bol , mock , we obtain L [2 -10 keV] , mock using the bolometric correction by Duras et al. (2020). With the calculated L [2 -10 keV] , mock , mock X-ray flux F [2 -10 keV] , mock is determined with the assumption of Γ = 1 . 8, the same assumption as in Section 2.3. Figure 6 (b) shows F [2 -10 keV] , mock distribution with the example parameter set of γ = 0 and σ µ = 0 . 3. \nWe also generate mock virial BH mass M vir , mock with the L bol , mock , M BH , mock , and the assumed FWHM distribution. We first calculate mock continuum and line luminosity L 5100 , mock , L 3000 , mock , and L H α, mock . For L 5100 , mock and L 3000 , mock , we used the bolometric correction by Netzer & Trakhtenbrot (2007) and Trakhtenbrot & Netzer (2012), respectively. For L H α, mock , we use the L 5100 , mock and scaling relation between L H α and L 5100 , mock by Jun et al. (2015). These bolometric correction and scaling relations are the same as those in Schulze et al. (2018). Then, we assume that FWHM of the emission line follows the log-normal distribution with the scatter of 0.17 dex (Shen et al. 2008), and we generate the observed FWHM of H α , H β , and MgII emission lines following Equations (1), (2), and (3). To consider the bias in single-epoch virial mass estimation, we add the bias with β bias = 0 . 6 in calculating FWHM. β bias represents the proportion by which ∆ L , the variation of luminosity from the mean luminosity ¯ L , affects the variation in FWHM. Thus, the FWHM is generated by a lognormal distribution with the scatter of 0.17 dex and the mean value corresponding to the luminosity of ¯ L + β ∆ L (c.f. Shen 2013; Li et al. 2021). Finally, the mock virial BH mass M vir , mock is calculated using the mock observed FWHM and the luminosity following Equations (1), (2), and (3). Figures 6 (c) and (d) show the FWHM H α, mock histogram and M vir , mock -M BH , mock distribution with the example parameter set of γ = 0 and σ µ = 0 . 3. \nConsidering possible systematic uncertainties from 2D decomposition analysis, we added an error based on a \nnormal distribution with 0.2 dex to the M ∗ , mock to consider the uncertainty of M ∗ derived from observations, resulting in the mock observed stellar mass ˜ M ∗ , mock . The resulted ˜ M ∗ , mock -M ∗ , mock distribution is shown in Figure 6 (e). \nFinally, we get the mock data set of the redshift z mock , observed stellar mass ˜ M ∗ , mock , and virial BH mass M vir , mock . Figure 6 (f) shows the final mock observed mass distribution ( M vir , mock -˜ M ∗ , mock ) with the example parameter set of γ = 0 and σ µ = 0 . 3. In Section 5, we discuss the mass relation, comparing our results with this mock catalog.', '4.1. Morphological parameters': 'Firstly, we examine the results obtained from the 2D decomposition. In Figure 7 (a)-(c), we show the distribution of n and r e (in the unit of pixel and kpc) for each filter using the top-5 stacked PSF. The distribution of n has a peak in the distribution at n ∼ 1 -2, clearly seen in the F277W and F444W filters. This is similar to studies of AGN hosts at high redshift, which show hosts characterized by disk-like morphology. \nIn Figure 7 (d), we compare H/T with S/N host . We can see a strong correlation between the reconstructed H/T and S/N host . High H/T means host galaxies dominate the AGN + host galaxy composite images, and we can easily detect host galaxies with high S/N host ; thus, our results indicate the validity of our analyses for the majority of the sample. Estimated morphological parameters and basic information for each host galaxy are reported in Table 3. In Appendix C, we confirm that galight can reconstruct r e and H/T correctly by running galight on mock galaxy images.', '4.2. Impact of different PSF models': "In Figure 8 (a)-(c), we compare morphological parameters, n , r e , and the host-to-total flux ratio H/T , in F444W obtained using each final PSFs. These parameters fall mostly along the y = x line, indicating a strong correlation. Thus, we can say that different PSF reconstruction methods do not significantly affect the results. \nIn Figure 8 (a), the size ( r e ) comparison shows more consistent results than n , indicating that it is minimally affected by different PSFs. On the other hand, in comparing n (Figure 8 (b)), regardless of the PSF, we can find a consistent estimation on the low n side ( n ≲ 3). However, at larger n ( ≳ 4), the scatter increases. This tendency is likely because larger n implies a compact S'ersic profile that resembles the PSF, making it challenging to distinguish from the PSF. We also find a ten- \nFigure 7. Summary of the decomposition results in each filter. Columns (a), (b), and (c) respectively show the distribution of the estimated n , log r e in the unit of pixel and kpc. In column (d), the relation between H/T and S/N host is shown. In each panel, red (gray) indicates detection (non-detection) in each filter. \n<!-- image --> \nthat PSFEx PSFs result in slightly larger n than the top-5 or top-75% stacked PSFs. Even when considering the other filters, we find highly consistent results, with a trend of increased scatter at higher n . \nRegarding H/T , we primarily see consistent results with strong correlations in Figure 8 (c). While H/T estimated using the top-5 and top-75% PSFs is largely consistent with each other, the PSFEx PSFs tend to result in slightly higher H/T than the other two PSFs. Zhuang & Shen (2023) suggested that using a narrower PSF than an exact PSF could overestimate the host flux and n . Thus, the above biases in n and H/T could be explained by the fact that PSFEx PSFs for F444W have a slightly narrower PSF than the other PSFs, as shown in Figure 4. A detailed comparison of the estimated H/T in other filters is summarized in Appendix B. \nIn conclusion, the decomposition results using different PSFs are generally highly consistent with each other. A comprehensive discussion of the technical and practical differences between PSF reconstruction methods will also be provided in Section 6.3. Nonetheless, these discussions are based on the comparisons between estimated values, and here, we cannot definitively determine the true exact value. Related to this, we provide the result of mock tests using different final PSFs in Appendix C. Also, note that all final PSFs are not single stellar images but stacked or modeled PSFs based on multiple stellar images. As mentioned in Section 3.1.3, single PSFs often have a wide range of sizes and shapes. Thus, using a single stellar image without testing other stellar images can risk misinterpreting the PSF image and giving different results. \nFigure 8. Comparison of the decomposition results ( r e in the unit of arcsec, n , and H/T ) based on the F444W filter using different PSFs (Columns a-c). Data points in red have satisfied our stringent criteria for host detection (Sec. 3.3). Column (d) compares M ∗ estimated by the SED fitting, with colors corresponding to whether each object was detected in more than two bands. Spearman's correlation coefficient ρ for each prior is shown in the lower right corner of each panel. High correlation coefficients and the distribution around y = x (black dashed line) suggest consistent results among the fitting with different PSFs. \n<!-- image -->", '4.3. Host images': "As seen in these high-quality host galaxy images, 2D decomposition analyses of JWST images open up the potential for a more detailed image-based galaxy analysis, such as studies traditionally conducted on inactive galaxies. Figure 9 shows three-color images (F277W, F150W, and F115W for RGB) of each host galaxy created by subtracting the PSF and the nearby Sersic component. \nFirstly, thanks to the high spatial resolution, deep observations, and meticulous decomposition analysis, we can access the highest-quality AGN-host galaxy images up to z ∼ 2 . 5, allowing us to identify substructures. Particularly, in the case of CID-273 ( z = 1 . 85) and CID307 ( z = 2 . 05), despite their redshifts ∼ 2, we can clearly \nidentify blue spiral arms with an overall diffuse red broad component. \nAdditionally, galaxies such as CID-54 ( z = 0 . 97), CID510 ( z = 1 . 12), CID-361 ( z = 1 . 18), CID-445 ( z = 1 . 26), and CID-452 ( z = 1 . 41) show more reddish colors at their centers than in the outer region, indicating the possibility of having a bulge-like structure or highly dustobscured region (e.g., Ito et al. 2024). Furthermore, there are cases with extended red structures (e.g., CID668; z = 0 . 97) which may indicate the presence of dust lanes as seen in the X-ray obscured (type-II) AGNs (Silverman et al. 2023). \n4.4. Estimated M ∗ and the comparison of different SED fitting methods \nFigure 9. Three-color cutout PSF-subtracted images (the F277W, F150W, and F115W for RGB) of all host galaxies detected in all F115W, F150W, and F277W in the order of redshift. We have not performed deconvolutions based on the PSF FWHM differences between each filter, and consequently, each filter image possesses a different-sized PSF. The target IDs and the redshifts are shown in the up-left corner of each image. The white bars in the lower left part are 1 '' in length. Thanks to the high spatial resolved deep observation of JWST and the careful decomposition analysis, we can clearly identify substructures, such as bulges, spirals, bars, and dust lanes. \n<!-- image --> \nAs described in Section 3.5, we perform SED fitting of the host galaxy photometry to estimate M ∗ . For five galaxies detected in less than two bands, we estimate M ∗ upper limits by assuming H/T = 0 . 2 and the aver- \nF277W photometry-toM ∗ ratio in the sample. The assumed H/T of 0 . 2 is because galaxies with H/T ≳ 0 . 2 are almost detected in our methods (Figure 7(d)). The inferred M ∗ for each host galaxy is reported in Table 3. \nFigure 10 shows four examples of the SED fitting results with the residuals. Generally, for the objects with z ≳ 1, the photometry or upper limit from F814W and F115W fall at a rest-frame wavelength shorter than 4000 ˚ A break and are important in constraining stellar population parameters of host galaxies. \nWe independently employ two distinct SED fitting codes, CIGALE and Prospector , as explained in Section 3.5. Both codes are run having as similar parameter settings as possible. In Figure 11, we compare the parameters obtained from both codes. As shown in Figure 11 (a), the results exhibit a significantly high positive correlation. However, we find an offset of approximately ∆ log M ∗ = +0 . 13 dex (corresponding to 1 σ ). This offset is not far from a common systematic M ∗ uncertainties among SED fitting methods reported in Pacifici et al. (2023). It remains challenging to determine whether M ∗ from CIGALE or Prospector is more accurate. In this study, we primarily use the CIGALE M ∗ in the main discussion to maintain consistency with previous studies for AGN-host galaxies (e.g., Zou et al. 2019; Ishino et al. 2020; Shen et al. 2020; Li et al. 2021; Koutoulidis et al. 2022; Zhuang et al. 2023; Li et al. 2024). \nFinally, in Figure 8 (d), we compare the estimated M ∗ with each final PSF. The estimated M ∗ are on the 1:1 line and show a strong correlation, suggesting that the estimated M ∗ with different final PSFs remain highly consistent. This consistency can be attributed to the fact that, as shown in Figure 8 (d), H/T or host flux from different final PSFs is also consistent. \nWe compare our CIGALE M ∗ with Zhuang et al. (2023), which also utilized CIGALE on the COSMOS-Web data, and find that these two measurements well agree with each other with a scatter of ∆ log M ∗ = +0 . 08 +0 . 20 -0 . 18 . This is very encouraging since their decomposition analysis is independent of our effort. \nIn Figures 11 (b)-(d), we compare results for the other output SED model parameters: A V , t age , and τ . While A V and t age exhibit large uncertainties, they show a positive correlation, with the median offset being close to zero within the range of uncertainties. Regarding τ , it is evident that significantly inconsistent values are observed around log ( τ/ Gyr) ∼ 0 . 5. Considering that t age is generally log ( t age / Gyr) ≲ 0 . 6 in our sample, this discrepancy can be attributed to the challenge of accurately determining SFH when τ ≫ t age . \nAdditionally, we confirmed a strong negative correlation ( ρ = -0 . 60, p ≪ 0 . 05) between ∆log τ and ∆log M ∗ . This relation is because larger τ indicates a longer-lasting SFH, i.e., star formation has been persisting more recently. Consequently, galaxies with larger \nτ tend to host more young stellar populations, resulting in a smaller mass-to-light ratio and smaller M ∗ . While these differences may be attributed to differences in stellar models or fitting strategies (i.e., Bayesian or nonBayesian), further investigation is omitted in this paper since it does not impact the main results of this study.", '5. M BH -M ∗ RELATION': "In the left panel of Figure 12, we plot our measurements of M BH as a function of M ∗ . Based on the large sample covering a broad range in both parameters, we find a weak positive correlation with a weighted Spearman's correlation coefficient 2 of ρ = 0 . 25. As done in previous local studies, we attempt to fit the observational data with a linear function of \nlog M BH = α log M ∗ + β. (10) \nThe red line represents the results with α ≃ 0 . 42 +0 . 12 -0 . 08 and β ≃ 4 . 08 +0 . 86 -1 . 26 , and the orange-shaded region showing the 1 σ confidence interval. The coefficient of determination R 2 for this linear fitting is 0.049, which indicates that this linear model does not sufficiently explain the data. We test the validity of this linear model by fitting the data with a model with the fixed α of 0 and compared the BIC values. Although the BIC for the α -free model is lower than the BIC for the α = 0 model, the difference in BIC is less than 10. This means the α -free model is not significantly better than the α = 0 model. We also run the Shapiro-Wilk test and got a p-value of 0.37. Therefore, at a 5% significance level, the null hypothesis cannot be rejected, and we cannot say that the observed data does not follow a Gaussian distribution. However, it is important to note that this result does not consider the selection bias and does not necessarily represent the intrinsic distribution on the M BH -M ∗ plane at high redshift. \nFor comparison, we also provide the linear fit to the local sample consisting of 30 inactive galaxies (Haring & Rix 2004) and 25 active galaxies (Bennert et al. 2011b). The fit to these 55 galaxies results in α local ≃ 0 . 97 +0 . 10 -0 . 11 and β local ≃ -2 . 48 +1 . 15 -1 . 11 . R 2 for this fitting is 0.63, and this suggests that the local M BH -M ∗ relation is well described by the linear model. Note that the local sample and our highz sample have different selection effects. Thus, we cannot directly compare α and β with α local and β local (see Sections 5.1 and 5.2 for the discussion with consideration of selection bias). \nFor investigating the redshift dependence of the mass relation, we calculate ∆ log ( M BH /M ⊙ ), the relative off- \nFigure 10. Representative examples of SED fits to PSF-subtracted host galaxies. Source ID and their estimated M ∗ with 1 σ confidence range are shown in the top right corner of each plot. The inverted triangle indicates the 3 σ upper limit due to the non-detection in each filter. The bottom colored shades show the transmission curves of F814W, F115W, F150W, F277W, and F444W from left to right. The lower plot in each panel shows the difference between the observational and best-fit model photometry scaled by the error of the data. \n<!-- image --> \nFigure 11. Comparison of SED fitting results with two different SED fitting codes. Panels (a) to (d) compare the estimated log M ∗ , A V , t age , and log τ . X and Y axis correspond to the result with CIGALE and Prospector. The black dashed line indicates the x = y line. The median and 1 σ confidence level of the difference between the two results (defined as ∆ = Prospector -CIGALE) and Spearman's correlation coefficient are shown in the lower right part in each panel. \n<!-- image --> \n∗ \n/circledot \nset of the black hole mass at given M ∗ from the local relation: \n∆log( M BH /M ⊙ ) = M BH -α local log ( M ∗ /M ⊙ ) -β local , (11) \nFor α local and β local , we use the above values from the local samples (Haring & Rix 2004; Bennert et al. 2011b), i.e., 0 . 97 and -2 . 48, respectively. We plot ∆log( M BH /M ⊙ ) as a function of z in the right panel of Figure 12. We then parameterize ∆log ( M BH /M ⊙ ) \nTable 3. Data summary of our sample. \nz \nFigure 12. ( left ) Observed M BH and M ∗ distribution as shown by the red circles. A linear fit is indicated by the red line with the shaded region indicating the 1 σ confidence range. The best-fit relation and Spearman's correlation coefficient are reported at the top of the panel. ( right ) ∆log ( M BH /M ⊙ ) as a function of redshift where zero corresponds to the local relation as marked by the gray band having a width representing the local dispersion. In both panels, samples from previous studies using HST are plotted: gray for lowz (Haring & Rix 2004; Bennert et al. 2011b), green for intermediatez (Jahnke et al. 2009; Bennert et al. 2011a; Cisternas et al. 2011; Schramm & Silverman 2013) and orange indicating those from Ding et al. (2020) at z ∼ 1 . 5. Red triangles indicate the M ∗ upper limit and ∆ log ( M BH /M ⊙ ) lower limit for five undetected targets. \n<!-- image --> \n<!-- image --> \n∗ \n/circledot \nto evolve with z as: \n∆log( M BH /M ⊙ ) = γ log (1 + z ) . (12) \nHere, we assume there is no redshift change in α and β and ∆log M BH can be described by only the evolution from the local relation (log M BH = α local log M ∗ + β local ). Fitting our data with Equation (12) without considering the selection bias results in γ = 1 . 33 +0 . 13 -0 . 14 , suggesting \npositive evolution. Although R 2 for this fitting is 0.050, the BIC for this fitting is significantly lower than the γ = 0 model (i.e., ∆ log ( M BH /M ⊙ ) = 0 with a scatter); thus, this model describes the data better than the γ = 0 model. However, because our sample is (X-ray) flux-limited, it's essential to consider the impact of selection bias when determining the mass relation (Ding et al. 2020; Li et al. 2021; Li et al. 2021, e.g.,). Thus, for \n∗ \nthe rest of this section, we measure the intrinsic slope of the mass relation ( α ) at z ∼ 1 . 5 and the redshift evolution parameter ( γ ) by comparing our results with mock catalogs as described in Section 3.6. This approach allows us to account for selection bias and measurement uncertainties to determine the intrinsic redshift evolution and dispersion of the mass relation.", '5.1. Intrinsic slope ( α ) of the mass relation at z ∼ 1 . 5': 'Past efforts to establish the evolution of the ratio between black hole and galaxy mass assume that there is a linear relation at higher redshifts and the slope of the relation matches the local relation (e.g., Ding et al. 2020; Li et al. 2021). This may not necessarily be the case. Here, we assess how well the parameters of a linear relation can be constrained. \nIn particular, we initially assumed that α and β , for constructing the mock samples (Equation (9), have constant values independent of the redshift, and the evolution is expressed simply as γ log (1 + z ). However, the results in the previous Section 5 may indicate a different α from the local value without considering selection biases. In this section, we examine whether the constant α assumption is valid by estimating the intrinsic value of α while considering selection bias and provide evidence for an intrinsic relation between M BH and M ∗ at z > 1 for the first time. \nSince the parameters α , β , and γ exhibit degeneracies, in this subsection, we constrain the z range to z = 1 -2 and perform fitting for α and β with an assumption of γ = 0 over this redshift range. Consequently, we cannot compare the estimated β with the values in the local relation directly. For the intrinsic dispersion of the mass relation σ µ , we assume 0.3 dex in this subsection, thus matching local studies. \nWith the mock observed data described in Section 3.6, we apply the selection criteria corresponding to our sample (Section 2.3) and compare them with the observed results to constrain the evolution parameters. We assume the observation thresholds as; \nF [2 -10 keV] > F [2 -10 keV] , lim , (13) \nFWHM line > 1000 km s -1 , (14) \nM ∗ > 10 10 M ⊙ (15) \nwhere the first and second conditions correspond to the detection limit of the X-ray observation and broad lines for single epoch M BH estimation. The third condition accounts for the detection limit of the host galaxies, and we confirm that galaxies with masses greater than 10 10 M ⊙ are successfully detected across all redshift ranges in our decomposition analysis. Thus, in comparing mock and real observations, galaxies with \nmasses below 10 10 M ⊙ are excluded to maintain consistency with this mock observation. As shown in Figure 2 (a), excluding the eight objects, all sources in our sample have F [2 -10 keV] above the XMM-Newton F [2 -10 keV] , lim . Note that all of the eight targets with F [2 -10 keV] smaller than XMM-Newton F [2 -10 keV] , lim are targets observed only with Chandra. We changed F [2 -10 keV] , lim depending on in which survey each target was detected. Because M BH of our real targets are based on single epoch estimation with three different lines (H α , H β , and MgII), we have three different selection thresholds on the mock galaxy using FWHM H α , FWHM H β , and FWHM MgII . \nFor each of our targets, we first select the mock galaxies with the corresponding selection bias, i.e., the selection condition using the same line information used to estimate M BH . Furthermore, we select the mock galaxies with a similar redshift; \| z -z mock \| < 0 . 1. Then, we calculate the probability that the mock galaxies with the similar z mock as the real galaxy would have the same ˜ M ∗ , mock with \| ∆ M ∗ \| < 0 . 1 and M vir , mock with \| ∆ M BH \| < 0 . 1, i.e., calculate the probability p following \np = N \| ∆ z \| < 0 . 1 , \| ∆ M ∗ \| < 0 . 1 , \| ∆ M BH \| < 0 . 1 N \| ∆ z \| < 0 . 1 . (16) \nWe then calculate the likelihood of our sample being observed for each parameter combination of α and β . Thereby, we estimate the probability distribution of these parameters using MCMC. In the sampling, we assume a uniform prior between 0.1 to 3 for α , and -25 to 8 for β . \nThe obtained α -β distribution is shown on the left panel of Figure 13, indicating a strong anti-correlated degeneracy between α and β . Because the 1 σ contour includes values ( α local , β local ) corresponding to the local relation, our results do not definitively reject the scenario where α and β do not evolve compared to the local relation up to z ∼ 2 . 5. In the right panels of Figure 13, we show the M BH -M ∗ distribution for mock observed galaxies (similar to Figure 6f) generated with manually sampled parameters on the ridge of the α -β degeneracy and the real observed galaxies ( α ∼ 0 . 2 -1 . 8). The plots suggest that the mock data exhibits a similar distribution to our sample within the comparable mass range. To break this degeneracy and improve the precision of determining α and β , a larger sample with a wider M BH range in future studies is essential. Even so, we demonstrate that a relation between M BH and M ∗ at highz is realized based on having a statistical sample afforded by the COSMOS-Web data set. As indicated by Figure 8 (d), the difference in the PSF reconstruction methods does not significantly affect the M ∗ estimation. \n<!-- image --> \n∗ \n/circledot \n∗ \n/circledot \n∗ \n/circledot \nFigure 13. ( left ) Posterior distribution in the α -β plane obtained through a comparison between mock observation and our sample. The posterior distributions for α and β are shown in the right and upper 1D histogram. Each contour indicates 1, 2, and 3 σ levels. The median and 1 σ confidence levels of α and β are shown in the top-right corner of each panel. ( right ) Comparison of mock observed masses generated with each α and β at z = 1 . 5 with the real observed data at z = 1 -2 shown in red circles. Contours show the distribution of mock observation, indicating 1, 2, and 3 σ levels. Each parameter set, α and β , is shown in the upper left corner of each panel. As shown, we cannot reject the scenario that α does not evolve up to z ∼ 2 . 5.Figure 14. Posterior distribution on the σ µ -γ plane obtained through a comparison between mock observation and our sample (the result with the top-5 stacked PSFs). Panels (a) to (c) correspond to the result using basic, Gaussian, and realistic prior settings, respectively. The posterior distribution for σ µ and γ are shown in the right and upper 1D histogram. Each contour (red to orange) indicates 1, 2, and 3 σ from inner to outer. The median and 1 σ confidence levels of γ and σ µ are shown in the top-right corner of each panel. Only in panel (b), we plot the contour from Li et al. (2021) in gray, and their γ and σ µ estimations are shown in the lower left corner based on SDSS quasars at 0 . 3 < z < 0 . 8 with Subaru HSC imaging. Regardless of the prior, the results suggest no or mild evolution at z < 2 . 5. \n<!-- image --> \nTherefore, the posterior distribution of α and β that are estimated based on M ∗ also shows no significant PSF dependency.', '5.2. Intrinsic evolution ( γ ) of the M BH /M ∗ relation': "To determine the evolution of mass relation with consideration of the selection bias and measurement uncertainties, we generate mock observed catalogs with free \nparameters of γ (evolution rate) and σ µ (intrinsic dispersion of the mass relation) and constrain them by comparing the mock catalogs with observational data in a similar manner to Section 5.1. Note that the assumption of γ = 0 and σ µ = 0 . 3 used in section 5.1 is not applied in the fitting performed in this subsection; both parameters are treated as free parameters during the fitting process here. In contrast to Section 5.1, we fixed α and β \nFigure 14 shows the estimated posterior distributions of γ and σ µ using each prior setting with the top-5 stacked PSF results. As evident in all panels by the orange contours, the intrinsic dispersion σ µ is strongly degenerate with the evolution rate γ where a smaller γ results in a larger σ µ as demonstrated in previous studies (Ding et al. 2020; Li et al. 2021). To reiterate, a smaller value for γ biases the mass relation towards relatively lower M BH values thus a larger σ µ is required to reproduce a certain set of observation data. \n<!-- image --> \n∗ \n/circledot \n∗ \n/circledot \nFigure 15. ( left ) Comparison of the observed sample and the mock observation of z ∼ 1 . 5 AGN on the M BH -M ∗ plane. The mock data at z ∼ 1 . 5 is constructed with the best-fit γ and σ µ (the result with the top-5 stacked PSFs assuming the 'Gaussian' prior) while assuming α local and β local . Small red circles with error bars and the red dashed lines indicate the observed data and the linear fit results (same with Figure 12 left ). Contours indicate a distribution of mock observed samples, showing 1, 2, and 3 σ from inner to outer. ( right ) Comparison of the intrinsic M BH -M ∗ relation at z ∼ 1 . 5 (the red line) to the local relation (the gray line). \nin Equation (9) to the values in the local relation ( α local , β local ) to consider the redshift evolution. As illustrated in Section 5.1, we cannot rule out the possibility for evolution of α . However, as described in Section 5.1, α , β , and γ are strongly degenerate, and obtaining physically meaningful results is challenging when all three parameters are left free for the sample being considered here. \nThere is still a possibility that σ µ depends on redshift. Nevertheless, as discussed later, imposing strong constraints on σ µ in our results is challenging due to the sample size and its uncertainties. Therefore, we set σ µ as a constant independent of redshift in the fitting. It means σ µ obtained through this method is considered to be an averaged value over z ∼ 0 . 68 -2 . 5. Even so, this σ µ estimation has the highest statistical significance for such a study at z ≳ 1. We discuss the redshift evolution of σ µ in Section 6.2. Finally, in this analysis, we assume no redshift-dependent parameters among the free parameters. Therefore, there is no need to restrict the redshift range within the data, as in Section 5.1. \nIn the fitting process, we assume a uniform prior distribution for γ between -1 < γ < 1. Then, we have three different prior settings for σ µ : a uniform distribution between 0 . 01 < σ µ < 1 . 0 (basic), a Gaussian distribution with a mean of 0.3 dex and a standard deviation of 0.1 dex (Gaussian), and a uniform distribution between 0 . 25 < σ µ < 1 . 0 with a prohibition of σ µ < 0 . 25 (realistic). \nConsidering the likelihood distribution for γ , the 'basic' prior setting shows a slight positive-to-no evolution with γ = 0 . 46 +0 . 32 -0 . 63 . Similarly, the 'Gaussian' prior setting results in γ = 0 . 48 +0 . 31 -0 . 62 . If we assume that the intrinsic dispersion should not be significantly smaller than the local dispersion, we limit the allowed range for σ µ to be above 0 . 25 ('Realistic' case). In this case, we find γ = 0 . 22 +0 . 39 -0 . 58 , closer to the case for no evolution ( γ = 0) than the results with the other priors. For the latter, the intrinsic dispersion is slightly higher at 0 . 38 +0 . 12 -0 . 09 . In all cases, our results are consistent with very mild or essentially a lack of evolution with respect to the local relation. \nThen, the left panel of Figure 15 compares mock observations using the median parameters ( γ = 0 . 48 and σ µ = 0 . 30) under the assumption of 'Gaussian' prior with the actual observational M BH -M ∗ distribution and relation. We can see that the mock data can explain the observed data well. The right panel of Figure 15 \ncompares the intrinsic relationship, i.e., the relation corrected for the selection bias, with the local relation. Again, the resulting intrinsic relation is consistent with the local relation within the range of errors.", '6. DISCUSSION': "In Section 5.2, our findings suggest a mild or lack of evolution of the mass relation from the local relation when considering selection biases and measurement uncertainties. In this section, we first compare the derived values of γ and σ µ from Section 5.2 with previous studies. Then, we also discuss the cosmic averaging scenario (Peng 2007; Hirschmann et al. 2010; Jahnke & Macci'o 2011).", '6.1. Comparison to other studies': "First, the conclusion of no or mild evolution with γ = 0 . 48 +0 . 31 -0 . 62 is consistent with studies based on 2D decomposition analysis (e.g., Ding et al. 2020; Li et al. 2021) and studies using a SED-fitting-based decomposition method (e.g., Sun et al. 2015; Suh et al. 2020). \nIn Figure 14, we compare our results on the estimated γ -σ µ distribution to those of Li et al. (2021). Our sample has higher redshift range than Li et al. (2021), and the change of ∆ log ( M BH /M ⊙ ) is also proportional to log (1 + z ) in our model. Therefore, to reproduce the observational results, when increasing γ , we need to decrease σ µ more than Li et al. (2021). In other words, the slope of the γ -σ µ degenerate relation is steeper in our study. As a result, while the sample size of this study is approximately six times smaller than Li et al. (2021), the uncertainty of γ is only ∼ 2 times larger than Li et al. (2021). On the other hand, due to the steep slope, imposing constraints on σ µ becomes challenging, and the uncertainty becomes ∼ 4 times larger than Li et al. (2021). Even so, our estimated value of σ µ is 0 . 30 +0 . 14 -0 . 13 , which is remarkably similar to Li et al. (2021) with σ µ =0 . 25 +0 . 03 -0 . 04 . \nIt is worth highlighting that the inference on the value of γ is very close to zero for the 'Realistic' case (Fig. 14c) where we assume that the intrinsic scatter ( σ µ ) cannot be lower than the local dispersion. Interestingly, if σ µ ( ∼ 0 . 4 -0 . 5) is actually higher than the local value, this would push the evolution parameter to negative values ( γ ∼ -0 . 5), thus presenting a scenario where the black holes have to catch up to their host galaxies by a bit.", '6.2. Scatter ( σ µ ) evolution and cosmic averaging': "When assuming a non-casual cosmic averaging scenario (Jahnke & Macci'o 2011), major mergers average and equalize the mass ratio M BH /M ∗ through cosmic history. Thus, σ µ should increase towards high redshift. \nTo test the cosmic averaging scenario with our data, we generate a mock sample (20,000 parameter sets of M BH and M ∗ ) at z ∼ 1 . 4, the median redshift of our observational sample. We generate M ∗ based on the SMF by Weaver et al. (2023), and calculate the M BH assuming the γ and σ sampled in the MCMC run with 'Gaussian' prior setting (Section 5.2). Each mock galaxy is assumed to undergo major mergers following the major merger rate from Rodriguez-Gomez et al. (2015) covering z ∼ 0 -1 . 4. We simulate the redshift evolution of M BH and M ∗ by summing them with those of merging partners. Then, we calculate σ µ at each redshift to trace the expected scatter evolution from the assumed conditions. In this simulation, we do not consider the accretion onto the black hole and star formation, i.e., both M BH and M ∗ are assumed to grow only through major mergers. We limit the mergers to those with mass ratios within ± 0 . 5 dex. \nFigure 16 compares the simulated redshift evolution of σ µ with the results from this study and Li et al. (2021). When assuming the evolution only through major mergers, the growth within the 1 σ uncertainty range significantly encompasses the results of Li et al. (2021). Moreover, our median redshift evolution is consistent with the results of Li et al. (2021). Therefore, the σ µ difference between the results from this study and Li et al. (2021) could be interpreted as the major merger-based scatter evolution. \nHowever, due to the large uncertainty in our results, we cannot draw any definitive conclusions regarding the redshift evolution of σ µ . Furthermore, as evident from Figure 16, our sample has a wide redshift range compared to Li et al. (2021). If σ µ varies with redshift, the sample should be binned in a narrower redshift range to trace the redshift evolution. Nevertheless, as mentioned in Sections 5.1 and 5.2, the degeneracy relation on the γ -σ µ plane tends to steepen toward higher redshifts, making it relatively challenging to impose constraints on σ µ . Future highz statistical studies will likely require samples of a similar size to Li et al. (2021) with N = 584, or even larger to address the redshift evolution of σ µ . Thus, it will be necessary to conduct comprehensive surveys of highz AGNs using next-generation survey data such as Euclid and Roman.", '6.3. General notes on PSF reconstruction methods': 'So far, various studies have performed decomposition of JWST images (e.g. Ding et al. 2022a; Ding et al. 2023; Stone et al. 2023; Yue et al. 2023; Harikane et al. 2023; Zhuang & Shen 2023; Zhuang et al. 2023; Stone et al. 2023). As discussed above or in the previous studies, the results of the AGN+host galaxy 2D decomposition \nFigure 16. Evolution of σ µ at z ≲ 2 based on a simple simulation of the cosmic averaging scenario. The vertical red error bars represent the scatter in σ µ within the z range of our sample using the basic prior. Faint grey lines correspond to each sampled parameter set while the red line and the orange filled region represent the median and 1 σ confidence level of σ µ redshift evolution. The green data denotes the constraint from Li et al. (2021) which is consistent within the 1 σ range of our results. \n<!-- image --> \ndepend significantly on the PSF reconstruction. Especially, Zhuang & Shen (2023) discussed the effect of different PSF on decomposition results. Zhuang & Shen (2023) compared three PSF modeling methods ( SWarp , photutils , and PSFEx ), but they did not compare them with χ 2 ν based methods directly. Thus, in this paper, we summarize the comparison when using different PSF reconstruction methods. \nIn this study, we compare three final PSFs: two obtained by χ 2 ν -based methods (a top-5 stacked PSF and a top-75% stacked PSF) and one from an empirically modeling method, PSFEx . As demonstrated in Figure 4, we find offsets in FWHM among different PSF reconstruction methods. These PSF variations, as indicated by Zhuang & Shen (2023), could potentially introduce biases in estimating n , r e , or H/T . This is due to a broader (narrower) PSF than in reality which tends to result in smaller (larger) n , larger (smaller) r e , and overestimation of host fluxes. However, as shown in Figure 8, morphological parameters such as n , r e , H/T generally exhibit consistent relations. Besides, as shown in Figure 8 (d), different PSFs have less impact on M ∗ estimation than on H/T . This could be due to the fact that M ∗ is estimated from SED fitting (Section 3.5) using multi- \nhotometry that averages the uncertainty in each band. If so, SED fitting with a smaller number of photometric bands (e.g., Ding et al. 2023; Yue et al. 2023), may lead to more severe effects from inaccurate PSF reconstruction. We also find each method has advantages and disadvantages from a technical aspect. Lastly, we summarize below the technical comparison between each method. \nModeling method: The approach of constructing an empirical PSF model from numerous stars, such as PSFEx in this study, has the advantage of being less influenced by noise compared to the χ 2 ν -based methods, as depicted in Figure 17. This method also allows flexible modeling, considering PSF as a function of position or brightness. \nHowever, a drawback is the requirement for many PSF candidates to model a local or flux-dependent PSF, which can be considered a trade-off. Furthermore, we also confirm that FWHM values of PSFEx PSFs depend on configuration parameters. When reconstructing PSFs, it is challenging to determine the best configuration parameters because the exact PSFs are not known. \nχ 2 ν -based selection method: Selecting PSFs based on χ 2 ν from a substantial number of stars, such as Top-5 or Top-75% PSFs, allows easy analysis considering the PSF uncertainties. Also, by fitting with various different single PSFs, the possibility for PSF mismatches is minimized. Notably, Yue et al. (2023) discussed the differences in broad-band PSFs attributed to variations in SED shapes between stars and AGNs. Our method using all single PSFs in the PSF library may consider this PSF uncertainty as a result of selecting lowχ 2 ν single PSFs with a matched shape. \nHowever, using fewer PSF candidates, like the top5 stacked PSF, might increase the noise of the final PSF, as observed in Figure 17. Note that this noise is generally smaller than the central main component; thus, it should not affect significantly except in the case with small H/T . Additionally, our approach involves visual inspection in PSF candidate selection, which might cause a bias. Also, as mentioned by Zhuang & Shen (2023), a lower χ 2 ν does not necessarily mean more correct PSF. Moreover, as a practical demerit, this method needs more time to create a PSF library with visual inspection in each filter and region and more computational cost for SED fittings with all single PSFs. \nGiven that we do not know the correct answers in this study, it is challenging to discuss which method produces the most accurate results. For some targets, decomposition is clearly successful with one final PSF, which can then be evaluated from the residual emission based on other PSFs. We also confirm that these failures \nin fitting can occur with all final PSFs. Therefore, we conclude that it is best to assess the impact on derived properties (e.g., n , r e , H/T ) by varying the method of PSF construction (Section 3.1) and place equal weight on assessing the uncertainties based on varying PSFs (Section 3.3) to obtain solid 2D decomposition results .', '7. CONCLUSIONS': "We performed a 2D decomposition analysis of highz ( z ∼ 0 . 68 -2 . 5) type-I AGNs using the COSMOS-Web (Casey et al. 2023) and PRIMER-COSMOS surveys to measure the black hole - stellar mass relation at highz . Our sample contains 107 targets that are X-rayselected, broad-line AGNs with single-epoch black hole mass estimates (log ( M BH /M ⊙ ) ∼ 6 . 9 -9 . 6) based on H β , H α , and Mg ii from previous spectroscopic surveys (e.g., Schulze et al. 2018). \nBy utilizing HST/ACS + JWST/NIRCam imaging that covers the rest-frame optical to near-infrared, we obtained multi-band information of the AGN host galaxies with unprecedented spatial resolution in which we can clearly identify substructures such as bars and spirals arms (Figure 9). Since AGN-host 2D decomposition is known to be sensitive to the PSF reconstruction methods, we compared the results with three final PSFs reconstructed using the modeling method PSFEx and a χ 2 ν -based selection method. Through a meticulous decomposition analysis using various PSFs, we successfully detected the host galaxies in more than two filters for over 90% of the entire sample. Then, we confirmed that host morphological parameters such as n , r e , and H/T remain relatively consistent regardless of the PSF reconstruction method used (Figure 8). Furthermore, given the high quality of the host galaxy images, this study is expected to serve as a crucial stepping stone for image-based spatially-resolved investigations of AGN host galaxies, such as double-S'ersic model fitting (decomposition fitting with AGN, bulge, and disk components) or parametric/non-parametric substructure analysis. \nWith AGN-subtracted photometry of the host galaxy in multiple bands, we estimate M ∗ by performing SED fitting and present the M BH -M ∗ relation at z ∼ 0 . 68 -2 . 5 (Figure 12). There is a weak positive correlation between M BH and M ∗ with the correlation coefficient of ρ = 0 . 25 ( p = 0 . 010). We fit the mass relation by a simple (log-)linear relation of log ( M BH /M ⊙ ) = α log ( M ∗ /M ⊙ )+ β with consideration of selection biases and measurement uncertainties. Our results show that the slope of the mass relation at z ∼ 2 ( α = 0 . 89 +0 . 61 -0 . 41 ) is consistent with the local relation ( α local = 0 . 97 +0 . 10 -0 . 11 ) (Figure 13). \nAssuming the redshift evolution term of the mass relation to be γ log (1 + z ), we further determine the evolution factor γ and the intrinsic scatter of the mass relation σ µ while considering selection biases and uncertainties based comparisons to mock catalogs (Figure 14). Even though the estimated probability distribution shows strong degeneracy between γ and σ µ , we find no or mild evolution with γ = 0 . 48 +0 . 31 -0 . 62 . If we assume that σ µ is not smaller than the local value, we obtain γ = 0 . 22 +0 . 39 -0 . 58 , which is more consistent with the no- or mild-evolution scenario. The estimated γ -σ µ distribution is largely consistent with Li et al. (2021) based on the HSC imaging of SDSS quasars at z < 0 . 8. Given the higher redshift range of our sample, the slope of the degeneracy relation between γ and σ µ is steeper than Li et al. (2021). Therefore, despite the sample being approximately six times smaller, the estimated γ uncertainty is just slightly larger than Li et al. (2021). \nFurthermore, the estimated value of the intrinsic scatter is σ µ = 0 . 30 +0 . 14 -0 . 13 which is consistent with the local relation and the recent estimate by Li et al. (2021). We show that this value σ µ at high-z may not be in contradiction to a cosmic averaging scenario (Figure 16) as recently put forward by Li et al. (2021) and Ding et al. (2022b) where AGN feedback is invoked to explain the constant level of dispersion with redshift. However, due to the small sample size, high redshift, and wide redshift range, our constraints on the redshift evolution of σ µ are weak. Thus, a larger sample size at z ∼ 1 -3 is needed, especially at highz . \nFuture large-scale surveys such as Euclid and Roman will significantly augment the sample size, along with deeper observations by JWST, to provide stronger constraints on SMBH and galaxy evolution. For future large imaging data sets, visual inspection for all multicomponent fits and manually exclusion of anomalous results will not be feasible; thus, improvements in 2D decomposition techniques or the imposition of more sophisticated conditions to confirm the robustness of host detection is needed. Additionally, by leveraging the high spatial resolution of JWST images, it is important to compare the morphology, substructures (shortly introduced in Section 4.3), and stellar populations of AGNhost galaxies with non-AGN galaxies, as well as to discuss the presence or absence of a bulge component and the M BH -M bulge relation. In order to address these challenges, it is imperative to enhance the 2D decomposition analysis by mitigating the uncertainties related to PSF reconstruction. This involves conducting a meticulous analysis of AGN PSFs, determining the validity of applying stellar PSFs to AGN with different SEDs than stars, identifying the most effective methods for accu- \nrate PSF reconstruction, and establishing a framework for evaluating the uncertainties in reconstructed PSFs. \n- This work is based on observations made with the 1\n- NASA/ESA/CSA James Webb Space Telescope. The 2\n- data were obtained from the Mikulski Archive for Space 3\n- Telescopes at the Space Telescope Science Institute, 4\n- which is operated by the Association of Universities 5\n- for Research in Astronomy, Inc., under NASA contract 6\n- NAS 5-03127 for JWST. These observations are associ7\n- ated with program IDs 1727 and 1837, and the specific 8\n- observations analyzed can be accessed via DOI. Nu9\n- merical computations were in part carried out on the 10\n- iDark cluster, Kavli IPMU. This work was made possi11\n- ble by utilizing the CANDIDE cluster at the Institut 12\n- d'Astrophysique de Paris, which was funded through 13\n- grants from the PNCG, CNES, DIM-ACAV, and the 14\n- Cosmic Dawn Center and maintained by Stephane 15\n- Rouberol. We thank Marko Shuntov, Kei Ito, and 16\n- Mingyang Zhuang for giving us useful advice for recon17\n- structing the PSFs. We thank Mingyang Zhuang, Yue 18 \n19 \nShen, and Junyao Li for giving us their measurements \n- for comparison. We thank the anonymous referee for 20\n- helpful feedback. Kavli IPMU is supported by World 21\n- Premier International Research Center Initiative (WPI), 22\n- MEXT, Japan. The Cosmic Dawn Center (DAWN) is 23\n- funded by the Danish National Research Foundation 24\n- under grant DNRF140. TT is supported by Forefront 25\n- Physics and Mathematics Program to Drive Transfor26\n- mation (FoPM), a World-leading Innovative Graduate 27\n- Study (WINGS) Program, the University of Tokyo. JS 28\n- is supported by JSPS KAKENHI (JP22H01262) and 29\n- the World Premier International Research Center Ini30\n- tiative (WPI), MEXT, Japan. This work was sup31\n- ported by JSPS Core-to-Core Program (grant number: 32\n- JPJSCCA20210003). BT acknowledges support from 33\n- the European Research Council (ERC) under the Eu34\n- ropean Union's Horizon 2020 research and innovation 35\n- program (grant agreement number 950533) and from the 36\n- Israel Science Foundation (grant number 1849/19). JR 37\n- is supported by JPL, which is run by Caltech for NASA. 38\n- GEM and SG acknowledge the Villum Fonden research 39\n- grant 13160 'Gas to stars, stars to dust: tracing star for40\n- mation across cosmic time,' grant 37440, 'The Hidden 41\n- Cosmos,' and the Cosmic Dawn Center of Excellence 42\n- funded by the Danish National Research Foundation un43\n- der the grant No. 140. 44 \nAPPENDIX", 'Facilities: JWST (NIRCam), HST (ACS)': 'Software: astropy (Robitaille et al. 2013; PriceWhelan et al. 2018; Collaboration et al. 2022), CIGALE (Boquien et al. 2019; Yang et al. 2022), FSPS (Conroy et al. 2009; Conroy & Gunn 2010), galight (Ding et al. 2020), lenstronomy (Birrer & Amara 2018; Birrer et al. 2021), matplotlib (Hunter 2007), wCorr (Bailey & Emad 2023), numpy (Harris et al. 2020), photutils (Bradley 2023), psfr (Birrer et al. in prep), Prospector (Leja et al. 2017; Johnson et al. 2021), jwst (Bushouse et al. 2023) \nTable 4. The number of the undetected host galaxies for each condition, PSF and filterTable 5. The number of host-galaxy detections for each final PSF \nTable 4 summarizes the number of undetected host galaxies based on the conditions described in Section 3.3 for each PSF and filter, while Table 5 then summarizes the number of detected hosts for each final PSF. Regarding the JWST filters, we have less galaxies that are classified as non-detection due to the BIC condition than due to the S/N host condition. The number of undetected hosts is lowest in F277W, followed by F150W and F444W. On the other hand, F115W and F814W show a larger number of undetected hosts, especially due to S/N host . This trend is because F814W is the HST observation with lower resolution and shallower depth than the JWST observations, and both F814W and F115W are in the shorter-wavelength side of the Balmer break, leading to a tendency for a smaller intrinsic H/T .', 'B. DETAILED COMPARISON OF DIFFERENT PSF RESULTS': "As mentioned in Section 3.1, 2D decomposition is significantly influenced by the differences in PSFs. In this study, we compare three different PSFs and discuss the PSF dependency of the results. Figure 17 (images on the left; a-c) shows each final PSF image for the four NIRCam filters. We fit each final PSF and each single PSF in the PSF library with the 2D Gaussian model, and measure the FWHM along a semi-major axis and ellipticity b/a (defined as FWHM minor /FWHM major ). Figures 17 (d) and (e) show the distribution of FWHM and b/a in each filter for an example AGN, CID-62, at z spec ∼ 1 . 92. As the number of stars used increases in the order of the top-5 stacked, the top-75% stacked, and PSFEx , we can see that the background noise is correspondingly lower. Regarding the FWHM distribution, the top-5 and the top-75% PSFs are consistent with the FWHM distributions of single PSFs within the PSF library. PSFEx have consistent FWHMs in F115W and F150W, and slightly smaller FWHMs in F277W and F444W, suggesting the possibility of bias between automatic PSF selection by PSFEx and semi-automatic PSF selection by galight . In addition, each final PSF tends to have a higher b/a than individual single PSFs. Regardless of whether χ 2 ν -based stacking or empirical modeling is employed, considering that the final PSF is a more reasonable \nPSF reconstruction than a single PSF, this suggests a potential bias towards a more elliptical PSF when using a single PSF. Next, Figure 18 compares the estimated H/T using each PSF and filter (the full-filter version of Figure 8 (c)). We can find strong positive correlations for all filters, with the distribution along the y = x relation. However, when comparing the distribution with y = x , it is evident that PSFEx tends to estimate larger H/T for F277W and F444W compared to top-5 and top-75% PSFs. As mentioned in Section 4.2, this trend can be interpreted by considering the differences in FWHM for each filter and PSF. As shown in Figure 4, PSFEx tends to exhibit smaller PSFs in F277W and F444W than the top-5 and top-75% PSFs. Zhuang & Shen (2023) suggest that different PSFs result in larger H/T in the 2D decomposition analysis. Therefore, it is conceivable that PSFEx shows larger H/T in F277W and F444W with the smaller PSFs than the top-5 and top-75% PSFs. \nFigure 17. Comparison of different methods to evaluate empirical models of the PSF. Columns (a) to (c) show images of the top-5 stacked PSF, the top-75% stacked PSF, and the PSFEx -PSF for an example galaxy, CID-62 ( z = 1 . 92), in F115W, F150W, F277W, and F444W, from top to bottom. FWHM along the semi-major axis measured from the 2D Gaussian fits is shown in the top-left corner of each image. The white bars in the lower left indicate a scale of 1 '' . Columns (d) and (e) show the distribution of the FWHM (semi-major axis) and ellipticity b/a . Gray histograms display the distribution of each single PSF in the library (Section 3.1.1). The values for the top-5 stacked PSF, the top-75% stacked PSF, PSFEx -PSF are marked by the blue, green, and red vertical lines, respectively. \n<!-- image -->", 'C. GALIGHT TEST WITH MOCK IMAGE': "We generate mock images of AGN and their host galaxies and perform 2D decomposition on these images to verify whether galight can accurately recover each parameter for the host galaxies. We use lenstronomy (Birrer & Amara 2018; Birrer et al. 2021) to generate 2,000 NIRCam/F150W mock galaxy images. These are constructed as a superposition of a PSF and a smooth analytic S'ersic model based on randomly sampled parameters: host-to-total flux ratio ( H/T ), S'ersic index ( n ), effective radius ( r e ), semi-major axis orientation ( θ ), and ellipticity ( b/a ). The software tools lenstronomy and galight have different definitions of r e , with the relation r e, galight = ( b/a ) -0 . 5 × r e, lenstronomy . \nFigure 19 (a-c) compares the estimated major parameters ( H/T , r e , and n , respectively) with the input values. First, focusing on H/T , we can find the distinct change in the trend around the input log ( H/T ) value log ( H/T ans ) ∼ -2. For log ( H/T ans ) ≲ -2, the estimated value log ( H/T est ) remains relatively constant around -1 . 5, deviating significantly from the y = x line. This implies that detecting weak galaxy components with log ( H/T ans ) ≲ -2 is practically challenging using the 2D decomposition analysis with galight . On the other hand, for log ( H/T ans ) ≳ -2, log ( H/T est ) exhibits a consistent trend along the y = x relation, while slightly larger than the input values. This trend is consistent \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 18. Comparison of the estimated H/T in each filter using different PSFs (Columns (a) to (d)). The gray and red colors correspond to whether they were detected in each filter. Spearman's correlation coefficient ρ for each prior is shown in the lower right corner of each panel. High correlation coefficients and the distribution around y = x (black dashed line) suggest consistent results among the fitting with different PSFs. \n<!-- image --> \nIn this section and consistent with the main part of this paper, all r e are unified under the definition in galight . The specified range for each parameter is log( H/T ) in [ ∼ -5 , 0], n in [1 , 6], r e in [ ∼ 0 . '' 05 , 1 . '' 6], b/a in [0 . 2 , 1 . 0], and θ in [0 , π ] (in radian). We assume the PSFEx PSF (Section 3.1.2) and that the center of host galaxy components are aligned with the center of PSF components. Additionally, we add Gaussian noise equivalent to the real NIRCam/F150W COSMOS-Web images. Using the generated mock images, we performed 2D decomposition with the lowχ 2 ν top-5 PSF reconstruction strategy (Section 3.1.1). Since the PSFEx PSF is assumed when generating the mock images, the results allow us to assess the performance in cases where the PSF may differ from the true PSF, similar to real data analysis. Note that we cannot conclude which PSF reconstruction method is superior from this analysis. In this test, we do not consider the effects of more complex morphological features than single S'ersic components, such as double S'ersic (disc+bulge) components, position offset between AGN and host, substructures (bars, spirals, and clumps), and nearby or merging objects. \nwith Zhuang & Shen (2023), who reported that different PSFs tend to overestimate H/T . As shown in Figure 7a, log ( H/T est ) for the real detected galaxies in this study are log ( H/T est ) ≳ -1. If we limit the mock results to log ( H/T est ) ≳ -1, the difference between the estimation and input values (defined as estimation - input values) is ∆log( H/T ) ∼ 0 . 06 ± 0 . 09 dex, suggesting sufficiently consistent estimation of H/T . Next, examining r e and n , we find a consistent distribution along the y = x relation, with an overall tendency to underestimate r e and overestimate n . We can also see that some objects hit the r e lower limit ( r e ∼ 0 . '' 05) and n upper limit ( n ∼ 7). If we limit the analysis to the above realistic cases (log ( H/T est ) ≳ -1), as shown in Figure 19b, r e shows consistent results with ∆ r = -0 . 14 ± 0 . 17 and less hitting to the r e lower limit. However, for n , many objects still exhibit the n upper limit even with log ( H/T est ) ≳ -1. If we additionally cut the upper limit object with n < 6 . 8, the difference between the estimation and the input values is ∆ n ∼ 1 . 4 ± 1 . 6, suggesting a consistent estimation with medium scatter. We confirm that largern objects tend to have larger ∆ n , indicating the increasing difficulty of accurately reconstructing n when they have larger n . Examining the residual H/T (defined as log ( H/T est ) -log ( H/T ans )) as a function of r e, ans (Figure 19d), we confirm that, with log ( H/T est ) > -1, the values of H/T can be recovered independently over a wide r e range of ∼ 0 . '' 1-1 . '' 2, effectively covering the typical r e of the sample (Figure 7b). \nIn this paper, we assume the PSF + single S'ersic model to fit the galaxy images. As discussed in section 3.2, single S'ersic model is a first-order approximation and cannot fully describe galaxy morphologies. For example, substructures such as bars and spiral arms are not considered, and both disk and bulge components cannot be described with a single S'ersic component. Even so, we expand the above mock tests using a PSF + double S'ersic model to address the effect of a bulge component on the PSF + single S'ersic model fitting. One of the double S'ersic components corresponds to a disk, and we assume r e, disk = 0 . '' 35 and n disk = 1 . 0. The other S'ersic component corresponds to a bulge, and we assume n bulge = 4 . 0. r e, bulge , H/T , bulge-to-host flux ratio B/H , θ disk , θ bulge , b/a disk , and b/a bulge is randomly sampled. The specified range for each parameter is log( H/T ) in [ ∼ -3 , 0], r e, bulge in [ ∼ 0 . '' 06 , 0 . '' 2], b/a disk in [0 . 6 , 1 . 0], b/a bulge in [0 . 8 , 1 . 0], θ disk in [0 , π ] (in radian), and θ bulge in [0 , π ] (in radian). Using these models and randomly sampled parameters, we make mock data by following the same method with the above single S'ersic mock data. Then, we performed 2D decomposition with the lowχ 2 ν top-5 PSF reconstruction strategy (Section 3.1.1), and the results are summarized in figure 20. Figure 20 compares the estimated H/T with the input values. Due to the inclusion of the bulge model, the deviation from the y = x line around log ( H/T ) ∼ -2 is larger than for the single S'ersic mock data (Figure 19a). However, similar to the results for the single S'ersic mock data, H/T is well reconstructed for log ( H/T est ) > -1. Figure 20b and c indicate the B/H dependence of n est and log ( H/T ) residual. From figure 20b, as B/H ans increases and the system becomes more bulge-dominated, the estimated n approaches values from 1 (= n disk ) to 4 (= n bulge ), indicating that the single S'ersic model fits the bulge for a bulge-dominated system. However, from figure 20c, H/T is well reconstructed independently on the B/H in the range of B/H ∼ 0 -0 . 6. Therefore, we conclude that we can reconstruct H/T , i.e., host photometry, well for systems where the bulge constitutes at most ∼ 50% of the total host galaxy flux.", 'REFERENCES': 'Akins, H. B., Casey, C. M., Lambrides, E., et al. 2024, \narXiv e-prints, arXiv:2406.10341, \ndoi: 10.48550/arXiv.2406.10341 \nBailey, P., & Emad, A. 2023, wCorr: Weighted \nCorrelations. \nhttps: \n//american-institutes-for-research.github.io/wCorr/ \nBaker, W. M., Tacchella, S., Johnson, B. D., et al. 2023. \nhttp://arxiv.org/abs/2306.02472 \nBeifiori, A., Courteau, S., Corsini, E. 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2023PhRvD.108j3040K	The atmospheric lepton fluxes play a crucial role in many particle and astroparticle physics experiments e.g. in establishing the neutrino signal and the muon background for neutrino oscillation measurements or the atmospheric background for astrophysical neutrino searches. The Matrix Cascade Equations MCEQ code is a numerical tool used to model the atmospheric lepton fluxes by solving a system of coupled differential equations for particle production interaction and decay at extremely low computational costs. Previously the MCEQ framework accommodated only longitudinal development of air showers an approximation that works well for neutrino and muon fluxes at high energies O 10 GeV and above. However for accurate calculations of atmospheric lepton angular distributions at lower energies the lateral component of hadronic cascades becomes significant. We introduce 2D MCEQ an efficient numerical approach for combined longitudinal and angular evolution of air showers that retains the low computational complexity. The accuracy of the 2D MCEQ is affirmed by its benchmark comparison with the standard Monte Carlo code CORSIKA. Our method can be used for twodimensional evolution of hadronic cascades in arbitrary media and paves the way for efficient threedimensional calculations of atmospheric neutrino fluxes.	2023-11-01T00:00:00Z	['2023arXiv230615263K', '10.1103/PhysRevD.108.103040', '10.48550/arXiv.2306.15263', 'arXiv:2306.15263', '2023PhRvD.108j3040K']	['Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Phenomenology']	Atmospheric lepton fluxes via twodimensional matrix cascade equations	2,023	191	0.22	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	0	https://arxiv.org/pdf/2306.15263.pdf	{'Atmospheric Lepton Fluxes via Two-Dimensional Matrix Cascade Equations': "Tetiana Kozynets, 1, ∗ Anatoli Fedynitch, 2, 3 and D. Jason Koskinen 1 \n1 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark 2 Institute of Physics, Academia Sinica, Taipei City, 11529, Taiwan 3 Institute for Cosmic Ray Research, University of Tokyo, \n5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8582, Japan \n(Dated: April 4, 2024) \nThe atmospheric lepton fluxes play a crucial role in many particle and astroparticle physics experiments, e.g. in establishing the neutrino signal and the muon background for neutrino oscillation measurements, or the atmospheric background for astrophysical neutrino searches. The Matrix Cascade Equations ( MCEq ) code is a numerical tool used to model the atmospheric lepton fluxes by solving a system of coupled differential equations for particle production, interaction, and decay at extremely low computational costs. Previously, the MCEq framework accommodated only longitudinal development of air showers, an approximation that works well for neutrino and muon fluxes at high energies ( O (10 GeV) and above). However, for accurate calculations of atmospheric lepton angular distributions at lower energies, the lateral component of hadronic cascades becomes significant. We introduce '2D MCEq ', an efficient numerical approach for combined longitudinal and angular evolution of air showers that retains the low computational complexity. The accuracy of the 2D MCEq is affirmed by its benchmark comparison with the standard Monte Carlo code corsika . Our method can be used for two-dimensional evolution of hadronic cascades in arbitrary media and paves the way for efficient three-dimensional calculations of atmospheric neutrino fluxes.", 'I. INTRODUCTION': "Interactions of cosmic rays with the atomic nuclei in the Earth's atmosphere produce cascades of secondary particles, referred to as the extensive air showers. These cascades have two components - electromagnetic (encompassing production and subsequent reinteraction of energetic electrons and photons) and hadronic (including production and subsequent reinteraction/decay of unstable mesons). One of the byproducts of hadronic cascades are neutrinos, which span the entire energy range from MeV to PeV and thereby form a broad landscape for probing fundamental physics. In particular, the GeV-scale neutrinos produced in the air showers constitute the main signal for atmospheric neutrino oscillation studies, including e.g. muon neutrino disappearance [1-4], tau neutrino appearance [5-9], and searches for physics beyond the Standard Model [10-13]. In addition, atmospheric neutrinos are an 'irreducible background' for astrophysical neutrino searches (e.g. [14-17]), which further strengthens the motivation for accurate modelling of neutrino production in the Earth's atmosphere. \nThe unoscillated neutrino fluxes depend on several main inputs: the primary cosmic ray flux (including composition and spectrum [18-21]); the hadronic interaction model (prescribing the probabilities of secondary particle yields in hadron-nucleus collisions; e.g. [22-27]); decay probabilities and branching ratios of unstable particles [18, 28]; and model for the atmospheric density as a function of altitude for specific geographical locations [29-32]. At O (GeV) and sub-GeV energies, the angular \ndistributions of the air shower secondary particles ('secondaries') are further affected by the Earth's magnetic field, which curves the trajectories of the charged cosmic ray primary particles ('primaries') and the secondary muons. In addition, the angular spread of the low-energy secondaries with respect to the primary particle axis becomes non-negligible: as the transverse momentum is Lorentz-invariant, the deflection angle grows with decreasing energy and can vary from a few degrees to tens of degrees at GeV-scale energies. Both of these effects are necessary to properly describe the angular evolution of individual air showers and the resulting full-sky angular distribution of the O (GeV) atmospheric neutrinos [33]. \nMonte Carlo simulations are the most natural approach to incorporate the many stages of the air shower modelling into a single computational framework [34, 35]. The Monte Carlo treatment implies that the generation and propagation of the cosmic ray primaries and the interactions and decays of the secondaries are executed on an event-by-event basis. These processes are stochastic and follow the probabilistic particle yield prescriptions of a given event generator . The most widely used realizations of this method to date include the general-purpose codes such as Geant4 [36], fluka [37], mcnp [38] and phits [39] for particle propagation in matter, as well as corsika [32] and aires [40] codes specialized in air shower evolution. The closed-source HKKMS [34] and Bartol [35] atmospheric neutrino flux models are also based on Monte Carlo simulations, employing event generators Jam + DPMJet [26, 41-43] and Target [44], respectively. The HKKMS model has been tuned to reproduce the muon flux data [41, 45] and is set out to incorporate the fixed-target experiment measurements of hadronic interaction yields in the future [46]. It is therefore commonly used as the baseline atmospheric neutrino \nflux model in experimental analyses by e.g. the SuperKamiokande and the IceCube collaborations, as well as in projections for the upcoming Hyper-Kamiokande, JUNO, and DUNE experiments [47, 48]. Both the HKKMS and the Bartol flux models include the geomagnetic effects and the deflection of the secondaries from the primary axis, and therefore are the standard reference for O (GeV) atmospheric lepton flux calculations. While the Monte Carlo approach for the inclusive atmospheric lepton flux calculations provides high level of detail as an inherent advantage, it is computationally expensive, fairly complex, and lacks sufficient flexibility for extraction of systematic uncertainties (e.g. those related to the cosmic ray flux model and the hadronic interaction model parameters [20, 49, 50]). \nAnother natural path towards the inclusive atmospheric neutrino flux modelling is via a solution to the cascade equations describing particle production, interaction, and decay in the atmosphere (see e.g. [30] for a broad introduction into this topic). Numerous studies have tackled these equations semi-analytically, with [51-53] being the latest developments. The semi-analytical method was further overtaken by the high-precision numerical solutions provided by the MCEq software [22, 54]. The MCEq cascade equation solver relies on the probabilities of the secondary particle yields in the interaction and decay processes extracted from event generators and stored as matrices. Avoiding the need to repeatedly run event generators within the user interface, MCEq enables computation of inclusive secondary particle fluxes on millisecond timescales, compared to several CPU-hours typically required by the Monte Carlo calculations. \nDespite the significant speedup over the Monte Carlo approaches and the flexibility to study the impact of the systematic parameters, the MCEq code could not be readily used to predict the angular distributions of the O (GeV) cascade secondaries. The reason for this constraint is that MCEq was originally written in the 1D approximation of the air shower development, i.e., under the assumption of strictly collinear (with respect to the primary cosmic ray axis) secondary particle production and propagation. \nIn this study, we are seeking to extend the MCEq framework with the angular evolution of the individual air showers. We develop the numerical technique and the practical implementation of a two-dimensional cascade equation solver, where the secondaries are allowed to deviate from the primary particle trajectory. Our code, '2D MCEq ' 1 , enables numerical computation of the resulting angular distributions of secondary particles. This advancement has broad applications in the analyses involving O (GeV) atmospheric leptons (and more generally, any hadronic cascade secondaries in arbitrary media) and contributes to future development of fully numerical \nor hybrid three-dimensional calculations of atmospheric neutrino fluxes and air showers. \nThis paper is structured as follows. In Sec. II A, we review the analytical cascade equations in the onedimensional approximation and further show how to incorporate the second (angular) dimension in Secs. II B and II C. The numerical (matrix) form of the 1D equations forming the basis of the MCEq code is reviewed in Sec. III A. We then derive the matrix form for the 2D equations by reformulating them in the frequency domain (Sec. III B). The pipeline of the 2D MCEq software is described in Sec. IV, which includes the steps to prepare the interaction/decay probability matrices (Secs. IV A and IVB) and the principles of the 2D cascade equation integrator (Secs. IV C and IV D). Finally, we compare the 2D MCEq angular distributions to those obtained with the corsika Monte Carlo in Sec. V, focusing on 2D cascades induced by a single cosmic ray primary.", 'A. One-dimensional cascade equations': "In this work, we employ the cascade theory to characterize the spatial development of the secondary particle showers induced by a single cosmic ray projectile. The mathematical basis of this theory is the system of coupled partial integro-differential cascade equations , which are a form of the Boltzmann transport equations for multiple particle species. For the secondary particle species h , we define the single-differential particle density n h with respect to the kinetic energy E : \nn h ( E ) = d N h d E , (1) \nwhich represents the number of particles N h per energy interval. In the one-dimensional cascade theory, this single-differential density evolves as a function of the atmospheric slant depth X : \nX ( h o ) = w h o 0 d lρ air ( l ) , (2) \nwhere h o is the observation altitude above the surface of the Earth, ρ air is the depth-dependent air density, and the integral is evaluated along the trajectory l of the shower core. With ρ air given in g cm -3 , and l taken in cm, the unit of X is g cm -2 . Then, the one-dimensional coupled \ncascade equations [22, 30, 54] read \nd n h ( E,X ) d X = -n h ( E,X ) λ int ,h ( E ) -n h ( E,X ) λ dec ,h ( E,X ) (3a) \n-∂ ∂E ( µ E n h ( E,X )) (3b) \n+ ∑ ℓ w ∞ E d E ℓ d N dec l ( E ℓ ) → h ( E ) d E n ℓ ( E ℓ , X ) λ int ,ℓ ( E ℓ ) (3c) \n+ ∑ ℓ w ∞ E d E ℓ d N int ℓ ( E ℓ ) → h ( E ) d E n l ( E ℓ , X ) λ dec ,ℓ ( E ℓ , X ) . (3d) \nThe 'sink' terms in Eq. 3a represent the decrease in the density of particle type h as the result of its interactions in the atmosphere after travelling the interaction length λ int ,h , or its decay after travelling the decay length λ dec ,h . Another sink term in Eq. 3b stands for the energy losses of the charged particles due to ionization, where µ E = -⟨ d E d X ⟩ is the average stopping power per unit length. The 'source' terms in Eq. 3c and Eq. 3d describe the increase of n h due to the interactions and decays of other particle species ℓ with energy E ℓ . The respective yields of the particle h are reflected in the differential production cross sections d N ℓ ( E ℓ ) → h ( E ) d E . The energy conservation constraint is given in the integral bounds ( r ∞ E ) of Eqs. 3c and 3d: it requires that the total energy E ℓ of the primary particle must be greater than, or equal to, the total energy E of the secondary particle.", 'B. Incorporating the second (angular) dimension': "In high-energy inelastic collisions or decays, the angular deflection θ ℓ → h of the secondary particles h from the primaries ℓ is minor ( ≪ 1 · at energies ≫ 10 GeV), justifying the use of the 1D approximation in the evolution of highenergy hadronic cascades [33]. In this regime, the velocity unit vector ˆ u ℓ → h of h translates to (0 , 0 , 1) ⊤ in a Cartesian coordinate system where the z axis aligns with ℓ . Lower energies necessitate consideration of the x and y components of ˆ u ℓ → h and explicit inclusion of the azimuthal angle φ ℓ → h . Then, the velocity vector becomes ˆ u ℓ → h = (sin θ ℓ → h cos φ ℓ → h , sin θ ℓ → h sin φ ℓ → h , cos θ ℓ → h ) ⊤ . To second order in θ , this can be approximated as \nˆ u ℓ → h = θ ℓ → h cos φ ℓ → h θ ℓ → h sin φ ℓ → h 1 -( θ ℓ → h ) 2 2 . (4) \nSimilarly, the initial particle ℓ can be assigned a unit velocity vector ˆ u ℓ = ( θ ℓ cos φ ℓ , θ ℓ sin φ ℓ , 1 -( θ ℓ ) 2 2 ) ⊤ in a fixed-frame Cartesian coordinate system, where θ ℓ is the angle between the direction of ℓ and the z axis of this system, and φ ℓ is the respective azimuthal angle. \nIn a Monte Carlo simulation, where the interactions or decays would be treated on an event-by-event basis, the direction ˆ u h of the secondary particle h in the fixed (lab) frame could be found via a simple addition of ˆ u ℓ and ˆ u ℓ → h . However, to incorporate angular evolution into the semi-analytical cascade theory, the distributions of the particle travel directions have to be formulated in terms of angular densities . Invoking the azimuthal symmetry, i.e., the invariance with respect to φ , we define the double-differential particle density η with respect to the energy and the polar angle as \nη h ( E,θ ) = 1 θ d 2 N h ( θ ) d θ d E , (5) \nwhich is normalized to the single-differential density: \nn h ( E ) = w θ max 0 η h ( E,θ ) θ d θ. (6) \nThroughout this study, we assume θ max = π/ 2, i.e. consider only forward-going particles, as well as the delta function-like angular distributions of the primaries. As the cascade develops, more secondaries will be produced off-axis and the angular distribution η h of the secondaries will evolve as a function of slant depth. On the other hand, the distribution of the relative angles between the primaries and the secondaries, θ l → h , is defined by the allowed phase space in a given interaction or decay process and constitutes a fixed convolution kernel . The angular distribution of the secondary particle h can then be obtained as a two-dimensional convolution of the primary angular density with the convolution kernel in the plane orthogonal to the primary particle direction. For the case of secondaries obtained in interactions, we denote this kernel as ς l → h and write \nη h ( E,θ ) = η ℓ ( E,θ ℓ ) ∗ ∗ ς ℓ → h ( E ℓ , E, θ ℓ → h ) , (7a) \n= w θ max 0 η ℓ ( θ ℓ ) ς ℓ ( E ℓ ,θ ℓ ) → h ( E,θ ) θ ℓ d θ ℓ , (7b) \nwhere the ' ∗∗ ' operator represents two-dimensional convolution. The convolution kernels for decays will be denoted as δ l → h throughout this study. Following the formalism of [55] for the convolution of two azimuthally symmetric functions, we have absorbed the integration over the azimuthal variables into the definition of ς ℓ → h and δ ℓ → h . To illustrate the 2D convolution principle, we consider as an example a proton-induced hadronic cascade, as shown in Fig. 1. In this simplified setup, a beam of protons with the energy density n enters the atmosphere at the slant depth X 0 aligned with the downward-pointing z axis, hence θ primary = 0 . The direction of this proton beam is represented by the unit vector ˆ u primary . In the 1D geometry, the velocity unit vector ˆ u secondary of ν µ is aligned with ˆ u primary , while in the 2D geometry, this does not hold beyond X 0 . As the proton interacts with the atmospheric nuclei at X 1 , the secondary products of the interaction (including the π + ) gain transverse momentum, \nand their angular distribution widens. This is represented by the convolution with the kernel ς p → π + . The angular distribution of muon neutrinos at X 2 further widens due to the convolution of the pion angular density with the decay kernel δ π + → ν µ . \nMathematically, the production of the particle h with the energy E by the interactions of the primary ℓ with the energy E ℓ leads to the following change in the angular density of h : \nd η int h ( θ ) d X = 1 λ int ,ℓ w π/ 2 0 η ℓ ( θ ℓ ) ς ℓ ( E ℓ ,θ ℓ ) → h ( E,θ ) θ ℓ d θ ℓ . (8) \nAn equivalent expression can be formulated for decays by replacing ς ℓ → h with δ ℓ → h . The appearance of the θ ℓ factor in the integrals of Eq. 8 is an important feature of the 2D convolution in the xy plane, where θ ℓ and θ are \ninterpreted as the radii of the ˆ u l and ˆ u h velocity vectors projected onto xy .", 'C. Two-dimensional cascade equations in the angular domain': 'Equation 8 and its equivalent for decays are the source terms in the two-dimensional cascade equations; they directly modify the angular densities of the secondaries, which evolve longitudinally as a function of the slant depth X in the atmosphere. At the same time, the sink terms in Eq. 3a and Eq. 3b do not change the angular distribution of the primaries and only contribute to the change in the overall normalization. With these two observations combined, we can write down the 2D version of Eq. 3 as follows: \nd η h ( E,X,θ ) d X = -η h ( E,X,θ ) λ int ,h ( E ) -η h ( E,X,θ ) λ dec ,h ( E,X ) -∂ ∂E ( µ E η h ( E,X,θ )) + ∑ ℓ w π/ 2 0 θ ℓ d θ ℓ w ∞ E d E ℓ ς ℓ ( E ℓ ,θ ℓ ) → h ( E,θ ) λ int ,ℓ ( E ℓ ) η ℓ ( E ℓ , X, θ ℓ ) (9) + ∑ ℓ w π/ 2 0 θ ℓ d θ ℓ w ∞ E d E ℓ δ ℓ ( E ℓ ,θ ℓ ) → h ( E,θ ) λ dec ,ℓ ( E ℓ , X ) η ℓ ( E ℓ , X, θ ℓ ) . \nThe longitudinal development of the secondary particle cascades is computed through the forward difference integration of Eq. 9. The new component in Eq. 9 compared to Eq. 3 is the angular development of the secondaries, which is taken care of via the 2D convolutions of the angular densities of the primaries with the interaction/decay convolution kernels.', 'III. MATRIX CASCADE EQUATIONS AND THE MCEQ CODE': 'The basic principle of the MCEq code is to evolve the hadronic and electromagnetic cascades in the atmosphere, given a cosmic ray primary flux and the probabilities of interactions and decays of all primary and secondary particles. With these inputs, MCEq solves the cascade equations for the densities of the secondaries of interest. We begin this section with a review of the numerical form of the one-dimensional equations (Eq. 3) based on the formalism derived in [22, 50, 54]. We further extend this numerical framework to two dimensions, building on [56].', 'A. Review of the matrix cascade equations in 1D': 'For a shower particle h which can interact with nuclei in the atmosphere (e.g. 14 N or 16 O), the interaction cross section is energy-dependent, as is the yield of the interaction products in an inelastic collision. Additionally, if the particle is unstable, the energy spectra of its decay products depend on the boost of the parent particle. It is therefore natural to discretize the transport equation in energy, i.e. to represent the particle densities in discrete energy bins E i , i ∈ [0 , N E -1]. The discrete one-dimensional cascade equation reads: \nd n h E i ( X ) d X = -n h E i ( X ) λ h -n h E i ( X ) λ h ( X ) \nint ,E i dec ,E i (10a) -∇ i [ µ h E i n h E i ( X )] (10b) + ∑ ℓ ∑ E ∗ k ≥ E ∗ i c ℓ ( E k ) → h ( E i ) λ ℓ int ,E k n ℓ E k ( X ) (10c) + ∑ ℓ ∑ E ∗ k ≥ E ∗ i d ℓ ( E k ) → h ( E i ) λ ℓ dec ,E k ( X ) n ℓ E k ( X ) , (10d) \nwhere we arranged the terms in the same order as in Eq. 3 to clarify the correspondences between the continuous and \n/uni0302 \n/uni0302 \n/uni0302 \n/uni03B8 \nFIG. 1. Schematic development of a hadronic cascade ( p → π + → ν µ ) in the 1D (longitudinal-only) and the 2D (longitudinal + angular) geometries. In this diagram, the longitudinal propagation is performed over three discrete steps along the slant depth X for illustrative purposes. At each step in X , the angular distribution of the primaries from the previous step is shown as the dotted line, and the current angular distribution of the specified particle as the solid line. The distributions of secondaries get wider further down the chain due to the convolution with the kernels ς p → π + and δ π + → ν µ (see text for details). \n<!-- image --> \nsin \n/uni0302 \n≈ \n/uni03B8 \nthe discrete equation versions. In Eq. 10c, we defined the coefficient c for the yield of particle h in interactions as \nc ℓ ( E k ) → h ( E i ) = d N ℓ ( E k ) → h ( E i ) d E ∣ ∣ ∣ E = E i ∆ E k , (11) \nwhich translates as the energy density of particles h with energy E i generated per primary ℓ within the energy bin E k . The decay coefficients d in Eq. 10d are defined in the same way. The equations for the different particle species are coupled through the yield coefficients c ℓ → h and d ℓ → h . The solution is obtained by solving Eq. 10 in X [22, 54] iteratively in the matrix form. The yield coefficients are derived from the event generators (e.g. UrQMD [25], DPMJet [26, 27], Sibyll [22, 23], or epos-lhc [24] for hadron-nucleus collisions, and Pythia [28] for decays) by histogramming the secondary particle yields as a function of the secondary particle kinetic energy, E i . In the 1D approximation, all secondary particle angles with respect to the primary particle direction of motion are contributing to the yield coefficient, thereby resulting in an angle-integrated interaction/decay probability.', 'B. 2D matrix cascade equations in the Hankel frequency domain': "Depending on the energy scales of hadronic interactions and unstable particle decays in the atmosphere, the widths of the angular distributions of the secondary particles can vary by orders of magnitude. For example, the average emission angle of 2 GeV pions produced in a collision of a 100 GeV is 10 · with respect to the primary proton direction, while 20 GeV pions deflect only by ∼ 1 · from the proton axis and produce ν µ at angles as small as 0 . 01 · relative to the pion direction. Due to the evolution of hadronic cascades over multiple generations (shower age), the total deflection is amplified by the number of generations even if the angular deflection in a single interaction/decay is small. As a result, Eq. 9 requires a 'universal' θ grid which could accommodate both large and small angular deflections. Making such a grid linear would imply an extremely fine discretization, and the numerical evaluation of the 2D convolution integrals would become prohibitively expensive. If the θ grid was logarithmic, the computation of the convolution integral would become more complicated due to the mis-alignment of the input and the output grids. While the techniques for convolving functions defined on logarithmic grids exist, they usually come with hyperparameters to be tuned \nangular densities \nby the user in order to keep the numerical errors to the minimum [57-59]. This extra freedom in the choice of hyperparameters could lead to unpredictable numerical behaviour in the integration of Eq. 9 over thousands of steps in X . \nTo avoid the complications of the 2D convolutions in the θ space (which we will also refer to as the 'real' space), we choose to operate in the spectral ('frequency') domain instead. This is motivated by the existence of the convolution theorem , which transforms the convolutions in the real space into multiplications in the frequency space. For the specific case of the 2D convolution of the azimuthally symmetric functions ς ℓ → h , δ ℓ → h , and η ( X,θ ℓ ), the correct transform enabling the use of the convolution theorem is the zeroth-order Hankel transform H [55]: \nH [ f ( θ )]( κ ) = w ∞ 0 f ( θ ) J 0 ( κθ ) θ d θ, (12) \nwhere f ( θ ) is a function of the continuous variable θ , κ is the spectral frequency mode ( κ ≥ 0), and J 0 is the zeroth-order Bessel function of the first kind. In the formal definition of H , the upper limit of the θ integral in Eq. 12 is ∞ , however we only consider the forward-going particles with θ ≤ π/ 2. \nThe convolution theorem states that, for the azimuthally symmetric functions f ( θ ) and g ( θ ), \nH [ f ( θ ) ∗ ∗ g ( θ )] = H [ f ( θ )]( κ ) · H [ g ( θ )]( κ ) , (13) \ni.e., the Hankel transform of the convolution result is a product of the Hankel transforms of the input functions in the frequency space [55]. Equation 13 is fully applicable to the two-dimensional cascade equations in Eq. 9. We therefore bring the convolution kernels and the angular densities of the cascade particles to the Hankel frequency space by defining their zeroth-order Hankel transforms as follows: \n˜ η h E i ( X,κ ) ≡ H [ η h E i ( X,θ )]( κ ); (14a) \n˜ ς ℓ ( E k ) → h ( E i ) ( κ ) ≡ H [ ς ℓ ( E k ) → h ( E i ) ( θ )]( κ ); (14b) \n˜ δ ℓ ( E k ) → h ( E i ) ( κ ) ≡ H [ δ ℓ ( E k ) → h ( E i ) ( θ )]( κ ) . (14c) \nThen, we can reformulate Eq. 9 as \nd˜ η h E i ( X,κ ) d X = -˜ η h E i ( X,κ ) λ h int ,E i -˜ η h E i ( X,κ ) λ h dec ,E i ( X ) -∇ i [ µ h E i ˜ η h E i ( X,κ )] + ∑ E ∗ k ≥ E ∗ i ∑ ℓ [˜ ς ℓ ( E k ) → h ( E i ) · ˜ η l E k ]( κ ) λ ℓ int ,E k (15) + ∑ E ∗ k ≥ E ∗ i ∑ ℓ [ ˜ δ ℓ ( E k ) → h ( E i ) · ˜ η ℓ E k ]( κ ) λ ℓ dec ,E k ( X ) , \nwhich is the main equation to be solved in '2D MCEq '. For practical applications, the κ grid is made discrete and integer-valued. This implies that in Eq. 15, the \nmultiplication of the Hankel-transformed convolution kernels and the Hankel-transformed angular densities of the primaries is performed elementwise with respect to the discrete frequency modes κ . The 1D MCEq equation (Eq. 10) is a special case of Eq. 15 for κ = 0, as J 0 (0) = 1 and Eq. 12 becomes equivalent to our earlier definition of the angular density normalization from Eq. 6. Therefore, Eq. 15 retains the computational complexity of Eq. 10, up to a linear scaling by the number of the frequency modes ( N κ ). One can then either choose to solve the N κ equations (one for each κ ) sequentially or in parallel, or to assemble the Hankel-transformed yield coefficients and angular densities into a more complex matrix structure. Our current implementation relies on the sequential solution of the N κ equations but can easily be adapted to the user's preference. In Secs. IV A and IV B, we explain how to arrive at the Hankel-transformed yield coefficients for the relevant interaction and decay channels, as well as provide further details on our choice of the κ grid where these coefficients are stored.", 'IV. 2D MCEQ PIPELINE AND SOLUTION SCHEME': 'The main computational advantage of the MCEq code compared to the Monte Carlo simulations comes from the pre-calculation of the particle yields, which is done outside of the user interface. The pre-tabulated interaction and decay coefficients are used to build the matrices for Eq. 10 (1D MCEq ) or Eq. 15 (2D MCEq ) during the code initialization. These coefficients are derived from the histogrammed kinematic properties of the secondary particles in hadronic interactions or decays, as evaluated via the chromo code [60]. The histograms for 1D MCEq include primary and secondary particle energies ( E prim and E sec ). For 2D MCEq , an additional Hankel mode κ is incorporated as a third dimension, which stores the angular densities of the secondaries in a compact form. Below, we describe how the output of any given event generator is used to populate the ( E prim , E sec , κ ) grid.', 'A. Compact representation of event information': "For the low-energy atmospheric neutrino flux calculations ( O (GeV) and below), we deploy a logarithmicallyspaced kinetic energy grid ranging from 10 MeV to 10 TeV 2 . This grid utilizes a bin width of ∆ log 10 E kin = 0 . 1, resulting in N E = 60 energy bins. The 2D MCEq code currently excludes electromagnetic cascades, deemed unnecessary for the evolution of hadronic cascades that \nproduce O (GeV) atmospheric leptons 3 . Thus, we apply Eq. 15 only to hadrons and leptons (excluding the τ lepton), totaling H = 21 particle species: 6 baryons ( p/ ¯ p, n/ ¯ n , and Λ 0 / ¯ Λ 0 ), 5 mesons ( π ± , K ± , K 0 L , ), and 10 leptons ( µ ± R / L , µ ± , ν e / ¯ ν e , and ν µ / ¯ ν µ ). Muons contribute 6 species, where each µ ± includes two polarizations: lefthanded 'L', right-handed 'R', and an unpolarized component (denoted as µ ± without a subscript). \nFor hadronic interactions, the chromo code [60] runs the following models: UrQMD [25], epos-lhc [24], Sibyll-2.3d [22, 23], and DPMJet-III 19.1 [26, 27]. Pythia 8.306 [28] is used for unstable particle decays. However, it cannot simulate the production of polarized muons in π ± and K ± two-body decays or the three-body decays of polarized muons, instead generating events in the spin-averaged phase spaces. Hence, muon polarization modeling is done separately, as outlined in Sec. IV E and detailed in Appendix A 3. \nOur event generation and histogramming scheme is consistent across all interaction/decay channels. For every primary in the kinetic energy bin k , we assign the logarithmic center ( E k = √ E k ' · E k '' for [ E k ' , E k '' ]) as the primary's initial energy. This primary particle enters the chosen event generator with four-momentum p µ prim = ( E k , 0 , 0 , √ E 2 k +2 E k m prim ) ⊤ , moving along the positive z axis. A stationary nitrogen nucleus ( 14 N) is the target for simulating hadronic interactions. Using other atmospheric nuclei like 16 O has little effect on secondary particle yields. For decays, we set the unstable particle at rest and boost its decay products to the lab frame, recording the daughter energies as in 1D MCEq . \nTo solve the 2D cascade equation (Eq. 15), we additionally need to compute the Hankel-transformed angular densities of secondary particles, i.e., ˜ ς l ( E k ) → h ( E i ) ( κ ). We illustrate this process using the muon neutrino production chain from Fig. 1, i.e., p + 14 N → π + + X ∗ → ν µ + µ + , where X ∗ denotes all other secondary particles and nuclear remnants. As angular distributions of all particles fill a discrete energy grid, we select the primary proton, secondary pion, and tertiary neutrino energy bins around 100GeV, 10GeV, and 4GeV, respectively. This illustrative choice reflects the characteristic energies for the low-energy neutrino production in air showers. \nThe simulation chain begins with protons incident on 14 N. We choose the epos-lhc hadronic model and compute the yield of the secondary π + . The top left panel of Fig. 2 shows the distribution of angles θ p → π + that the secondary pions make with the primary proton axis. The number of entries in the histogram, \nn π + ≡ n p · c p → π + , is equal to the total secondary pion yield in this interaction. Each pion contributes a delta function δ ( θ -θ j p → π + ) , j ∈ [0 , n π + -1] , to the angular density. The Hankel transform of the delta function has an analytical representation, H [ 1 a δ ( θ -a ) ] ( κ ) = J 0 ( κa ) 4 , which can populate the κ -grid immediately after the event generation. Summation of these Hankel-transformed delta functions approaches the Hankel transform of the secondary pions' underlying angular density. This density can be expressed via inverse Hankel transform as: \nς p → π + ( θ ) = 1 n p H -1 [ ˜ ς p → π + ( κ ) ] , \n= 1 n p H -1 n π + ∑ j =1 J 0 ( κθ j p → π + ) ( θ ) . \n(16a) (16b) \nEquations 15 and 16 could theoretically extend to an infinite number of modes κ . In practice, a truncated κ -grid with 24 logarithmically spaced integer modes between 0 and 2000 suffices to accurately represent the angular distributions of GeV-scale atmospheric leptons. In the example shown in Fig. 2, two key observations validate this approach. Firstly, the inverse Hankel transform from Eq. 16 effectively represents the original pion angular distribution, thus demonstrating the utility of the Hankel transform for compacting angular densities of secondary particles. Secondly, the amplitudes of the higher-frequency modes with κ ≥ 100 are negligible, indicating a sufficiently broad angular distribution of the GeV-scale pions produced in proton14 N interactions. For sharper-edged distributions, like in the pion decay to muon neutrinos (middle panel of Fig. 2), the truncated κ grid may not sufficiently reconstruct the angular density, resulting in a characteristic 'ringing'. However, in a realistic air shower, the effect of this artefact is minimal due to the wider pion angular distribution. \nThe final ν µ angular distribution from the Monte Carlo simulation chain is successfully reconstructed through the inverse transform of the convolution result in the Hankel space: \nς p → ν µ ( θ ) = H -1 [ ˜ ς p → π + ( κ ) · ˜ δ π + → ν µ ( κ ) ] ( θ ) . (17) \nThis demonstrates the application of the convolution theorem from Eq. 13. 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GLYPH<246>mftkGLYPH<246>q \nadenqd+ sgqntfg sgd enqvGLYPH<246>qc \nqfx nmdr vhkk mns cdGLYPH<222>dbs eqnl \nd knv,dmdqfx rdbnmcGLYPH<246>qhdr vhkk \nGLYPH<246>jdm \nbGLYPH<246>qd \nne uhGLYPH<246> \np- 5- Sgd mdv bnlonmdms hm \nkx- Etqsgdqlnqd+ sgd dunktshnm \nGLYPH<246>c \nGLYPH<246>ants \nhshdr ne sgd oqhlGLYPH<246>qhdr vhsg sgd \nhr sgd GLYPH<246>mftkGLYPH<246>q \nhydGLYPH<246>akd \nrkGLYPH<246>ms \nqfx nmdr vhkk mns cdGLYPH<222>dbs eqnl \nc sgd \nf \nhmcdw hm \n+ sghr dptGLYPH<246>shnm fhudr sgd dun, \njdqmdkr- \nGLYPH<246>jdm \nbGLYPH<246>qd \nne uhGLYPH<246> \nkGLYPH<246>q \nchrsqhatshnmr ne rdbnmcGLYPH<246>qhdr \nc sgd \nf \nhmcdw hm \nkx- Etqsgdqlnqd+ sgd dunktshnm \nhshdr ne sgd oqhlGLYPH<246>qhdr vhsg sgd \nm hm GLYPH<246> \nrhmfkd hmsdqGLYPH<246>bshnm.cdbGLYPH<246>x \ndmdqfx cdmrhsx ne GLYPH<246>kk oGLYPH<246>qshbkd \nd rkGLYPH<246>ms \ncdosg \nhydGLYPH<246>akd \nrkGLYPH<246>ms \njdqmdkr- \nDp- 5 cdlGLYPH<246>mcr \n+ sghr dptGLYPH<246>shnm fhudr sgd dun, \nq ne oGLYPH<246>qshbkdr ne sxod \nkGLYPH<246>q \nchrsqhatshnmr ne rdbnmcGLYPH<246>qhdr \nlncGLYPH<246>sd \nansg kGLYPH<246>qfd \ndmdqfx cdmrhsx ne GLYPH<246>kk oGLYPH<246>qshbkd \nd dptXshnmr hm sgd GXmidk \nbx cnlXhm \nm hm GLYPH<246> \nrhmfkd hmsdqGLYPH<246>bshnm.cdbGLYPH<246>x \nπ \nhmf rtbg GLYPH<246> \ne \nglyph[triangleleft] \nDp- 5 cdlGLYPH<246>mcr \nh \nGLYPH<213> \nNglyph[triangleright] \nδ \n( \nd rkGLYPH<246>ms \ncdosg \nd dptXshnmr hm sgd GXmidk \n× \nbx cnlXhm \nq ne oGLYPH<246>qshbkdr ne sxod \nrbqdshyGLYPH<246>shnm+ GLYPH<246>mc \nlncGLYPH<246>sd \nansg kGLYPH<246>qfd \nπ \nktshnm hmsdfqGLYPH<246>kr vntkc adbnld \ne \nglyph[triangleleft] \nhr \nΛ \ne \nglyph[triangleleft] \nhmf rtbg GLYPH<246> \nh \nGLYPH<213> \nN \n( \nrbGLYPH<246>kdr \nne gGLYPH<246>cqnmhb hmsdqGLYPH<246>bshnmr \nh \nGLYPH<213> \nNglyph[triangleright] \nδ \n( \nm sgd nsgdq gGLYPH<246>mc+ he sgd \nr hm sgd GLYPH<246>slnrogdqd+ sgd vhcsg \n× \nm ne sgd rdbnmcGLYPH<246>qx \nrbqdshyGLYPH<246>shnm+ GLYPH<246>mc \ndms ne sgd rdbnmcGLYPH<246>qx oGLYPH<246>qshbkd \noqnctbs ne bnmunktshnm vntkc \nhr \nΛ \ne \nglyph[triangleleft] \nGLYPH<213> \nN \n( \nrbGLYPH<246>kdr \nne gGLYPH<246>cqnmhb hmsdqGLYPH<246>bshnmr \nd knv,dmdqfx rdbnmcGLYPH<246>qhdr vhkk \nktshnm hmsdfqGLYPH<246>kr vntkc adbnld \nadenqd+ sgqntfg sgd enqvGLYPH<246>qc \nGLYPH<246>r \nsgd ch \nΦ \nr hm sgd GLYPH<246>slnrogdqd+ sgd vhcsg \nGLYPH<246>c \nGLYPH<246>ants \nh \np- 5- Sgd mdv bnlonmdms hm \nm sgd nsgdq gGLYPH<246>mc+ he sgd \ndmrhsx+ dudm he sgd fqhcr ne sgd \ndms ne sgd rdbnmcGLYPH<246>qx oGLYPH<246>qshbkd \nhr sgd GLYPH<246>mftkGLYPH<246>q \noqnctbs ne bnmunktshnm vntkc \nm ne sgd rdbnmcGLYPH<246>qx \nqfx nmdr vhkk mns cdGLYPH<222>dbs eqnl \nadenqd+ sgqntfg sgd enqvGLYPH<246>qc \nGLYPH<246>jdm \nbGLYPH<246>qd \nne uhGLYPH<246> \nsgd sdbgmhptdr enq bnmunkuhmf \nd knv,dmdqfx rdbnmcGLYPH<246>qhdr vhkk \nkx- Etqsgdqlnqd+ sgd dunktshnm \nGLYPH<246>r \nsgd ch \nΦ \np- 5- Sgd mdv bnlonmdms hm \nhshdr ne sgd oqhlGLYPH<246>qhdr vhsg sgd \ndmrhsx+ dudm he sgd fqhcr ne sgd \nhr sgd GLYPH<246>mftkGLYPH<246>q \nGLYPH<246>c \nGLYPH<246>ants \nhydGLYPH<246>akd \nrkGLYPH<246>ms \nqfx nmdr vhkk mns cdGLYPH<222>dbs eqnl \nkGLYPH<246>q \nchrsqhatshnmr ne rdbnmcGLYPH<246>qhdr \nsgd sdbgmhptdr enq bnmunkuhmf \njdqmdkr- \nGLYPH<246>jdm \nbGLYPH<246>qd \nne uhGLYPH<246> \nhshdr ne sgd oqhlGLYPH<246>qhdr vhsg sgd \nkx- Etqsgdqlnqd+ sgd dunktshnm \nm hm GLYPH<246> \nrhmfkd hmsdqGLYPH<246>bshnm.cdbGLYPH<246>x \nhydGLYPH<246>akd \nrkGLYPH<246>ms \nDp- 5 cdlGLYPH<246>mcr \njdqmdkr- \nd dptXshnmr hm sgd GXmidk \nkGLYPH<246>q \nchrsqhatshnmr ne rdbnmcGLYPH<246>qhdr \nlncGLYPH<246>sd \nansg kGLYPH<246>qfd \nm hm GLYPH<246> \nrhmfkd hmsdqGLYPH<246>bshnm.cdbGLYPH<246>x \nhmf rtbg GLYPH<246> \nbx cnlXhm \nDp- 5 cdlGLYPH<246>mcr \nd dptXshnmr hm sgd GXmidk \nrbqdshyGLYPH<246>shnm+ GLYPH<246>mc \nbx cnlXhm \nlncGLYPH<246>sd \nansg kGLYPH<246>qfd \nktshnm hmsdfqGLYPH<246>kr vntkc adbnld \nrbGLYPH<246>kdr \nne gGLYPH<246>cqnmhb hmsdqGLYPH<246>bshnmr \nhmf rtbg GLYPH<246> \nm sgd nsgdq gGLYPH<246>mc+ he sgd \nr hm sgd GLYPH<246>slnrogdqd+ sgd vhcsg \nrbqdshyGLYPH<246>shnm+ GLYPH<246>mc \noqnctbs ne bnmunktshnm vntkc \nm ne sgd rdbnmcGLYPH<246>qx \nrbGLYPH<246>kdr \nne gGLYPH<246>cqnmhb hmsdqGLYPH<246>bshnmr \nktshnm hmsdfqGLYPH<246>kr vntkc adbnld \nGLYPH<246>r \nsgd ch \nΦ \nd knv,dmdqfx rdbnmcGLYPH<246>qhdr vhkk \nr hm sgd GLYPH<246>slnrogdqd+ sgd vhcsg \nm sgd nsgdq gGLYPH<246>mc+ he sgd \ndmrhsx+ dudm he sgd fqhcr ne sgd \nGLYPH<246>c \nGLYPH<246>ants \nm ne sgd rdbnmcGLYPH<246>qx \noqnctbs ne bnmunktshnm vntkc \nGLYPH<246>r \nsgd ch \nvd bgnnrd sn nodqGLYPH<246>sd hm sgd rodbsqGLYPH<246>k GLYPH<213>[eqdptdmbxGLYPH<248>( cnlGLYPH<246>hm \nSn GLYPH<246>unhc sgd bnlokhbGLYPH<246>shnmr ne sgd 1C bnmunktshnmr hm \nπ \nΓ \nk \nΓ \nglyph[triangleleft] \nk \nΓ \nglyph[triangleleft] \ni \nk \nh \ni \nk \nhmrsdGLYPH<246>c- \nHm oGLYPH<246>qshbtkGLYPH<246>q+ \nvd vhrg sn sqGLYPH<246>mrenql sgd ch \nΦ \ntrhnm \nκ \n( \nk \nems \nπ \nglyph[triangleleft] \ncda \nπ \nglyph[triangleleft] \nsgd \nδ \nroGLYPH<246>bd GLYPH<213>vghbg vd vhkk GLYPH<246>krn qdedq sn GLYPH<246>r sgd [qdGLYPH<246>kGLYPH<248> roGLYPH<246>bd(+ \nπ \nΓ \n≥ \ni \ne \nΓ \nglyph[triangleleft] \ni \nk \nglyph[triangleleft] \nk \ni \nglyph[triangleleft] \nπ \ni \nGLYPH<213> \nNglyph[triangleright] \nδ \nk \n( \nδ \nk \n( \nδ \nk \nc \nδ \nk \nc \nδ \nk \nk \nglyph[triangleright] \njdqmdkr GLYPH<246>mc sgd oqhlGLYPH<246>qx oGLYPH<246>qshbkd GLYPH<246>mftkGLYPH<246>q \ncdmrhshdr hm rtbg \nvd bgnnrd sn nodqGLYPH<246>sd hm sgd rodbsqGLYPH<246>k GLYPH<213>[eqdptdmbxGLYPH<248>( cnlGLYPH<246>hm \nκ \nGLYPH<213> \nN \n( \nsgd cdbqdGLYPH<246>rd hm sgd nudqGLYPH<246>kk mnqlGLYPH<246>khyGLYPH<246>shnm- \nBnlahmhmf sghr \nhmrsdGLYPH<246>c- \nHm oGLYPH<246>qshbtkGLYPH<246>q+ \nvd vhrg sn sqGLYPH<246>mrenql sgd ch \nΦ \ntrhnm \nGLYPH<246> \nvGLYPH<246>x \nsgGLYPH<246>s \nsgdhq bnmunktshnm hm sgd qdGLYPH<246>k roGLYPH<246>bd hr qdokGLYPH<246>bdc \ni \nk \ncda \nπ \nglyph[triangleleft] \nκ \n4 \nk \n( \ne \nΓ \nglyph[triangleleft] \nh \nπ \nΓ \n( \nk \nglyph[triangleleft] \n≥ \nnardquGLYPH<246>shnm vhsg Dp- 3+ vd bGLYPH<246>m vqhsd cnvm sgd 1C udqrhnm \ni \nGLYPH<213> \nN \n( \nπ \ni \nGLYPH<213> \nNglyph[triangleright] \nδ \nk \n( \nδ \nk \nc \nδ \nk \nglyph[triangleright] \nax ltkshokhbGLYPH<246>shnm hm sgd eqdptdmbx roGLYPH<246>bd- Hm nsgdq vnqcr+ \n<!-- image --> \nrbqdshyGLYPH<246>shnm+ GLYPH<246>mc \nsgd mtldqhbGLYPH<246>k \nqfx nmdr vhkk mns cdGLYPH<222>dbs eqnl kx- Etqsgdqlnqd+ sgd dunktshnm hydGLYPH<246>akd rkGLYPH<246>ms cdosgr bGLYPH<246>m gGLYPH<246>ud GLYPH<246> kGLYPH<246>q chrsqhatshnmr ne rdbnmcGLYPH<246>qhdr m hm GLYPH<246> rhmfkd hmsdqGLYPH<246>bshnm.cdbGLYPH<246>x Dp- 5 cdlGLYPH<246>mcr GLYPH<246> [tmhudqrGLYPH<246>kGLYPH<248> lncGLYPH<246>sd ansg kGLYPH<246>qfd GLYPH<246>mc rlGLYPH<246>kk hmf rtbg GLYPH<246> fqhc khmdGLYPH<246>q vntkc hm oqGLYPH<246>bshbd GLYPH<246>mc sgdqdenqd qdrsqhbs ntq GLYPH<246>ssdmshnm sn δ ≡ ν ↪ 1=ookxhmf sgd GGLYPH<246>mjdk sqGLYPH<246>mrenql sn ansg φ k ≥ e ↪Γ k ≥ e GLYPH<246>mc π GLYPH<213> Nglyph[triangleright] δ k (+ vd bGLYPH<246>m cd ff md GLYPH<208> π e glyph[triangleleft] h GLYPH<213> Nglyph[triangleright] θ ( × ∈ W π e glyph[triangleleft] h GLYPH<213> Nglyph[triangleright] δ (GLYPH<210>GLYPH<213> θ ( GLYPH<246>mc GLYPH<208> φ k Γ glyph[triangleleft] i ( ≥ e Γ glyph[triangleleft] h ( GLYPH<213> θ ( × ∈ W φ k Γ glyph[triangleleft] i ( ≥ e Γ glyph[triangleleft] h ( GLYPH<213> δ (GLYPH<210>GLYPH<213> θ (- Sgdm+ trhmf sgd bnmunktshnm sgdnqdl+ vd qdenqltkGLYPH<246>sd Dp- 5 GLYPH<246>r c GLYPH<208> π e glyph[triangleleft] h GLYPH<213> Nglyph[triangleright] θ ( c N ; -GLYPH<208> π e glyph[triangleleft] h GLYPH<213> Nglyph[triangleright] θ ( κ e -GLYPH<208> π e glyph[triangleleft] h GLYPH<213> Nglyph[triangleright] θ ( κ e GLYPH<213> N ( d knv,dmdqfx rdbnmcGLYPH<246>qhdr vhkk GLYPH<246>c GLYPH<246>ants sgd oqhlGLYPH<246>qx oGLYPH<246>qshbkd qfx nmdr vhkk mns cdGLYPH<222>dbs eqnl kx- Etqsgdqlnqd+ sgd dunktshnm hydGLYPH<246>akd rkGLYPH<246>ms cdosgr bGLYPH<246>m gGLYPH<246>ud GLYPH<246> kGLYPH<246>q chrsqhatshnmr ne rdbnmcGLYPH<246>qhdr m hm GLYPH<246> rhmfkd hmsdqGLYPH<246>bshnm.cdbGLYPH<246>x Dp- 5 cdlGLYPH<246>mcr GLYPH<246> [tmhudqrGLYPH<246>kGLYPH<248> cd ff mhshnm ne ∈ + sgd toodq khlhs ne sgd δ hmsdfqGLYPH<246>k hm Dp- 6 hr /similarequal + gnvdudq vd nmkx bnmrhcdq sgd enqvGLYPH<246>qc,fnhmf oGLYPH<246>qshbkdr hm oqGLYPH<246>bshbd GLYPH<246>mc sgdqdenqd qdrsqhbs ntq GLYPH<246>ssdmshnm sn δ ≡ ν ↪ 1=ookxhmf sgd GGLYPH<246>mjdk sqGLYPH<246>mrenql sn ansg φ k ≥ e ↪Γ k ≥ e GLYPH<246>mc π GLYPH<213> Nglyph[triangleright] δ k (+ vd bGLYPH<246>m cd ff md GLYPH<208> π e glyph[triangleleft] h GLYPH<213> Nglyph[triangleright] θ ( × ∈ W π e glyph[triangleleft] h GLYPH<213> Nglyph[triangleright] δ (GLYPH<210>GLYPH<213> θ ( GLYPH<246>mc GLYPH<208> φ k Γ glyph[triangleleft] i ( ≥ e Γ glyph[triangleleft] h ( GLYPH<213> θ ( × ∈ W φ k Γ glyph[triangleleft] i ( ≥ e Γ glyph[triangleleft] h ( GLYPH<213> δ (GLYPH<210>GLYPH<213> θ (- Sgdm+ trhmf sgd bnmunktshnm sgdnqdl+ vd qdenqltkGLYPH<246>sd Dp- 5 GLYPH<246>r sgd sdbgmhptdr enq bnmunkuhmf glyph[triangleleft] -i · glyph[triangleleft] -h k cda π glyph[triangleleft] i Φ trhnm jdqmdk GLYPH<246>mc sgd dmrhsx+ dudm he sgd fqhcr ne sgd sgd sdbgmhptdr enq bnmunkuhmf ) ∣ glyph[triangleleft] -i · glyph[triangleleft] -h ∣ k W GLYPH<208> Γ k Γ glyph[triangleleft] i ( ≥ e Γ glyph[triangleleft] h ( -GLYPH<208> π k glyph[triangleleft] i GLYPH<210>GLYPH<213> θ ( κ k cda π glyph[triangleleft] i GLYPH<213> N ( glyph[triangleright] FIG. 2. Top left : Angular distribution of secondary pions in p + 14 N → π + + X , derived from the epos-lhc event generator (solid blue line), and the inverse Hankel transform result (dashed red line) of the top right panel. Top right : Hankel transform of the secondary pions' angular distribution, calculated using Eq. 16. Middle : Similar to the top panels but evaluated for daughter ν µ in π + → µ + + ν µ decay. Bottom : Demonstrating the convolution theorem on the p → π + → ν µ chain: the inverse transform of the product of the Hankel transforms (top right and middle right) reproduces the angular density of the tertiary neutrinos (bottom left). The dotted black line in the bottom left panel indicates the properly normalized angular distribution of secondary pions for comparison. Energy settings for all considered particles are found in Table A1. \nlncGLYPH<246>sd \nansg kGLYPH<246>qfd \nGLYPH<246>mc \nrlGLYPH<246>kk \nktshnm hmsdfqGLYPH<246>kr vntkc adbnld \nhmf rtbg GLYPH<246> \nfqhc khmdGLYPH<246>q \nvntkc \nm sgd nsgdq gGLYPH<246>mc+ he sgd \nδ \nfqhc \nrbqdshyGLYPH<246>shnm+ GLYPH<246>mc \nsgd mtldqhbGLYPH<246>k \noqnctbs ne bnmunktshnm vntkc \nktshnm hmsdfqGLYPH<246>kr vntkc adbnld \nGLYPH<246>r \nsgd ch \nΦ \ntrhnm jdqmdk GLYPH<246>mc sgd \nm sgd nsgdq gGLYPH<246>mc+ he sgd \nδ \nfqhc \nc \nπ \njdqmdkr GLYPH<246>mc sgd oqhlGLYPH<246>qx oGLYPH<246>qshbkd GLYPH<246>mftkGLYPH<246>q \ncdmrhshdr hm rtbg \nne Dp- 0 GLYPH<246>r enkknvr9 \nsgd cdbqdGLYPH<246>rd hm sgd nudqGLYPH<246>kk mnqlGLYPH<246>khyGLYPH<246>shnm- \nBnlahmhmf sghr \ne \nglyph[triangleleft] \nh \nGLYPH<213> \nNglyph[triangleright] \nθ \n( \nc \nN \nGLYPH<208> \ne \nglyph[triangleleft] \nems \nπ \nglyph[triangleleft] \nh \nκ \ni \nGLYPH<213> \nNglyph[triangleright] \nθ \n( \ne \nems \nπ \nglyph[triangleleft] \n· \nglyph[triangleleft] \n- \nh \nh \nh \nk \n∣ \n∣ \nGLYPH<208> \nπ \nglyph[triangleleft] \n- \ncda \nπ \nglyph[triangleleft] \ne \nglyph[triangleleft] \nGLYPH<208> \nπ \nκ \ni \nk \nΓ \nglyph[triangleleft] \nWGLYPH<208> \nφ \n- \nWGLYPH<208> \nW \nφ \nk \nΓ \nglyph[triangleleft] \nΓ \nk \nΓ \nglyph[triangleleft] \nGLYPH<208> \n( \ne \ncda \nπ \nglyph[triangleleft] \ni \n( \ni \nh \nGLYPH<213> \nNglyph[triangleright] \nθ \n( \n≥ \nh \ne \nΓ \nglyph[triangleleft] \nh \nk \nGLYPH<213> \nN \n( \nems \nπ \nglyph[triangleleft] \nh \n( \nh \n( \n- \ni \n- \nκ \n≥ \n( \ne \nΓ \nglyph[triangleleft] \ne \nΓ \nglyph[triangleleft] \nκ \nk \nh \n( \n; \n- \n) \n) \nπ \nGLYPH<208> \nk \nglyph[triangleleft] \nk \nglyph[triangleleft] \nGLYPH<208> \nGLYPH<208> \nπ \nπ \nk \ni \ni \nglyph[triangleleft] \ni \nGLYPH<210>GLYPH<213> \nθ \n( \nGLYPH<210>GLYPH<213> \nθ \n( \nGLYPH<210>GLYPH<213> \nθ \n( \nk \nh \nΓ \nΓ \nπ \nΓ \n( \nπ \nGLYPH<213> \nNglyph[triangleright] \nδ \nGLYPH<213>5( \n4 \nGLYPH<213>7( \nGLYPH<213>7(", 'B. 2D MCEq matrix production': 'We generate 10 million events per primary species and energy bin on the MCEq grid, following the methodology in Sec. IV A. Since different hadronic models have different valid energy ranges, we create interpolated matrices which allow to smoothly transition between the hadronic model choices. This emulates the approach used in codes such as corsika and is explained in Appendix A 2. \nThe generation of secondary particle yields involves H = 21 primary particles in 2D MCEq . The yield of the secondaries is stored as a function of the secondary and primary kinetic energies (on a grid of length N E ) and the Hankel frequency mode κ . The resulting dimension of the 2D MCEq matrices is N κ × ( N E · H ) × ( N E · H ), equating to 24 × (1260 × 1260). These are treated as 24 separate sparse matrices when solving the 2D cascade equation in the Hankel frequency domain.', 'C. Solution in the Hankel space': "̸ \nThe convolution theorem transforms the 2D convolution into multiplication in the Hankel frequency domain. The amplitude of the primary angular distribution corresponding to the mode κ is multiplied by the amplitude of the interaction/decay kernel corresponding to exactly the same mode, i.e., the modes κ 1 and κ 2 are not coupled if κ 1 = κ 2 . This allows us to treat each of the N κ = 24 equations of the 2D MCEq completely independently and solve them using the strategy analogous to that of the 1D MCEq [22, 54]. This involves applying the forward Euler integrator to Eq. 15, i.e., performing the longitudinal evolution of the Hankel-transformed angular densities of the cascade secondaries in discrete slant depth steps ∆ X . This approach is best summarized in the matrix form: \n˜ η ( X t +1 , κ ) = ˜ η ( X t , κ ) -∇ E [diag( µ ) · ˜ η ( X t , κ )] +∆ X t [ ( -I + C k ) Λ int + 1 ρ ( X t ) ( -I + D k ) Λ dec ] ˜ η ( X t , κ ) , (18) \nwhere X t and X t +1 = X t +∆ X t are two consecutive slant depth values, and C κ and D κ are the slices of the yield coefficient 'cubes' ς l → h ( κ ) and δ l → h ( κ ) at the frequency mode κ . Following [22, 54], we also construct the diagonal matrices Λ int and Λ dec from interaction and decay lengths. Each diagonal entry corresponds to a specific particle h and energy bin E i , arranged similarly to the particle density vector ˜ η ( X,κ ) that we seek to evolve. After ˜ η ( X final , κ ) is computed, the final step is to reconstruct the angular densities of η ( X final , θ ) via the inverse Hankel transform.", 'D. Reconstruction of the real-space solutions': "After the final integration step, the 2D MCEq solver returns the state vector ˜ η ( X final , κ ). This includes the Hankel frequency space amplitudes for all cascade particles across the MCEq energy grid. The inverse Hankel transform enables the reconstruction of the secondary particle densities as a function of the angle θ relative to the primary particle axis: \nη ( X , θ ) = H -1 [ ˜ η ( X , κ )] ( θ ) , \nfinal final (19a) ≡ w ∞ 0 ˜ η ( X final , κ ) J 0 ( κθ ) κ d κ, (19b) ≃ w κ max 0 ˜ η ( X final , κ ) J 0 ( κθ ) κ d κ. (19c) \nAlthough the κ grid in 2D MCEq is discrete with logarithmically spaced modes, accurate computation of the integral in Eq. 19c requires a quasicontinuous κ range, which is achieved via spline interpolation of ˜ η ( X final , κ ). This can be done at any X t (0 ≤ X t ≤ X final ) along the integration path, should the solution at a particular slant depth/altitude be of interest to the user. \nThe reconstruction of angular densities of high-energy secondaries (with kinetic energies of 10 GeV and above), as well as those created very early in the cascade evolution, must be treated with care if the starting angular distribution of the primaries is narrow (e.g. delta function-like). Direct application of Eq. 19 may lead to 'ringing' in the reconstructed angular densities for such secondary particles. Thus, it is recommended to apply Eq. 19 for secondary particles with energy < ∼ 10GeV at several kilometers into the atmosphere, and to use the 1D approximation at high energies/altitudes.", 'E. Modeling of muon transport': "Muons play a crucial role in air shower development and atmospheric neutrino flux calculations. They either decay in-flight, generating muon and electron (anti)neutrinos, or survive until the surface level, creating a source of background in neutrino detection. \nMuon polarization occurs as atmospheric muons are produced in the two-body decays of π ± and K ± . In the parent meson rest frame, the muons are fully polarized, with their momenta perfectly aligned or antialigned with their spin direction. This affects the angular distributions and the energy spectra of neutrinos originating from muon decays [53]. A simplified representation of the muon population across the entire continuous range of helicities is achieved by including only six muon species into the 2D MCEq cascade equations (see Sec. IV A). \nAnother phenomenon affecting muon transport is multiple scattering, which modifies the trajectories of the muons as they scatter with atmospheric nuclei. We implement a Gaussian approximation to the Moli'ere formalism for describing this effect [32, 61]. This results in an overall \nwidening of the muon angular distributions compared to the case without multiple scattering. \nAppendix A3 details the implementation of muon polarization and multiple scattering, as well as their impact on the results of 2D MCEq .", 'A. Simulation setup': "To validate our solutions to the two-dimensional matrix cascade equations via 2D MCEq , we use the corsika v7.75 Monte Carlo code [32] as a benchmark. We aim to compare the angular distributions of the GeV-scale atmospheric neutrinos and muons generated in the cosmicray induced air showers. All of our simulations are run for a single angle of incidence of the cosmic ray primary, and the secondary particle angular distributions are computed with respect to the primary particle axis. In the terminology of Fig. 1, we are comparing the distributions of arccos(ˆ u primary · ˆ u secondary ) between 2D MCEq and corsika . \nTo ensure a fair comparison, we equalize the configuration settings between MCEq and corsika to the extent possible. Most importantly, we match the choice of hadronic interaction models by using UrQMD [25] as the low-energy model 5 and epos-lhc [24] as the highenergy model in both corsika and 2D MCEq . The transition energy between models is set to 150 GeV 6 . In Appendix B, we provide a comprehensive list of other relevant physics settings in MCEq and corsika . \nOur setup for the benchmarking of lepton densities consists of a proton primary incident onto the Earth's atmosphere at an inclination angle θ 0 with respect to the negative z axis (downward direction). We test both vertical ( θ 0 = 0 · ) and inclined showers (30 · ≤ θ 0 ≤ θ max = 80 · ). The energy of the proton either is fixed or follows a spectrum with a power-law dependence (e.g. ∝ E -2 . 7 ). For each considered initial condition, we simulate ∼ 1 million events in corsika with different random seeds. This lets us gather enough statistics for the low-energy muons and neutrinos at several observation altitudes. The corresponding binned angular distributions are compared directly to the angular densities obtained with 2D MCEq by solving Eq. 15.", 'B. Results': "To provide a representative example of how the 2D MCEq solutions compare to the corsika Monte Carlo outputs, we choose the case of a 100 GeV proton primary incident at θ 0 = 30 · . In Fig. 3, we show the angular densities of neutrinos ( ν µ + ¯ ν µ ; ν e + ¯ ν e ) and muons ( µ -+ µ + ) at low energies (up to 5 GeV). \nFrom Fig. 3, we find that the angular distributions of O (few GeV) leptons with respect to the proton primary axis are in a very good agreement between 2D MCEq and corsika . For neutrinos, the differences between the two codes are predominantly statistical and reach at most 10% in the tails of the distributions. This level of agreement holds across all altitudes and energy bins considered. For muons, a characteristic tilt of the corsika -toMCEq angular distribution ratio is observed at all altitudes, reaching ∼ 20% in the distribution tails. This is indicative of a bias of the corsika angular distribution towards smaller angles or the MCEq angular distributions towards larger angles. One possible explanation for this pattern is that all particles in 2D MCEq travel exactly the same distance to reach a specific depth, including those that deflect by as much as 20-30 · from the primary axis. However, muons with such large deflection angles naturally travel longer distances than those at 0 · , introducing a factor of 1 / cos θ increase in the integration step length. This means that muons with large deflections from the primary direction must lose more energy than currently modelled in 2D MCEq and migrate from the energy bins shown in Fig. 3 to the bins of lower energy. Qualitatively, this explains why corsika could have fewer muons at large angles. However, since the amount of energy lost is directly proportional to the distance travelled (∆ E ≃ ⟨ d E d X ⟩ ∆ X ), any discrepancies related to the energy loss are expected to accumulate with distance. This shows to a small degree in Fig. 3, where the tilt in the corsika -toMCEq ratio of the muon angular densities develops mildly as a function of altitude. While we cannot definitively connect the angle-dependent discrepancy between 2D MCEq and corsika with the simplified treatment of the angle-dependent propagation step length in 2D MCEq , the latter remains a relevant feature to be implemented in future iterations of the MCEq code. At present, we point out that the 2D MCEq angular densities still provide a very good overall representation of the corsika angular distributions, which can be seen from the agreement of the distribution moments ( ⟨ θ ⟩ and ⟨ θ 2 ⟩ ) in Fig. 4. The sub-degree level of difference observed in the angular distribution moments will not be possible to resolve under any realistic experimental resolution at GeV-scale energies, implying a negligible impact on experimental analyses. \nWhile the main objective of the 2D MCEq code is to evolve angular distributions of the secondaries in addition to the energy spectra already provided by 1D MCEq , we compare the energy spectra from 2D MCEq to those from corsika to provide further validation to our approach. \nN \n2 \nd \nFIG. 3. Angular distributions of atmospheric leptons in a proton-induced air shower ( E 0 = 100GeV, θ 0 = 30 · ), as computed numerically in 2D MCEq (solid lines) and simulated in the corsika Monte Carlo (filled histograms with errorbars). The angle θ on the x axis is the angle a given secondary makes with the direction of the primary proton. The different colors correspond to the different energy bands, and the bottom sub-panel in each plot shows the ratio of corsika ('C') to MCEq ('M'). \n<!-- image --> \nTo obtain the energy spectra from 2D MCEq , we extract the κ = 0 mode from the Hankel-space solutions, which is equivalent to the angle-integrated particle densities as per Sec. III B. As seen in Fig. 5, the spectra from MCEq and corsika agree within a few % in the 1-10 GeV region, which is the main energy range of interest in this study. Above 10 GeV, the difference between the two codes grows as a function of energy, reaching ∼ 20% at the maximum neutrino energies available from 100 GeV primary showers. The energy dependence of the corsika -toMCEq ratio \ncould point to the difference in the treatment of hadronic interactions, e.g. the hadron yields between the different UrQMD model versions. The same level of agreement is observed when the primaries have power law-like spectra, as demonstrated in Figs. C.1 and C.2 in Appendix C. \nFor shower inclinations less than 60 · , our tests demonstrate the same level of agreement as in Figs. 3 and 5. In Figs. D.1 and D.2 in Appendix D, a comparison between 2D MCEq and corsika is further presented for highly inclined showers (80 · ), revealing up to 25% differences in \n+ \nFIG. 4. Comparison of the first ( ⟨ θ ⟩ , or the distribution mean) and the second ( √ ⟨ θ 2 ⟩ , or the distribution width) moments of the angular distributions of atmospheric leptons at the Earth's surface as computed in corsika ('C') and MCEq ('M') for the same initial conditions as in Fig. 3. The bottom panel shows the difference between the corsika and the MCEq estimates. \n<!-- image --> \nthe angular distributions at large angles of deflection from the primary particle axis. While it is possible that the different implementations of muon energy losses or muon propagation geometries are contributing to the observed mismatch, the precise impact of these factors has not been quantified in this study. However, we emphasize that the level of agreement of the angular distributions and spectra in the 1-10 GeV energy region is still satisfactory even for highly inclined showers, and the mild angle-dependent discrepancy observed for single-primary showers will be smeared out by the integration of the secondary particle fluxes across the full sky. \nFinally, we note that both the 2D MCEq solutions and the corsika Monte Carlo outputs are subject to systematic uncertainties due to the choice of the hadronic interaction model used to describe the particle yields in the hadron-nucleus inelastic collisions. For the energy spectra and angular distributions of the low-energy leptons, the choice of the low-energy hadronic interaction model ( E primary ≤ 150 GeV) has the largest impact. We test two hadronic interaction models internally within 2D MCEq in Figs. E.1 and E.2 in Appendix E, finding up to 20% differences between the models at lepton energies below 10 GeV.", 'VI. SUMMARY AND OUTLOOK': "In this work, we have detailed the development and application of 2D MCEq , an extended version of the 1D MCEq software. The 2D MCEq code provides an efficient numerical approach to angular evolution of hadronic cascades with broad particle physics applications - in particular, in atmospheric lepton flux modelling. This \ntool considers all crucial aspects of hadronic and leptonic physics, such as inelastic interactions of hadrons with atmospheric nuclei, decays of unstable particles, energy losses, muon polarization, and muon multiple scattering. \nValidation of 2D MCEq was performed against the standard Monte Carlo code, corsika . The results display agreement within 1-10 % for neutrino angular distributions in air showers up to medium inclinations. Larger differences between the two codes were observed in the distribution tails (corresponding to large angles and high energies). \nGiven the very high level of agreement with corsika and a significant computational superiority over the Monte Carlo approach, 2D MCEq provides a very appealing option for atmospheric lepton flux calculations. The computational cost of the 2D MCEq calculations is between several CPU-seconds for vertical showers and 1 CPUminute for the near-horizontal showers, compared to multiple CPU-hours to gather sufficient statistics for inclusive flux calculations via the Monte Carlo simulations. Our tool therefore opens the pathway to fast exploration of the systematic uncertainties on the angular distributions of atmospheric leptons, including those associated with the hadronic interaction models and the cosmic ray primary flux. The 2D MCEq code can further be utilized within hybrid air-shower calculation frameworks, such as the integration of corsika and conex [64], with the added feature of explicit angular dependence. \nFuture enhancements will involve the integration of three-dimensional calculations, accounting for factors such as the Earth's spherical geometry, the initial angular distribution of cosmic ray primaries, the geomagnetic cutoff for these primaries, and the deflection of cascade secondaries within the geomagnetic field.", 'p ( E =100GeV) at 30 inclination': "FIG. 5. Energy spectra of atmospheric leptons in a proton-induced air shower ( E 0 = 100GeV, θ 0 = 30 · ), as computed numerically in 1 MCEq (solid lines) and simulated in the corsika Monte Carlo (filled markers). Here, ' MCEq ' corresponds to the κ = 0 slice of the 2D MCEq solution. The bottom sub-panel in each plot shows the ratio of corsika ('C') to MCEq ('M'). The shaded gray band represents the region where the MCEq solution is numerically unstable (see the caption of Fig. A.3 for details). \n<!-- image -->", 'ACKNOWLEDGEMENTS': "The authors, T.K. and D.J.K., acknowledge the support from the Carlsberg Foundation (project no. 117238). The author, A.F., is grateful for the supportive environment provided by Prof. Hiroyuki Sagawa's group at the ICRR as a recipient of the JSPS International Research Fellowship (JSPS KAKENHI Grant Number 19F19750). \nThe authors acknowledge the invaluable computational resources provided by the Academia Sinica Grid-Computing Center (ASGC), which is supported by Academia Sinica.", '1. Event generation chain example': 'TABLE A1. Energy grid settings for the event generation chain example in Fig. 2. The three columns (from left to right) correspond to the left bin edge, the bin center, and the right bin edge of the respective particles on the MCEq kinetic energy grid.', '2. Hadronic model interpolation': "The Sibyll-2.3d and epos-lhc generators are the common choice for the high-energy hadronic interaction model in the air shower codes such as corsika . They are nominally valid down to the primary energies of E thresh = 80GeV [32]. At E primary < E thresh , the UrQMD or DPMJet-III 19.1 models are recommended. In our tests, the transition threshold is lifted to E thresh = 150GeV, where the low-energy UrQMD model is still valid [62, 63]. To transition between the two energy regimes, we run both 'high' and 'low' energy models in chromo and interpolate across the two energy bins adjacent to E thresh (one on each side of E thresh ). In this case, 'linear spline' refers to the order-1 spline interpolation on the logarithmic MCEq energy grid. An example result of the interpolated angular distributions of the secondary pions in the p + 14 N → π + + X ∗ process is shown in Fig. A.1.", 'p (141GeV) + (4GeV)': "1 \nFIG. A.1. Angular density of the secondary pions obtained in the p + 14 N → π + + X ∗ process, as simulated in the UrQMD (orange) and the epos-lhc event generators. Assuming the threshold of 150 GeV in the low-/high-energy model transition, the protons in the MCEq energy bin centered at ∼ 140 GeV fall into the 'intermediate' energy regime, where we use a model linearly interpolated between UrQMD and epos-lhc . \n<!-- image -->", '3. Muon production and propagation': 'Along with the generic hadronic cascade development, the cascade equations need to account for muon-specific phenomena - namely, muon polarization and muon multiple scattering - for accurate GeV and sub-GeV neutrino flux predictions. The implementation of these phenomena in 2D MCEq is outlined below.', 'a. Muon polarization': "Most atmospheric muons are produced in the two-body decays of π ± and K ± . In the rest frame of the decaying mesons, the muons are completely polarized, i.e. have their momenta perfectly aligned or antialigned with their spin direction. This is a direct consequence of the angular momentum conservation. For example, µ -always has a negative helicity (projection of the muon spin onto its momentum) in the π -/K -rest frame, since ¯ ν µ generated in the same decay must necessarily be right-handed (i.e., have a positive helicity). A similar argument holds for µ + and ν µ , with the helicity assignments flipped. In the lab frame, the µ -helicity in the two-body decay of a meson M -reads [53]: \nP ( β M , θ ∗ ) = 1 β µ · (1 -r M ) + (1 + r M ) cos θ ∗ β M (1 + r M ) + (1 -r M ) cos θ ∗ β M , (A1) \nwhere r M = ( m µ /m M ) 2 , β is the Lorentz velocity factor ( β X ≡ v X /c ), and θ ∗ is the angle of the muon emission in the decaying meson rest frame (defined e.g. with respect to the z axis). For each individual muon, the helicity defined in Eq. A1 is a continuous quantity spanning the range between -1..1. To simplify the solution of the cascade equations, where the helicity expectation values of the population of muons rather than the spin states of individual muons are of interest, we switch to the helicity basis , i.e. the basis of purely right-handed muons µ ± R and purely left-handed muons µ ± L . Then, on average, the probability of finding µ -in the right-/left-handed state is \nP R , L ( β M , θ ∗ ) = 1 2 [1 ±P ( β M , θ ∗ )] , (A2) \nwhere '+' corresponds to the right-handed state and '-' - to the left-handed state. For µ + , the correspondence between the right-/left-handedness and the +/- sign in Eq. A2 is flipped. In practice, to find the probabilities of the polarized muon production, we let π ± and K ± decay at rest in Pythia and select the two-body decay events. This immediately gives us the muon emission angle θ ∗ and, after boosting to the lab frame, the velocity factors β µ and β M . Then, we compute the probabilities of the right-/left-handedness according to Eq. A2 and assign the respective helicity to each generated muon. We also keep track of the muon lab frame energies and their emission angles with respect to the primary meson boost axis. Finally, we fill in the 'cubes' of the polarized muon yield coefficients in the Hankel frequency space as outlined before in Sec. IV A. \nTo compute and pre-histogram the neutrino yields in the decays of polarized muons, we use the whizard Monte Carlo code [65, 66], which can take the spin direction of the parent muon as an input and sample from the final three-particle phase space simultaneously. This is in contrast with the analytical expressions for the daughter lepton momenta in the polarized muon decay provided in [53] and [67], which are given separately for each daughter and marginalized over momenta of the other two decay products. Thus, we prefer the whizard Monte Carlo approach for simplicity and efficiency of implementation. For each of µ -and µ + decaying at rest, and each of the two choices of spin configurations (ˆ s µ = ⟨ 0 , 0 , ± 1 ⟩ ), we generate 10,000,000 three-body decay events. We then boost the daughter neutrinos to the muon energies corresponding to the kinetic energy grid of MCEq . This gives us the angular distributions and the energy spectra of neutrinos originating from the decays of the left-helical and the right-helical muon states µ ± L and µ ± R . The superposition of the latter, as prescribed by Eq. A2, gives an accurate representation of the muon population across the entire continuous range of helicities accessible to the polarized muons via Eq. A1. This justifies the inclusion of only six muon species ( µ ± L , µ ± R , as well as unpolarized muons µ ± originating from the three-body decays of K ± ) into the MCEq cascade equations. \nIn Fig. A.2, we show example angular distributions the electron antineutrinos resulting from the µ -→ ν µ + ¯ ν e + e -decay as computed in whizard . For example, at E µ = 5GeV and E ¯ ν e ≤ 2GeV, we find that both the shape of the neutrino angular distributions and their normalization differs depending on muon polarization. While [22, 54] already included the muon polarization effects in the one-dimensional approximation via the analytical prescription of [53], this simple example further illustrates the importance of the muon polarization treatment for the two-dimensional solver. \nIn Figs. A.3 and A.4, we show the impact of the muon polarization on the energy spectra and angular distributions of atmospheric neutrinos produced in a full proton-induced air shower. In this example, the primary proton has a fixed energy of 100 GeV and a 30 · inclination. The neutrino fluxes with muon polarization enabled/disabled are obtained via 2D MCEq as the solutions to Eq. 15 at the sea level. This corresponds to X ≈ 1196 g cm -2 in the US Standard \n1 \nFIG. A.2. Comparison of the ¯ ν e angular distributions with E ¯ ν e ≤ 2GeV in the decay of a 5 GeV muon for three muon polarization cases: left-handed (yellow), right-handed (purple), and unpolarized (red). The displayed events were generated with the whizard Monte Carlo code [65, 66]. \n<!-- image --> \nFIG. A.3. Impact of muon polarization on the energy spectra of the sea-level atmospheric neutrino fluxes in a 100 GeV proton air shower with 30 · inclination. The dashed lines refer to the case of all atmospheric muons being treated as unpolarized. The solid lines correspond to the muon polarization treatment as described in the main text of the present section. The shaded gray band represents the region where the MCEq solution is unstable due to the numerical implementation of the delta function-like initial condition on the discrete MCEq energy grid (see [68] for the discretization details). \n<!-- image --> \natmosphere [32]. We find that muon polarization has the largest impact on the ν e fluxes and energy spectra. Assuming that all muons are unpolarized can lead to 10-30% error in the ν e spectrum normalization and the nearly the same bias in the angular density. The spectrum and the angular distribution of ν µ are affected at the level of a few percent; ¯ ν µ experience an up 10% effect growing towards higher energies due to the energy-dependent π + /π -and µ + /µ -ratios. The muon normalizations and angular distributions remain unchanged as expected.", 'b. Muon multiple scattering': "An additional effect modifying the muon angular distributions is their Coulomb scattering with atmospheric nuclei, e.g. 14 N or 16 O. The effect of multiple scattering is described by the Moli'ere theory [61] if the number of scatters is \nFIG. A.4. Impact of muon polarization on the angular distributions of the sea-level atmospheric neutrinos with E ν ≃ 2GeV. The initial conditions of the air shower are the same as in Fig. A.3. \n<!-- image --> \nlarge ( ≫ 20 across a layer of matter with a given thickness), and by the Poissonian probability of scattering with the Rutherford cross section if the number of scatters is small [32, 69]. We implement a Gaussian approximation to the Moli'ere formalism, which is one of the multiple scattering handling options provided in the corsika Monte Carlo [32]. In the Gaussian approximation, the probability P of a muon deflecting by the space (unprojected) angle θ after traversing ∆ X of the atmospheric slant depth is defined as follows: \nP ( θ, ∆ X ) = 1 πθ 2 s ∆ X · exp [ -θ 2 ∆ Xθ 2 s ] , (A3) \nwhere θ 2 s = 1 λ s ( E s E µ, lab β µ ) 2 , E s = 0 . 021GeV, λ s = 37 . 7gcm -2 , E µ, lab is the total muon energy in the lab frame, and β µ - its lab-frame Lorentz velocity factor [32, 69, 70]. In Fig. A.5, we show several representative angular densities computed according to Eq. A3. For illustration, we use ∆ X = 1gcm -2 . In general, however, ∆ X varies with X in response to the longitudinal atmospheric density variations, and the width of the muon multiple scattering kernel is variable. While in just 1 g cm -2 the expected muon deflection is small ( O (0 . 1 · ) at GeV energies), this effect accumulates with the slant depth and results in a noticeable shift of the sea-level muon angular distribution, especially in horizontal showers. \nTo incorporate this additional convolution kernel into the solution of Eq. 15, we need to Hankel-transform Eq. A3. In the Hankel frequency space, the muon multiple scattering kernel reads: \n˜ P ( κ, ∆ X ) = exp [ -κ 2 ∆ Xθ 2 s 4 ] , (A4) \nwhich is scaled so that the overall normalization of the muon angular distribution (represented by the κ = 0 mode) remains unchanged. We can then directly multiply Eq. A4 by the Hankel amplitudes of the muon angular distributions after each integration step ∆ X . This way, the simplified muon multiple scattering model becomes a natural part of the matrix cascade equations. The treatment of multiple scattering is identical for all of the muon species, i.e. µ ± L , µ ± R , and µ ± . \nIn Fig. A.6, we show the impact of the muon multiple scattering on the sea-level muon angular distributions in a proton-induced air shower, given the same initial conditions as in Figs. A.3 and A.4. For a representative example, we focus on the muons with E µ ≃ 2GeV (i.e., the parents of O (GeV) electron and muon neutrinos). We find that the cumulative effect of muon multiple scattering is a ∼ 1 · shift of the angular density peak, compared to the air shower evolved without muon multiple scattering. We confirmed that the 'tilt' seen in the lower panel of Fig. A.6 grows with the distance traversed by the muons. The effect on the neutrino angular distributions was, however, found to be negligible, introducing at most O (1%) bias at the sea level if muon multiple scattering was not included in the cascade equations. \n1 \n2 \nFIG. A.5. Gaussian approximation of the muon deflection angles due to their multiple scattering on atmospheric nuclei (see [32, 61, 69] and Eq. A3). The assumed slant depth traversed by the muons is ∆ X = 1gcm -2 . As expected, lower-energy muons (e.g. 1 GeV; yellow line) get deflected more than the higher-energy muons (e.g. red and purple lines for 2 GeV and 5 GeV, respectively). \n<!-- image --> \n1e 3 \nFIG. A.6. Impact of muon multiple scattering on the angular distributions of the sea-level atmospheric muons with E µ ≃ 2 GeV. The initial conditions of the air shower are the same as in Fig. A.3. \n<!-- image -->", 'Appendix B: Comparison of the configuration settings between CORSIKA and MCEq': "The geomagnetic field and the respective curving of the charged particle trajectories are not implemented in 2D MCEq at the time of writing. We therefore effectively disable the geomagnetic field in corsika by setting B x = B z = 10 -5 µ T ( corsika requires \| B \| > 0). In addition, since the 2D MCEq code currently excludes electromagnetic cascades, we also disable the electromagnetic interactions in corsika by setting all electromagnetic flags (' elmflg ') to false. While the choice of the hadronic models is matched between corsika in MCEq , the switch between the low- and the high-energy regimes means a sharp transition between the two hadronic interaction models in corsika and a smooth interpolation between the models in 2D MCEq (see Appendix A 2 for details). Thus, when both energy regimes are covered in a simulated case, small discrepancies between the two codes are possible due to the different implementations of the low-energy/high-energy model transition. When comparing the simulation outputs at the energies below the transition threshold, one further has to be mindful of the different low-energy model versions. We expect the differences due to the low-/high-energy transition implementation and the internal model versions to be smaller than due to a full change of the low-energy interaction model to a different one, which is investigated in Appendix E. \nIn corsika , the azimuthal angle of the primary particle incidence is fixed at φ 0 = 0. The height of the first possible interaction of the proton with atmospheric nuclei is set to 112 . 8 km in both MCEq and corsika . The atmospheric density as a function of the slant depth X is modelled according to the Linsley parametrization of the US Standard atmosphere [32]. In MCEq , the average stopping power of the charged particles due to ionization, bremsstrahlung and pair production is taken from tables provided by the Particle Data Group [18, 22], whereas the energy derivative ∂ ∂E is approximated as a five-point stencil. In corsika , the average stopping power is calculated analytically via the Bethe-Bloch prescription [69, 71], and is directly used to reduce the energy of the charged particles between two propagation steps. The Gauss approximation is employed in both codes for muon angular deflections due to multiple scattering.", 'Appendix C: MCEq-CORSIKA benchmarking for a power law cosmic ray spectrum': 'FIG. C.1. Angular distributions of atmospheric leptons in proton-induced air showers with a power law starting spectrum (see title). \n<!-- image -->', 'p ( np E 2.7 ) at 30 inclination': 'FIG. C.2. Energy spectra of atmospheric leptons in proton-induced air showers with a power law starting spectrum (see title). \n<!-- image --> \np ( E =100GeV) at 80 inclination \nFIG. D.1. Angular distributions of atmospheric leptons in a proton-induced air shower ( E 0 = 100GeV, θ 0 = 80 · ), as computed numerically in 2D MCEq (solid lines) and simulated in the corsika Monte Carlo (filled histograms with errorbars). \n<!-- image -->', 'p ( E =100GeV) at 80 inclination': "N \nd \nFIG. D.2. Energy spectra of atmospheric leptons in a proton-induced air shower ( E 0 = 100GeV, θ 0 = 80 · ), as computed numerically in 1D MCEq (solid lines) and simulated in the corsika Monte Carlo (filled markers). \n<!-- image --> \n2 \nFIG. E.1. Angular distributions of low-energy atmospheric leptons ( E ≃ 2 GeV) in a proton-induced air shower, as computed numerically in 2D MCEq using two different low-energy hadronic interaction models ( DPMJet -III 19.1 ('D') and UrQMD 3.4 ('U')). The bottom sub-panel in each plot shows the DPMJet / UrQMD solution ratio. \n<!-- image --> \nD / U \nFIG. E.2. Energy spectra of atmospheric leptons in a proton-induced air shower ( E 0 = 100GeV, θ 0 = 30 · ), as computed numerically in 2D MCEq using two different low-energy hadronic interaction models. The legend follows that of Fig. E.1. The shaded gray band represents the region where the MCEq solution is numerically unstable (see the caption of Fig. A.3 for details). \n<!-- image --> \n- [1] M. G. Aartsen et al. (IceCube Collaboration), Determining neutrino oscillation parameters from atmospheric muon neutrino disappearance with three years of IceCube DeepCore data, Phys. Rev. D 91 , 072004 (2015).\n- [2] M. G. Aartsen et al. (IceCube Collaboration), Measurement of atmospheric neutrino oscillations at 6-56 GeV with IceCube DeepCore, Phys. Rev. Lett. 120 , 071801 (2018).\n- [3] K. Abe et al. (Super-Kamiokande Collaboration), Atmospheric neutrino oscillation analysis with external constraints in Super-Kamiokande I-IV, Phys. Rev. D 97 , 072001 (2018).\n- [4] P. Adamson et al. 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2024arXiv240906389N	Local vortexstructured auroral spiral and a largescale transpolar arc TPA were contemporaneously observed by the Polar ultraviolet imager UVI when a substorm almost recovered. The TPA grew along the dawnside auroral oval from the nightside to the dayside ovalaligned TPA and a chain of multiple auroral spots and spiral were located azimuthally near the poleward edge of the nightside auroral oval. Contemporaneous appearances of the TPA and the auroral spiral can be seen after the spiral appeared alone. Polar also detected the ovalaligned TPA and another dawnside TPA with the nightside end distorted toward the premidnight sector Jshaped TPA before and after the spirals formation respectively. To examine these associated magnetotail structures we performed global magnetohydrodynamic MHD simulations based on two different types of code BATSRUS and improved REPPU and examined how the fieldaligned current FAC profiles varied in association with changes of the auroral form to TPA andor auroral spiral. Global MHD simulations with the two different types of code can reproduce the TPAs and the associated FAC structures in the magnetotail. The auroral spiral and its nightside FAC profile however were not formed in both simulations suggesting that its formation process cannot be treated within an MHD framework but is closely related to some kinetic process. When the Jshaped TPA and the auroral spiral contemporaneously appeared the two MHD simulations could not reproduce the TPA spiral and their associated magnetotail FAC structures also advocating that a kinetic effect related to the spiral formation might prevent the TPA occurrence.	2024-09-01T00:00:00Z	['2024arXiv240906389N', 'arXiv:2409.06389', '10.48550/arXiv.2409.06389']	['Physics - Space Physics', 'Astrophysics - Solar and Stellar Astrophysics', 'Physics - Plasma Physics']	Contemporaneous Appearances of LocalScale Auroral Spiral and GlobalScale Transpolar Arc Changes of Auroras and FieldAligned Current Profiles Before a Substorm and After Its Recovery Phase	2,024	192	0.26	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.06389.pdf	{'Key Points:': '- 1. After the substorm recovery phase, a nightside distorted transpolar arc and the auroral spiral can appear simultaneously.\n- 2. Auroral spiral cannot be treated within the magnetohydrodynamics framework, but their formation is closely related to some kinetic process.\n- 3. The transpolar arc could not be formed in magnetohydrodynamic simulations when it appeared simultaneously with the auroral spiral.', 'Running Title: Concomitance of Auroral Spiral and TPA': 'Keyword: Transpolar arc; Auroral spiral; Field-aligned currents, Substorm, Magnetohydrodynamic simulations; Kinetic effects', 'Abstract': "Local vortex-structured auroral spiral and a large-scale transpolar arc (TPA) were contemporaneously observed by the Polar ultraviolet imager (UVI), when a substorm almost recovered. The TPA grew along the dawnside auroral oval from the nightside to the dayside (ovalaligned TPA), and a chain of multiple auroral spots and spiral were located azimuthally near the poleward edge of the nightside auroral oval. Contemporaneous appearances of the TPA and the auroral spiral can be seen after the spiral appeared alone. Polar also detected the oval-aligned TPA and another dawnside TPA with the nightside end distorted toward the premidnight sector (Jshaped TPA) before and after the spiral's formation, respectively. To examine these associated magnetotail structures, we performed global magnetohydrodynamic (MHD) simulations, based on two different types of code, BAT-S-RUS and improved REPPU, and examined how the fieldaligned current (FAC) profiles varied in association with changes of the auroral form to TPA and/or auroral spiral. Global MHD simulations with the two different types of code can reproduce the TPAs and the associated FAC structures in the magnetotail. The auroral spiral and its nightside FAC profile, however, were not formed in both simulations, suggesting that its formation process cannot be treated within an MHD framework but is closely related to some kinetic process. When the J-shaped TPA and the auroral spiral contemporaneously appeared, the two MHD simulations could not reproduce the TPA, spiral and their associated magnetotail FAC structures, also advocating that a kinetic effect related to the spiral formation might prevent the TPA occurrence.", 'Plain Language Summary': 'The auroral spiral, local-scale vortex-structured auroral phenomenon, and global-scale transpolar arc (TPA) bridging the polar cap between the nightside and dayside sectors of the auroral oval have been discussed to understand how the physical processes occurring in the magnetotail can be relevant to their formations within a framework of solar wind-magnetosphere-ionosphere coupling system. In this study, we find that the TPA appears prior to and after the auroral spiral, and sometimes the spiral and TPA were concomitant. Based on magnetohydrodynamic (MHD) computer simulations with two different types of simulation code, we tried to reproduce and investigate concomitance of TPA and auroral spiral and the associated field-aligned current (FAC) structures on the magnetic equatorial plane of the nightside magnetosphere. The TPA prior to and after the spiral and the associated FAC profile can be reproduced by global MHD computer \nConfidential manuscript submitted to Journal of Geophysical Research: Space Physics \nsimulations, but the auroral spiral and its magnetotail FACs cannot. Even when the spiral and the TPA were concomitant, neither of the auroras can be formed in the MHD simulations, suggesting that auroral spiral formation might not be addressed within an MHD framework but would closely be related with some kinetic process.', '1. Introduction': "The auroral phenomena that locally exhibit a vortex structure and mainly appear as azimuthallyaligned spot emissions in the poleward region of the nightside auroral oval are referred to as auroral spiral (e.g., Elphinstone et al., 1995; Nowada et al., 2023; Partamies et al., 2001a). The auroral spirals were distributed over the nightside ionosphere from 18 h to 5 h magnetic local time (MLT), while their magnetic latitude (MLat) was concentrated around 65° (Partamies et al., 2001a) and/or between 70° and 80° (Davis and Hallinan, 1976). Statistical diameter distributions of the auroral spirals are generally 25-75 km (Partamies et al., 2001a, 2001b), or 20-1,300 km (Davis and Hallinan, 1976). Recently, it has been reported that the auroral spiral appeared during various substorm phases: expansion phase (Keiling et al. 2009a, 2009b) and late substorm recovery phase (Nowada et al. 2023). Depending on the substorm phase of spiral occurrence, the fundamental spiral features, such as its scale, duration, and formation mechanism, should be different (e.g., Nowada et al. 2023). Keiling et al. (2009a, 2009b) examined auroral spirals that occurred from the substorm onset to the expansion phase, and estimated its scale at 200-300 km. In contrast, the auroral spiral observed during the late stage of the substorm recovery phase had smaller core scale from 150 to 250 km (Nowada et al., 2023). \nIn the early days of auroral spiral studies, Davis and Hallinan (1976) and Hallinan and Davis (1970) showed that in addition to the spiral, curl- and fold-type auroras were accompanied by vortex structures. Spirals are the auroral phenomena with the largest vortex whose scale is larger than 50 km. Curls are small-scale auroral phenomena with vortices of less than 10 km, which are embedded in auroral arcs. Folds are distorted arcs with an intermediate scale of about 20 km. These vortical auroras, however, can appear contemporaneously in the same auroral form and could not be easily distinguished. Recently, based on the Polar ultraviolet imager (UVI) and all-sky camera imager data, Nowada et al. (2023) followed the auroral spiral formation process after substormassociated auroral bulge distributed azimuthally in the nightside ionosphere subsided. After the substorm entered the recovery phase, several poleward-elongated auroral stream structures appeared in the nightside auroral oval. Finally, the auroral spiral was formed during the late stage of the substorm recovery phase. Nowada et al. (2023) also estimated the source region of the auroral spiral on the nightside magnetotail equatorial plane using an empirical magnetic field model. They, however, could not physically reveal how the spiral was formed, based on the \nrelations with the solar wind conditions and the corresponding magnetotail processes. \nAt present, the formation mechanism of the auroral spiral remains unclear. Keiling et al. (2009b) and Voronkov et al. (2000) proposed that the spiral might be formed by shear flow ballooning instability which is treated within a magnetohydrodynamic (MHD) regime and is driven by a pressure gradient in the magnetotail. Furthermore, the auroral spiral formation cannot be yet addressed from the viewpoint of solar wind-magnetosphere-ionosphere coupling system. Lysak and Song (1996), however, proposed a spiral formation model based on magnetosphere-ionosphere coupling within a magnetohydrodynamics (MHD) scheme. The current sheet instability (CSI) in the nightside ionosphere plays a crucial role in formation of the auroral spiral. CSI is quite similar to the Kelvin-Helmholtz instability (KHI) caused by a velocity shear, but a magnetic shear generated by an upward (from the ionosphere to the magnetosphere) field-aligned current (FAC) plays a significant role in its growth. Lysak and Song (1996) concluded that an auroral spiral can be formed by CSI because of the conductance difference between the ionosphere and the magnetosphere. Hallinan (1976) and Partamies et al. (2001b) also pointed out that enhancement of an upward FAC filament structure drives the magnetic shear and the resulting CSI to form the auroral spiral. Although the two groups of Lysak and Song (1996), and Hallinan (1976) and Partamies et al. (2001b) independently proposed the models to explain the auroral spiral formation, they all emphasized that the CSI due to an upward FAC is the major mechanism of the auroral spiral formation. At present, however, we do not have the observational evidence for their models and the CSI reproductions, based on global-scale computer simulations using MHD code, suggesting that the auroral spiral features, including the formation process, cannot yet be explained completely within the MHD regime. \nTranspolar arc (TPA) is a 'crossbar' type auroral phenomenon bridging the polar cap from the nightside to the dayside auroral oval and is also a part of θ-like-shaped aurora (theta aurora). The formation process and fundamental physical features of TPA have been investigated and discussed (see reviews by Fear and Milan, 2012a, 2012b) since a theta aurora was detected by the Dynamics Explorer (DE) 1 spacecraft (Frank et al., 1982). As a representative TPA formation model, Milan et al. (2005) built a formation model based on nightside magnetic reconnection, which successfully explained the formations in several TPA cases (Fear and Milan, 2012a, 2012b; Kullen et al. 2015; Nowada et al. 2018, and references therein). The development of TPA from the nightside to the dayside poleward edge of the auroral oval can be explained by continuous formation of newly \nclosed field lines generated by nightside reconnection retreating its site toward the farther tail. Nowada et al. (2020) proposed a possible formation model of the dawnside (duskside) TPA whose nightside end gets distorted toward premidnight (postmidnight) sector, nightside distorted (J- and L-shaped) TPAs. They concluded that the source of the nightside distorted TPAs is FACs induced by the plasma velocity shear between a fast plasma flow caused by nightside magnetic reconnection and the slower background magnetotail plasma flow. The conventional TPA has a straight bar shape which connects the nightside and dayside sectors of the auroral ovals, but significant distortions at the nightside end as seen in the J (L)-shaped TPA comes from significant magnetic field line twisting and magnetotail deformation due to the IMF-BY component which had already been revealed by many researchers (e.g., Cowley, 1981, 1994; Gosling et al. 1990; Tsyganenko and Fairfield, 2004; Tsyganenko and Sitnov, 2005; Tsyganenko et al. 2015). \nMany researches successfully reproduced TPAs, based on global MHD simulations (e.g., Kullen and Janhunen, 2004; Tanaka et al., 2004; Watanabe et al., 2014; and references therein). This suggests that an auroral phenomenon of TPA itself can be treated and discussed within an MHD framework. TPA is actually identified as global-scale aurora because it connects dayside and nightside polar cap regions, and has significant conjugacy between the Northern and Southern Hemispheres (e.g., Carter et al. 2017; Nowada et al., 2020). \nIn this study, we examined a unique case observed by the Polar UVI on 10 January 1997 that the auroral morphologies temporally and drastically changed before and after a substorm. Before the substorm growth phase, the TPAs aligned with the dawnside auroral oval were observed. Even though the substorm headed for complete recovery, the auroral morphology drastically changed; that is, a nightside distorted TPA (J-shaped TPA) appeared and remained, and even auroral spiral appeared simultaneously. In particular, focusing on the interval of contemporaneous appearances of the local-scale auroral spiral and the global-scale J-shaped TPA, we examined how associated magnetotail structures were varying and how detailed ionospheric FAC profiles were, and argued how the auroral spiral and the TPA are physically related, based on two different global MHD simulations, together with the Polar UVI observations. \nThe instrumentation is described in Section 2. In Section 3, we show the solar wind conditions, the results of the Polar UVI observations and global MHD simulations based on two different types of code, and the comparison between the observations and the simulations. The summary and discussion of this study are described in Section 4. Finally, we described the conclusions of this \nstudy in Section 5.", '2. Instrumentation of the Ultraviolet Auroral Imager': "The ultraviolet imager (UVI) onboard Polar, which was launched on 24 February 1996, provides global auroral imaging data in ultraviolet range (Torr et al., 1995). We used the UVI images in altitude adjusted corrected geomagnetic (AACGM; Baker and Wing, 1989) and geographic coordinates. The UVI image data are degraded in the direction perpendicular to the track of Polar by the satellite's wobble motion (e.g, Parks et al., 1997). Because this wobble is, however, predictable (Parks et al., 1997), we used the UVI image data from which the wobble effects were mostly removed.", '3-1. An Overview of Observation': 'Figure 1 shows a fine example of the auroral spiral (pointed out with a thick yellow arrow) and the nightside distorted transpolar arc (J-shaped TPA) contemporaneously observed by the Polar UVI (panels a and b) and the all-sky camera (ASC) installed at the Longyearbyen station (75.32° magnetic latitude and 111.0° magnetic longitude, Figure 1c) at 21:08 UT on 10 January 1997. The long-term movie of the auroral spiral detected by the Longyearbyen ASC is available in the supporting information of Nowada et al. (2023). The auroral spiral clearly identified by both Polar UVI and the Longyearbyen ASC was only one (pointed out with thick orange arrows). We, hereafter, discuss this auroral spiral identified by satellite- and ground-based observations, and the other auroral signatures azimuthally neighboring on the auroral spiral are referred to as auroral spots (auroral spiral and spots). The auroral spiral was seen to the southwest region of Svalbard Island (Figures 1b and 1c). The dawnside TPA had a nightside end distorted toward the midnight sector, and extended from the poleward edge of the postmidnight auroral at ~ 1h MLT and ~ 72° MLat to the prenoon auroral oval across the dawnside of the north pole (Figure 1b; see also the supporting information of Nowada et al., 2023). This type of TPA is referred to as J-shaped TPA (see Nowada et al., 2020).', '3-2. Solar Wind Conditions': "Figure 2 shows the OMNI solar wind parameters and geomagnetic indices for the 6 h interval \nbetween 17:00 UT and 23:00 UT on 10 January 1997. From top to bottom, the panels show the SMU and SML indices (Newell and Gjerloev., 2011), the Y and Z components of the interplanetary magnetic field (IMF-By and -Bz) in geocentric solar magnetospheric (GSM) coordinates, the IMF clock angle derived with arctan(IMF-By/IMF-Bz), the Akasofu-Perreault parameter (εAP, a measure of the solar wind energy input rate; Perreault and Akasofu, 1978), and the solar wind dynamic pressure (Pd), velocity (VSW), and proton density (NP). The values of Kp are shown at the bottom of the figure. The intervals of the transpolar arc (growing to the dayside sector aligned with the dawnside auroral oval, oval-aligned TPA; denoted by TPA at the top of Figure 2), the auroral spiral (AS), the contemporaneous appearances of the J-shaped TPA and the auroral spiral (J-TPA+AS), and the J-shaped TPA (J-TPA) were identified by visual inspection, based on the images from Polar UVI and the all-sky camera (ASC) at the Longyearbyen station. Again, note that AS denotes only the auroral spiral that could be identified by both satellite- and ground-based observations. Each time interval is bracketed a pair of broken cyan, black, violet, or yellow lines. The detailed durations of these auroral signatures and the corresponding polarities of the IMF-BZ component are also summarized in Table 1. \nDuring this interval, the whole cycle of a substorm from onset to recovery phases, which was accompanied by moderate geomagnetic disturbances with a Kp range from 3+ to 4, can be seen. Namely, SML shows a sharp, large decrease from ~18:50 UT (the substorm onset) to ~19:05 UT, and then SML recovered to ~ 0 nT at ~22:15 UT. Small negative bay variations were seen in SML during the time intervals of AS and J-TPA+AS. On the contrast, SMU was almost constant during the presented time interval. \nDuring the J-TPA+AS interval, clear jumps were seen in the IMF-By and -Bz components, and the IMF clock angle at 21:10 UT. In particular, the IMF-Bz component, which remained weakly southward for at least 4 h, showed a sharp polarity change from southward (negative) to northward (positive), while IMF-By had a dominantly dawnward (negative) component, that is, the sign changes were not seen. Due to this abrupt IMF-Bz jump, the IMF clock angle increased from -90° to -45°. The ε parameter shows a significant decrease, associated with the clock angle's jump, which was preceded by a gradual decrease after the oval-aligned TPA disappeared. During the other auroral intervals, significant changes of the IMF, the IMF clock angle, and the solar wind plasma parameters were absent, although small excursions in the solar wind were seen. Taking a look at the solar wind conditions and the corresponding auroral morphologies, all TPA-type arcs \nand auroral spiral were observed even when no significant perturbations of SML and SMU occurred.", '3-3. Comparison of Polar UVI Auroral Observations with Magnetohydrodynamic (MHD) Simulations in the Polar Cap': 'Figure 3 shows representative examples of the oval-aligned TPA, the auroral spiral, contemporaneous appearances of the J-shaped TPA and the spiral, and the J-shaped TPA that were detected by the Polar UVI (Figures 3a to 3d), and contour plots of the ionospheric conductance determined by the corresponding magnetotail processes that were reproduced by the improved REProduce Plasma Universe (REPPU) code (Tanaka, 1994; Nakamizo et al., 2009; Figures 3e to 3h). The improved REPPU code applies a grid system with angularly unstructured and increasing radial spacing. The computation using this grid system is more effectively stabilized than the other codes, that is, the solutions are hard to diverge, because there is no apparent singularity. Hence, we can expect that the structures of the global-scale TPA and the local-scale spiral are expected to be reproduced in such a grid system. According to Nakamizo et al. (2009), in the improved REPPU code, the order number of grid splitting can be increased/decreased freely, depending on the systems discussed. Changing the order number for grid splitting, we can perform the computations with higher temporal and spatial resolutions. In this MHD simulation, the ionospheric conductance can be used as a proxy of the FAC intensity. \nFigures 3a - 3d are long Lyman-Birge-Hopfield emission (LBHL; ∼ 170 nm) images from Polar UVI with an integration time of 36 s in the same format as Figures 1a. Polar detected the TPA growing from the poleward edge of the nightside auroral oval near 3h MLT to the dayside, which was also aligned with the dawnside auroral oval (Figure 3a). This TPA was stable and had no significant dawn-dusk motion within the polar cap. Such a static TPA near the dawn (or dusk) auroral oval, so-called oval-aligned arc, had already been reported and its fundamental characteristics, such as the relation to the IMF orientations, was discussed by Murphree and Cogger (1981) and Kullen et al. (2002, 2015). The reproduction of the oval-aligned TPA, based on the improved REPPU MHD simulation can be clearly seen in Figure 3e, suggesting that the oval-aligned TPA and the associated field-aligned structure in the magnetotail can be addressed in an MHD regime. \nIn Figure 3b, azimuthally-chained four auroral spots, including an auroral spiral identified by the Longyearbyen all-sky camera (Nowada et al., 2023), were seen at the poleward edge of the \nnightside auroral oval, but they were not reproduced by the MHD simulation as shown in Figure 3f. Even during the concomitant interval of the auroral spiral and spots, and J-shaped TPA (Figure 3c), the auroral spiral and spots were not reproduced at the poleward edge of the nightside auroral oval (Figure 3g). This result suggests that the auroral spiral might not be reproduced within an MHD framework. Interestingly, for this event, not only the spiral but also the J-shaped TPA that should be treated in an MHD regime were not reproduced by the simulation. Comparison of Figure 3d with Figure 3h supports that the J-shaped TPA is a phenomenon which can be discussed within an MHD scale. From these observations and global MHD simulations, we suggest that when the spiral and the TPA appear contemporaneously, the auroral spiral might impact on the formation process of the TPA basically discussed within an MHD framework.', 'Simulations': "To investigate the source regions of the TPA and the auroral spiral in the nightside magnetosphere, we mapped each pixel data of the Polar UVI images onto the magnetic equatorial plane, based on the Tsyganenko 96 empirical magnetic field model (T96; Tsyganenko and Stern, 1996). The technical details on mapping are described in Nowada et al. (2023). Figure 4 shows the projections of the Polar UVI images of the oval-aligned TPA (Figure 4a, highlighted with a magenta broken oval), the auroral spiral (Figure 4b), and the J-shaped TPA (Figure 4c, surrounded with an orange broken curve) onto the GSM-X-Y nightside magnetic equatorial plane. Figures 4d-4f show the Polar UVI plots in the nightside ionosphere from 18 h to 6 h MLT. Note that the nightside equatorial plane mapping of the auroral images sensitively depends on magnetic field models (Lu et al., 2000). \nBased on the mapping onto the magnetic equatorial plane, the source regions of the oval-aligned TPA had the structures elongating tailward to ~ 55 RE at Y ~11 RE to 12 RE (in GSM), as shown with a magenta broken oval in Figure 4a. It is interesting that the oval-aligned TPA's source region lay on the dawnside magnetotail equatorial plane. In Figure 4c, the J-shaped TPA's source region extended from the duskside at X ~ -16 RE to the dawnside at X ~ -90 RE, over ~20 RE in the dawndusk direction across midnight. Note that this projection of the J-shaped TPA using the T96 field line model might be different from the actual mapping, because the field line model does not take into account magnetotail deformation and the associated field line twisting due to the intense IMF- \nBY effect that is necessary for forming a J-shaped TPA (Nowada et al., 2020). \nThe projection of the auroral spiral (Figure 4b) also elongated tailward, and its scale was ~ 20 RE and ~4 RE in the X and Y directions in GSM, respectively, even though the spiral had the form of a spot in the ionosphere. This elliptic form is consistent with the discussions by Kaufmann et al. 1990) for quiet time magnetosphere and Lu et al. (2000) during a substorm. The auroral spiral had almost the same scale as that seen at different times on the same day (Nowada et al., 2023). \nTo discuss the detailed formation process of these auroral phenomena, we also performed global magnetohydrodynamic (MHD) computer simulations using the Block-Adaptive-Tree-Solar windRoe-Upwind-Scheme (BAT-S-RUS) code provided by the Community Coordinated Modeling Center (CCMC) at NASA Goddard Space Flight Center (GSFC). Figure 5 shows the distributions of the FACs at the magnetic equatorial plane of the magnetotail (Figures 5a-5c) and the nightside ionosphere (Figures 5d-5f) reproduced by the BAT-S-RUS MHD simulation. Each time label is almost the same as that in Figure 4. \nIn all Figures 5a-5c, the field-aligned current (FAC) sheet structures were clearly generated in the dawn sector (Figures 5a and 5b) and in the dawn to midnight sector (Figure 5c). The FAC sheets seen in Figures 5a and 5c correspond to the nightside TPA projections shown in Figures 4a and 4c. The FAC sheet profile of the J-shaped TPA in the magnetotail reproduced by the MHD simulation (Figure 5c) is totally different from that of the Polar observation (Figure 4c), because, in the Polar observation, each UVI pixel data were simply traced back along the T96 model field lines without considering the magnetotail twisting due to significant IMF-BY effects. Taking a look at Figure 5f, however, the J-shaped TPA can be reproduced in the ionosphere with the simulation, and the dawndusk distributed FAC structure that extended tailward as shown in Figure 5c might agree with the actual J-shaped-TPA-associated FAC profile. According to a J-shaped TPA formation model derived by the magnetotail observation (Nowada et al., 2020; particularly see their Figure 7), as the J-shaped TPA is growing to the dayside polar cap, the corresponding TPA's FAC structure should extend toward the magnetotail, deformed by the IMF-BY. Hence, the simulation result can explain well this J-shaped TPA formation model proposed by Nowada et al. (2020). On the contrast, the TPA that was growing straightforwardly to the dayside (Figure 5a) can be reproduced and explained well by the MHD simulation. The ion velocity vectors show large-scale clockwise vortices around these intense FAC sheets (blueish colored region) at X ~ -13.5 to -45 RE and Y ~ -2.0 to -18.0 RE (Figure 5a) or Y ~ 4.5 to -18.0 RE (Figure 5c), as indicated with dotted magenta \novals. These large vortices should be the source of the FACs from the magnetotail to the ionosphere (downward FACs) associated with the TPAs. \nAlthough the FAC sheet structure can also be seen in Figure 5b, Polar did not detect this structure in the corresponding region (Figure 4b). Furthermore, the tailward elongating structures that corresponded to the auroral spiral and spots as detected by Polar were reproduced neither by this MHD simulation nor by the improved REPPU code, as shown in Figures S1a and S1b in supporting information. \nIn Figures 5d and 5f, the FAC profiles corresponding to the oval-aligned and J-shaped TPAs, respectively, observed by the Polar UVI were seen in the nightside ionosphere. The auroral spiral, however, was not reproduced at the poleward edge of the nightside auroral oval (Figure 5e) because of absence of the tailward elongating structures on the magnetic equatorial plane. Hence, neither BAT-S-RUS nor improved REPPU global MHD simulation code can reproduce the auroral spiral in the nightside ionosphere. \nIn the previous sections, we showed that the TPA was reproduced by the two different types of global MHD code, while the auroral spiral (and auroral spots) at the poleward edge of the nightside auroral oval was (were) not. These results suggest that the global-scale TPA can be treated within an MHD framework, while the spiral cannot. To investigate this point in more detail, we discuss the case in which the global-scale TPA and the local-scale auroral spiral contemporaneously appeared. Figure 6 shows a Polar UVI snapshot (Figure 6a) and its projection onto the magnetic equatorial plane (Figure 6b) that were taken when the J-shaped TPA and the auroral spiral were concomitant at 21:08:40 UT. At this time, the auroral spiral was also detected by the all-sky camera installed at Longyearbyen (Nowada et al., 2023). \nInterestingly, the magnetic equatorial projections of both auroras had tailward elongating structures: in the dawn sector in -80 < Xgsm < -11 RE and -20 < Ygsm < -3 RE for the J-shaped TPA, and in the dusk side sector in -75 < Xgsm < -38 RE and 6 < Ygsm < 9 RE for the auroral spiral. The X range of the J-shaped TPA was ~32 RE, longer than that of the spiral, and the Y scale of the Jshaped TPA was about 6 times larger than that of the spiral. \nWith the BAS-T-RUS code, we also tried to simulate the FAC structures in the magnetotail and \nthe auroral profiles in the ionosphere. Figure 7 shows the comparison of the Polar UVI observation with the BAT-S-RUS MHD simulation results. The magnetic equatorial map of the Polar UVI image is shown in Figure 7a, and Figures 7b and 7c show the distributions of FACs (J// and J//Z, respectively) on the magnetic equatorial plane. Interestingly, neither the FAC profile in the source region of the J-shaped TPA (highlighted with magenta ovals) nor that of the auroral spiral (orange and green ovals) was reproduced by the global MHD simulation, although the intense J// and weak J//Z were distributed at the J-shaped TPA source region in the dawn sector. The clear tailwardelongating FAC sheet structure associated with the oval-aligned TPA and J-shaped TPA, as seen in Figures 4a, 4c, 5a, 5c, 6 and 7a, was not be observed in Figures 7b and 7c. For the spiral source, the profiles of J// and J//Z could not be found. \nTo examine these FAC profiles in more detail, we zoomed in the FAC profiles at the auroral spiral, changing the color scale. Figure 8 shows the zoomed-in plots of the J// (Figure 8a) and J//Z (Figure 8b) profiles at the source region of the auroral spiral identified with the Polar UVI observations. Both J// and J//Z components did not show characteristic FAC structures at the spiral source region, although only very weak background current distributions in the out-of-plane sense can be found even though the color scale was changed. The flow velocities show no clear vortices or vortex-like structures that induce the FACs associated with the auroral spiral. These results suggest that the global MHD simulation may not reproduce the precise FAC profile associated with the auroral spiral in the magnetotail. \nSimilarly to the BAT-S-RUS simulation, the improved REPPU MHD simulation also did not reproduce and hence cannot explain in detail concomitance of the FAC sheet structure associated with the J-shaped TPA and the tailward elongating structure that corresponds to the auroral spiral (see Figures S1c and S1d).", '3.6 Comparison of the Results from the Two Different Types of MHD Code': 'Figure 9 shows the comparison between the MHD reproductions of the nightside ionosphere during contemporaneous appearances of the J-shaped TPA and the auroral spiral, based on the BAT-S-RUS code (panel a) and the improved REPPU code (panels b and c). Both simulations reproduced well fundamental ionospheric FAC systems from the ionosphere (magnetosphere) to the magnetosphere (ionosphere) as indicated by blue (red). Neither the J-shaped TPA nor the auroral spiral at the poleward edge of the auroral oval was, however, reproduced, indicating that \nthe TPA treated within a global-scale MHD regime may be affected by the local-scale auroral spiral that was not reproduced by the MHD simulations as shown in Figures 5, 7 and 8. These results suggest that an auroral spiral and its formation process should be discussed within not an MHD regime but a kinetic framework.', '3.7 Views from Ground-Based Observations: Spiral-Associated Geomagnetic Field Variations': 'We investigated and argued the case of contemporaneous appearances of the J-shaped TPA and the auroral spiral, based on the Polar UVI observations and the global MHD simulations with the BAT-S-RUS and improved REPPU code. Besides, using equivalent ionospheric currents (EICs) derived from geomagnetic field variations from the IMAGE (International Monitor for Auroral Geomagnetic Effects) ground observatory network (Tanskanen, 2009), we can estimate a profile of the FACs associated with the spiral (and the TPA). Figure 10 shows the EIC vectors every 1min. from 21:05 UT to 21:10 UT. EICs were reconstructed by fitting spherical elementary currents to the measured magnetic field after decomposing the pure ionospheric current contribution and the telluric current component from the raw IMAGE geomagnetic field data. This EIC reconstruction technique was designed by Vanhamäki and Juusola (2020). The plotted EIC vectors roughly correspond to derivations of the ionospheric horizontal magnetic field components (local magnetic north-south and east-west components) by 90˚ clockwise. During the present time interval including the time when the spiral and the J-shaped TPA appeared contemporaneously (particularly, 21:08 UT), counter-clockwise rotations of the EIC vectors are seen at the spiral, shown with curved green arrows. These EIC profiles enable us to estimate that FACs were flowing upward from the ionosphere to the magnetotail associated with the spiral (as highlighted by bluish color). These FAC and EIC senses are consistent with that of the FACs in a local MHD simulation by Lysak and Song (1996) and a series of spirals (Davis and Hallinan, 1976; Partamies et al., 2001a). Because most part of the TPAs observed before, during, and after the spiral were lying over the Arctic Ocean, the magnetometers to measure the geomagnetic field deviations over the whole TPA were absent. \nFigure 11 shows deviations of BN component (local magnetic north-south direction) and its power spectrograms at three ground magnetometer observatories near the auroral spiral (indicated with the three magenta dots in the UVI plot) during the auroral spiral and contemporaneous appearances \nof the TPA and the spiral from 19:59 UT to 21:23 UT. The BN deviations were derived by subtracting the magnetic field DC component from the observed magnetic field, and the power spectra were calculated from the wavelet analysis (Torrence and Compo, 1998). The frequency band of Pc5 ultra-low-frequency (ULF: 1.67 mHz-6.67 mHz) waves is bracketed by green horizontal broken lines. The two times of interest of 20:16:32 UT (fine auroral spiral) and 21:08:40 UT (contemporaneous appearances of the J-shaped TPA and the auroral spiral) are indicated by sky blue and orange vertical broken lines, respectively. The plots of deviations of BE (local magnetic east-west direction) and BZ (vertical up-down direction) and their power spectrograms during the same time interval as Figure 11 are shown in Figure S2 in supporting information. \nAlthough the BN component and the other two components show the perturbations (or waves) in the Pc5 ULF frequency range during the present interval, clear ULF waves were not always seen. Hence, we cannot conclude whether or not Pc5 ULF waves themselves play a role in auroral spiral formation, which would be the case in, for instance, auroral particle acceleration due to field line resonance (Rankin et al., 2005).', '4. Summary and Discussions': 'The auroral spiral observed during the late stage of the substorm recovery phase on 10 January 1997 appeared after the oval-aligned TPA, followed by contemporaneous appearances of the nightside distorted TPA (J-shaped TPA) and the auroral spiral and then by only the J-shaped TPA. Nowada et al. (2023) focused on the auroral morphological changes after the substorm expansion phase, particularly auroral spiral formation, for the present event and clarified how the auroral spiral and its source in the magnetotail were formed. On the other hand, in this study, we tried to explain how a local-scale spiral can affect a global-scale TPA because these two auroral phenomena with different scales contemporaneously appeared. \nTo examine the FAC profiles in the magnetotail and ionospheric processes during concomitance of the J-shaped TPA and the spiral, we performed global MHD simulations using two different types of MHD code of BAT-S-RUS and improved REPPU. These simulations reproduced the global-scale TPA and the associated magnetotail FAC structure, but the local-scale auroral spiral and its concomitance with the TPA were not generated in the polar cap region and the magnetotail. An interesting point here is that the J-shaped TPA that should be treated within an MHD regime could not be reproduced by global MHD simulations, when it appeared together with the local- \nscale auroral spiral. This result suggests that the formation process of the auroral spiral, which might not be discussed within an MHD regime, that is, can be related with some kinetic effect, could influence the MHD-scale TPA formation. Table 1 summarizes the temporal transitions of the auroral type, corresponding IMF-BZ conditions, and auroral reproductions by the global MHD simulations. \nDepending on the substorm phases, the corresponding magnetotail dynamics, including the magnetic field and plasma variations, totally differ. Therefore, it might be natural that the formation process of the auroral spirals during the substorm expansion phase as addressed within an MHD regime (shear flow ballooning instability) and as examined by Keiling et al. (2009b) and Voronkov et al. (2000) does not apply to our auroral spiral during the late stage of a substorm recovery phase. If we consider that Huang et al. (2022) succeeded to reproduce the spiral by the Particle-In-Cell (PIC) simulation, it can be valid to conclude that the auroral spiral formation is more effective in a kinetic process than in an MHD effect, although their spiral formation region and main mechanism are totally different from ours. \nThe perturbations (rather than waves) in the ULF Pc5 band were also observed, so their role in auroral spiral formation should be considered in future works by performing again global MHD simulations with higher temporal and spatial resolutions, that is, simulations with the improved REPPU code with a high order number grid splitting. Furthermore, based on computer simulations using the PIC and hybrid-Vlasov (Vlasiator; Palmroth et al., 2018) code, we will examine whether or not the auroral spiral (and spots) at the poleward edge of the nightside auroral oval and its (their) source region(s) in the nightside magnetosphere can be reproduced from a kinetic point of view.', 'Acknowledgments': 'M.N. enjoyed fruitful and constructive discussions with Qiu-Gang Zong, Alexander William Degeling, Timo Pitkänen, and Jong-Sun Park and was supported by a grant of the National Natural Science Foundation of China (NSFC 42074194). Y.M. was supported by basic research funding from Korea Astronomy and Space Science Institute (KASI2024185002). N.P. was supported by the Norwegian Research Council (NRC) under CoE contract 223252. Q.Q.S. was supported by NSFC 41731068, 41961130382, and 41974189. We thank George K. Parks for providing the Polar UVI data and Kan Liou for processing the data. We thank the institutes that maintain the IMAGE Magnetometer Array: Tromsø Geophysical Observatory of UiT, the Arctic University of Norway \n(Norway), Finnish Meteorological Institute (Finland), Institute of Geophysics, Polish Academy of Sciences (Poland), GFZ German Research Centre for Geosciences (Germany), Geological Survey of Sweden (Sweden), Swedish Institute of Space Physics (Sweden), Sodankylä Geophysical Observatory of the University of Oulu (Finland), Polar Geophysical Institute (Russia), DTU Technical University of Denmark (Denmark), and Science Institute of the University of Iceland (Iceland).', 'Data Accessibility': 'Polar ultraviolet imager (UVI) level-1 data can be accessed from https://cdaweb.gsfc.nasa.gov/pub/data/polar/uvi/uvi\\_level1/. Data for calibrating the level-1 data and calculating the position of the UVI images can be accessed from https://doi.org/10.6084/m9.figshare.5197084.v1 (Uritsky and POLAR UVI team, 2017). All IMAGE magnetometer numerical data used in Figure 10 can be downloaded from the IMAGE website (https://space.fmi.fi/image/www/?page=user\\_defined). The numerical data from the magnetometers used in Figure 11 as well as the SML and SMU indices can be downloaded from the SuperMAG website (https://supermag.jhuapl.edu/mag/). The OMNI solar wind magnetic field and plasma data can be acquired from Coordinated Data Analysis Web (https://cdaweb.gsfc.nasa.gov/pub/data/omni/omni\\_cdaweb/hro\\_1min/1997/), which is administrated by GSFC/NASA. The Kp index is provided by the Helmholtz Centre Potsdam - GFZ German Research Centre for Geosciences (https://kp.gfz-potsdam.de/en/). We also gratefully acknowledge the SuperMAG collaborators (https://supermag.jhuapl.edu/info/?page=acknowledgement) for using the SML and SMU indices. The BAT-S-RUS MHD simulation results can be referred to the three runs of Motoharu\\_Nowada\\_112620\\_1 (https://ccmc.gsfc.nasa.gov/results/viewrun.php?domain=GM&runnumber=Motoharu\\_Nowada\\_ 112620\\_1), Motoharu\\_Nowada\\_090923\\_3 (https://ccmc.gsfc.nasa.gov/results/viewrun.php?domain=GM&runnumber=Motoharu\\_Nowada\\_ 090923\\_3), and Motoharu\\_Nowada\\_090523\\_1 (https://ccmc.gsfc.nasa.gov/results/viewrun.php?domain=GM&runnumber=Motoharu\\_Nowada\\_ 090523\\_1) in the CCMC. All numerical data of the REPPU global MHD simulation results in this study are available in Nowada et al. (2024).', 'References': "Angelopoulos, V., Cruce, P., Drozdov, A. et al. (2019). The Space Physics Environment Data Analysis System (SPEDAS). Space Sci. Rev., 215, 9. https://doi.org/10.1007/s11214-018-05764. \nBaker, K. B., and Wing, S. (1989). A new magnetic coordinate system for conjugate studies at high latitudes. J. Geophys. 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Res. -Space Physics-, 128, e2023JA031400. https://doi.org/10.1029/2023JA031400. \nNowada, M., Miyashita, Y., Nakamizo, A., Partamies, N., and Shi, Q. -Q. (2024). Numerical data of the REPPU global MHD simulation results (1 st version published on 6 th September, 2024) \n[Dataset]. Science Data Bank. https://www.scidb.cn/en/anonymous/RlZGN3p1. \nPalmroth, M., Ganse, U., Pfau-Kempf, Y., Battarbee, M., Turc, L., Brito, T., Grandin, M., Hoilijoki, S., Sandroos, A., and von Alfthan, S. (2018). Vlasov methods in space physics and astrophysics. Living Rev. Comput. Astrophys., 4(1). https://doi.org/10.1007/s41115-018-0003-2. \nParks, G., Brittnacher, M., Chen, L. J., Elsen, R., McCarthy, M., Germany, G., and Spann, J. (1997). Does the UVI on Polar detect cosmic snowballs? Geophys. Res. Lett., 24(24), 3,109 - 3,112. https://doi.org/10.1029/97GL03005. \nPartamies, N., Kauristie, K., Pulkkinen, T. I., and Brittnacher, M. (2001a). Statistical study of auroral spirals. J. Geophys. Res. -Space Physics-, 106(A8), 15,415 -15,428. https://doi.org/10.1029/2000JA900172. \nPartamies, N., Freeman, M. P., and Kauristie, K. (2001b). On the winding of auroral spirals: Interhemispheric observations and Hallinan's theory revisited. J. Geophys. Res. -Space Physics-, 106(A12), 28,913 - 28,924. https://doi.org/10.1029/2001JA900093. \nPerreault, P., and Akasofu, S. -I. (1978). Study of geomagnetic storms. Geophys. J. R., Astron. Soc., 54, 3, 547 - 573. https://doi.org/10.1111/j.1365-246X.1978.tb05494.x. \nRankin, R., K. Kabin, J. Y. Lu, I. R. Mann, R. Marchand, I. J. Rae, V. T. Tikhonchuk, and E. F. Donovan (2005). Magnetospheric field-line resonances: Ground-based observations and modeling. J. Geophys. Res. -Space Physics-, 110, A10S09. https://doi.org/10.1029/2004JA010919. \nShue, J. -H., Song, P., Russell, C. T., Steinberg, J. T., Chao, J. K., Zastenker, G., Vaisberg, O. L., Kokubun, S., Singer, H. J., Detman, T. R., and Kawano, H. (1998), Magnetopause location under extreme solar wind conditions, J. Geophys. Res. -Space Physics-, 103(A8), 17691 - 17700, https://doi.org/10.1029/98JA01103. \nTanaka, T. (1994). Finite volume TVD scheme on an unstructured grid system for threedimensional MHD simulation of inhomogeneous systems including strong background potential fields. J. Comput. Phys., 111, 2, 381 - 389. https://doi.org/10.1006/jcph.1994.1071. \nTanaka, T., Obara, T., and Kunitake, M. (2004). Formation of the theta aurora by a transient convection during northward interplanetary magnetic field. J. Geophys. Res., -Space Physics-, 109, A09201. https://doi.org/10.1029/2003JA010271. \nTanskanen, E. I. (2009). A comprehensive high-throughput analysis of substorms observed by IMAGE magnetometer network: Years 1993-2003 examined. J. Geophys. Res. -Space Physics-, \n114, A05204, J. Geophys. Res., 114, A05204. https://doi.org/10.1029/2008JA013682. \nTorr, M. R., Torr, D. G., Zukic, M., Johnson, R. B., Ajello, J., Banks, P., Clark, K., Cole, K., Keffer, C., Parks, G., Tsurutani, B., and Spann, J. (1995). A far ultraviolet imager for the International Solar-Terrestrial Physics Mission. Space Sci. Rev., 71 (1-4), 329 - 383. https://doi.org/10.1007/BF00751335. \n- Torrence, C., and Compo, G. P. (1998). A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc., 79, 1, 61 - 78, https://doi.org/10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2.\n- Tsyganenko, N. A., and Fairfield, D. H. (2004). Global shape of the magnetotail current sheet as derived from Geotail and Polar data. J. Geophys. Res. -Space Physics-, 109(A03218). https://doi.org/10.1029/2003JA010062. \nWatanabe, M., Sakito, S., Tanaka, T., Shinagawa, H., and Murata, K. T. (2014). Global MHD modeling of ionospheric convection and field-aligned currents associated with IMF By triggered theta auroras. J. Geophys. Res., -Space Physics-, 119, 6,145 -6,166. https://doi.org/10.1002/2013JA019480. \nFigure 1 . Fine snapshots of contemporaneous appearances of the auroral spiral (pointed out by thick orange arrows) and the nightside distorted transpolar arc (J-shaped TPA, indicated with thick magenta arrows) observed by Polar ultraviolet imager (UVI) and Longyearbyen all-sky camera (ASC) are shown. Panels a and b are Lyman-Birge-Hopfield short (LBHS) emission images with an integration time of 36 s in altitude adjusted corrected geomagnetic (AACGM; Baker and Wing, 1989) and geographic coordinate systems, respectively, at 21:08:40 UT on 10 January 1997. Panel a is oriented such that the bottom, right, top, and left correspond to midnight (0 h magnetic local time; MLT), dawn (6 h MLT), noon (12 h MLT), and dusk (18 h MLT), respectively. The white circles are drawn every 10° from 60° to 80° magnetic latitude (MLat) for panel a and from 50° to 80° geographic latitude for panel b. The white straight lines are drawn every 2 h in MLT. The color code is assigned according to the auroral brightness in units of Rayleigh. Panel c shows an image of the auroral spiral taken from the ASC installed at Longyearbyen for the nearest observational time of panels a and b. \n<!-- image --> \nFigure 2 . Plots of solar wind parameters and geomagnetic activity indices during the 6 h interval from 17:00 UT to 23:00 UT on 10 January 1997 are shown. From top to bottom: the SMU and SML indices which were derived, based on the high-latitude geomagnetic field data obtained from the SuperMAG geomagnetic observatory network (Newell and Gjerloev, 2011; Gjerloev, 2012); the IMF-BY and -BZ components; the IMF clock angle (arctan(IMF-BY/IMF-BZ)); the Akasofu- \n<!-- image --> \nPelleaut parameter (εAP) computed with VSWBt 2 sin 4 (θCLOCK/2)(4πL0 2 /μ0), where VSW is the solar wind velocity, Bt is the magnetic field intensity as calculated by sqrt(IMF-BX 2 + IMF-BY 2 + IMFBZ 2 ), and L0 = 7.0 RE; the solar wind dynamic pressure (Pd); solar wind speed (VSW), and proton number density (NP). The Kp index is indicated at the bottom of the figure. The intervals of ovalaligned transpolar arc (TPA), auroral spiral, contemporaneous appearances of the J-shaped TPA and the auroral spiral, and the J-shaped TPA are bracketed with cyan, black, violet, and orange broken vertical lines, respectively. \nFigure 3. Representative snapshots of LBHL 36 s images of the four types of aurora and the corresponding snapshots of aurora-associated field-aligned current (FAC) density reproduced by a global MHD simulation using the improved REProduce Plasma Universe (REPPU) code are shown. The imager data plots of (a) oval-aligned TPA, (b) auroral spiral, (c) contemporaneous appearances of the TPA and the auroral spiral, and (d) J-shaped TPA are displayed. The color code is assigned according to the logarithm of auroral brightness in units of Rayleigh. The white circles are drawn every 10° from 60° to 80° magnetic latitude (MLat), and the white straight lines are drawn every 2 h in MLT. The color contours in panels e to h show the intensity of ionospheric conductance associated with auroras (a proxy of the FAC intensity), which is determined by magnetotail conditions. The reddish (bluish) color indicates the FACs toward (away from) the ionosphere. The magenta circles are drawn every 5° from 65° to 90° MLat, and the magenta straight lines are drawn every 2 h in MLT. \n<!-- image --> \nFigure 4: Magnetotail magnetic equatorial projections of Polar UVI auroral pixel data (panels ac) and the original Polar UVI nightside ionospheric snapshots (panels d-f) for the oval-aligned TPA at 18:20:00 UT (panels a and d), the auroral spiral at 20:16:32 UT (panels b and e), and the J-shaped TPA at 21:48:32 UT (panels c and f) are shown. The format of panels d-f is the same as those of Figures 1a and 3a-3d, except that only the nightside in 18 to 6 h MLT is shown. The color code range is the same as that shown in Figure 3. Each UVI pixel is traced to the magnetic equatorial plane, based on the Tsyganenko 96 geomagnetic field model (Tsyganenko and Stern, 1996), using the calculation routines implemented in the Space Physics Environment Data Analysis Software (SPEDAS) 4.1 package (Angelopoulos et al., 2019). Magenta and orange broken ovals highlight the oval-aligned and J-shaped TPAs detected by Polar. The dashed black curves in panels a-c indicate the model magnetopause locations derived from Shue et al. (1998). \n<!-- image --> \nFigure 5: The density of field-aligned current (FAC) on the magnetic equatorial plane over the ranges of X = 20 to -100 RE and Y = 15 to -20 RE (panels a-c) and in the nightside ionosphere (panels d-f) in the oval-aligned TPA, the auroral spiral, and the J-shaped TPA cases are shown. These FAC structures are reproduced by the Block Adaptive Tree-Solar wind-Roe Upwind Scheme (BAT-S-RUS) code provided by the Community Coordinated Modeling Center (CCMC). In panels a to c, the color code is assigned according to the Z component (vertically up-down component to the X-Y plane) of the FAC structures (J//Z) on the magnetic equatorial plane in units of μA/m 2 , and the ionospheric FACs (J//) are shown in panels d-f. The vectors in panels a-c indicate the plasma flow velocity (VXY) projected onto the magnetic equatorial plane in units of km/s. \n<!-- image --> \nFigure 6. Polar UVI snapshot (panel a) and its magnetic equatorial projection (panel b) for contemporaneous appearances of the J-shaped TPA and the auroral spiral at 21:08:40 UT are shown. The format of the Polar UVI plot is the same as those of Figures 1a and 3a-3d, but the bottom, left, top, and right correspond to dusk (18 h in MLT), noon (12 h in MLT), dawn (6 h in MLT), and midnight (0 h in MLT), respectively. Yellow (orange) and magenta ovals indicate the auroral spiral detected by the Polar UVI and the ASC at Longyearbyen, and the J-shaped TPA by Polar, respectively. The color code shows the auroral brightness in units of Rayleigh. The dashed black curve indicates the model magnetopause location derived from Shue et al. (1998). \n<!-- image --> \nFigure 7. Three panels show the Polar UVI magnetic equatorial projection of the contemporaneous appearances of the J-shaped TPA and the auroral spiral at 21:08:40 UT (panel a), and the corresponding BAT-S-RUS global MHD simulation results on the equatorial plane over the ranges of X = 10 to -100 RE and Y = 15 to -20 RE (panels b and c). In panels b and c, the color code is assigned according to the FAC (J//) and the Z component of the FAC (J//Z), respectively, in units of μA/m 2 . The vectors indicate the plasma flow velocity (Vxy) on the magnetic equatorial plane in units of km/s. \n<!-- image --> \nBAT-S-RUS Code Velocity \nFigure 8: Zoom-in plots of the FAC (J//) (panel a) and the FAC Z-component (J//Z, panel b) at the auroral spiral source region at 21:08 UT are shown. The source region is determined by the Polar UVI observations, and these FAC distributions were reproduced by BAT-S-RUS global MHD simulations. The figure format is the same as those of Figures 5 and 7. \n<!-- image --> \n[Re] \nFigure 9: Comparison of the FACs reproduced by BAT-S-RUS (panel a) with the auroral conductance dependent on the magnetospheric conditions (a proxy of FACs) obtained from the improved REPPU code (panels b and c) around the times when the J-shaped TPA and auroral spiral were contemporaneously observed is shown. \n<!-- image --> \nFigure 10: Counter-clockwise rotational vortex-like equivalent ionospheric electric current (EIC) structures, shown with curved green arrows, derived from the observations of the IMAGE ground observatory network (Tanskanen, 2009) near the auroral spiral region from 21:05 UT to 21:10 UT are shown with 1 min time step. These EIC vectors (black vectors) are derived by extracting the pure ionospheric current contribution by removing the telluric components from the measured geomagnetic field data, based on the technique proposed by Vanhamäki and Juusola (2020). The color code is assigned according to the FAC intensity (in unit of A/km 2 ) and orientation estimated by the EIC vectors. \n<!-- image --> \nFigure 11: Map that shows the locations of three ground magnetometer stations relative to the auroral spiral on and near Svalbard is shown in the left panel. Deviations of the N component magnetic field (BN, local magnetic north-south component) and their power spectrograms at HRN(HOR), HOP, and BJN stations during the auroral spiral intervals on and near the Svalbard Island from 19:59 UT to 21:23 UT are shown in the right panels. The power spectra of the BN deviations are calculated, based on the wavelet analysis using the Morlet function (see Torrence and Compo, 1998 for details). In the power spectrograms, the confidence level higher than 93% for the wave power intensity is surrounded by cyan solid ovals. The frequency range of the Pc5 waves (1.67 - 6.67 mHz) is bracketed by two horizontal green broken lines. The two representative times of 20:16:32 UT (fine auroral spiral) and 21:08:40 UT (contemporaneous appearances of the J-shaped TPA and the auroral spiral) are indicated by sky blue and orange vertical broken lines, respectively. \n<!-- image --> \nTable 1. Transition of auroral shapes, IMF-Bz conditions, the results of whether or not the observed auroras were reproduced by global MHD simulations, and the duration of the auroras seen during about 5 h interval from 17:46 UT to 22:55 UT are summarized, based on the Polar UVI observations. \nFigure S1 shows the improved REProduce Plasma Universe (REPPU) simulation results of the field-aligned current (FAC) map on the magnetic equatorial plane in the nightside magnetosphere and the distribution of the ionospheric conductance determined by the magnetotail conditions, that is, a proxy of the ionospheric FAC intensity. The simulation results at the nearest times when Polar observed the fine auroral spiral (one of the auroral spots seen at the poleward edge of the nightside auroral oval) and the contemporaneous appearances of the nightside distorted transpolar arc (J-shaped TPA) and the auroral spiral are shown. In Figure S2, the deviations of the two geomagnetic field components (BE and BZ) and their power spectrograms at the three ground magnetometer stations on the Svalbard island which were located near the auroral spiral are shown to examine what perturbations (or waves) in an ultra-low-frequency band can be observed in association with the auroral spiral. \n<!-- image --> \nJournal of Geophysical Research: Space Physics \nSupporting Information for \nContemporaneous Appearances of Local-Scale Auroral Spirals and Global-Scale Transpolar Arc: Changes of Auroras and Field-Aligned Current Profiles Before a Substorm and After Its Recovery Phase \nMotoharu Nowada 1 *, Yukinaga Miyashita 2,3 , Aoi Nakamizo 4 \n, Noora Partamies 5 , and Quan-Qi Shi 1 \n- 1: Shandong Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and Physics, Institute of Space Sciences, Shandong University, Weihai, Shandong, People's Republic of China.\n- 2: Korea Astronomy and Space Science Institute, Daejeon, South Korea.\n- 3: Korea University of Science and Technology, Daejeon, South Korea.\n- 4: National Institute of Communications and Technology, Koganei, Tokyo, Japan.\n- 5: Department of Arctic Geophysics, The University Centre in Svalbard, Norway.", 'Contents of this file': 'Figures S1 and S2', 'Introduction': 'Figure S1. Snapshots of the results of the global MHD simulation, based on the improved REProduce Plasma Universe (REPPU) code (Tanaka, 1994; Nakamizo et al., 2009) when the auroral spots including the auroral spiral appeared at the poleward edge of the nightside auroral oval (~20:17 UT) and when the J-shaped TPA and the auroral spiral contemporaneously appeared (~21:10 UT) are shown. Panels a and c show the colored field-aligned current (FAC) contours projected on the magnetic equatorial plane. Panels b and d show the auroral conductance (a proxy of the FAC intensity, color contours) in the nightside ionosphere from 18 h to 6 h MLT at the nearest times of panels a and c. \n<!-- image --> \nFigure S2. Deviations of the BE (local magnetic east-west direction) and the BZ (vertical up-down direction) components and their power spectrograms at the HRN (HOR), HOP, and BJN stations during the time intervals of auroral spiral on and near the Svalbard island from 19:59 to 21:23 UT are shown. The power spectra of these geomagnetic field fluctuations were calculated, based on the wavelet analysis using the Morlet function (Torrence and Compo, 1998). The vertical axis gives the frequency of the AC components of BE and BZ, and the color code is assigned to the logarithmic values of the wave power intensity. The confidence level higher than 93% for the wave power intensity is surrounded by cyan solid curves. The frequency range of Pc5 waves (1.67 - 6.67 mHz) are bracketed by two horizontal green broken lines. The two times of interest of 20:16:32 UT (fine auroral spiral) and 21:08:40 UT (contemporaneous appearances of the J-shaped TPA and auroral spiral) are indicated by sky blue and orange vertical broken lines, respectively. \n<!-- image -->'}
2022JHEP...07..128O	Double holography plays a crucial role in recent studies of Hawking radiation and information paradox by relating an intermediate picture in which a dynamical gravity living on an endoftheworld brane is coupled to a nongravitational heat bath to a much betterunderstood BCFT picture as well as a bulk picture. In this paper causal structures in generic double holographic setups are studied. We find that the causal structure in the bulk picture is compatible with causality in the BCFT picture thanks to a generalization of the GaoWald theorem. On the other hand consistency with the bulk causal structure requires the effective theory in the intermediate picture to contain a special type of superluminal and nonlocal effect which is significant at long range or IR. These are confirmed by both geometrical analysis and commutators of microscopic fields. Subregion correspondences in double holography are discussed with the knowledge of this nonlocality. Possible fundamental origins of this nonlocality and its difference with other types of nonlocality will also be discussed.	2022-07-01T00:00:00Z	['2022JHEP...07..128O', '10.48550/arXiv.2107.01219', '2021arXiv210701219O', 'arXiv:2107.01219', '10.1007/JHEP07(2022)128']	['AdS-CFT Correspondence', 'Boundary Quantum Field Theory', 'Brane Dynamics in Gauge Theories', 'High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']	Causal structures and nonlocality in double holography	2,022	192	0.35	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	39	https://arxiv.org/pdf/2107.01219.pdf	{'Hidetoshi Omiya a and Zixia Wei b': 'a Department of Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan b Center for Gravitational Physics, \nYukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan', 'Abstract': 'Double holography plays a crucial role in recent studies of Hawking radiation and information paradox by relating an intermediate picture, in which a dynamical gravity living on an end-of-the-world brane is coupled to a non-gravitational heat bath, to a much better-understood BCFT picture as well as a bulk picture. In this paper, causal structures in generic double holographic setups are studied. We find that the causal structure in the bulk picture is compatible with causality in the BCFT picture, thanks to a generalization of the Gao-Wald theorem. On the other hand, consistency with the bulk causal structure requires the effective theory in the intermediate picture to contain a special type of superluminal and nonlocal effect which is significant at long range or IR. These are confirmed by both geometrical analysis and commutators of microscopic fields. Subregion correspondences in double holography are discussed with the knowledge of this nonlocality. Possible fundamental origins of this nonlocality and its difference with other types of nonlocality will also be discussed.', '1 Introduction': 'The information loss problem in black hole evaporation [1-3] has puzzled physicists for decades and recently gets a great development [4, 5]. Starting from a pure initial state and tracking the time evolution from the formation to the evaporation of a black hole, the entanglement entropy between its interior and exterior is expected, from unitarity, to start increasing from zero and finally return back to zero again. This behavior is known as a Page curve [2]. Hawking\'s original computation [1] was performed with a local quantum field theory living on a classical spacetime with a black hole. By simply factorizing the Hilbert space into interior and exterior on the classical spacetime, Hawking\'s computation suggests that the entanglement entropy monotonically grows and leads to a breakdown of unitarity, i.e. loss of information. \nRecent studies have resolved this problem by coupling a gravitational region containing a black hole to a non-gravitational region working as a heat bath, and studying the entanglement entropy between them. It is found that another saddle point which Hawking did not count dominates at late time and reproduces the expected Page curve [4, 5]. The existence and dominance of this saddle point is justified both by a class of doubly holographic models [6] and by gravitational path integral [7,8]. \nConsider a d -dimensional AdS gravity living on Q interacting with a d -dimensional CFT living on Σ through a ( d -1)-dimensional interface ∂Q = ∂ Σ. Double holography relates the current setup to two different but equivalent theories. One is a boundary CFT (BCFT) on Σ, which can be obtained by applying the AdS/CFT correspondence to Q and regarding it as a ( d -1)-dimensional CFT living on ∂ Σ. The other one is an AdS d +1 gravity with an end-ofthe-world brane floating in it. The asymptotic boundary and the end-of-the-world brane are identified with Σ and Q , respectively. In this paper, we call the latter two equivalent pictures the BCFT picture and the bulk picture respectively. At the same time, we call the original setup the intermediate picture, in the sense that it can be regarded as an intermediate process when jumping between the BCFT picture and the bulk picture. See figure 1 for a sketch. \nAlthough the terminology \'double holography\' is relatively new, the correspondence between the three pictures has been known for a long time since [9,10]. In particular, the duality between the intermediate picture and the bulk picture is often called the Karch-Randall type brane-world holography. On the other hand, the duality between the bulk picture and the BCFT picture is further explored in [11,12] and called the AdS/BCFT correspondence. With \ndouble holography, the second saddle point in Hawking radiation can be thought to come from boundary OPE in the BCFT picture and minimal surfaces ending on the end-of-the-world brane in the bulk picture [13-15]. On the other hand, from a gravitational path integral point of view, this saddle point comes from spacetime configurations with higher topology [7,8]. \nFigure 1: The three equivalent pictures of a doubly holographic model. The asymptotic boundary Σ and the end-of-the-world brane Q are shown in grey and blue, respectively. 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sha1\\_base64="DHYFmP/awVKbI1YkrbQQsDf+MsY=">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</latexit> \nWhile dynamics in double holography plays a crucial role in recent studies of Hawking radiation, discussions in the Lorentzian signature [16-18] are limited and mostly focusing on specific spacetime configurations. One of the most important ingredients in a Lorentzian theory is the causal structure. In the following of this paper, we study the causal structures in double holography for generic spacetime dimensions and configurations. \nThe causal structure in holography was firstly discussed by Gao and Wald [19]. They studied the AdS/CFT correspondence [20-22] and proved a theorem which implies that the causal structure in AdS is compatible with causality (the property that one cannot send a \nsignal outside of the light cone) in CFT. As for double holography, we will firstly show that the causal structure in the bulk picture is compatible with causality in the BCFT picture, as a generalization of the Gao-Wald theorem. On the other hand, we will see that, the intermediate picture is expected to have a unique causality violation, for it to be compatible with the bulk causal structure. More specifically, a signal propagating within the gravitational region Q or within the non-gravitational region Σ cannot travel faster than light, while superluminal information propagation is allowed when sending the signal from Q to Σ and vice versa. These features will also be confirmed by computing commutators of microscopic fields. See [23] for a previous study of causality within the end-of-the-world brane in the context of Randall-Sundrum type brane-world holography [24,25]. \nThis causal structure implies that the effective theory in the intermediate picture is a nonlocal one. To our knowledge, this property has not been explicitly examined in previous works. The nonlocality prompts us to reconsider the notion of domain of dependence which is usually discussed in local theories. Based on this, the subregion correspondence in double holography will be discussed. In ordinary AdS/CFT or AdS/BCFT correspondence, a subregion in the (B)CFT picture is equivalent to its entanglement wedge in the bulk picture [26-28]. We will introduce an analogy of the entanglement wedge for intermediate subregions instead of BCFT subregions and call it the tentative entanglement wedge. It will be shown that the tentative entanglement wedge is not equivalent to the corresponding intermediate subregion, in contrast to what one may expect at first glance. \nThe causality violation and nonlocality in the intermediate picture becomes significant when zooming out to IR and neglectable when zooming in to UV. Going beyond double holography, features in the intermediate picture suggest a possibility that effective theories of quantum gravity show similar nonlocality in general. We will argue that the effect from spacetime configuration with higher topology is a possible origin of such nonlocality and discuss its difference from nonlocality known in early studies of quanutm gravity [29,30]. \nThis paper is organized as follows. In section 2, we present a careful review of a doubly holographic model starting from the AdS/BCFT correspondence. After that, we review compatibility of causality in holographic dualities and summarize the main technical results about causal structures in double holography found in this paper. In section 3, we focus on the vacuum configuration of double holography as a simple concrete example and verify our results by explicitly writing down the geodesics. In section 4, we show that causality in the BCFT picture is compatible with the bulk causal structure for generic configurations with \nreasonable assumptions, and discuss its physical consequences in AdS/BCFT. In section 5, we study the causal structure in the intermediate picture and discuss its relation with other holographic models. In section 6, we compute the commutator of a light primary operator in the intermediate picture using a holographic computation and confirm that it is consistent with the results shown in the previous sections. In section 7, we discuss physical consequences coming from the nonlocality in the intermediate picture. After clarifying confusing notions of subregions and reduced states associated to them, and examining the correspondence between subregions in the three pictures, we discuss more fundamental features of nonlocality in quantum gravity. In section 8, we summarize our results and discuss future directions. In appendix A, basics of (globally) AdS spacetime are reviewed. \nWe would like to finish the introduction with a reading guide. Section 4, 5, 6 and 7 are relatively independent from each other. Therefore, readers can go straight to each of them after checking the basic results in section 2 and 3. \nNote added: While we were writing this paper, we got aware of an interesting work [31], where subregion correspondence in double holography is independently considered in a setup similar to ours discussed in section 7.3, but from a different perspective.', '2 Preliminaries and Summary of Technical Results': 'In this section, we firstly review the setup of double holography and causality compatibility in the AdS/CFT correspondence. Then we will summarize our results about the causal structures in double holography.', '2.1 Review of Double Holography and Related Topics': 'We present a review of double holography in this subsection. Readers who are familiar with this topic may skip to the next subsection after catching a glimpse of table 1 and figure 1 for the notations using in this paper. \nThe notion of double holography 1 , which is proposed in [6] and further explored in [1315], arises naturally when considering a CFT defined on a d -dimensional manifold (Σ , γ ij ) with a time-like boundary ∂ Σ. When the boundary maximally preserves the conformal symmetry, the theory is called a boundary conformal field theory (BCFT). The gravity \ndual of a holographic BCFT can be constructed in a bottom-up way called AdS/BCFT correspondence [11,12]. We will consider the Lorentzian signature throughout this paper. \nAccording to AdS/BCFT, the gravity dual of a holographic BCFT is given by a portion of a ( d +1)-dimensional asymptotically AdS spacetime (AAdS) ( M , g µν ). The boundary of M is given by ∂ M = Σ ∪ Q with ∂ Σ = ∂Q . Here, Σ is the ordinary asymptotic boundary of M on which Dirichlet boundary conditions are imposed to the bulk fields. On the other hand, ( Q,h ab ) is an end-of-the-world brane extended to the bulk from ∂ Σ. In contrast with the asymptotic boundary Σ, Neumann boundary conditions are imposed to the bulk fields on the end-of-the-world brane Q . To be more concrete, the bulk action is given by \nI bulk = 1 16 πG N ∫ M √ -g ( R -2Λ) + 1 8 πG N ∫ Σ √ -γB + 1 8 πG N ∫ Q √ -h ( K -T ) + I matter , (2.1) \nwhere the four terms are the Einstein-Hilbert term in the bulk M , the Gibbons-Hawking term on the asymptotic boundary Σ, the Gibbons-Hawking term on the brane Q , and the action for the bulk matter fields. The metric of M , Σ and Q are denoted by g µν , γ ij and h ab , respectively. B ij ( K ab ) is the extrinsic curvature of Σ ( Q ), and T is the tension of Q . 2 Variation of the gravity sector at the vicinity of the asymptotic boundary Σ is given by \nδI bulk = 1 16 πG N ∫ Σ √ -γ ( B ij -Bγ ij ) δγ ij . (2.2) \nDirichlet boundary condition δγ ij = 0 is imposed on Σ. In contrast with this, although the variation at the vicinity of the end-of-the-world brane Q is similarly given by \nδI bulk = 1 16 πG N ∫ Q √ -h ( K ab -Kh ab + Th ab ) δh ab , (2.3) \nthe boundary condition imposed on it is chosen to be the Neumann type \nK ab -Kh ab -Th ab = 0 . (2.4) \nAt leading order of G -1 N , the gravity dual of a holographic BCFT defined on Σ is an on-shell configuration of (2.1). \nHere, we note that there is no matter field localized on the brane Q in (2.1). We would like to take this as default in this paper, while consequences caused by such a matter field will be discussed in section 5.2. \nNow, we have two equivalent pictures for a holographic BCFT. We will call them the BCFT picture and the bulk picture, respectively. We use T BCFT Σ to denote the theory in the BCFT picture, and T bulk M to denote the theory in the bulk picture. The novelty of double holography is that there exists another intermediate picture by simply applying holography only to the ( d -1)-dimensional boundary ∂ Σ of the BCFT but not to the ambient region. After doing this, naively, we will get a theory composed of two parts. One is a gravitational theory living in an AAdS d whose asymptotic boundary is identified with ∂ Σ. 3 Another part is the original holographic CFT d living in the ambient region without gravity. This intermediate picture plays an important role in recent progress of information paradox, since it couples a gravitational region to a non-gravitational heat bath to which Hawking radiation can escape, and allows us to use better-understood BCFT picture and bulk picture to study the information flow. Let us just call it the intermediate picture. \nAlthough we have intuitively explained how the intermediate picture arises from the BCFT picture, it can also be derived from the bulk picture, long known as Karch-Randall brane-world holography [9, 10, 35]. In that context, it is known that the bulk theory (2.1) in M is equivalent to a theory on its boundary ∂ M = Σ ∪ Q where Q has a dynamical gravity 4 due to the Neumann boundary condition (2.4) while Σ does not due to the Dirichlet boundary condition. Besides, there is a common holographic CFT living on both Q and Σ as a matter theory, and the boundary condition for this CFT on ∂Q = ∂ Σ is transparent [37]. These together give the intermediate picture. 5 See [38,39] for an effective action on the endof-the-world brane (also called the Karch-Randall brane in this context) Q . We use T int Σ ∪ Q to denote the theory in the intermediate picture. \nSo far, we have introduced a bunch of notations associated to the manifolds considered \nin this paper. We summarize them in table 1 so that the readers can refer to it easily. \nIn short, the doubly holographic model introduced here has three equivalent pictures: the BCFT picture, the bulk picture and the intermediate picture. 6 The relationship between these three pictures are shown in figure 1. \nTable 1: Notations for the manifolds considered in this paper.', '2.2 Compatibility of Causality in the AdS/CFT Correspondence': 'If two theories are equivalent via a duality or correspondence, a physical process in one theory can be translated into another equivalent physical process in the other theory, at least in principle. As a result, if a physical process is not achievable in one theory, so should not be its dual process in the dual theory. This fact, while being a matter of course, is in general non-trivial and can give powerful constraints when studying correspondences which are not completely understood. \nLet us take the compatibility of causality in the AdS/CFT correspondence as an example. In standard AdS/CFT correspondence, a holographic CFT defined on a d -dimensional Lorentzian manifold Σ is equivalent to a gravitational theory in an AAdS d +1 manifold whose asymptotic boundary is given by Σ. Here, we consider the case where Σ has no boundary. On the CFT side, one cannot send a signal from p ∈ Σ to q ∈ Σ if p and q are space-like separated on Σ, assuming the CFT is local and unitary. The corresponding process in the bulk theory is to send a signal from p ∈ Σ to q ∈ Σ through the bulk M , after the GKP-W \nrelation [21, 22]. Therefore, we expect the latter process in the bulk M should also be impossible, i.e. p and q should also be space-like separated in the bulk M , for the bulk causal structure to be compatible with causality in CFT. In other words, no shortcut should be allowed in M when considering signal propagations on Σ. The situation is shown in figure 2. Let us summarize this in a more clear and standard way.', 'Statement A (Compatibility of Causality in AdS/CFT) .': 'In the AdS/CFT correspondence, for any two points p, q ∈ Σ = ∂ M , if p and q are not causally connected on the asymptotic boundary Σ , then they are not causally connected in the bulk M either. \nFigure 2: No shortcut should be allowed in the bulk M (shaded) when considering signal propagations on the asymptotic boundary Σ (shown in grey). The solid line shows a null geodesic on Σ. The dashed line shows a time-like/null geodesic in M connecting two spacelike separated (with respect to Σ) points on Σ, which should not have existed. \n<!-- image --> \nOne can check that the statement A holds in pure AdS by explicitly writing down the geodesics. This point will be reviewed in section 3.1. \nFor more general configurations, the Gao-Wald theorem [19] guarantees that the statement A holds under several reasonable assumptions. 7 The Gao-Wald theorem is summarized as \n<latexit sha1\\_base64="N1pYycQS1zD7fQt0Ui9HHsUnBUE=">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</latexit> \n<latexit 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sha1\\_base64="NRndlfqGam0mc8rgB2fam8GHEPo=">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</latexit> \nfollows:', 'Theorem A.0 (Gao-Wald theorem) .': 'Let ( M , g µν ) be a spacetime with a time-like boundary ∂ M = Σ at asymptotic infinity. Suppose ( M , g µν ) satisfies the following four conditions: \n- 1. M is a solution of the Einstein equation in which the matter sector satisfies averaged null energy condition (ANEC),\n- 2. Null generic condition,\n- 3. ¯ M≡M∪ Σ is strongly causal (strong causality condition),\n- 4. J + ¯ M ( p ) ∩ J -¯ M ( q ) is compact for any p, q ∈ M . \nLet A Σ ( p, M ) be \nA Σ ( p, M ) ≡ { r ∈ Σ \| there exists a future directed causal curve λ which starts from p and ends at r satisfying λ -( p ∪ r ) ⊂ M } , (2.5) \nthen for p ∈ Σ and ∀ q ∈ ∂A Σ ( p, M ) , q satisfies q ∈ J + ¯ M ( p ) \\ I + ¯ M ( p ) . Moreover, any causal curve connecting p and q lies entirely in Σ , and hence it is also a null geodesic on (Σ , γ ij ) , i.e. q ∈ J + Σ ( p ) \\ I + Σ ( p ) . \nHere, J + N ( p ) ( I + N ( p )) is the causal (chronological) future of p on the manifold N . We briefly comment on the four conditions above. The first one, ANEC, states that the energy-stress tensor T µν satisfies \n∫ l T µν k µ k ν ≥ 0 , (2.6) \nfor any null curve l . Here, ∫ l stands for the integration along l , and k µ is its tangent vector. The second condition, null generic condition, states that every null geodesic in M must contain a point at which \nk µ k ν k [ ρ R σ ] µν [ α k β ] = 0 , (2.7) \nglyph[negationslash] \nother physical conditions in the AdS/CFT correspondence [40,41]. \nwhere k is the tangent vector of the null geodesic 8 . This condition together with the ANEC ensures the existence of the conjugate points for any null geodesic [42, 43], which is one of the most essential points in the proof of the Gao-Wald theorem. The third condition, strong causality condition, roughly means that no causal curve in ¯ M can be almost closed. More rigorously 9 , for any point p ∈ ¯ M , every neighborhood of p contains another neighborhood of p which no causal curve can intersect more than once. This condition and the fourth condition guarantee that bulk spacetime under consideration has a sensible causal structure. \nLet us then explain how the Gao-Wald theorem guarantees that statement A holds. First of all, for points on the asymptotic boundary Σ, ∂A Σ ( p, M ) bounds the region which can receive a signal through the bulk M , and J + Σ ( p ) \\ I + Σ ( p ) bounds the region which can receive a signal through the asymptotic boundary Σ. Since one consequence of the Gao-Wald theorem is \n∂A Σ ( p, M ) ⊆ J + Σ ( p ) \\ I + Σ ( p ) , (2.8) \nstatement A follows straightforwardly from the Gao-Wald theorem, for configurations satisfying the four conditions. \nFrom now on, we are going to consider the causal structures in the three pictures of the doubly holographic model introduced in section 2.1. 10', '2.3 Summary of Results: Causal Structures in Double Holography': 'Consider an on-shell configuration M of (2.1) with Dirichlet boundary condition imposed on the asymptotic boundary Σ and Neumann boundary condition (2.4) imposed on the end-ofthe-world brane Q . M has the following properties under some reasonable assumptions such as ANEC. \nStatement B. Let Σ be compatible with a BCFT. For any two points p, q ∈ Σ , if p and q are not causally connected on the asymptotic boundary Σ , then they are not causally connected in the bulk M either. \n10 It is worth noting that the notion of quantum tasks [44] provides a good framework for using the consistency of more general physical processes to study holography [18, 45-47]. The physical process we consider can also be regarded as one of the most simple versions of a quantum task, but we will not go deep into it. \nStatement C. For any two points p, q ∈ Q , if p and q are not causally connected on the end-of-the-world brane Q , then they are not causally connected in the bulk M either. \nStatement D. Two points p ∈ Σ and q ∈ Q can be causally connected in the bulk M , even if they are not causally connected on ∂ M = Σ ∪ Q . \nIn the following few sections, we are going to firstly give a concrete example of M and check that it satisfies the statements above by explicitly writing down the null geodesics in section 3. Then we will show that statement B holds for more general configurations in section 4. After that, we will prove statement C for general configurations in section 5. Although the statement C looks similar to statement B, the mathematical mechanism is actually totally different. We will see that Neumann boundary condition (2.4) plays the most important role in this consequence. \nThe mathematical facts above lead to many important physical consequences. We can see them by picking up two pictures in double holography and comparing them with each other.', 'Compatibility of causality in AdS/BCFT': 'Comparing the bulk picture and the BCFT picture, statement B implies that, in the AdS/BCFT correspondence, the causal structure in the bulk picture is compatible with causality in the BCFT picture. This gives a further consistency check to AdS/BCFT.', 'Causality Structure in the Intermediate Picture': 'Comparing the bulk picture and the intermediate picture, statement B, C, D imply that the effective theory T int Σ ∪ Q in the intermediate picture has a special causal structure to be compatible with the causal structure in the bulk picture. First of all, superluminal information propagation is not allowed within Σ or Q . However, it is allowed when sending a signal from Q to Σ or vice versa. This point will also be confirmed by computing the commutators of quantum fields in section 6. \nThe fact that superluminal effects can be observed only when one can access both Q and Σ implies that the intermediate theory T int Σ ∪ Q has a nonlocality which is significant at longrange scale. Physical consequences coming from this nonlocality will be discussed in section 7.', '3 Vacuum Configuration as an Example': 'In this section, we focus on the vacuum configuration of (2.1) as the most simple example of M . This configuration is a portion of pure AdS spacetime. After giving the configuration, we will write down the null geodesics in the pure AdS and our configuration. Finally, we will check that statement B, C, and D hold in this specific case.', '3.1 Vacuum Configuration and Null Geodesics': 'For simplicity, let us focus on AdS 3 . The discussion can be straightforwardly extended to higher dimensions.', 'Vacuum configuration': 'The metric of a pure AdS 3 can be written as \nds 2 = -( r 2 +1) dτ 2 + dr 2 r 2 +1 + r 2 dφ 2 (3.1) \nin the global coordinate. Here, -π < φ ≤ π . The vacuum configuration of (2.1) is a portion of pure AdS 3 with an end-of-the-world brane Q . The location of Q is given by \ncos φ = T √ 1 -T 2 1 r . (3.2) \nWe let Q intersect with the asymptotic boundary at φ = ± π/ 2. Note that the tension T is restricted to -1 < T < 1. For positive (negative) T , the larger (smaller) portion of global AdS 3 is identified as the bulk region M . Q approaches the asymptotic boundary at \| T \| → 1. See the left panel of figure 3. \nLet us introduce another useful coordinate by performing the following transformation. \nr cos φ = sinh ρ, (3.3) \n1 + r 2 = cosh 2 ρ cosh 2 η. (3.4) \nThe metric turns out to be \nds 2 = cosh 2 ρ ( -cosh 2 η dτ 2 + dη 2 ) + dρ 2 . (3.5) \nIn this case, the location of the brane is given by \nρ = arctanh( T ) ≡ ρ ∗ . (3.6) \nTherefore, Q is a pure AdS 2 . See the right panel of figure 3. \nρ \nFigure 3: A time slice τ = const . of the vacuum configuration in the ( τ, r, φ ) coordinate (left) and the ( τ, η, ρ ) coordinate (right) respectively. The asymptotic boundary Σ is shown in grey and the end-of-the-world brane Q is shown in blue. The dotted curve shows the asymptotic boundary of global AdS 3 . The bulk region M is surrounded by M = Σ ∪ Q . Here, tension T is positive in this figure so that the larger portion of pure AdS 3 is identified as the bulk M . \n<!-- image -->', 'Null geodesics in global AdS 3': 'Let us firstly write down the null geodesics in global AdS 3 . Since any null geodesic in global AdS 3 intersects the asymptotic boundary, it is sufficient to consider those launching from ( τ, r, φ ) = ( -π/ 2 , ∞ , -π/ 2). The location of such a geodesic can be simply expressed in the ( τ, η, ρ ) coordinate as \nρ = const . , (3.7) \nτ = 2arctan ( tanh η 2 ) . (3.8) \nIt is straightforward to see that all geodesics launching from ( τ, r, φ ) = ( -π/ 2 , ∞ , -π/ 2) reach ( τ, r, φ ) = ( π/ 2 , ∞ , π/ 2). Note that ρ →±∞ gives the null geodesic on the asymptotic boundary. Therefore, a null geodesic on the asymptotic boundary is also a null geodesic in the bulk in global AdS 3 . See figure 4. \n<latexit sha1\\_base64="7vJU+LEQEsB7jn0ER9B/DmefLlI=">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</latexit> \n<latexit 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All such geodesics reach q : ( τ, r, φ ) = ( π/ 2 , ∞ , π/ 2). The right panel shows the projection onto a time slice. \n<!-- image --> \nAs a warm-up, let us check that the no-shortcut statement holds in pure AdS 3 . Thanks to the symmetries, it is sufficient to check the following statement. \nStatement A.1. Consider pure AdS 3 . Let p be a spacetime point with p : ( τ, r, φ ) = ( τ p , ∞ , φ p ) and R be an observer localized at ( r, φ ) = ( ∞ , φ R ) on the asymptotic boundary. Consider sending a signal at the speed of light from p to the observer R . Then ∆ τ asym = ∆ τ bulk , where ∆ τ asym ( ∆ τ bulk ) is the required time on the asymptotic boundary (in the bulk) for receiver R to observe the emitted light ray. \nglyph[negationslash] \nTo confirm this, we can divide the situations into two cases. If p and R reside at the antipodal points, i.e. φ R = φ p -π , there are infinitely many paths in the bulk which take the same time as the null geodesic on the asymptotic boundary but no shorter path exists. As a result, ∆ τ asym = ∆ τ bulk = π . If φ R = φ p -π , the null geodesic on the asymptotic boundary is the only shortest path. Since this null geodesic is at the same time a bulk null geodesic, ∆ τ asym = ∆ τ bulk . 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Noticing that the location of a bulk null geodesic (3.7) and the location of the end-of-theworld brane Q (3.6) have the same form, we find that null geodesics on Q are just given by null geodesics in the bulk M with ρ = ρ ∗ . \nLet us then check that statement B, C and D hold in this case. Again, thanks to the symmetries, it is sufficient to check the following statements respectively. \nStatement B.1. Let p be a spacetime point with p : ( τ, η, ρ ) = ( τ p , η p , -∞ ) and R be an observer located at ( η, ρ ) = ( η R , -∞ ) on the asymptotic boundary Σ . 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sha1\\_base64="09ToqoKYXGa3Vc0OkbxE/Z0FFjQ=">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</latexit> \nFigure 5: Projection of the null geodesics connecting p and the observer R on to a time slice for different situations. The orange dotted curves are the the fastest paths in the bulk M . The green dotted curves are the fastest paths on the boundary ∂ M = Σ ∪ Q . The shortest path on the boundary ∂ M is omitted if it coincides that in the bulk M . \n<!-- image --> \nsignal at the speed of light from p to R . Then ∆ τ Σ = ∆ τ M , where ∆ τ Σ ( ∆ τ M ) is the required time on the asymptotic boundary Σ (in the bulk M ) for R to receive the signal. \nStatement C.1. Let p be a spacetime point with p : ( τ, η, ρ ) = ( τ p , η p , ρ ∗ ) and R be an observer located at ( η, ρ ) = ( η R , ρ ∗ ) on the end-of-the-world brane Q . Consider sending a signal at the speed of light from p to R . Then ∆ τ Q = ∆ τ M , where ∆ τ Σ ( ∆ τ M ) is the required time on the brane Q (in the bulk M ). \nStatement D.1. Let p be a spacetime point with p : ( τ, η, ρ ) = ( τ p , η p , ρ ∗ ) on the end-of-theworld brane Q and R be an observer located at ( η, ρ ) = ( η R , -∞ ) on the asymptotic boundary Σ . Consider sending a signal at the speed of light from p to R . Then ∆ τ Σ ∪ Q > ∆ τ M , where ∆ τ Σ ∪ Q ( ∆ τ M ) is the required time on Σ ∪ Q (in the bulk M ). \nTo check these three statements, we consider the following four cases shown in figure 5. \n- · (The upper left panel of figure 5) If p and R are both on the brane Q but not antipodal points, the null geodesic on the brane Q is the only shortest path, and it is a null geodesic in the bulk M at the same time. As a result, ∆ τ Q = ∆ τ M .\n- · (The upper right panel of figure 5) p and R are both on the asymptotic boundary Σ but not antipodal points. Similar to the previous case, ∆ τ Σ = ∆ τ M .\n- · (The lower left panel of figure 5) p and R are antipodal points. In this case, p and R are on both Σ and Q . As we have seen above, there are infinitely many paths in the bulk which take the same time as the null geodesic on the asymptotic boundary but no shorter path exists. As a result, ∆ τ Q = ∆ τ Σ = ∆ τ M .\n- · (The lower right panel of figure 5) p is on Q \\ ∂Q and R is on Σ \\ ∂ Σ. As one can see from the figure, there is a shortcut in the bulk. We can prove the existence of such a shortcut by applying the proposition 4.5.10 of [42], which states that two points joined by a causal curve γ which is not a null geodesic, can also be joined by a time-like curve. In this statement, \'a causal curve γ which is not a null geodesic\' means that either the acceleration vector of γ is non-zero and not parallel to its tangent vector on some open interval, or γ has some point on which the tangent vector is discontinuous. Consider a null geodesic on ∂ M connecting p and R (the green dotted curve of lower right panel). Since there is a co-dimension two defect ∂ Σ = ∂Q , the tangent vector in the bulk is discontinuous here. Then proposition 4.5.10 of [42] tells us that these two points at p \nand R can be joined by a time-like curve in the bulk. Therefore, we can use the null geodesic in the bulk to send information faster than the geodesic on ∂ M . As a result, ∆ τ Σ ∪ Q > ∆ τ M . \nThese four cases cover all possible situations. Therefore, we have confirmed that the statement B.1, C.1 and D.1 hold. \nNote that in the confirmation of statement D.1, we did not use any property of the specific configuration, except for the existence of a codimension-2 defect at ∂ Σ = ∂Q . Since double holography naturally introduces such a codimension-2 defect, causality violation in the intermediate picture universally occurs in more general configurations.', 'Comments on higher dimensions': 'The results above can be straightforwardly extended to higher dimensions. The metric of a pure AdS d +1 can be written as \nds 2 = -( r 2 +1) dτ 2 + dr 2 r 2 +1 + r 2 d Ω 2 (3.9) \nwhere d Ω 2 is the line element of S d -1 . All the discussions in AdS 3 can be similarly repeated in AdS d +1 by regarding the S 1 parameterized by φ in (3.1) as a equator of S d -1 .', "3.2 Zooming in to the Poincar'e Patch": 'Although we have already checked the statements hold for the vacuum configuration of (2.1), one deficiency in our discussion so far is that we did not evaluate how faster a signal can be sent in the bulk than on the boundary in statement D.1. \nIn this subsection, we restrict our attention to the Poincar\'e patch. See figure 6. The Poincar\'e patch does not cover the whole global AdS spacetime, so it cannot be used to discuss the global causal structure. However, many analytic calculations become extremely simple in it. 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The shaded region shows the Poincar\'e patch. \n<!-- image --> \nPerforming the following coordinate transformation to global AdS d +1 (3.9) \nsinh ρ = r Ω d -1 , y = √ 1 + ( r Ω d -1 ) 2 √ 1 + r 2 cos τ -r Ω d , t = √ 1 + r 2 sin τ √ 1 + r 2 cos τ -r Ω d , ξ i = r Ω i √ 1 + r 2 cos τ -r Ω d , (3.10) \nwe get a new coordinate ( t, y, ρ ) which describes the Poincar\'e patch. Here, the unit vector Ω = (Ω 1 , . . . , Ω d ) denotes a point on the ( d -1)-dimensional sphere S d -1 and i = 1 , . . . , d -2. The resulting metric is \nds 2 = dρ 2 +cosh 2 ρ ( dy 2 -dt 2 + d ξ 2 y 2 ) . (3.11) \nThe end-of-the-world brane Q locates at \nρ = arctanh ( T d -1 ) ≡ ρ ∗ . (3.12) \nand the asymptotic boundary Σ locates at ρ = -∞ . 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The asymptotic boundary Σ is shown in grey and the brane Q is shown in blue. \n<!-- image --> \nby coordinate transformation \nz = y cosh ρ , x d -1 = y tanh ρ , x i = ξ i , ( i = 1 , . . . , d -2) . (3.14) \nLet us introduce one more coordinate, which is important when calculating the correlation function on the brane by holography. With a new parameter µ satisfying \nsinh ρ = tan µ, (3.15) \nthe metric becomes \nds 2 = dµ 2 cos 2 µ + 1 cos 2 µ ( dy 2 -dt 2 + d ξ 2 y 2 ) . (3.16) \nWhen θ ≡ π/ 2 -µ glyph[lessmuch] 1, the metric can be expanded as \nds 2 ∼ 4 dθ 2 θ 2 + 4 θ 2 ( dy 2 -dt 2 + d ξ 2 y 2 ) . (3.17) \nThese coordinates are shown on a time slice of AdS 3 in figure 7. \nNow we calculate the time advance in statement D.1 by considering sending a signal at the speed of light from spacetime point p ∈ Q to an observer R residing on Σ. Let ∆ t Σ ∪ Q \nand ∆ t M be the time taken by the fastest null geodesic to travel from p to R through the ∂ M and through M\\ ∂ M , respectively. Here, ∆ t is measured at asymptotic infinity in the t of Poincar\'e coordinate. \nWe would like to firstly calculate ∆ t Σ ∪ Q . We use ( t, y, ξ ) to denote the coordinate on Σ ∪ Q , let the y > 0 half be Σ, and let the y < 0 half be Q . Consider sending a signal from p : ( t, y, ξ ) = (0 , -y p , 0 ) on Q to a receiver R located at ( y, ξ ) = ( y R , ξ R ) on Σ. Now the light ray travels through Q and Σ with different geometries. Accordingly, the geodesic on ∂ M can be divided into two pieces: one on Q and one on Σ. Time needed for the light ray to reach R is the sum of the time for it to travel through Q from y = -y p to y = 0 and the time for it to travel through Σ from y = 0 to y = y R : \n∆ t 2 Σ ∪ Q = ( y p + y R ) 2 + ξ 2 R . (3.18) \nLet us then consider sending information through the bulk. Using the bulk Poincar\'e coordinate ( t, z, ξ , x d -1 ), p is given by ( t, z, ξ , x d -1 ) = (0 , y p cos µ ∗ , 0 , -y p sin µ ∗ ) and R located at ( z, ξ , x d -1 ) = (0 , ξ R , y R ). Using the expression for null geodesic (see appendix A.2 for details), \n∆ t 2 M = y 2 p + y 2 R +2 y p y R sin µ ∗ + ξ 2 R = ( y p + y R ) 2 + ξ 2 R -2 y p y R (1 -sin µ ∗ ) . (3.19) \nWe can again confirm that \n∆ t M < ∆ t Σ ∪ Q . (3.20) \nThe (squared) time saved by taking a shortcut in the bulk is ∆ t 2 Σ ∪ Q -∆ t 2 M = 2 y p y R (1 -sin µ ∗ ).', '4 Compatibility of Causality in AdS/BCFT': 'In this section, we show that, under the same assumptions as the Gao-wald theorem A.0 plus one more reasonable assumption on Σ, any on-shell configuration of the bulk action (2.1) satisfies statement B. This guarantees that the bulk picture obtained from the AdS/BCFT construction is compatible with causality in the BCFT picture. In the end of this section, we will comment on CFTs defined on a general Σ with possibly non-conformal boundaries as well as their gravity duals.', '4.1 An Assumption on Σ': 'In the discussions below, we make the following assumption. \nAssumption 1. (Σ , γ ij ) can be mapped to R 1 ,d -2 × [0 , ∞ ) or R 1 ,d -2 × [0 , l ] via a Weyl transformation \nγ ij ( x ) → Ω 2 ( x ) γ ij ( x ) , (4.1) \nwhere l is a constant. \nThis assumption restricts the geometry to a very narrow class. Indeed, we will see that this assumption, though sufficient, is not a necessary condition to prove statement B later. However, let us explain why this assumption is made.', 'BCFT vs. Locally CFT with Boundaries': 'First of all, let us distinguish two different concepts: boundary conformal field theory (BCFT) and locally CFT with boundaries (LCFTB). \nBCFT is used to refer a theory which is not only defined on a manifold with boundaries, but also maximally preserves the conformal symmetry so that CFT techniques including conformal Ward identity can be applied for its analysis [48]. Therefore, both Σ and the boundary condition on ∂ Σ are highly restricted. \nOn the other hand, there exist manifolds which are not compatible with conformal boundary conditions but can still let a locally CFT 11 live on them with no contradiction. We will call a locally CFT living on a manifold with boundaries a locally CFT with boundaries (LCFTB). By definition, BCFTs are LCFTB while the inverse is not true. \nTo our knowledge, it is still unsolved to give the most general class of manifolds which are compatible with a BCFT. However, Lorentzian BCFT with time-like boundaries is usually discussed in R 1 ,d -2 × [0 , ∞ ) or R 1 ,d -2 × [0 , l ] and should be straightforwardly generalized to any manifold satisfying assumption 1. For example, the corresponding boundary condition for a BCFT defined on such a manifold is examined for d = 2 in [17]. Therefore, we would like to just make assumption 1 to restrict our discussions to a class of Σ which are compatible with a BCFT.', '4.2 Showing Statement B for Generic Configurations': "Now we proceed to prove statement B. We have already seen in section 3 that this is satisfied when the bulk M is a vacuum configuration given by a portion of global AdS. Therefore, in the following, we would like to focus on the case in which the bulk geometry is away from a vacuum configuration by adding matters and gravitational excitations in it. \nWe would like to assume that M satisfies the four conditions given in the Gao-Wald theorem A.0 as well as assumption 1. \nConsider the asymptotic boundary Σ. The region which can receive a signal through the bulk M from p ∈ Σ is A Σ ( p, M ), and the region which can receive a signal through Σ is its causal future J + Σ ( p ). To prove statement B, it is sufficient to show \n∂A Σ ( p, M ) ⊆ J + Σ ( p ) . (4.2) \nLet us firstly see how far we can go without assumption 1. First of all, we can perform an embedding Σ ⊂ Σ ' , M⊂M ' which satisfy \n- · dim(Σ ' ) = dim(Σ) and dim( M ' ) = dim( M ).\n- · M ' is a spacetime with a time-like asymptotic boundary M ' = Σ ' .\n- · M ' satisfies the four conditions given in the Gao-Wald theorem A.0. \nThen the Gao-Wald theorem is applicable to M ' . A consequence is \nA Σ ' ( p, M ' ) ∪ ∂A Σ ' ( p, M ' ) ⊆ J + Σ ' ( p ) , (4.3) \nfor any point p ∈ Σ ' . See [19] for details. \nLet us try to derive (4.2) from (4.3). The way of embedding implies \nA Σ ( p, M ) ⊆ ( A Σ ' ( p, M ' ) ∩ Σ) . (4.4) \nThen we have \n∂A Σ ( p, M ) ⊆ ( A Σ ' ( p, M ' ) ∩ Σ) ∪ ∂ ( A Σ ' ( p, M ' ) ∩ Σ) = ( A Σ ' ( p, M ' ) ∩ Σ) ∪ ( ∂A Σ ' ( p, M ' ) ∩ Σ) ∪ ( A Σ ' ( p, M ' ) ∩ ∂ Σ) = [ A Σ ' ( p, M ' ) ∩ (Σ ∪ ∂ Σ)] ∪ ( ∂A Σ ' ( p, M ' ) ∩ Σ) = ( A Σ ' ( p, M ' ) ∩ Σ) ∪ ( ∂A Σ ' ( p, M ' ) ∩ Σ) = ( A Σ ' ( p, M ' ) ∪ ∂A Σ ' ( p, M ' )) ∩ Σ ⊆ ( J + Σ ' ( p ) ∩ Σ ) . (4.5) \nHere, we used the property \nR 1 ⊆ R 2 = ⇒ ∂R 1 ⊆ R 2 ∪ ∂R 2 (4.6) \nin the first line, the identity relation \n∂ ( R 1 ∩ R 2 ) = ( ∂R 1 ∩ R 2 ) ∪ ( R 1 ∩ ∂R 2 ) (4.7) \nin the second line, and \n( X ∩ Z ) ∪ ( Y ∩ Z ) = ( X ∪ Y ) ∩ Z (4.8) \nin the third line and the fifth line. We also used ∂ Σ ⊂ Σ in the fourth line and (4.3) in the last line. \nSince J + Σ ( p ) ⊆ ( J + Σ ' ( p ) ∩ Σ), in order to derive (4.2) from (4.5), we need to show J + Σ ( p ) = ( J + Σ ' ( p ) ∩ Σ). However, this does not necessarily hold for general Σ. This is because two points p, q ∈ Σ which can be causally connected through Σ ' is not always causally connected through Σ. \nSo far, we have not used assumption 1 yet. Let us then see how assumption 1 can rescue the situation. \nConsider p, q ∈ ∂ Σ connected by a null geodesic γ on ∂ Σ, whose tangent vector is given by u . The Gauss-Weingarten equation relates the covariant derivative ˜ D ˜ m on ∂ Σ and the covariant derivative D i on Σ by \nu i D i u j = ( u ˜ m ˜ D ˜ m u ˜ n ) e j ˜ n -k ˜ m ˜ n u ˜ m u ˜ n n j , (4.9) \nwhere n j is the normal vector to ∂ Σ, pointing toward the ambient space. For a geodesic on ∂ Σ \nu ˜ m ˜ D ˜ m u ˜ n = 0 , (4.10) \nand hence \nu i D i u j = -k ˜ m ˜ n u ˜ m u ˜ n n j . (4.11) \nThis will be a null geodesic in Σ if \nk ˜ m ˜ n u ˜ m u ˜ n = 0 . (4.12) \nThen we will show that (4.12) holds under assumption 1. For R 1 ,d -2 × [0 , ∞ ) or R 1 ,d -2 × [0 , l ], this is trivial. Now consider the Weyl transformation (4.1). Then the extrinsic curvature k ˜ m ˜ n transforms as \nk ˜ m ˜ n → k ' ˜ m ˜ n = 1 Ω k ˜ m ˜ n +( n i ∂ i Ω) h ˜ m ˜ n = ( n i ∂ i Ω) h ˜ m ˜ n . (4.13) \nHere, k ' ˜ m ˜ n is the extrinsic curvature defined on the manifold after the weyl transformation and h ˜ m ˜ n is the induced metric on ∂ Σ. It turns out that the extrinsic curvature is proportional to the induced metric. Therefore, (4.12) holds, i.e. any null geodesic on ∂ Σ is also a null geodesic on Σ. \nSuppose two points p, q ∈ Σ are connected in Σ ' by a causal curve λ p,q which has an overlap with Σ ' \\ Σ, then there exist p ' , q ' ∈ ∂ Σ such that \nλ p,q = λ p,p ' + λ p ' ,q ' , + λ q ' ,q , (4.14) \nand \nλ p,p ' , λ q,q ' ∈ Σ . (4.15) \nFurther more, there exists a causal curve λ ' p ' ,q ' ∈ ∂ Σ which connects p ' and q ' , otherwise q ' would be outside of the light cone of p ' and lead to a contradiction with the fact that any null geodesic on ∂ Σ is also a null geodesic on Σ. In one word, any two points p, q ∈ Σ which are causally connected through Σ ' must be causally connected through Σ, and hence \nJ + Σ ( p ) = ( J + Σ ' ( p ) ∩ Σ) . (4.16) \nCombining (4.16) with (4.5), we obtain (4.2). Therefore, statement B is shown.", '4.3 Comments on More General Σ': 'We have seen that causality compatibility between the bulk picture and the BCFT picture can be proven for Σ satisfying assumption 1. Let us exclude this assumption here and see what happens. Inspired by the proof given in section 4.2, it is convenient to consider the following two cases. \n- · Case 1: For any null vector u ˜ m on ∂ Σ, k ˜ m ˜ n u ˜ m u ˜ n ≤ 0.\n- · Case 2: There exists a null vector u ˜ m on ∂ Σ such that k ˜ m ˜ n u ˜ m u ˜ n > 0. \nAs we have already seen in section 4.2, Σ which satisfies assumption 1 saturates the condition in case 1. \nFigure 8: Time slices of (2+1)D static Σ with different extrinsic curvatures on ∂ Σ. The left, middle, right figure show k ˜ m ˜ n u ˜ m u ˜ n = 0, k ˜ m ˜ n u ˜ m u ˜ n < 0 and k ˜ m ˜ n u ˜ m u ˜ n > 0, respectively. Consider sending a signal from ∂ Σ to another location on ∂ Σ. The shortest paths are shown by red arrows. When k ˜ m ˜ n u ˜ m u ˜ n = 0, the shortest path is a null geodesic on ∂ Σ which is also a null geodesic in Σ. When k ˜ m ˜ n u ˜ m u ˜ n < 0, the shortest path is a null geodesic in the ambient Σ \\ ∂ Σ. When k ˜ m ˜ n u ˜ m u ˜ n > 0, the shortest path is a null geodesic on ∂ Σ which is not a geodesic in Σ. \n<!-- image --> \nIn general, the causal structure of Σ is determined by null geodesics in Σ and those on ∂ Σ. Figure 8 shows one simple example of Σ for each of k ˜ m ˜ n u ˜ m u ˜ n = 0, k ˜ m ˜ n u ˜ m u ˜ n < 0 and k ˜ m ˜ n u ˜ m u ˜ n > 0. \nIn case 1, one can show that any two points on any null geodesic on ∂ Σ can be connected by a time-like or null geodesic in Σ. Therefore, the causal structure of Σ is fully determined by null geodesics in Σ. As a result, one can simply apply the Gao-Wald theorem to show statement B as in section 4.2. \nIn case 2, however, there exist two points on ∂ Σ which can be connected by a null geodesic on ∂ Σ but not by any time-like or null geodesic in Σ. In this case, Gao-Wald theorem is not directly applicable and more inputs may be necessary to prove statement B.', 'Towards Gravity Dual for Locally CFT with General Boundaries': 'The AdS/BCFT construction, i.e. constructing a gravity dual for a BCFT defined on a given Σ via finding an on-shell configuration of (2.1) satisfying the Neumann boundary condition (2.4) on the end-of-the-world brane Q , is originally proposed for Σ compatible with \na BCFT [11,12]. A natural question is whether this construction can be extended to Σ with general ∂ Σ. \nThe answer is no. It is pointed out in [49] that, assuming Σ is a portion of R d ( d ≥ 3) with one connected piece of boundary, the only configurations of ∂ Σ compatible with the above construction are R d -1 and S d -1 . The reason is because the Neumann boundary condition (2.4) is too strong. \nTherefore, the Neumann boundary condition (2.4) should be modified to be compatible with more general ∂ Σ. This is reasonable since (2.4) is originally proposed to maximally preserve the corresponding symmetries in the bulk, and there is no reason to expect it for less symmetric cases. There are two totally different recipes on how to change (2.4) for it to be compatible with more general Σ. One is to introduce matter fields on the brane Q [32] by hand. Another one is to impose a mixed type of boundary conditions on Q [50, 51]. \nTo check which one (either of the existing recipes or a new one) should be picked, we propose that the causality compatibility between Σ and M should be regarded as a principle to rule out inappropriate recipes. We will not go deep into it, but this is expected to be an interesting future direction.', '5 Causal Structure in the Intermediate Picture': 'In this section, we discuss the causal structure in the intermediate picture for general configurations. Since the spacetime geometry associated to the intermediate picture is composed from the asymptotic boundary Σ and the brane Q , we need to study the causal structure for three cases: sending a signal within Σ, within Q , and across Σ and Q . The results for these three cases are given by statement B, C, D, respectively. \nStatement B has already been studied in section 4.2. As for statement D, we have explicitly checked that the bulk introduces superluminal propagation in pure AdS case in section 3. Besides, we have shown the existence of superluminal propagation without using the specific properties of the vacuum configuration. The only ingredient needed is the existence of the co-dimension 2 defect, which is naturally introduced by ∂ Σ. Thus, presence of superluminal propagation in the intermediate picture, i.e. statement D, has already been shown for general configurations. \nTherefore, what we need to do is to show statement C for general configurations. The analysis given here is parallel to that in [23], where causality in brane universes is discussed. \nAfter that, we will summarize the causal structure in the intermediate picture of double holography, and discuss its relations to other holographic setups.', '5.1 Showing Statement C for General Configurations': 'Here, we show that any null geodesic on Q is also a null geodesic in M , under the Neumann boundary condition (2.4). Then, it follows straightforwardly that 12 two spacetime points which are not causally connected on Q are not causally connected in M either (statement C). \nConsider an arbitrary curve on the brane Q and use u a to denote its tangent vector. Then the covariant derivative on the brane D a and that in the bulk ∇ µ can be related by the Gauss-Weingarten equation as \nu µ ∇ µ u ν = ( u a D a u c ) e ν c -K ab u a u b n ν (5.1) \nwhere n ν is the normal vector to Q . Here, K ab is the extrinsic curvature of the brane. Taking the trace of (2.4) and substituting it back, we have \nK ab = -1 d -1 Th ab . (5.2) \nFor a geodesic on the brane, we have \nu a D a u b = 0 . (5.3) \nMoreover, for a null geodesic, K ab u a u b ∝ h ab u a u b = 0 which follows from Eq. (5.2), and hence \nu µ ∇ µ u ν = 0 . (5.4) \nIn other words, a null geodesic on the brane Q is also a null geodesic in the bulk M . Note that the argument so far is extremely powerful since the Neumann boundary condition (2.4) is almost the only requirement.', '5.2 Causal Structure and Relations to Other Setups': 'So far, we have shown that all three statements B , C, and D hold for generic configurations in double holography. Combining the three statements, to be compatible with the bulk picture, the effective theory in the intermediate picture should have the following property: \n- · Let p and q be two points which are space-like separated on Σ ∪ Q . Then influencing q by adding perturbations on p is possible if and only if p ∈ Σ ∧ q ∈ Q or p ∈ Q ∧ q ∈ Σ. \nThis is the causal structure of the intermediate picture in double holography. \nIn the following, the relations and differences from causal structures in other holographic models are discussed.', 'v.s. the Gao-Wald theorem': 'Although statement C and statement B are both important elements in the causal structure of the intermediate theory and look similar to each other, the mechanisms behind them are very different. \nAs we have seen in the section 5.1, the key ingredient which guarantees the causality within brane Q and the bulk M is the Neumann boundary condition (2.4) imposed on Q . We would like to emphasize that neither restrictions on the bulk matter nor the on-shell condition of M is required. \nOn the other hand, the Gao-Wald theorem which guarantees the no-shortcut statement in AdS/(B)CFT has a mechanism distinct from this. With Dirichlet boundary condition imposed on Σ, the most significant point of the Gao-Wald theorem is that it requires M to be on-shell and the bulk matter to satisfy ANEC.', 'v.s. the T ¯ T -deformed CFT/cutoff AdS correspondence': 'Pulling the boundary theory into the bulk in holographic duality causes a nonlocality in general. One of the most famous examples is the cutoff AdS. Consider a global AdS 3 , make a finite radial cutoff and regard the cutoff surface as the manifold on which the boundary theory is defined. Then one can find that there exist two points which are space-like separated on the boundary manifold while causally connected in the bulk [52-54]. Therefore, the corresponding boundary theory should contain superluminal information propagations. It is proposed in [52] that the boundary theory corresponding to a cutoff AdS 3 is a T ¯ T -deformed \nCFT [55-57] which indeed show a superluminal propagation pattern compatible with the bulk side. \nIn this sense, it is not surprising that the intermediate theory in the double holography has a nonlocal causal structure since the boundary ∂ M = Σ ∪ Q in this case is also pulled into the bulk. Instead, the truly special point is that no short cut can be made when sending a signal from Q to Q , even if Q is pulled into the bulk. This is what the Neumann boundary condition (2.4) causes. Contrast to this, in cutoff AdS, Dirichlet boundary condition is imposed on the cutoff surface.', 'v.s. dS/flat brane-world holography': 'In this paper, we are focusing on the Karch-Randall brane Q , i.e. the asymptotically AdS brane intersecting the asymptotic boundary Σ at a time-like corner. The tension for such a brane is restricted to -( d -1) /L < T < ( d -1) /L . On the other hand, the analysis in section 5.1 has no restrictions on T . Therefore, the statement C is also applicable to asymptotically dS brane realized by \| T \| > ( d -1) /L , and asymptotically flat brane realized by \| T \| = ( d -1) /L . Implications for the brane-world cosmology is discussed in [23].', 'v.s. brane with matter localized on it': 'We would like to comment on what happens if one put matters on the brane Q by hand. The presence of matters changes the Neumann boundary condition (2.4) as 13 \nK ab -Kh ab -Th ab +8 πG N T Q ab = 0 . (5.5) \nHere, T Q ab is the (localized) matter stress-energy tensor on the brane. Taking the trace of (5.5), we obtain \nK = -d d -1 T -8 πG N d -1 T Q (5.6) \nwith T Q = T Q ab h ab . Substituting it back to (5.5), we have \nK ab = -1 d -1 Th ab -8 πG N ( T Q ab -T Q d -1 h ab ) . (5.7) \nNow we impose null energy condition on the brane \nT Q ab u a u b ≥ 0 . (5.8) \nHere u a is an arbitrary null vector on the brane. Then we have \nK ab u a u b = -8 πG N T Q ab u a u b ≤ 0 , (5.9) \nwhich implies that brane is concave in null direction (see the middle panel of figure 8 for example). Therefore, there exists a shortcut in the bulk. A physical interpretation of this consequence is that massive objects on the brane bends the brane and this creates shortcuts. This is discussed in [23] as the solution to the horizon problem in the context of brane cosmology.', '6 Commutators in the Intermediate Picture': 'Until now, we have studied causal structures determined from the geometry of the background manifold. We have found that, the bulk causal structure is compatible with BCFT causality. On the other hand, a causality violation is expected in the intermediate theory T int Σ ∪ Q for it to be compatible with the bulk causal structure. In this context, causality of a theory T N defined on manifold N is a property that a signal cannot be sent between two space-like separated points in T N . This notion of causality does not explain the fundamental mechanism underlying this property. \nOn the other hand, in the context of quantum many body systems (including QFTs, spin systems, etc.), there is a more fundamental notion of causality called microcausality. Microcausality of a theory T N is a property that (gauge invariant) operators commute outside the lightcone: \n[ O ( p ) , O ( p \' )] = 0 if p and p \' are spacelike separated . (6.1) \nIn this context, the previous notion of causality is often called macrocausality. Note that both macrocausaility and microcausality depends on the details of the theory. The difference is that microcausality says more about the details. \nIn this section, we examine the commutator of the operators in the intermediate picture living on ∂ M = Σ ∪ Q . Here, we fix the bulk configuration M to be the one considered \nin section 3, i.e. a portion of pure AdS d +1 with an end-of-the-world brane, as a concrete example. We calculate the vacuum expectation value of the commutator \n〈 [ O ( p ) , O ( p \' )] 〉 int ∂ M = 〈O ( p ) O ( p \' ) 〉 int ∂ M -〈O ( p \' ) O ( p ) 〉 int ∂ M , (6.2) \nand see its behavior. In the following, we will use the ( z, t, x ) coordinate introduced in (3.13) and the ( µ, t, y, ξ ) coordinate introduced in (3.16) to describe a point in the bulk M . Accordingly, the ( t, x ) coordinate and the ( t, y, ξ ) are used to describe a point on Σ or Q . \nWe would like to compute (6.2) using the holographic dictionary. In the usual AdS/CFT correspondence, the extrapolate dictionary [58] \n〈O ( t, x ) O ( t \' , x \' ) 〉 CFT ∝ lim z,z \' → 0 z -∆ z \'-∆ 〈 φ ( z, t, x ) φ ( z \' , t \' , x \' ) 〉 bulk , (6.3) \nis often used for calculation of the correlation functions. Here, O is a primary operator in the boundary CFT with scaling dimension ∆, and φ is its corresponding field in the bulk. The left-hand side is the Wightman function of O in the boundary CFT and the right-hand side is the corresponding Wightman function in the bulk. In the following, we consider φ to be a bulk scalar field with mass m for simplicity. In this case, the scaling dimension is given by ∆ = d/ 2 + √ d 2 / 4 + m 2 . \nOn the other hand, in our setup, the bulk M is a portion of pure AdS, and its boundary ∂ M is composed by an end-of-the-world brane Q placed at µ = µ ∗ = π/ 2 -θ ∗ plus a portion of the asymptotic boundary, instead of just a single asymptotic boundary as in the usual AdS/CFT. See figure 7 for a sketch. A holographic dictionary applicable for our setup has already been proposed in [38, 39]. For example, a correlation function between p ∈ Σ and p \' ∈ Q can be computed by \n〈O ( p ) O ( p \' ) 〉 int ∂ M ∝ lim z → 0 z -∆ θ \'-∆ 〈 φ ( z, t, x ) φ ( µ \' , t \' , y \' , ξ \' ) 〉 bulk M \| θ \' = θ ∗ , (6.4) \nif we neglect the back reaction by the scalar field and look at the limit where the brane Q is close to the asymptotic boundary ( θ ∗ glyph[lessmuch] 1). Here, θ = π/ 2 -µ (see equation (3.17)). Of course, we can also consider the case in which both of the two points live on Σ or Q . This dictionary, together with the usual one (6.3), implies that computing the bulk Wightman function is sufficient to obtain the correlation function in the intermediate picture. \nTo calculate the bulk Wightman function, we introduce the conformal invariant \nζ = 2 〈 X -X \' , X -X \' 〉 +2 = 2 zz \' z 2 + z \' 2 -( t -t \' ) 2 +( x -x \' ) 2 . (6.5) \n<latexit sha1\\_base64="Mj4Ed1RqgPpLVzr7Jq2uc7aNpAM=">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</latexit> \n<latexit sha1\\_base64="lvugvfJ5A6IOj0e0R2g/daXWtiE=">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</latexit> \n<latexit sha1\\_base64="0CYflWJvD7ghGAv/IkhnKzQ6Ai0=">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</latexit> \n<latexit sha1\\_base64="AGJQ2qIOV41dp/1oQSKcFurS9ko=">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</latexit> \n<latexit sha1\\_base64="yXDgnQqurZZbLZmoS4+jO7WQ2kc=">AAACaHichVFNLwNBGH66vqo+WhwQl4YQp5r2gIhDExdHbS1NWpHdNZh0u7vZnTZpGz+AiyPiRCIifoaLP+DQn4AjiYuDt7tNBME7mZlnnnmfd56Z0R1TeJKxZkjp6Ozq7gn3Rvr6BwajsaHhDc+uuAZXDdu03byuedwUFlelkCbPOy7XyrrJN/XSSmt/s8pdT9jWuqw5fKus7VliVxiaJEot1rnUtmNTLMH8iP8EyTaYSs89Lh++RmbX7Ng1itiBDQMVlMFhQRI2ocGjVkASDA5xW2gQ5xIS/j7HASKkrVAWpwyN2BKNe7QqtFmL1q2anq826BSTukvKOKbZA7thL+ye3bJH9v5rrYZfo+WlRrMeaLmzHT0ay739qyrTLLH/qfrTs8QuFn2vgrw7PtO6hRHoq/WTl9xSdroxwy7ZM/m/YE12Rzewqq/GVYZnzxGhD0h+f+6fYCOVSM4nUhn6iXUEEcYEJjFL772ANFaxBpXOFTjGKc5CT0pMGVXGg1Ql1NaM4Esokx+/lY8Q</latexit> \nFigure 9: The analytic structure of the bulk correlation function (6.6). There are two branchcuts on the real axis with ζ > 1 and ζ < 0, respectively. The branch-cut with ζ > 1 is introduced by the hypergeometric function 2 F 1 and the branch-cut with ζ < 0 is introduced by ζ ∆ . Using equation (6.5), we can show that region ζ > 1 or ζ < 0, 0 < ζ < 1, ζ = 1 correspond to time-like, space-like, null separated two points in the bulk, respectively. \n<!-- image --> \nHere, 〈· , ·〉 is the inner product associated to the embedding space R 2 ,d and X A ( X \' A ) is the coordinate in R 2 ,d (see appendix A.1 for details). Using the conformal invariant simplifies the derivation of correlation functions in the bulk and the final result is [59] \n〈 φ ( z, t, x ) φ ( z \' , t \' , x \' ) 〉 bulk M ∝ ζ ∆ 2 F 1 (∆ / 2 , 1 / 2 + ∆ / 2 , 1 + (∆ -d/ 2); ζ 2 ) , (6.6) \nwhere ∆ = d/ 2 + √ d 2 / 4 + m 2 . Figure 9 shows the analytic structure of (6.6) on ζ plane. We let the branch-cut of ζ ∆ run in the range ( -∞ , 0) as in [60]. This analytic structure is important for determining the microscopic causal structure in the bulk [60]. \nLet us calculate the commutator for three cases: p, p \' ∈ Σ; p, p \' ∈ Q ; and p ∈ Σ , p \' ∈ Q . We introduce the following quantity for later use, \n∆ β ( p, p \' ) ≡ -( t -t \' ) 2 +( x -x \' ) 2 ( p, p \' ∈ Σ) -( t -t \' ) 2 +( ξ -ξ \' ) 2 +( y -y \' ) 2 ( p, p \' ∈ Q ) -( t -t \' ) 2 +( ξ -ξ \' ) 2 +( y + y \' ) 2 ( p ∈ Σ , p \' ∈ Q ) . (6.7) \nWith such a ∆ β , we have \n∆ β > 0 ⇐⇒ p and p \' are space -like separated on ∂ M . ∆ β = 0 ⇐⇒ p and p \' are null separated on ∂ M . (6.8) ∆ β < 0 p and p \' are time like separated on ∂ . \nwhere ∂ M = Σ ∪ Q . \n⇐⇒ -M \nThe case p, p \' ∈ Σ is the same as the usual AdS/CFT and can be calculated straightforwardly. From the definition of ζ (6.5), we have ζ → 0 near the asymptotic boundary region z, z \' ∼ 0. Now the asymptotic behavior of the hypergeometric function tells us that the behavior of the bulk correlator around the asymptotic boundary is like \n〈 φ ( z, t, x ) φ ( z \' , t \' , x \' ) 〉 bulk M → ζ ∆ = ( 2 zz \' z 2 + z \' 2 -( t -t \' ) 2 +( x -x \' ) 2 + iglyph[epsilon1] ) ∆ . (6.9) \nHere, we put iglyph[epsilon1] ( glyph[epsilon1] glyph[lessmuch] 1) in this way to correctly obtain the Wightman function 14 . Then the dictionary (6.3) leads to \n〈O ( p ) O ( p \' ) 〉 int p,p \' ∈ Σ ∝ ( 1 -( t -t \' ) 2 +( x -x \' ) 2 + iglyph[epsilon1] ) ∆ . (6.10) \nUsing the above expression of Wightman function, we obtain the commutator for p, p \' ∈ Σ as \n〈 [ O ( p ) , O ( p \' )] 〉 int p,p \' ∈ Σ ∝ ( 1 -( t -t \' ) 2 +( x -x \' ) 2 + iglyph[epsilon1] ) ∆ -( 1 -( t -t \' ) 2 +( x -x \' ) 2 -iglyph[epsilon1] ) ∆ . (6.11) \nWe use the fact that 〈O ( p \' ) O ( p ) 〉 is given by the complex conjugate of 〈O ( p ) O ( p \' ) 〉 . Since the branch-cut of ζ ∆ runs on the negative half of the real axis, (6.11) vanishes for \n∆ β = -( t -t \' ) 2 +( x -x \' ) 2 > 0 . (6.12) \nIn other words, the commutator vanishes for space-like separated points in Σ. Therefore, microscopic causality is preserved within Σ. \nNow we move on to the case p, p \' ∈ Q . In this case, it is more convenient to use the ( µ, t, y, ξ ) coordinate. When the two bulk points are sitting at µ = µ ∗ = π/ 2 -θ ∗ , the conformal invariant ζ turns out to be \nζ = 2 cos 2 ( π 2 -θ ∗ ) yy \' ∆ β +2cos 2 ( π 2 -θ ∗ ) yy \' , (6.13) \nwhere \n∆ β = -( t -t \' ) 2 +( ξ -ξ \' ) 2 +( y -y \' ) 2 . (6.14) \nThen the dictionary (6.4) gives the Wightman function on the brane as \n〈O ( p ) O ( p \' ) 〉 int p,p \' ∈ Q ∝ ( 2 yy \' ∆ β + iglyph[epsilon1] +2 yy \' sin 2 θ ∗ ) ∆ × 2 F 1 ( ∆ / 2 , 1 / 2 + ∆ / 2 , 1 + (∆ -d/ 2); ( ∆ β + iglyph[epsilon1] 2 yy \' sin 2 θ ∗ +1 ) -2 ) ≡ G Q + ( t, y, ξ , t \' , y \' , ξ \' ) (6.15) \nSubstituting to equation (6.2), we obtain \n〈 [ O ( p ) , O ( p \' )] 〉 int x,x \' ∈ Q ∝ G Q + ( t, y, ξ , t \' , y \' , ξ \' ) -[ G Q + ( t, y, ξ , t \' , y \' , ξ \' ) ] ∗ . (6.16) \nThe behavior of this function is determined by the analytic structure of (6.15), which is the same as that of the bulk Wightman function (see figure 9). Using (6.13), relation between the value of ζ and the causal relation of two points p, p \' on the brane (time-like for ∆ β < 0, space-like for ∆ β > 0, and null for ∆ β = 0) is \n∆ β < 0 → ζ > 1 , ζ < 0 , (6.17) \n∆ β = 0 → ζ = 1 , (6.18) \n∆ β > 0 → 0 < ζ < 1 . (6.19) \nCombining with figure 9, we observe that the commutator vanishes for ∆ β > 0, i.e. two space-like separated points on the brane. Thus microscopic causality is preserved within Q . \nFinally, let us consider the commutator for p ∈ Σ , p \' ∈ Q . In this case, the conformal invariant ζ can be written as \nζ = 2 zy \' cos ( π 2 -θ ∗ ) z 2 +∆ β -2 yy \' (1 -sin ( π 2 -θ ∗ ) ) . (6.20) \nwhere \n∆ β = -( t -t \' ) 2 +( ξ -ξ \' ) 2 +( y + y \' ) 2 . (6.21) \nSubstituting this to (6.9) and using the dictionary (6.4) by taking z → 0, we obtain \n〈O ( p ) O ( p \' ) 〉 int p ∈ Σ ,p \' ∈ Q ∝ ( 1 ∆ β -2 yy \' (1 -cos θ ∗ ) + iglyph[epsilon1] ) ∆ . (6.22) \nNote that the contribution from the hypergeometric function disappears thanks to the z → 0 limit, unlike the p, p \' ∈ Q case. Since 〈O ( p \' ) O ( p ) 〉 is the complex conjugate of 〈O ( p ) O ( p \' ) 〉 , commutator (6.2) is given by \n〈 [ O ( p ) , O ( p \' )] 〉 int p ∈ Σ ,p \' ∈ Q ∝ ( 1 ∆ β -2 yy \' (1 -cos θ ∗ ) + iglyph[epsilon1] ) ∆ -( 1 ∆ β -2 yy \' (1 -cos θ ∗ ) -iglyph[epsilon1] ) ∆ . (6.23) \nRecall that we take the branch-cut of ζ ∆ to run in the range ( -∞ , 0). Thus, the commutator vanishes in the region \n∆ β -2 yy \' (1 -cos θ ∗ ) > 0 , (6.24) \nwhich is strictly smaller than the space-like region indicated by the geometry of ∂ M \n∆ β > 0 . (6.25) \nTherefore, the commutator does not necessarily vanish even when the two points are spacelike separated on ∂ M = Σ ∪ Q . This clearly shows a breakdown of the microscopic causality and an existence of some nonlocal interaction 15 between Q and Σ in the intermediate picture. \nComparing (6.24) and (3.19), we can see that they are in exactly the same form, and it is straightforward to see that \n〈 [ O ( p ) , O ( p \' )] 〉 int vanishes p and p \' cannot communicate through the bulk . (6.26) \n⇐⇒ M \nThis fact implies that the origin of the microscopic causality violation (or emergence of nonlocal interaction between Q and Σ) is the causal structure of the bulk M . \nWe will use \'violation of causality\' and \'nonlocality\' interchangeably when talking about the effective theory in the intermediate picture, T int Σ ∪ Q .', '7 Nonlocality and Subregions in Double Holography': 'So far, we have seen that the effective theory in the intermediate picture should have a violation of causality and hence nonlocality to be compatible with the causal structure in the bulk picture. \nIn this section, we are going to discuss physical consequences and features of the nonlocality observed in the intermediate picture. Starting from a discussion on the notion of domain of dependence when nonlocality is involved, we will review typical subregions in the usual AdS/CFT correspondence, such as causal wedges and entanglement wedges. After that, we will examine the relation between some typical subregions in double holography. In particular, we will see that a straightforward analog of subregion/subregion duality in the AdS/CFT correspondence does not serve as a subregion/subregion duality in the intermediate/bulk correspondence. \nIn the end of this section, we will point out that the nonlocality appearing in the intermediate picture is sensitive at IR (or long range). We will compare it with other known nonlocality in quantum gravity, and give an intuitive explanation about how it can arise in the gravitational path intergral language.', '7.1 Breakdown of Domain of Dependence': 'The nonlocal nature appearing in the intermediate picture prompts us to reconsider the notion of domain of dependence . To proceed, let us split the usual notion of domain of dependence into two. In the following, we use N to denote a d -dimensional spacetime and suppose there is a field theory T N living on it. \nFor the same subregion A , consider the set consisting of all points p ∈ N with the property that every local observable at p can be determined from the initial condition on A . Let us call it the effective domain of dependence of A and denote it as D E ( A ). \nFor a space-like subregion A ⊂ N , consider the set consisting of all points p ∈ N with the property that every causal curve through p intersects A . Let us call it the geometrical domain of dependence of A and denote it as D G ( A ). \nNote that D G ( A ) only depends the geometry of N , while D E ( A ) also depends on the effective theory T N living on it. If T N is a local relativistic theory, D G ( A ) and D E ( A ) degenerate to the conventional domain of dependence. In general, however, D G ( A ) = D E ( A ).', 'D G ( A ) = D E ( A ) in the intermediate picture': "glyph[negationslash] \nThe mismatch between the geometrical domain of dependence and the effective domain of dependence also occurs in the intermediate picture of double holography. Let us see it using the same configuration as in section 3 and section 6. See figure 3, 5 and 7 for sketches. In this case, the manifold we consider is Σ ∪ Q and the effective theory on it T int Σ ∪ Q should be \nglyph[negationslash] \ncompatible with the causal structure in the bulk picture. Here, the index 'int' stands for the intermediate picture. \nFor simplicity, we focus on d = 2 and use the ( ρ, t, y ) coordinate introduced in (3.11) to parameterize the bulk M . Accordingly, ∂ M = Σ ∪ Q can be parameterized by ( t, y ). In the following, we will use y > 0 to parameterize Σ, and y < 0 to parameterize Q . This is the same as what we did when computing (3.18) and (3.19). \nLet us take the subregion A as a finite interval of Σ given by 0 < a ≤ y ≤ b on time slice t = 0. Obviously, the geometrical domain of dependence D int G ( A ) is the region surrounded by t = ± ( y -a ) and t = ± ( y -b ). \nLet us then move on to determine the effective domain of dependence D int E ( A ). Note that while the geometrical domain of dependence D int G ( A ) is determined by geometrical inputs from the boundary manifold Σ ∪ Q , the effective domain of dependence D int E ( A ) should be determined by geometrical inputs from the bulk manifold M , since we expect the effective theory in the intermediate picture is equivalent to a local theory in the bulk picture. For any point q ∈ (Σ ∪ Q ) \\ A , one can always find a point p ∈ Q such that p can communicate with 16 q through the bulk M but not with any point on A . Let us explicitly give such a p for different types of q for the the vanishing tension T = 0 case. 17 The coordinates of p and q are denoted as ( t p , y p ) and ( t q , y q ), respectively. It is sufficient to consider t q ≥ 0 without loss of generality. \n- · For 0 < t q ≤ y q , ( t p , y p ) = ( t q -y 2 q /t q , t q -y 2 q /t q ) can send a signal to q but cannot communicate with any point on A .\n- · For \| y q \| ≤ t q , ( t p , y p ) = (0 , 0) can send a signal to q but cannot communicate with any point on A .\n- · For y q < 0 < t q ≤ \| y q \| or t q = 0, ( t p , y p ) = ( t q , y q ) can send a signal to q but cannot communicate with any point on A . \nIn other words, any point not living on A cannot be determined only from A . Therefore, D int E ( A ) = A . \nAs a result, for the finite interval considered above, D int G ( A ) ⊃ D int E ( A ) = A . Note that D int E ( A ) is codimension-1 while D int G ( A ) is codimension-0 with respect to Σ ∪ Q . The breakdown of the notion of conventional domain of dependence in the intermediate picture prompts us to reconsider naive discussions assuming D int G ( A ) = D int E ( A ) including state matching with other pictures, causal wedge reconstruction and entanglement wedge reconstruction. These topics cannot be totally split from each other, but we would like to start by comparing A in the intermediate picture, A in the BCFT picture and the corresponding region in the bulk picture.", '7.2 Subregions and States Associated to A': 'In this subsection, we focus on a single interval A on Σ \\ ∂ Σand consider the physics associated to it in the three pictures of double holography. \nTo proceed, let us distinguish two notions. Consider a manifold N and a theory T N living on it. A subregion D ⊆ N is a geometrical object, while the information contained in D should be determined by also taking the theory living on it, T N , into account. Let us use ( D , T N ) to denote the information contained in D . Although D is often used to refer to ( D , T N ), we would like to make a distinction here to avoid confusion.', 'Subregions and states in the BCFT picture': 'With this in mind, let us consider the domain of dependence of A in the BCFT picture. First of all, the BCFT picture has a BCFT T BCFT Σ living on Σ. Since T BCFT Σ is local and relativistic, D BCFT G ( A ) and D BCFT E ( A ) coincide. Therefore, we just use D BCFT ( A ) to denote it and call it the domain of dependence. Almost by definition, the reduced density matrix ρ BCFT A contains the same information as ( D BCFT ( A ) , T BCFT Σ ) . Let us denote this as \nρ BCFT A glyph[similarequal] ( D BCFT ( A ) , T BCFT Σ ) (7.1) \nwith a slight abuse of notation.', 'Subregions in the bulk picture': 'Let us then go to the bulk picture and consider subregions associated to A . First of all, the causal wedge of A , C ( A ), is defined as the intersection of the causal future and the causal past of D BCFT ( A ) in the bulk M : \nC ( A ) = J + ( D BCFT ( A ) ) ∩ J -( D BCFT ( A ) ) . (7.2) \nSecond, the entanglement wedge [62,63] of A , E ( A ), is defined as the bulk geometrical domain of dependence of a space-like bulk region surrounded by A and its Hubeny-RangamaniTakayanagi (HRT) surface [64-66] γ ( A ). Here, the HRT surface of A , γ ( A ), is a space-like bulk surface which satisfies the following conditions: \n- · It is codimension-2 with respect to the bulk manifold M .\n- · It shares boundaries with A , i.e. ∂γ ( A ) = ∂A .\n- · It is homologous to A , i.e. one can bring A to γ ( A ) by a continuous deformation in the bulk. In particular, the end-of-the-world brane Q is regarded as trivial.\n- · It is an extremal surface.\n- · If there are multiple candidates satisfying these conditions, γ ( A ) is given by the one with the minimal area. \nDue to the treatment of the end-of-the-world brane Q , there is a chance that γ ( A ) touches Q and E ( A ) intersects Q , as shown in the right panel of figure 10. In this case, there exists a space-like region I ( A ) ⊂ Q such that D int G ( I ( A )) coincides E ( A ) ∩ Q . This is guaranteed by statement C. Such an I ( A ) is called the island of A [6, 13, 14]. \nIt is known that the causal wedge is contained in the entanglement wedge [62,63,67], i.e. \nC ( A ) ⊆ E ( A ) , (7.3) \nTherefore, information in the causal wedge can be recovered from the information in the entanglement wedge. Let us denote this as \n( C ( A ) , T bulk M ) glyph[precedesequal] ( E ( A ) , T bulk M ) . 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sha1\\_base64="4Rp8bCxxpZGJIkas74cUsvotfEA=">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</latexit> \nFigure 10: Two examples of entanglement wedges for a single interval A (shown in red). For simplicity, here we show the projection on a time slice in AdS 3 . The asymptotic boundary Σ is shown in grey, and the end-of-the-world brane Q is shown in blue. The HRT surface γ ( A ) is shown in orange, and the entanglement wedge E ( A ) is shaded in yellow. In the left panel, γ ( A ) does not touch the brane Q . In the right panel, γ ( A ) touches the brane Q and hence an island I ( A ) (shown in green) arises. \n<!-- image --> \nA', 'Subregions and states in the intermediate picture': "Returning to the intermediate picture, geometrically \nD int G ( A ) ⊃ D int E ( A ) = A. (7.5) \nTherefore, one can recover ( D int E ( A ) , T int Σ ∪ Q ) from ( D int G ( A ) , T int Σ ∪ Q ) . On the other hand, by definition of D int E ( A ), ( D int G ( A ) , T int Σ ∪ Q ) cannot be recovered from ( D int E ( A ) , T int Σ ∪ Q ) . Let us denote this relation as \nρ int A glyph[similarequal] ( D int E ( A ) , T int Σ ∪ Q ) ≺ ( D int G ( A ) , T int Σ ∪ Q ) , (7.6) \nwhere the ' glyph[similarequal] ' follows from the definition of D int E . \nNow we have introduced several subregions associated to A in the three pictures. Let us see how they correspond to each other.", 'BCFT picture v.s. Bulk picture: Entanglement wedge reconstruction': 'Subregions in the BCFT picture and the bulk picture are related by the entanglement wedge reconstruction, i.e. \nρ BCFT A glyph[similarequal] ( D BCFT ( A ) , T BCFT Σ ) glyph[similarequal] ( E ( A ) , T bulk M ) glyph[similarequal] ρ bulk E ( A ) . (7.7) \nSee [28,68,69] for proofs and methods for entanglement wedge reconstruction. \n<latexit sha1\\_base64="moqheDmBSlhhXYe3LV8dJJZngkA=">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</latexit> \n<latexit 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Bulk picture: An assumption': 'Since the correspondence between the intermediate picture and the bulk picture is not an ordinary AdS/CFT duality, we do not have a standard way to specify the correspondence between subregions in the two pictures. 18 However, we would like to make the following assumption and argue that it holds. \nAssumption 2. The entanglement wedge E ( A ) in the bulk picture is equivalent to its intersection with the asymptotic boundary and the brane, D int G ( A ) ∪D int G ( I ( A )) , in the intermediate picture, i.e. \n( E ( A ) , T bulk M ) glyph[similarequal] ( D int G ( A ) ∪ D int G ( I ( A )) , T int Σ ∪ Q ) . (7.8) \nNote that I ( A ) can be an empty set. \nIn other words, this assumption argues that a doubly holographic structure holds also for entanglement wedges. Since an entanglement wedge has already provided a correspondence between a gravitational theory in the bulk and a field theory on the asymptotic boundary, the assumption above is just as justified as many other doubly holographic models. In this way, we argue that this assumption holds.', 'Intermediate picture v.s. BCFT picture': 'Let us then compare the intermediate picture and the BCFT picture. Geometrically, \nD BCFT ( A ) = D int G ( A ) ⊃ D int E ( A ) = A. (7.9) \nThe next question is whether ( D BCFT ( A ) , T BCFT Σ ) and ( D int G ( A ) , T int Σ ∪ Q ) contain the same information. Combining (7.7) and (7.8), we have \nρ BCFT A glyph[similarequal] ( D BCFT ( A ) , T BCFT Σ ) glyph[similarequal] ( E ( A ) , T bulk M ) glyph[similarequal] ( D int G ( A ) ∪ D int G ( I ( A )) , T int Σ ∪ Q ) glyph[followsequal] ( D int G ( A ) , T int Σ ∪ Q ) . (7.10) \nMoreover, combining this with (7.6), we have \nρ BCFT A glyph[follows] ρ int A . (7.11) \n(7.11) is an important consequence. It implies that, even for the same subregion A on Σ, the associated state in the BCFT picture and that in the intermediate picture are different from each other. This consequence is consistent with the arguments that ρ BCFT A is a fine-grained state while ρ int A corresponds to a coarse-grained state [70]. Note that (7.11) holds even when A does not have an island, i.e. I ( A ) = ∅ .', '7.3 A Tentative Subregion Duality and its Breakdown': 'In the previous subsection, we argue that a subregion duality between the intermediate picture and the bulk picture, assumption 2, is expected to hold. However, this argument does not cover all the intermediate subregions. For example, for a subregion A ⊂ Σ with a non-vanishing island I ( A ) ⊂ Q , assumption 2 tells us that the bulk dual for the intermediate subregion D int G ( A ∪ I ( A )) is the entanglement wedge E ( A ), but nothing about what the bulk dual of the intermediate subregion A or that of D int G ( A ) is. 19 \nIn the following, we will give a straightforward analogy of the entanglement wedge in the intermediate/bulk correspondence and test if it is qualified as a dual bulk subregion to an intermediate subregion. \nTo clarify the statement, let us introduce some new concepts and terminologies. \nFor a space-like intermediate subregion R ⊂ Σ ∪ Q = ( ∂ M ) which is codimendion-1 with respect to Σ ∪ Q , the tentative HRT surface γ T ( R ) is the minimal extremal surface which is codimension-2 with respect to M , ending on ∂R , and homologous to R . The tentative entanglement wedge E T ( R ) is the bulk domain of dependence of a space-like region surrounded by R and γ T ( R ). See figure 11 for some examples. \nIn other words, tentative HRT surfaces and tentative entanglement wedges are given by applying the definitions of HRT surfaces and entanglement wedge directly to the intermediate picture instead of the (B)CFT picture. We use \'tentative\' to imply that we still need to investigate whether they satisfy important properties owned by conventional HRT surfaces and entanglement wedges. \nNote that a main difference between a tentative HRT surface and a HRT surface is the treatment of brane Q . In the definition of tentative HRT surfaces, Q is treated as a part of the \'boundary\'. Therefore, γ ( R ) cannot touch Q unless R has an overlap with Q . The \ndifference can be clearly observed by comparing figure 10 and figure 11. \nFigure 11: Four examples of tentative entanglement wedges for an intermediate subregion R (shown in red). For simplicity, here we show the projection on a time slice in AdS 3 . The asymptotic boundary Σ is shown in grey, and the end-of-the-world brane Q is shown in blue. The tentative HRT surface γ T ( A ) is shown in orange, and the tentative entanglement wedge E T ( A ) is shaded in yellow. \n<!-- image --> \nSince in the bulk/BCFT correspondence, ( E ( A ) , T bulk M ) glyph[similarequal] ( D BCFT ( A ) , T BCFT Σ ) glyph[similarequal] ρ BCFT A works as a subregion duality for any A ⊂ Σ, there are two natural guesses one may make at first glance: \n- · ( E T ( R ) , T bulk M ) and ( D int E ( R ) , T int Σ ∪ Q ) glyph[similarequal] ρ int A may be equivalent for any intermediate subregion R ⊂ Σ ∪ Q ,\n- · ( E T ( R ) , T bulk M ) and ( D int G ( R ) , T int Σ ∪ Q ) may be equivalent for any intermediate subregion R ⊂ Σ ∪ Q . \nHowever, we can easily see that the first guess fails by considering S, R ⊂ Σ such that ∂S = ∂R and hence E T ( S ) = E T ( R ) while D int E ( S ) = D int E ( R ). 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sha1\\_base64="+RaCL9RWXg/4WinbNg+tKeRAiyQ=">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</latexit> \n<latexit 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sha1\\_base64="+hxP48WAW118YWjvTUw4vREVsFA=">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</latexit> \nFigure 12: An example in which E T ( S ) = E T ( R ) while D int E ( S ) = D int E ( R ). The figure shows the intermediate picture in the vacuum configuration with d = 2. Taking S (green) and R (red) as spatial intervals on Σ with ∂S = ∂R but S = R , we have E T ( S ) = E T ( R ) from the definition of tentative entanglement wedges, while D E ( S ) = S = R = D E ( R ) from the discussion in section 7.1. \n<!-- image --> \nglyph[negationslash] \nglyph[negationslash] \nglyph[negationslash] \nOn the other hand, we have even already seen a support to the second guess: assumption 2 tells us this should hold at least for R = A ∪ I ( A ). In this case, E T ( A ∪ I ( A )) = E ( A ). \nHowever, in spite of all the \'justifications\', the second guess does not works for arbitrary R ⊂ Σ ∪ Q either. To see this, let us firstly review some important properties of conventional entanglement wedges which protect the subregion duality in the usual AdS/CFT correspondence. Then, we will show that these properties are not satisfied by tentative entanglement wedges.', 'C ⊆ E and nesting for entanglement wedges': 'C ⊆ E [26, 63, 71] and the entanglement wedge nesting [62, 63] are known as properties \nthat entanglement wedges should satisfy for the entanglement wedge reconstruction to work in the ordinary AdS/CFT or AdS/BCFT correspondence. \nC ⊆ E states that the causal wedge C ( A ) is completely contained in the entanglement wedge E ( A ) for any A ⊂ Σ. Since C ( A ) is a bulk region which can causally communicate with D (B)CFT ( A ), E ( A ) should at least contain C ( A ) to be dual to D (B)CFT ( A ) [26, 63, 71]. \nEntanglement wedge nesting states that, for any B ⊂ A ⊂ Σ, E ( B ) ⊂ E ( A ). If entanglement wedge nesting was not satisfied, the subregion duality for A \n( E ( A ) , T bulk M ) glyph[similarequal] ( D BCFT ( A ) , T (B)CFT Σ ) (7.12) \nand the subregion duality for B \n( E ( B ) , T bulk M ) glyph[similarequal] ( D (B)CFT ( B ) , T (B)CFT Σ ) (7.13) \nwould not be able to hold at the same time [62,63]. \nC ⊆ E and entanglement wedge nesting are not independent from each other. It is discussed in [67] that entanglement wedge nesting implies C ⊆ E implies locality of the boundary theory T (B)CFT Σ , i.e. \nentanglement wedge nesting = ⇒C ⊆ E = ⇒ locality of T (B)CFT Σ . (7.14)', 'Breakdown of C ⊆ E T and nesting for tentative entanglement wedges': 'Let us then go back to the intermediate/bulk correspondence. Accordingly, we can define C ⊆ E and nesting for tentative entanglement wedges, by simply replacing A ⊂ Σ with R ⊂ Σ ∪ Q in the previous discussions. Let us call the two corresponding statements C ⊆ E T and tentative entanglement wedge nesting, respectively. \nMost straightforwardly, we have already known that T int Σ ∪ Q , the boundary theory in this case, is nonlocal. Therefore, by simply applying (7.14), we have \nnonlocality of T int Σ ∪ Q = ⇒ breakdown of C ⊆ E T = ⇒ breakdown of tentative entanglement wedge nesting . (7.15) \nWe can also confirm the breakdown of C ⊆ E T and tentative entanglement wedge nesting directly. See figure 13 and 14 for a simple example. As a result, ( E T ( R ) , T bulk M ) is not equivalent to ( D int G ( R ) , T int Σ ∪ Q ) in general. \nAs a summary of this subsection, for an arbitrary intermediate subregion R , its tentative entanglement wedge ( E T ( R ) , T bulk M ) is dual to neither its effective domain of dependence \n( D int E ( R ) , T int Σ ∪ Q ) nor its geometrical domain of dependence ( D int G ( R ) , T int Σ ∪ Q ) . More should be investigated to answer the question whether there is a bulk subregion dual to ( D int E ( R ) , T int Σ ∪ Q ) or ( D int G ( R ) , T int Σ ∪ Q ) for an arbitrary intermediate subregion R . 20 We would like to leave this as a future problem. \n<!-- image --> \n<latexit sha1\\_base64="9fic6A8G+DmZBGiKEjiO1nNUJwk=">AAACZHichVHLSsNAFD2Nr1qrrRZBEKRYFFflRgTFlejGpbX2AVVKEkcNTZOQpIVa/AHdKi5cKYiIn+HGH3DRHxDEZQU3LrxNA6Ki3mFmzpy5586ZGdU2dNcjaoWknt6+/oHwYGQoOjwSi4+O5V2r5mgip1mG5RRVxRWGboqcp3uGKNqOUKqqIQpqZa2zX6gLx9Utc8tr2GKnquyb+p6uKR5TmcNyPEVp8iP5E8gBSCGIDSt+g23swoKGGqoQMOExNqDA5VaCDILN3A6azDmMdH9f4AgR1tY4S3CGwmyFx31elQLW5HWnpuurNT7F4O6wMokZeqRbatMD3dEzvf9aq+nX6Hhp8Kx2tcIux44nsm//qqo8ezj4VP3p2cMelnyvOnu3faZzC62rrx+et7PLmzPNWbqiF/Z/SS265xuY9VftOiM2LxDhD5C/P/dPkJ9Py5SWMwupldXgK8KYxDTm+L0XsYJ1bCDH5wqc4BRnoScpKiWk8W6qFAo0CXwJaeoD+kyJ+g==</latexit> \n<latexit 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sha1\\_base64="+hxP48WAW118YWjvTUw4vREVsFA=">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</latexit> \nFigure 13: A simple example showing the breakdown of C ⊆ E T . Consider a d = 2 vacuum setup with a vanishing tension T = 0. The left figure shows the t = 0 slice, and the right figure shows Σ ∪ Q . In this case, Q is perpendicular to Σ, and z -coordinate coincides the minus direction of y -coordinate. Taking subregion R (red) as -4 ≤ y ≤ 8 and t = 0. The tentative entanglement wedge E T ( R ) is shown in yellow. The geometrical domain of dependence D int G ( R ) is the region surrounded by the bold dotted lines in the right picture and goes outside of E T ( R ). 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Consider the identical setup with figure 13 and take a subregion S 1 (shown in pink) such that ∂S 1 = ∂R . By definition, E T ( S 1 ) = E T ( R ) and its intersection with Σ ∪ Q is shown in yellow. Consider S 2 ⊂ S 1 shown in dark green. Its entanglement wedge E T ( S 2 ) intersects Σ ∪ Q at the region shown in light green. Clearly, E T ( S 2 ) glyph[negationslash]⊆ E T ( S 1 ) and hence tentative entanglement nesting breaks down. \n<!-- image -->', '7.4 IR-sensitive Nonlocality and Quantum Gravity': 'Until now, we have discussed the causal structure in double holography, pointed out that the effective theory in the intermediate picture T int Q ∪ Σ should allow (effectively) superluminal information propagation when sending a signal from Q to Σ, and thus contain a nonlocality, in order to be compatible with the bulk picture. This was verified from both the geometrical causal structure (i.e. geodesics) and the microscopic causal structure (i.e. commutators). \nAs a result, the conventional notion of domain of dependence breaks down, and no longer serves as a region associated to a density matrix of a spatial subregion. Accordingly, conventional subregion duality also breaks down. \nAlthough our explicit examples are mostly given in the vacuum background for simplicity, these results hold for any on-shell bulk configurations. Double holography was originally proposed in the study of Hawking radiation. In that context, the intermediate picture, where a black hole lives on Q and can communicate with the heat bath Σ, can be straightforwardly \nregarded as a setup of black hole evaporation. The benefits of double holography is that this \'black hole communicating with heat bath\' setup in the intermediate picture is related to well-defined BCFT picture and classical bulk picture. In this context, the intermediate picture can be regarded as an effective theory of quantum gravity defined on a classical geometrical background Q ∪ Σ. This fact implies that, when defined on a classical geometrical background, the effective theory of quantum gravity should have a nonlocal effect, at least for the intermediate picture in double holography. \nLet us emphasize one key feature of this nonlocal effect in T int Q ∪ Σ . The fact that information propagation within Q or Σ cannot be superluminal implies that the nonlocal effect can be ignored if we zoom in to the UV. On the other hand, if we zoom out to the IR, the nonlocal effect becomes significant. For the latter feature, we say that the intermediate picture is IR-sensitive . See figure 15. There is, however, a special region in which the nonlocal effect is significant at any length scale. This region is ∂ Σ = ∂Q . \nFigure 15: The IR-sensitive nonlocality in the intermediate picture. The left figure shows a time slice of the intermediate picture where the arrow shows a superluminal propagation from Q to Σ induced from a shortcut in the bulk. When zooming in to UV, one can only see Σ, and no nonlocality will be observed within Σ. \n<!-- image --> \n<latexit sha1\\_base64="18rLAvR6igLSbRLNibjmTFEM6sQ=">AAACZHichVG/S8NAGH2Nv2qttiqCoIhYKk7li4viVHRxNK21gkpJ4qmhaRKStFCL/4CuioOTgoj4D7i7+A84uDkJ4ljBxcGvaUG0qN9xd+/efe+7d3eaYxqeT/QYkjo6u7p7wr2Rvmj/QCw+OLTm2WVXFzndNm13XVM9YRqWyPmGb4p1xxVqSTNFXisuNfbzFeF6hm2t+lVHbJXUXcvYMXTVZ0pRCvEEpSiIyXYgt0AiPX6rPAFYseNX2MQ2bOgoowQBCz5jEyo8bhuQQXCY20KNOZeREewLHCDC2jJnCc5QmS3yuMurjRZr8bpR0wvUOp9icndZOYkkPdA11emebuiFPn6tVQtqNLxUedaaWuEUYoej2fd/VSWefex9qf707GMH84FXg707AdO4hd7UV/ZP69mFTLI2TRf0yv7P6ZHu+AZW5U2/VETmDBH+APnnc7eDtdmUTClZkRPpRTQjjDFMYYbfew5pLGMFOT5X4AjHOAk9S1FpWBpppkqhlmYY30Ka+ATQ0YwT</latexit> \n<latexit 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sha1\\_base64="MBK72G11ed/Dbpqd1piqT4JOAyg=">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</latexit> \nWe would like to understand this IR-sensitive nonlocality as a nature of quantum gravity. However, before proceeding, we would like to note that this behavior is very different from the nonlocality known in string theory [29,30]. Intuitively, the extending structure of strings causes a non-vanishing commutator even outside of the light cone. This effect is significant at the string scale, and is expected to be invisible at large length scale in which strings look like point particles. Accordingly, we can say that this is a UV-sensitive nonlocal effect. See figure 16. Note that the IR-sensitive nonlocality found in the intermediate picture and the \nUV-sensitive nonlocality known in string theory do not contradict with each other, but they are expected to be caused by different mechanisms. \nFigure 16: The UV-sensitive nonlocality known in the string theory. When zooming in a field theory (left) which has a string origin to UV, nonlocality caused by the extension of strings becomes significant. \n<!-- image --> \nLet us go back and consider what kind of effect in quantum gravity can give rise to an IR-sensitive nonlocality. A gravitational path integral [72-74] can be written as: \nZ QG = ∑ topology of N ∫ D g ∫ D φ exp ( iI [( N , g ) , φ ]) (7.16) \nwhere φ denotes matter fields and I [( N , g ) , φ ] denotes a local gravitational action: \nI [( N , g ) , φ ] = 1 16 πG N ∫ N √ -g ( R -2Λ) + ∫ N √ -g L [ φ ] + (boundary terms) . (7.17) \nAt weak gravity limit G N → 0, it is straightforward to expect that the effective theory is a local QFT on a classical spacetime ( N 0 , g 0 ) determined from the on-shell condition. In this case, the partition function can be written as: \nZ QG ∼ ∫ D φ exp ( iI [( N 0 , g 0 ) , φ ]) . (7.18) \nOn the other hand, a full quantum gravitational picture should include corrections from other geometries. In particular, it is natural to expect that geometries containing spacetime wormholes (and hence with higher topology), though suppressed at O ( e -1 /G N ) order, should \nbe also included as quantum corrections. Accordingly, scattering amplitude on such (possibly off-shell) geometry should also be counted unless there are strong evidences showing that it vanishes. As a result, if we use an effective theory defined on the dominant geometry to include corrections coming from higher topology, this effective theory is expected to be nonlocal and include superluminal phenomena. We conjecture that the mechanism depicted above can be thought as a fundamental origin of the nonlocality appearing in the intermediate picture. See (7.19) for example. \n<!-- image --> \nIn this equation, we show time slices of the intermediate picture. We can see that the nonlocal effects in the effective theory arises from an \'ensemble\' of non-dominant geometries with spacetime wormholes. This \'ensemble\' also gives an effective extra-dimension. 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Especially, we have found that the bulk causal structure is compatible with causality in the BCFT picture, while it requires a violation of causality in the intermediate picture. We kept spacetime dimensions and configurations as general as possible. Therefore, all the doubly holographic models considered recently, e.g. [13, 14, 16, 17, 31, 33, 36, 39, 70, 75-77] are covered by this paper. \nBesides a specific case study for the vacuum configuration, we have extended the original Gao-Wald theorem to generic AdS/BCFT setups, and found that the bulk causal structures are compatible with causality in the BCFT picture. We have also discussed the possibility of using the compatibility of causality as an input to restrict constructions of bulk duals for general locally CFT with boundaries. \nOn the other hand, we have found that being compatible with the bulk structure requires a violation of causality in the intermediate picture. More specifically, superluminal propagations exist if and only if we consider the communication between the gravitational region Q and the non-gravitational region Σ. We have also confirmed this point in the microscopic causal structure by computing the commutators for a light primary field in the vacuum configuration. The breakdown of both macrocausality and microcausality indicates that there should be a nonlocal interaction in the intermediate picture, when considering it as an effective theory defined on Q ∪ Σ. \nWe discussed the physical consequences of the emergent nonlocality found in the intermediate effective theory. We have shown that the existence of nonlocality results in the breakdown of the conventional notion of domain of dependence. Based on this, the correspondences between subregions in the three pictures were reconsidered. Especially, we have shown that a subregion in the BCFT picture and the same geometrical subregion in the intermediate picture contain different information. More specifically, the latter one can be reconstructed from the former one, while the inverse is impossible, as shown in (7.11). Another intriguing consequence is that, a straightforward analog of entanglement wedges to the intermediate/bulk correspondence, which we call tentative entanglement wedges, do not \nserve as duals of corresponding intermediate subregions. \nWe have also commented that the nonlocality found in this paper is significant at IR, i.e. at long range. This feature is similar to that of a spacetime wormhole. Therefore, a possible quantum gravitational origin of this sort of nonlocality is contributions coming from wormhole configurations in the gravitational path integral. \nThere are many possible future directions of our work. Here, we list some of them and give brief comments. \nAlthough we have concluded that the effective theory in the intermediate picture should include a nonlocal interaction, the effective action is not given. In double holography, the bulk allows shortcuts and works like a traversable wormhole. Therefore, it is natural to expect that the nonlocal term is given by a bilocal deformation such as double trace deformation [78,79], at least at the leading order, though more efforts are needed to examine the explicit form. \nIn this paper, we have discussed causal issues in double holography by considering sending a signal from one point to another point on on-shell bulk geometries. Recently, causal puzzles and their resolutions associated with codimension-1 subregions [80] and off-shell geometries [81] are discussed in the AdS/CFT correspondence. It would be interesting to study similar aspects in the framework of double holography. \nAnother promised direction is to derive the nonlocality in more general gravity-coupledto-heat-bath setups without using holography. Inspired by the relation between the two methods for deriving the island formula: one by double holography [6] and the other by gravitational path integral [7, 8], it is natural to expect that the nonlocality comes from the wormhole configurations, as also commented in the main text. However, we need a quantitative analysis to justify this intuition and make it more clear. Also, it would be interesting to understand its relation with early discussions on spacetime wormholes [72-74]. \nLast but not least, the implication of the IR-sensitive nonlocality in the context of black hole physics, especially the information loss problem, is also interesting. Although we did not focus on black holes, the results obtained in this paper are general enough to apply to these configurations. There have been many discussions on whether long-range nonlocality should be involved in a black hole evaporation process [82,83]. Results in this paper indicate that long-range nonlocality indeed exists in doubly holographic models, but the role it plays is not clear so far. It would also be important to readdress the firewall problem [31, 84] in double holography, with the IR-sensitive nonlocality involved.', 'Acknowledgements': 'We are grateful to Ibrahim Akal, Bowen Chen, Yuya Kusuki, Tatsuma Nishioka, Tadashi Takayanagi, Kotaro Tamaoka, Takahiro Tanaka, Tomonori Ugajin and Zi-zhi Wang for useful discussions. We would like to thank Hao Geng, Andreas Karch and Federico Piazza very much for valuable comments on this paper. ZW is supported by the ANRI Fellowship and Grant-in-Aid for JSPS Fellows No. 20J23116.', 'A AdS d +1 spacetime': 'In this appendix, we summarize coordinates and geodesics for AdS spacetime considered in this paper.', 'A.1 Coordinates': "Anti-de Sitter (AdS) spacetime is a spacetime with a constant negative curvature in the Lorentz signature. This is given by the hypersurface \n〈 X,X 〉 ≡ -( X 0 ) 2 -( X d +1 ) 2 + X 2 = -L 2 (A.1) \nin the R 2 ,d , with its inner product given by 〈· , ·〉 . Here, coordinate in R 2 ,d is defined by ( X 0 , X d +1 , X ) and its metric is given by \nds 2 = -( dX 0 ) 2 -( dX d +1 ) 2 + d X 2 . (A.2) \nThe quantity L is AdS radius, which is set to L = 1 in the following. There are several convenient coordinates in AdS spacetime used in this paper. One is the global coordinate ( τ, r, Ω ), defined by \nX 0 = √ 1 + r 2 cos τ , X d +1 = √ 1 + r 2 sin τ , X = r Ω . (A.3) \nHere, Ω is the point on S d . Then the metric is given by \nds 2 = -(1 + r 2 ) dτ 2 + dr 2 1 + r 2 + r 2 d Ω 2 . (A.4) \nNote that we should take covering space to avoid the closed time like curve. \nThe coordinate we are most interested in is the Poincar'e coordinate. This is defined by \nX 0 = z 2 ( 1 + 1 -t 2 + x 2 z 2 ) , X d +1 = t z , X d = z 2 ( 1 + -1 -t 2 + x 2 z 2 ) , X i = x i z . (A.5) \nwhere x = ( x 1 , . . . , x d -1 ). The metric is \nds 2 = dz 2 -dt 2 + d x 2 z 2 (A.6) \nIn this coordinate, asymptotic boundary of the AdS spacetime is given by z = 0.", 'A.2 Geodesics': 'Geodesics in the AdS spacetime can be easily obtained by using the fact that they are embedded in R 2 ,d . Suppose that M with dim M = d + 1 (in our case AdS d +1 spacetime) is embedded in the N with dim N = d + 2 (in our case R 2 ,d spacetime). Then, GaussWeingarten equation relates the covariant derivative of N (which is denoted by D ) and the covariant derivative of the M (which is denoted ∇ ) by \nu A D A v B = ( u µ ∇ µ v ν ) e B ν -K µν u µ v ν n B . (A.7) \nHere, e A µ is the basis of T ( M ), K µν is the extrinsic curvature of M and n A is the normal vector to M . Therefore, an equation for the affine parametrized geodesic on M can be written in terms of covariant derivative on N as \nu A D A u B = -( g µν u µ u ν ) n B . (A.8) \nwhere g µν is the metric of AdS spacetime. Here, we used the fact that the extrinsic curvature of AdS spacetime embedded in R 2 ,d is given by 21 \nK µν = g µν . (A.12) \nR ABCD e A µ e B ν e C ρ e D σ = R µνρσ -( K µσ K νρ -K µρ K νσ ) . (A.9) \nOf course, Riemann curvature R ABCD of R 2 ,d is zero, and we know that the curvature of the AdS spacetime is given by \nR µνρσ = -( g µρ g νσ -g µσ g νρ ) . (A.10) \nLet us write a geodesic as γ ( λ ) = ( X 0 ( λ ) , X d +1 ( λ ) , X ( λ )). Its tangent vector u is given by \nu ( λ ) ≡ dX dλ . (A.13) \nThen, the geodesic equation of AdS spacetime is given by \nd 2 X ( λ ) dλ 2 = -( g µν u µ ( λ ) u ν ( λ )) X ( λ ) . (A.14) \nNote that we chose n = X in (A.7). This is possible, because if we regard a point on the AdS as a vector in R 2 ,d , they are always orthogonal to the hypersurface (A.1). This can be seen by taking derivative of the definition of the AdS spacetime (A.1). This gives \nd dλ 〈 X ( λ ) , X ( λ ) 〉 = 2 〈 u, X 〉 = 0 , (A.15) \nwhere 〈· , ·〉 is the usual flat metric on R 2 ,d . \nNow the geodesic equation (A.14) can be easily solved. For time-like geodesics ( g µν u µ u ν = -1), \nX ( λ ) = Y cosh λ + Z sinh λ , (A.16) \nfor space-like geodesics ( g µν u µ u ν = 1), \nX ( λ ) = Y cos λ + Z sin λ , (A.17) \nand for null geodesics ( g µν u µ u ν = 0) \nX ( λ ) = Y + Zλ . (A.18) \nHere, Y and Z denotes the initial condition. From equation (A.15), they must satisfy \n〈 Y, Z 〉 = 0 . 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2022ApJ...938..110B	We present constraints on cosmological parameters from the Pantheon analysis of 1701 light curves of 1550 distinct Type Ia supernovae SNe Ia ranging in redshift from z 0.001 to 2.26. This work features an increased sample size from the addition of multiple crosscalibrated photometric systems of SNe covering an increased redshift span and improved treatments of systematic uncertainties in comparison to the original Pantheon analysis which together result in a factor of 2 improvement in cosmological constraining power. For a flat CDM model we find SUB M SUB 0.334 0.018 from SNe Ia alone. For a flat w SUB0SUBCDM model we measure w SUB0SUB 0.90 0.14 from SNe Ia alone H SUB0SUB 73.5 1.1 km sSUP1SUP MpcSUP1SUP when including the Cepheid host distances and covariance SH0ES and w SUB0SUB 0.9780.0310.024 when combining the SN likelihood with Planck constraints from the cosmic microwave background CMB and baryon acoustic oscillations BAO both w SUB0SUB values are consistent with a cosmological constant. We also present the most precise measurements to date on the evolution of dark energy in a flat w SUB0SUB w SUB a SUBCDM universe and measure w SUB a SUB 0.12.00.9 from Pantheon SNe Ia alone H SUB0SUB 73.3 1.1 km sSUP1SUP MpcSUP1SUP when including SH0ES Cepheid distances and w SUB a SUB 0.650.320.28 when combining Pantheon SNe Ia with CMB and BAO data. Finally we find that systematic uncertainties in the use of SNe Ia along the distance ladder comprise less than onethird of the total uncertainty in the measurement of H SUB0SUB and cannot explain the present Hubble tension between local measurements and early universe predictions from the cosmological model.	2022-10-01T00:00:00Z	['10.48550/arXiv.2202.04077', 'arXiv:2202.04077', '2022ApJ...938..110B', '2022arXiv220204077B', '10.3847/1538-4357/ac8e04']	['Cosmology', 'Dark energy', 'Dark matter', 'Type Ia supernovae', 'Cosmological models', 'Expanding universe', '343', '351', '353', '1728', '337', '502', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	The Pantheon Analysis Cosmological Constraints	2,022	192	0.67	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	621	https://arxiv.org/pdf/2202.04077.pdf	{'No Header': '13 \nDraft version November 16, 2022 \nTypeset using L A T X twocolumn style in AASTeX63 \nE', 'The Pantheon+ Analysis: Cosmological Constraints': "Dillon Brout, 1, 2 Dan Scolnic, 3 Brodie Popovic, 3 Adam G. Riess, 4, 5 Anthony Carr, 6 Joe Zuntz, 7 Rick Kessler, 8, 9 Tamara M. Davis, 6 Samuel Hinton, 6 David Jones, 10, 2 W. D'Arcy Kenworthy, 5 Erik R. Peterson, 3 Khaled Said, 6 Georgie Taylor, 11 Noor Ali, 12 Patrick Armstrong, 13 Pranav Charvu, 3 Arianna Dwomoh, 3 Cole Meldorf, 9 Antonella Palmese, 14 Helen Qu, 15 Benjamin M. Rose, 3 Bruno Sanchez, 3 Christopher W. Stubbs, 16, 1 Maria Vincenzi, 3 Charlotte M. Wood, 17 Peter J. Brown, 18, 19 Rebecca Chen, 3 Ken Chambers, 20 David A. Coulter, 10 Mi Dai, 5 Georgios Dimitriadis, 21 Alexei V. Filippenko, 22 Ryan J. Foley, 10 Saurabh W. Jha, 23 Lisa Kelsey, 24 Robert P. Kirshner, 25, 1 Anais Moller, 26, 27 Jessie Muir, 28 Seshadri Nadathur, 29 Yen-Chen Pan, 30 Armin Rest, 4 Cesar Rojas-Bravo, 10 Masao Sako, 15 Matthew R. Siebert, 10 Mat Smith, 31 Benjamin E. Stahl, 22 and Phil Wiseman 32 \n1 Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 2 NASA Einstein Fellow \n3 Department of Physics, Duke University, Durham, NC, 27708, USA \n4 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA \n- 5 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 USA \n6 \nSchool of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia \n- 7 Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, United Kingdom\n- 8 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA\n- 9 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA \n10 Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 92064, USA \n- 11 Research School of Astronomy and Astrophysics, Australian National University, Canberra, Australia 12 Ume˚a University, 901 87, Ume˚a, Sweden \nMt. Stromlo Observatory, The Research School of Astronomy and Astrophysics, Australian National University, ACT 2601, Australia \n14 \nDepartment of Physics, University of California, Berkeley, CA 94720-7300, USA \nDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA \n16 Department of Physics, Harvard University 17 Oxford Street, Cambridge, MA 02138, USA \n17 \nDepartment of Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA \n18 Department of Physics and Astronomy, Texas A&M University, 4242 TAMU, College Station, TX 77843, USA \nGeorge P. and Cynthia Woods Mitchell Institute for Fundamental Physics & Astronomy, College Station, TX 77843, USA \n20 \nInstitute of Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA \n21 \nSchool of Physics, Trinity College Dublin, The University of Dublin, Dublin 2, Ireland \n22 Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA \n23 Department of Physics and Astronomy, Rutgers, the State University of New Jersey, Piscataway, NJ 08854, USA 24 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK 25 Gordon and Betty Moore Foundation, Palo Alto, CA 94304, USA \n26 \n29 \nCentre for Astrophysics & Supercomputing, Swinburne University of Technology, Victoria 3122, Australia \n27 \nLPC, Universit'e Clermont Auvergne, CNRS/IN2P3, F-63000 Clermont-Ferrand, France \nPerimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada \n28 \nDepartment of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK \n30 \nGraduate Institute of Astronomy, National Central University, 32001 Jhongli, Taiwan \nUniversit'e de Lyon, Universit'e Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, F-69622, Villeurbanne, France \n32 School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK \nAccepted to the Astrophysical Journal \[email protected] \n19 \n31 \n15", 'ABSTRACT': "We present constraints on cosmological parameters from the Pantheon+ analysis of 1701 light curves of 1550 distinct Type Ia supernovae (SNe Ia) ranging in redshift from z = 0 . 001 to 2.26. This work features an increased sample size from the addition of multiple cross-calibrated photometric systems of SNe covering an increased redshift span, and improved treatments of systematic uncertainties in comparison to the original Pantheon analysis which together result in a factor of 2 × improvement in cosmological constraining power. For a FlatΛCDM model, we find Ω M = 0 . 334 ± 0 . 018 from SNe Ia alone. For a Flat w 0 CDM model, we measure w 0 = -0 . 90 ± 0 . 14 from SNe Ia alone, H 0 = 73 . 5 ± 1 . 1 kms -1 Mpc -1 when including the Cepheid host distances and covariance (SH0ES), and w 0 = -0 . 978 +0 . 024 -0 . 031 when combining the SN likelihood with Planck constraints from the cosmic microwave background (CMB) and baryon acoustic oscillations (BAO); both w 0 values are consistent with a cosmological constant. We also present the most precise measurements to date on the evolution of dark energy in a Flat w 0 w a CDM universe, and measure w a = -0 . 1 +0 . 9 -2 . 0 from Pantheon+ SNe Ia alone, H 0 = 73 . 3 ± 1 . 1 kms -1 Mpc -1 when including SH0ES Cepheid distances, and w a = -0 . 65 +0 . 28 -0 . 32 when combining Pantheon+ SNe Ia with CMB and BAO data. Finally, we find that systematic uncertainties in the use of SNe Ia along the distance ladder comprise less than one third of the total uncertainty in the measurement of H 0 and cannot explain the present 'Hubble tension' between local measurements and early-Universe predictions from the cosmological model. \nKeywords: supernovae, cosmology", '1. INTRODUCTION': "Type Ia supernovae (SNe Ia) anchor the standard model of cosmology with their unmatched ability to map the past 10 billion years of expansion history. SNe Ia provided the first evidence of the accelerating expansion of the Universe (Riess et al. 1998; Perlmutter et al. 1999), and they remain invaluable because they are (1) bright enough to be seen at large cosmic distances, (2) common enough to be found in large numbers, and (3) can be standardized to ∼ 0 . 1 mag precision in brightness or ∼ 5% in distance per object. \nStatistical leverage from large samples of SNe Ia has grown rapidly over the last 3 decades, and wellcalibrated and standardized compilations of these samples have facilitated measurements of the relative expansion history across the redshift range 0 < z < 1 characterized by the equation-of-state parameter of dark energy ( w = P/ ( ρc 2 )), and the measurement of the Hubble constant H 0 , the current expansion rate determined from absolute distances. Measurements of w are constrained from the comparison of standardized SN Ia magnitudes over a wide range of redshifts obtained from different surveys with different observing-depth strategies. Measurements of H 0 require very nearby ( < 50 Mpc, ∼ 1 discovered per year) SNe Ia found by multiple surveys in galaxies that host calibrated primary distance indicators [e.g., Cepheids, tip of the red giant branch (TRGB)] which are then compared to SNe in the Hubble flow, often from the same surveys. \nHowever, simply combining several subsamples into a large sample of SNe Ia does not provide meaningful gains \nwithout rigorous cross-calibration, self-consistent analysis of their light curves and redshifts, and characterization of their numerous sources of related uncertainties or covariance. As samples and compilations grow, ever greater attention must be paid to the control of systematic uncertainties which would otherwise dominate sample uncertainties. \nThis analysis, Pantheon+, is the successor to the original Pantheon analysis (Scolnic et al. 2018b) and builds on the analysis framework of the original Pantheon to combine an even larger number of SN Ia samples and include those that are in galaxies with measured Cepheid distances in order to be able to simultaneously constrain parameters describing the full expansion history (e.g., Ω M , w 0 , w a ) with the local expansion rate (H 0 ). The original Pantheon compilation of 1048 SNe Ia was used to measure a value (from SNe Ia alone) of w = -1 . 090 ± 0 . 220. Riess et al. (2016), in their measurement of the local expansion rate H 0 , used a prerelease version of Pantheon based on Scolnic et al. (2015) and further augmented the sample as Pantheon did not extend to reach the low redshifts of the primary distance indicators at z < 0 . 01. \nAlthough there was significant overlap in data and analysis between the Pantheon measurement of w and the H 0 measurement of Riess et al. (2016), Riess et al. (2016) included several Cepheid calibrator SNe Ia that were not included in Pantheon and the fitting for H 0 and parameters describing the expansion history were done independently rather than simultaneously. Dhawan et al. (2020) later established a framework for consid- \nering the covariance between SNe in primary distance indicator hosts and SNe in the Hubble flow. We build on that framework, which was developed originally for a redshift-binned Hubble diagram, and in this paper we create the first unbinned sample with covariance extending down to z = 0 . 001 that can be used to propagate correlated systematics for simultaneous measurements of H 0 , Ω M , w 0 , and w a . We analyze the largest set of cosmologically viable SN Ia light curves to date, include low-redshift samples to extend the lower bound in redshift to 0 . 001 which contains the primary distance indicators (SNe in SH0ES Cepheid host galaxies), propagate systematic uncertainties for both primary distance indicators and higher-redshift SNe simultaneously, and leverage the large strides made in the field of SN Ia cosmology since the original Pantheon. \nThis paper is the culmination of a series of papers that comprise the Pantheon+ analysis. A graphic of an overview of the numerous Pantheon+ supporting analyses, on which this paper heavily relies, is shown in Fig. 1. Details of each paper pertinent to this analysis are described in Section 3. In short, these papers include (Scolnic et al. 2021, hereafter S22), which describes the sample of 1701 cosmologically viable SN Ia light curves of 1550 distinct SNe, which we will refer to as 'the Pantheon+ sample.' The redshifts and peculiar velocities of the SNe used here are given by Carr et al. (2021) and a comprehensive analysis of peculiar velocities is presented by Peterson et al. (2021). The crosscalibration of the different photometric systems used in this analysis can be found in (Brout et al. 2021, hereafter Fragilistic), and calibration-related systematic uncertainty limits are determined by Brownsberger et al. (2021). The underlying SN Ia populations describing the dataset are given by Popovic et al. (2021b). The model for intrinsic brightness variations was developed by Brout & Scolnic (2021) and then improved and evaluated by Popovic et al. (2021a). The novel systematic framework for simultaneous measurement of H 0 and cosmology is developed by Dhawan et al. (2020), and improved methodology for systematic uncertainties is described by Brout et al. (2021). \nIn this work we discuss briefly the aforementioned papers in the context of their use in this analysis, evaluate several additional systematic uncertainties not addressed in these works, measure cosmological parameters, examine additional signals in the Hubble diagram, and compile systematic uncertainty budgets on cosmological parameters. A companion paper by the SH0ES Team (Riess et al. 2021, hereafter R22) combines from this work 277 Hubble flow (0.023 < z < 0.15) SNe Ia and 42 SNe Ia in Cepheid-calibrator hosts, their relative dis- \ntances, and their covariance, with the absolute distances of primary distance anchors (Cepheids and TRGB) from R22 in order to measure H 0 under the assumption of FlatΛCDM. Similarly, in this work we utilize the full Pantheon+ sample of 1550 SNe Ia in combination with the R22 Cepheid host distances to show the impact of cosmological models with more freedom than those used in R22 as well as the impact of SN-related systematic uncertainties on inferences of H 0 . \nAn important aspect of this work is the public release of the data and simulations used here that allow for the reproduction of multiple different stages of this analysis. In Appendix C, we present the numerous products that will be made available, including SN distances, redshifts, uncertainties, covariance, and extensive SNANA simulations (Kessler et al. 2009) of the data that model astrophysical effects, cosmological effects, and the observation/telescope effects of each survey down to the level of cadence, weather history, etc. We encourage the community to validate alternate analyses of the publicly released Pantheon+ sample on these simulations. \nThe structure of the paper is as follows. In Section 2, we describe the methodology from fitting SN light curves to constraining cosmological parameters. Section 3 summarizes all of the inputs to the analysis including the data sample, calibration, and redshifts. In Section 4, we describe the cosmological results. Sections 5 and 6 are our discussions and conclusions, respectively.", '2.1. Measuring Distances to SNe Ia': "To standardize the SN Ia brightnesses we fit light curves using SNANA with the SALT2 model as originally developed by Guy et al. (2010) and updated in Brout et al. 2021, hereafter SALT2-B22. For each SN, the SALT2 light-curve fit returns four parameters: the lightcurve amplitude x 0 where m B ≡ -2 . 5log 10 ( x 0 ); x 1 , the stretch parameter corresponding to light-curve width; c , the light-curve color that includes contributions from both intrinsic color and dust; and t 0 , the time of peak brightness. Extinction due to Milky Way dust is accounted for in the SALT2 light-curve fitting. From the parameters m B and x 1 , c , we standardize the SN brightnesses and infer distance modului ( µ ), used in the Hubble diagram, with a modified version of the Tripp (1998) distance estimator. Following (Kessler & Scolnic 2017, hereafter BBC), the distance modulus is defined as \nµ = m B + αx 1 -βc -M -δ bias + δ host , (1) \nwhere α and β are global nuisance parameters relating stretch and color (respectively) to luminosity. M is the \nFigure 1. Analysis roadmap of this work and supporting/complementary Pantheon+ and SH0ES papers. Components of the analysis here are shown in blue. The companion paper R22, which provides a constraint on H 0 , requires the Hubble diagram and covariance computed in this work. Likewise, measurements of H 0 in this work require the R22 Cepheid distance and covariance. Supporting papers are shown in gray boxes. \n<!-- image --> \nfiducial magnitude of an SN Ia, which can be calibrated by setting an absolute distance scale with primary distance anchors such as Cepheids. δ bias is a correction term 1 to account for selection biases that is determined from simulations following Popovic et al. (2021b), described in detail in Appendix A. δ host is the luminosity correction (step) for residual correlations between the standardized brightness of an SN Ia and the host-galaxy mass, \nδ host = γ × (1 + e ( M glyph[star] -S ) /τ Mglyph[star] ) -1 -γ 2 , (2) \nwhere γ is the magnitude of the SN Ia luminosity differences between SNe in high ( M glyph[star] > 10 10 M glyph[circledot] ) and low ( M glyph[star] < 10 10 M glyph[circledot] ) stellar mass galaxies and where 'hostless' SNe have been assumed to reside in galaxies with low stellar mass. M glyph[star] is the inferred stellar mass measured in units of solar mass ( M glyph[circledot] ) from spectral energy \ndistribution (SED) fitting to the photometry of each host galaxy, S is the step location (nominal analysis assumes S = 10 10 M glyph[circledot] ), and τ Mglyph[star] describes the width of the step. \nThe total distance modulus error, σ µ , for SN i is described as \nσ 2 µ,i = f ( z i , c i , M glyph[star],i ) σ 2 meas ,i + σ 2 floor ( z i , c i , M glyph[star],i ) + σ 2 lens ,i + σ 2 z,i + σ 2 vpec ,i , (3) \nwhere σ meas is the measurement uncertainty of SALT2 light-curve fit parameters and their associated covariances (see Eq. 3 of Kessler & Scolnic 2017) resulting from photometric uncertainties. The measurement uncertainty is scaled by f ( z i , c i , M glyph[star],i ) specific to each survey in order to account for selection effects that can reduce the observed scatter at the limits of each sample. The uncertainty contribution from gravitational lensing as given by Jonsson et al. (2010) is σ lens = 0 . 055 z . We note that, as discussed by Kessler et al. (2019a), the correct lensing distribution is utilized in simulations. The nominal distance modulus uncertainty contribution due \nto the combination of redshift measurement uncertainty ( σ z ) and peculiar velocity uncertainty ( σ vpec ) have both been converted to distance modulus uncertainty under the assumption of a cosmological model. Chen et al. (2022) note that the optimal way to characterize redshift measurement uncertainty at high redshifts (e.g., the DES sample, z > 0 . 3) is to float the redshift and use the uncertainty in redshift as a prior in the lightcurve fit. However, following previous analyses we fix the redshift and include the associated distance modulus uncertainty σ z in Eq. 3, which is a correct estimate at low redshifts ( z < 0 . 1). Lastly, σ floor represents the floor in standardizability owing to intrinsic unmodeled variations in SNe Ia such that \nσ 2 floor ( z i , c i , M glyph[star],i ) = σ 2 scat ( z i , c i , M glyph[star],i ) + σ 2 gray , (4) \nwhere σ 2 scat ( z i , c i , M glyph[star],i ) is determined from a model that describes intrinsic brightness fluctuations and σ 2 gray is a single number representing a gray (color independent) floor in standardizability for all SNe Ia; σ 2 gray determined after the BBC fitting process in order to bring the Hubble diagram reduced χ 2 to unity. The details of σ 2 scat ( z i , c i , M glyph[star],i ), its model dependence, and its contribution to systematic uncertainties are discussed in further detail in Section 3.3.2 and Appendix A. \nTo determine the distance modulus values of all the SNe, we follow the BBC fitting process with updates to increase the dimensionality of bias corrections in Popovic et al. (2021b). The likelihood (as given in Eq. 6 of Kessler & Scolnic 2017) results in a cosmologyindependent minimization of the free parameters ( α , β , γ , σ gray ) that minimize the scatter in the Hubble diagram. While the BBC process was designed for utility for photometric cosmology analyses and uses SN Ia classification probabilities, the data analyzed here are a spectroscopically confirmed SN Ia sample, and therefore we set the non-Ia SN probabilities to 0 for the whole sample.", '2.2. The Covariance Matrix': "Following Conley et al. (2011), we compute covariance matrices C stat & C syst to account for statistical and systematic uncertainties and expected correlations between the SN Ia light curves in the sample when analyzing cosmological models. BBC produce both a redshift-binned and an unbinned Hubble diagram, enabling both binned and unbinned covariance matrices. For the original Pantheon (Scolnic et al. 2018b), JLA (Betoule et al. 2014), and DES3YR (Brout et al. 2019a), C stat and C syst were redshift-binned matrices (or smoothed as a function of redshift) citing computational limitations. Following Brout et al. (2021), in this work we utilize the unbinned \nHubble diagrams to create unbinned covariance matrices. The Pantheon+ sample (Scolnic et al. 2021) also includes 'duplicate SNe Ia,' SNe Ia that have been observed simultaneously by numerous different surveys, so that statistical covariance C stat is computed as \nglyph[negationslash] \nC stat ( i, j ) = σ 2 µ i = j σ 2 floor + σ 2 lens + σ 2 z + σ 2 vpec i = j & SN i = SN j , (5) \nwhere each row of the matrix corresponds to an SN light curve , the diagonal of C stat is the full distance error ( σ 2 µ ) of the i th light curve, and where measurement noise from components other than the light curve itself are included as off-diagonal covariance between entries corresponding to light curves of the same SN (SN i = SN j ) observed by two different surveys. \nSystematic uncertainties can manifest in three key places in the analysis: (1) from changing aspects affecting the light-curve fitting (e.g., survey photometry, calibration, SALT2 model), (2) from changing redshifts that propagate to changes in distance modului relative to a cosmological model, and (3) from changes in the astrophysical or survey-dependent assumptions in the simulations used for bias corrections. For each of these categories we examine all of the known significant sources of systematic uncertainty ( ψ ) with sizes S ψ which result in residuals in the Hubble diagram relative to our baseline analysis ( µ BASE ). In order to compute the effect of systematics, we first define \n∆ µ i ψ ≡ µ i ψ -µ i BASE -( µ ref ( z ψ ) -µ ref ( z BASE )) , (6) \nwhere µ i ψ is the set of distances for systematic ψ . For systematics that affect redshift, we have included new methodology in Eq. 6 that utilizes a reference cosmological model distance µ ref ( z ) corresponding to FlatΛCDM (Ω M = 0 . 3 , Ω Λ = 0 . 7). The µ mod ( z ψ ) and µ mod ( z BASE ) are the cosmological model distances corresponding to redshifts z ψ and z BASE . In order to propagate redshift effects into a distance × distance covariance matrix, the additional component µ mod ( z ψ ) -µ mod ( z BASE ) accounts for the difference in inferred model distance. \nAssuming linearity between ∆ µ ψ and ψ , we compute the derivative for each ψ in order to build a 1701 × 1701 systematic covariance matrix as, \nC ij syst = ∑ ψ ∂ ∆ µ i ψ ∂ S ψ ∂ ∆ µ j ψ ∂ S ψ σ 2 ψ , (7) \nwhich denotes the covariance between the i th and j th light-curve fit summed over the different sources of sys- \ntematic uncertainty ( ψ ) with uncertainty σ ψ (see Section 3 for details). As shown by Brout et al. (2021), the σ ψ serve as priors on the known size of systematic uncertainties, but the data itself can constrain the impact of each systematic under the condition that information has not been collapsed by binning/smoothing (as was done for the original Pantheon, JLA, and DES3YR). \nFluctuations of the sample of light curves that pass the sample quality cuts (Table 2 of S22) for different systematics result in an ill-defined covariance matrix. To have a well-defined unbinned covariance matrix requires a subtle treatment in order to ensure that the sample is consistent in both the light-curve fitting and BBC stages across all systematics in the analysis. Quality cuts at the light-curve stage are only applied to the set of SNe based on their values found in the baseline analysis, and this SN sample is used for all systematic tests. We perform the BBC process twice - the first iteration to identify the subset of < 1% of SNe for which bias corrections are unable to be computed, and a second iteration using only the common set of SNe that have valid bias corrections in all systematic variants. The final cosmology sample of 1701 light curves that satisfy all criteria is described in detail in S22 (see the 'Systematics' row in Table 2 of S22). \nFinally, the statistical and systematic covariance matrices are combined and used to constrain cosmological models: \nC stat+syst = C stat + C syst . (8)", '2.3. Cosmology': "Constraining cosmological models with SN data using χ 2 has been used in previous SN Ia cosmology analyses (e.g., Riess et al. 1998; Astier et al. 2006) and first included systematic covariance in Conley et al. (2011). Here we follow closely the formalism of Conley et al. (2011) where cosmological parameters are constrained by minimizing a χ 2 likelihood: \n-2ln( L ) = χ 2 = ∆ glyph[vector] D T C -1 stat+syst ∆ glyph[vector] D, (9) \nwhere glyph[vector] D is the vector of 1701 SN distance modulus residuals computed as \n∆ D i = µ i -µ model ( z i ) , (10) \nand each SN distance ( µ i ) is compared to the predicted model distance given the measured SN/host redshift ( µ model ( z i )). The model distances are defined as \nµ model ( z i ) = 5 log( d L ( z i ) / 10 pc) , (11) \nwhere d L is the model-based luminosity distance that includes the parameters describing the expansion history \nH ( z ). For a flat cosmology (Ω k = 0) the luminosity distance is described by \nd L ( z ) = (1 + z ) c ∫ z 0 dz ' H ( z ' ) , (12) \nwhere d L ( z ) is calculated at each step of the cosmological fitting process, and the parameterization of the expansion history (used in Eq. 12 and therefore in the likelihood Eq. 9) in this work is defined as \nH ( z ) = H 0 √ Ω M (1 + z ) 3 +Ω Λ (1 + z ) 3(1+ w ) . (13) \nSee Hogg (1999) for the forms of the expansion history H ( z ) used in the case that the assumption of flatness is relaxed. \nThe parameters M (Eq. 1) and H 0 (Eq. 13) are degenerate when analyzing SNe alone. However, we also present constraints that include the recently released SH0ES Cepheid host distance anchors (R22) in the likelihood which facilitates constraints on both M and H 0 . \nWhen utilizing SH0ES Cepheid host distances, the SN distance residuals are modified to the following: \n∆ D ' i = µ i -µ Cepheid i i ∈ Cepheid hosts µ i -µ model ( z i ) otherwise , (14) \nwhere µ Cepheid i is the Cepheid calibrated host-galaxy distance provided by SH0ES and where µ i -µ Cepheid i is sensitive to the parameters M and H 0 and is largely insensitive to Ω M or w . We also include the SH0ES Cepheid host-distance covariance matrix ( C Cepheid stat+syst ) presented by R22 such that the likelihood becomes \n-2ln( L ' ) = ∆ glyph[vector] D ' T ( C SN stat+syst + C Cepheid stat+syst ) -1 ∆ glyph[vector] D ' , (15) \nwhere C SN stat+syst denotes the SN covariance. \nWe evaluate the likelihoods with the PolyChord (Handley et al. 2015) sampler in the CosmoSIS package (Zuntz et al. 2015) using 250 live points, 30 repeats, and an evidence tolerance requirement of 0.1. This resulted in converged chains containing 1000-3000 independent samples. We verified the SN-only results with CosmoMC (Lewis & Bridle 2002) and with the fast cosmology grid-search program in SNANA . The likelihood for Pantheon+ and R22 Cepheid host distance samples will be made available in the public version of CosmoSIS. In this work we also utilize the additional public likelihoods in CosmoSIS in order to combine with and assess agreement with external cosmological probes: Planck (Collaboration et al. 2018) and baryon acoustic oscillations (BAO, likelihoods discussed in Section 4). \nIn this work we investigate four cosmological parameterizations: \n- · FlatΛCDM: Ω M is floated and we fix w = -1 and Ω M +Ω Λ = 1.\n- · ΛCDM: Ω M and Ω Λ are floated and we fix w = -1.\n- · Flat w CDM: w and Ω M are floated and we fix Ω M + Ω Λ = 1.\n- · Flat w 0 w a CDM: w = w 0 + w a (1 + z ), Ω M , w 0 , w a are floated and we fix Ω M +Ω Λ = 1. \nWe blind our analysis in two ways simultaneously. First, we blind the binned distance residuals output by the BBC fit as cosmological parameters could be inferred visually from simply looking at the Hubble diagram. Secondly, in order to prevent accidental viewing of the cosmological parameters themselves, the CosmoSIS chains were shifted by unknown values following the formalism of Hinton (2016).", '3. DATA AND ANALYSIS INPUTS': 'Here we review each component of the dataset and analysis. We discuss the fundamental purpose , the baseline treatment in this analysis, and the systematic uncertainties associated with each aspect (if applicable). The impact of systematics in both distance and cosmological inference is shown in Section 4. We provide a brief overview of this section here.', 'Data': 'Sec. 3.1.1: SN Ia Light Curves \nSec. 3.1.2: Redshifts \nSec. 3.1.3: Peculiar Velocities \nSec. 3.1.4: Host-Galaxy Properties', 'Calibration and Light-Curve Fitting': 'Sec. 3.2.1: Calibration \nSec. 3.2.2: SALT2 Model \nSec. 3.2.3: Milky Way Extinction', 'Simulations': 'Sec. 3.3.1: Survey Modeling \nSec. 3.3.2: Intrinsic Scatter Models \nSec. 3.3.3: Uncertainty Modeling \nSec. 3.3.4: Validation \n3.1. Data', '3.1.1. SN Ia Light Curves': 'Purpose: The flux-calibrated light-curve photometry is fit to determine the SALT2 parameters used in standardization (Eq. 1). \nBaseline: The light-curve data is described in detail by S22 and references therein. The full set of spectroscopically classified photometric light curves is compiled from \n18 different publicly available and privately released samples. In total, 2077 SN light-curve fits converged using SALT2; after quality cuts are applied (Table 2 of S22), this results in 1701 SN light curves of 1550 unique SNe Ia usable for cosmological constraints. The sample includes a 3.5 σ Hubble residual outlier cut to remove 5 potential contaminants that are likely non-normal Type Ia or misidentified redshifts. The sample of cosmologically viable light curves includes 81 light curves of 42 SNe used to calibrate Cepheid brightnesses as utilized by R22. The survey SN photometry compiled in Scolnic et al. (2021) and analyzed here is from DES 1 (Brout et al. 2019b; Smith et al. 2020a), Foundation 1 (Foley et al. 2018), PS1 (Scolnic et al. 2018b), SNLS (Betoule et al. 2014), SDSS (Sako et al. 2011), HST (Gilliland et al. 1999; Riess et al. 2001; Suzuki et al. 2012; Riess et al. 2018, 2004, 2007), Lowz (grouped together as LOSS 1 1 , Ganeshalingam et al. 2010; LOSS 2 1 , Stahl et al. 2019; SOUSA 12 , Brown et al. 2014; CNIa0.02 1 , Chen et al. 2020; CSP, Krisciunas et al. 2017b; CfA1, Riess et al. 1999; CfA2, Jha et al. 2006; CfA3, Hicken et al. 2009; CfA4, Hicken et al. 2012, and numerous smaller low-redshift samples 1 of 1-2 SNe given by Burns et al. 2018, Burns et al. 2020, Milne et al. 2010, Krisciunas et al. 2017a, Stritzinger et al. 2010a, Gall et al. 2018, Zhang et al. 2010, Tsvetkov & Elenin 2010, and Kawabata et al. 2020.) \nSystematics: See Calibration Section 3.2.1.', '3.1.2. Redshifts': 'Purpose: The peculiar-velocity corrected CMB-frame redshift of each SN/host is required to compare the inferred distance to a distance predicted by a cosmological model, as given in Eq. 10. Additionally, heliocentric redshifts are required in the SALT2 light-curve fits in order to shift the model spectrum to match the data. \nBaseline: The redshifts for all of the SNe (and their host galaxies, depending on what is available) are provided by Carr et al. (2021), who performed a comprehensive review of redshifts for the Pantheon+ samples and made numerous corrections. Carr et al. (2021) report the heliocentric redshifts for each SN and convert the redshift into the CMB frame. The redshifts of the Pantheon+ sample cover a range of 0 . 001 < z < 2 . 3. While redshifts of the 42 Cepheid host calibrator SNe are included, they are not used in the comparison of SN Ia magnitudes to the Cepheid distance scale and are only provided for reference and for SALT2 fitting. \nSystematics: Following Carr et al. (2021), we apply a \ncoherent shift to each redshift of +4 × 10 -5 . This was conservatively stated by Calcino & Davis (2017) for the potential size of a local void bias and by Davis et al. (2019) as a potential measurement bias.', '3.1.3. Peculiar Velocities': "Purpose: Peculiar motions of galaxies arise from coherent flows, motion of halos, inflow into clusters or superclusters, and intragroup motion. Corrections are applied to the observed redshifts (after light-curve fitting) based on peculiar-velocity maps derived from independent large spectroscopic galaxy surveys. \nBaseline: The nominal peculiar velocities used for this analysis were determined by Peterson et al. (2021) from a comparison of multiple treatments of peculiar-velocity maps and group catalogs. Corrections were applied by Carr et al. (2021) for the Pantheon+ sample. The baseline corrections are based on 2M++ (Carrick et al. 2015) with global parameters found in Said et al. (2020) and combined with group velocities estimated from Tully (2015) group assignments. The σ vpec in Eq. 3 is found using 240 km s -1 after accounting for uncertainties propagated into the covariance matrix described below. This σ vpec floor is in agreement with what was used in Peterson et al. (2021) and for the SNe between 0 . 001 < z < 0 . 02 it is likely a conservative estimate as Kenworthy et al. (2022) found for the most nearby SN calibrators a floor of 155 ± 25 kms -1 . This apparent reduction at the lowest redshifts may be due to the peculiar velocity maps having higher fidelity at these redshifts and because Pantheon+ has relatively better virial-group information at these redshifts. \nSystematics: Peterson et al. (2021) discuss multiple viable alternatives for the treatment of peculiar velocities. The first approach is to use the 2M++ corrections (Carrick et al. 2015) integrating over the line-of-sight relation (iLOS) between distance and the measured redshift. We take this variation as the first systematic with σ 2 ψ = 0 . 5. The second approach is to use the 2MRS (Lilow & Nusser 2021) peculiar-velocity map; however, differences between 2MRS and 2M++ at very low redshift ( z < 0 . 01) cause numerical stability issues for offdiagonal C syst elements. We incorporate only the diagonal differences between 2MRS and 2M++ into C syst with σ 2 ψ = 0 . 5. As a numerically stable estimate of the off-diagonal terms, we use the 2M++ velocities transformed by the slope and offset difference between the 2M++ and 2MRS maps found in Peterson et al. (2021). The two approaches added in quadrature result in an effective σ 2 ψ = 1 . 0. \n3.1.4. Host-Galaxy Properties \nPurpose: The observed host-galaxy mass versus SN luminosity relation is used to standardize the SN Ia brightnesses in two ways. First, simulations of the dataset include correlations between SN color and SN stretch and host properties such as dust as a function of host mass following Popovic et al. (2021a). Second, a further residual correction is applied in the Tripp Eq. 1 where the 'mass step' γ is fit in the BBC stage. \nBaseline: The host-galaxy stellar masses are presented by S22 and references therein. Masses are determined for all host galaxies, and star-formation rates and morphologies are also included the lowz sample. In the baseline analysis we apply the mass step at 10 10 M glyph[circledot] following Pantheon and DES3YR. \nSystematics: Several independent analyses (Sullivan et al. 2010; Childress et al. 2013; Kelsey et al. 2020) have suggested that the optimal location of the mass step could range between 10 9 . 8 M glyph[circledot] and 10 10 . 2 M glyph[circledot] . We therefore include a systematic uncertainty where the mass step occurs at 10 10 . 2 M glyph[circledot] .", '3.2.1. Calibration': 'Purpose: Photometric calibration of each passband in each survey is needed to fit light curves and facilitate comparison of the brightnesses of SNe across different telescopes/instruments/filters. Photometric calibration is also important to homogenize spectrophotometric datasets used in the SALT2 model training. \nBaseline: The calibration of all 25 photometric systems used in this work is discussed in Fragilistic (Brout et al. 2021). The outputs of Fragilistic are a best-fit calibration solution for each of the 105 passbands and a joint 105 × 105 covariance matrix that describes the covariance between the zeropoint calibrations of all passbands that arise from using a single common stellar catalog to tie all surveys together (PS1). \nSystematics: The systematics due to calibration and their impact are discussed in detail in Fragilistic. We estimate the impact of the correlated filter zero-point and central wavelength uncertainties by refitting SALT2 light curves (with retrained SALT2 models; see next SALT2 Model ) using 9 realizations of the 105 zeropoints. For each of the 9 realizations a value of σ 2 ψ = 1 / 9 is adopted such that they add in quadrature to ∼ 1. The uncertainty in modeling the spectrum of the HST primary standard star C26202 has been tripled to account for the recent update in Bohlin et al. (2020); it is now set to 15 mmag over 7000 ˚ A ( σ ψ = 3 for a systematic of 5 mmag over 7000 ˚ A). Lastly, an additional conservative systematic is included only for the CSP SNe to account \nfor the 2% recalibration in CSP tertiary stellar magnitudes from Stritzinger et al. (2010b) to Krisciunas et al. (2017b) ( σ ψ = 1).', '3.2.2. SALT2 Model': 'Purpose: The trained SALT2 model is required to fit light curves and determine the light-curve parameters ( m b , c , x 1 ) for each SN used in Eq. 1. \nBaseline: We use the Fragilistic calibration solution and newly trained SALT2-B22 model 3 which was developed following the formalism of Guy et al. (2010) and Taylor et al. (2021). The SALT2 model includes a component of training statistical uncertainty which is incorporated in the fitted light-curve parameters \nSystematics: For each of the 9 correlated realizations of Fragilistic filter zero-points and central wavelengths discussed above (for Calibration ) we simultaneously retrain the SALT2 model. Additionally, to conservatively account for a possible systematic from the redevelopment of the SALT2 model training process itself, we adopt an additional systematic by fitting the dataset with the SALT2 model trained by Betoule et al. (2014) and applying a scaling of σ ψ = 1 / 3 (See Section 5 and Fig. 15 for impact).', '3.2.3. Milky Way Extinction': "Purpose: Values of the Milky Way (MW) Galactic dust extinction, E ( B -V ) MW , are applied to the SALT2 model spectra during both the model training process and during the data light-curve fitting process. The 'extinction curve' describes the relation between the amount of reddening and extinction as a function of wavelength. \nBaseline: We account for MW extinction using maps from Schlegel et al. (1998), with a scale of 0.86 following Schlafly et al. (2010). We assume the MW extinction curve from Fitzpatrick (1999) with R V = 3 . 1. \nSystematics: Similarly to Pantheon, we adopt a global 5% uncertainty scaling of E ( B -V ) MW based on the fact that Schlafly & Finkbeiner (2011a), in a reanalysis of Schlafly et al. (2010), derive smaller values of reddening by 4%, despite using a very similar SDSS footprint ( σ ψ = 1). While Schlafly & Finkbeiner (2011b) found that their results prefer the Fitzpatrick (1999) extinction curve, we conservatively include an additional systematic uncertainty in the MW extinction curve and analyze the data (training and light-curve fit) using the Cardelli et al. (1989) and apply a systematic scaling of σ ψ = 1 / 3 \nTable 1. References for inputs to SNANA simulations used for this analysis. We give references for the 'Cadence,' which describes the observing history; the 'DETEFF,' which describes the detection efficiency based on the signal-to-noise ratio (SNR); and the 'SPECEFF,' which describes the spectroscopic selection efficiency as a function of SN magnitude. \nas this reflects the preference of Fitzpatrick (1999) over Cardelli et al. (1989). \nFigure 2. Comparison between observed data (black points) and simulations (blue lines) for the largest subsamples in this analysis: DES, HST , SDSS, SNLS, PS1, LOWz , Foundation (FOUND). We compare three key distributions: the SALT2 light-curve-fit parameters x 1 and c are shown as well as the measured redshift. \n<!-- image -->", '3.3. Simulations': "3.3.1. Survey Modeling \nPurpose: We utilize catalog-level simulations of large samples of SN Ia ( > 1 , 000 , 000 per survey) light curves. SNANA simulations specific to each survey in our analysis are prescribed by each aspect of acquiring an SN Ia sample. As detailed in Figure 1 of Kessler et al. (2019a), the simulations require three main sets of inputs: \n- A Source Model for generating SNe with realistic astrophysical properties and applying cosmological effects such as redshifting, dimming, lensing, peculiar velocities, and MW extinction.\n- A Noise Model , unique to each survey, for applying instrumental and atmospheric noise to determine a detection efficiency ('DETEFF').\n- A Trigger Model , unique to each survey, that includes the observing cadence and describes an efficiency as a function of B -band peak magnitude for detecting SNe and obtaining a spectroscopic confirmation ('SPECEFF').. \nThese simulations for each survey are combined and used to forward model the underlying populations of the SN properties (Popovic et al. 2021a,b) and to determine the expected biases in measured SN distances that follow from the known selection effects. These biases are corrected in the δ bias term of Eq. 1. \nBaseline: Depicted in Fig. 2 are the distributions of the key observables ( z , x 1 , c ) for both data and simulations of each survey used in this analysis. We find good agreement between the data and simulations, as described in detail by Popovic et al. (2021a) and Popovic et al. (2021b). We note that the agreement in the redshift dimension is achieved despite not explicitly tuning the redshift distribution of surveys. \nWe simulate SNe in LOWz and Foundation down to z = 0 . 001. Novel for this work specifically are the simulations of primary distance indicator hosts of SNe in the range 0 . 001 < z < 0 . 01 which are assumed to have the same color and stretch populations as those of their respective surveys (LOWz and FOUND), and specifically over this redshift range they are assumed to be complete with flat spectroscopic selection efficiency. These simulations facilitate bias corrections to the Cepheid calibrator SNe and thus the propagation of modeling systematics to the SNe used in the companion SH0ES analysis (R22). \nThe simulation inputs for survey cadence, DETEFF, and SPECEFF functions have been evaluated in many analyses over the past decade. Table 1 shows a summary of where we obtain these inputs for each survey. Survey metadata is used to model the cadence and instrumental properties, if available, such as for FOUND, SDSS, PS1, DES, and SNLS. LOWz data do not provide such metadata, and thus the cadence and noise \nproperties are extracted from the data as described in Section 6 of Kessler et al. (2019a) following the procedure developed by Scolnic et al. (2018b), which assumes that the LOWz subset of SNe is magnitude-limited. These are simulations of the CfA and CSP samples, but not of the newer samples included in this work (LOSS, SOUSA, CNIa0.02), thereby implicitly assuming that the CfA and CSP samples have similar selection effects and therefore distance biases as the newer additions. To simulate SN-host correlations, a catalog of host-galaxy properties and specifically their stellar-mass distributions are taken from Popovic et al. (2021b). The simulations used for bias corrections for all surveys are performed in ΛCDM ( w = -1 . 0, Ω M = 0 . 3, Ω Λ = 0 . 7) with the SALT2-B22 model. \nSystematics: We increase the SNR of each simulation by 20%, resulting in all survey simulated distributions changing by more than 1 σ , as a single conservative systematic in the determination of the selection biases. Kessler & Scolnic (2017) showed that the sensitivity of the bias corrections to the input cosmology is relatively weak; this was confirmed by Brout et al. (2019a) and found to be a negligible contribution to SN Ia uncertainty budgets. We therefore do not include this as an additional systematic.", '3.3.2. Intrinsic Scatter Models': "Purpose: A model of the intrinsic SN brightness variations, called 'intrinsic scatter,' is needed to account for the observed residual variation in SN Ia standardized luminosities that exceeds expectations from measurement uncertainties alone. In addition, models of the true ('parent') populations of SN Ia SALT2 parameters c and x 1 are required for the Source Model in SNANA . The intrinsic scatter model is utilized in the bias-correction simulations. \nBaseline: We utilize the Brout & Scolnic 2021, hereafter BS21 model that prescribes SN Ia scatter into two color-dependent components: a standard cosmological color law specific to SNe Ia and additional dust-based color laws and dust extinctions that vary with each galaxy/SN. This approach is preferred because of its novel replication of the observed relationships between SN color and residual Hubble diagram scatter as well as its ability to replicate the 'mass step' as a function of SN Ia color. We use the scatter model parameters from BS21 with improvements from Popovic et al. (2021a) in our baseline bias-correction simulations; because the BS21 model includes within it the parent c population, we also utilize the separate parent population for x 1 derived by Popovic et al. (2021b). Improving upon Scolnic \n& Kessler (2016), Popovic et al. (2021b) fit for parent populations in bins of mass to account for host-SN Ia relationships. Popovic et al. (2021b) split their populations into high- and low-redshift groups, and notably for low-redshift surveys the x 1 populations are fitted with a two-Gaussian model to recreate the observed double peak in the x 1 distribution. \nSystematics: We include two categories of systematics for the intrinsic scatter model and parent populations: (1) different models of intrinsic scatter, and (2) determination of parameters for the BS21 model. For the former, we use two additional scatter models from Kessler et al. (2013) that have been used in previous cosmology analyses (JLA, Pantheon, DES3YR). These are (1) the 'G10' model based on Guy et al. (2010) which describes ∼ 70% of the excess Hubble scatter from 'gray' variations and the remaining scatter from wavelengthdependent variations, and (2) the 'C11' model based on Chotard (2011) which describes ∼ 30% of the excess Hubble scatter from coherent variations, and the remaining scatter from wavelength-dependent variations. For the G10 and C11 scatter models, bias corrections are performed in 7-D as given by Popovic et al. (2021b). For the systematic uncertainty in the determination of the BS21 model parameters we adopt three different viable sets of dust and intrinsic SN populations from Popovic et al. (2021a). These populations are the best-fit (maximum likelihood) parameters (hereafter P21), the mean posterior set of parameters, and a set that represent a 1 σ fluctuation in the uncertainty. Lastly, while the BS21 and P21 models impact the simulated bias corrections, the SALT2 training and light-curve fitting has not been altered. The choice of scatter model is propagated through the simulations used for the bias corrections applied in Eq. 1 and for the uncertainty modeling in σ scat of Eq. 4.", '3.3.3. Distance-Modulus Uncertainty Modeling': 'Purpose: To match the reported SN distance-modulus uncertainties (Eq. 3) to the scatter in distance that is observed in the data. \nBaseline: The BS21 model parameters have been fit to the observed scatter in the dataset. We can utilize large BS21 simulations to determine σ scat ( z, c, M glyph[star] ) after accounting for selection effects. The efficacy of this method is shown in Fig. 3, which demonstrates good agreement between the observed RMS of the Hubble residuals and the uncertainties of the distance-modulus values. \nSystematics: To conservatively account for how SN cosmology was done in the past (JLA, Pantheon), in Eq. 3 we set σ scat ( z, c, M glyph[star] ) = 0 and allow only a single σ gray \nFigure 3. Pantheon+ distance-modulus uncertainties (shown as dashed lines with mean σ µ and split on host mass) in comparison to the observed root-mean square (RMS) of the distance-modulus residuals (shown as solid lines as RMS µ split on host mass), as a function of color. This shows that the distance errors are adequately modeled (Eq. 4) as a function of SN color and host stellar mass. In previous analyses, the uncertainties were roughly flat as a function of color. \n<!-- image --> \nparameter to replicate the methodology used with historic intrinsic scatter models (G10 and C11). However, we note that for G10 and C11, the trends in RMS seen for the data in Fig. 3 do not match the reported uncertainties.', '3.3.4. Validation': 'Purpose: To verify that our analysis can recover input values in data-sized simulated samples and does not produce biases. Such tests are sensitive to the light-curve fitting and BBC technique (as well as implementation and coding errors); however, they are not sensitive to certain aspects of the analysis such as the assumption of the SALT2 model or photometric calibration. \nBaseline: We perform an end-to-end test of our baseline analysis pipeline from survey photometry catalog-level simulations. We create 20 realizations of each survey in an arbitrary cosmological model ( w = -1): 10 with the BS21 scatter model and 10 with the G10 scatter model. We perform light-curve fitting, apply bias corrections, compile into 10 Hubble diagrams, and maximize the cosmological likelihoods (Eq. 9) using a fast cosmology grid-search program in SNANA (Kessler et al. 2009), with approximate priors from CMB measurements (Planck Collaboration et al. 2018) to obtain best-fit cosmological parameters and uncertainties. For the BS21 model simulations we recover a mean best-fit w = -1 . 012 ± 0 . 011 \nTable 2. Standardization Parameters and Results \nNotes : The nuisance parameters, as defined in Eq. 1 and 3 are given here for different assumptions about the intrinsic scatter model, as described in Sec. 3 (Intrinsic Scatter Model). That σ int ∼ 0 and γ ∼ 0 for the BS21 and P21 models are due to modeling the scatter and mass-step as part of the BBC process, which is discussed in further detail in Appendix A. The BS21 is the baseline choice for intrinsic scatter. The RMS is given in units of mag. The Hubble diagram likelihood values for each model ( L ) include an uncertainty normalization term. \nand for the G10 model simulations we recover a mean best-fit w = -0 . 983 ± 0 . 015; both are within ∼ 1 σ of the input cosmology. The 20 realizations are made available publicly 4 along with bias-correction simulations.', '4.1. Standardization Parameters': 'The standardization nuisance parameters α , β , γ , and σ gray defined in Eq. 1 and 3 are shown in Table 2 for each of the scatter models used in this work. The best-fit α are similar across scatter models to within ∼ 1 σ . The best-fit β values differ across models owing to different treatments of SN Ia color; however, the values for the baseline dust model (BS21) and the P21 dust model are self-consistent. \nAs shown in Table 2, the additional σ gray term for the BS21 and P21 models is found to be zero. As discussed in Sec. 2, this is consistent with the expectation that if the simulations correctly model the intrinsic scatter and noise of the data, the σ scat ( z, c, M glyph[star] ) term of Eq. 3 is sufficient to describe the distance-modulus uncertainties with σ gray = 0. As discussed in Appendix A, for our G10 and C11 systematic treatment, σ scat ( z, c, M glyph[star] ) is set \nto 0, and therefore σ gray ≈ 0 . 10 approximates the scatter, though it does not account for the observed color dependence. \nTable 2 also shows that the best-fit host stellar mass corrections ( γ ) are consistent with zero for BS21 and P21. This is in agreement with the findings of Popovic et al. (2021a), that modeling the intrinsic scatter in bias-correction simulations with correlations that match those in the observed data removes the need for ad hoc corrections in intrinsic brightness (i.e., γ = 0). This can also be seen in Fig. 5. For the bias correction based on the G10 and C11 models that do not include any mass dependence, the resulting γ is ∼ 0 . 05 found at 7 σ confidence.', '4.2.1. The Hubble Diagram': 'The Pantheon+ Hubble diagram of 1701 SN Ia light curves compiled from 18 different surveys and ranging in redshift from 0.001 to 2.26 is shown in the top panel of Fig. 4. In the bottom panel of Fig. 4 are the residuals to the best-fit cosmology (Eq. 10). Best-fit cosmological parameters will be presented in the following subsections. \nShown in Table 2 is the total observed scatter (RMS) in the Hubble diagram residuals to the best-fit model (bottom of Fig 4) for different scatter models. The BS21 model results in the lowest Hubble diagram RMS and χ 2 , a > 5 σ improvement determined from the difference in likelihoods relative to the G10 and C11 scatter models. The observed scatter of ∼ 0 . 17 mag is larger than seen in the original Pantheon because Pantheon+ extends to lower redshifts and thus is more impacted by scatter induced by peculiar velocities. If we set the minimum redshift to 0.01, the total scatter is reduced to 0.15 mag, matching that of Pantheon. Finally, compared to the original BS21 analysis, P21 uses a more rigorous fitting process that is optimized to better characterize SN Ia colors and intrinsic scatter in addition to Hubble residuals. For this reason, the improvements of P21 are not solely described by the cosmological model likelihood L of Table 2. We therefore have included the use of P21 population parameters as a systematic uncertainty.', '4.2.2. The Very Nearby Hubble Diagram': "We note from Fig. 4 that in the very nearby Universe, z < 0 . 008 ( v < 2400 kms -1 ), the mean of the Hubble diagram residuals is positive by ∼ 5% at ∼ 2 σ significance. This is seen after the use of peculiar-velocity maps from either 2M++ or 2mrs. A similar signal is also seen in \nFigure 4. Top panel : The Pantheon+ 'Hubble diagram' showing the distance modulus µ versus redshift z . The 18 different surveys are each given different colors. Bottom panel : The distance-modulus residuals relative to a best-fit cosmological model with binned data for reference (black points). Both the data errors and the binned data errors include only statistical uncertainties. At z < 0 . 01, the sensitivity of peculiar velocities is very large, and the uncertainties shown reflect this uncertainty. Dashed line is the predicted Hubble residual bias stemming from biased redshifts due to volumetric effects in the very nearby universe (assuming 250 km s -1 uncorrected velocity scatter). \n<!-- image --> \nFigure 5. Pantheon+ sample Hubble diagram residuals (teal) to the best-fit cosmology ( µ -µ model ) for the baseline analysis as a function of SALT2 c , SALT2 x 1 , and hostgalaxy stellar mass M glyph[star] . Distances ( µ ) follow Eq. 1 and include α , β , δ bias , and δ host corrections. Binned data are shown for reference (black). No significant residual correlations are seen. \n<!-- image --> \nFigure 6. The systematic covariance matrix as defined in Eq. 7. To show the inherent structure, the dataset is sorted by survey and within each survey (colored boxes), by redshift. 'CALIB' are the set of 81 SN light curves in the SH0ES Cepheid-calibrator galaxies. The shading corresponds to the size of the covariance in magnitudes. \n<!-- image --> \n- \n0 \n. \n0020 \nthe Hubble residuals of the Cepheid distances (Kenworthy et al. 2022). A bias of roughly this size and direction is expected in the presence of measurement errors and unmodeled peculiar velocities which scatter more objects down from higher redshifts and greater volume than from the reverse. This effect is significant only for the most nearby galaxies ( z < 0 . 008). In Fig. 4, we include the prediction (dashed line) for this bias assuming 250 km s -1 uncorrected velocity scatter (not a fit). \nFigure 7. Visualizing the impact of a number of the top systematic uncertainties in this analysis. The µ residuals are described by Eq. 6. Each of these systematics is explained in Sec. 3 and are combined to form the covariance matrix shown in Fig. 6. Fragilistic provides 9 systematic sets of trained SALT2 models, zero-point solutions, and filter central wavelengths. Here we show the impact on distance of just the first 3. \n<!-- image --> \nIn the the 3-rung distance ladder utilized to measure H 0 by the SH0ES Team (R22) and in Eq. 14 in this work, the nearby ( z < ∼ 0 . 01) Hubble diagram is not used. Rather, only the distance moduli from such nearby SNe are used in the SN-Cepheid absolute distance calibration in the 2 nd rung. Furthermore, in the R22 measurement of the Hubble flow, only SNe with redshifts z > 0 . 023 are used in the 3 rd rung to limit sensitivity to peculiar velocities. This approach is insensitive to the volumetric redshift scatter effects and there is no resulting impact on the R22 H 0 . However, more local measurements of H 0 from, for example, a 2-rung distance ladder using primary distance indicators like Cepheids and TRGB and their host redshifts (mostly at z ≤ 0 . 01) are more sensitive to peculiar velocities and the volumetric bias they induce, and are likely to be biased low at the few percent level if not appropriately accounting for this expected bias (Kenworthy et al. 2022). For measurements of other cosmological parameters (e.g., w or Ω M ) with Pantheon+ described in the following subsections, the mean Hubble residual bias of the Low-z and Foundation sample is ∼ 2 mmag and ∼ 1 mmag (respectively), and is considered to be negligible. \n4.2.3. The Distance Covariance Matrix \nBuilt following Eq. 7, the 1701 × 1701 systematic distance covariance matrix is shown in Fig. 6. The sample is sorted by survey and redshift to help visualize the co- \nariances. The Hubble diagram residuals (Eq. 10) that are used to build the covariance matrix are shown in Fig. 7 for several example sources of systematic uncertainty. As discussed in Appendix C, the information used to create the Hubble diagram as well as the covariance matrix is publicly available 5 and tools to read in this information are in CosmoSIS. The SDSS subsample contributions to the covariance matrix (Fig. 6) stand out visually due to their strong spectroscopic selection function.", '4.3. Constraints on Cosmological Parameters From Pantheon+ and SH0ES': 'Parameter constraints from the Pantheon+ SNe Ia and SH0ES Cepheid host absolute distances are shown in Table 3 for FlatΛCDM, ΛCDM, Flat w CDM, and Flat w 0 w a CDM. Unless otherwise stated, constraints on cosmological parameters include both statistical and systematic uncertainties. From the Pantheon+ SNe Ia, for a FlatΛCDM model we find Ω M = 0 . 334 ± 0 . 018. We note that SH0ES (R22) utilizes Pantheon+ SNe at z < 0 . 8 to constrain the deceleration parameter and find q 0 = -0 . 51 ± 0 . 024. In a flat universe q 0 = 3Ω M 2 -1, which gives Ω M = 0 . 326 ± 0 . 016, consistent with the result for Ω M reported in this work. Results for H 0 from the inclusion of the SH0ES Cepheid host distances are discussed below. \nThe constraints on Ω M and Ω Λ for a ΛCDM model are shown in Fig. 8. We find Ω M = 0 . 306 ± 0 . 057 and Ω Λ = 0 . 625 ± 0 . 084; a flat universe is within the 68% confidence region and Ω M = 0 and Ω Λ = 0 are together rejected at 4.4 σ using only the SNe. \nFor a Flat w CDM model, from the SNe Ia alone (not including SH0ES Cepheid calibration) we find Ω M = 0 . 309 +0 . 063 -0 . 069 and w = -0 . 90 ± 0 . 14 as shown in the third row of Table 3 and in the blue contour of Fig. 9. This result is consistent within 1 σ of the cosmological constant ( w = -1). \nFor a Flat w 0 w a CDM model, from the SNe Ia alone (not including SH0ES Cepheid calibration) we find w 0 = -0 . 93 ± 0 . 15 and w a = -0 . 1 +0 . 9 -2 . 0 as shown in the fourth row of Table 3 and in Fig. 10. These results are again consistent with a cosmological constant. \nUsing distances and a stat+syst covariance matrix that extends to the Cepheid calibrators (Eq. 15) and combining the Pantheon+ SNe with the SH0ES Cepheid host distance calibration, we are able to robustly and simultaneously constrain H 0 and other cosmological parameters describing the expansion history. While we use SH0ES Cepheid data and covariance in this', 'Pantheon+ ACDM Constraints': 'Figure 8. Confidence contours at the 68% and 95% level for the Ω M and Ω Λ cosmological parameters for the ΛCDM from the Pantheon+ dataset, as well as from the Planck and combined BAO datasets. The constraints from including both the statistical and systematic uncertainties (shaded red) are shown as well as when only statistical uncertainties are propagated (unfilled dashed). We include two lines for reference: one for a flat universe, where Ω M + Ω Λ = 1 and the other that indicates an accelerating universe. \n<!-- image --> \nwork, likewise Pantheon+ distances and covariance are used in Section 5.2 of R22 in order to fit H 0 and q 0 in FlatΛCDM. As shown in the top Pantheon+ & SH0ES section of Table 3, for ΛCDM, Flat w CDM, and Flat w 0 w a CDM we find H 0 = 73 . 4 ± 1 . 1, 73 . 5 ± 1 . 1, and 73 . 3 ± 1 . 1 kms -1 Mpc -1 , respectively. We note that more complex models do not result in decreased H 0 constraining power from the SNe Ia + Cepheids, while this is not necessarily true for other cosmological probes (Sec. 4.4).', '4.4. Constraints on Cosmological Parameters From Multiple Probes': "In this work we combine the Pantheon+ SNe with external cosmological probes: CMB from Planck (Collaboration et al. 2018) TTTEEE-lowE and baryon acoustic oscillations (BAO) from SDSS MGS (Ross et al. 2015), SDSS BOSS (Alam et al. 2017), SDSS eBOSS LRG (Bautista et al. 2020), SDSS eBOSS ELG (Bautista et al. 2020), SDSS eBOSS QSO (Hou et al. 2020), SDSS eBOSS Lya (du Mas des Bourboux et al. 2020), all of which have been implemented in CosmoSIS. The afore- \nFigure 9. 68% and 95% confidence contours for Flat w CDM for cosmological parameters Ω M , H 0 , and w . The contours from the Pantheon+ (red), Pantheon+ & SH0ES combined dataset (teal), Planck Collaboration et al. (2018) TTTEEE-lowE constraints (gray). The combination of Planck and Pantheon+ (blue) is also shown, which is consistent with a cosmological constant. Planck constraints are bounded by 0 . 2 < Ω M < 0 . 4 for computational speed. The histograms depict marginalized relative probabilities between probes. \n<!-- image --> \nTable 3. Results for Cosmological Models \nNotes: Summary of marginalized parameter constraints for Pantheon+ and other external probes. The mean and 68% confidence limit are provided for each cosmological parameter. A blank value indicates a parameter not used in the cosmological fit. \nFigure 10. Constraints for Flat w 0 w a CDM from the Pantheon+ dataset in combination with SH0ES, Planck TTTEEE-lowE. \n<!-- image --> \nFigure 11. Constraints for Flat w 0 CDM from the Pantheon+ dataset in combination with Planck & galaxyBAO or Planck & allBAO. \n<!-- image --> \nFigure 12. Constraints for Flat w 0 w a CDM from the Pantheon+ dataset in combination with Planck & galaxyBAO or Planck & allBAO. \n<!-- image --> \nmentioned BAO constraints are denoted 'allBAO'; we also provide constraints from the combination of spectroscopic redshift galaxy-only subset of BAO probes denoted 'galaxyBAO.' We report constraints in Table 3 for combinations of datasets that are deemed compatible and discussed below. \nFor a Flat w CDM model when combining Pantheon+ and Planck we find w = -0 . 982 +0 . 022 -0 . 038 and Ω M = 0 . 325 +0 . 010 -0 . 008 , and when further including allBAO we find w = -0 . 978 +0 . 024 -0 . 031 and Ω M = 0 . 316 +0 . 005 -0 . 008 , both of which are consistent with the cosmological constant at ∼ 3% (Fig. 11). As can be seen in Fig. 9, we do not include SH0ES in combinations with Planck because these measurements are incompatible (R22). \nFor a Flat w 0 w a CDM model when combining Pantheon+ and Planck we find w 0 = -0 . 851 +0 . 092 -0 . 099 and w a = -0 . 70 +0 . 49 -0 . 51 , and when combining Pantheon+, Planck, and BAO we find w 0 = -0 . 841 +0 . 066 -0 . 061 and w a = -0 . 65 +0 . 28 -0 . 32 , which is moderately consistent (2 σ ) with a cosmological constant (Fig. 12). We note that this result is not driven by any single probe. In Fig. 10 we show constraints for Planck alone and for the combination of Planck & Pantheon+. While the broader model freedom of the Flat w 0 w a CDM allows the Planck alone H 0 to be consistent with 73 km s -1 Mpc -1 owing to degeneracy between H 0 and w a (see Fig. 10), after combining Planck with Pantheon+, the H 0 /w a degeneracy is broken (H 0 = 67 . 4 +1 . 1 -1 . 2 kms -1 Mpc -1 ). Therefore, the inclusion of SH0ES with Planck & Pantheon+ results in a Bayesian evidence ratio of -9, and we deem this set of probes incompatible and do not include them in Fig. 10 nor Table 3.", '4.5. Impact of Systematics on Cosmological Parameter Fits': 'To understand the impact of systematic uncertainties, in Table 4 we group the systematics investigated in this work into four main categories: Calibration/SALT2, Redshifts, Astrophysics, and Modeling. The baseline, systematic treatments ( S ψ ) and scaling priors ( σ ψ ) (as described in detail in Section 3) are summarized for each source. The final three columns of Table 4 relate to fits of the sample when combined with Planck Collaboration et al. (2018) in a Flat w CDM model when isolating that systematic. We define both the change in best fit (∆ w sys ) and the systematic uncertainty contribution to w ( σ sys w ) as follows: \n∆ w sys = w sys -w stat (16) \nσ sys w = √ σ 2 w tot -σ 2 w stat , (17) \nwhere w sys and σ w tot are the cosmological constraints when utilizing C stat+sys and where w stat and σ w stat are the statistical-only constraints when utilizing C stat . \nWe find that the final systematic uncertainty in w ( σ sys = 0 . 019) is comparable yet smaller ( ∼ 80%) than the statistical uncertainty, suggesting that the measurement is not systematics dominated. The largest contribution to the systematic error budget (0.011) is due to the potential for redshift-measurement bias. This is followed by the uncertainties in the Fragilistic calibration offsets and the resulting propagation to SALT2 modeltraining uncertainties and light-curve fitting uncertainties (0.009). Additionally important is the conservative uncertainty that was applied owing to the usage of the new SALT2 training methodology (0.008) as well as the uncertainty in the MW extinction maps (0.008). \nInterestingly, numerous systematic uncertainties are found to be negligible (e.g., BS21 Parameters, G10 versus C11) in the cosmological parameter budget. While certain systematics cause redshift-dependent trends as shown in Fig. 7, they also change the relative scatter of the Hubble residuals. This can most easily be seen for the cosmological likelihood values ( L ) for the distances with different intrinsic scatter models shown in Table 2. If the baseline analysis is significantly preferred (larger L ) by the data over one of the analysis variants, the impact of that systematic on cosmological constraints will be reduced, as is the case for intrinsic scatter. \nAs we have built a covariance matrix that includes the Cepheid calibrators, we can measure H 0 with and without systematic uncertainties. For FlatΛCDM, we find H 0 = 73 . 6 ± 1 . 1 kms -1 Mpc -1 , and when considering only statistical uncertainties from the SNe alone (excluding Cepheid and physical distance calibration uncertainties) σ stat+syst H 0 = 0 . 7 kms -1 Mpc -1 , and σ syst H 0 = 0 . 29 kms -1 Mpc -1 . This suggests that SN systematic uncertainties are not dominating the constraint on H 0 and cannot explain the ∼ 7 kms -1 Mpc -1 difference between Planck and SH0ES. \nIn Figure 13 we show deviations to the best-fit H 0 for each individual source of systematic uncertainty relative to the baseline analysis and assuming ΛCDM. For reference we also show the full SN contribution to the H 0 error bar (dashed). The deviations from the baseline (∆H 0 ) are small and add in quadrature to 0.32 km s -1 Mpc -1 . We note that when assessing redshift-specific systematics, because model redshifts are not used for the SN-Cepheid calibration in Eq. 14, they mainly impact the Hubble-flow SNe (third rung of the distance ladder). \nFinally, to help visualize the impact of systematic uncertainties, we show in Fig. 8 the constraints when in- \nFigure 13. The impact on recovery of H 0 , as explained in Sec. 2, of the systematic uncertainties described in Table 4. The units of these measurements are km s -1 Mpc -1 . The dashed lines are given at ∆H 0 of 0.7, which is the entire contribution of the uncertainty in R22 from SN measurements. \n<!-- image --> \ncluding either statistical-only uncertainties or the combined statistical and systematic uncertainties. Error budgets for different cosmological parameterizations can be generated with the delineated files for systematics provided as part of this release.', '4.6. Local Structure in the SN Ia Hubble Diagram': 'Large compilations of SN distances have provided impetus for searches of local structure, over/underdensities, and proper motion (e.g., Mathews et al. 2016; Soltis et al. 2019; Hu et al. 2020). As an initial study, we create sky maps of the SN Hubble diagram residuals (see Fig. 14) and examine two specific areas on the sky that have been documented in the literature and have sufficient SN statistics in the Pantheon+ sample for study.', '4.6.1. The CMB Kinematic Dipole': 'The motion of the Milky Way and Solar System relative to the CMB rest frame ( v = 369 . 82 kms -1 ) is corrected for following Carr et al. (2021) and Peterson et al. (2021). The effect of the CMB dipole motion can be seen in the z HEL sky map (middle right of Fig. 14), where z HEL is the heliocentric redshifts. The z CMB skymap (middle left of Fig. 14) has the CMB dipole-causing peculiar redshift removed following Eq. 7 of Peterson et al. (2021). The direction of the CMB dipole, l = 264 · and b = 48 · (red o in Fig. 14), is shown for reference as well as its antipole (red x). \nTable 4. Sources of Uncertainty \nNotes : A summary of the systematic uncertainties and the baseline component of the analysis as described in Sec. 3, the size of the systematic S Ψ used to determine the impact of that systematic, the scaling of the systematic σ ψ as constrained in this analysis, and the contribution to the total uncertainty in w CDM (can be compared to statistical uncertainty of 0.03), and the shift when allowing the uncertainty on the best-fit cosmological parameter. The last column shows the simplistic change in best-fit cosmology if a perturbation of size σ ψ is applied with statistical-only uncertainties. The amount shown is different than seen for the combined shift for best-fit and increase of uncertainty given in the previous columns due to the self-calibration as explained by Brout et al. (2021). \nAs discussed in Section 3.1.3, we examine different velocity reconstructions due to local structure that include estimates of the bulk flow; these are the 2M++ (Carrick et al. 2015) and 2MRS (Lilow & Nusser 2021) corrections and are shown in the top row of Fig. 14. These corrections also include the CMB dipole correction. Peterson et al. (2021) show that the peculiar velocity corrections overall reduce the Hubble residual scatter by \n∼ 10%, and this is qualitatively confirmed in our maps. The heliocentric map shows a strong dipole as expected; the z CMB map shows the dipole somewhat removed but with an overcorrection (as expected at lowz because local galaxies share some of our motion); and both z HD maps show that the peculiar velocity corrections have removed most of the overcorrection. \nFigure 14. Healpix (NSIDE=16) Hubble residual sky maps (colorbar is residual magnitudes) with 20 degree 2D-Gaussian kernel smoothing, and Hubble residuals for two selected apertures. z > 0 . 01 is applied. Dots show the locations of the SNe in the Pantheon+ sample, with white dots showing the nearby SNe ( z < 0 . 15) and black dots the distant SNe ( z > 0 . 15). Top left: ) Hubble diagram corresponding to the baseline analysis utilizing both z CMB dipole corrections and 2M++ peculiar velocity corrections. The circled regions designate the 20 degree regions centred on the negative CMB dipole (red) and CMB cold spot directions (blue). The small circle in the top right (and x in bottom left) of each figure represent the direction (and opposite direction) of the motion causing the CMB dipole Top right: ) same as top left but instead using 2MRS peculiar-velocity corrections Middle left: ) same as top left but instead not applying any peculiar-velocity corrections Middle right: ) same as top left but instead not applying either peculiar-velocity corrections nor the CMB dipole correction. Bottom left: ) 20 degree region aligned with the (opposite) CMB dipole velocity depicting Hubble diagram residuals as a function of redshift. Bottom right: ) same as bottom left but with aperture centered at the CMB cold spot ( l = 209 · , b = 57 · ), and over a higher redshift range. \n<!-- image --> \nHowever, both reconstructions produce a small signal that can be seen in the maps in the direction opposite the motion causing the CMB dipole. This signal is found to be local, at z < 0 . 02, and grows with decreasing redshift until z ≈ 0 . 01 (bottom left of Fig. 14). A possible reason that there is a residual signal in the negative dipole direction in both the z CMB and peculiar velocity corrected redshifts is that the MW motion is coupled with the motion of nearby galaxies in a way that is not yet sufficiently modelled. It is also likely that this is due to low-number statistics (this is only a 1 σ deviation) and the uneven sky coverage (the SNe in this region are mostly clustered in Stripe-82). Lastly we note that the positive residuals are driven by SNe at z < 0 . 02, and thus are not included in the SH0ES (Riess et al. 2021) sample and inference of H 0 .', '4.6.2. The CMB Cold Spot': "The 'CMB cold spot,' a 5 · region of -70 µK centered at ( l ∼ 209 · , b ∼ -57 · ), was first detected in data from the Wilkinson Microwave Anisotropy Probe (Vielva et al. 2004; Cruz et al. 2006), and subsequently in Planck data (Gurzadyan et al. 2014). Evidence for an underdensity aligned with the CMB cold spot was presented by Rudnick et al. (2007). Szapudi et al. (2015) and Kov'acs et al. (2021) subsequently found the Eridanus supervoid in the direction of the cold spot at z ≈ 0 . 15. However, it is not clear if the alignment of Eridanus and the CMB cold spot is causal or coincidental. \nWe find a signal in the Pantheon+ Hubble diagram when examining SNe within a 20 · radius of the location of the CMB cold spot ( blue circle region in top-left Fig. 14). The difference in Hubble diagram residuals as a function of redshift is shown in the bottom-right panel of Fig. 14. There are 9 SNe in this region of the sky with redshifts on the near side (0 . 12 < z < 0 . 15) and there are 14 SNe on the far side (0 . 15 < z < 0 . 20) of the proposed void at z = 0 . 15, and there is a Hubble residual difference of -0 . 15 ± 0 . 06 mag between these two sets of SNe. For an estimate of the significance, we examine 1000 randomly selected 20 · apertures across the sky with at least 8 SNe in each of the near and far redshift ranges split on redshifts between 0.08 and 0.20, and find that deviations with a similar significance occur only 0.2% of the time. We note however, that there are not many independent regions that satisfy the selection criteria and the vast majority of the SNe in the cold-spot selection come from the small deep-field patch within that region. Taking 100 random samples of 10 degree radius from the largest densely sampled region in Pantheon+ (Stripe-82 region) we find no other \npatch has a significance that exceeds 1.6 σ , making the Eridanus patch the most significant step at that redshift in our data.", '5. DISCUSSION': "This analysis is the latest in a series of papers that attempt to both grow the compilation of measured SN Ia light curves and improve on the systematic floor. The two most recent compilations and analyses are those of JLA and Pantheon, which respectively included ∼ 40% and ∼ 60% of the SN light curves analyzed here. As seen in Fig. 1 of Scolnic et al. (2021), the majority of the statistical increase for Pantheon+ is in the addition of numerous low-redshift samples extending down to z = 0 . 001. However, the largest differences in the Hubble diagram are not solely the result of statistical increase, but rather due to improvements in our methodology. \nWe show in Fig. 15 the difference in inferred distancemodulus values (marginalized over M ) for the Pantheon+ sample relative to the assumptions used in the JLA analysis, for the three most significant improvements presented in this work. First is the update in the flux cross calibration to the Fragilistic solution, which impacts both the training of the SALT2 model and the zero-points used in light-curve fitting. Second is the impact from updating the MW extinction curve used in JLA (Cardelli et al. 1989) to the Fitzpatrick (1999) relation that is used here. Third is the change resulting from improved modeling of the SN Ia intrinsic scatter; while in this work we adopt the BS21 model, we include the models developed for JLA (G10 and C11) as systematics. Each of these changes has been motivated externally by previous works (e.g., Brout et al. 2021; Schlafly & Finkbeiner 2011b; Brout & Scolnic 2021); however, they nonetheless cause shifts in d µ/ d z of ∼ 0 . 05, or ∼ 0 . 04 in w . Finally, because all of three of these changes have the same sign of d µ/ d z slope, rather than canceling each other, when combined in this work they result in a ∼ 0 . 1 difference in the constraint on w relative to JLA (after combining with CMB). \nAs discussed by Scolnic et al. (2019), the constraining power of large samples of SNe Ia extends beyond inferences of H 0 and w/ Ω M . Large compilations of lowz SNe Ia enable precision measurements of the local growth-of-structure, typically parameterized by fσ 8 (e.g., Huterer et al. 2017; Stahl et al. 2021). Work is ongoing for this measurement using the Pantheon+ sample (Boruah et al., in prep.), which will include validation with simulations as well as propagation of the covariance matrix, which previously would have limited effect on σ 8 calculations owing to smoothing/binning over redshift. \nFigure 15. Largest differences in analysis compared to Betoule et al. (2014) and Scolnic et al. (2018b). (Top panel) Updating the extinction curve used in the light-curve fitting from CCM to F99; (Middle panel) Updating the SALT2 model, as discussed in Brout et al. (2021); (Bottom panel) Changing the baseline assumption for the intrinsic scatter to the P21, G10, and C11 models. \n<!-- image --> \nWhile in Sec. 4 we show a Healpix map of Hubble residuals across the sky, there are additional and related tests of anisotropy that can be performed with these data. Previous analyses of the first Pantheon sample (e.g., Colin et al. 2019; Soltis et al. 2019; Andrade et al. 2018; Brownsberger et al. 2019) typically search for radial or hemispherical residuals across the sky. The addition of statistics in the low-redshift sample and improved accounting in Pantheon+ would particularly strengthen these types of studies. A search for \nmatter over/underdensities was performed by Colg'ain (2019), which varied the minimum and maximum redshift in the original Pantheon sample and redetermined cosmological constraints. Colg'ain (2019) found for Pantheon that Ω M could be < 0 for a low maximum z of ∼ 0 . 15, though with only ∼ 2 σ difference compared to the value of Ω M from the full sample. We show a similar test in Fig. 16 and find relatively stable values of Ω M with no signs of the underdensity seen by Colg'ain (2019). \nFigure 16. Constraints on Ω M in FlatΛCDM when the bounds of the redshift range of the sample are changed. In the top panel, the minimum redshift is varied. The nominal minimum redshift is 0.01 for Pantheon+ cosmology fits without SH0ES. In the bottom panel, the maximum redshift is varied. The nominal maximum redshift is 2.4 for all fits. \n<!-- image --> \nThe main goal of this work, constraints from SNe Ia alone for a Flat w CDMmodel, results in stat+syst uncertainties of +0 . 058 -0 . 063 and 0.13 for Ω M and w , respectively. This represents a factor of 2 improvement in figure of merit over the original Pantheon (stat+syst uncertainties 0.072 and 0.22 for Ω M and w ). This cannot be explained solely by statistical improvements, but rather is also due to a leap in systematics methodology over the original Pantheon and JLA. As shown by Brout et al. (2021), cosmology uncertainty budgets are improved by a factor of ∼ 1 . 5 when not binning or smoothing data and covariance. In Appendix B we discuss and show a binned error budget for comparison and find a similar factor of 1.5 improvement from this choice alone. In examining the unbinned error budget in Table 4, it can \nbe seen that several systematics are no longer impacting SN Ia cosmology analyses as strongly as had previously been thought. One such example is the negligible size of the parent population systematic despite including three additional sources of scatter model uncertainty, as was also seen by Popovic et al. (2021a). This, as well as the reduction of a number of other systematics in comparison to their size in binned analyses (also shown in Appendix Table 6), is due to the power of the large datasets themselves to self-constrain the size of systematic uncertainties when the systematic itself is not solely degenerate with the cosmological model parameterization. This is especially important because it brings this work from potentially being dominated by systematics to rather being dominated by statistical uncertainties. Furthermore, as shown by Brout et al. (2021), as datasets grow in size, many systematics will continue to shrink without any additional effort. Lastly, it is important to note that approaches such as the Approximate Bayesian Computation method given by Jennings et al. (2016) will not be able to make use of this self-constraining benefit unless additional parameters are included to allow the data themselves to scale the input sizes of the systematic uncertainties ( S sys in Brout et al. 2021). \nWhile the SN Ia mass step has received much attention in the last decade, we find here that its contribution to the error budget is exceedingly small. Unlike previous analyses, the mass-step treatment in this work is based on a SN color and dust-dependent model (BS21). We find that this more physical model results in smaller scatter in the Hubble diagram (Table 2) and better χ 2 relative to cosmological models which then results in smaller systematic uncertainties. We note that properties of SN Ia host galaxies other than stellar mass have been seen to correlate with SN Ia Hubble diagram residuals. Star-formation rate, specific star-formation rate (sSFR), stellar-population age, and metallicity have all been shown to correlate to varying degrees with the distance-modulus residuals after standardization (Sullivan et al. 2010; Lampeitl et al. 2010; Childress et al. 2013; Rose et al. 2019; Rigault et al. 2013). For this reason, using sSFR values presented by S22, we also examined the size of a sSFR step in the subset of the Pantheon+ sample for which we have obtained sSFR measurements ( z < 0 . 2). Without applying any bias corrections, we find a significant step in sSFR (across the median sSFR) of 0 . 031 ± 0 . 011. However, after applying the nominal set of dust and mass-based bias corrections (BS21) used in this analysis, we find a step in sSFR of 0 . 008 ± 0 . 011, consistent with zero. This is likely due to galaxy properties (i.e., stellar mass) being linked to dust properties, and that applying a dust-mass correction is \naccounting for most, if not all, of the correlations with sSFR and is also tracing the dust distribution. \nGoing forward, statistical constraints on w and Ω M from SNe will improve significantly owing to upcoming datasets from SN programs of the Dark Energy Survey (D'Andrea et al. 2018), Zwicky Transient Facility (ZTF; Dhawan et al. 2022), Young Supernova Experiment (YSE; Jones et al. 2021), Legacy Survey of Space and Time (LSST; The LSST Dark Energy Science Collaboration et al. 2018; S'anchez et al. 2021), Nancy Grace Roman Telescope (Hounsell et al. 2018), etc. It is likely that these future datasets will improve the statistical precision by a factor of 100 (Scolnic et al. 2018a). \nThe size of systematic errors on cosmological parameter estimates matched the statistical errors for JLA and the original Pantheon. Systematic uncertainties in this work have been reduced in comparison to Pantheon, and while their impact is still significant, it is no longer the dominant component of the total uncertainty. With the coming surveys, systematics will also likely improve alongside the increase in statistics, as has been the case for previous analyses over the last two decades , and as expected from the impact of systematic 'self-calibration' described in Brout et al. (2021). \nAs shown in the systematics error budget Table 4, the dominant sources of systematic uncertainty are now from 1) the combination of SALT2 training and calibration of surveys, 2) potential redshift measurement biases, and 3) Milky Way dust systematics. Fortunately there are paths forward for each of these. For survey flux calibration, dedicated programs are needed and there are currently multiple paths underway to improve the fundamental calibration of SN Ia samples and how they are tied to various other samples (e.g., Regnault et al. 2015; Narayan et al. 2019; Stubbs & Brown 2015). There is also ongoing work (Taylor et al. in prep) to train the SALT2 model with more photometric systems which has already shown promising improvements to systematic uncertainties and the ability to constrain rest frame U-band. The systematic from the redshift measurement floor has the potential to be reduced using improved cosmology fitting methodology, although the extent to which the data itself can constrain the size of this floor remains unproven. Alternatively, future large surveys can use multiple spectroscopic instruments and redshifting codes to mitigate potential sources of redshift measurement bias. The Pantheon+ sample is especially sensitive to Milky Way dust systematics because of the differences in the samples used for low and high redshift. At low redshift, to obtain sufficient statistics in a volume limited sample, we have used SNe across the sky and with up to 0.2 in MWEBV, whereas the high redshift \nsurveys have been carried out in low extinction regions of the sky (MWEBV < 0.05). Future surveys of larger volumes will be able to mitigate this with a plethora of both low and high redshift in low MW extinction regions on the sky. \nThroughout this work, there are a number of upstream components of this analysis that impact downstream analysis steps: i.e. new calibration (or MWEBV Maps/Color law) motivates new SALT2 training, which motivates new fitting of the SN parent populations, which motivates new bias corrections. The Pippin framework (Hinton & Brout 2020), used extensively in this work, was intentionally developed to automate and asynchronize this multistep type of analysis; however, it has yet to incorporate aspects such as the SALT2 retraining (Taylor et al. 2021) or population fitting (Popovic et al. 2021a). Likely, this framework will need to expand for future analyses. \nThere is an alternate approach to obtaining cosmology constraints from SNe that has been gaining traction over the last decade. Bayesian Hierarchical Models (BHM) have been developed that utilize bias-corrected observables (Shariff et al. 2016) and that incorporate selection effects directly into the model (Rubin et al. 2015) or likelihood (Hinton et al. 2019). However, unlike BBC in combination with CosmoSIS, these methods have not been validated with large realistic simulations. As noted in Appendix C, we release as part of this analysis 10 realistic simulations of the Pantheon+ dataset for such validations. \nWhile constraints on w should easily improve with upcoming large SN samples, the road to improving constraints on H 0 is more challenging. There are a limited number of SNe Ia that will explode in the near future within a ∼ 40 Mpc radius, a constraint due to HST discovery limits of Cepheids. At roughly one SN Ia per year, it will take several decades to double the current sample of 42 SNe calibrated by SH0ES Cepheid hosts. Fortunately, we find that the systematics in the measurement of H 0 from the SNe are at the scale of 0.3 km s -1 Mpc -1 as shown in Fig. 13. This is consistent \nwith the general finding of Brownsberger et al. (2021), who showed how robust H 0 is to systematic uncertainties in comparison to the relatively calibration-sensitive constraints of w 0 or Ω M . Lastly, there is ongoing work that combines the progress used here by Peterson et al. (2021) and applies it to a 'two-rung' distance-ladder analysis, in which SNe are excluded from the distance ladder (Kenworthy et al. 2022).", '6. CONCLUSION': "This work is the culmination of a number of supporting analyses as part of the Pantheon+ effort. In this work, we summarize the various inputs and analyses required to combine the supporting works and ultimately measure distances and cosmological parameters. For the first time we are able to measure the cosmic expansion history and the local distance ladder H 0 simultaneously. We combine our results with additional external probes. Importantly, we release a number of data and analysis products to facilitate reproducing our work by the community. This includes a joint covariance of SNe used for measurements of H 0 and w . \nFor our main results, we find Ω M = 0 . 334 ± 0 . 018 in FlatΛCDM from SNe Ia alone. For a flat w 0 CDM model, we measure w 0 = -0 . 90 ± 0 . 14 from SNe Ia alone and w 0 = -0 . 978 +0 . 024 -0 . 031 when combining SNe with constraints on the CMB and allBAO; both are consistent with a cosmological-constant model of dark energy. We also present the most precise measurements to date on the evolution of dark energy in a Flat w 0 w a CDM universe, and measure w a = -0 . 1 +0 . 9 -2 . 0 from Pantheon+ alone and w a = -0 . 65 +0 . 28 -0 . 32 when combining with CMB and BAO data. Finally, while nominal constraints on H 0 are presented in a companion paper by the SH0ES team (R22), we perform joint constraints of H 0 with expansion history and find H 0 = 73 . 5 ± 1 . 1 in Flat w CDM, and we show how systematic uncertainties in measurements of the SN component of the distance ladder cannot account for the current level of the 'Hubble tension.'", 'REFERENCES': "Abbott, T. M. C., Allam, S., Andersen, P., et al. 2019, \nApJL, 872, L30, doi: 10.3847/2041-8213/ab04fa Alam, S., Ata, M., Bailey, S., et al. 2017, Monthly Notices of the Royal Astronomical Society, 470, 2617-2652, doi: 10.1093/mnras/stx721 \nAndrade, U., Bengaly, C. A. P., Santos, B., & Alcaniz, J. S. 2018, The Astrophysical Journal, 865, 119, doi: 10.3847/1538-4357/aadb90 \nAstier, P., Guy, J., Regnault, N., et al. 2006, A&A, 447, 31, doi: 10.1051/0004-6361:20054185 \nBautista, J. E., Paviot, R., Vargas Maga˜na, M., et al. 2020, Monthly Notices of the Royal Astronomical Society, 500, 736-762, doi: 10.1093/mnras/staa2800 \nBetoule, M., Kessler, R., Guy, J., et al. 2014, A&A, 568, A22, doi: 10.1051/0004-6361/201423413 \nTully, R. B. 2015, The Astronomical Journal, 149, 171, doi: 10.1088/0004-6256/149/5/171 \nVielva, P., Mart'ınez-Gonz'alez, E., Barreiro, R. B., Sanz, \nJ. L., & Cay'on, L. 2004, ApJ, 609, 22, \ndoi: 10.1086/421007 \n- Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods\n- Zhang, T., Wang, X., Li, W., et al. 2010, PASP, 122, 1, doi: 10.1086/649851\n- Zuntz, J., Paterno, M., Jennings, E., et al. 2015, Astronomy and Computing, 12, 45-59,\n- doi: 10.1016/j.ascom.2015.05.005 \nTable 5. Distance Bias (and Uncertainty) Estimation for Scatter Models \nNotes: Formalism for 4d and 7d bias corrections are described by Popovic et al. (2021b) that depend on the intrinsic scatter model assumed - either G10/C11 or BS21/P21. The statistical and intrinsic scatter uncertainties from Eq. 3 are shown here; the other uncertainty components from Eq. 3 are independent of the scatter model.", '7. ACKNOWLEDGEMENTS': "D.S., D.B., and A.R. thank the John Templeton Foundation for their support of grant #62314. D.B. acknowledges support for this work provided by NASA through NASA Hubble Fellowship grant HST-HF2-51430.001 awarded by the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. D.S. is supported by DOE grant DE-SC0010007, the David and Lucile Packard Foundation, and NASA under Contract No. NNG17PX03C issued through the WFIRST Science Investigation Teams Programme. We acknowledge the generous support of Marc J. Staley, whose fellowship partly funded B.E.S. whilst contributing to the work presented herein as a graduate student. A.V.F. is grateful for support from the TABASGO Foundation, the Christopher R. Redlich Fund, the U.C. Berkeley Miller Institute for Basic Research in Science (in which he was a Miller Senior Fellow), and many individual donors. S.N. thanks the STFC Ernest Rutherford Fellowship for support via grant ST/T005009/1 L.K. thanks the UKRI Future Leaders Fellowship for support through the grant MR/T01881X/1. This work was completed in part with resources provided by the University of Chicago's Research Computing Center. The Katzman Automatic Imaging Telescope (with which the LOSS samples were obtained) and its ongoing operation were made possible by donations from Sun Microsystems, Inc., the Hewlett-Packard Company, AutoScope Corporation, Lick Observatory, the NSF, the University of California, the Sylvia & Jim Katzman Foundation, and the TABASGO Foundation. Research at Lick Observatory is partially supported by a generous gift from Google. \nSimulations, light-curve fitting, BBC, and cosmology pipeline are managed by PIPPIN (Hinton & Brout 2020). Contours and parameter constraints are generated using the ChainConsumer package (Hinton 2016). Plots are generated with Matplotlib (Hunter 2007). We use astropy (Price-Whelan et al. 2018), SciPy (Virtanen et al. 2020), and NumPy (Oliphant 2006). Analysis and visualisations provided in part by https://github.com/bap37/Midwayplotter. Brout thanks his spouse Isabella and their future daughter for their support as the due date is rapidly approaching!", 'A. ADDITIONAL FORMALISM FOR DISTANCE AND UNCERTAINTY ESTIMATION': 'As shown in BS21, SN Ia scatter has both a color and host-mass dependence (increasing scatter) and a redshift dependence that arises from selection effects (decreasing scatter). In this work we introduce a new method of accounting for the uncertainties using the scatter model predictions. We include σ scat ( z, c, M glyph[star] ) from simulations as an additive uncertainty inside Eq. 3 rather than the multiplicative uncertainty f ( z, c, M glyph[star] ) on the computed σ meas that has been used in past analyses. The σ scat ( z, c, M glyph[star] ) term is computed from simulations that use the choice of scatter model. The BBC process, after correcting distances for selection effects, determines the magnitude of σ scat ( z, c, M glyph[star] ) in each z, c, M glyph[star] bin by requiring that the observed-simulated distance reduced χ 2 in each bin is unity. If the simulations using a model of intrinsic scatter fully describe the observed scatter in the data, the uncertainty modeling term in Eq. 3, σ scat ( z, c, M glyph[star] ), will cause σ gray to be 0. \nIn the case of the decrease in observed scatter at high redshift arising from only intrinsically bright/blue events being selected at the limits of the telescope (Kessler et al. 2015), we instead apply as a downscaling of f ( z, c, M glyph[star] ) of the reported measurement uncertainty and set σ scat ( z, c, M glyph[star] ) = 0. Conversely, for bins of z, c, M glyph[star] with χ 2 greater than \nunity, the necessary σ scat ( z, c, M glyph[star] ) is applied and f is set to 1. The resulting f ( z, c, M glyph[star] ) and σ scat ( z, c, M glyph[star] ) found from simulations are applied to the Pantheon+ data. \nThe method and dimensionality for the application of bias corrections is dependent on the adopted scatter model. Table 5 summarizes the differences between the two main methods used in this work, the first of which is applied when assuming the BS21/P21 scatter model, and the other when assuming the G10 or C11 scatter model. The main difference between these groups of scatter models, as discussed in Sec. 3, is whether the intrinsic scatter is driven by diversity in the reddening ratios R V of the light curves, which affects the application of bias corrections. For both analysis paths, we follow the methodology introduced by Popovic et al. (2021b).', 'B. BINNED SYSTEMATIC ERROR BUDGET': 'In Table 6 we show a systematic error budget that is nearly identical to what was performed in Table 4, except that the dataset (∆ D ) and covariance matrix ( C stat+syst ) are binned in 20 redshift bins. This error budget is similar to the methodology performed in the most recent SN cosmology analyses where binned covariance matrices were used (e.g., Pantheon, DES3YR Brout et al. 2019a) and where smoothed data vectors and matrices (which were shown to be equivalent to binned) were used (JLA). The total systematic error when binning is a factor of 1.5 larger (0.029) than when not binning the dataset (0.019). \nSystematics that improve the most with unbinned matrices are those with smaller σw unbinned sys /σw binned sys . Binned analyses collapse valuable information in the Hubble diagram down to a single dimension, redshift. We find that as expected, the redshift bias systematic does not improve much at all. This is because systematics that only exhibit redshift dependence are degenerate with cosmological model parameters and cannot be self-constrained by the data as easily. Systematics that exhibit dependence in other parameters (such as SN color) can be drastically reduced in SN Ia cosmological parameter error budgets when not performing binned analyses.', 'C. PRODUCTS': 'The following data products that are provided in part by the full suite of Pantheon+ supporting papers are released publicly in machine readable format 6 at pantheonplussh0es.github.io and as part of SNANA and CosmoSIS (where noted). \n- · Light-Curve Photometry, Redshifts, and Host-Galaxy Properties; from S22 and Carr et al. (2021)\n- · Trained SALT2-B22 Model; from Brout et al. (2021)\n- · SALT2 Fit Parameters; from S22\n- · 10 Catalog Level Simulations of Pantheon+ Light-Curve Fit Parameters; this work\n- · SN/Host Redshifts and Peculiar Velocities; from Carr et al. (2021)\n- · SN Distance Modulii and Redshifts; this work, Carr et al. (2021) , see Table 7 7\n- · SN Distance Covariance; this work\n- · Cepheid Host Distances; from R22\n- · Cepheid Host Distance Covariance; from R22\n- · SN Ia + Cepheid Host Cosmology Likelihood; this work\n- · SN Cosmology Chains; this work \nTable 6. Comparison of Binned and Unbinned Systematic Error Budgets \n- a Constraints are combined with Planck prior.\n- b LCFIT denotes zero-points and filter central wavelengths have been varied during light-curve fitting.\n- c Due to implementation methodology of this systematic, it has not been performed in the binned case. \nTable 7 . The Pantheon+ Hubble Diagram \nFull table available in machine readable format at \nhttps://iopscience.iop.org/article/10.3847/1538-4357/ac8e04.'}
2024arXiv240907729C	We present an integralbased technique IBT algorithm to accelerate supernova SN radiative transfer calculations. The algorithm utilizes integral packets which are calculated by the path integral of the MonteCarlo energy packets to synthesize the observed spectropolarimetric signal at a given viewing direction in a 3D timedependent radiative transfer program. Compared to the eventbased technique EBT proposed by Bulla et al. 2015 our algorithm significantly reduces the computation time and increases the MonteCarlo signaltonoise ratio. Using a 1D spherical symmetric type Ia supernova SN Ia ejecta model DDC10 and its derived 3D model the IBT algorithm has successfully passed the verification of 1 spherical symmetry 2 mirror symmetry 3 cross comparison on a 3D SN model with directcounting technique DCT and EBT. Notably with our algorithm implemented in the 3D MonteCarlo radiative transfer code SEDONA the computation time is faster than EBT by a factor of 1030 and the signaltonoise SN ratio is better by a factor of 510 with the same number of MonteCarlo quanta.	2024-09-01T00:00:00Z	['2024arXiv240907729C', 'arXiv:2409.07729', '10.48550/arXiv.2409.07729']	['Astrophysics - High Energy Astrophysical Phenomena']	An IntegralBased Technique IBT to Accelerate the MonteCarlo Radiative Transfer Computation for Supernovae	2,024	192	0.44	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.07729.pdf	{'An Integral-Based Technique (IBT) to Accelerate the Monte-Carlo Radiative Transfer Computation for Supernovae': 'Xingzhuo Chen , 1 Lifan Wang, 1 and Daniel Kasen 2 \n1 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics & Astronomy, \nTexas A. & M. University, Department of Physics and Astronomy, 4242 TAMU, College Station, TX 77843, USA 2 Astronomy Department and Theoretical Astrophysics Center, University of California, Berkeley, Berkeley, CA 94720, USA', 'ABSTRACT': "We present an integral-based technique (IBT) algorithm to accelerate supernova (SN) radiative transfer calculations. The algorithm utilizes 'integral packets', which are calculated by the path integral of the Monte-Carlo energy packets, to synthesize the observed spectropolarimetric signal at a given viewing direction in a 3-D time-dependent radiative transfer program. Compared to the event-based technique (EBT) proposed by Bulla et al. (2015), our algorithm significantly reduces the computation time and increases the Monte-Carlo signal-to-noise ratio. Using a 1-D spherical symmetric type Ia supernova (SN Ia) ejecta model DDC10 and its derived 3-D model, the IBT algorithm has successfully passed the verification of: (1) spherical symmetry; (2) mirror symmetry; (3) cross comparison on a 3-D SN model with direct-counting technique (DCT) and EBT. Notably, with our algorithm implemented in the 3-D Monte-Carlo radiative transfer code SEDONA, the computation time is faster than EBT by a factor of 10 -30, and the signal-to-noise (S/N) ratio is better by a factor of 5 -10, with the same number of Monte-Carlo quanta. \nKeywords: Supernova", '1. INTRODUCTION': "Type Ia supernovae (SNe Ia) have been widely applied in cosmological studies (e.g. Abbott et al. (2019); Brout et al. (2022)) based on the empirical relation between the light curve properties and the luminosities (e.g. Phillips (1993)). However, the explosion mechanism of SNe Ia is still a mystery; studies of the ejecta structure rely on simulations that include hydrodynamics, nucleosynthesis, and radiative transfer. \nRadiative transfer simulations of supernovae calculate the specific intensity and plasma excitation states to obtain spectra that can be compared to optical spectroscopic and photometric observations. Many radiative transfer codes assume that SNe Ia are spherical symmetric and reasonable agreement have been found among the simulation results of different codes (Blondin et al. 2022). \nCorresponding author: Lifan Wang \[email protected] \nHowever, imaging of nearby SN Ia remnants (e.g. Ferrazzoli et al. (2023)) and hydrodynamic simulations (e.g. Gamezo et al. (2004); Kromer et al. (2013)) suggest that SNe Ia have an asymmetric 3-D structure. Given the distance to most SNe Ia, this 3-D structure cannot be resolved, but can be probed by spectropolarimetric observations (e.g., Wang et al. 2003; Cikota et al. 2019; Yang et al. 2022). The polarization from SN Ia is primarily due to Thomson scattering of photons, and a non-zero linear polarization signal can arise in asymmetric SN ejecta due to the incomplete cancellation of polarization from photons scattered from different geometrical structures (Wang & Wheeler 2008). \nMonte-Carlo methods, which use energy packets to emulate the propagation of photons in the SN plasma, provide a flexible way to treat complicated physical processes (Lucy 2002, 2003) and have been applied in 3D time-dependent radiative transfer codes such as SEDONA (Kasen et al. 2006), and ARTIS (Kromer & Sim 2009). 3-D Monte-Carlo radiative transfer (MCRT) simulations can be computationally expensive, as many photon packets must be propagated to reduce the statistical noise in the synthesized observables. This espe- \ncially poses a challenge for modeling SN polarization, as the expected signals are typically only at the 1% level or less. To reduce the noise in SN MCRT simulations, Bulla et al. (2015) introduced an 'event-based technique' (EBT) for spectropolarimetry synthesis, which has been applied to simulations of SNe Ia (Bulla et al. 2016a,b), superluminous SNe (Inserra et al. 2016), and kilonovae (Bulla et al. 2021). \nIn this paper, we introduce a new integral-based technique (IBT), which is implemented in the MCRT code SEDONA, to efficiently construct observable spectra from a 3-D MCRT computation. We find that IBT is 10 -30 times faster and 5 -10 times less Monte-Carlo noise than EBT, with the same number of Monte-Carlo quanta. Moreover, IBT can be applied to the ultraviolet and infrared wavelength, and early phase of SNe, which was not calculated with EBT (Bulla et al. 2015). Section 2 reviews the direct-counting technique (DCT) and event-based technique (EBT) that have been used in previous MCRT codes. Section 3 presents the IBT method. Section 4 uses several toy models to verify the spectropolarimetry results from IBT. Section 5 compares the Monte-Carlo simulation error and computation time of the three algorithms: DCT, EBT, and IBT. Section 6 summarizes our main conclusions.", '2. POLARIZED SPECTRUM EXTRACTION': 'In the SEDONA code, an optical photon packet (OP) represents a collection of photons with the same frequency, spatial coordinate, and propagation direction. The OPs undergo interaction events due to Thomson scattering, bound-bound transitions, bound-free transitions, and free-free transitions which can change their frequency and direction. \nThe polarization information of an OP is stored as a dimensionless Stokes vector \ns OP = 1 q u v , (1) \nand the Stokes vector can be constructed with s OP and the energy of the OP E OP . \nTypically, OPs are generated with zero initial polarization as expected for thermal emission. During the propagation of an OP, the linear polarization state changes in a Thomson scattering event according to the Rayleigh scattering phase matrix (Section 1.17, equation 217 of (Chandrasekhar 1960)). In bound-bound, bound-free, and free-free interactions, the OPs are typically assumed to be fully depolarized. The current version of SEDONA does not include magnetic fields, and \nthe V parameter in the Stokes vector, which represents the circular polarization, is set to zero.', '2.1. Direct Counting Technique': 'A straightforward method of spectral synthesis in MCRT is the direct counting technique (DCT), which sums the energy of OPs that escape the simulation domain within a certain frequency, time, and viewing angle bin. The observed flux is then calculated as \n I Q U V = 1 4 πr 2 ∆ t ∆ ν ∆Ω ∑ i E OP,i s OP,i (2) \nwhere s OP,i is dimensionless Stokes vector of the i -th OP, E OP,i is the lab frame energy of the i -th OP, ∆ t is the size of the arrival time bin, ∆ ν is the size of the frequency bin, r is the distance to the SN center, ∆Ω is the viewing direction bin size. The summation is over all the OPs in the arrival time interval [ t -∆ t/ 2 , t +∆ t/ 2), the frequency interval [ ν -∆ ν/ 2 , ν + ∆ ν/ 2), and the viewing direction interval [ φ -∆ φ/ 2 , φ + ∆ φ/ 2); [ θ -∆ θ/ 2 , θ +∆ θ/ 2). \nIn the DCT method, the signal-to-noise ratio (S/N) values of the synthetic spectra depend on the number of OPs simulated and the choice of the bins of arrival time, frequency, and viewing direction. Therefore, a higher resolution in viewing angle results in fewer OPs per bin and a lower S/N. The DCT has been the default implementation in the SEDONA code.', '2.2. Virtual Packets': "Several techniques have been suggested to improve the S/N in MCRT simulations. The idea of a 'virtual packet' (VP) in the SN simulation context is discussed by Kerzendorf & Sim (2014) and implemented in the 3D simulation by Bulla et al. (2015). Generally, VPs are used as follows \n- · An observer viewing direction is specified at the start of the MCRT simulation.\n- · VPs are created in the SN plasma to represent the emitted and the scattered photons at the corresponding volume-time-frequency bin pointing to the viewing direction.\n- · The energy of each VP is reduced by the optical depth along the path to the observer.\n- · Finally, the escaped VPs are used to synthesize the spectrum for the specified observer viewing angle. \nEvent-based techinque (EBT), proposed by Bulla et al. (2015), utilizes VPs to increase the S/N relative to DCT. In the EBT calculation, when an OP has undergone a physical event, a VP is created at the same coordinate and time with the same co-moving frame frequency. If the interaction event is Thomson scattering, then the co-moving frame Stokes vector of the VP is calculated from the Rayleigh scattering rule (Chandrasekhar 1960): \n¯ S V P = ¯ S OP P ( ¯ θ OP , ¯ φ OP , ¯ θ, ¯ φ ) , (3) \nwhere ¯ S OP is the Stokes vector of the OP in the comoving frame, ( ¯ θ OP , ¯ φ OP ) is the OP propagating direction in the co-moving frame, ( ¯ θ, ¯ φ ) is the viewing direction in the co-moving frame, P is the combination of the rotation matrix and the scattering matrix (See Equation 10 in Bulla et al. (2015)). In the following, comoving frame quantities are denoted with a bar, whereas non-bar quantities refer to the lab frame. A detailed expression of the P matrix is given in Appendix A. \nIf the interaction event is a bound-bound transition, bound-free transition, or free-free transition, then an depolarized VP propagating towards the viewing direction is created. The energy of the VP is \n¯ E V P = 1 4 π ¯ E OP . (4) \nAfter creation, the VPs propagate along the viewing direction with the speed of light, and are not considered in the subsequent computation of the plasma excitation and ionization states. The total optical depth along the VP trajectory is integrated as \nτ tot = ∑ j d j α tot,j , (5) \nwhere d j is the length of the j -th line segment of the VP trajectory, α tot,j is the total extinction coefficient at the j -th line segment. The lab frame extinction coefficient is calculated from the co-moving frame value \nα tot,j ( ν ) = (¯ ν/ν )¯ α tot,j (¯ ν ) , (6) \nwhere ¯ ν and ν are the co-moving frame frequency and the rest-frame frequency respectively. \nThe VP is moved in small line segments, the length of which are determined by finding the minimum distance among the following \n- · The VP reaches the boundary of a volume cell.\n- · The VP reaches the end of a time step.\n- · The co-moving frame frequency of the VP reaches the boundary of the frequency grid of α tot . \nThe final spectrum is constructed by summing the VPs \n I Q U V = 1 4 πr 2 ∆ t ∆ ν ∑ i E V P,i s V P,i e -τ tot,i , (7) \nthe summation is over all the VPs in the frequency bin [ ν -∆ ν/ 2 , ν + ∆ ν/ 2), and arrival time interval [ t -∆ t/ 2 , t +∆ t/ 2). \nThe EBT can be accelerated by only emitting VPs with frequencies within a desired spectral range (e.g., 3500 -10000 ˚ A ) for the synthetic observables, and by removing the VP once the integrated optical depth reaches a critical value τ tot = 10. We do not apply these optimizations in the present calculations, but Bulla et al. (2015) find that they can decrease the computational time by a factor of ∼ 4 while not affecting accuracy.", '3. THE INTEGRAL-BASED TECHNIQUE': 'Our proposed IBT uses integrated packets (IPs), which are an improved version of VPs that can be used to more efficiently construct spectra. Compared to VP, the major changes of IP are: (1) A VP stores the energy and polarization at a single frequency with a four-vector S V P , while an IP stores the energy and polarization at multiple frequency bins with a 4 × N tensor of cell radiance R ([ I, Q, U, V ]; [ ν 1 , ..., ν N ]); (2) The creation of a VP is based on the physical interactions of OPs, whereas an IP is created in each volume cell for each time step.', '3.1. The Integrated Packets': 'The IP computation is based on a universal logarithmically spaced frequency grid [ ν 0 , ν 1 ...ν N ]. The frequency ratio between the adjacent pixels is a constant C , satisfying \nν n +1 /ν n = C. (8) \nUsing this frequency grid, the Doppler shift of the variables could be simplified to the shifting of the index of the frequency bin, which saves computation time. \nWe define the cell radiance R ( ⃗x, t, ν ) in the lab frame for a cell located at coordinate ⃗x and time t as the energy radiated per frequency bin at frequency ν toward the viewing direction, the units of R ( ⃗x, t, ν ) is erg Hz -1 sr -1 . The formal solution of the cell radiance is: \nR ( ⃗x, t, ν ) = ∆ v ∆ t s ( ¯ ν ν ) -2 ( ¯ j emi + ¯ j sc ) (9) \n¯ j emi = ¯ j emi ( ⃗x, t, ¯ ν ) 1 0 0 0 (10) \n¯ j sc = α T ∫ Ω ¯ I ( ⃗x, t, ¯ θ in , ¯ φ in , ¯ ν ) P ( ¯ θ in , ¯ φ in , ¯ θ, ¯ φ ) d Ω , (11) \nwhere ¯ I is the specific intensity as a four-vector of the Stokes parameters; ∆ t s is the size of the simulation time step; ¯ j emi and ¯ j sc are the four-vectors of the emission terms due to depolarized isotropic emission and scattered light from Thomson scattering, respectively; the ¯ j emi function is the depolarized isotropic emission term, which includes all the emission from bound-bound, bound-free, and free-free transitions; the P ( ¯ θ in , ¯ φ in , ¯ θ, ¯ φ ) function is a combination of the Rayleigh scattering phase matrix and rotation matrix discussed in Appendix A; α T is the extinction coefficient for Thomson scattering; the integral is over all the incident directions ( ¯ θ in , ¯ φ in ); the (¯ ν/ν ) -2 term is the correction of the Doppler effect (Castor e.g., 1972, eq. (1-3); see also Mihalas 1978, p. 31,33,495-496). \nFor each IP, the cell radiance R is calculated on the universal frequency grid and stored as a 4 × N vector. During the propagation of an IP, R is reduced by the extinction coefficient on the trajectory. The spectral synthesis is the summation of R over the IPs at a specific arrival time. Section 3.2 and Section 3.3 illustrate the computation of R in a MCRT simulation during the creation of an IP. Section 3.4 illustrates the update of R during the propagation of IP and spectral synthesis.', '3.2. Isotropic Emission': 'The isotropic emission term, including bound-bound, bound-free, and free-free interactions, can be calculated by the summation of the OPs from the same physical events \n∆ v ∆ t s ( ¯ ν ν ) -2 ¯ j emi ( ⃗x, t, ¯ ν ) = ( ¯ ν ν ) -2 ∑ i ¯ E OP,i 4 π ∆¯ ν , (12) \nwhere the summation is over all the OPs within the frequency bin [¯ ν -∆¯ ν/ 2 , ¯ ν +∆¯ ν/ 2), the volume cell ∆ v , and the time step bin [ t s , t s +∆ t s ). Only OPs that are newly created or undergone the same physical events are taken into account. \nIn the present SEDONA calculation, we include bound-free and free-free transitions and adopt the line \nexpansion opacity approximation (Kasen et al. 2006, eq. (8)) for bound-bound transitions which are treated as purely absorptive. Therefore, the isotropic emission term is calculated from the current plasma state as ¯ j emi ( ⃗x, t, ¯ ν ) = B (¯ ν )¯ α abs ( ⃗x, t, ¯ ν ), where B (¯ ν ) is the blackbody spectrum and ¯ α abs is the total extinction coefficient including bound-bound, bound-free, and freefree transitions (Thomson scattering is not included in this term).', '3.3. The Path Integral': 'The cell radiance from Thomson scattering is calculated from the path integral of the OPs \n∆ v ∆ t s ( ¯ ν ν ) -2 ¯ j sc = ∑ i,j ¯ E i ¯ d i,j ¯ α T ¯ s i,j P ( ¯ θ i,j , ¯ φ i,j , ¯ θ, ¯ φ ) 4 π ∆¯ ν (13) \nwhere ¯ d i,j is the co-moving frame length of the j -th line segment of the i -th OP trajectory in the volume cell and in the time step, the summation is over all the trajectories within the frequency bin [¯ ν -∆¯ ν/ 2 , ¯ ν +∆¯ ν/ 2), the volume cell ∆ v , the time step bin [ t s , t s +∆ t s ). \nFor each volume cell and at each time step, an IP is created. The initial coordinate of the IP is randomly chosen in the volume cell, and the initial time of the IP is randomly chosen in the time step. The cell radiance R is calculated on the universal frequency grid and stored in IP as a 4 × N tensor.', '3.4. The Propagation of Integral Packets': 'After creation, the IPs propagate toward the viewer with the speed of light, and the cell radiance R is reduced during propagation. Similar to the propagation of the VPs, the trajectory of an IP is split into line segments by the edges of time steps and volume cells. Moreover, the trajectory is further split to satisfy the criterion: \n∣ ∣ ∣ ∣ ¯ ν begin -¯ ν end ¯ ν end ∣ ∣ ∣ ∣ ≤ C -1 , (14) \nwhere ¯ ν begin is the co-moving frame frequency at the beginning of the line segment, ¯ ν end is the co-moving frame frequency of the same lab frame frequency ν at the ending of the line segment, the constant C is defined in Equation 8. \nIn the calculation of VP in Equation 5 and 7, the total opacity is accumulated over the trajectory, which computationally requires one more double precision floating point number per VP to store the opacity. This process reduces the computation time on exponentials at the cost of memory space. While in the propagation of IPs, the cell radiance is updated at every line segment on the \ntrajectory without accumulating the total opacity over the trajectory \nR i,j +1 ( ν ) = R i,j ( ν ) e -d i,j α tot,i,j ( ν ) , (15) \nwhere d i,j is the length of the i -th IP at the j -th line segment over the trajectory, α tot,i,j ( ν ) is the total extinction coefficient of the i -th IP at the j -th line segment and at lab frame frequency ν . The final spectrum is the sum of the IPs: \n I Q U V = 1 4 πr 2 ∆ t ∑ i R i,final , (16) \nwhere R i,final is the final cell radiance when the i -th IP escapes the simulation domain. The summation is over all IPs in the arrival time interval [ t -∆ t/ 2 , t +∆ t/ 2). Note that the grid size of arrival time ∆ t should be much larger than the time step size ∆ t s to avoid grid mismatch issue, while this issue is not significant in DCT or EBT calculations.', '4. MODEL VALIDATION': 'In this section, we prepare two toy models to validate the IBT computation results. The first model is the SN Ia delayed-detonation model DDC10. DDC10 is calculated from hydrodynamic simulations in Blondin et al. (2013) and adopted as a benchmark in Blondin et al. (2022) to evaluate the differences between radiative transfer codes. We remap the 1-D model into a 60 × 60 × 60 3-D Cartesian grid. retaining the spherical symmetry and with the outer boundary velocity limited to 25,000 km/s. Figure 1 shows the abundance of several elements and the density profile of the DDC10 model. The second model is an ellipsoidal model based on the DDC10 model; the y and z coordinates are reprojected with y new = 1 . 3 y and z new = z/ 1 . 3, in order to introduce asphericity. This model is also realized on a 60 × 60 × 60 Cartesian grid, with a boundary velocity of 32,500 km/s. The geometry of the model is shown in the right panel of Figure 1. \nIn the following text, the spherical symmetric DDC10 model is denoted as D1D, and the modified ellipsoidal 3Dmodel is denoted as D3D. Both the models are treated with 3-D time-dependent radiative transfer calculations in SEDONA starting from 0.976 days after the explosion. Before 6.6 days, the simulation time step is a logarithmic with ∆ t s /t s = 0 . 03. After 6.6 days, the simulation time step is ∆ t s = 0 . 2 days. The arrival time bin size is ∆ t = 1 day, which satisfies ∆ t ≥ 5∆ t s . The frequency grid ranges from 1 × 10 14 Hz to 5 × 10 15 Hz (30,000 ˚ A to 600 ˚ A ) with ∆ ν/ν = 0 . 002.', '4.1. Spherical Symmetry': 'Because the linear polarization components cancel out, a spherical symmetric model should produce zero polarization. Using IBT, we calculate the spectropolarimetry of the D1D model to test if the algorithm recovers zero polarization. Figure 2 shows the spectra and the polarization percentage at different times after the explosion. At each time step, 6 × 10 6 OPs are generated to represent the energy release from radioactive decay and gamma-ray scattering. We find that the polarization signal at all times and wavelengths is consistent with the expected zero polarization. Moreover, the simulation error between 7 days and 31 days and between 2000 and 10000 ˚ A wavelength range (within which most SNe Ia spectropolarimetry observations are made (Cikota et al. 2019)) is as low as ∼ 0 . 2%. Notably, the spectra in the ultraviolet wavelength between 7 days and 21 days also show high S/N, despite the emergent flux being low in this band. In the early phase of SNe Ia, most of the OPs are generated below the photosphere, therefore the cell radiance R above the photosphere has low S/N. As a result, the spectrum shows low S/N at 34 days after the explosion. Moreover, the S/N in the infrared wavelength is lower than that in the optical wavelength, because most of the OPs are in the optical wavelength. Due to the limited number of OPs in the ultraviolet wavelength at late phase, the ultraviolet spectrum at 30-31 days and after is dominated by Monte-Carlo noise. In the IBT simulation of this paper, this phenomenon is only observed below 2000 ˚ A and can be alleviated by increasing the number of OPs.', '4.2. Mirror Symmetry': 'In this section, we use IBT to calculate the spectropolarimetry at two mirror symmetric viewing directions of the D3D model. At each time step we generate 1 × 10 7 OPs. Based on the mirror symmetry of the D3D model in the X-Z, Y-Z, and X-Y planes, spectropolarimetry in the viewing directions ( cos ( θ ) = 0 . 51122 , φ = π/ 6) and ( cos ( θ ) = 0 . 51122 , φ = 11 π/ 6) should have the same Stokes Q terms and inverted U terms. Figure 3 shows the spectropolarimetry at the two symmetric viewing directions. We notice the simulation results are consistent with the theoretical expectations of the symmetry at above 2000 ˚ A wavelength from early phase (7-8 days after the explosion) to late phase (50-51 days after explosion). Similar to the spherical symmetric validation results in Figure 2, the ultraviolet spectropolarimetry below 2000 ˚ A after ∼ 25 days are dominated by MonteCarlo noise. Moreover, ultraviolet spectropolarimetry at 7-8 days and 15-16 days shows an exceptional high S/N. \nFigure 5 shows the computation time of particle propagation at each time step as a function of the time after explosion using the EBT and IBT methods as described in Section 4.3, and the corresponding DCT computation time with 4 × 10 6 OPs per time step, the same number of OPs as the EBT and IBT results. The time needed to compute the plasma state and opacity does not change drastically at different time steps. In contrast, the computation time for packet propagation increases in the first ∼ 3 days, because the optical depth is high and the OPs cannot easily escape the SN ejecta and undergo repeated interaction events. With the expansion of the SN ejecta, the optical depth decreases, and the accumulated OPs escape the SN ejecta, therefore the computation time decreases. In particular, the computation time for EBT is a factor of ∼ 30 higher than DCT around 25 days after the explosion, and is a factor of ∼ 10 higher than DCT around 60 days. In contrast, the introduction of IBT only increases the computation time by a factor of 0.3-0.5 of the DCT computation time, throughout the time after the SN explosion, to reach a comparable S/N as EBT. \n<!-- image --> \nFigure 1. Left panel: the mass fractions of several isotopes in the DDC10 model. Middle panel: the density structure of the DDC10 model. Homologous expansion is assumed, so the radius is equal to velocity times time. Right panel: the geometry of the modified DDC10 model with 3-D structures (D3D model). \n<!-- image -->', '4.3. Cross Comparison': 'In this section, we compare the simulation results from the EBT, IBT, and DCT methods using the D3D model. Figure 4 shows the spectra and linear polarization time sequence calculated by the three methods. In the DCT calculation, the viewing direction is split into a 10 × 10 bin and 4 . 6 × 10 8 OPs are generated per time step to achieve a reasonable S/N. For the EBT and IBT calculations, we only generate 4 × 10 6 OPs per time step. Figure 4 shows that the IBT calculation produced spectra and polarization with a S/N comparable to a DCT calculation that used ∼ 115 times more OPs. This noise reduction holds from 6 days to 33 days after the explosion and from ultraviolet to infrared wavelengths. \nDue to the high opacity at the early phase of SN, the OPs undergo multiple scattering events near the core of SN, which results in a rapidly growing number of VPs in the EBT calculation and an increasing memory usage expense. Therefore, in this example EBT calculation we started the calculation at 25 days after the explosion due to the limit of hardware.', '5.1. Computational Speed Comparison': 'The computational speed of the three methods is measured on one 48-core compute node of Grace supercomputer in TAMU HPRC 1 . Each node has 2 sockets of Intel Xeon 6248R CPUs, in total 48 cores, and has 384 GBs of RAM memory. The majority of the computation time in a 3-D time-dependent radiative transfer computation consists of two components: (1) the computation of the plasma state, level population, and opacity of the grid; (2) the propagation of energy packets including OPs, VPs, or IPs. The choice of spectral synthesis \nmethods only affects the propagation time of the energy packets.', '5.2. Signal to Noise Ratio': 'In an MCRT simulation, the output spectral flux error scales roughly as σ ∝ √ N , where N is the number of photon packets. We therefore use the simulation results on the D1D model to measure the simulation error of the three spectral retrieval methods. First, we split the output spectra into ultraviolet wavelength range (10002000 ˚ A ), optical wavelength range (2000-10000 ˚ A ), and infrared wavelength range (10000-25000 ˚ A ). Second, we measure the simulation root mean squared error (RMSE) of the Stokes parameters Q and U using the following equation: \nFigure 6 shows the relationship between the number of OPs per time step and the resulting RMSE at 1011 days after the explosion using DCT or IBT, and at 37-38 days after the explosion using DCT, EBT, or IBT. We notice the result from IBT could reproduce the RMSE ∝ N -0 . 5 relation in most of the time and \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 2. The spectral flux ( I ) and the polarization percentage ( Q / I , U / I ) of the spherical symmetric D1D model at different times. The spectra are calculated by IBT with 6 × 10 6 OPs per simulation time step. The arrival time bin relative to the explosion time is labeled in the title of each subplots. The polarization percentages of Q / I and U / I are averaged using a 7-pixel bin. \n<!-- image --> \nRMSE = √ √ √ √ √ ∑ N ν ( Q ( ν ) I ( ν ) -0 ) 2 + ( U ( ν ) I ( ν ) -0 ) 2 N ν , (17) \nwhere N ν is the number of frequency indicies in the selected wavelength range, the summation is over all frequency indicies in the selected wavelength range. Note that the theoretical solution of the D1D model is no polarization which results in the zero term in the equation. The RMSE observes the inverse square root relation with the number of packets, RMSE ∝ N -0 . 5 . \nwavelength ranges. The DCT result also reproduces the relation RMSE ∝ N -0 . 5 in the optical and infrared wavelength ranges when the number of packets is large enough. However, in the ultraviolet wavelength, and in the optical wavelength with the number of packets per step smaller than ∼ 4 × 10 5 , the inverse square root relation is not observed due to the large simulation error. The result from EBT also reproduces the RMSE ∝ N -0 . 5 relation in optical ind infrared wavelength ranges at 37-38 days after the explosion. In the ultraviolet wavelength range at 37-38 days after the explosion, none of the methods could simulate good S/N spectra to reproduce the RMSE ∝ N -0 . 5 relation. \nA direct comparison of the computation efficiency of the three methods could be made using the RMSE values measured in the optical and infrared wavelength range. The RMSE of IBT is smaller than that of DCT by a factor of 20 -30, leading to a factor of 400 -900 less \n<!-- image --> \nFigure 3. The spectropolarimetry of the D3D model at the two symmetric viewing directions ( cos ( θ ) = 0 . 51122 , φ = π/ 6) and ( cos ( θ ) = 0 . 51122 , φ = 11 π/ 6). The arrival time bin at each panel is labeled in the title of figures. In the sub-figures of Stokes U / I term, the flipped values at one of the viewer points is shown as dashed line, in order to make direct comparisons. The polarization percentages of Q / I and U / I are averaged using a 7-pixel bin. \n<!-- image --> \npackets needed to calculate to reach similar S/N. The RMSE of EBT is smaller than that of DCT by a factor of 5 -10, resulting in a factor of 25 -100 less packets to calculate to achieve a comparable S/N. The accuracy comparison between EBT and IBT is consistent with the reports in Table 1 of Bulla et al. (2015).', '6. CONCLUSION': 'We present IBT, an algorithm to efficiently synthesize the spectropolarimetry signal from 3-D MCRT simulations. Comparing to EBT (Bulla et al. 2015), the present IBT method is upgraded in the following aspects: \n- · In EBT, a VP is created at each scattering event of an OP, which could lead to an uncontrolled number of VPs and limit the application of EBT on high-opacity plasma models (i.e. SNe Ia at ∼ 3 days after the explosion). However, IBT uses the cell radiance R ( ⃗x, t, ν ) as a proxy of all VPs in a volume cell of the plasma model, and limits the number of IPs to save memory and computation time in tracing the particles.\n- · The frequency grid in IBT is logarithmically spaced (Equation 8), which simplifies the Doppler shift of cell radiance R ( ⃗x, t, ν ) to re-indexing the pixel. Therefore, the computation of Equation 15 is simplified to vector operations. This upgrade further accelerates the computation of IP propagation. \nIn the tests on the D3D model, both IBT and EBT have successfully reproduced the spectral time sequence with spectropolarimetry from 1000 ˚ A to 25000 ˚ A, and the S/N is comparable to DCT with ∼ 115 more MonteCarlo OPs. The computation time for IBT is ∼ 0.3 of DCT with the same number of OPs per viewing direction, while the computation time for EBT is a factor of 10 -30 of DCT per viewing direction. With an opacity limit and wavelength limit to accelerate the computation, the EBT computation time is still about a factor of two of DCT per viewing direction(Bulla et al. 2015). Moreover, IBT has successfully calculated the spectral time sequence from 1 day to 60 days after the explosion, while EBT failed before day 25 due to the limited memory space of the computing facility. \n<!-- image --> \n<!-- image --> \nFigure 4. The spectra and linear polarization of the D3D model at the viewing direction cos ( θ ) = 0 . 5, φ = 3 . 4455, using DCT (black), EBT (green), and IBT (red) methods. The DCT calculation is performed with 4 . 6 × 10 8 OPs per time step (which is 115 times more than EBT or IBT, marked in the legend), while EBT and IBT calculation generates 4 × 10 6 OPs per time step. The arrival time bin is shown on top of each panel. Due to the memory limit, the EBT calculation is only started at 25 days after the explosion, and the EBT results are not shown at 6-7 days and 18-19 days. The Monte-Carlo error bar of the DCT result is estimated by the number of escaped OPs at each bin. Due to the lack of OPs, the spectrum between 1000 ˚ A and 1500 ˚ A at 32-33 days from DCT is missing. The polarization percentage is binned with 7 pixels. \n<!-- image --> \nFigure 5. Comparison of particle propagation computation time at different time steps using the DCT (black), EBT (green), and IBT (red) methods. Each time step generates 4 × 10 6 OPs in the three simulations. The computation time of the plasma state and opacity is also shown in the figure with ultramarine color. Note that the EBT calculation is not activated until 25 days after the explosion, and the green curve before day 25 is overlapped to the black curve. \n<!-- image --> \nIn the tests on the D1D model, IBT has successfully reproduced the zero polarization signal as the theoretical predictions from the spherical symmetry. Using the same number of Monte-Carlo OPs, the S/N of the IBT spectrum is ∼ 30 times higher than the DCT spectrum S/N and 5 -10 times higher than the EBT spectrum. \nChen et al. (2020, 2024) developed the Artificial Intelligence Assisted Inversion (AIAI) method which enables theoretical models of SNe Ia to be derived using the observational data as input constraints. The published AIAI is based on 1-D models. The studies demonstrate that it is possible to combine empirical models of SNe Ia with detailed radiative transfer code to improve the use of SNe Ia as cosmological probes. In the future, we will extend the 1-D models to 3-D time-dependent models. The proposed IBT is the ideal algorithm to efficiently generate a time sequence of spectrophotometric and spectropolarimetric data bases which can be used to train AI models, this can be an important step in assimilating the extensive observational data of SNe Ia into studies of the physics of thermonuclear explosions and the use of SNe Ia as cosmological distance candles. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 6. The RMSE of DCT, EBT, and IBT simulation results on the D1D model, as a function of the number of OPs generated per time step. The polarization spectra are binned with 7 pixels. Upper panel shows the RMSE in the time bin of 10-11 days after the explosion, lower panel shows the RMSE in the time bin of 37-38 days after the explosion. From left to right, the wavelength range is ultraviolet, optical, and infrared. The dashed line shows the theoretical inverse square root relation between RMSE and the number of packets ( RMSE ∝ N -0 . 5 ) for comparison. \n<!-- image --> \nSoftware: SEDONA(Kasen et al. 2006) \n- X.C. is supported by the grant NSF-AST 1817099. 1\n- The authors would like to thank Prof. J. C. Wheeler 2\n- from University of Texas at Austin and Prof. David 3\n- Jeffery from University of Nevada at Las Vegas for sup4\n- portive discussions. Portions of this research were con5\n- ducted with the advanced computing resources provided 6\n- by Texas A&M High Performance Research Comput7\n- ing. This work used FASTER super computer at TAMU 8\n- HPRC through allocation PHY240215 from the Ad9\n- vanced Cyberinfrastructure Coordination Ecosystem: 10\n- Services & Support (ACCESS) program, which is sup11\n- ported by National Science Foundation grants 2138259, 12\n- 2138286, 2138307, 2137603, and 2138296(Boerner et al. 13\n- 2023). 14', 'A. RAYLEIGH SCATTERING': 'The Thomson scattering between free electron and photon is calculated by the Rayleigh scattering phase matrix: \nP R (Θ) = 3 4 cos 2 Θ+1 cos 2 Θ -1 0 0 cos 2 Θ -1 cos 2 Θ+1 0 0 0 0 2 cos Θ 0 0 0 0 2 cos Θ , (A1)', 'Chen et al.': "where Θ is the angle between the incident light and the emergent light. Before the calculation of Rayleigh scattering phase matrix, the reference axis of the incident light should be rotated into the scattering plane, and the polarization state is changed by the rotation matrix: \nL ( ϕ in ) = 1 0 0 0 0 cos 2 ϕ in sin 2 ϕ in 0 0 -sin 2 ϕ in cos 2 ϕ in 0 0 0 0 1 , (A2) \nwhere ϕ in is the rotation angle between the observer's reference axis and the scattering plane reference axis. A similar rotation matrix of L ( π -ϕ out ) is also applied to the emergent light, in order to convert the polarization state to the observer's reference axis. \nWe define ¯ S as the multiplication of the packet energy and the dimensionless Stokes vector: \n¯ S = ¯ I ¯ Q ¯ U ¯ V = ¯ E ¯ s . (A3) \nTherefore, the full Rayleigh scattering formula in the SN radiative transfer simulation is written as: \n¯ S out = L ( π -¯ ϕ out ) P R ( ¯ Θ) L ( ¯ ϕ in ) ¯ S in . (A4) \nNote that the Rayleigh scattering happens in the co-moving frame, and all the variables in the above equation are measured in the co-moving frame. In Equation 3, Equation 9, and Equation 13, the operator is defined as P ( ¯ θ in , ¯ ϕ in , ¯ θ out , ¯ ϕ out ) = L ( π -¯ ϕ out ) P R ( ¯ Θ) L ( ¯ ϕ in ). \nReplacing the axis rotation angles and scatter angle ( ¯ ϕ in , ¯ ϕ out , ¯ Θ) with the propagating direction of the incident OP ( ¯ θ in , ¯ φ in ) and the propagating direction of the emergent VP/IP ( ¯ θ , ¯ φ ), the expression of the operator P ( ¯ θ in , ¯ ϕ in , ¯ θ, ¯ ϕ ) becomes (Chandrasekhar 1960): \n¯ I out = 3 8 π (( ( l, l ) 2 +( l, r ) 2 ) ¯ I in + ( ( r, l ) 2 +( r, r ) 2 ) ¯ Q in +(( l, l )( r, l ) + ( l, r )( r, r )) ¯ U in ) (A5) \n¯ Q out = 3 8 π (( ( l, l ) 2 -( l, r ) 2 ) ¯ I in + ( ( r, l ) 2 -( r, r ) 2 ) ¯ Q in +(( l, l )( r, l ) -( l, r )( r, r )) ¯ U in ) (A6) \n¯ U out = 3 8 π ( 2( l, l )( l, r ) ¯ I in +2( r, r )( r, l ) ¯ Q in +(( l, l )( r, r ) + ( r, l )( l, r )) ¯ U in ) (A7) \n¯ V out = 3 8 π (( l, l )( r, r ) -( r, l )( l, r )) ¯ V in , (A8) \nwhere ( l, l ), ( r, l ), ( l, r ), ( r, r ) are: \n( l, l ) = sin ( ¯ θ ) sin ( ¯ θ in ) + cos ( ¯ θ ) cos ( ¯ θ in ) cos ( ¯ φ in -¯ φ ) (A9) \n( r, l ) = cos ( ¯ θ ) sin ( ¯ φ in -¯ φ ) (A10) \n( l, r ) = -cos ( ¯ θ in ) sin ( ¯ φ in -¯ φ ) (A11) \n( r, r ) = cos ( ¯ φ in -¯ φ ) . (A12)", 'REFERENCES': 'Abbott, T. M. C., Allam, S., Andersen, P., et al. 2019, \nApJL, 872, L30, doi: 10.3847/2041-8213/ab04fa \nBlondin, S., Dessart, L., Hillier, D. J., & Khokhlov, A. M. \n2013, MNRAS, 429, 2127, doi: 10.1093/mnras/sts484'}
2024arXiv240906844H	The reconstruction of an inflationary universe considering the parametrization of the scalar spectral index as a function of the number of efolds in the framework of a modified Friedmann equation is analyzed. In this context we examine the possibility of reconstructing the Hubble parameter together with the effective potential considering a modified Friedmann equation specified by mathcalFHpropto rho where mathcalFH corresponds to an arbitrary function of the Hubble parameter H and rho denotes the energy density associated with the matter in the universe. To reconstruct the background variables during the inflationary scenario we develop a new methodology by expressing the spectral index in terms of the Hubble parameter and its derivatives. Thus we obtain a general formalism for the reconstruction of the inflation using the slow roll approximation together with the parametrization of the scalar spectral index as a function of the number of efolds N. As specific examples we consider the simplest attractor ns12N together with different functions mathcalFH associated to the modified Friedmann equation to rebuild the Hubble parameter and the effective potential in terms of the scalar field phi. Additionally we examine the reheating epoch by considering a constant equation of state parameter in which we determine the temperature and the number of efolds during this epoch using the background variables found during the reconstruction of the different mathcalFHmodels studied. Besides we constrain the different parameters associated with the reconstructed inflationary mathcalFHmodels during the epochs of inflation and reheating using current astronomical data from Planck and BICEPKeck results.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.06844', 'arXiv:2409.06844', '2024arXiv240906844H']	['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Theory']	Reconstructing inflation and reheating in the framework of a generalized mathcalFH Friedmann equation	2,024	192	0.17	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.06844.pdf	{'Reconstructing inflation and reheating in the framework of a generalized F ( H ) Friedmann equation': "Ram'on Herrera 1, ∗ and Carlos R'ıos 2, † \n1 Instituto de F'ısica, Pontificia Universidad Cat'olica de Valpara'ıso, \nAvenida Brasil 2950, Casilla 4059, Valpara'ıso, Chile. 2 Departamento de Ense˜nanza de las Ciencias B'asicas, Universidad Cat'olica del Norte, Larrondo 1281, Coquimbo, Chile. \nThe reconstruction of an inflationary universe considering the parametrization of the scalar spectral index as a function of the number of e -folds in the framework of a modified Friedmann equation is analyzed. In this context, we examine the possibility of reconstructing the Hubble parameter together with the effective potential considering a modified Friedmann equation specified by F ( H ) ∝ ρ , where F ( H ) corresponds to an arbitrary function of the Hubble parameter H and ρ denotes the energy density associated with the matter in the universe. To reconstruct the background variables during the inflationary scenario, we develop a new methodology by expressing the spectral index in terms of the Hubble parameter and its derivatives. Thus, we obtain a general formalism for the reconstruction of the inflation, using the slow roll approximation together with the parametrization of the scalar spectral index as a function of the number of e -folds N . As specific examples, we consider the simplest attractor n s -1 = -2 /N together with different functions F ( H ), associated to the modified Friedmann equation, to rebuild the Hubble parameter and the effective potential in terms of the scalar field ϕ . Additionally, we examine the reheating epoch by considering a constant equation of state parameter, in which we determine the temperature and the number of e-folds during this epoch, using the background variables found during the reconstruction of the different F ( H ) -models studied. Besides, we constrain the different parameters associated with the reconstructed inflationary F ( H ) -models during the epochs of inflation and reheating, using current astronomical data from Planck and BICEP/Keck results.", 'I. INTRODUCTION': "It is widely recognized that during the early universe, the introduction of the inflationary stage, or inflation, remains a possible solution to many long-standing problems of the hot big bang model see e.g., Refs.[1-3]. However, the most significant characteristic of inflation is that it provides a causal explanation for the origin of the observed anisotropy of the cosmic microwave background (CMB) radiation and the distribution of the large-scale structure observed today [4-6]. \nTo describe the inflationary era during the early universe, various inflationary models have been proposed in the context of the theory of general relativity (GR) as well as in modified theories of gravity or alternatives to Einstein's general relativity. The implementation of inflationary models is based on the introduction of a homogeneous scalar field ϕ associated to the matter of the universe. The evolution of this scalar field is governed by the Klein-Gordon equation and together with the Friedmann equation, and they constitute the simplest set of field equations utilized to study the inflationary dynamics of the early universe in the framework of a Friedmann Roberson Walker (FRW) metric. In relation to the modified gravity, we can distinguish those models that utilize a modified Friedmann equation to describe the early universe. In this sense, considering a spatially flat FRW metric, the modified Friedmann equation can be written as \nF ( H ) = κ 3 ρ, (1) \nwhere F ( H ) > 0 is an arbitrary function associated to the Hubble parameter defined by H = ( da/dt ) /a , where a ( t ) denotes the scale factor. Besides, the quantity κ = 8 πG = M -2 p where M p represents the Planck mass and ρ corresponds to the energy density relates to the matter of the universe. Thus, we have that the dimension associated to the function F ( H ) corresponds to M 2 p = κ -1 . Also, we note that Eq.(1) is reduced to the standard Friedmann equation when the function F ( H ) = H 2 . \nThe motivation for considering this modification of the Friedmann equation, defined by Eq.(1) arises from the fact that various models in the literature, have analyzed this modification to describe the early and present universe. Thus, \nsome examples are the Friedmann equations in one anti de Sitter bulk with a Gauss Bonnet (GB) term, in which we have three regimes for the history of the brane universe compacted from the function F ( H ) ∝ H β [7-10]. In this case, when we consider the value β = 3, we have the GB regime and then the GB term dominates gravity at highest energies. The situation in which we have β = 1 corresponds to the high energy limit of the brane world cosmology, Randall-Sundrum (RS) regime[11] and the case β = 2 is the standard Friedmann equation. Besides, this class of function of the form power-law type associated to the Hubble parameter is found from the entropy considerations as Tsallis Entopic Proposal or Fractional Entropy (see Refs. [12, 13]), in which we have F ( H ) ∝ H β . \nIn the framework of the deformed Ho˘ r ava-Lifshitz gravity from entropic force, we have that the function F ( H ) is given by F = H 2 + αH 4 [3 -2 ln(4 πM 2 p /H 2 )], where the inverse of the parameter α corresponds to the parameter of Ho˘ r ava-Lifshitz [14]. Besides, corrections to the Friedmann equation inspired by Kaniadakis entropy in which adopting the thermodynamics-gravity conjecture was obtained in Ref.[15]. Here the function F ( H ) becomes F = H 2 -γH -2 , where γ (with dimensions of M 4 p ) is associated to Kaniadakis parameter[16], and the Kaniadakis entropy (or Kentropy), is one-parameter extension of the classical Boltzmann-Gibbs-Shannon entropy. It emerges from a coherent and self-consistent relativistic statistical framework, maintaining the fundamental aspects of standard statistical theory while reclaiming it under specific conditions[17]. Another type of function F ( H ) found in the literature corresponds to F ( H ) = H 2 + αH 4 , where α is an arbitrary parameter with dimension of M -2 p . Here the flat Friedmann equation can be derived by considering the Clausius relation to the apparent horizon of FRW universe, in which entropy is defined to be proportional to its horizon area plus a logarithmic correction associated with this area[18], see also Ref.[74]. Also, this function F ( H ) = H 2 + αH 4 or modified Friedmann equation arises when considering an AdS-Schwarzschild black hole via holographic renormalization, incorporating mixed boundary conditions that correspond to the Einstein field equations in four dimensions [20]. Additionally this modified Friedmann equation can be found in the framework of a Chern-Simons type of theory [21, 22], in which the parameter α can be positive or negative[23]. In this sense, there are several other functions F ( H ) or modifications to the Friedmann equation that can be found in the literature, see e.g., Refs.[18, 24-26]. \nOn the other hand, in the literature several authors have analyzed the reconstruction of the background variables, particularly the effective potential associated with a scalar field in the context of inflation, using observational data, such as, the scalar power spectrum, scalar spectral index n s , and the tensor to scalar ratio r [27-30]. In this respect, an attractive approach to reconstruct the effective potential of the scalar field using the slow roll approximation, is to consider the parametrization of the observational parameters in terms of the number of e -folds N . Specifically, by utilizing the scalar spectral index as a function of N i.e., n s = n s ( N ), often referred to as an attractor, it is possible to rebuild the effective potential as a function of the scalar field [31]. In this context, the simplest attractor given by n s -1 = -2 /N for largeN (with N ∼ O (10) ∼ O (10 2 )), aligns with the Planck data, when the number of e -folds N is set to N = 60 [32]. By assuming this attractor in the framework of the GR, it is feasible to reconstruct an effective potential[33] with different limits[1, 34, 35]. In relation to warm inflation, the reconstruction of the effective potential and the dissipation coefficient in terms of the scalar field, was necessary to consider two attractors; the scalar spectral index and the tensor to scalar ratio in terms of the number of e -folds N [36]. Here in the weak dissipative regime, considering the attractors n s -1 ∝ N -1 together with r ∝ N -2 , it was found that the reconstruction of the effective potential and the dissipation coefficient as functions of the scalar field depends on hyperbolic functions. During the strong regime was obtained that the potential and the dissipation coefficient as functions of the scalar field exhibit a power-law behavior under certain conditions[36]. Analogously, for the construction of the background variables associated with Galilean inflation or G-inflation was necessary to utilize two observational parameters [37]. \nIn addition, different analysis for the reconstruction of the background variables using another parametrizations in terms of the number of e -folds N in the context of the slow roll approximation can be found in the literature. Thus, for example, we have the reconstruction of the effective potential as a function of the scalar field using the parametrization on the slow roll parameter ϵ = ϵ ( N ) [38-40]. Besides, considering the two slow roll parameters ϵ ( N ) and η ( N ) as a function of the number of e -folds N , it is possible to rebuild the effective potential and the tensor to scalar ratio in terms of the scalar field[41, 42]. For a review of other methodologies to rebuild the background variables during the inflationary stage, see Refs.[43-50]. \nOn the other hand, at the end of the inflationary stage, the universe undergoes a reheating phase to connect with the standard big bang model [3, 51]. During this reheating scenario, matter and radiation are produced through the decay of the inflaton field or other fields. As a result, the temperature of the universe rises, eventually leading to the radiation-dominated era and then connecting with the standard hot big bang model. To explain the reheating scenario during the early universe, various reheating models are employed to increase the temperature in this period. One such mechanism involves the perturbative decay of the scalar field through an oscillatory process at the minimum of the effective potential after the end of inflation [52]. Additionally, there are mechanisms associated with non-perturbative analysis, such as; the parametric resonance decay of the inflaton, the instant reheating model [53] or another field [54]. Furthermore, some inflationary models do not involve the inflaton field oscillating around the minimum of the potential; these are known as non-oscillating models (NO models). In such cases, reheating occurs through the decay \nof another field known as the curvaton [55, 56]. For further details, see also Ref.[57] for the reheating from tachyonic instability and other reheating models in Ref.[58]. \nDuring the reheating scenario, several important parameters characterize this stage, including the reheating temperature T reh , the equation of state (EoS) defined as ω reh = p/ρ , which describes the matter content during the reheating era, and the duration of this stage, characterized by the number of e -folds N reh . Here, the quantities p and ρ denote the pressure and energy density of the matter-associated fluid during the reheating scenario. We mention that in the case of the reheating temperature, there is a lower bound imposed by the primordial nucleosynthesis (BBN), given by T BBN ∼ 10 MeV[59]. On the other hand, with regard to the EoS parameter, various numerical calculations have been developed to analyze its dynamics based on specific interactions involving the inflaton and other fields related to matter[60]. In this sense, the dynamics evolution in the cosmological time of the EoS parameter depends on the interaction of the inflaton and the another fields. In particular, the authors of Ref.[61] determined that numerically, the EoS parameter exhibited a slow increase from a value of ω reh = 0 at the end of inflation to ω reh ∼ 0 . 3 during the reheating epoch. In the case of a massive field, it was found that numerically, the EoS parameter increases from a negative value at the end of inflation in which ω reh = -1 / 3, to a value ω reh ≃ 0 [62, 63]. In this sense, as a first approximation and in order to find analytical expressions to the reheating parameters, such as, the temperature T reh and the number N reh , we can consider that the EoS parameter during the reheating stage remains approximately constant throughout the this epoch[64]. \nThe goal of this study is to rebuild inflation in the framework of the generalized Friedmann equation, through the parametrization of an observational parameter in terms of the number of e -folds N . In particular, we will consider an attractor from the parametrization of the scalar spectral index n s as a function of the number of e -folds N , i.e., n s = n s ( N ). Here we will develop a new methodology based on rewriting the spectral index in relation to the Hubble parameter and its derivatives. In this, sense, we study how using different modified Friedmann equations, these affect the reconstruction of the Hubble parameter together with the effective potential, in terms of the scalar field. We will also establish a general methodology in the context of the slow roll approximation to build the Hubble parameter and the effective potential, by considering as attractor the scalar spectral index n s = n s ( N ). In this form, settling on a specific scalar spectral index n s ( N ), we will determine the possibility of rebuilding the Hubble parameter as well as the potential as a function of the scalar field considering different modified Friedmann equations thought various functions F ( H ). \nBesides, we will study the reheating era and how the parameters associated to this period, such that, the reheating temperature and number of e -folds are changed from the reconstruction of the background variables obtained in the inflationary period (under the slow roll approximation). Thus, from these reheating parameters, we will analyze the reheating temperature and the duration of the reheating in terms of the observational parameter n s , from Planck data. In addition, we will determine how these reheating quantities are constrained considering different EoS parameters ω reh on the plane T reh = T reh ( n s ) and N reh = N reh ( n s ), respectively. \nWe organize our paper as follows: In Section II we give a brief analyze of the inflationary phase in a generalized Friedmann equation F ( H ). Thus, in this section we show the basic equations under the slow roll approximation during the inflationary epoch to rebuild the background variables. Beside, we present the observational parameters such as the scalar spectral index, power spectrum together with the tensor to scalar ratio in this generalized Friedmann equation. Additionally, we obtain under a general formalism, an expression for the Hubble parameter (differential equation) in terms of the number of e -folds N to find the reconstruction from any parametrization related to the scalar spectral index n s ( N ). In Section III, we study the reheating stage under a general formalism in the framework of the modified Friedmann equation. In this section, we express for any function F ( H ), the reheating temperature T reh together with the number of e -folds N reh during this era. \nIn Section IV, we assume a specific attractor for the scalar spectral index n s = n s ( N ) given by n s = 1 -2 /N , in order to rebuild both the Hubble parameter H ( ϕ ) as the effective potential V ( ϕ ) in terms of the scalar field ϕ . In this sense, we consider different functions F ( H ) to reconstruct the inflationary scenario and the reheating epoch. During the inflationary scenario, we find the different constrains on the parameter-space from Planck data. In relation to the reheating scenario, we determine the reheating temperature together with the number of e -folds in this epoch. In this way, we find these quantities on the plane T reh = T reh ( n s ) and N reh = N reh ( n s ) for various values of the EoS parameters to constraint the parameters of our modified model. Finally, in Section V we give our conclusions. We chose units in which c = ℏ = 1.", 'II. RECONSTRUCTING INFLATION IN A MODIFIED FRIEDMANN EQUATION': 'In this section, we will present a brief analysis of the implications of considering a modified Friedman equation characterized through function F ( H ) associated with the Hubble parameter H . To describe the matter, in the generalized Friedmann equation F ( H ) ∝ ρ , we introduce that the energy density ρ is associated to the inflaton field \nϕ . In this way, we can write that the energy density associated to scalar field becomes \nρ = 1 2 ˙ ϕ 2 + V ( ϕ ) , (2) \nwhere V ( ϕ ) represents the effective potential related to the inflaton field. Besides, the pressure related to the scalar field is defined by \np = 1 2 ˙ ϕ 2 -V ( ϕ ) . (3) \nHere we have considered that the scalar field ϕ is a homogeneous scalar field i.e., ϕ = ϕ ( t ). Also, in the following the dots mean derivatives with respect to the time. \nFurther, the continuity equation can be written as \n˙ ρ +3 H ( ρ + p ) = 0 , (4) \nand replacing equations (2) and (3) in Eq.(4), we obtain that the dynamics of the scalar field can be written as \n¨ ϕ +3 H ˙ ϕ + V ϕ = 0 , (5) \nwhere V ϕ represents the derivative of V ( ϕ ) with respect to the ϕ field, i.e., V ϕ = ∂V/∂ϕ . Besides, in the following, we will use that the notation F H corresponds to F H = d F /dH , F HH denotes F HH = d 2 F /dH 2 , H ϕ = dH/dϕ , H ϕϕ = d 2 H/dϕ 2 , etc. \nFrom Eqs.(1), (2) (3) and (5), we find that the speed of the scalar field ˙ ϕ results \n˙ ϕ = -F H κ ( H ϕ H ) . (6) \nOn the other hand, introducing the number of e -folds N , provides a way to quantify the amount of inflation required during the expansion of the universe during the inflationary stage. Thus, the number of e -folds N defined between two different values of the time; t (or scalar field ϕ ) and t end becomes \n∆ N = N -N end = ln [ a ( t end ) a ( t ) ] = ∫ t end t Hdt = κ ∫ ϕ end ϕ ( H ˙ ϕ ) dϕ, (7) \nwhere N end and t end denote the number of e -folds and the time at the end of the inflationary era. \nOn the other hand, introducing the parameters ϵ H and η H equation we have [26] \n, in the context of a theory with a generalized Friedmann \nϵ H = -˙ H H 2 = -d ln H d ln a = 1 κ F H H ( H ϕ H ) 2 , and η H = -H H ˙ H = -d ln H ϕ d ln a = 1 κ F H H H ϕϕ H . (8) \nHere we have used the equation for the speed of the scalar field given by Eq.(6). \nWe note that these parameters coincide with the parameters given by the standard slow roll parameters ϵ and η , when F ( H ) = H 2 , under the slow roll approximation. In this way, during slow roll approximation we have that the quantities ϵ SR and η SR when F ( H ) = H 2 are reduced to \nϵ SR = ϵ H ≃ 1 2 κ ( V ϕ V ) 2 , η SR ≃ 1 κ ( V ϕϕ V ) , and η H = η SR -ϵ SR . (9) \nLet us observe that an inflationary scenario of the universe occurs when a > 0, which implies that the parameter ϵ H < 1. Furthermore, the end of the inflationary era takes place when a = 0 or equivalently when the parameter ϵ H = 1. \nIn relation to the cosmological perturbations, the general perturbed metric about the flat FRW becomes ds 2 = -(1 + 2 A ) dt 2 +2 a ( t ) B ,i dx i dt + a ( t ) 2 [(1 -2 ψ ) δ ij +2 E ,i,j +2 h ij ] dx i dx j , where A , B , ψ and E correspond to the scalar type metric perturbations and the quantity h ij denotes the traverse-traceless tensor type perturbation. In this sense, following Ref.[26], the equation for the Fourier modes associated to the scalar perturbations can be written as \nd 2 u k dη 2 + ( k 2 -1 z d 2 z dη 2 ) u k = 0 , (10) \nwhere u k corresponds to the Mukanov variable defined as u = z R , in which the quantity z = a ˙ ϕ/H . Besides, the variable η denotes the conformal time and the quantity R is the gauge invariant comovil curvature perturbation. \nThe term (1 /z )( d 2 z/dη 2 ) associated to Eq.(10) in the framework of the modified Friedmann equation can be written as [26] \n1 z d 2 z dη 2 = 2 a 2 H 2 [ 1 + 1 2 ϵ H ( 5 -3 H F HH F ) -3 2 η H + 1 2 η 2 H + ϵ 2 H ( 1 + H 2 2 F HHH F -H F HH F \n-1 ϵ H η H ( 3 -2 H F HH ) + O ( ˙ ϵ H , ... ) . (11) \nH H H ) 2 F H \nIn the particular case in which the function F ( H ) = H 2 , the different terms given by Eq.(11) are reduced to those obtained in Ref.[65]. \nIn this form, following the approach outlined in Ref.[26], the primordial curvature perturbation A s produced during inflation in the context of a generalized Friedmann equation can be written as \nA s ( k ) = ( κH 2 F H H ϕ ) 2 ( H 2 π ) 2 . (12) \nIn addition, the power spectrum of the scalar perturbations A s given by Eq.(12) can be rewritten as \nA s = ( κH 2 F H H N )( H 2 π ) 2 , (13) \nwhere we have utilized the relation ϕ = F H H N /κ . number k and it is evaluated for a particular mode, when the cosmological scale exits the horizon i.e., when k = aH \nBy using the primordial curvature perturbation, we can determine the so-called scalar spectral index n s n s -1 = d ln A s /d ln k . Thus, from Eq.(12) the expression for the scalar spectral index is given by[26] \n˙ 2 Here the perturbation defined by Eq.(12) is a function of the wave . , defined as \nn s -1 = 2 η H -2 ( 3 -H F HH F H ) ϵ H . (14) \nWe note that in the special case in which F ( H ) = H 2 , Eq.(14) reduces to the standard expression n s -1 = 2 η SR -6 ϵ SR . Additionally, the tensor perturbation (transverse-traceless) during the inflationary scenario would produce gravitational waves, where the tensor perturbations amplitude is denoted by A T . Thus from these perturbations we can define an important observational quantity called the tensor to scalar ratio r , defined as r = A T /A s . In this form, we have that the tensor to scalar ratio can be written as [26] \nr = 8 F H H ϵ H . (15) \nWe note that Eq.(13) and Eq.(15) are reduced to the standard expressions A s = κH 2 / ( 8 π 2 ϵ SR ) and r = 16 ϵ SR when we choose the function F ( H ) = H 2 (standard Friedmann equation).', 'A. Reconstructing inflation: General reconstruction of H ( ϕ ) and V ( ϕ ) from n s ( N )': 'In this subsection we consider the methodology to rebuild the background variables, such as, the Hubble parameter and the effective potential in terms of the scalar field, assuming as attractor the scalar spectral index in terms of the number of e -folds N . In this context, we will utilize a new methodology to reconstruct the background variable considering the Hubble parameter and its derivatives in terms of the number of e -folds N . Firstly, we will rewrite the scalar spectral index given by Eq.(14) in terms of the Hubble parameter, the function F and the derivatives as a function of the number of e -folds N through the parameters ϵ H and η H , respectively. In this way, giving the attractor n s = n s ( N ), we should first find the Hubble parameter and the effective potential in terms of the number of e -folds N . Subsequently, using the expression given by Eq.(7), we should find the e -folds N in terms of the scalar field ϕ . Finally, considering these equations, we can rebuild the Hubble parameter as a function of the scalar field i.e., H ( ϕ ), and subsequently we can determine the effective potential V ( ϕ ). \nFrom this methodology, we start by rewriting the parameters ϵ H and η H associated to the Hubble parameter in terms of the number of e -folds. Thereby, we can now rewrite the Hubble parameter and its derivatives in terms of the number N considering the relation between N and the scalar field from the relation \ndN = -Hdt = -( H ˙ ϕ ) dϕ, such that N ϕ = dN dϕ = -H ˙ ϕ . (16) \nIn this way, we obtain that \nH ϕ = -( H ˙ ϕ ) H N . (17) \nNow, using Eq.(6) we find that the Eq.(17) becomes \nH 2 ϕ = κH 2 H N F H . (18) \nHere we note that the ratio H N / F H is a positive quantity. Besides, as the quantity ˙ H = -HH N < 0, then we have that H N > 0 and it suggests that the derivative F H > 0. \nAdditionally, we can rewrite the quantity H ϕϕ in terms of the derivatives of the number of e -folds N as \nH ϕϕ = κH 2 F H ( H NN 2 H N + H N H -F HH H N 2 F H ) . (19) \nThus, from these relations we can rewrite the parameters ϵ H and η H , respectively. In this context, by replacing Eqs.(18) and (19) into Eq.(8), we obtain that these parameters as a function of F , H and its derivatives with respect to the number N result \nϵ H = H N H , and η H = ϵ H + H NN 2 H N -F HH H N 2 F H . (20) \nUsing previous results we can rewrite the spectral index n s given by Eq.(14) as \nn s -1 = H NN H N -4 H N H + F HH H N F H = d ln H N dN + g ( H ) d ln H dN , (21) \nwhere we have defined the function g ( H ) as \ng ( H ) = H F HH F H -4 . (22) \nIn order to obtain the Hubble parameter in terms of the number of e -folds N , we can solve Eq.(21) to find a first integral given by \nH N exp[ G ( H )] = exp [∫ ( n s -1) dN ] , (23) \nwhere the new function G ( H ) is given by \nG ( H ) = ∫ g ( H ) H dH. (24) \nIn this way, the Hubble parameter H = H ( N ) can be obtained from the differential equation given by Eq.(23) assuming a specific attractor for the scalar spectral index n s = n s ( N ). \nIn order to find the effective potential in terms of the number of e -folds i.e., V ( N ), we can consider the modified Friedmann equation (1) obtaining \nV ( N ) ≃ 3 κ F ( H ) . (25) \nHere we have considered the slow roll approximation in which the scalar potential V ≫ ˙ ϕ 2 / 2 during the inflationary stage. \nAdditionally, to determine a relationship between the number of e -folds N and the scalar field ϕ , we can use the Eqs.(6), (16), and (18) to obtain \nN ϕ = √ κ F H H N H. (26) \nIn this form, integrating Eq.(26) we can determine the relation between N = N ( ϕ ) for a specific attractor n s = n s ( N ). Finally, replacing the solution given by Eq.(26) into Eq.(25), we will rebuild the effective potential as a function of the scalar field ϕ , i.e., V = V ( ϕ ).', 'III. REHEATING: GENERAL DESCRIPTION': "In this section, we will analyze the reheating era in the framework of the generalized Friedmann equation in a general description. In this sense, we will utilize the expressions associated to the background variables to find the reheating parameters, such as, the reheating temperature, as well as, the number of e -folds during this epoch. To start with this study, we can assume that the physical scale cross the horizon during inflation when the wave number k is equal to k = a k H k . In addition, we can assume that the physical scale crosses the horizon at the current epoch when k 0 = a 0 H 0 . Here, the subscript ' k ' denotes that the quantities are evaluated when k = a k H k , and the subscript '0' represents the physical quantities evaluated at the present time. Besides, the ratio between the wave numbers k/k 0 can be written as \nk k 0 = a k H k a 0 H 0 = ( a k a end )( a end a reh )( a reh a eq )( a eq H eq a 0 H 0 )( H k H eq ) . (27) \nAs before, here we have used that the 'end' subscript means that the variable is evaluated at the end of inflation. In addition, the notation 'reh' corresponds to the reheating era and 'eq' denotes the of radiation-matter equality. \nIn fact, we can write the number of e -folds N in each stage (duration) as a function of the scale factor a as follows \nN k = ln ( a end a k ) , N reh = ln ( a reh a end ) and N RD = ln ( a eq a reh ) , \nrespectively. Here, the notation N reh corresponds to the number of e -folds during reheating era and N RD denotes to the number of e -folds in the radiation dominance (RD). By using these different e -folds in each era, we rewrite Eq.(27) as \nln ( k a 0 H 0 ) = -N k -N reh -N RD +ln ( a eq H eq a 0 H 0 ) +ln ( H k H eq ) . (28) \nBesides, we can utilize the EoS parameter ω reh related to the reheating regime to express the ratio between the energy density at the end of inflation ρ end and the energy density during the reheating stage ρ reh . In this way, the ratio ρ reh /ρ end can be written as \nρ reh ρ end = e -3 N reh (1+ ω reh ) . (29) \nFor this expression we have considered that during the reheating stage the energy density ρ has a dependence with the scale factor a as; ρ ∝ a -3(1+ ω reh ) , in which the parameter ω reh is assumed a constant. \nIn order to determine the enrgy density of the field at the end of inflation ρ end in Eq.(29), we can first rewrite the parameter ϵ H from Eq.(8) as \nϵ H = 2 κ H F H ( ˙ ϕ 2 2 ) = 2 κ H F H ( ρ -V ) , (30) \nwhere we have used Eqs.(1) and (2), respectively. Considering that the end of the inflationary era occurs when ϵ H = 1 (or equivalently a = 0), we can find from the above equation that the energy density at the end of the inflationary era ρ end can be obtained from the relation \nρ end -1 2 κ ( d F d ln H )∣ ∣ ∣ ∣ end = V end , (31) \nwhere we mention that the second term of Eq.(31) is a function of ρ end and it depends of the function F ( H ). \nIn the particular case in which the function \nF \n( \nH \n) corresponds to the standard Friedmann equation i.e., \nF \n( \nH \n) = \nH \nwe have that the term ( \nd \nF \n/d \nln \nH \n) \n\| \nknown result for \nρ \nend \ngiven by \nρ \nend \nend \n= (3 \n/ \n2) \nV \n= (2 \nκ/ \n3) \nρ \nend \nend \n. \nThus, substituting this result into Eq.(31) we find the well \n[64, 66, 67]. \nOn the other hand, to find the reheating temperature T reh , we can consider the entropy conservation, in which the entropy generated during reheating is preserved in the CMB together with the neutrino background at the current epoch [67]. In this context, following Ref.[67] we can write this conservation as \ng s,reh a 3 reh T 3 reh = a 3 0 ( 2 T 3 0 + 21 4 T 3 ν, 0 ) , (32) \nwhere the quantity g s,reh corresponds to the effective number of relativistic degrees of freedom for entropy at reheating, the temperature T 0 denotes the present CMB temperature i.e., T 0 ≃ 2 . 7K and the quantity T ν, 0 is the present neutrino temperature. Following Ref.[67] we can consider that the relation between the temperatures T ν, 0 and T 0 is given by T ν, 0 = (4 / 11) 1 / 3 T 0 , and then from Eq.(32) we can relate the scale factors during the reheating scenario and at the current era from the expression a reh /a 0 = [43 / (11 g s,reh )] 1 / 3 T 0 /T reh . \nIn addition, we can consider that the energy density at the end of reheating ρ reh corresponds to the hot radiation ρ reh ∝ T 4 reh , with which we can write \nρ reh = π 2 30 g ⋆ ,reh T 4 reh , (33) \nwhere the quantity g ⋆ ,reh represents the effective number of relativistic degrees of freedom at the end of reheating scenario. \nIn this form, utilizing the above expressions, we find that the reheating temperature T reh in terms of the parameters ρ end , ω reh and N reh becomes \nT reh = exp [ -3 4 (1 + ω reh ) N reh ][ 30 ρ end g ⋆ ,reh π 2 ] 1 / 4 . (34) \nBesides, we find that the duration of the reheating epoch characterized by the number of e -folds N reh results \nN reh = 4 1 -3 ω reh [ -N k -ln ( k a 0 T 0 ) -1 3 ln ( 11 g s,reh 43 ) -1 4 ln ( 30 κ 2 ρ end g ⋆ ,reh π 2 ) + 1 2 ln ( π 2 rA s 2 )] . (35) \nHere we note that to obtain the energy density at the end of the inflationary scenario ρ end , we need to solve the Eq.(31) for the different functions F ( H ), and then to write the density ρ end in terms of the potential at the end of inflation i.e., ρ end = ρ end ( V end ). In this sense, we have analyzed the reheating scenario for any F ( H ) -model under one general description. In the following, we will apply our methodology for different functions F ( H ) for the simplest attractor for the scalar spectral index in terms of the number of e -folds given by n s = 1 -2 /N .", 'IV. RECONSTRUCTION FROM THE SCALAR SPECTRAL INDEX n s = n s ( N )': "In this section we will make use of the methodology earlier described, considering as attractor one specific parametrization for the scalar spectral index as a function of the number of e -folds N . In this context, assuming different functions F ( H ), we will reconstruct the Hubble parameter together with the effective potential in terms of the scalar field. In addition, in this section we will analyze the reheating era, considering the background variables obtained in the preceding section from of different functions F ( H ). \nIn order to give a scalar spectral index n s as a function of the number of e -folds N , we consider the simplest parametrization (or attractor) n s = n s ( N ) defined by [31] \nn s ( N ) = 1 -2 N , (36) \n̸ \nwith N = 0. Here as we mentioned before, this parametrization is in agreement with Planck's measurements, when the number of e -folds becomes N ≃ 60 [32]. \nIn addition, to test the modified Friedmann model, we will assume that the reconstruction of the background variables will be developed in light of three types of functions F ( H ) associated to the Hubble parameter H . First, we will consider that the function F ( H ) of the power-law type F ( H ) = α λ H β . As a second example, we will assume that the function of type F ( H ) = H 2 -γH -2 and finally we will choose a function of type F ( H ) = H 2 ± θH 4 to study the reconstruction and the reheating of these F ( H ) -models. \n2 \n, \nJ \n( \nϕ \n) = \n(4 \n- \nβ \n) \n(3 \n- \nβ \n) \nB \nβ \n- \n3 \nβ \n- \n4 \n1 \nκ/ \n( \nα \nλ \nA \n1 \nβA \n1 \n) \nϕ \n+˜ \nc \n1 \n2 F 1 [ 1 , 1 + 1 β -4 , 2 + 1 β -4 , 1 + A 1 B 1 ( √ κ/ ( α λ βA 1 ) ϕ +˜ c 1 ) ] , (42) \nwhere ˜ c 1 is a new constant of integration, and the function 2 F 1 corresponds to the hypergeometric function [68]. \nThus, from Eqs.(39) and (42), we find that the reconstruction of the Hubble parameter as a function of ϕ can be written as \nH ( ϕ ) = [ (4 -β ) ( A 1 J -1 ( ϕ ) + B 1 )] 1 / ( β -4) . (43) \nIn this way, using now the Eq.(1) in the slow roll approximation, in which V ≃ 3 α λ H β /κ (see Eq.(25)), we obtain that the reconstruction of the the potential V ( ϕ ) is given by \nV ( ϕ ) = 3 α λ κ [ (4 -β ) ( A 1 J -1 ( ϕ ) + B 1 )] β/ ( β -4) . (44) \n( \n) \n( \nB \n1 \n+ \n√ \n) \n×", 'A. Reconstruction example I: F ( H ) = α λ H β': "The first example that we will consider to reconstruct the background variables is the case in which F ( H ) is a function of the power-law type. In this sense, we will analyze the function defined as \nF ( H ) = α λ H β , (37) \nwhere the parameters α λ and β are positive constants. We mention that the dimension of the parameter α λ is [ α λ ] = M 2 -β p and β is a dimensionless quantity. In particular, for the standard Friedmann equation in which F ( H ) = H 2 , we have that β = 2 and α λ = 1. Besides, as we mentioned before this class of function associated to the Hubble parameter F ( H ) ∝ H β is obtained in different limits in a GB theory, as well in entropy considerations. \nTo rebuild the Hubble parameter in terms of the number of e -folds N , firstly we can replace Eq.(37) into Eq.(22) to obtain that the function g ( H ) becomes \ng ( H ) = H F HH F H -4 = β -5 = constant . (38) \nNow, introducing this result into Eq.(24) and using Eqs.(23) and (36), we find that the Hubble parameter as a function of the number of e -folds N yields \n̸ \nH ( N ) = [ (4 -β ) ( A 1 N + B 1 )] 1 / ( β -4) > 0 , with β = 4 , (39) \nand the quantities A 1 and B 1 are two arbitrary integration constants which have units of M β -4 p , since the Hubble parameter has dimension of M p . In general these integration constants are positives, negatives o zero. However, in what follows, we will assume for simplicity that the integration constant B 1 > 0, while the constant A 1 must be positive, as we will see later. \nTo rebuild the Hubble parameter in terms of the scalar field H = H ( ϕ ), we have to find the number of e -folds N as a function of the scalar field ϕ i.e., N = N ( ϕ ). Thus, from Eq.(26) we get \ndN dϕ = N ϕ = √ κ α λ βA 1 ( N H ) = √ κ α λ βA 1 N [ (4 -β ) ( A 1 N + B 1 )] -1 / ( β -4) , (40) \nand solving this differential equation, we obtain that the solution for the number N ( ϕ ) can be written as \nN ( ϕ ) = J -1 ( ϕ ) , (41) \nwhere the quantity J -1 ( ϕ ) represents the inverse of the function J ( ϕ ) defined by \n( \nβ \n- \n3 \nβ \n- \n4 \n) \nHere we note that in the special case in which the parameter β = 2 and then α λ =2 = 1, the reconstruction of the effective potential in terms of the scalar field defined by Eq.(44) is reduced to standard expressions obtained in GR (T-model) [31] in which \nV ( ϕ ) = 3 2 κB 1 tanh 2 [ 1 2 ( √ κB 1 A 1 ϕ +¯ c 1 )] . (45) \nOn the other hand, in order to constraint the free parameters of our model, we will consider the amplitude of the power spectrum of the scalar perturbations given by Eq.(13) together with the tensor-scalar ratio given by Eq.(15). In this form, we have that the amplitude of the power spectrum of the scalar perturbations results \nA s = κ 4 π 2 N 2 ( α λ βA 1 ) ⇒ Λ = ( α λ βA 1 ) = κ 4 π 2 N 2 A s . (46) \nHere we note that since that the parameters α λ and β are positives quantities, then from the above expression we have that the integration constant A 1 > 0. If we consider the case in which the number of e -folds during the crossing epoch is N k = 60 and A s = 2 . 2 × 10 -9 , we obtain from Eq.(46) that A 1 ≃ 4 . 14 × 10 10 M -2 p /α λ β . In particular for the standard Friedmann equation in which the parameter β = 2 and α 2 = 1, we obtain the constraint on the parameter A 1 ≈ 2 × 10 10 M -2 p . \nOn the other hand, the tensor to scalar ratio r is calculated as follows \nr = 8Λ H 2 N 2 = 2 κ π 2 A s [ (4 -β ) ( A 1 N + B 1 )] 2 / ( β -4) , (47) \nwhere we have used Eq.(39). Since that the tensor to scalar ratio r is a real and positive quantity, we can constraint the parameter β to 0 < β < 4. \nBesides, using Eq.(47) we will obtain a constraint for the parameter B 1 . To do this, we consider the fact that r evaluated at the time of crossing is constrained by the value r ( k ) \| k = a k H k = r k < 0 . 039. Thus, from this observational parameter we have \nB 1 > 1 4 -β ( 2 κ π 2 A s r k ) (4 -β ) / 2 -κN k 4 π 2 α λ βA s . (48) \nSince B 1 > 0, we can obtain a lower bound for the parameter α λ given by \nα λ > (4 -β ) κN k 4 π 2 βA s ( 2 κ π 2 A s r k ) ( β -4) / 2 . (49) \nAdditionally, to find the number of e -folds at the end of the inflationary epoch N end , we can consider that the parameter ϵ H ( N ) given by Eq.(8) can be rewritten as \nϵ H = H N H = A 1 (4 -β ) N 2 ( A 1 N + B 1 ) -1 = 1 (4 -β )(1 + µN ) N , (50) \nwhere we have used Eq.(39), and we have defined the dimensionless parameter µ = B 1 /A 1 . Thus, the above equation can be evaluated at the end of inflation, where ϵ H = 1 (or equivalently a = 0). In this way, we find the number of e -folds at the end of the inflationary era N end becomes \nN end = √ 1 + ( 4 4 -β ) µ -1 2 µ . (51) \nHere we note that for the special case in which the parameter µ ≪ 1, the number of e -folds at the end of inflation tends to zero, i.e., N end ∼ 0. While in the situation in which µ ≫ 1 the number of e -folds N end ∼ [(4 -β ) µ ] -1 / 2 . \nIn Fig.1 shows the evolution of the number of e -folds (left panel) and the effective potential (right panel) as a function of the scalar field for two different values of the parameter β . In addition, when β = 1 (or α λ =1 = 1 ), we show three values for the brane tension σ , associated to the parameter α λ =1 from the relation α λ =1 = (2 κσ/ 3) 1 / 2 . In this form, for β = 1, we consider the values; σ = 10 -9 M 4 p , σ = 10 -8 M 4 p and σ = 10 -7 M 4 p , which give the values α 1 = 2 . 58 × 10 -5 M p (purple curve), α 1 = 8 . 16 × 10 -5 M p (blue curve) and α 1 = 2 . 58 × 10 -4 M p (green curve), respectively. \nBesides, we have obtained for σ = 10 -9 M 4 p the values A 1 = 1 . 61 × 10 15 M -3 p and B 1 = 1 . 15 × 10 13 M -3 p associated to the purple curve, for the brane tension σ = 10 -8 M 4 p the values A 1 = 5 . 08 × 10 14 M -3 p and B 1 = 2 . 98 × 10 13 M -3 p (blue curve) and the values A 1 = 1 . 61 × 10 14 M -3 p and B 1 = 3 . 56 × 10 13 M -3 p for σ = 10 -7 M 4 p (green curve). In addition, we have considered the special case β = 2 in which α 2 = 1 (red curve), which as mentioned before represents the standard Friedmann equation. For this situation we have found the values A 1 = 2 . 07 × 10 10 M -2 p and B 1 = 8 . 35 × 10 8 M -2 p . In these plots we have considered that the constant ¯ c 1 = 0 for simplicity and the number of e -folds N k = 60. In relation to the right panel of this figure, we note that the reconstructed effective potential in terms of the scalar field shows a maximum value given by a flat region for largeϕ ( ϕ > 15 M p ) in which the number of e -folds N is also large ( N > 30), see left panel. In addition, from the right panel, we can note that scalar field begins to roll from the maximum value of the potential (flat region) towards values of the scalar field ϕ ∼ 0 where the number of e -folds at the end of inflation N end is approximately zero. \nIn Fig.2 we show the contours curves associated to the tensor to scalar ratio r , and different combinations of the parameters α 1 and B 1 from Eq.(47) for the special case in which N = 60. From this plot, given a value of the tensor to scalar ratio from the vertical column we can constrain the parameter space of B 1 -α 1 . \n) \n( \nFIG. 1: Evolution of the number of e -folds (left panel) and the effective potential (right panel) versus the scalar field for two different values of the parameter β . In particular for the case in which β = 1, we have considered three values on the brane tension σ . Also, in both panels we have used that the integration constant ¯ c 1 = 0 and the number N k = 60. \n<!-- image --> \nFIG. 2: Contour plot for the tensor to scalar ratio r in terms of the integration constant B 1 and the parameter associated to brane tension α 1 . Here we have fixed the number of e -folds N k = 60 and ¯ c 1 = 0. \n<!-- image --> \nOn the other hand, we will study the reheating epoch using the reconstruction of the background variables derived during the inflationary scenario. To analyze the reheating era, we find that the energy density at the end of inflation, from Eq.(31) yields \nρ end = ( 1 1 -β/ 6 ) V end . (52) \nHere as mentioned before, we note that if β = 2, we recover the standard case for the Friedmann equation and the energy density at the end of inflation is reduced to ρ end = (3 / 2) V end . By using Eq.(52) and replacing into Eqs.(36) \nand (47), we find that the reheating temperature T reh and the number of e -folds during the reheating N reh , given by Eqs.(34) and (35) as a function of the spectral index n s become \nT reh ( n s ) = exp [ -3 4 (1 + ω reh ) N reh ( n s ) ][ 30 V end g ⋆ ,reh π 2 (1 -β/ 6) ] 1 / 4 , and (53) \nN reh ( n s ) = 4 1 -3 ω reh [ 2 n s -1 -ln ( k a 0 T 0 ) -1 3 ln ( 11 g s,reh 43 ) -1 4 ln ( 30 κ 2 V end g ⋆ ,reh π 2 (1 -β/ 6) ) + ˜ r 1 ( n s ) ] , (54) \n̸ \nwhere ω reh = 1 / 3 and we have defined the function ˜ r 1 ( n s ) as \n˜ r 1 ( n s ) = ln { κ 1 / 2 [ (4 -β ) ( (1 -n s ) 2 A 1 + B 1 )] 1 / ( β -4) } . (55) \nThe Fig.3 shows the evolution of the temperature T reh ( n s ) and the number of e -folds N reh ( n s ) during the reheating epoch versus the scalar spectral index from Eqs.(53) and (54), respectively. Four graphs have been plotted: three plots represent different values of the brane tension σ in which β = 1, while one corresponds to β = 2 reflecting the standard Friedmann equation. In all the panels different values have been used for the EoS parameter ω reh related to the reheating epoch and these values are; ω reh = {-1 / 3 , 0 , 2 / 3 , 1 } . In this figure, the different regions are; the light blue shaded region corresponds to the maximum and minimum bounds of the scalar spectral index n s = 0 . 9649 ± 0 . 0042 from Planck data at 1 σ limits. Also, the blue shaded denotes to a projected sensitivity from the central value of the scalar spectral index ( n s = 0 . 9649) with an uncertainly of ± 10 -3 in the test analyzed by the authors in Ref.[69]. Besides, the pink shaded region is the electroweak scale where the temperature T EW ∼ 100GeV and the purple shaded region corresponds to a temperatures below 10 MeV, and this region is disapproved by primordial nucleosynthesis. \nIn the upper left panel of Fig.3, we have considered the parameter β = 1 in which the Friedmann equation is H ∝ ρ , and we have chosen that the tension of the brane σ = 10 -9 M 4 p (or equivalently α 1 = 2 . 58 × 10 -5 M p ). Here we have determined that the values of the parameters A 1 , B 1 and N end correspond to; A 1 = 1 . 61 × 10 15 M -3 p , B 1 = 1 . 15 × 10 13 M -3 p and N end = 0 . 33. In the upper right panel, we have also β = 1 and the for the tension of the brane the value σ = 10 -8 M 4 p ( or α 1 = 8 . 16 × 10 -5 M p ). In this situation, we have found that the values for the parameters are given by A 1 = 5 . 08 × 10 14 M -3 p , B 1 = 2 . 98 × 10 13 M -3 p and N end = 0 . 33, respectively. In the lower left plot we have again used β = 1 and for the brane σ = 10 -7 M 4 p (or α 1 = 2 . 58 × 10 -4 M p ), together with the values A 1 = 1 . 61 × 10 14 M -3 p , B 1 = 3 . 56 × 10 13 M -3 p and N end = 0 . 31. In the lower right plot, we have considered the standard Friedmann equation in which the parameter β = 2 i.e., H 2 ∝ ρ , and then α 2 = 1. For this case, we have obtained that the values for the parameters A 1 , B 1 and N end become A 1 = 2 . 07 × 10 10 M -2 p , B 1 = 8 . 35 × 10 8 M -2 p and N end = 0 . 49, respectively. \nWe mention that the stage of instantaneous reheating is given when the number of e -folds at the end of inflation N reh approaches zero. In these panels, this corresponds to the point in which all lines converge to the value N reh ≃ 0. From this plot, we note that the scenario of instantaneous reheating the model presents the maximum temperature of the reheating on the order of T reh ∼ 10 16 GeV for β = 2 and for the standard Friedmann equation a little less T reh ∼ 10 15 GeV. Besides, we observe that the stage of instantaneous reheating (when N reh ≃ 0) does not depend on the EoS parameter ω reh , since all curves associated to the different ω reh converge to the same point. Besides, we can note from the different panels that the highest value of the number of e -folds during the reheating takes place when the EoS parameter ω reh = 2 / 3 independently of the value of β , and this value of N reh becomes N reh ≲ 40. \nIn relation to the reheating temperature T reh , we note that that the models associated to β = 1 and β = 2, present a good compatibility with Planck's 1 σ bounds on the index n s , for the distinct values of the parameter ω reh , excluding the value ω reh = -1 / 3, when the temperature T reh < 10 +12+13 GeV. In particular, when the EoS parameter takes the value ω reh = -1 / 3, the compatibility with Planck's 1 σ bounds is disadvantaged when the temperature T reh < 10 +13+14 GeV for the case in which β = 2, and when the temperature T reh < 10 +12+13 GeV for the standard Friedmann equation ( β = 1). \nThe summary of the parameter-space of the model F = α λ H β can be consulted in the Table.I. Here the permitted values obtained for the integration constants A 1 and B 1 together with the number of e -folds at the end of inflation N end for two different values of the parameter or power β = 1 , 2 are shown in Table I. In particular , for the situation in which β = 1, we have considered three different values of the tension for the brane σ . Additionally, in this table we have utilized the values N k = 60, A s k = 2 . 2 × 10 -9 and r k = 0 . 039. Also, from this table we can observe that the ratio between the integration constants A 1 and B 1 is the order of O (10 2 ). Besides, we note from Table I that the number of e -folds at the end of the inflationary era N end ∼ O (1) independently on the value of the power β . \nFurthermore, we observe that the values associated to the integration constants A 1 and B 1 are very similar for the cases in which the brane tension σ = 10 -8 M 4 p and σ = 10 -7 M 4 p . Additionally, we note that when we decrease the value of the power β from the values β = 2 to β = 1, the constraints on the integration constants A 1 and B 1 decrease by several orders of magnitude ∼ O (10 4 ). \nTABLE I: This table summarizes the parameter-space of the modified Friedmann equation F = α λ H β , for the specific cases in which the power β takes the values β = 1 and β = 2, respectively. In particular for the situation in which β = 1 we have considered three different values for the brane tension σ . \nFIG. 3: The figure shows the number of e -folds during the reheating scenario N reh (upper panels) and the reheating temperature T reh (lower panels) versus the scalar spectral index n s for different values of the parameter β . In particular for the special case in which the power β = 1, we have considered three different values for the brane tension σ . For the case β = 1 and the brane tension σ = 10 -9 M 4 p corresponds to the upper left panel. The case β = 1 and σ = 10 -8 M 4 p is for the upper right panel. Again the case β = 1 and σ = 10 -7 M 4 p corresponds to the lower left panel and the situation in which β = 2 i.e., the standard Friedmann equation is the lower right panel. Besides, in all the panels we have assumed different values of the EoS parameter ω reh associated to the reheating epoch; ω reh = -1 / 3 , 0 , 2 / 3 and ω reh = 1, respectively. Also, in each of the panels, we have utilized the parameters shown in Table I. \n<!-- image -->", 'B. Reconstruction example II: F ( H ) = H 2 -γH -2': "As a second example that we will consider to rebuild the background variables corresponds to the modified Friedmann equation in which the function F ( H ) motivated by the Kaniadakis Entropy [15] is given by \nF ( H ) = H 2 -γH -2 , (56) \nwhere γ is a positive constant with dimensions of M 4 p . As we comment in the Introduction, the Kaniadakis entropy (or K-entropy), is a one-parameter extension of the classical Boltzmann-Gibbs-Shannon entropy[15]. Besides, we note that in order to obtain an energy density ρ > 0, we have to satisfy that the Hubble parameter H > γ 1 / 4 . \nFor this type the function F ( H ), we find from Eq.(22) that the function g ( H ) becomes \ng ( H ) = -7 γ +3 H 4 γ + H 4 . (57) \nThus, introducing this result into Eq.(24) and using the attractor n s = n s ( N ) given by Eq.(36), we find that from Eq.(23), the equation for the Hubble parameter in terms of the number of e -folds N can be written as \nγ 3 + H 4 = 2 ( A 2 N + B 2 ) H 6 , (58) \nwhere the quantities A 2 and B 2 are two integration constants. As we will see later the constant A 2 > 0 and for simplicity in the following we will consider that integration constant B 2 > 0. By making a change of variable, such that x = H 2 in the previous equation, then we obtain a cubic equation, whose real solution for the Hubble parameter H = H ( N ) is given by \nH ( N ) = x 1 / 2 = [ 1 3 f ( 1 + A 1 / 3 + A -1 / 3 ) ] 1 / 2 , (59) \nwhere we have assumed that the function A = A ( N ) in terms of the number of e -folds N is a quantity positive and it is defined as \nA = A ( N ) = 1 2 [ 2 + 9 γf 2 + √ (2 + 9 γf 2 ) 2 -4 ] where f = f ( N ) = 2 ( A 2 N + B 2 ) . (60) \nAs before, in order to obtain the reconstruction of the background variables, we need to determine the relation between the number of e -folds and the scalar field, i.e., N = N ( ϕ ). In this way, we can use Eq.(26) to find the number of e -folds in terms of the scalar field ϕ from the differential equation given by \ndN dϕ = N ϕ = √ κ 2 A 2 ( N H ) = √ 3 κ 2 A 2 √ 2 N ( 1 + A 1 / 3 + A -1 / 3 ) -1 / 2 ( A 2 N + B 2 ) 1 / 2 . (61) \nHere we note that this differential equation has no analytical solution. In this context and in order to solve the Eq.(61), we can assume that during the inflationary scenario the ratio between the integration constants A 2 /B 2 ≪ N . Thus, using this approximation, we obtain that the Hubble parameter as a function of the number of e -folds N i.e., H = H ( N ) can be approximate to the expression \nH ( N ) ≃ C 1 -C 2 N + O ( N -2 ) , (62) \nwhere the constants C 1 and C 2 = ( ¯ C 2 + ¯ C 3 ) / ¯ C 4 are given by \nC 1 = ( 1 / √ 6 B 2 ) [ 1 + ( 1 + 2 B +2 √ B ( B +1) ) 1 / 3 + ( 1 + 2 B +2 √ B ( B +1) ) -1 / 3 ] 1 / 2 , (63) \n¯ C 2 = A 2 [ 1 + ( 1 + 2 B +2 √ B ( B +1) ) 1 / 3 + ( 1 + 2 B +2 √ B ( B +1) ) -1 / 3 ] (64) \n¯ C 3 = 2 A 2 3 [ √ B ( B +1) + 2 B ( 1 + B + √ B ( B +1) )] [ 1 -( 1 + 2 B +2 √ B ( B +1) ) 2 / 3 ] ( B +1) ( 1 + 2 B +2 √ B ( B +1) ) 4 / 3 (65) \n¯ C 4 = 2 √ 6 B 3 / 2 2 [ 1 + ( 1 + 2 B +2 √ B ( B +1) ) 1 / 3 + ( 1 + 2 B +2 √ B ( B +1) ) -1 / 3 ] 1 / 2 , (66) \nwith the quantity B defined as B = 9 γB 2 2 . \nUnder the approximation in which the ratio A 2 /B 2 ≪ N , we have assumed that the early universe suffers an inflationary expansion from a nearly exponential expansion of the scale factor, in which the Hubble parameter is practically a constant and given by Eq.(62), (for a quasi-de Sitter inflation, see e.g., Refs.[70, 71]). \nThus, considering the approximation for the Hubble parameter given by Eq.(62), we find that the differential equation given by (61) can be approximate to \nN ϕ = √ κ 2 A 2 ( N H ) ≃ √ κ 2 A 2 ( N C 1 -C 2 /N ) . (67) \nIn this form, from Eq.(67), we obtain that the number of e -folds in terms of the scalar field ϕ i.e., the solution N = N ( ϕ ) can be written as \nN ( ϕ ) ≃ -C 2 C 1 1 ProductLog { -C 2 C 1 exp [ -1 C 1 (√ κ 2 A 2 ϕ -˜ c 2 )]} , (68) \nwhere ˜ c 2 corresponds to a new integration constant. Besides, here the ProductLog function corresponds to a product logarithm, also called the Omega function or Lambert W function, see e.g., Ref.[72]. \nThus, we find that the reconstruction of the Hubble parameter as a function of the scalar field becomes \nH ( ϕ ) ≃ C 1 ( 1 + ProductLog { -C 2 C 1 exp [ -1 C 1 (√ κ 2 A 2 ϕ -˜ c 2 )]}) . (69) \nBesides, the reconstruction of the effective potential in terms of the scalar field under the slow roll approximation yields \nV ( ϕ ) ≃ 3 κ [ H 2 ( ϕ ) -γ H 2 ( ϕ ) ] ≃ 3 C 2 1 κ [ ( 1 + ProductLog { -C 2 C 1 exp [ -1 C 1 (√ κ 2 A 2 ϕ -˜ c 2 )]}) 4 -γ/ C 4 1 ] ( 1 + ProductLog { -C 2 C 1 exp [ -1 C 1 (√ κ 2 A 2 ϕ -˜ c 2 )]}) 2 . (70) \nOn the other hand, we can find a constraint on the integration constant A 2 , using the expression for the power spectrum of the scalar perturbation. In this sense, we have that the amplitude of the power spectrum of the scalar perturbation, given by Eq.(13) becomes \nA s = κ 8 π 2 N 2 A 2 ⇒ A 2 = κ 8 π 2 N 2 A s . (71) \nThus, from Eq.(71) and assuming that the number of e -folds N k = 60 and the power spectrum of the scalar -9 10 -2 \nperturbation corresponds to A s = 2 . 2 × 10 we find that the integration constant A 2 = 2 . 07 × 10 M p . \nIn addition, using Eq.(62), we find that the tensor to scalar ratio in terms of the number of e -folds N yields \nr = 16 A 2 H 2 N 2 ≃ 2 κ π 2 A s ( C 1 -C 2 N ) 2 . (72) \nHere we note that using Eq.(72), we will numerically obtain a constraint on the second integration constant B 2 , since the ratio r is evaluated at the horizon exit during the inflationary epoch where it is constrained by r ( k ) \| k = a k H k = r k < 0 . 039 together with the number N k = 60. Thus, numerically we will determine the value of the integration constant B 2 . \nBesides, by considering that the function F ( H ) = H 2 -γH -2 > 0 or H 4 > γ , then combining Eqs.(71) and (72) we obtain an upper bound for the parameter γ given by \n( r N 2 16 A 2 ) 2 > γ. (73) \nIn particular considering the upper limit for the tensor to scalar ratio r k = 0 . 039 together with the number of e -folds N k = 60 and the constraint found on the integration constant A 2 = 2 . 07 × 10 10 M -2 p we have \n1 . 80 × 10 -19 M 4 p > γ. (74) \nIn Fig.4, the left panel shows the reconstructed effective potential in terms of the scalar field given by Eq.(70) for different values of the parameter γ , when we fix the tensor to scalar ratio r k = 0 . 039. From this panel, we note that for the different values of the parameter γ , which satisfy the upper bound given by Eq.(74) in which 10 -19 M 4 p > γ , the shape of the reconstructed effective potential V ( ϕ ) does not change, since the different lines associated to γ are overlaid one after the other. This suggests that due to that the parameter γ ≪ 1, the reconstruction of the effective potential V ( ϕ ) is not strongly affected for this parameter. To build this panel, we have used three different values of the parameter γ where we fix the value of the tensor to scalar ratio r k = 0 . 039. The purple curve corresponds to γ = 10 -22 M 4 p , and the values obtained of B 2 , C 1 and C 2 result B 2 = 5 . 95 × 10 8 M -2 p , C 1 = 2 . 90 × 10 -5 M p and C 2 = 5 . 05 × 10 -4 M p , respectively. To obtain the blue curve we have used γ = 10 -24 M 4 p , and then B 2 = 5 . 94 × 10 8 M -2 p , C 1 = 2 . 90 × 10 -5 M p and C 2 = 5 . 06 × 10 -4 M p . Finally, the green curve is obtained using γ = 10 -26 M 4 p , B 2 = 5 . 94 × 10 8 M -2 p , C 1 = 2 . 90 × 10 -5 M p and C 2 = 5 . 06 × 10 -4 M p . It is useful to remember that the value of the parameter B 2 is obtained numerically from Eq.(72), fixing the tensor to scalar ratio r for the different values of γ . In particular for the tensor to scalar ratio r k = 0 . 039, we find that of the integration constant B 2 does not change significantly for the different values of γ utilized. \nIn the right panel of the Fig.4, we also show the reconstructed effective potential V as a function of the scalar field ϕ . Here three curves have been drawn for different values of the tensor to scalar ratio at the time of the crossing, when we fix the parameter γ to the value γ = 10 -24 M 4 p . In particular, the purple curve corresponds to the value r k = 0 . 039, and then we numerically find that the integration constant B 2 becomes B 2 = 5 . 94 × 10 8 M -2 p and therefore the quantities C 1 and C 2 result C 1 = 2 . 90 × 10 -5 M p and C 2 = 5 . 06 × 10 -4 M p , respectively. In the case of the blue curve in which r k = 0 . 02, we find that the parameters are B 2 = 1 . 90 × 10 9 M -2 p , C 1 = 1 . 62 × 10 -5 M p and C 2 = 8 . 85 × 10 -5 M p . Finally, the green curve corresponds to r k = 0 . 01, and then the values are given by B 2 = 4 . 24 × 10 9 M -2 p , C 1 = 1 . 09 × 10 -5 M p and C 2 = 2 . 65 × 10 -5 M p , respectively. \nFIG. 4: The left panel shows the reconstructed effective potential in terms of the scalar field for three different values of the parameter γ , when we fix the tensor to scalar ratio to the upper bound r k = 0 . 039. The right panel also shows the reconstructed effective potential as a function of the scalar field for three different values of the tensor to scalar ratio r , when we now set the parameter γ = 10 -24 M 4 p . In both panels we have used that the constant ˜ c 2 = 0 for simplicity. \n<!-- image --> \nOn the other hand, we will analyze the reheating era for our second example, in which the function F ( H ) associated to the modified Friedmann equation is given by F ( H ) = H 2 -γH -2 . In the same way as in the example I, we solve the Eq.(31) using this function F ( H ) to find that the energy density at the end of the inflationary epoch becomes \nρ end = 3 8 V end ( 3 ± √ 1 + 32 γ κ 2 V 2 end ) . (75) \nIn order to obtain the standard result of the above expression, in which ρ end → (3 / 2) V end , when γ → 0, it is necessary to consider the positive sign of the solution for ρ end given by Eq.(75). Thus, in the following we will consider the positive sign of the solution for ρ end of Eq.(75). \nBesides, as before using Eqs.(34), (35) and (36) and considering Eqs.(72) and (75), we can rewrite the reheating temperature T reh and the number of e -folds N reh during the reheating epoch as a function of the spectral index n s . \nIn this way, we find that T reh and N reh in terms of the scalar spectral index can be written as \nT reh ( n s ) = exp [ -3 4 (1 + ω reh ) N reh ( n s ) ] 45 V end ( 3 + √ 1 + 32 γ κ 2 V 2 end ) 4 g ⋆ ,reh π 2 1 / 4 , and (76) \nN reh = 4 1 -3 ω reh 2 n s -1 -ln ( k a 0 T 0 ) -ln 3 ( 11 g s,reh 43 ) -ln 4 45 κ 2 V end ( 3 + √ 1 + 32 γ κ 2 V 2 end ) 4 g ⋆ ,reh π 2 + ˜ r 2 ( n s ) , (77) \n̸ \nwhere ω reh = 1 / 3 and the function ˜ r 2 ( n s ) is defined as \n˜ r 2 ( n s ) = ln { κ 1 / 2 [ C 1 -(1 -n s ) 2 C 2 ]} . (78) \nThe Fig.5 shows the number of e -folds (upper panel) and the temperature (lower panel) during the reheating scenario as functions of the spectral index for the modified Friedmann equation given by Eq.(56). In these panels we have used different values for the tensor to scalar ratio as well different values associated to the EoS parameters ω reh in each case. Thus, three curves have been plotted, one for each value of r , for each EoS parameter ω reh with values ω reh = {-1 / 3 , 0 , 1 , 2 / 3 } . As before, we note that the instantaneous reheating of the model takes place when N reh ∼ 0 and temperature corresponds to a maximum valor. Besides, we note that any temperature between the BBN bound and the instantaneous reheating value is allowed inside the Planck's 1 σ bound, independently of the EoS parameter. However, we note that the curve associated with the upper limit for the tensor to scalar ratio r k ∼ 0 . 039 (purple curve) is more centered around the value within the Planck's 1 σ . Besides, we observe that the model predicts from Planck data at 1 σ limits ( n s = 0 . 9649 ± 0 . 042) a small number of e -folds during the reheating epoch N reh < 35 for the different temperatures T reh . \n] \n[ \nFIG. 5: The upper panel shows the number of e -folds and the lower panel shows the temperature during the reheating stage versus the scalar spectral index, for different values of the tensor to scalar ratio. In these panels the different curves and shaded regions are similar to the Fig.(3). Besides, we have fixed the parameter γ = 10 -24 M 4 p . \n<!-- image --> \nThe Fig.6 shows the three-dimensional plot for the number of e -folds in terms of the EoS parameter and the scalar spectral index, given by Eq.(77). From this figure, we note that the maximum values of the number of e -folds occur for values of the EoS parameter in the range 0 < ω reh < 1 within the Planck's 1 σ bound. Besides, from this 3-D plot we observe that the number of the e -folds, during the reheating epoch becomes smaller, when we consider negative values of the EoS parameter ω reh . \nFIG. 6: Three-dimensional plot for the number of e -folds versus the EoS parameter ω reh and the scalar spectral index n s . Here we have used that the tensor to scalar ratio corresponds to the upper bound r k = 0 . 039 and N k =60. \n<!-- image -->", 'C. Reconstruction example III: F ( H ) = H 2 ± θH 4': "As a final example to reconstruct the inflationary scenario using the scalar spectral index n s = n s ( N ), we will analyze the modified Friedmann equation given by \nF ( H ) = H 2 ± θH 4 , (79) \nwhere θ is a positive constant with dimension M -2 p . This type of modification to the Friedmann equation comes for example, from the Chern-Simons theories of gravity [22] or quantum corrections to the entropy of the apparent horizon [18] as well black hole entropy in loop quantum gravity[73, 74] or due to correction to black hole entropy due to thermal equilibrium fluctuation or quantum fluctuation, see also Refs.[75, 76]. \nIn relation to the Eq.(79) and the parameter θ , we need to ensure that the term H 2 should be comparable to the term \| θ \| H 4 . In this form, we can assume that the parameter θ must be of the order \| θ \|∼ H -2 for both terms to be significant during the early universe. \nTo begin the reconstruction of the background variables from the modified Friedmann equation defined by Eq.(79), we determine the expression of the function g ( H ) given by Eq.(22) yields \ng ( H ) = -(3 ± 2 θH 2 ) (1 ± 2 θH 2 ) . (80) \nNow using Eqs.(23), (24) and (36), we find that the first order differential equation for the Hubble parameter as a function of the number of e -folds N can be written as \n( 1 ± 2 θH 2 ) H N H 3 = A 3 N 2 , (81) \nwhere A 3 is a constant of integration and as we will see later this constant is positive. In the following, by simplicity, we will consider the positive sign of Eq.(79) in our analysis. \nFrom Eq.(81) we obtain that the solution for the Hubble parameter as a function of N becomes \nH ( N ) = 1 √ 2 θ [ ProductLog ( 1 2 θ e 1 θ ( A 3 N + B 3 ) )] -1 / 2 , (82) \nwhere B 3 corresponds to a new integration constant. In the following, we will assume that this constant B 3 > 0 for simplicity. \nTo rebuild the background variables, we need to find the number of e -folds N as a function of ϕ . Thus, from Eq.(26), we have that the first order differential equation for N = N ( ϕ ) becomes \nN ϕ = √ κ 2 A 3 ( N H ) = √ κθ A 3 N [ ProductLog ( 1 2 θ e 1 θ ( A 3 N + B 3 ) )] 1 / 2 . (83) \nNevertheless, we cannot analytically solve this first order differential equation to obtain N = N ( ϕ ). In order to find an analytical solution to Eq.(83), we can consider that during inflation the ratio between the integration constants A 3 /B 3 ≪ N . By using this approximation, we obtain that the Hubble parameter in terms of the number of e -folds can be approximate to an expansion quasi de Sitter given by \nH ( N ) ≃ D 1 -D 2 N + O ( N -2 ) , (84) \nwhere D 1 and D 2 correspond to two constants defined as \nD 1 = 1 √ 2 θ [ ProductLog ( 1 2 θ e B 3 /θ )] -1 / 2 , and (85) \nD 2 = A 3 2 √ 2 θ 3 / 2 [ ProductLog ( 1 2 θ e B 3 /θ )] -1 / 2 [ 1 + ProductLog ( 1 2 θ e B 3 /θ )] -1 , (86) \nrespectively. In this context, the differential equation given by Eq.(83) is reduced to \nN ϕ ≃ √ κ 2 A 3 ( N D 1 -D 2 /N ) , (87) \nand we find that the solution for the number of e -folds N as a function of the scalar field ϕ yields \nN ( ϕ ) ≃ -D 2 D 1 ( ProductLog { -D 2 D 1 exp [ -1 D 1 (√ κ 2 A 3 ϕ -˜ c 3 )]}) -1 , (88) \nwhere ˜ c 3 is a new integration constant. In this form, replacing Eq.(88) into Eq.(84), we obtain that the reconstruction of the Hubble parameter H = H ( ϕ ), during the inflationary epoch results \nH ( ϕ ) ≃ D 1 ( 1 + ProductLog { -D 2 D 1 exp [ -1 D 1 (√ κ 2 A 3 ϕ -˜ c 3 )]}) . (89) \nBesides, the reconstructed effective potential in terms of the scalar field can be written as \nV ( ϕ ) ≃ ( 3 D 2 1 /κ ) ( 1 + ProductLog { -D 2 D 1 exp [ -1 D 1 (√ κ 2 A 3 ϕ -˜ c 3 )]}) 2 [ 1 -θ D 2 1 ( 1 + ProductLog { -D 2 D 1 exp [ -1 D 1 (√ κ 2 A 3 ϕ -˜ c 3 )]}) 2 ] -1 . (90) \nAdditionally, from the power spectrum of the scalar perturbations defined by Eq.(12), we can determine a constraint on the parameter A 3 given by \nA s = κ 8 π 2 N 2 A 3 = ⇒ A 3 = κ 8 π 2 N 2 A s > 0 . (91) \nAlso, we find that the tensor to scalar ratio r in terms of the number of e -folds N becomes \nr = 16 A 3 H 2 N 2 ≃ 2 κ π 2 A s ( D 1 -D 2 N ) 2 , (92) \nFrom Eq.(92), different values of the parameter B 3 can be numerically obtain for various values of the parameter θ . In Fig.7 the left panel shows the reconstruction of the Hubble parameter versus the scalar field, for different values of the parameter θ . In order to chose the different values of the parameter θ , we have considered that θ ∼ H -2 to ensure that the terms H 2 and θH 4 are the order. Besides, in the right panel we show the reconstruction of the \neffective potential in terms of the inflaton field, for various values of the parameter θ . In the two panels we have fixed the tensor to scalar ratio r k = 0 . 039. Also, in both panels the purple curve is obtained using the following parameters; for θ = 10 8 M -2 p , we determine that the constants B 3 , D 1 and D 2 are given by B 3 = 2 . 86 × 10 9 M -2 p , D 1 = 2 . 58 × 10 -5 M p and D 2 = 3 . 15 × 10 -4 M p . The blue curve is obtained for the special value of θ = 10 9 M -2 p , in which B 3 = 2 . 24 × 10 10 M -2 p , D 1 = 2 . 25 × 10 -5 M p and D 2 = 1 . 18 × 10 -4 M p . Finally, the green curve is built with the values; θ = 10 10 M -2 p , B 3 = 2 . 17 × 10 11 M -2 p , D 1 = 2 . 09 × 10 -5 M p and D 2 = 1 . 94 × 10 -5 M p , respectively. In relation to the left panel of this figure, we note that the reconstructed Hubble parameter in terms of the scalar field presents an approximately quasi de Sitter behavior (flat region) for largeϕ ( ϕ > 10 M p ). In addition, from the right panel, we note that the reconstructed effective potential in terms of the scalar field V = V ( ϕ ) shows a maximum value given by a flat region for values of ϕ > 10 M p , in which the scalar field begins to roll towards values of the scalar field ϕ ∼ 0 in which the inflationary epoch ends. \nFIG. 7: The figure shows the reconstructions of the Hubble parameter (left panel) and the effective potential (right panel) versus the scalar field, for different values of the parameter θ . In both panels we have considered that the tensor to scalar ratio r k = 0 . 039 and the number of e -folds N k =60. \n<!-- image --> \nOn the other hand, to analyze the reheating epoch in our modified Friedmann equation F ( H ) = H 2 + θH 4 , we will proceed as in the previous examples. To determine the energy density at the end of the inflationary scenario, we need to calculate the second term of the Eq.(31). In this way, we have that this term becomes \n1 2 κ ( d F d ln H )∣ ∣ ∣ ∣ end = 1 κ H 2 ( 1 + 2 θH 2 )∣ ∣ end . (93) \nIntroducing this expression into the Eq.(31), we find that the energy density at the end of the inflationary stage results \nρ end = 3 κθ ( 1 + κθV end ± √ 1 + κθV end ) . (94) \nHere we note that in order to obtain the standard result from Eq.(94) in which ρ end = (3 / 2) V end , when θ → 0 , it is necessary to consider the negative sign of this equation. In the following, we will use the negative sign of the expression given by Eq.(94). \nTo study the reheating epoch, as before, we need to determine the functional form of the reheating parameters T reh and N reh in terms of the scalar spectral index n s for our F ( H )-model. In this context, using Eqs.(34), (35) and (36) and now replacing the new expressions given by (92) and (94), we find that the temperature and the number of e -folds as a function of the n s during the reheating epoch result \nT reh ( n s ) = exp [ -3 4 (1 + ω reh ) N reh ( n s ) ] [ 90 ( 1 + κθV end -√ 1 + κθV end ) κθπ 2 g ⋆ ,reh ] 1 / 4 , and (95) \nN reh = 4 1 -3 ω reh [ 2 n s -1 -ln ( k a 0 T 0 ) -1 3 ln ( 11 g s,reh 43 ) -1 4 ln ( 90 κ ( 1 + κθV end -√ 1 + κθV end ) θπ 2 g ⋆ ,reh ) + ˜ r 3 ] , (96) \n̸ \nwith the EoS parameter ω reh = 1 / 3 and the function ˜ r 3 = ˜ r 3 ( n s ) defined as \n˜ r 3 ( n s ) = ln { κ 1 / 2 [ D 1 -(1 -n s ) 2 D 2 ]} . (97) \nThe Fig.8 exhibits the number of e -folds (upper panel) and the temperature (lower panel) during the reheating epoch versus the spectral index for the modified Friedmann equation F ( H ) = H 2 + θH 4 . In these plots we have considered various values for the parameter θ , as well various values associated to the EoS parameters ω reh in each case. In the left panel, we have assumed two different values of the parameter θ for each EoS parameter ω reh with values of ω reh = {-1 / 3 , 0 , 1 , 2 / 3 } . In the right panel we have utilized the specific case in which the parameter θ = 10 10 M -2 p , also for various values of the EoS parameter ω reh . \nIn this context, for each barotropic index ω reh = {-1 / 3 , 0 , 1 , 2 / 3 } three sets of curves have been drawn considering different values of θ . In the left panel, the purple curve has been obtained by using the following set of parameters; for θ = 10 8 M -2 p , we have found that the constants B 3 , D 1 and D 2 together with the number of e -folds at the end of the inflationary era are given by B 3 = 2 . 86 × 10 9 M -2 p , D 1 = 2 . 58 × 10 -5 M p , D 2 = 3 . 15 × 10 -4 M p and N end = 13 . 1, respectively. The blue curve corresponds to the value θ = 10 9 M -2 p , in which B 3 = 2 . 24 × 10 10 M -2 p , D 1 = 2 . 25 × 10 -5 M p , D 2 = 1 . 18 × 10 -4 M p and N end = 6 . 08. In the right panel, the green curve corresponds to the values of θ = 10 10 M -2 p , B 3 = 2 . 17 × 10 11 M -2 p , D 1 = 2 . 09 × 10 -5 M p , D 2 = 1 . 94 × 10 -5 M p and N end = 1 . 53, respectively. \nFIG. 8: In both plots; the upper panel shows the number of e -folds and the lower panel exhibits the temperature during the reheating stage versus the scalar spectral index, for different values of the parameter θ , as well as various values of the EoS parameter ω reh . In the left panel, we show two different values of θ ; θ = 10 8 M -2 p and θ = 10 9 M -2 p , while the right panel shows the situation in which θ = 10 10 M -2 p . In these panels the different curves and shaded regions are as the Fig.(3). Beside, in these plots, we have fixed the tensor to scalar ratio r k = 0 . 039 together with N k =60. \n<!-- image --> \n/uni03C9 \nFIG. 9: Three-dimensional plot for the reheating temperature T reh (logarithmic scale) in terms of the scalar spectral index n s and the EoS parameter ω reh . Here we have used that the tensor to scalar ratio corresponds to the upper bound r k = 0 . 039 and the parameter θ = 10 8 M -2 p . \n<!-- image --> \nAs before, we observe that the instantaneous reheating takes place when N reh ∼ 0 and reheating temperature presents a maximum valor. Besides, we notice that any temperature between the BBN bound and the instantaneous temperature is allowed inside the observational bound, independently of the value of ω reh . In relation to the two plots, we note that from the left panel, the purple and blue curves are more centered around the value within the Planck's 1 σ , in relation to the right panel, in which we have considered the value of the parameter θ = 10 10 M -2 p . Also, we \nobserve from the right panel that increases the value of the parameter θ , the model predicts a large number of e -folds during the reheating epoch ( N reh < 40), for the different reheating temperatures. \nIn Fig.9 we show the three-dimensional plot for the reheating temperature T reh (logarithmic scale) in terms of the EoS parameter and the scalar spectral index. From this figure, we note that the maximum value of the reheating temperature or instantaneous reheating takes place for all values of the EoS parameter ω reh within the Planck's 1 σ limit. Also, we note that for positive values of the EoS parameter the reheating temperature can achieve lower values of the reheating temperature within Planck data 1 σ limit, in comparison when we consider negative values of the EoS parameter in which the reheating temperatures are higher than T reh > 10 7 GeV, (see also Fig.(8)).", 'V. CONCLUDING REMARKS': "In this article we have studied the reconstruction of the background variables during the early universe, in the framework of a generalized Friedmann equation given by the function F ( H ) associated to the Hubble parameter H . In this reconstruction we have utilized as cosmological parameter, the scalar spectral index n s = n s ( N ), where N denotes the number of e -folds during the inflationary scenario. \nUnder the slow roll approximation, we have developed a new general formalism for reconstructing background variables. This involves a new methodology that rewrites the scalar spectral index as a function of the Hubble parameter and its derivatives. In this new analysis of this general formalism, we have determined from the attractor given by the scalar spectral index n s = n s ( N ), an integral relation for a first order differential equation associated to the Hubble parameter in terms of the number of e -folds N , see Eq.(23). Additionally, we have obtained a relation between the scalar field and the e -folds N , to reconstruct the background variables, such as, the Hubble parameter H = H ( ϕ ) and the effective potential V = V ( ϕ ), as functions of the scalar field ϕ . \nTo rebuild the background variables, we have considered the simplest parametrization (attractor) for the scalar spectral index n s = n s ( N ) as a function of the number of e -folds N given by n s = 1 -2 /N for largeN . By assuming this parametrization for the scalar spectral index, we have applied our methodology for three different modified Friedmann equations associated with the various functions F ( H ). Thus, in the reconstruction of the background variables, we have considered the functions; F ( H ) ∝ H β as a first example, the function F ( H ) = H 2 -γH -2 , as a second application and finally the function F ( H ) = H 2 + θH 4 , as example III. In this sense, using the new methodology for reconstructing the background variables with different functions F ( H ), we have found that these reconstructed background variables depend on the sign of the integration constants that arise from the solutions of the differential equations, which are solved in terms of the number of e -folds N , for each of the chosen functions F ( H ). In addition, for the different F ( H ) -models analyzed, we have constrained these integration constants from the observational parameters, such as the power scalar spectrum and the tensor to scalar ratio, in the particular case in which the number of e -folds corresponds to N = 60. In Figs. 1, 4 and 7, we show the evolution of the reconstructed effective potential in terms of the scalar field for the three different F ( H ) -models studied. In all cases analyzed, we have found that the scalar field begins to roll from the maximum value of the potential (approximately) towards values of the scalar field close to zero, where the number of e -folds at the end of inflationary epoch is N end ∼ 0, see e.g., the left panel of Fig.1. \nIn relation to analyze of the reheating era for our different F ( H ) -models, we have obtained that it is possible to quantify this epoch in terms of the reheating parameters such as; the temperature, the number of e -folds and the EoS parameter during the reheating of the universe. From the reconstruction of the background variables found during the inflationary epoch from the attractor n s = n s ( N ), we have been able to obtain these reheating parameters. In particular for the first function studied, we have found that the reheating temperature associated to the power β = 1 and β = 2 present a good compatibility with Planck's 1 σ bound on the scalar spectral index, for the different values of the EoS parameter ω reh , excluding the negative value ω reh = -1 / 3. For the second function F ( H ), we have found that the reheating temperature for the different values of the EoS parameter does not depend of the parameter γ ≪ 1, since the reconstruction of the Hubble parameter and effective potential are not affected strongly for this parameter. In relation to the third function F ( H ), we have obtained that reheating temperature increases when we increase the parameter θ . In addition, we have found that the reheating temperature for positive values of the EoS parameter can achieve lower values in comparison with values of ω reh < 0. \nRegarding the number of e -folds during the reheating for these F ( H ) -models, we have observed that this number presents a high value when the EoS parameter ω reh is positive (the highest value corresponds to ω reh = 2 / 3), see the respective figures associated to the reheating. In the different F ( H )-models, we have found that the stage of instantaneous reheating is given when the number of e -folds at the end of inflation N reh approaches zero. 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2024arXiv240902172K	Dwarf galaxies have historically posed challenges to the cold dark matter CDM model and while many of the socalled dwarf galaxy problems have been mitigated by incorporating baryonic processes the observed diversity of dwarf galaxy rotation curves remains a contentious topic. Meanwhile the growing observational samples of active galactic nuclei AGN in dwarf galaxies have prompted a paradigm shift in our understanding of dwarf galaxy evolution traditionally thought to be regulated by stellar feedback. In this study we explore the potential role of AGN feedback in shaping dark matter distributions and increasing the diversity of dwarf galaxy rotation curves using a new suite of cosmological zoomin simulations of dwarf galaxies with the FIRE3 model. Our findings indicate that the presence of active black holes BHs in dwarf galaxies can lead to diverse outcomes ranging from cuspier to more corelike profiles. This variability arises from the dual role of BHs in providing additional feedback and regulating the extent of stellar feedback. Consistent with previous research we find that AGN feedback is most impactful when cosmic ray CR modelling is included with CRs from any source significantly influencing dark matter profiles. Overall our results highlight that the interplay between stellar feedback BHs and CRs produces a broad spectrum of dark matter density profiles which align with observed correlations between rotation curve shapes and baryonic dominance. This underscores the importance of including the full range of baryonic processes in dwarf galaxy simulations to address the persistent smallscale challenges to the CDM paradigm.	2024-09-01T00:00:00Z	['2024arXiv240902172K', '10.48550/arXiv.2409.02172', 'arXiv:2409.02172']	['Astrophysics - Astrophysics of Galaxies', 'Astrophysics - High Energy Astrophysical Phenomena']	Diverse dark matter profiles in FIRE dwarfs black holes cosmic rays and the cuspcore enigma	2,024	192	0.61	['EPRINT_HTML', 'EPRINT_PDF']	1	https://arxiv.org/pdf/2409.02172.pdf	{'No Header': ', 1-22 (2024)', 'Diverse dark matter profiles in /f.pc/i.pc/r.pc/e.pc dwarfs: black holes, cosmic rays and the cusp-core enigma': "Sophie Koudmani, 1 , 2 , 3 ★ Douglas Rennehan, 3 Rachel S. Somerville, 3 Christopher C. Hayward, 3 Daniel Anglés-Alcázar, 4 , 3 Matthew E. Orr, 3 Isabel S. Sands 5 and Sarah Wellons 6 \n- 1 St Catharine's College, University of Cambridge, Trumpington Street, Cambridge CB2 1RL, UK\n- 2 Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\n- 3 Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA\n- 4 Department of Physics, University of Connecticut, 196 Auditorium Road, U-3046, Storrs, CT, 06269, USA\n- 5 TAPIR, California Institute of Technology, Mailcode 350-17, Pasadena, CA 91125, USA\n- 6 Department of Astronomy, Van Vleck Observatory, Wesleyan University, 96 Foss Hill Drive, Middletown, CT 06459, USA \nSubmitted to MNRAS", 'ABSTRACT': "Dwarf galaxies have historically posed challenges to the cold dark matter (CDM) model and, while many of the so-called 'dwarf galaxy problems' have been mitigated by incorporating baryonic processes, the observed diversity of dwarf galaxy rotation curves remains a contentious topic. Meanwhile, the growing observational samples of active galactic nuclei (AGN) in dwarf galaxies have prompted a paradigm shift in our understanding of dwarf galaxy evolution, traditionally thought to be regulated by stellar feedback. In this study, we explore the potential role of AGN feedback in shaping dark matter distributions and increasing the diversity of dwarf galaxy rotation curves, using a new suite of cosmological zoom-in simulations of dwarf galaxies with the /f.pc/i.pc/r.pc/e.pc-3 model. Our findings indicate that the presence of active black holes (BHs) in dwarf galaxies can lead to diverse outcomes, ranging from cuspier to more core-like profiles. This variability arises from the dual role of BHs in providing additional feedback and regulating the extent of stellar feedback. Consistent with previous research, we find that AGN feedback is most impactful when cosmic ray (CR) modelling is included, with CRs from any source significantly influencing dark matter profiles. Overall, our results highlight that the interplay between stellar feedback, BHs, and CRs produces a broad spectrum of dark matter density profiles, which align with observed correlations between rotation curve shapes and baryonic dominance. This underscores the importance of including the full range of baryonic processes in dwarf galaxy simulations to address the persistent 'small-scale challenges' to the CDM paradigm. \nKey words: methods: numerical - galaxies: active - galaxies: dwarf - galaxies: evolution - galaxies: formation - dark matter", '1 INTRODUCTION': "Dwarf galaxies are pivotal in the Lambda - Cold Dark Matter ( Λ CDM) model, serving as the building blocks of hierarchical structure formation. Defined as galaxies with stellar masses below 3 × 10 9 M ⊙ (similar to the mass of the Large Magellanic Cloud), dwarf galaxies are crucial astrophysical probes in near-field cosmology and galaxy formation studies. \nDue to their shallow potential wells, dwarf galaxies are ideal testbeds for studying galactic feedback processes ('baryonic physics'). They are also highly dark-matter dominated, making them valuable for testing dark matter models. However, their sensitivity to both dark matter and baryonic processes has led to several controversies, collectively known as the 'dwarf galaxy problems', which highlight discrepancies between observations and Λ CDM predictions from N-body simulations (see Sales et al. 2022, for a recent review). Key issues include the missing satellites problem, which \nnotes a deficit in the observed number of Milky Way satellites compared to theoretical predictions (e.g. Kauffmann et al. 1993; Klypin et al. 1999; Moore et al. 1999), and the too-big-to-fail problem, where the central masses of the largest observed satellites do not match those of the most massive simulated subhaloes (e.g. BoylanKolchin et al. 2011). Additionally, the cusp-core problem refers to the observation that dwarf galaxy dark matter halo profiles are less centrally peaked or 'cuspy' than predicted by the CDM model with some dwarf galaxies instead exhibiting core-like profiles (e.g. Flores &Primack 1994; Moore 1994). \nThese discrepancies have led to explorations of alternative dark matter models, such as warm dark matter (WDM) (Blumenthal et al. 1982). WDMmodels,whichinclude a free-streaming cut-off at dwarf galaxy scales, can address the missing satellite problem and reduce central halo densities, thus partially resolving the too-big-to-fail problem(Lovell et al. 2012). However, WDM struggles with the cuspcore problem, as the resulting cores are not large enough (e.g. Shao et al. 2013). Another approach involves self-interacting dark matter (SIDM), where particles scatter elastically, creating cored density \nprofiles and reducing halo ellipticity (see Tulin & Yu 2018, for a review). Constraints from strong lensing measurements of galaxy clusters suggest a velocity-dependent cross section that decreases from dwarf to cluster scales (e.g. Meneghetti et al. 2001; Randall et al. 2008; Vogelsberger et al. 2012). Additionally, fuzzy dark matter (FDM), which proposes extremely low-mass particles like ultralight axions, produces cored central profiles and suppresses small-scale structures in the dwarf regime due to their long de Broglie wavelengths (see Niemeyer 2020, for a review). \nHowever, there has also been a large body of theoretical work focusing on improving the baryonic physics modelling which may largely alleviate the 'dwarf galaxy problems' observed in pure CDM simulations. In particular, star formation suppression from reionization could significantly decrease the number of bright dwarf satellites (e.g. Efstathiou 1992; Okamoto et al. 2008; Fitts et al. 2017; Katz et al. 2020) thereby resolving the missing satellite problem. Furthermore, stellar feedback (e.g. Navarro et al. 1996; Governato et al. 2010; Parry et al. 2012; Brooks & Zolotov 2014; Hopkins et al. 2014, 2018; Chan et al. 2015; Kimm et al. 2015; Emerick et al. 2018; Smith et al. 2018, 2019; Gutcke et al. 2021; Jackson et al. 2024) may significantly decrease the central dark matter densities of dwarf galaxies. In particular, it has been demonstrated that cyclic supernova (SN) bursts play an important role in creating cores in dwarf galaxies (e.g. Pontzen & Governato 2012) and that late-time star formation is crucial for maintaining these cores (e.g. Read et al. 2019; Muni et al. 2024). \nNotwithstanding this progress in alternative dark matter models and improved baryonic physics, some prominent discrepancies between simulations and observations still appear difficult to resolve without fine-tuning the models. In particular, theoretical models significantly struggle with explaining the observed diversity of dwarf rotation curves (see e.g. Santos-Santos et al. 2020, for a recent compilation). Due to the self-similar nature of structure formation in the Λ CDMuniverse, the full mass profile of a halo can be characterised by a single parameter such as the virial mass (Navarro et al. 1997). Hence dwarf galaxy rotation curve shapes are predicted to be almost identical for similar maximum circular velocities - and yet a great diversity is observed ranging from extremely cuspy profiles to large cores for the same inferred dark matter halo mass (see Oman et al. 2015). Galaxy formation simulations based on CDM have traditionally struggled to reproduce this observed diversity of dwarf rotation curves (Dutton et al. 2016; Sawala et al. 2016; Garrison-Kimmel et al. 2019; Applebaum et al. 2021). Some groups find that SIDM may be able to improve these discrepancies (e.g. Ren et al. 2019; Kaplinghat et al. 2020), whilst others find no preference between SIDM and CDM simulations with stellar feedback (e.g. Santos-Santos et al. 2020; Zentner et al. 2022). \nThe most fundamental issue for CDM simulations lies in obtaining a balance in the burstiness of star formation histories, with some simulation set-ups predominantly producing smooth (or quenched) star formation histories and cusps (e.g. Vogelsberger et al. 2014; Sawala et al. 2016; Gutcke et al. 2022) whilst others predominantly produce bursty star formation histories and cores (e.g. Navarro et al. 1996; Read & Gilmore 2005; Governato et al. 2010; Pontzen & Governato 2012; Chan et al. 2015; Garrison-Kimmel et al. 2019). \nOne potential solution could be additional feedback processes in the dwarf galaxy regime interacting with SNe and thereby modulating the star formation histories, leading to a natural diversity in rotation curves (see discussion in Garrison-Kimmel et al. 2019). Recently, the role of active galactic nuclei (AGN) in resolving the remaining dwarf galaxy problems has garnered significant interest. Traditionally, AGN feedback was only considered in massive galax- \nies, with stellar feedback thought to dominate in low-mass galaxies. However, these models have been called into question by the growing observational samples of AGN in dwarfs, with detections spanning the whole electromagnetic spectrum from X-ray (e.g. Schramm et al. 2013; Baldassare et al. 2015, 2017; Lemons et al. 2015; Miller et al. 2015; Mezcua et al. 2016, 2018a; Pardo et al. 2016; Aird et al. 2018; Birchall et al. 2020, 2022; Latimer et al. 2021) to optical (e.g. Greene &Ho2004,2007;Reineset al. 2013; Chilingarian et al. 2018; Molina et al. 2021; Polimera et al. 2022) to IR (e.g. Satyapal et al. 2014; Sartori et al. 2015; Marleau et al. 2017; Kaviraj et al. 2019) and to radio observations (e.g. Greene et al. 2006; Wrobel & Ho 2006; Wrobel et al. 2008; Nyland et al. 2012, 2017; Reines & Deller 2012; Reines et al. 2014; Mezcua et al. 2018b, 2019; Reines et al. 2020; Davis et al. 2022). Intriguingly, recent targeted surveys indicate high AGN occupation fractions, on the order of 10 per cent, in dwarf galaxies, suggesting there may not be a drastic decline of AGN activity in the dwarf regime (e.g. Baldassare et al. 2015; Birchall et al. 2022; Bichang'a et al. 2024; Mezcua & Domínguez Sánchez 2024). \nThese tantalising observations have motivated theorists to investigate this largely unexplored regime. Analytical models point towards favourable AGN energetics in the dwarf regime (compared to stellar feedback, see e.g. Dashyan et al. 2018) and suggest that AGN activity may be able resolve all of the remaining 'dwarf galaxy problems' (Silk 2017). However, hydrodynamical simulations have painted a more complex picture. AGN activity in dwarf galaxies may be significantly suppressed by stellar feedback evacuating gas from the central region (e.g. Dubois et al. 2015; Anglés-Alcázar et al. 2017b; Habouzit et al. 2017; Trebitsch et al. 2018; Hopkins et al. 2022a; Byrne et al. 2023a) and black holes (BHs) may be wandering in dwarf galaxies due to their shallow potential wells (e.g. Bellovary et al. 2021; Ma et al. 2021; Sharma et al. 2022; Beckmann et al. 2023) further decreasing their accretion efficiency. The modelling of BH growth is another crucial aspect as the fiducial Bondi model suppresses the growth of low-mass BHs compared to gas-supplylimited or torque-driven schemes due to its strong dependence on BH mass (e.g. Anglés-Alcázar et al. 2013, 2015; Koudmani et al. 2022; Wellons et al. 2023; Gordon et al. 2024). The cosmic environment also has a crucial influence on AGN activity in the dwarf regime with minor mergers triggering AGN episodes whilst longterm residence in dense environments is detrimental to BH growth (e.g. Kristensen et al. 2021). \nDespite these caveats, various groups have identified physical regimes where AGN in dwarfs can accrete efficiently in accordance with observed samples. These dwarf AGN significantly influence their host galaxies by driving powerful outflows (e.g. Koudmani et al. 2019, 2022) and suppressing star formation (e.g. Barai & de Gouveia Dal Pino 2019; Sharma et al. 2020, 2022, 2023; Koudmani et al. 2021, 2022; Wellons et al. 2023; Arjona-Galvez et al. 2024). Given this potentially significant impact by AGN on the baryon cycle in dwarf galaxies, could AGN feedback also impact the dark matter distributions in dwarfs as predicted by analytical models? \nThe interaction between AGN feedback, star formation, and dark matter in dwarf galaxies has not been thoroughly explored, though sometrends have been noted, such as AGN mildly suppressing central dark matter densities (see Koudmani et al. 2022; Arjona-Galvez et al. 2024), hinting at a possible role of AGN in driving cusp-to-core transformations in dwarfs. However, the central density suppressions by the AGN in these simulations were mostly minor and did not change the qualitative nature of the central profiles. This is likely due to the effective equation of state (Springel & Hernquist 2003) employed for modelling the interstellar medium (ISM) in these simulations, which leads to smoother star formation histories alongside less bursty AGN \nfeedback at low redshifts, often operating as 'maintenance-mode' feedback (also see discussion in Koudmani et al. 2022). Recently, Arora et al. (2024) examined the role of BH feedback in core formation within the NIHAO simulations and found negligible effects in low-mass systems, largely due to inefficient BH growth in this mass regime, constrained by the Bondi accretion scheme. In contrast, they found significant impacts of BH feedback on core formation in more massive galaxies, aligning with earlier studies that show efficient AGN activity can create cores in these galaxies (e.g. Martizzi et al. 2013; Peirani et al. 2017; Macciò et al. 2020). \nIt is therefore timely to investigate whether efficient AGN activity can induce cusp-to-core transformations in dwarf galaxies and potentially account for the observed diversity in dwarf rotation curves. Clearly, within this context it is crucial to model the multi-phase nature of the ISM but also to consider the role of non-thermal components such as magnetic fields or cosmic rays (CRs) (e.g. Uhlig et al. 2012; Booth et al. 2013; Hanasz et al. 2013; Salem & Bryan 2014; Pakmor et al. 2016; Farber et al. 2018; Holguin et al. 2019; Dashyan & Dubois 2020; Martin-Alvarez et al. 2020; Steinwandel et al. 2020). In particular, it has been shown that CRs may drive core formation (e.g. Hopkins et al. 2020b; Martin-Alvarez et al. 2023) though their impact in the dwarf regime remains controversial as gas-rich mergers may re-establish cores in these systems (see discussion in Martin-Alvarez et al. 2023). What is more, it has been found that CRs may significantly boost the efficiency of AGN feedback (e.g. Su et al. 2021, 2024; Byrne et al. 2023b; Wellons et al. 2023) by suppressing cold gas accretion from the CGM due to enhanced CR pressure support. Therefore, it is important to also assess the role of CRs when investigating cusp-core transformations with AGN. \nThe aim of this paper is to systematically investigate the impact of AGNfeedback and CRs on dark matter profiles in dwarf galaxies and to assess whether incorporating these additional baryonic processes may be the key to resolving the diversity of rotation curve problem. To this end, we perform simulations with and without CRs and/or AGN of three different dwarf haloes with explicit multi-phase ISM physics within the /f.pc/i.pc/r.pc/e.pc-3 galaxy formation framework and contrast the resulting dark matter profiles, their cosmic evolution and their connection with different feedback processes. \nThe remainder of this paper is structured as follows. In Section 2, we introduce our three dwarf simulation set-ups and describe the different variations of the /f.pc/i.pc/r.pc/e.pc-3 galaxy formation model explored in this work. The results of our analysis are presented in Section 3. Firstly, we focus on the stellar and BH properties of our simulated dwarfs, including visualisations of the stellar distributions in Section 3.1, comparisons with observed nearby low-mass galaxies (see Section 3.2) and observed BH masses in dwarfs (see Section 3.3). We then analyse how our different feedback set-ups influence the dark matter profiles at 𝑧 = 0 in Section 3.4 and examine the driving forces behind the diversity of dark matter profiles in 3.5. We investigate the cosmic evolution of the central dark matter densities in Section 3.6 and assess whether this could explain the observed diversity of rotation curves in Section 3.7. We discuss our results and caveats to our modelling in Section 4 and conclude in Section 5.", '2.1 General set-up': "Our simulation suite is based on a set of 3 initial conditions (ICs) of dwarf haloes selected from the /f.pc/i.pc/r.pc/e.pc ICs set, spanning a range of halo masses in the 'classical dwarf' regime. We select two low-mass \ndwarfs: m10q and m10y , with 𝑧 = 0 halo masses of 𝑀 vir = 8 × 10 9 M ⊙ and 𝑀 vir = 1 . 4 × 10 10 M ⊙ , respectively. Our third halo, m10z , is an intermediate-mass dwarf with 𝑧 = 0 halo mass 𝑀 vir = 3 . 5 × 10 10 M ⊙ . All of these ICs were generated using the /m.pc/u.pc/s.pc/i.pc/c.pc IC generator (Hahn & Abel 2011) and have been thoroughly explored with the /f.pc/i.pc/r.pc/e.pc-2 model (Hopkins et al. 2018). In the standard /f.pc/i.pc/r.pc/e.pc-2 runs, m10q has a quiescent growth history and is relatively isolated, m10y forms early and has a large core, and m10z is an ultra-diffuse galaxy. \nTo simulate these systems, we use the /g.pc/i.pc/z.pc/m.pc/o.pc hydrodynamics+gravity code, employing the mesh-free finite mass method (Lanson & Vila 2008a,b; Gaburov & Nitadori 2011; Hopkins 2015, 2017) that tracks constant-mass gas resolution elements in a Lagrangian fashion. Our simulations include both baryons and dark matter, with mass resolutions of 𝑀 gas = 262 M ⊙ and 𝑀 DM = 1303 M ⊙ , respectively. The gravitational softening for the dark matter, stars and BH particles is set to 𝜖 soft = 0 . 0625 ckpc ℎ -1 , whereas the softening for the gas resolution elements is set adaptively. This allows us to easily resolve core formation as demonstrated in previous work (e.g. Chan et al. 2015; Hopkins et al. 2018).", '2.2 Galaxy formation model': 'For the galaxy formation physics, we use the newly updated /f.pc/i.pc/r.pc/e.pc-3 model(Hopkins et al. 2023a) which tracks the state of the multi-phase ISM and multiple forms of stellar feedback, including feedback from SNee 1 of Type Ia and II, stellar winds from OB and AGB stars as well as multi-wavelength photo-heating and radiation pressure. Star formation occurs in gas that is molecular, dense, self-gravitating, self-shielding, and Jeans-unstable. All simulations include magnetic fields, using the magneto-hydrodynamics methods from Hopkins & Raives (2016) and Hopkins (2016). The /f.pc/i.pc/r.pc/e.pc-3 model also has new optional physics modules compared to the previous /f.pc/i.pc/r.pc/e.pc (Hopkins et al. 2014) and /f.pc/i.pc/r.pc/e.pc-2 (Hopkins et al. 2018) models. Two of the most important optional features are a novel sub-grid BH model, including seeding, accretion and feedback, as well as the inclusion of CR physics. To separate the impacts of CRs and BHs, we perform four simulations for each set of ICs, including runs without CRs or BHs, with both CRs and BHs and with only CRs or BHs, respectively. For the runs without BHs but with CRs, no BHs are seeded and the CRs only stem from SNe. For the runs without CRs and with BHs, we incorporate all AGN feedback channels except for CR injection and SNe do not inject CRs either. See Table 1 for an overview of the simulation suite. \nThe modelling of BHs and CRs remains highly uncertain. Several studies have investigated the impact of different modelling assumptions for CRs (e.g. Hopkins et al. 2020b, 2023b; Butsky et al. 2023) and BHs (Anglés-Alcázar et al. 2017b, 2021; Su et al. 2021, 2024; Byrne et al. 2023a,b; Wellons et al. 2023) within the /f.pc/i.pc/r.pc/e.pc model. For our study, we base our BH model on the extensive study by Wellons et al. (2023) who identified a modelling space that allows for efficient BH feedback whilst still matching observational constraints, in particular with regards to star formation distributions. We summarise these BH modelling choices in Section 2.2.1 and point the interested reader to Wellons et al. (2023) for more details and an in-depth parameter study of alternatives to our choices. For the CR modelling, we employ the numerically-efficient subgrid CR model introduced \n1 We couple the SN momentum following Hopkins et al. (2023a) and do not yet include the updates to the momentum coupling from Hopkins (2024). Though we note that these changes mainly affect high-mass galaxies at low resolution, whilst the impact on high-resolution dwarfs is very limited. \nTable 1. Overview of dwarf zoom-in simulation suite. We list the simulation names (first column), ICs (second column), and whether BHs (third column) and CRs (fourth column) are included. We also provide the central dark matter densities at 𝑟 = 150 pc (fifth column), the ratios between the central dark matter density for the hydro runs and dark matter only simulations (sixth column), and the shape of the central dark matter density profile (seventh column) as shown in Figure 8. All density estimators are rounded to two significant figures. \nin Hopkins et al. (2023b) allowing us to explore several different IC set-ups to 𝑧 = 0 whilst minimising the computational overhead costs that would arise from explicit CR transport. We summarise the key characteristics of the CR subgrid model in Section 2.2.2 and refer the interest reader to Hopkins et al. (2023b) for a more detailed overview of this approach.', '2.2.1 BH modelling': "For our BH modelling, we largely adopt the default /f.pc/i.pc/r.pc/e.pc-3 BH implementation (see Hopkins et al. 2023a), except for the AGN feedback injection where we use the continuous wind mode (rather than particle spawning) and for the CR injection where we employ the subgrid model from Hopkins et al. (2023b). Below we summarise the salient details as well as our modifications to the /f.pc/i.pc/r.pc/e.pc-3 BH model and refer the interested reader to Hopkins et al. (2023a) and Wellons et al. (2023) for a more in-depth description and parameter studies. \nBHs may be seeded into the simulation from any star-forming gas particles that satisfy physically-motivated properties for the formation of low-mass seeds. In particular, the gas particles must have extremely high densities ( Σ gas ≳ 5000 M ⊙ pc 2 ) and low metallicities ( 𝑍 gas ≲ 0 . 001 Z ⊙ ) in order to spawn BH particles of mass 𝑀 BH = 100 M ⊙ . With a DM particle mass of 𝑀 DM ∼ 10 3 M ⊙ , we therefore do not fully resolve the dynamical friction forces that would drive the orbits of our seed BHs towards the centre of the galaxy. Hence we follow Wellons et al. (2023) and employ a subgrid prescription where the BHs are 'drifted' towards the local binding energy extremum of the stars, dark matter and BH particles so that they are not artificially ejected (also see Hopkins et al. 2023a). Similarly, since we cannot resolve the dynamics of binary BHs, two BHs are merged if their interaction kernels and force softenings overlap and they are gravitationally bound to one another. \nOnce a BH particle is present, it grows via accretion by draining gas from the nearest 𝑁 ∼ 256 gas cell neighbours. Wellons et al. (2023) parameterized the accretion rate as / 𝑀 acc ≡ 𝜂 acc 𝑀 gas Ω , where 𝜂 acc is the accretion efficiency, 𝑀 gas is the gas mass within the BH kernel, and Ω = √︁ 𝐺𝑀 tot / 𝑅 is the orbital frequency determined at the kernel size 𝑅 , based on the total mass enclosed with 𝑅 , 𝑀 tot . /f.pc/i.pc/r.pc/e.pc-3 modulates the BH accretion rate via a gas reservoir that represents an accretion disc from which the BH particle grows at a mass growth rate / 𝑀 BH = 𝑀 𝛼 disc / 𝑡 dep , where 𝑀 𝛼 disc is the mass of the accretion disc and 𝑡 dep = 42 Myr ( 1 + 𝑀 BH / 𝑀 𝛼 disc ) 0 . 4 is the depletion time. \nWhile Wellons et al. (2023) tested various distinct accretion models (i.e. variations in 𝜂 acc ), their best-fitting models favoured accretion that was powered by gravitational torques. Physically, these torques result from asymmetries in a galaxy's gravitational potential, or from interactions between (a) the dark matter and gas, (b) the stellar component and gas, or (c) the self-interaction of the gas with itself via shocks and dissipation (Hopkins & Quataert 2010, 2011). Theaccretion efficiency in the case of gravitational torque-dominated accretion comes from the work in Hopkins & Quataert (2011), \n𝜂 acc = 𝐶 ([ 𝑀 BH + 𝑀 d ]/ 𝑀 d ) 1 / 6 1 + 3 𝑀 1 / 3 d , 9 ( 𝑀 gas / 𝑀 d ) , (1) \nwhere 𝐶 is a resolution-dependent constant, 𝑀 BH is the BH mass, 𝑀 d is the mass of angular-momentum support material, and 𝑀 d , 9 = 𝑀 d / 10 9 M ⊙ . The model has been studied extensively in the context of cosmological zoom-in simulations in Anglés-Alcázar et al. (2013, 2017b,a, 2021) and Hopkins et al. (2016). It is important to note that we choose a value of 𝐶 = 0 . 1 in order to have reasonable BH masses within our three simulated dwarf galaxies. However, the parameter 𝐶 cannot take into account all sub-grid processes that may occur below the resolution scale. \nOne important piece of (partially) unresolved physics is the impact of stellar feedback on the accretion flow in the vicinity of the BH. Wellons et al. (2023) introduce a scaling factor 𝑓 acc (such that 𝜂 acc → 𝑓 acc 𝜂 acc ) that depends on the local gravitational acceleration, 𝑎 g , eff = 𝐺𝑀 ( < 𝑅 )/ 𝑅 , as \n𝑓 acc = 𝑎 g , eff 𝑎 g , eff + 𝑎 g , crit , (2) \nwhere 𝑎 g , crit ≈ 10 -7 cms -2 . Below 𝑎 g , crit (an effective surface density of Σ tot ∼ 3000 M ⊙ pc 2 ), a significant fraction of gas is expelled due to stellar feedback (Grudić et al. 2019; Hopkins et al. 2022a). However, we emphasise that in addition to this stellar feedback effect on unresolved scales, our simulations explicitly capture the impact of stellar feedback on larger resolved scales (comparable to the size of the BH kernel). \nThe accretion process itself drives powerful outflows, jets and radiation that influence the properties of the galaxies hosting the BHs, collectively coined BH feedback. The /f.pc/i.pc/r.pc/e.pc-3 BH feedback model includes three channels: radiative feedback, mechanical feedback, and CR feedback. Each mode injects energy at a rate proportional to the BH accretion rate / 𝑀 BH . \nAsgasfalls into a BH, it loses gravitational energy that is converted into radiation, with an efficiency that is dependent on the location of the innermost stable circular orbit of the BH. While the location of that orbit depends on the spin of the BH, the typical efficiency used in the literature is 𝜖 r = 0 . 1, implying that 10% of the rest massenergy accretion rate is converted into radiation. Given that we do not track the BH spin evolution in our simulations, we also adopt this fiducial radiative efficiency value in our work. In /f.pc/i.pc/r.pc/e.pc-3, that radiation is treated in the same manner as the stellar feedback with multiband radiation transport and metallicity-dependent opacities using the LEBRON method (see Hopkins et al. 2020a), except using a quasar template spectrum (Shen et al. 2020). The radiation momentum flux depends on the luminosity absorbed ( 𝐿 abs ) in the gas as / 𝑝 = 𝐿 abs / 𝑐 . \nThere are other outflows generated from the accretion process besides radiation-driven winds. For example, there may be jets or hydromagnetic winds from the accretion disc itself. The physics that drives these processes occurs on scales that are much below the resolution of our simulations and, therefore, need to be parameterized into a sub-grid model. The /f.pc/i.pc/r.pc/e.pc-3 BH model assumes that these winds have mass outflow rates proportional to the accretion rate. We assume a mass loading of one, such that / 𝑀 out = / 𝑀 BH . The kinetic energy rate in the wind itself depends on the wind velocity at launch, 𝑣 wind , such that / 𝐸 mech = / 𝑀 BH 𝑣 2 wind / 2. Wellons et al. (2023) recast the mechanical luminosity in terms of the new mechanical efficiency, 𝜂 mech ≡ 𝑣 2 wind /( 2 𝑐 2 ) , such that / 𝐸 mech = 𝜂 mech / 𝑀 BH 𝑐 2 . In this work, we choose 𝑣 wind = 10 , 000 km s -1 (see e.g. Fiore et al. 2017, compiled in fig. 3 of Rennehan et al. 2024) and, hence, have 𝜂 mech = 5 . 6 × 10 -4 . \nThe /f.pc/i.pc/r.pc/e.pc-3 model also includes a CR model and, therefore, it also includes a BH feedback channel that accounts for relativistic massive particles accelerated by the AGN jets. We describe the CR propagation model in Section 2.2.2, but note here that the CRs are injected at a rate / 𝐸 CR = 𝜂 CR / 𝑀 BH 𝑐 2 , where 𝜂 CR is the efficiency of CRfeedback. In this work, we use an efficiency of 𝜂 CR = 0 . 01 which was identified as yielding realistic galaxy properties in the Wellons et al. (2023) parameter study. \nFor the mechanical and radiative feedback, we use the solid angle weighting to inject the AGN feedback (akin to SN feedback injection, see Hopkins et al. 2018, for details). Following Su et al. (2021), we inject the majority of the AGN energy within a relatively narrow opening angle by weighting the AGN wind energy and radiation pressure received by each gas cell as \n𝑤 ( 𝜃 ) = 𝜖 jet ( 𝜖 jet + cos 2 𝜃 ) ( 𝜖 jet + 1 )( 𝜖 jet + ( 1 -cos 2 𝜃 )) . (3) \nWe set 𝜖 jet = 0 . 35 which leads to an effective base opening angle of ∼ 7 ° . The angle weighting is applied with respect to the total gas angular momentum accreted by the BH, 𝐽 gas , which is measured in the centre-of-mass frame of the BH - accretion disc sink particle. This allows for AGN energy injection without significantly disrupting the small-scale gas supply in the central region which favours repeated bursts and/or continuous AGN activity allowing us to probe the 'physically interesting' regime for AGN influencing cusp to core transformations. As discussed in Su et al. (2021), the large-scale collimation properties of the jet are not hugely sensitive to the small-scale opening angle since the feedback energy chooses the path of least resistance effectively recollimating even for large opening angles, also see Koudmani et al. (2019). Long-range radiation transport and CRs are treated via the LEBRON approximation and injected isotropically (see Hopkins et al. 2020a, 2023b). \n2.2.2 CR modelling \nThe /f.pc/i.pc/r.pc/e.pc-3 model includes a comprehensive CR model that explicitly follows the multi-species/multi-spectral CR dynamics (Hopkins et al. 2021a,b,c, 2023a) with the entire distribution function (Hopkins et al. 2022b) integrated with the magnetohydrodynamics (MHD) solver. However, there are also optional, simplified models that make a series of approximations valid in our regime of interest - dwarf galaxies. In particular, we use the sub-grid CR model described in (Hopkins et al. 2023b) and briefly describe our motivation for this model below. \nTheprimary assumption is that the CR energy spectrum is spatially constant, and that CRs above ≳ 1 GeV dominate the CR pressure (e.g. Chanet al. 2019). This 'single-bin' model only evolves the integrated CRenergy density, drastically reducing the computational time while providing a reasonable approximation of the CR pressure, which is the dominant variable that impacts galaxy evolution. The study in Chan et al. (2019) showed that higher values of the isotropic diffusion coefficient 𝜅 are necessary to explain 𝛾 -ray luminosities in dwarf and 𝐿 ∗ -galaxies. Importantly, the values of the diffusion coefficient that best match the observations leads to most CRs escaping from galaxies with low gas density (i.e. dwarfs), while in starbursting galaxies with dense gas, the CR energy is lost almost entirely to collisions. Once the CRs escape the dense gas, the adiabatic gain/loss terms from the 𝑃 d 𝑉 work on the low density gas also becomes negligible. \nGiven that several of the CR physical processes are seemingly subdominant in the dwarf regime, Hopkins et al. (2023b) made a series of further simplifying assumptions that are applicable in a 'streaming+diffusion' limit. In particular, the reduced sub-grid model assumes that the CR energy equation is in steady state ( 𝜕 𝑡 𝑒 cr → 0) and that the magnetic fields below the resolution scale are isotropically 'tangled'. The latter assumption allows the anisotropic diffusion tensor 𝜿 \| \| to be replaced with an isotropic diffusion coefficient 𝜅 , averaged over the resolution scale (and spatio-temporally constant). The dominant loss term for CR energy is assumed to be a combination of hadronic/pionic, Coulomb, and ionization loses - therefore, neglecting losses from diffusive reacceleration (Hopkins et al. 2022b). Streaming contributions from advective and Alfvén velocities are also ignored, leading to a two parameter model depending on the isotropic diffusion rate 𝜅 and a streaming velocity 𝑣 𝜅 . An approximate solution for the spatial distribution of CR energy density for such a model is presented in Hopkins et al. (2023b), and depends on an integral of an exponentially-declining function of the loss function. \nThe resultant CR model is solved using a LEBRON-type approximation (see Hopkins et al. 2018, for details). The 'optical depth' of the CRs is equivalent the loss function we describe above, and the CR energy at a given distance from a source (i.e. SNe or AGN) is solved using the gravity tree in GIZMO . In /f.pc/i.pc/r.pc/e.pc-3, 10 per cent of the SN energy is converted to CRs and, as we mention above, 10 per cent of the BH luminosity is in the form of CRs. In this work, we use the values from Hopkins et al. (2023b); 𝜅 = 5 × 10 28 cm 2 s -1 and 𝑣 𝜅 = 20 km s -1 .", '3.1 Overview of the simulation suite': "We begin our analysis of the simulation suite by inspecting visualisations of our simulated dwarfs' stellar distributions. Figure 1 shows mock u/g/r Hubble images of the stars in each dwarf set-up, made by ray tracing through the gas: for each pixel, we sum up the luminosities \nFigure 1. Mock u/g/r Hubble images showing the stellar distributions of our simulated dwarf galaxies. The side length of each projection is 10 kpc. The rows depict our three different IC set-ups, whilst the columns represent the different galaxy formation physics variations explored. The stellar sizes and colours are clearly impacted by the addition of CRs and/or AGN feedback hinting at a potential influence by these processes on dark matter distributions. \n<!-- image --> \nof the star particles in each band along the line of sight. Dust associated with the gas elements attenuates the light, assuming a Milky Way-like attenuation curve in each band and a fixed dust-to-metals ratio. The side length of each projection is 10 kpc. \nThe three rows correspond to our three different ICs whilst the different columns represent our four physics variations based on the /f.pc/i.pc/r.pc/e.pc-3 physics modules: the fiducial /f.pc/i.pc/r.pc/e.pc-3 set-up (without CRs or AGN), /f.pc/i.pc/r.pc/e.pc-3 with AGN, /f.pc/i.pc/r.pc/e.pc-3 with CRs and /f.pc/i.pc/r.pc/e.pc-3 with both CRs and AGN. We note that there are several uncertainties associated with modelling AGNandCRsingalaxyformationsimulationsandherewe only explore a small subset of this parameter space that was identified by Wellons et al. (2023) as producing realistic galaxy properties. Furthermore, our simulations likely represent an upper limit for the impact of CRs since we are using an effective subgrid model for the CR population (Hopkins et al. 2023b) which only accounts for local losses and assumes a constant diffusion and effective streaming speed. See Section 2.2.2 and Hopkins et al. (2023b) for more details. \nKeeping these caveats in mind, we note that both CRs and AGN have a significant impact on the stellar distributions. Generally, the addition of CRs leads to more compact stellar distributions, whereas the simulations without CRs also have an extended diffuse component. This is partly due to CRs suppressing star formation as the additional pressure support prevents gas accretion from the CGM onto the galaxy (also see Hopkins et al. 2020b, 2023b; Farcy et al. 2022; DeFelippis et al. 2024; Thomas et al. 2024). For the runs with AGN feedback, the picture becomes even more complex. The runs without CRs but with AGN appear qualitatively similar to their non- \nAGN counterparts if not slightly more diffuse. The runs with CR, however, display systematic differences with added AGN feedback. This is consistent with the analysis from Wellons et al. (2023) where they find that CR-driven feedback is among the most efficient AGN feedback channels. In all cases, the runs with AGN are more compact and redder, pointing towards older stellar populations. Overall, these visualisations demonstrates that both AGN and CRs have a significant non-linear impact on dwarf galaxy evolution.", '3.2 Stellar assembly': 'Before assessing the impact of our baryonic modelling choices on dark matter distributions, it is vital to confirm that our feedback implementations produce realistic dwarf galaxy properties. To this end, we assess the stellar properties of our simulated dwarfs and compare these with observational constraints in Fig. 2, which shows the stellar mass - halo mass (SMHM) relation, galaxy colours and stellar half mass radii. Note that in this and later plots the symbol style indicates the model variations explored in our simulation suite, as indicated by the legend.', '3.2.1 SMHM relation': "Firstly, we examine if the simulated dwarf galaxies align with observational expectations for the SMHM relation. The left panel of Fig. 2 shows our 12 dwarf zoom-in simulations in 𝑀 halo -𝑀 stellar space at redshift 𝑧 = 0. \n<!-- image --> \n<!-- image --> \nFigure 2. Integrated stellar properties of our simulations in comparison with observational constraints. The symbol style in this and later plots indicates the model variations. Left panel: Stellar mass - halo mass (SMHM) relation. SMHM relations based on empirical models as well as individual observed galaxies are plotted for comparison as indicated by the legend. Middle panel: (Intrinsic) simulated galaxy colours against stellar mass. The colours from the SDSS galaxies are indicated as grey contours and the integrated photometry of local dwarfs is plotted as grey squares (Mateo 1998) and grey diamonds (Carlsten et al. 2022), for comparison. Right panel: Stellar half-mass radii against stellar mass with the data from local dwarfs (Carlsten et al. 2022) shown as grey diamonds for comparison. Overall , though the addition of AGN and/or CRs introduces significant scatter in the simulated distributions, our simulated stellar properties are in good agreement with the observations for all simulation set-ups explored. \n<!-- image --> \nFor comparison, we plot several SMHM relations from the literature, including Behroozi et al. (2013, 2019) and Moster et al. (2018). The low-mass end of the SMHM relation remains largely unconstrained by observations, evident from the significant differences between these models. The Behroozi et al. (2013, 2019) models extend down to log ( 𝑀 halo / M ⊙ ) = 10 . 0, and the Moster et al. (2018) relation was shown to also be valid down to the dwarf regime, at least to log ( 𝑀 halo / M ⊙ ) = 10 . 0, by O'Leary et al. (2023). We plot the extrapolated relations by Behroozi et al. (2013, 2019) and Moster et al. (2018) as dashed and dotted lines, respectively. Extrapolating SMHM relations below the minimum mass considered has pitfalls, especially regarding physical processes like reionization suppression, which are effective at low masses but not included in most of the models. The scatter is especially difficult to model in this regime since more bursty star formation histories at lower stellar masses are expected to increase the scatter whilst high-redshift quenching due to reionization suppression may act to substantially reduce the scatter in the dwarf regime (O'Leary et al. 2023). \nOur simulations with the m10q ICs, the least massive halo, are firmly in the extrapolated regime, whilst m10y and m10z are at the lower end of validity for the Behroozi et al. (2013, 2019) and Moster et al. (2018) relations. Hence for our simulations comparing to SMHM relations constructed for low-mass galaxy samples, rather than the fiducial SMHM relations for the general galaxy population, is more instructive. We include SMHM relations derived for dwarf galaxies by Read et al. (2017), Jethwa et al. (2018), and Nadler et al. (2020) as shaded regions. The differences in these relations arise due to different assumptions for surface-brightness incompleteness correction, see discussion in Behroozi et al. (2019). We also show various observed data points from the SPARC data base (Li et al. 2020) to indicate the scatter in the observations. \nWe note that the empirical relations as well as the SPARC data base use different definitions for the halo mass, either employing the virial mass as defined by the mass enclosed in a sphere whose mean density is 200 times the critical density of the Universe at 𝑧 = 0 or the halo mass based on the collapse of a spherical top-hat perturbation following Bryan & Norman (1998). We have checked for our \nsimulated dwarfs that the difference between these two definitions is at most 0.1 dex, and plot the simulated halo masses as the average value between these two definitions. The simulated stellar mass is calculated within twice the stellar half mass radius. Given the significant uncertainty and scatter in SMHM relations, we do not adjust the stellar masses for different radial cut-offs or initial mass functions. \nFrom this comparison with the empirical relations and SPARC observations, none of our models can be ruled out based on their integrated stellar masses. All our simulations fall within the range of the empirical SMHM relations and are well within the scatter of the observed data. The simulation set-ups with CRs (indicated by squares and diamonds) are generally less efficient at forming stars than the simulations without CRs (indicated by stars and crosses). Most simulations fall towards the upper end of the predicted SMHM relations, with the m10zCR+AGN simulation the only set-up resulting in an 'undermassive' galaxy. Though we note that even this set-up is still within the scatter of the most-up-to-date dwarf galaxy SMHM relation by Nadler et al. (2020).", '3.2.2 Galaxy colours': 'Next, we compare the galaxy colours of our simulated dwarfs with observational data. The middle panel of Fig. 2 presents the 𝑔 -𝑟 colour versus stellar mass for our simulated dwarfs. For comparison, we display the colours of SDSS galaxies from the NASA Sloan Atlas (NSA) as a grey contours, the colours of Milky-Way dwarf satellites from Mateo (1998) as grey squares and the ELVES dwarf satellites, a nearly volume-limited sample of Milky Way-like hosts in the Local Volume, from Carlsten et al. (2022) as grey diamonds. \nWe use the Bruzual & Charlot (2003) stellar population synthesis model to calculate galaxy colours as a function of stellar age and metallicity, assuming a Kroupa (2001) IMF, including all star particles within twice the stellar half-mass radius. For the MW dwarf satellites, we convert the 𝐵 -𝑉 colours using the transformation equations from Jester et al. (2005), noting that while these equations are for stars, they should approximate galaxies well unless strong emission lines are present. One important caveat is that our simu- \ngalaxy colours do not account for dust attenuation. However, we checked that, as expected for dwarf galaxies, our simulations have relatively low metallicities and therefore dust attenuation is expected to be minimal. \nAll simulated dwarfs are overall in good agreement with both the local dwarf galaxy constraints and the colour distribution of SDSS galaxies. Although we note that SDSS is very incomplete in the mass regime we consider here and the ELVES dwarfs were selected to be satellites while our dwarfs are field objects. The SDSS galaxies cluster around 𝑔 -𝑟 ∼ 0 . 35, whilst most of our dwarfs are offset towards somewhat redder colours, clustering around 𝑔 -𝑟 ∼ 0 . 45. Accounting for dust attenuation would likely make this offset more severe. However, this discrepancy is also inherently linked with completeness bounds in SDSS in this mass range, with blue galaxies likely overrepresented. \nInterestingly, the m10qCR set-up has a similar stellar mass to its AGNcounterpart m10qCR+AGN , but is shifted towards bluer colours due to a late-time burst in star formation which is also visible in the stellar projections in Figure 1. Conversely, the m10q set-up has a lower stellar mass and is redder than its AGN counterpart, indicating a complex interplay between CRs and AGN activity. For m10y , neither CRs nor the AGN have a strong impact on the integrated galaxy colours, though there are small, yet systematic, offsets with the CR runs being redder than the no CR runs, and the AGN runs being redder than their counterparts without AGN. Finally, for m10z , the set-up with neither CRs nor AGN yields the bluest galaxy colour, m10z , whilst the three other set-ups all have relatively similar colours, despite spanning an order of magnitude in stellar mass, indicating that all three of these formed the majority of their stars early in cosmic history.', '3.2.3 Stellar sizes': 'We now turn towards examining the stellar sizes which provide crucial clues to both the star formation and dynamical histories of our dwarfs. We plot the stellar half mass radii of our simulated dwarfs at 𝑧 = 0 as colour-coded symbols. For comparison, we also plot the observed effective radii of local dwarf galaxies as grey diamonds based on the ELVES catalogue from Carlsten et al. (2022), which also includes the Milky Way dwarf measurements from McConnachie (2012) as a subsample. To make this comparison more consistent, we calculate the simulated stellar half mass radii based on the averages of 2D projections along random sightlines. Again there is appreciable scatter in the observations and our simulated dwarfs fall well within this scatter. \nAs we found from our visual inspection of the stellar distributions in Fig. 1, the stellar distributions are generally more compact with CRs. The impact of AGN feedback on the stellar sizes is more complex, leading to both more extended distributions (e.g., compare m10yCR with m10yCR+AGN ) or more compact distributions (e.g., compare m10qCR with m10qCR+AGN ). The differences between the AGN and no-AGN set-ups are more pronounced if CRs are present, in agreement with the findings from Wellons et al. (2023) that CRs enhance the efficacy of the AGN feedback in /f.pc/i.pc/r.pc/e.pc. \nOverall our stellar sizes are in good agreement with the observations, though the m10qCR+AGN simulation is slightly more compact than local dwarfs at similar stellar masses. However, we note that the observed radii are based on stellar light whilst we are calculating the simulated radii based on the stellar mass distribution. Observational effects can add significant scatter leading to both more compact half-light radii (e.g. Klein et al. 2024) or more extended observed distributions (e.g. Parsotan et al. 2021), with the stellar sizes also \nFigure 3. BHmass - stellar mass scaling relations compared to observational constraints. We show the redshift evolution of the dwarf zoom-in simulations from 𝑧 = 4 to 𝑧 = 0 and plot data from observed dwarfs and globular clusters as indicated by the figure legends. We also show the observational BH mass - stellar scaling relations from the literature by Reines & Volonteri (2015) and Greene et al. (2020), greyed out in the extrapolated regime, as well as the high-redshift relation from Pacucci et al. (2023) and NSC mass - stellar mass relation from Neumayer et al. (2020), which can be taken as an upper limit. Our simulated BHs are generally overmassive compared to the (heavily) extrapolated local relations though they are well within the limits set by the NSC relation and their offsets are much smaller than those uncovered by JWST at high redshifts. \n<!-- image --> \nbeing significantly dependent on the wavelength (e.g. Cochrane et al. 2023).', '3.3 BH assembly': "Having analysed the stellar properties, we turn to analyse the evolution of the BHs in our simulations in the context of observational constraints. \nFig. 3 shows our simulated dwarfs in stellar mass - BH mass space. The evolutionary tracks of the dwarf simulations from 𝑧 = 4 to 𝑧 = 0 are plotted as dotted lines, colour-coded by simulation setup. Note that the BHs are generally seeded around 𝑧 ∼ 8-10 with the exact seeding time depending on the metallicity and density conditions. However, we here focus on later redshifts as by 𝑧 = 4, the most massive progenitor dwarf hosts a BH for all model variations so that we can track these alongside the stellar mass growth of the main progenitor. Integer redshifts are marked with the marker symbols for the respective simulation set-ups as indicated in the legend. \nWe note that there is substantial uncertainty at the low-mass end of the BH - galaxy scaling relations due to the difficulty in reliably measuring the masses of intermediate-mass BHs (IMBHs) which have a much smaller gravitational sphere of influence and generally lower luminosities. Indeed there are no BH mass measurements for dwarf galaxies in the stellar mass range covered by our simulations, 10 6 M ⊙ ≲ 𝑀 stellar ≲ 10 8 M ⊙ . For more massive dwarfs there are only very limited measurements and we show these observed BH masses in massive dwarfs for reference, including virial BH mass \nestimates for active dwarf galaxies from Baldassare et al. (2020) and Wasleske & Baldassare (2024) as dark-grey squares and stars, respectively, and measurements from Greene et al. (2020) (based on dynamics, reverberation mapping or scaling from the stellar velocity dispersion in the specific case of TDE events that seem to have reliable determinations of the host galaxies) as dark-grey hexagons and triangles for mean values and upper limits, respectively. Error bars are omitted for clarity. We note that there are no dynamical nuclear IMBH mass measurements below 𝑀 stellar ∼ 10 9 M ⊙ and, excluding upper limits, the lowest BH mass estimate based on TDEs stems from a host galaxy of stellar mass 𝑀 stellar = 2 . 5 × 10 8 M ⊙ . Belowthis mass, we are firmly in the extrapolated regime and indicate this by greying out the local BH - galaxy scaling relations below this mass. \nTo extend our comparison to lower-mass systems, we include IMBH candidates from globular clusters around the Milky Way and M31, compiled by Greene et al. (2020), plotted in light-grey. Some globular clusters, like 𝜔 Cen, are hypothesised to be remnants of dwarf galaxies disrupted by the Milky Way's tidal field (e.g. Freeman 1993; Bekki & Freeman 2003; Meza et al. 2005). The presence of IMBHs in globular clusters is controversial, with significant uncertainty in BH mass values. Thus, we show a range for each cluster, indicating the lowest and highest BH mass values reported in the literature, with crosses for mean values and triangles for lower/upper limits. In addition, we plot an updated firm lower limit for the BH mass in 𝜔 Cen which was recently determined by Häberle et al. (2024) and a new IMBH detection in M31's most massive globular cluster which may also originate from a stripped dwarf galaxy (Pechetti et al. 2022), both indicated in olive-green. \nWe also show several observed BH mass - stellar mass scaling relations from the literature to give an indication of the scatter in the extrapolated relations. The Greene et al. (2020) relations are based on dynamical measurements by Kormendy & Ho (2013) as well as more recent dynamical measurements and upper limits in the lowmass regime. We plot the relation for the total galaxy sample as a solid black line, and the early-type and late-type galaxy relations as loosely and densely dashed black lines, respectively. The latetype relation has a significantly lower normalization, likely due to a weaker correlation between disc mass and BH mass (e.g. Kormendy & Richstone 1995). We also show the Reines & Volonteri (2015) scaling relation, based on local AGN, including dwarfs, as a dasheddotted black line. This relation is significantly shallower than that of Greene et al. (2020), leading to larger BH masses in the low-mass dwarf regime. \nFor the classical dwarf mass range covered by our simulations, 𝑀 stellar ≲ 10 8 M ⊙ , we would expect a flattening of the scaling relations, with the transition mass dependent on the seeding mechanism (see e.g. Greene et al. 2020), which sets a lower limit for BH masses. The extrapolated BH - galaxy scaling relations may hence be considered lower limits for BH masses in the dwarf regime. We also plot the high-redshift scaling relation derived by Pacucci et al. (2023) from overmassive BHs uncovered by JWST to indicate the high BH mass to stellar mass ratios that can be reached with optimal growth conditions in the early Universe (modulo observational uncertainties). To estimate upper limits, we show the nuclear star cluster (NSC) - galaxy scaling relations. NSCs and massive BHs often coexist in galaxies, with NSCs being the dominant central mass component in dwarfs 2 . Neumayer et al. (2020) compile NSC mass measurements \nextending to low-mass dwarfs with stellar masses ≲ 10 6 M ⊙ , making the relation valid for the entire 𝑧 = 0 galaxy stellar mass range considered here. \nKeeping these observational caveats in mind, we find that all of our dwarf AGN end up being overmassive compared to the extrapolated local BH - galaxy scaling relations. Most runs are offset by about an order of magnitude with the exception of m10zAGN which closely follows the early-type relation from Greene et al. (2020) and is only minimally offset from the Reines & Volonteri (2015) relation. Again, we emphasise that these offsets are not necessarily unexpected given that we are in the heavily extrapolated regime for these relations. Indeed the offsets are not as severe as the intrinsic high-redshift offsets inferred from recent JWST observations by Pacucci et al. (2023) and well within the bounds provided by the nuclear star cluster relation. \nFor the redshift evolution, we plot the BH mass of the most massive progenitor for redshifts 𝑧 = 1 , 2 , 3 , 4. We note that in all cases the set-ups without CRs have more significant stellar and BH mass growth due to the weaker AGN feedback. The difference is especially significant at low redshifts. At late times, between 𝑧 = 1 and 𝑧 = 0, all simulations with CRs only have minimal stellar mass growth. For m10yCR+AGN and m10zCR+AGN , there is still significant BH growth during this time indicating that whilst the CR pressure has shut down star formation, the BH is still able to accrete. For m10qCR+AGN BH growth is already shut down from 𝑧 = 3, however, between 𝑧 = 4 and 𝑧 = 3, this run similarly experiences a rapid BH growth phase whilst star formation is already quenched. \nWe also investigated the role of BH mergers versus accretion in growing the BHs in our dwarf simulations and find that in all cases BH growth is predominantly driven by gas accretion. In particular, we verified that merging with other BH seeds represents a negligible growth channel with a maximum of two, five and nine seed BHs forming in the m10q , m10y and m10z simulations, respectively. With a seed mass of 100 M ⊙ , this represents only a very small fraction of the final BH mass in all cases.", '3.4 Dark matter distributions': "Fig. 4 shows the spherically averaged dark matter density profiles as a function of radius. The three panels display these profiles for our three sets of ICs, respectively. For these profiles, we centre the dark matter distributions on the minimum potential of the main halo 3 . In each case, we plot the mean 𝑧 = 0 profiles, averaged over Δ 𝑇 = 500 Myr, for our four different galaxy formation model variations as colourcoded solid lines as indicated by the legend. We also performed additional dark-matter only (DMO) runs and we plot these as solid black lines. In each cases the temporal dispersion of these profiles is indicated by shaded regions which show the 1 𝜎 scatter in the densities over Δ 𝑇 = 500 Myr. We note that for the majority of the simulation runs, the central density is not very sensitive to the temporal bin width as the evolution of the central density is relatively steady over the last Gyr. The m10zCR simulation, however, experiences significant fluctuations in the central density from ∼ 12-13 Gyr due to mergers with substructures, so for our 𝑧 = 0 profiles we choose the bin width \nmass, and BHs only start to dominate at stellar masses above a few times 10 10 M ⊙ . \nFigure 4. Spherically averaged dark matter density profiles as a function of radius. The panels display simulated profiles based on our three sets of ICs, with colour-coding indicating the galaxy formation model variations as listed in the legend. The runs without CRs are shown as dotted lines for clarity. Density profiles from dark-matter-only (DMO) simulations are shown as solid black lines. The simulated profiles are averaged over Δ 𝑇 = 500 Myr and the 1 𝜎 temporal scatter is shown by the colour-coded shaded regions. The expected profiles for an NFW halo are plotted as grey-shaded regions indicating the 2 𝜎 scatter due to varying halo concentrations. The dark matter softening length is marked as a grey dashed line, and the Power radius (enclosing the central 2000 dark matter particles, see Power et al. 2003) is indicated by colour-coded dashed lines for each of the simulation set-ups. Flattening of the profiles within these radii may be partly due to numerical artefacts. CRs and/or AGN feedback cause significant diversity in dark matter profiles, ranging from cuspy to various degrees of cored profiles. Central densities at 𝑟 = 150 pc, a characteristic radius for distinguishing cusps from cores in observations (Read et al. 2019), are highlighted by marker symbols corresponding to the different galaxy formation physics configurations (with the circle symbol representing the DMO simulations). \n<!-- image --> \nto be small enough as to not be affected by these transient features. For all of the simulated profiles, we highlight the central dark matter density at 𝑟 = 150 pc, a characteristic radius for distinguishing cusps from cores in observations (Read et al. 2019), with colour-coded symbols. \nFurthermore, we indicate the gravitational softening length at 𝑟 = 62 . 5 pc and the Power radius 4 (enclosing the nearest 2000 neighbours, see Power et al. 2003) as grey and colour-coded dashed vertical lines, respectively. Within these radii, the dark matter profiles may be impacted by numerical heating effects. In addition, we show the expected NFW profile based on the virial mass as grey shaded regions indicating the 2 𝜎 scatter from the halo mass - concentration relation (Dutton & Macciò 2014) to indicate the typical variations in profiles due to varying halo concentrations. The DMO runs are in good agreement with the NFW expectations, with m10y having slightly higher and m10z having slightly lower concentration than the mean relation. \nFirst of all, we find that the introduction of AGN and CRs introduces a significant amount of scatter. The simulations with the default /f.pc/i.pc/r.pc/e.pc-3 model, without CRs or AGN, all lead to distinct cores at 𝑟 = 150 pc. The addition of CRs and/or AGN, however, leads to a variety of outcomes from stronger cores to weakened cores to cusps at the same inner radius 5 . \nFor simulations without CRs, the addition of an AGN enhances the core size and further suppresses the central density for all three cases. Though it should be noted that the magnitude of the additional density suppression varies and is significantly weaker for m10y at 𝑟 = 150 pc. \nFor simulations with CRs but without AGN, the profiles become cuspier than their no-CRs counterpart for m10q and m10y , whilst for m10z there is small suppression of the central densities with CRs. The addition of an AGN to these CR-based set-ups leads to a further increase of the central densities for m10q , whilst the central densities for m10y are minimally decreased. For m10z there is a much stronger impact, with the central density notable enhanced, yet still in the core regime. \nOverall, there is a variety of outcomes when introducing CRs and/or AGN to dwarf simulations which mostly do not follow clear trends due to the dual role of these processes as an additional source of feedback that may both suppress star formation (and therefore suppress core formation) or enhance core formation via the additional energy input. Hence great care has to be taken to untangle the contribution from these additional baryonic processes and their interplay with star formation. A further consideration is that small perturbations in hydrodynamical simulations (e.g. floating-point round-off or random number generators) can lead to chaotic-like behaviour with the small perturbations growing over time and manifesting as macroscopic differences in galaxy properties (e.g. Genel et al. 2019; Keller et al. 2019; Borrow et al. 2023). However, the variability we observe here is significantly larger than the variability expected from this 'numerical butterfly effect' which typically leads to central density fluctuations of at most 10 to 15 per cent (see Keller et al. 2019) whereas we observe variations ranging from 70 to 90 per cent for our different sets of ICs and galaxy formation physics configurations (see Table 1). Furthermore, as we show in the next Section, these central density trends are relatively steady as a function of cosmic \nFigure 5. Central dark matter densities as a function of cosmic time. The dark matter densities are binned over Δ 𝑇 = 500 Myr and the temporal 1 𝜎 scatter in each bin is shown by the shaded regions. The colour-coding indicates the respective simulation set-ups, as listed in the legend. For clarity, we plot the simulations without CRs as dotted lines. We measure the central densities at the inner radius 𝑟 = 500 pc to ensure that the central densities are well-resolved for all set-ups considered. For the m10q and m10y runs, the central densities are relatively steady at low redshifts, whilst the m10z runs still shows significant fluctuations due to a higher number of mergers and strong feedback events at late times. \n<!-- image --> \ntime which would not be expected if the variability resulted from the amplification of small numerical perturbations. Overall, this then points to a physical origin of the diversity of dark matter profiles that we observe in our simulations.", '3.5 Driving forces of core formation': 'Our analysis from Section 3.4 indicates that our different feedback physics configurations significantly affect the central dark matter distributions of dwarf galaxies. In this Section, we explore the driving forces behind these trends by contrasting the cosmic evolution of the central dark matter densities with SN and AGN energy injection histories. \nFig. 5 shows the central dark matter densities at 𝑟 = 500 pc as a function of cosmic time. We choose this radius as it is small enough to showcase the differences between the models explored whilst being large enough to ensure we are not being dominated by numerical heating effects, i.e. we choose a radius that is at least the size of the Power radius for all set-ups explored. Again we average the dark matter densities over time bins of Δ 𝑇 = 500 Myr and indicate the mean densities as solid lines and the standard deviation in each bin by the shaded regions. \nThe m10q set-up is shown in the first panel. Reflecting its early formation history, the dark matter density trends we found at 𝑧 = 0 in Fig. 4 have persisted for the past 6 Gyr. The central densities are relatively steady with only m10qAGN showing significant evolution after redshift 𝑧 = 1 with the central density of this run showing a slow decline. \nSimilarly for m10y (middle panel), the 𝑧 = 0 trends have persisted for the past ∼ 5 Gyr. Both of the CR-based runs, m10yCR and m10yCR+AGN , do not show any significant evolution at low redshifts, whilst the runs without CRs, m10y and m10yAGN show a steady decline from 𝑧 ∼ 1 onwards. Note that the density fluctuations for these runs are also much stronger, reflecting the potential perturbations characteristic of core formation (e.g. Pontzen & Gov- \n2; Martizzi et al. 2013). Focusing on early times between redshifts 𝑧 = 4 and 𝑧 = 1, we find that most of the runs are quite similar, except for m10yCR which exhibits a strong core during that period. We also note that at 𝑡 ∼ 5 Gyr, the m10y system experiences a major merger. For the DMO run this significantly raises the central density, and the strong early core for the m10yCR set-up is erased by this merger. \nFor the m10z system (right panel), there is significantly more latetime evolution as well as stronger fluctuations as reflected in its more irregular morphology, see Fig. 1. Indeed, the general central density trends we observe at 𝑧 = 0 have only been in place for the past ∼ 2 Gyr and the m10zCR run experiences strong fluctuations for most of this period. We have verified that these potential fluctuations are driven by late-time mergers with substructures that pass through the centre and are independent of the slice width chosen or the method used to determine the halo centre when calculating the central density. Between redshifts 𝑧 = 2 and 𝑧 = 0 . 5, the set-ups m10zAGN , m10zCR , and m10zCR+AGN , all show a relatively similar redshift evolution of their central densities. The m10z set-up, however, closely follows the DMO run and even surpasses it from 𝑡 = 7-9 Gyr, indicating strong baryonic cooling at the centre. At 𝑧 ∼ 0 . 5 a merger induces density fluctuations for all of our m10z set-ups, but most prominently for the CR-based runs m10zCR and m10zCR+AGN . After this merger, the two CR-based set-ups have relatively steady central densities. On the other hand, the runs without CRs, m10z and m10zAGN experience significant density suppressions at late times. \nIn the following two subsections, we turn to investigate the origin of these cosmic evolution patterns in the dark matter central densities, examining SN and AGN energy injection as a function of time.', '3.5.1 SN feedback': 'Fig. 6 shows the injected SN energy as a function of cosmic time with the instantaneous energy shown as dashed lines and the cumulative energy shown as solid lines, both binned over 250 Myr for clarity. \nFigure 6. Instantaneous (dashed lines) and cumulative (solid lines) SN energy injected as function of cosmic time. The three columns display simulations based on our three sets of ICs, with colour-coding indicating the galaxy formation physics configurations as listed in the legends. We also indicate the 𝑧 = 0 binding energy of the halo as dotted lines. The cumulative SN energy injected compared to the binding energy is a strong predictor of core formation as well as the central dark matter density more generally. Some deviations from this pattern are introduced due to the important role of late-time star formation bursts in preserving cores. \n<!-- image --> \nThe three columns correspond to our three sets of ICs. The no-AGN and AGN runs are shown in the top and bottom row, respectively. \nWe focus on the SN energy injected in the central region of the halo within a fixed 1 kpc aperture (in physical coordinates). This choice is motivated by the fiducial inner radius considered by observers for determining the shapes of dwarf galaxy rotation curves (see Section 3.7) and it also corresponds to the scale of the half-mass radius for most of our systems (see Fig. 2). \nThe SN energy injection rates are recalculated from the stellar ages and metallicities based on S/t.pc/a.pc/r.pc/b.pc/u.pc/r.pc/s.pc/t.pc99 tables for a Kroupa (2001) IMF. We focus here on the SN II rates, which constitute the majority of SN events (see Hopkins et al. 2023a), for simplicity. For reference, we also indicate the 𝑧 = 0 binding energy of the respective haloes as dotted lines, which we estimate as 𝑓 bar 𝑀 vir 𝑉 2 vir , where 𝑓 bar is the universal baryon fraction and 𝑉 vir is the virial velocity of the halo (also see Wellons et al. 2023). \nFirstly, we note that both CRs and AGN have a significant impact on star formation, and therefore the SN energy histories. As discussed in previous works (Sparre et al. 2017; Su et al. 2018; Genel et al. 2019; Keller et al. 2019; Hopkins et al. 2023c), due to their shallow \npotential wells, dwarf systems are very sensitive to baryonic feedback processes so that even small changes can significantly affect star formation histories. \nThis highly variable behaviour is quite clearly illustrated by the m10q set-up in the first column. With CRs, the system experiences a higher initial burst which is powerful enough to quench the dwarf system from 𝑧 ∼ 6. Hence both m10qCR and m10qCR+AGN end up with cuspy profiles, even though the cumulative injected SN energy for m10qCR+AGN slighly exceeds the 𝑧 = 0 binding energy, as the absence of late-time star formation means that a cusp can be reestablished quite easily after the initial burst. Note that m10qCR has a slighly reduced central density at 𝑧 = 0 compared to m10qCR+AGN due to low-level late-time star formation activity; however, these bursts are not powerful enough to induce a core. Without CRs, the initial star formation burst is less powerful, allowing for continued star formation activity. For the m10q set-up, there are a few more powerful bursts at intermediate redshifts, which then quench the systems at much later times from 𝑧 ∼ 1 . 5. For the m10qAGN setup, the AGN regulation means that the intermediate redshift star formation bursts are less intense so that star formation can continue \nuntil 𝑧 = 0. Both m10q and m10qAGN end up with cumulative SN energy almost an order of magnitude above the 𝑧 = 0 binding energy leading to cored profiles; however, m10qAGN has a more extreme core due to the continued star formation activity (also see Muni et al. 2024). \nFor m10y , there is a similar picture albeit not as extreme as for the m10q halo. With CRs, high-redshift star formation is somewhat burstier, and the more powerful bursts then also mean that most of the SN activity is restricted to 𝑧 ≳ 1. This is still sufficient to (cumulatively) exceed the 𝑧 = 0 binding energy and induce a core for both of the CR-based runs; however, the cores are significantly weaker and less extended than for the runs without CRs. Both m10y and m10yAGN have continued star formation activity until 𝑧 = 0, higher cumulative injected SN energies, and therefore much stronger cores. Adding in AGN only has a small impact for both m10yAGN and m10yCR+AGN compared to their no-AGN counterparts. Again the bursts at high redshift are somewhat weaker, allowing for more SNactivity at lower redshifts and therefore somewhat stronger cores. However, as discussed in Sections 3.4 and 3.6, the differences induced by AGN feedback are only minimal for the m10y profiles at 𝑧 = 0. \nFor m10z , we obtain the most complex interplay between star formation, AGN activity and CRs. Here, the set-ups without CRs have more powerful initial star formation bursts (underlining that the initial intensity of the burst is not necessarily directly dictated by the galaxy formation physics configuration). Overall, m10z , m10zCR , and m10zAGN all experience episodes of star formation for most of cosmic time, although for m10z , those are shifted to lower redshifts, whilst for the m10zCR simulation SN activity is more concentrated at higher redshifts. m10zAGN experiences bursty star formation throughout cosmic time, leading to the highest cumulative injected SN energy and the most pronounced core. Likewise, the cumulative SN energy in the m10z and m10zCR set-ups also exceeds the 𝑧 = 0 binding energy and both of these result in cored profiles. Interestingly, however, star formation in the m10zCR+AGN set-up is significantly suppressed, and powerful bursts are mostly restricted to high redshifts ( 𝑧 > 2), so that the total injected SN energy remains well below the 𝑧 = 0 binding energy by a factor of three. This indicates that for the m10zCR+AGN set-up, the core formation may be driven by the AGN. \nThis analysis underlines that CRs and AGN primarily affect cuspto-core transformation, and the diversity of dark matter profiles, by regulating star formation histories in our simulations. The only setup where the AGN feedback is actively driving core formation is m10zCR+AGN , where star formation is too heavily suppressed to drive core formation. In the cases of m10q and m10y , the AGN is not efficient at suppressing star formation, leading to similar or higher levels of SN activity compared to the equivalent no-AGN setups. However, in these scenarios, the AGN may still enhance core formation; we examine this possibility in the next section.', '3.5.2 AGN feedback': "In the /f.pc/i.pc/r.pc/e.pc-3 model, AGN feedback is injected in several channels including mechanical feedback, radiative feedback and CRs (see Section 2.2.1 for details). Comparing this multi-channel AGN energy to the SN energy that is predominantly injected mechanically is not straightforward, especially since we are most concerned with feedback-induced gravitational potential perturbations. Therefore, we need to assess how the AGN energy will couple dynamically. Fully tracking and analysing coupling efficiencies of different AGN feedback modes is beyond the scope of this paper, so we make some simplifying assumptions based on previous literature. \nFor the AGN, only a very small fraction of the injected energy is directly coupled in the form of mechanical winds, with only ∼ 0 . 6 per cent of the AGN luminosity injected as wind energy (see Section 2.2.1). \nFor the radiative feedback, on the other hand, the full luminosity is injected into the simulation domain using the LEBRON method (see Hopkins et al. 2020a) for radiation transport. This radiation transport then predicts the radiation pressure forces, i.e. momentum flux, from the absorbed photon luminosity. Overall only a small fraction of the luminosity will therefore be energetically coupled to the gas as a radiatively driven wind. The LEBRON method has been most thoroughly tested for stellar feedback. Whilst AGN radiation has a different spectrum, we can still draw some tentative conclusions from these studies for our estimates. For the stellar feedback, only 0 . 4 𝐿 / c of the photon momentum couples to the gas in dwarfs, whilst the remainder escapes the galaxy, and the vast majority of this photon coupling occurs in the single-scattering regime (see analysis in Hopkins et al. 2020a). We here assume that a similar fraction of the luminosity would couple to the gas for radiative AGN feedback. Following (King & Pounds 2015), a fraction of approximately 5 per cent of that availabile AGN luminosity is then coupled as mechanical energy in the form of a radiatively driven wind in the single scattering regime, yielding an overall mechanical energy efficiency of 2 per cent compared to the AGN luminosity for the radiative feedback channel. \nFor the AGN-driven CR feedback, 1 per cent of the accreted restmass energy, i.e. 10 per cent of the luminosity with the assumed radiative efficiency of 10 per cent, is injected as CRs. For CR transport, we employ the subgrid model from Hopkins et al. (2023b), which interpolates between the limit in which CRs escape the galaxies with negligible losses and that in which CRs lose most of their energy catastrophically before escaping, using a formalism akin to the LEBRON method. For dwarf galaxies, we are generally far from the proton calorimetric limit where most energy is lost before CRs escape the dense gas, and we assume that (as with the radiation) 40 per cent of the CRs couple before escaping - which likely provides an upper limit. This then yields an overall mechanical energy efficiency of 4 per cent compared to the AGN luminosity for the CR feedback channel. \nKeeping in mind that these assumed efficiencies may be overestimating the actual coupled AGN energy, we show the 'effective' injected AGN energy based on our assumed efficiencies for the different channels as a function of cosmic time in Fig. 7. Again the cumulative energy is shown as the solid lines and the instantaneous energy is shown as dashed lines binned over 250 Myr for clarity. The AGN energy is based on the accretion rate of the most massive BH in the main halo at 𝑧 = 0 and its most massive progenitor halo at higher redshifts. The three different panels represent our three different ICs and the colour coding of the lines indicates the different physics configurations as listed in the legends. \nOverall, we note that AGN energy injection is more steady than SN energy injection, with the AGN in most set-ups being active continuously, except for m10qCR+AGN where the AGN activity, just like the star formation, is quenched at early times and is then mostly quiescent at late times apart from a relatively weak late-time burst. In all cases, the injected AGN energy matches or exceeds the binding energy though we caution that due to the extremely uncertain mechanical coupling efficiencies this has to be taken with a grain of salt and the AGN energy that is actually coupled mechanically to the gas may be lower. Furthermore, we note that whilst the SN feedback is distributed throughout the galaxy the AGN energy injection is only focussed at the centre, with radiation and mechanical winds injected \nFigure 7. Instantaneous (dashed lines) and cumulative (solid lines) effective AGN energy injected as a function of cosmic time. The three panels display simulations based on our three sets of ICs, with colour-coding indicating the galaxy formation physics configurations as listed in the legends. The 𝑧 = 0 binding energy of the halos is indicated by colour-coded dotted lines. AGN feedback is injected through three channels: radiation, mechanical winds, and, for some, CRs. The overall 'effective' energy is calculated based on reported couplings from the literature. Runs with CRs show more efficient AGN feedback regulation, with m10zCR+AGN experiencing efficient bursts that induce a core despite low star formation activity. Without CRs, BHs grow more efficiently, leading to higher energy injection rates and increased core sizes for all set-ups. \n<!-- image --> \nin a collimated fashion, and therefore AGN feedback is likely less efficient than SN feedback in affecting the ISM at a fixed energy injection rate. Indeed, past simulations have demonstrated that a significant fraction of the AGN wind energy escapes the ISM once the winds have opened up a central cavity (e.g. Bourne et al. 2015; Koudmaniet al. 2019; Torrey et al. 2020; Mercedes-Feliz et al. 2023). \nFor m10q , in the CR-based run m10qCR+AGN , the AGN does not have a significant impact on the central densities since the late-time activity, which is crucial for core formation, is mostly suppressed. Without CRs, for the m10qAGN set-up, there is a sharp AGN burst at 𝑡 ∼ 8 Gyr which is also associated with pronounced central dark matter density fluctuations and followed by a steady decline in the central dark matter density suggesting that the AGN may be contributing to the formation of the core with this set-up. \nFor m10y , the m10yAGN set-up has a strong AGN burst at 𝑡 ∼ 5 . 5 Gyr, which is again associated with strong central density fluctuations indicating gravitational perturbations as well as a drop in the star formation rate. With the CRs, the AGN energy injection is again significantly suppressed (though not completely shut down like in the m10qCR+AGN set-up) and the relatively low levels of activity do not have a significant impact on core formation. \nFor m10z , contrary to the other two sets of ICs, the energy injected from the CR-based run, m10zCR+AGN , exceeds the AGN energy injected with the equivalent no-CR set-up m10zAGN . This is mostly driven by a significant burst in the m10zCR+AGN set-up at 𝑡 ∼ 9 Gyr, which takes the overall injected AGN energy budget above the binding energy of the halo and is followed by strong density fluctuations. Most notably, even though central star formation is completely suppressed from 𝑡 ∼ 10 Gyr in the m10zCR+AGN simulation, the AGN activity persists at relatively high levels, maintaining the core in the absence of SN-driven winds. This is driven by two factors in our AGN modelling. Firstly, at late times the ratio between BH mass and subgrid accretion disc mass in the m10zAGN run is quite large ( 𝑀 BH / 𝑀 d ∼ 10 4 ), which leads to depletion timescales of the order of ∼ 1 Gyr (also see Section 2.2.1), allowing for persistent AGN activity even after the inflows onto the BH - accretion disc particle have sub- \nlst the gas density is significantly decreased in the central region (and therefore the gas is not star-forming), the BH is still able to accrete at low rates from this supply since there is no explicit density criterion for the BH accretion. This likely represents an optimistic estimate of the BH accretion rate and we discuss the implications and caveats of these modelling choices in more detail in Section 4.3.2. For the m10zAGN set-up the AGN has a significant burst at 𝑡 ∼ 12 Gyr which again is associated with strong central density fluctuations suggesting that here the AGN may also enhance the formation of the core. \nOverall, we note for most of our AGN set-ups, including m10qAGN , m10yAGN , m10yCR+AGN , m10zAGN , and m10zCR+AGN , the equivalent no-AGN set-up already has a SNinduced core. For all of these cases, apart from m10zCR+AGN , the addition of an AGN further decreases the central densities enhancing the existing core. However, it is difficult to disentangle whether this additional decrease is driven directly by AGN energy injection or indirectly by suppressing high-redshift star formation and 'delaying' SNactivity to later times favouring core formation. Indeed, it is likely that both of these factors contribute to the enhancement of cores. \nThe m10zCR+AGN set-up is our only simulation where the AGN is efficient in globally suppressing star formation compared to its no-AGN counterpart (also see Fig. 2). Here, there is no SN activity for the last ∼ 3 . 5 Gyr from 𝑧 ∼ 0 . 5 and only weak SN activity between 𝑧 = 2 and 𝑧 = 0 . 5, yet the core is maintained. Indeed, the central density significantly decreases from 𝑧 = 2 to 𝑧 = 1 and is then maintained at low levels despite the late-time merger at 𝑧 ∼ 0 . 5, providing strong evidence for AGN-driven core formation. \nOverall, we conclude that AGN feedback can induce significant scatter in dark matter density profiles, predominantly indirectly by regulating star formation histories but also in some cases directly by driving powerful potential fluctuations. \nFigure 8. Central dark matter densities at 𝑟 = 150 pc as a function of virial mass ( left panel ), stellar mass to virial mass ratio ( middle panel ), and BH mass to virial mass ratio ( right panel ). Simulated 𝑧 = 0 data points are highlighted by marker symbols corresponding to the respective galaxy formation physics configurations as indicated by the legend. The binned redshift evolution from 𝑧 = 3 to 𝑧 = 0 is represented by dotted lines. The left and middle panels include various observed central dark matter densities from Read et al. (2019) for comparison. In the middle panel, a dashed grey line marks the stellar mass to virial mass ratio where previous studies suggest stellar-feedback-driven core formation may become inefficient. Our simulations broadly agree with this theory, except for m10zCR+AGN , which forms a core despite a decreasing stellar mass to virial mass ratio, though has an increasing BH mass to virial mass ratio, indicating AGN-driven core formation. \n<!-- image -->", '3.6 Cosmic evolution of central dark matter densities in observational context': 'We further analyse the origin of the diversity of our simulated dark matter profiles by examining the central densities in the context of observations. Fig. 8 shows the central dark matter densities at 𝑟 = 150pc as a function of virial mass (left panel), stellar mass to virial mass ratio (middle panel) and BH mass to virial mass ratio (right panel). The dotted lines indicate the (binned) redshift evolution of our simulated dwarfs from 𝑧 = 3 to 𝑧 = 0, again binned over 500 Myr. For clarity we only highlight the 𝑧 = 0 data point with the marker symbol corresponding to our different physics configurations, as indicated by the legend. In the left panel we also show the expected central densities for an NFW profile and a cored profile following the /c.pc/o.pc/r.pc/e.pcNFW profile from Read et al. (2016) as grey lines with the 2 𝜎 concentration scatter indicated by dark-grey and light-grey shaded regions, respectively. Furthermore, we show the observed central densities of 16 observed nearby dwarf galaxies, including classical dwarf spheroidals and gas-rich dwarf irregulars from Read et al. (2019). \nThe first panel demonstrates that our simulations are generally in good agreement with the observed central densities of nearby dwarfs, keeping in mind that the sample sizes are limited in both cases (16 observed dwarfs versus 12 simulated dwarfs). Two of the m10z simulations, m10zCR and m10zAGN , lead to very strong cores, with the central densities being more severely suppressed than in the observations. Our simulations do not reproduce the observed strong cusps with central densities above the NFW mean relation. However, we note that the vast majority of our dwarfs have late star formation histories, with only the CR set-ups for m10q and m10y resulting in an early truncation of star formation, as defined by Read et al. (2019), with no activity for the past 6 Gyr. These dwarfs with early truncation are associated with cuspy profiles hinting that we would require more early-forming set-ups to reproduce these strong cores. \nThe middle panel shows the simulated and observed central densities as a function of stellar mass to virial mass ratio 𝑀 stellar / 𝑀 vir . As discussed in Read et al. (2019), the observed dwarfs show a clear trend with higher 𝑀 stellar / 𝑀 vir being associated with lower central dark matter densities. Moreover, in the observations, as well as in previous theoretical studies, it is found that no stellar-feedback-driven \ncores form below a critical ratio of ( 𝑀 stellar / 𝑀 vir ) crit = 5 × 10 -4 since the integrated SN energy is insufficient to drive the required potential fluctuations (see discussion in Di Cintio et al. 2014). We also find a strong correlation between 𝑀 stellar / 𝑀 vir and central density suppression for our simulated dwarfs. Indeed, the correlation is significantly tighter than for the observed dwarfs likely due to increased scatter induced by measurement uncertainties in the observations. Looking at the redshift evolution tracks of our simulated dwarfs, we find that most systems also follow these trends in a temporal sense with two notable exceptions. The CR-based runs with the m10q ICs show a slight decrease as the 𝑀 stellar / 𝑀 vir decreases. However, this is mostly a by-product of these runs being quenched at very high redshifts so that the central dark matter density evolution is dominated by the general assembly history of the halo with the virial mass increasing and the halo concentration decreasing (whilst the stellar mass remains constant) at late times. The second exception to the 𝑀 stellar / 𝑀 vir trend, however, is more notable: for the m10zCR+AGN set-up the central densities as a function of 𝑀 stellar / 𝑀 vir are steadily decreasing so that a core forms despite the 𝑧 = 0 𝑀 stellar / 𝑀 vir ratio being below the critical ratio identified by Di Cintio et al. (2014) and Read et al. (2019). This provides further evidence that this run could be experiencing AGN-driven core formation. \nWe further examine this possibility in the third panel where we plot the simulated central densities (as well as their binned redshift evolution from 𝑧 = 3) for the AGN-based simulation set-ups as a function of BH mass to virial mass ratio 𝑀 BH / 𝑀 vir . For the runs without CRs, m10qAGN , m10yAGN , and m10zAGN , the central densities again steadily decrease as a function of 𝑀 BH / 𝑀 vir . However, we caution that this does not necessarily indicate that the BHs are contributing to core formation due to the strong correlation between 𝑀 BH and 𝑀 stellar (see Fig. 3 for scaling relations of our simulated dwarfs). The m10qCR+AGN set-up is the only simulation where the central density ratio decreases as 𝑀 BH / 𝑀 vir decreases, as with the stellar mass to halo mass ratio, and this mainly reflects that the central density is unaffected by baryonic processes at late times. The m10yCR+AGN set-up has a very shallow gradient again reflective of the relatively quiescent late-time evolution. Finally, the m10zCR+AGN set-up exhibits a significant decrease in central densities as the 𝑀 BH / 𝑀 vir increases (opposite trend from 𝑀 stellar / 𝑀 vir ) providing further support for the BH-driven core formation scenario. \nFigure 9. Circular velocity profiles at 𝑧 = 0 as a function of radius including all matter (solid lines) and baryons only (dashed-dotted lines). The profiles are averaged over Δ 𝑇 = 500 Myr with the shaded regions indicating the 1 𝜎 scatter. The first three columns display simulations based on our three sets of ICs with the no-AGN runs (top row) and AGN runs (bottom row) plotted separately for clarity. The markers highlight the circular velocity at the fiducial radius which is used as a reference point in observational studies of rotation curves. The fourth column translates these velocity curves to the rotation curve shape - baryon dominance parameter space, also showing observed dwarf galaxies in a similar mass range from the SPARC sample as compiled in Santos-Santos et al. (2020). Cosmic environment, CRs and BH feedback can all influence the rotation curve shape parameter, with our simulations broadly reproducing the observed trends with baryonic dominance. \n<!-- image -->', '3.7 The diversity of rotation curves': "As discussed in Section 1, one of the most prominent remaining dwarf galaxy 'problems' pertains to the observed diversity of dwarf galaxy rotation curves. From our analysis in the previous sections, we have found that AGN and CRs may significantly increase the scatter in dark matter density profiles and hence may contribute to resolving this on-going controversy. Ultimately, to make statistical predictions for dwarf galaxy rotation curve shapes, we would need a much larger simulation sample spanning a larger range of environments and halo masses. Nevertheless, we can use our simulation suite to examine the broad trends for rotation curve shapes and assess the potential for more sophisticated galaxy formation models including the impact of AGN and CRs at the low-mass end to resolve the long-standing diversity of rotation curves problem. \nIn Fig. 9, we present circular velocity curves for all our dwarf simulations. As for the dark matter profiles, we average the velocity curves over Δ 𝑇 = 500Myrandindicatethe1 𝜎 scatter with the shaded regions. Following the methodology presented in Santos-Santos et al. (2020), we plot circular velocity profiles for the total matter content of our dwarfs (dark matter, gas, stars, and BHs) as solid lines and for baryons (gas, stars, and BHs) as dotted-dashed lines. \nThe total circular velocity curves for each set of ICs converge between 1 to 10 kpc, with the extent of the matter deficits in the central regions reflecting the core sizes from Fig. 4. For the baryonic circular velocity curves, however, there are significant differences. We also examine the gas and stellar distributions (not shown here) and find that both of these are significantly modified for galaxies with pronounced cores, which tend to have more extended stellar distributions (also see Figure 1). For the cored dwarfs this then leads to slowly rising rotation curves for both dark matter and stars. Dwarfs that are quiescent at 𝑧 = 0, and generally have cuspier profiles, are more gas poor (especially at the centre), leading to overall lower baryonic circular velocities in the central region. \nFollowing Santos-Santos et al. (2020), we quantify the cusp versus core nature of the profiles by calculating characteristic velocity ratios, comparing the inner 'fiducial' rotation velocity 𝑉 fid with the asymptotic flat rotation velocity 𝑉 max . The fiducial rotation velocity \nis measured at the fiducial inner radius defined as: \n𝑟 fid = GLYPH<18> 𝑉 max 35 km s -1 GLYPH<19> kpc . (4) \nThe rotation curve shape parameter 𝜂 rot can then be defined as \n𝜂 rot = 𝑉 fid 𝑉 max = 𝑉 ( 𝑟 fid ) 𝑉 max . (5) \nRapidly rising rotation curves have 𝜂 rot ≲ 1, with the NFW profile yielding a rotation curve shape parameter of 𝜂 rot ∼ 0 . 65. Cored profiles have 𝜂 rot ≪ 1; see Santos-Santos et al. (2020) for a detailed discussion of this parameter. \nAsin Santos-Santos et al. (2020), we also inspect a second velocity ratio - the baryonic importance parameter, which is defined as \n𝜂 bar = GLYPH<18> 𝑉 b , fid 𝑉 fid GLYPH<19> 2 = GLYPH<18> 𝑉 b ( 𝑟 fid ) 𝑉 ( 𝑟 fid ) GLYPH<19> 2 , (6) \nwhere 𝑉 b , fid is the contribution from baryons (gas, stars, BHs) to the circular velocity curve. Under the assumption of spherical symmetry, 𝜂 bar represents the mass fraction of baryons within the fiducial radius. \nThe two velocity ratios 𝜂 rot and 𝜂 bar can be used to plot the 'cuspiness' of the rotation curves against the contribution from baryons in the inner region of the galaxy. We plot this distribution in the rightmost panel in Fig. 9. For comparison, we also show observed data as collated by Santos-Santos et al. (2020). We restrict the observed galaxy sample to the classical dwarf regime with 𝑉 max ≤ 60 km s -1 to compare this more easily and directly with our simulated dwarfs. The observational data stems mainly from the SPARC sample and, in all cases, the rotation curves were inferred from high-resolution HI and/or H 𝛼 velocity fields. Given that all of the observational rotation curves are inferred from the baryonic contributions, the dwarfs with extremely low baryonic contributions are missing from the observed samples. Similarly to Santos-Santos et al. (2020) we therefore show our low𝜂 bar simulated dwarfs at 𝜂 bar = 0 . 05 for clarity. \nFocussing first on the observations, we find that, as discussed in Santos-Santos et al. (2020), the baryonic importance parameter appears to be a poor predictor of rotation curve shapes with a significant amount of scatter in the data. Specifically, this plot highlights \nthe appearance of profiles ranging from cusps to weak cores at low baryonic dominance and cusps to strong cores at high baryonic dominance in the observations. This large amount of scatter may naively seem at odds with the theory that baryons are driving cusp to core transformations and has been extremely difficult to reproduce with simulations. Despite the high scatter, there are nevertheless some interesting observational trends where strong cores in dwarfs tend to be associated with higher baryonic importance and strong cusps tend to be associated with lower baryonic importance. However, both of these aspects seem counter-intuitive if cores are associated with strong outflows and cusps with baryons cooling at the centre. \nOur simulations cover almost all observed scenarios: cusps and weak cores at low baryonic dominance (e.g., m10qCR and m10zCR+AGN ) and cusps and strong cores at high baryonic dominance (e.g., m10y and m10zAGN ). The only missing scenario are the 'ultra cuspy' dwarfs at both low and high baryonic dominance, which might be influenced by environmental factors, suggesting our sample might be too small to cover the required range of assembly histories. To reproduce these extremely cuspy dwarfs with CDM-based simulations, we would need strong cosmic inflows so that feedback becomes inefficient leading to overcooling (see e.g. Smith et al. 2019). Another possible pathway for inferring ultra-cuspy dwarfs from gas rotation curves constitutes strong, transient perturbation to the gas, usually resulting in an offset between the gas, stars and dark matter. However, in the fiducial /f.pc/i.pc/r.pc/e.pc-3 model (as well as in the SPARC sample), these transient ultra-cuspy dwarfs are generally associated with more massive systems ( 𝑀 halo ≳ 10 11 M ⊙ and 𝑉 max ≳ 50 kms -1 ), see Sands et al. (2024) for details. \nNevertheless, even with our small sample, we obtain a range of profiles at fixed baryonic importance ranging from cusps to cores, a feat that has been historically difficult to reproduce in simulations (e.g. Oman et al. 2015; Garrison-Kimmel et al. 2019; Santos-Santos et al. 2020). We are able to reproduce these trends due to the interplay of the different feedback processes resulting in varying overall feedback efficiencies for the same baryon content, either enhancing or reducing the overall feedback impact compared to simulations that do not include CRs or BHs. Cusps are generally associated with highly concentrated stellar distributions in gas-poor dwarfs, whilst cores are created by galactic fountain activity in gas-rich dwarfs that produces a core through cyclic feedback events but does not expel the baryons. In this manner, we are able to match both the large observed scatter and the weak observed trends with baryonic dominance.", '4.1 Comparison with previous theoretical work': "We have investigated the impact of BH feedback and CRs on the cusp versus core problem in dwarf galaxies with a special focus on whether including additional baryonic feedback channels may help explain the observed diversity of rotation curves in dwarf galaxies. \nSeveral groups have investigated the role of stellar feedback in driving cusp to core transformations in galaxy formation simulations with mixed results (e.g. Governato et al. 2010; Parry et al. 2012; Pontzen & Governato 2012; Hopkins et al. 2014, 2018; Kimm et al. 2015; Emerick et al. 2018; Smith et al. 2019; Gutcke et al. 2021). Most simulations tend to either predominantly produce cusps or predominantly produce cores which appears to be closely related to the star formation threshold employed and, more crucially, whether the cold, dense ISM phase is resolved in the simulations (see discussion in Jahn et al. 2023). Nevertheless, for a given ISM model it has proven \nvery challenging for simulations to reproduce the observed diversity of dwarf rotation curves (e.g. Oman et al. 2015; Garrison-Kimmel et al. 2019; Santos-Santos et al. 2020). The past decade has unveiled growing observational samples of dwarf galaxies with AGN. These observations raise the question of whether active BHs may increase the diversity in star formation histories as well as directly impact central dark matter distributions and thereby increase the diversity of dwarf rotation curves in simulations. \nPrevious studies found systematic yet small-scale suppressions of central dark matter densities by AGN feedback in dwarfs (Koudmani et al. 2022; Arjona-Galvez et al. 2024), yet in both cases, the simulations did not explicitly resolve the multiphase ISM. Cusp-to-core transformations with AGN feedback were also explored by Arora et al. (2024). However, they employed the fiducial Bondi accretion model for BH growth, which suppresses the growth of low-mass BHs such that AGN activity in dwarfs is insignificant in these simulations and does not affect dark matter distributions. \nIn our study, we investigate, for the first time, the impact of AGN feedback on dark matter profiles in dwarfs with a resolved multiphase ISM model and a BH growth scheme that does not suppress AGNactivity in dwarf galaxies. Several works exploring BH physics within the /f.pc/i.pc/r.pc/e.pc model have found that CRs appear to be a key ingredient for effective AGN feedback (e.g. Su et al. 2021; Wellons et al. 2023; Byrne et al. 2023b), so we also investigate the impact of CR feedback on the interplay between baryons and dark matter in dwarfs. We find that efficient AGN feedback can actively drive core formation and, together with CRs, significantly enhances the diversity of dwarf rotation curves by leading to more varied star formation histories. Indeed, in many cases our runs including CRs (with or without AGN) lead to cuspier profiles due to suppressed star formation activity. This is in contrast with recent findings from Martin-Alvarez et al. (2023), who find that CRs are prone to enhance core formation. We note that for one of our set-ups (compare m10z to m10zCR ), we also observe this behaviour, and indeed it is plausible that CRs could act to either suppress or enhance cores depending on their interaction with the ISM and the assembly history of the given dwarf galaxy. Our most important finding is that BHs and CRs as additional baryonic processes lead to varying overall feedback efficiencies that can lead to both cuspier or more cored profiles for a given assembly history. \nWe also demonstrate the promising potential for these additional baryonic processes to resolve the diversity of dwarf galaxy rotation curves problem: our simulations exhibit a wide variety of profiles, including cuspy profiles at low baryonic dominance and strong cores at high baryonic dominance, thereby reproducing the puzzling relationship between rotation curve shapes and the gravitational importance of baryons from the observations. The former is associated with highly concentrated stellar distributions in gas-poor dwarfs, whilst the latter is a signature of galactic fountain activity in gas-rich dwarfs that produces a core through cyclic feedback events but does not expel the baryons. \nOne remaining discrepancy with the observations lies in the 'extremely' cuspy dwarfs which are even denser than NFW expectations. Within our relatively modest sample size of simulations we are unable to reproduce these. Apart from the obvious need to explore more halo masses and environments, as these objects may be linked to strong cosmic inflows which lead to overcooling (see e.g. Smith et al. 2019), in future work, it would also be important to explore observational uncertainties in rotation curve measurements as well as theoretical uncertainties in the modelling of BHs and CRs, and we discuss both of these aspects in the following sections.", '4.2 Observational uncertainties': "In observations, the dark matter distribution of dwarf galaxies may be inferred from the rotation curves of gas or stars, yet it has been shown that in practice these rotation curves may be highly unreliable tracers of the 'true' underlying circular velocity distribution. In particular, various dynamical perturbations may lead to large discrepancies of 50 per cent or more (see Pineda et al. 2017; Roper et al. 2023; Downing & Oman 2023; Sands et al. 2024, for some recent studies). Processes that are associated with inducing AGN activity, including mergers and strong cosmic gas inflows, lead to inaccurate estimates of the matter distributions in dwarfs. Interestingly, all of these studies find that observational uncertainties are more likely to lead to an underestimation of the true circular velocity, i.e. overestimation of the occurrence of cored profiles in observations, whilst the inverse error is less common though may still occur in the central few kiloparsecs due to dynamical phenomena, especially for more massive dwarfs (see discussion in Sands et al. 2024). \nApart from the observational uncertainties in inferring dark matter mass distributions, it also remains extremely challenging to determine the drivers behind core formation from observations. In particular, there are no observational constraints on the potential role of AGNincusp-to-core transformations. This is mainly due to the number of observed AGN in nearby dwarf galaxies being very limited. To further complicate matters, AGN are associated with disturbed gas rotation curves - see e.g. Manzano-King & Canalizo (2020), who find a strong association between AGN activity and disturbed gas kinematics in observed dwarf rotation curves, making it even more difficult to establish a link between AGN activity and central dark matter densities. JWST may partly alleviate these issues by disentangling rotational and outflow components with high-resolution NIRSpec-IFU observations (e.g. Bohn et al. 2023). From our simulations, we predict that if cored dwarf galaxies were found below the critical mass ratio of GLYPH<16> 𝑀 stellar 𝑀 vir GLYPH<17> crit = 5 × 10 4 this would be a strong indicator of AGN-driven core formation in dwarfs.", '4.3 Theoretical uncertainties': "The diversity of models presented in this work (with or without CRs and with or without AGN) are meant to represent different outcomes of 'the same underlying physics' under different conditions. In particular, the impact of AGN feedback is expected (and observed) to be highly variable in the dwarf regime. The AGN activity levels in dwarfs are highly dependent on the seeding model assumed (with some dwarfs possibly not hosting any BHs at all) and whether the BH is off-centre for the majority of the dwarf's history, i.e. the no-AGN model here may represent the same underlying physics as the AGN model but in a situation where no BH seeded or only an extremely weakly accreting off-centre BH is present. Hence just seeding and BH dynamics could drive substantial variability in addition to other effects such as varying feedback efficiencies depending on BH spin evolution. Similarly, the CR subgrid model we employ here represents a case where losses within the galaxy are minimal and hence the impact of CRs in our dwarfs likely represents an upper limit. With full CR transport the outcome will likely lie somewhere inbetween the no-CR and CR simulations presented in this work. It would require significantly higher resolution simulations (and correspondingly more detailed BH modelling) and explicit CR transport to depict the full variability in dark matter distributions introduced by baryonic physics within one galaxy formation model. Our simulations provide strong motivation for exploring such detailed models \nin future work. Below we discuss the caveats of our CR and BH modelling in more detail.", '4.3.1 CR modelling': 'Whilst the focus of our paper lies on the interplay between AGN feedback and dark matter distributions in dwarfs, we also assess the impact of CRs on the cusp versus core problem since CR injection has been identified as an important AGN feedback channel. For this, we take advantage of the CR subgrid model by Hopkins et al. (2023b), which allows for the inclusion of the impact of CRs on galaxy formation without the significant computational overhead associated with full CR transport. \nAs discussed in previous sections, for dwarf galaxies, this subgrid model presents a good approximation since, in this mass regime, we are far away from the calorimetric limit for protons. Consequently, we can safely assume that losses within the galaxy only play a secondary role. Nevertheless, with this subgrid model, we may still be somewhat overestimating the effects of CR feedback at late cosmic times, as we are not fully capturing the inhomogeneity of CRs in the CGM, which may modify thermal instabilities (Butsky et al. 2020). In particular, we find that this implementation of CR feedback suppresses star formation so strongly that the addition of CRs generally leads to cuspier profiles. Indeed Hopkins et al. (2023b) also find in their validation simulations with a 10 11 M ⊙ halo that the subgrid CR model leads to suppressed star formation and a somewhat weaker core compared to an equivalent simulation with full CR transport, though these differences are small ( ∼ 1 per cent) and both CR models lead to a weaker core compared to their no-CR set-up. Our results should therefore be regarded as an upper limit for the impact of CR feedback and, whilst exploring alternative CR implementations is beyond the scope of this paper, our work provides strong motivation for investigating cusp-to-core transformations with AGN and full CR transport in future work.', '4.3.2 BH modelling': "The modelling of AGN feedback in cosmological simulations remains subject to significant theoretical uncertainties due to the vast dynamic range of relevant scales spanning (at least) 14 orders of magnitude from the event horizon ( ∼ 10 -6 pc for Sgr A*) to the cosmic web ( ∼ 10 8 pc). Hence BH growth, feedback, and dynamics cannot be modelled ab-initio and need to be included as 'subgrid' processes in cosmological simulations. \nHere, we model the BH growth based on the torque-based accretion model (Hopkins & Quataert 2011), which allows for efficient BH growth in the dwarf regime because it does not suffer from the strong BH mass dependence inherent to the widely used Bondi model (also see Anglés-Alcázar et al. 2013, 2015; Koudmani et al. 2022; Wellons et al. 2023). This leads to relatively high BH masses compared to the extrapolated scaling relations - although we emphasise again that this comparison should be interpreted with caution because we would expect a flattening of the scaling relations in this mass regime (Greene et al. 2020). Nevertheless, from an accretion perspective, our simulations may be exploring an upper limit on the impact of BHs in the dwarf regime. \nThe inflow rates provided by the torque-based model are then coupled to a gas reservoir representing the accretion disc, with the depletion time of this reservoir set by the thin 𝛼 -disc model. However, there are a few caveats to this approach. Firstly, the 𝛼 -disc is generally only applicable in the radiatively efficient accretion regime. \nSecondly, this approach only tracks the mass flow rates through the disc and does not follow angular momentum flow rates. This in turn means that the BH spin evolution cannot be tracked self-consistently; therefore we have to assume constant efficiencies for the disc and jet luminosities, which otherwise could be directly inferred from the disc and BH properties (see e.g. Beckmann et al. 2019; Sala et al. 2021; Talbot et al. 2021, 2022, 2024; Huško et al. 2022, 2024; Koudmani et al. 2024). \nFor the AGN feedback, we only explore one set of parameters for the three AGN channels (mechanical winds, radiation and CRs) that allows us to match the main observed characteristics of nearby dwarf galaxies; see Fig. 2. However, as discussed in Wellons et al. (2023), there are several other 'plausible' AGN feedback set-ups within the /f.pc/i.pc/r.pc/e.pc model, leaving a large parameter space to be explored. In particular, we assume collimated injection for winds and radiation; however, the cusp versus core problem may be sensitive to alternative injection geometries, as discussed in Zhang et al. (2024) for the case of stellar feedback. This may allow us to also have a direct impact on dwarf core formation at lower halo masses, as in the m10zCR+AGN set-up, without compromising in terms of producing realistic stellar properties. \nThe final caveat in our model is that we do not have sufficient resolution to self-consistently model BH dynamical friction. Hence we employ a drift force to model the unresolved dynamical friction, which generally ensures that the BHs remain close to the centre of the halo. However, in dwarf galaxies, off-centre BHs may actually be physical (see e.g. Pfister et al. 2019; Bellovary et al. 2021; Ma et al. 2021) and in particular for massive BH binaries, the BHs orbiting around the centre could be an important contribution to core formation via core scouring. This has mainly been investigated in the context of massive elliptical galaxies (see e.g. Rantala et al. 2018, 2019), however, these mechanisms may also translate to the dwarf regime and, in turn, the central dark matter densities of dwarfs are expected to affect binary shrinking times (see Tamfal et al. 2018). To also account for the dynamical effects of massive BHs on core formation in future work, it will be important to include the unresolved effects of dynamical friction as accurately as possible following dynamical friction estimators based on high-resolution N-body simulations (e.g. Ma et al. 2023; Genina et al. 2024; Partmann et al. 2024).", '5 CONCLUSIONS': "In this work, we have investigated the impact of feedback from massive BHs on cusp-to-core transformations and the associated 'diversity of rotation curves problem' in dwarf galaxies with a new suite of high-resolution cosmological zoom-in simulations based on the /f.pc/i.pc/r.pc/e.pc-3 galaxy formation model. Our suite includes three different dwarf haloes spanning a range of masses and environments within the classical dwarf regime of 8 × 10 9 M ⊙ < 𝑀 halo < 4 × 10 10 M ⊙ . For each of these haloes, we investigate simulations with and without AGN as well as with and without CRs yielding four physics variations for each of the three haloes. We note that these different set-ups may represent different outcomes of the same underlying galaxy formation physics under different conditions, with the efficiency of both AGN and CRs expected to be highly variable in dwarf galaxies. Our main conclusions are as follows: \n- (i) AGNmaydrive core formation directly as an additional source of feedback dynamically heating the central dark matter distribution; see the m10zCR+AGN set-up. \n- (ii) AGNmayalsoenhance core formation indirectly by suppressing high-redshift star formation and shifting SN activity to later times, which favours core formation at 𝑧 = 0; see e.g. the m10qAGN and m10zAGN set-ups.\n- (iii) CRs may also indirectly affect cusp-to-core transformations by regulating star formation histories - for most of our simulations, this leads to suppressed star formation and therefore suppressed core formation; see e.g. the m10qCR and m10yCR set-ups.\n- (iv) Our simulation suite is in good agreement with observed dark matter central density distributions, broadly following the trends from the nearby dwarfs in Read et al. (2019). In particular, cores are generally more pronounced for higher stellar-to-halo mass ratios, 𝑀 stellar / 𝑀 vir . The only exception to this trend comes from the AGN-driven core in the m10zCR+AGN set-up, which has a strongly suppressed 𝑀 stellar / 𝑀 vir ratio and a cored profile. This parameter space is a key target for observational searches for cores induced by AGN feedback.\n- (v) BH feedback and CRs create a variety of cusps and cores in circular velocity profiles, with correlations between rotation curve shapes and baryonic influence that align with observations, potentially helping to resolve the diversity of rotation curves problem in dwarf galaxies. \nOverall, our findings suggest that AGN in dwarf galaxies can significantly impact central dark matter densities. BHs influence dwarf galaxy rotation curves through two key mechanisms: directly driving core formation via AGN feedback and indirectly regulating core formation by altering star formation histories and SN activity. CRs further contribute to this regulation by enhancing AGN feedback and suppressing star formation, thereby influencing core formation. The combined effects of AGN and CRs lead to a spectrum of central dark matter densities, reflecting the varying levels of AGN activity, star formation, CR influence, and their non-linear interaction. Due to the bursty star formation and accretion histories in dwarfs, AGN feedback is likely to occur in a highly stochastic manner; this may explain the wide range of dark matter central densities inferred from dwarf observations at fixed halo mass. Linking surveys of dwarf rotation curves with searches for AGN signatures will be crucial for future observations to elucidate the impact of BHs on dark matter profiles and to determine whether the diversity of rotation curve problem may be explained by baryonic processes.", 'ACKNOWLEDGEMENTS': "The authors are grateful for helpful discussions with Vasily Belokurov, Martin Bourne, Jenny Greene, Vid Irsic and Sergio MartinAlvarez. The authors would also like to thank Isabel Santos-Santos for providing the observational rotation curve data. The simulations presented in this work were run on the Flatiron Institute's research computing facilities (the Iron compute cluster), supported by the Simons Foundation. SK has been supported by a Flatiron Research Fellowship and a Junior Research Fellowship from St Catharine's College, Cambridge. The Flatiron Institute is supported by the Simons Foundation. DAA acknowledges support by NSF grant AST-2108944, NASA grant ATP23-0156, STScI grants JWSTGO-01712.009-A and JWST-AR-04357.001-A, Simons Foundation Award CCA-1018464, and Cottrell Scholar Award CS-CSA-2023028 by the Research Corporation for Science Advancement. Support for ISS was provided by NSF Collaborative Research Grant 2108318. SW received support from the NASA RIA grant 80NSSC24K0838.", 'DATA AVAILABILITY': 'The data underlying this article will be shared on reasonable request to the corresponding author.', 'REFERENCES': 'This paper has been typeset from a T E X/L A T E X file prepared by the author.'}
2023arXiv230607320L	The James Webb Space Telescope JWST is revolutionizing our knowledge of zgt5 galaxies and their actively accreting black holes. Using the JWST Cycle 1 Treasury program Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization UNCOVER in the lensing field Abell 2744 we report the identification of a sample of little red dots at 3 lt zrmphot lt 7 that likely contain highlyreddened accreting supermassive black holes. Using a NIRCamonly selection to F444Wlt27.7 mag we find 26 sources over the sim45 arcmin2 field that are blue in F115WF200Wsim0 or betarm UVsim2.0 for flambda propto lambdabeta red in F200WF444W 14 betarm opt sim 2.0 and are dominated by a pointsource like central component. Of the 20 sources with deep ALMA 1.2mm coverage none are detected individually or in a stack. For the majority of the sample SED fits to the JWSTALMA observations prefer models with hot dust rather than obscured starformation to reproduce the red NIRCam colors and ALMA 1.2mm nondetections. While compact dusty star formation can not be ruled out the combination of extremely small sizes langle re rangleapprox50 pc after correction for magnification red restframe optical slopes and hot dust can by explained by reddened broadline active galactic nuclei AGNs. Our targets have faint Mrm 1450 approx 14 rm to 18 mag but inferred bolometric luminosities of Lrm bol 10431046 ergs reflecting their obscured nature. If the candidates are confirmed as AGNs with upcoming UNCOVER spectroscopy then we have found an abundant population of reddened luminous AGN that are at least ten times more numerous than UVluminous AGN at the same intrinsic bolometric luminosity.	2023-06-01T00:00:00Z	['arXiv:2306.07320', '10.48550/arXiv.2306.07320', '2023arXiv230607320L']	['Astrophysics - Astrophysics of Galaxies']	UNCOVER Candidate Red Active Galactic Nuclei at 3ltzlt7 with JWST and ALMA	2,023	192	0.68	['EPRINT_HTML', 'EPRINT_PDF']	131	https://arxiv.org/pdf/2306.07320.pdf	{'UNCOVER: Candidate Red Active Galactic Nuclei at 3 < z < 7 with JWST and ALMA': "Ivo Labbe, 1 Jenny E. Greene, 2 Rachel Bezanson, 3 Seiji Fujimoto, 4, ∗ Lukas J. Furtak, 5 Andy D. Goulding, 2 Jorryt Matthee, 6 Rohan P. Naidu, 7, 8, † Pascal A. Oesch, 9, 10 Hakim Atek, 11 Gabriel Brammer, 10 Iryna Chemerynska, 11 Dan Coe, 12, 13, 14 Sam E. Cutler, 15 Pratika Dayal, 16 Robert Feldmann, 17 Marijn Franx, 18 Karl Glazebrook, 1 Joel Leja, 19, 20, 21 Michael Maseda, 22 Danilo Marchesini, 23 Themiya Nanayakkara, 1 Erica J. Nelson, 24 Richard Pan, 23 Casey Papovich, 25, 26 Sedona H. Price, 3 Katherine A. Suess, 27, 28 Bingjie Wang ( 王冰洁 ), 19, 20, 21 John R. Weaver, 15 Katherine E. Whitaker, 15, 10 Christina C. Williams, 29, 30 and Adi Zitrin 5 \n1 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Melbourne, VIC 3122, Australia 2 Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544 3 Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA 4 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 5 Physics Department, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 8410501, Israel 6 Department of Physics, ET+ H Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland 7 Center for Astrophysics \| Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 8 MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Ave., Cambridge, MA 02139, USA 9 Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland 10 Cosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen, Jagtvej 128, København N, DK-2200, Denmark 11 Institut d'Astrophysique de Paris, CNRS, Sorbonne Universit'e, 98bis Boulevard Arago, 75014, Paris, France 12 Space Telescope Science Institute (STScI), 3700 San Martin Drive, Baltimore, MD 21218, USA 13 Association of Universities for Research in Astronomy (AURA), Inc. for the European Space Agency (ESA) 14 Center for Astrophysical Sciences, Department of Physics and Astronomy, The Johns Hopkins University, 3400 N Charles St. Baltimore, MD 21218, USA 15 Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA 16 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands 17 Institute for Computational Science, University of Zurich, Winterhurerstrasse 190, CH-8006 Zurich, Switzerland 18 Leiden Observatory, Leiden University, P.O.Box 9513, NL-2300 AA Leiden, The Netherlands 19 Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA 20 Institute for Computational & Data Sciences, The Pennsylvania State University, University Park, PA 16802, USA 21 Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA 22 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706 USA 23 Department of Physics and Astronomy, Tufts University, 574 Boston Ave., Medford, MA 02155, USA 24 Department for Astrophysical and Planetary Science, University of Colorado, Boulder, CO 80309, USA 25 Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA \n- 26 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA \n27 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA 28 Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA 29 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Avenue, Tucson, AZ 85719, USA 30 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA \n(Dated: June 2023)", 'ABSTRACT': 'The James Webb Space Telescope (JWST) is revolutionizing our knowledge of z > 5 galaxies and their actively accreting black holes. Using the JWST Cycle 1 Treasury program Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization (UNCOVER) in the lensing field Abell 2744, we report the identification of a sample of little red dots at 3 < z phot < 7 that likely contain highly-reddened accreting supermassive black holes. Using a NIRCam-only selection to F444W < 27 . 7 mag, we find 26 sources over the ∼ 45 arcmin 2 field that are blue in F115W -F200W ∼ 0 (or β UV ∼ -2 . 0 for f λ ∝ λ β ), red in F200W -F444W = 1 -4 ( β opt ∼ +2 . 0), and are dominated by a point-source like central component. Of the 20 sources with deep ALMA 1.2-mm coverage, none are \ndetected individually or in a stack. For the majority of the sample, SED fits to the JWST+ALMA observations prefer models with hot dust rather than obscured star-formation to reproduce the red NIRCam colors and ALMA 1.2 mm non-detections. While compact dusty star formation can not be ruled out, the combination of extremely small sizes ( ⟨ r e ⟩ ≈ 50 pc after correction for magnification), red rest-frame optical slopes, and hot dust can be explained by reddened broad-line active galactic nuclei (AGNs). Our targets have faint M 1450 ≈ -14 to -18 mag but inferred bolometric luminosities of L bol = 10 43 -10 46 erg/s, reflecting their obscured nature. If the candidates are confirmed as AGNs with upcoming UNCOVER spectroscopy, then we have found an abundant population of reddened luminous AGN that are at least ten times more numerous than UV-luminous AGN at the same intrinsic bolometric luminosity. \nKeywords: Active galactic nuclei (16), High-redshift galaxies (734), Intermediate-mass black holes (816), Early universe (435)', '1. INTRODUCTION': "Accreting supermassive black holes (BHs) can be extremely luminous beacons of galaxy formation in the early universe, and over the past twenty years we have been able to detect rare and massive ( ∼ 10 9 M ⊙ ) accreting BHs up to z ∼ 7 . 5 (i.e. UV-luminous quasars Mortlock et al. 2011; Ba˜nados et al. 2018; Fan et al. 2019; Wang et al. 2021; Harikane et al. 2022; Fan et al. 2022). These rare objects require very wide-area red imaging to find, and have number densities many orders of magnitude lower than typical supermassive BH populations today. Making such monsters from scratch at such early times is a long-standing challenge that has intrigued astronomers for decades (see reviews by Inayoshi et al. 2020; Greene et al. 2020). If we could constrain the luminosity function of high-redshift accreting BHs down to low luminosity, that would provide crucial new constraints on when supermassive BHs form and grow (e.g., Ricarte & Natarajan 2018; Haiman et al. 2019; Dayal et al. 2019; Volonteri et al. 2021; Natarajan 2021; Matsuoka et al. 2022; Haidar et al. 2022; Zhang et al. 2023). \nThe faint-end slope of the quasar luminosity function at z ≳ 6 has important implications for the cosmic sources of reionization (e.g., Glikman et al. 2011; Madau & Haardt 2015; Matsuoka et al. 2018; Giallongo et al. 2019; Dayal et al. 2020), for the number density of low-mass black holes (e.g., Matsuoka et al. 2018; Shen et al. 2019; Onoue et al. 2019), and for scaling relations between black holes and their host galaxies (e.g., Piana et al. 2021; Ding et al. 2022). Deep optical surveys like the Hyper Suprime-Cam Survey have made progress quantifying the luminosity function down to luminosities of M 1450 ≈ -22 mag or L Bol ≈ 10 46 erg/s at z > 5 \n(Matsuoka et al. 2018; Akiyama et al. 2018; Matsuoka et al. 2022). X-ray surveys reach a bit further, to bolometric luminosities of L bol ∼ 10 45 erg/s at z ≈ 6 (e.g., Nanni et al. 2017; Vito et al. 2019; Shen et al. 2020; Wang et al. 2021; Wolf et al. 2022). Thus current limits reach the Eddington limit for M BH ∼ 10 8 M ⊙ , but are not deep enough to probe expected AGN from most drop-out galaxies at these redshifts (Volonteri & Reines 2016). \nIn addition to selection via blue color (e.g., Richards et al. 2002), AGN can be selected in many complementary ways. X-rays are less sensitive to obscuration than the UV, and have provided the most complete census of AGN to date at all redshifts (e.g., Ueda et al. 2014; Aird et al. 2018; Vito et al. 2019; Giallongo et al. 2019). Mid-infrared selection, likewise, should select sources independent of viewing angle, and has been extremely fruitful in recent years thanks to the combination of Spitzer and WISE (e.g., Stern et al. 2005; Goulding et al. 2014; Assef et al. 2018; Zou et al. 2022; Ishikawa et al. 2023). At the very luminous and rare end are the Hot Dust Obscured Galaxies (HotDOGS Dey et al. 2008; Tsai et al. 2015; Wu et al. 2018) and Extremely Red Quasars (ERQs; Ross et al. 2015; Zakamska et al. 2016; Hamann et al. 2017). Finally, there are a class of reddened broad-line objects that are challenging to select photometrically (e.g., Croom et al. 2001), but have been identified as an important sub-population particularly at the luminous end (e.g., Glikman et al. 2012; Banerji et al. 2015). For a nice review, see Hickox & Alexander (2018). \nIn the era of JWST +NIRCAM, we have an amazing new resource for selecting active galaxies. Because of the combination of resolution and sensitivity in the near-infrared bands, we can select high redshift compact sources that are UV-faint but emerging in the rest-frame optical as possible AGN candidates (although there were hints from HST and ALMA: Morishita et al. 2020; Fuji- \n. \n0 \n± \n0 \n.X \n9 \nIRCam photometry of the Furtak et al. (2022a) triply-lensed AGN-candidate. site broad-line AGN fits can both explain the NIRCam photometry well. For the a low-mass A V = 0 component and a high-mass reddened component is needed. ombination of a luminous L bol = 10 46 erg/s reddened component is needed, in V = 0 component emerging as scattered AGN light. Figure 1. Schematic model (right) and photometric fit (left) for compact red sources like CEERS 3210. The model consists of a reddened, but not completely obscured, broad-line AGN template ( red ) representing a direct view of the accretion disk and Broad Line Region, a small contribution from the same AGN template without reddening ( blue ) that we posit represents scattered light, and (3) an unreddened Narrow Line Region with emission line strengths coupled to the broad Hα of the red component following typical correlations for broad-line AGN (Stern & Laor 2012a,b). This fit not only provides a decent fit to the photometry, but predicts both broad and narrow line strengths very similar to those observed with NIRSpec. \n<!-- image --> \nmoto et al. 2022; Endsley et al. 2023). NIRCam imaging has already revealed a number of intriguing photometric prospects for low luminosity accreting BHs (e.g., Onoue et al. 2023; Ono et al. 2022; Furtak et al. 2022a; Endsley et al. 2022; Larson et al. 2023). \ne NIRCam + ALMA SEDs. In each case three types of models are fit: a) stellar nts (light blue), b) stellar population only with ALMA (blue), 3) AGN only with odel is composed of three elements: 1) unreddened star forming component, 2) escent component, each with independent ages. The best fit overshoots the ALMA fits including ALMA are poor despite the large amount of freedom and can not ALMA photometry. The AGN-only fits, including a hot dust component of the th the faint ALMA fluxes, providing a satisfactory explanation of the full SED. ly-lensed source presented in (Furtak et al. 2022a) (see also Figure 7). Note the ), much bluer than expected for normal stellar population models. luded for clarity for 23778. XXX add zoom-in panel on NIRCam showing poor fit Stunning spectroscopic confirmation of the AGN nature of these sources is now also becoming available (Kocevski et al. 2023; Harikane et al. 2023; Oesch et al. 2023; Ubler et al. 2023; Barro et al. 2023). Onoue et al. (2023) identified a point-like source as a likely z = 5 . 2 AGN from early Cosmic Evolution Early Release Science (CEERS) Survey data, subsequently found to be a M BH ∼ 10 7 M ⊙ black hole radiating at ∼ 10% of its Eddington luminosity at z > 5 from NIRSpec follow-up (Kocevski et al. 2023). Even more intriguing is the source CEERS 3210. Like CEERS 1670, it has a very compact morphology. However, the spectral energy distribution (SED) changes from blue in NIRCam short-wavelength (SW), with F115W-F200W ∼ 0, to red in NIRCam long-wavelength (LW), with F277WF444W ∼ 2 . 0. The object is so red that the rise resembles a Balmer break at z > 7 (Labb'e et al. 2023). NIRSpec observations also revealed the source to be a broad-line AGN at z = 5 . 6 (Kocevski et al. 2023). Harikane et al. (2023) and Barro et al. (2023) present an even larger sample of compact red sources in CEERS, although not all show concrete evidence for broad H α and the nature of these is not always clear. Finally, Matthee et al. (2023, submitted) have identified a sample of 'Little Red Dots' with broad H α in JWST grism data (Oesch et al. 2023). \nThe central goal of this paper is to photometrically identify more compact red sources with \nJWST /NIRCam imaging, and then use spectral synthesis models to examine the source of the red continuum and UV excess, be it dusty star formation or reddened light from an AGN 1. We will propose that the red restframe optical continuum likely arises from a reddened broad-line AGN, while the flat blue continuum comes either from an additional scattered component, or some star formation in the host galaxy. Objects like this have been seen very locally (e.g., Mrk 231, Veilleux et al. 2016), and at a range of redshifts z < 3 (e.g., Assef et al. 2020; Pan et al. 2021; Noboriguchi et al. 2022; Glikman et al. 2023), but it is hard to construct complete samples of reddened broad-line AGN (e.g., Glikman et al. 2012). We should note that there are many possible origins for the UV light, including star-formation in the host galaxy or emission from an outflow (Veilleux et al. 2016), but what matters most for our discussion is the origin of the red rest-frame optical continuum. \nAn If we could systematically identify this low-luminosity population from deep NIRCam imaging, it would provide critical new insight into an important population of partially obscured AGN that may constitute a significant fraction of high-redshift AGN. We present a preliminary sample of 26 candidates in the ∼ 45 arcmin 2 Abell 2744 field. Thanks to the large lensing area afforded by Abell 2744, the observations can reach up several magnitudes deeper than in blank fields. \nThroughout, we assume a concordance cosmology with H 0 =70, Ω Λ = 0 . 7, Ω M = 0 . 3 (Hinshaw et al. 2013).", '2. SEARCH FOR RED COMPACT SOURCES': "The UNCOVER data are very well-suited to a search for more compact red sources. We have nearly complete \nFigure 2. Color-color (left) and color-compactness (right) selections. These panels show all the sources selected with the 'red2' color and compactness criteria (red circles), which is specifically designed to select z > 6 AGN similar to those found in (Furtak et al. 2022a; Kocevski et al. 2023). Sources selected to be PSF-dominated are highlighted with black dots. In greyscale, we show the entire UNCOVER catalog, with numbers of objects indicated by the color-bar. The compact red sources are clear outliers in color-color-compactness space. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nF277 \nF444 \nwavelength coverage from 1 -4 µ m to uniform depth, along with deep ALMA data to constrain the presence of cold dust (Fujimoto et al. 2023).", '2.1. The UNCOVER Survey': "Our search is performed using the JWST Cycle 1 Treasury program Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization (UNCOVER; Bezanson et al. 2022). UNCOVER imaging was completed in November 2022, comprising ultradeep (29 -30 AB mag) imaging over 45 arcmin 2 in the galaxy cluster Abell 2744. This well-studied Frontier Field cluster (Lotz et al. 2017) at z = 0 . 308 has one of the largest high-magnification areas of known clusters, and thus made an excellent target for deep (4-6 hours per filter) imaging across seven NIRCam filters (F115W, F150W, F200W, F277W, F356W, F410M, F444W). The nominal depth of ∼ 30 mag can reach sources as faint as 31.5 mag with the help of magnification. Photometric catalogs (Weaver et al. 2023) including existing HST data have been made available to the public, and the lens model is also publicly available (Furtak et al. 2022b). The initial selection of objects is based on the UNCOVER Data Release DR1 images and catalogs (2023). \nFujimoto et al. (2023) present deep ALMA 1.2 mm continuum imaging in Abell 2744. A wider, deeper 1.2 mm map of the full NIRCam UNCOVER area was newly obtained in Cycle 9 (#2022.1.00073.S; S. Fujimoto in prep), reaching continuum r.m.s. sensitivity of 33 µJy in the deepest areas. Prior-based photometry is extracted for all sources by measuring the ALMA flux in the natural resolution map (beam ≈ 0 . 7 -0 . 8 '' ) at the NIRCam positions. \nIn late July 2023, deep (2.7-19 hour) spectroscopic follow-up with the NIRSpec/PRISM will constitute the second phase of UNCOVER, but there are already many \nTable 1. Sample Selection \nNote -Asimple flow chart of our selection. In the first step, we apply the combined color and compactness cut ( § 2.2) to select the Parent sample. Then we fit all sources in two dimensions and keep only those well-fit by a point source, which forms the Main sample ( § 2.4). Finally, we generate an SED sample from those galaxies ( § 3.3). \nexciting things to find within the imaging (e.g., Furtak et al. 2023; Atek et al. 2023).", '2.2. Color and Morphology Selection': "Inspired by the compact red broad-line AGN in CEERS, we devise a color+morphology search through the UNCOVER catalog for systems with both a moderate blue continuum and a rising red continuum, with the continuum break occurring between F150W and F356W for the typical source in our sample. This break will move into F444W for z ≈ 8, limiting the redshift range of the sample. The spirit of the selection is to find objects with a red continuum slope in the rest-frame optical, which also shows significant emission in the rest-UV. The color selection requires red colors in adjacent pairs of NIRCam bands to avoid selecting SEDs have blue continua throughout but with one band simply boosted by strong emission lines. We do not use photometric redshift or stellar population models, to avoid being limited by the models or templates used. \nThe initial Parent sample is selected with the following color selection. Starting with sources that are \nC \nwell-detected in the F444W band, with SNR (444) > 14 & m 444 < 27 . 7 mag, we select sources that are ( red1 \| red2 ) & compact , where \nred1 = (115 -150 < 0 . 8) (200 -277 > 0 . 7) (200 -356 > 1 . 0) \nOr \nAnd: \nred2 = (150 -200 < 0 . 8) \n(277 -356 > 0 . 7) \n(277 -444 > 1 . 0) \ncompact = f 444 (0 . 4 '' ) /f 444 (0 . 2 '' ) < 1 . 7 \n. \nWe find that extended dusty galaxies and inclined spirals over a wide range of redshift are contaminants when applying the color cuts alone. While such galaxies are certainly interesting (e.g., Nelson et al. 2022; Barrufet et al. 2023), we explicitly incorporate a compactness criterion, such that extended dusty galaxies are excluded and compact sources remain. The compactness criterion selects sources with a flux ratio up to 30% higher than objects on the stellar locus. The precise limit does not matter and is mostly to reduce the sample size, as we will fit 2-D models to the Parent sample in § 2.4 to search for sources that are likely dominated by a point source. \nThe UNCOVER survey is located in a gravitational lensing field, with a high density of bright galaxies with extended wings. This can cause the photometric aperture in the UNCOVER DR1 catalog to pick up light from nearby foreground sources and bias both the colors and the total magnitudes (the latter are based on a Kron-like autoscaling aperture). This is particularly an issue for the highest magnification sources (see e.g., Furtak et al. 2023). To reduce the impact of contamination, we extract PSF-photometry by fitting 10-20 nearby isolated, bright, unsaturated stars to each source, fitting the position and flux in the F444W image, and then rescaling the flux in the other filters. This PSF photometry produces excellent agreement with the aperture photometry for isolated sources, with the benefit of strongly reduced contamination, especially in the bluer NIRCam SW filters, for sources with bright neighbors. For sources with additional complex extended structure the PSF-photometry is also more reflective of the nuclear component. It is clear from these initial fits that many sources are indistinguishable from a PSF (see e.g., appendix Figure 14). \nFigure 3. The F444W magnitude and photometric redshift distributions of our sample, colored by the log of the magnification. Most objects in the sample have magnification ∼ 1 . 5, but there are three strongly lensed systems, one of which Furtak et al. (2023) has three images and so appears three times in this figure. We show the Parent sample, selected as ( red 1 \| red 2) & compact , as well as the 'Main' sample. The Main sample were selected to be point-source dominated, as described in § 2.4. \n<!-- image --> \nFrom the initial color and compactness criteria, we select the Parent sample of 40 AGN candidates out of a catalog of 50,000 (see Table 1). Next we will fit the sources with a point-source model to explore whether there is any evidence for extended light ( § 2.4), and then examine the SEDs in detail ( § 3.1). The number of targets at each step is summarized in Table 1. The basic sample properties are summarized in Table 2.", '2.3. Complementary Red Extended Sample': 'If we select all galaxies that satisfy ( red1 \| red2 ) but have compact > 1 . 7, we have a sample of 16 galaxies. These galaxies span a wide range of redshift 2 < z < 6 and appear typically powered by dusty star-formation taking place in an extended region, and they are nearly all detected in ALMA. Below, in § 3.1, we will contrast their far-infrared-to-mm SEDs with the compact red sources as one additional argument that we are likely identifying sources with hotter dust than expected from dusty star formation.', "2.4. Joint PSF+S'ersic fits": "To refine the compactness criteria, we perform a series of two-dimensional fits to the light profile of each galaxy in F444W using Galfit (Peng et al. 2002, 2010). The main purpose is to determine if a source is unresolved, or alternatively dominated by a point-source component, taking into account variations in the PSF with position \nTable 2. Sample \nNote -Table of objects that satisfy ( red 1 \| red 2) and compact . a are the three images of the lensed compact red object presented in Furtak et al. (2023). Column (1): UNCOVER ID. Column (2): R.A. Column (3): Dec. Column (4): Photometric redshift z phot from the AGN-only best-fit. Column (5): Magnification ( µ ). Column (6): Flag for galaxies falling in the Main sample, consistent with being point sources ( § 2.4). Column (7): Flag for galaxies in the SED sample, with F444W emission dominated by AGN light ( § 3.1). Column (8): F444W mag. Column (9): F277W-F356W color (mag). Column (10): F277W-F444W color (mag). Column (11): r e (milliarcsec) from the single-component S'ersic fit. Column (12): F444W magnitude from the S'ersic part of the two-component PSF+S'ersic fit. Column (13): F44W magnitude of the PSF component of the PSF+S'ersic fit. The Main sample are selected to have F444W Sers > F444W PSF . \nand magnitude. We choose to model the sizes in the F444W filter. While the resolution is higher in NIRCam SW, the sources have much higher SNR ratio in F444W owing to the very red colors. In addition, the size at the longest wavelength is less affected by dust and thus closer to the true distribution of luminosity or stellar mass. Bright, isolated, unsaturated stars are used as PSF models, as it is known that the simulated PSFs from WebbPSF (Perrin et al. 2014) are too narrow compared to empirical PSFs Ding et al. (e.g., 2022); Weaver et al. (e.g., 2023). Stars are selected based on a cut in flux ratio between D = 0 . 2' and D = 0 . 4' apertures and an inspection of SEDs to retain sources with stellar SEDs. There are 10-20 nearby suitable PSF stars for each source. Before fitting, a local background is subtracted at scales of 40 pixels (1 . 6 '' ) by growing all segmentation maps by 2 pixels and subtracting the median of the background pixels. \nWe start with single-component S'ersic fits. We fit each galaxy using one of the nearby stars as a PSF model. Uncertainties are determined by taking the bestfit model, placing it in the residual map of another randomly drawn source in the sample, and re-fitting the source with a randomly drawn nearby PSF star. This process is repeated 200 times using unique combinations of PSFs and residuals, reflecting both systematic and random uncertainties. To determine whether the sources are consistent with point sources, the same process was performed on stars. For each source a random nearby star was drawn, scaled to the same SNR, placed in the residual map, and then modeled with a single S'ersic model using another nearby star as the PSF. This process was repeated 200 times as well, using different combinations of stars, PSFs, and residual maps. We then examine the median and upper 90% size of the fits as a function of their F444W magnitude and signal-tonoise ratio (SNR; Figure 5). Any source that is smaller than the 90% percentile of the stellar fits is considered consistent with a PSF and therefore unresolved. This is the case for 17 out of the original 40 sources. \nSome, however, show more complex morphology with extended structure in addition to a bright point-like source, as might be expected for an AGN embedded in a host galaxy. We then perform a two-component PSF+S'ersic fit. From the two-component fits, we identify an additional 9 sources that are dominated by a PSF component, where more than half the light comes from the point-source component. This leaves a total of 26 sources that the two-dimensional fits deem to be point-source dominated. Of the 40 in the Parent sample, 26 targets are PSF-dominated based on our joint PSF+S'ersic fits - the 'Main' sample. See Figure 6 for \nimages that demonstrate the compact nature of these red sources. \nWe can translate our two-dimensional fits into size limits for the sample. At the magnitude limit F444W=27 . 5 AB of our sample, we can measure sizes to a bit under one pixel (Figure 5), or R e ≲ 0 . 03 '' at 90% confidence. At the bright end sizes can be measured to better than R e ≈ 0 . 01 '' . At the median distance of the objects redshift z ∼ 5 these limits correspond to ∼ 140 pc and ∼ 40 pc respectively, where a simple correction by µ -1 / 2 is applied to correct for lensing. The median size of the sample is ⟨ r e ⟩ = 54 +33 -10 pc, with the uncertainties derived from bootstrap resampling. We note that the majority of the sample are only moderately lensed - 24/26 have µ < 3 and are far away from caustics with low shear and modest relative uncertainties on the magnification (Figure 3). In the two cases of highly magnified sources (ID=8296, 21860), the size limits are considerably tighter (Furtak et al. 2023) but more detailed modeling taking into account the lens model uncertainties and shear is required. \nIt is tempting to assume that these small sizes point to an AGN origin for the emission. However, very compact star-forming galaxies are known to exist at lower redshift (e.g., Geach et al. 2018) and we do not have strong constraints on the mass-size relation for massive star-forming galaxies at 4 < z < 7. Some very compact massive galaxies have been found in early JWST data (e.g., Carnall et al. 2023; Robertson et al. 2023; Morishita & Stiavelli 2023; Williams et al. 2023), and extremely small sizes < R e > ≈ 150 pc are reported for candidate massive galaxies 7 < z < 9 (Baggen et al. 2023). Apart from a strong triply-lensed system with size limits < 30 pc (Furtak et al. 2023) (also in this sample, with ID=8296, 9992, 10712), the morphologies alone cannot distinguish between a stellar and accretion origin for the bulk of the sources. In the following section, we will present SED fits, and further isolate the 'SED' sample of 17 objects that are likely to be AGN-dominated based modeling of the spectral energy distributions (SEDs).", '3. SPECTRAL ENERGY DISTRIBUTIONS AND NUMBER DENSITIES OF THE COMPACT RED SOURCES': "In this section, we model the spectral properties of the new sample of compact red sources. We fit the seven NIRCam UNCOVER bands, along with HST photometry and ALMA 1.2 mm data where available. By selection, the sources have blue continua in the NIRCam SW or rest-frame UV, typically β UV ≈ -2, and rising red slopes in NIRCam LW filters corresponding to the rest- \n<!-- image --> \nF444W \nmodel \nresidual \nF444W \nmodel \nresidual \n<!-- image --> \n<!-- image --> \nFigure 4. Examples of Sersic profile fits to the F444W image. From left to right: NIRCam SW color image (based on F115W, F150W, F200W), NIRCam LW color image (F277W,F356W,F444W), F444W image, GALFIT best-fit model, residual, and a star scaled to the same flux placed in the residual map. The top row shows an unresolved source modeled with a single Sersic component that is indistinguishable from a star, the middle row shows a source that is better modeled by a two-component PSF+Sersic model, but is dominated by the PSF component. The bottom row shows an extended source without evidence for a bright point source. These are removed from the sample. \n<!-- image --> \nframe optical, with β opt ≈ 2 , f λ ∝ λ β opt (e.g., Figure 1). There is a clear inflection or break between F150W and F356W depending both on the intrinsic A V and on the redshift. Such composite SEDs can be challenging to model with any single galaxy or AGN model. Because of their composite color components, we model the SEDs as composites as well, with either multiple galaxy components (young, dusty star-forming, and evolved stellar populations), or with AGN components included. \nIt is useful to examine one example in a bit more detail (and see also discussion in Barro et al. 2023). Furtak et al. (2022a) presented a highly magnified compact red source with three images in the UNCOVER field. Because of the high magnification, this source has a particularly stringent limit on its size of r e ∼ 35 pc, making stellar origins for the emission seem even less likely than the typical source in our sample. Furtak et al. present three possible galaxy models to fit the SED. Prospector -β (Wang et al. 2023) prefers to fit the red color with a massive evolved galaxy and strong Balmer break, but cannot simultaneously fit the UV emission. EAZY (Brammer et al. 2008) is well suited to handle different components with different reddening, but still is unable to satisfactorily fit the strong red upturn. Modeling software that includes optical/UV AGN emission also struggles to find multi-component solutions: Furtak et al. present fits from both CIGALE (Boquien et al. 2019) and Beagle+AGN (Chevallard & Charlot \n2016; Vidal-Garc'ıa et al. 2022). The latter only includes narrow emission lines from an AGN and thus attempts to reproduce the red NIRCam LW colors by extreme narrow-emission lines to boost the fluxes in the broad band filters. This seems unlikely to arise from a 35 pc radius and the need to boost the NIRCam filters by a factor × 10 leads to extreme equivalent widths > 10 , 000 ˚ A. In the context of the compact red sources in CEERS that have broad H α in their spectra at much lower equivalent widths ( ∼ 400 ˚ A), the continuum is arguably AGN-dominated, with the steep red continuum due to a reddened broad-line AGN continuum. The UV component might arise from scattered light (e.g., Assef et al. 2020; Glikman et al. 2023), patchy transmission, or star formation. That possibility is shown in the AGNonly fits in Figure 6. \nWe have thus performed custom fitting to handle our need for multiple components that each may have different reddening and AGN contributions. We will describe the general properties of the fits in this section, but refer to the Appendix for more details.", '3.1. Composite model fits': "In the AGN-only fits we model the SED with a twocomponent AGN model. Each AGN component is comprised of the Vanden Berk et al. (2001) AGN template in the optical/UV, combined with the template from Glikman et al. (2006) extending to the rest-frame near- \nstar \n<!-- image --> \nTable 3. SED Fits \nNote -Column (1): Galaxy ID. Column (2): ALMA 1.2 mm flux based on forced photometry (units of 10 µ Jy). Column (3): ALMA flux errors (10 µ Jy). Column (4): AGN fraction from the combined fit at F444W. Column (5): Stellar masses from galaxy-only fits including ALMA ( M ⊙ ). Column (6): Stellar masses from galaxy-only fits excluding ALMA ( M ⊙ ). Column (7): Bolometric luminosity (erg/s) inferred from AGN-only fits, magnification-corrected and assuming that L bol = 10 L 5100 . Column (8): M 1450 (mag) magnification-corrected. Column (9): χ 2 of the best-fit AGN-only model including ALMA. Column (10): The χ 2 for the best-fit galaxy-only model, without ALMA. Column (11): The χ 2 for the best-fit galaxy-only model including ALMA. In cases with strong ALMA constraints, the χ 2 is dramatically larger due to the inclusion of the ALMA upper limit. Column (12): Flag for galaxies in the SED-selected sample. \ninfrared. In addition to the unreddened AGN, we also include a reddened version of the same template, with reddening ranging from A V = 0 . 5 -5 applied using the Calzetti (2001) reddening law. The reddened component is the primary component in terms of luminosity and must be heavily extincted to fit the red slope. The blue component is unreddened but weak, and we interpret this as scattered AGN light at a few percent of the luminosity of the reddened component. This component is generally well-fit by an unreddened AGN template, as expected for electron scattering, but we highlight some outliers at the end of this section. In some cases we may well be seeing star formation from the host; we note that in a small number of objects, the rest-UV emission does appear slightly extended (Figure 6). \nWhen our fits include the ALMA 1.2 mm point, we model the mid-to-far infrared emission with the CLUMPY torus models (Nenkova et al. 2008a,b). Details are in the Appendix (see also Conroy et al. 2009; Leja et al. 2018), but note that we fix the clump properties that determine the shape of the torus model and we fix the inclination to 40 degrees, such that we only incorporate mid-to-far infrared emission and not rest-frame UV/optical emission from the accretion disk. Also key is that we scale the infrared emission to the obscured optical luminosity, thus enforcing energy balance. More detailed constraints on the torus parameters would require longer-wavelength data. In addition, because we are using fixed empirical AGN templates, we do allow for small differences in the emission-line strengths, applying \n<!-- image --> \nFigure 5. The effective radii R e of the sample (green circles), compared to the effective radii measured on stars scaled to the same SNR, to determine if a source is resolved or consistent with being dominated by a point source. The best-fit R e of stars are shown by dark gray symbols with the 90% upper envelope based on bootstrap resampling shown in light gray. Sources are considered point-source dominated if the R e falls below the 90% envelope for stars (solid circles), or if a two-component PSF + Sersic fit indicates most of the light is in a point source (open circles). \n<!-- image --> \na Gaussian prior with a width of 0.3 dex. The goodness of fit for the AGN-only models is included in Table 3. Finally, in a minority of cases (six of the 26), the UV continuum is steeper than the AGN or star-forming galaxy template, and in these cases we additionally include a UV power-law component. Thus, we are fitting a total of eight free parameters including the redshift. The \ndetails of the fitting routine are described in Appendix A. \nThe galaxy-only fits are performed in the same spirit, but rather than AGN templates we include star-forming templates with a range of A V = 0 -5 (Calzetti 2001), and a quiescent galaxy component. When we jointly model the ALMA data, we utilize the dust emission models as implemented in FSPS (see description in Leja et al. 2017). As with the infrared emission from the AGN, we fix the parameters that set the shape of the dust emission SED since we have minimal longwavelength constraints. We also enforce energy balance between the reddened UV/optical continuum and the dust emission. \nThe photometric redshift is a free parameter in our AGN-only fits. We find decent agreement between our z phot and the catalog value derived from EAZY (Brammer et al. 2008), with a median ⟨ δz/ (1 + z ) ⟩ = 0 . 04 and scatter σz/ (1 + z ) ≈ 0 . 15. There is a minority of cases where the AGN-only fits return a redshift δz ≈ 1 -2 lower than EAZY, due to a degeneracy between a featureless red slope (in the case of a reddened AGN) and a strong Balmer/4000 ˚ A break (CEERS 3210 is one example; Labb'e et al. 2023; Kocevski et al. 2023), in particular when the optical constraints are weak. When we have deep optical photometry from HST , it can be possible to rule out the higher-z solution through the presence of a Lyman Break. Overall, the different fits yield reasonable agreement on their photometric redshifts. \nIn Figure 6 (third column) we show example galaxyonly fits. To model the SEDs typically requires an unreddened (blue) star-forming galaxy template and a star-forming but heavily dust-obscured component. In a small number of cases, there are sources that can be fit by a combination of an evolved population with a strong break and a star-forming component. Indeed, there is at least one spectroscopic example of a galaxy-AGN composite with a strong break and a broad H α line (Carnall et al. 2023), but in general our modeling does not find strong evidence of 4000 ˚ A breaks, although in principle dust-reddened galaxies with strong breaks could also fit the observed SEDs (see also Kocevski et al. 2023; Chworowsky et al. 2023; Barro et al. 2023). Rather, the very red continuum slope is better fit by dusty starformation. \nWhen we exclude the ALMA photometry, we are generally able to fit the SEDs with either composite galaxy or composite AGN models (Figure 7 left, Table 3). Figure 7 (left) illustrates that the galaxy-only fits without ALMA constraints nearly always prefer to fit the red end of the spectrum with dusty star formation, which can have a very similar slope to a reddened AGN con- \nFigure 6. Color image stamps and SED fits to the SED sample of compact red sources (see details in § 3.3). The color coding is based on the NIRCam imaging in F115W (blue), F150W (green), F200W (red) in the left column and F277W, F356W, F444W in the right column. The images convey how compact the sources are, and that there is both a prominent red and prominent blue component to the SED. The SED fits are in all cases composite. In the third columns, we show the galaxy-only fits, while in the fourth column are displayed AGN-only fits. The data points are shown in filled black, while the model photometry are in open circles. Total model is in black, while reddened and unreddened components are in blue and red respectively. The green component is an additional blue power-law that is required to fit the SEDs of 6 of the SED-selected targets. Details of the fitting are discussed in § 3.1. \n<!-- image --> \ntinuum. The difference in χ 2 between the AGN-only and galaxy-only fits are not significant in general. Barro et al. (2023) also could not clearly distinguish between dusty star-formation and reddened AGN activity from their comprehensive spectral synthesis modeling to the compact red sources in CEERS. However, as we will show next, the predicted ALMA 1.2 mm fluxes for our adopted dusty star forming models are ruled out by the ALMA data, for typical star-forming galaxy dust templates.", '3.2. ALMA Constraints': "We now perform SED fits including the ALMA photometry. The majority of the objects in the PSFselected sample (20/26) have constraints from ALMA (see Table 3). The 1.2 mm measurement falls fairly close to the peak dust temperature for a dusty starburst at z ∼ 5. Even with substantial uncertainty on the dust peak temperature due to star formation, the ALMA point is constraining. Fitting to NIRCam only, the bestfit galaxy-only SED fits prefer a dusty star-forming component which over-predicts the ALMA 1.2 mm 2 σ upper limits by a factor of a few to > 100. Inclusion of the ALMA data leads to a significantly worse fit: ALMA limits the amount of dusty star formation and a quiescent stellar population struggles to reproduce the red NIRCam colors. The χ 2 for the galaxy fit including the ALMA data is significantly worse than the galaxy-only fit excluding ALMA in all SED-selected objects, ranging from ∆ χ 2 = 6 -340. Dust-reddened AGN can also produce red NIRCam rest-frame optical colors, but are characterized by much hotter dust, peaking at an observed wavelength of 100 µ m (rest wavelength 20 µ m, e.g., Kirkpatrick et al. 2012) and at least two orders of magnitude fainter at ALMA 1.2 mm than a dusty starburst with the same rest-optical spectrum (Figure 7). \nIn Figure 8 right, we show the individual and stacked SEDs for the PSF-selected sample. The UV/optical SEDs are well-described by the dusty star-forming template from the template fits from Eazy . The ALMA points (no detections, shown are 2 σ upper limits) fall below the predicted ALMA flux by a factor of a few to > 10 per object, with the stacked 84% upper flux limit > 30 × lower. In contrast, we also consider the ALMA measurements of the complementary star-forming sample from § 2.3. These galaxies obey the same color cuts as the compact red sources but the sizes are extended rather than compact. We see that the NIRCam SEDs for both sources are very similar (by selection). However, the dusty star-forming galaxies are detected in ALMA in nearly all cases. Thus, the ALMA non-detections suggest that the compact red sources are not typical dusty \nstar-forming galaxies, despite the galaxy-only fits being well fit with dusty star-formation in the rest-frame UVoptical. \nWe should mention the following caveats to our current dust templates. We fit with a fixed dust temperature distribution, corresponding to the default settings of FSPS T d ≈ 20 K (e.g., B'ethermin et al. 2015; Jin et al. 2019). Compact star-forming galaxies at high redshift may have hotter dust temperatures (e.g., B'ethermin et al. 2020; Sommovigo et al. 2022; Mauerhofer & Dayal 2023). At these redshifts, the CMB temperature is also approaching these typical dust temperatures, and thus provides an additional source of heating (da Cunha et al. 2015; Jin et al. 2019). As a test we reran the analysis with a dust temperature of T d ∼ 40K, finding no significant changes in the results (see the appendix for more details). Unfortunately, we do not have enough longwavelength information to perform more detailed dust modeling at present.", '3.3. Best-fit Models and the SED-selected Sample': "The SED modeling including ALMA upper limits prefers the AGN explanation because we do not detect the dust peak expected from cold dust in typical dusty star-forming galaxies. As a final test, we ask the fit to decide between AGN and galaxy templates, including star-forming (reddenend and unreddened), quiescent galaxies, and reddened and unreddened AGN templates. While the components are degenerate, these fits allow us to investigate in which cases the continuum can be fit with some combination of galaxy models without invoking any AGN. In general, the sources that have ALMA constraints cannot be fit without a dominant AGN component in the red. \nBased on these galaxy+AGN fits, we define the 'SED' sample as the objects that have AGN-dominated fits at F444W. Specifically, in 17 sources, the AGN component comprises more than twice the flux of the galaxy component within the fit to the F444W band. In all but one case, the SED-selected targets are also the sources with ALMA constraints. \nFrom the reddened AGN component, we derive an implied bolometric luminosity from the dereddened 5100 ˚ A luminosity, assuming that L bol = 10 L 5100 (e.g., Richards et al. 2006; Shen et al. 2020). While imperfect, this model-dependent value represents our best guess for the intrinsic luminosity of the AGN in the context of our preferred AGN model. The inferred bolometric luminosities for the compact red sources range from L bol ≈ 10 43 -10 46 erg/s (Table 3).", '3.4. Noteworthy Objects': 'Figure 7. Three examples of SED fits to the NIRCam + ALMA SEDs. The observed photometry are shown as black dots and the 1 σ errors are shown, including for the ALMA non-detections. Three types of models are fit: a) galaxy-only, without using the ALMA constraints (light blue), b) galaxy-only, with ALMA (blue), 3) AGN-only with hot dust (red), including ALMA. The left panel shows the HST+JWST SED, while the middle panel shows the same fits including the IR-to-mm. The galaxy-only model is composed of three elements: 1) dust-free star forming component, 2) obscured star forming component, and 3) quiescent component, each with independent ages. Without ALMA the flexible stellar population model fits can reproduce the JWST SED well, preferring highly reddened star formation. These fits overshoot the ALMA measurements by orders of magnitude in these cases. When including ALMA, the stellar only models lead to poor fits. ALMA limits the amount of dusty star formation, leaving only the old stellar population to reproduce the NIRCam colors. The AGN-only fits also require large amounts of dust, but the hot dust component of the AGN is fully consistent with the faint ALMA fluxes. The middle source is the brightest of the triply-lensed source presented in (Furtak et al. 2022a). Note the extremely blue UV slope of 23778 ( β ∼ -4 . 0), bluer than expected for normal stellar population models or AGN. \n<!-- image --> \n<!-- image --> \nFigure 8. Composite rest-frame SED of the 20/26 candidate AGNs with ALMA coverage (right) versus extended sources selected with the same color criterion but without requiring compactness (left). The extended sources are dusty star-forming galaxies at comparable redshifts. Red points indicate the individual HST+JWST photometry, shifted to the rest-frame using the photometric redshift. The red model shows the best-fit FSPS-based templates from Eazy where the IR SED is the predicted cool dust emission from star formation using the Draine & Li (2007) dust emission model. The blue points are ALMA 1.2mm measurements with 2 σ upper limits indicated as downward triangles. The large blue points are the stacked ALMA fluxes. Vertical lines denote 1 σ uncertainties. The dusty SF sample has ALMA fluxes consistent with expectations for dust emission from star formation. In contrast, the compact candidate AGN sample has similar NIRCam colors, but is undetected in ALMA, with average fluxes ≳ 30 times lower than expected for dust-obscured star formation. \n<!-- image --> \nThere are a couple types of compact red sources worth calling out specifically. \nBesides the Furtak object, we identify a strongly lensed source with µ = 40 and a photometric redshift of z = 5 . 5 (Figure 9). We do not identify additional images for this object, as it lies just outside the multiply imaged region. Notably, however, the object lies in a highly sheared region. The tangential magnification for it is much higher than the radial one. Given that it appears to be a point source, this level of magnification could further tighten the constraints on its size once the redshift is secure. Also, if confirmed spectroscopically, this source would have an implied bolometric luminosity of L bol < 10 43 erg/s, making it one of the least-luminous AGN known at z > 4. It would also be a strong candidate for a black hole with a mass M BH < 10 6 M ⊙ . Black holes in this mass range are challenging to find locally, and only a handful of candidates exist at z > 0 . 3 (e.g., Schramm et al. 2013; Mezcua 2017, 2019; Halevi et al. 2019; Greene et al. 2020). \nWe also identify a handful of compact red sources that have extremely blue UV continuum slopes, with β UV ≈ -4 where f λ ∝ λ β . An example of this subset of objects is shown in Figure 7 (right; but see other sources with a green component in Figure 6). No quasar composite spectra have been observed to be this blue (e.g., Vanden Berk et al. 2001; Davis et al. 2007; Bonning et al. 2007; Ivashchenko et al. 2014; Temple et al. 2021; Saccheo et al. 2023), with typical α opt = -0 . 5 where f ν ∝ ν α opt , corresponding to β opt = -1 . 5 where f λ ∝ λ β opt . These sources are thus simultaneously remarkably red and remarkably blue. Only spectra will be able to confirm our estimated photometric redshifts and (hopefully) determine the nature of these objects. \nIt is possible that the blue continuum reflects a scattered light component. Electron scattering would not change the color of the AGN spectrum, but dust scattering could, depending on the composition and geometry of the dust (e.g., Kishimoto et al. 2001; Draine 2003; Zakamska et al. 2005). Some indirect evidence exists for dust scattering in more luminous obscured AGN (Kishimoto et al. 2001; Alexandroff et al. 2018). In terms of starlight, in theory very young and low-metallicity stars may be able to get to β UV = -4 (e.g., Schaerer 2003), in practice the observed slopes of UV-selected galaxies have been closer to β UV ≈ -2 (e.g., Bouwens et al. 2014)', '3.5. Number Densities': 'It is notoriously challenging to calculate effective volumes involving strong lensing (e.g., Zitrin et al. 2015; Bouwens et al. 2017; Atek et al. 2018; Ishigaki et al. 2018). Here, we adopt the lens model from Furtak et al. \nFigure 9. A highly magnified compact red source. If the photometric redshift of z ≈ 5 . 5 is confirmed, this source has an implied bolometric luminosity of L bol < 10 43 erg/s. This low luminosity could correspond to a very low black hole mass. As above, the photometric data are shown as black dots and the vertical lines are the ± 1 σ uncertainties, the complete model is shown in black, and the model photometry is shown as open circles. The dominant sub-components for this fit are a blue galaxy component dominating the UV light and a heavily reddened AGN component dominating the rest-frame optical light. The ALMA limit severely limits the contribution from dusty star formation. \n<!-- image --> \n(2022b) and the approach outlined in Atek et al. (2018). For each source, given the unlensed absolute magnitude, we calculate the effective volume of the survey by integrating over the area across the map with strong enough magnification to ensure we could detect that source. We do not account for varying incompleteness as a function of magnitude or photometric redshift, since the uncertainties about the intrinsic SEDs of these sources makes that modeling tricky. We do note that the sources are relatively bright ( > 14 σ ) in F444W, which mitigates the magnitude incompleteness, and their compactness makes us less sensitive to surface brightness incompleteness. We do quantify systematic uncertainties in selection by comparing the number per bin for the Main sample of 26 objects and the SED-selected sample of 17 objects. \nMore specifically, our error bars include the following components: (1) Uncertainties from counting errors calculated using Gehrels (1986); (2) Uncertainties from sample selection encapsulated by the difference in number between the main and SED-selected samples. When \nthe SED-selected sample has no objects in a bin, we plot the 3 σ upper limit on zero sources as the upper error bar in that bin; (3) for the three bins with strongly lensed sources, we conservatively recalculate the volume in that bin with the lensed source removed. \nWe do not try to capture other systematic uncertainties associated with the lens modeling. In the Hubble Frontier Fields with the HST -based lens models, there were multiple lens models, which afforded some kind of estimate of the impact of different decisions on lens model uncertainties. We do not yet have that suite of lens models based on the JWST Abell 2744 data. We note that 22/26 sources have fairly modest magnifications µ = 1 -4, and lensing errors are not expected to impact the derived LFs. \nWe present number densities in the rest-UV with M 1450 , which is the standard for blue quasars (Figure 10). The UV luminosities are low compared to UVselected samples, and we find much higher number densities than a naive extrapolation of the bright end from SDSS (e.g., McGreer et al. 2013; Finkelstein & Bagley 2022; Fan et al. 2022) and at lower luminosity from the Hyper-Suprime Camera sample (Matsuoka et al. 2018; Akiyama et al. 2018; Matsuoka et al. 2023). Interestingly, Laporte et al. (2017); Morishita et al. (2020); Fujimoto et al. (2022) also find hints that the numberdensity of UV-faint AGN could be much higher at low UV luminosity. Our number densities fall below the galaxy luminosity function (here from Bouwens et al. 2015), and are consistent with the number-densities from broad-line AGN that have been spectroscopically identified in CEERS (Harikane et al. 2023; Barro et al. 2023). If we deredden the UV luminosities, they would extend to M 1450 ≈ -23 mag. They would overlap with the faint-end of the AGN UV luminosity function (e.g., Akiyama et al. 2018; Matsuoka et al. 2018, 2023). At ∼ 10 -5 Mpc -3 mag -1 , the compact red sources presented here would be more than ten times more numerous than known UV-selected faint quasars. \nIt is hard to compare with the X-ray number counts, since they differ from survey to survey (e.g., Aird et al. 2015; Vito et al. 2018; Giallongo et al. 2019). Our numbers are nominally consistent with, but on the high side of, these X-ray studies. It is worth saying that we may not be probing the same objects, since the existing X-ray counts require HST cross-matches to enable photometric redshifts.', '4. NATURE OF COMPACT RED SOURCES': "We have presented a sample of 26 point-source dominated sources that are red in the rest-frame optical but have a UV component. From the \nHST +NIRCam/ JWST +ALMA photometry alone, there are three primary interpretations of these sources. First, they could be red due to a strong Balmer/4000 ˚ A break, and we could be seeing the cores of massive galaxies today (e.g., Labb'e et al. 2023; Baggen et al. 2023). In some cases, the compact red sources have NIRCam LW colors that are too red to be fit with a 4000 ˚ A break. Instead, galaxy-only templates tend to prefer a second solution with the red continuum arising from dusty star formation. These models tend to overpredict the ALMA upper limits by factors of 10 or more. The easiest way to have consistency with the data is to posit quite hot dust ( ≳ 100K). Since the sources are very compact, it seems possible in principle that they have abnormally hot dust powered by star formation. The UV component could still be a weak UV-emitting AGN, and the red component dominated by dusty star formation (e.g., Kocevski et al. 2023). \nWe have argued for a third possibility, based on the compact morphology and SED properties. We suggest that these sources are possibly dominated by reddened broad-line AGN with 3 < z < 7. Rest-UV continuum could arise either from some star formation or through percent-level scattering of the unabsorbed AGN continuum. In the following two sections, we explore more quantitatively the implications should the compact red sources be dominated by dusty star formation or AGN emission respectively.", '4.1. Dusty Star formation Interpretation': "If the galaxies do not contain AGN but the observed SED is the result of dust-obscured star formation, the implications for the formation of massive galaxies could be significant. Few of these systems are bright enough to have been included in previous HST -based surveys, or did not appear massive due to the shallow nature of the Spitzer/IRAC photometry (e.g., Labb'e et al. 2023). From our galaxy-only fits, several sources have inferred stellar masses of > 10 10 -10 -11 M ⊙ (Table 3). Their number densities are on the order of ∼ 10 -4 -10 -5 Mpc -3 implying a substantial population that was overlooked at rest-UV/optical wavelengths before the advent of JWST , echoing the results inferred from infrared/sub-mm studies (Williams et al. 2019; Casey et al. 2021; Zavala et al. 2021; Algera et al. 2023; Fujimoto et al. 2023) and HST -dark galaxies (Wang et al. 2019; Barrufet et al. 2023). Recent JWST NIRCam/MIRI observations report the identification of point-source like candidate massive > 10 10 M ⊙ galaxies at z ∼ 8 (Akins et al. 2023), one of which may be detected at 2 mm, indicating the presence of a dusty starburst (Akins et al. 2023). If these galaxies are all mas- \nFigure 10. Left : UV luminosity function for compact red sources with 3 < z < 5, as compared with other samples from the literature. Compact red source luminosities have been corrected for magnification using the lensing model of Furtak et al. (2022b). The point below M 1450 = -15 is from the single lensed source shown in Figure 9 and due to the strong lensing, the volume is quite uncertain. The UV AGN data are taken from Akiyama et al. (2018, grey crosses), the X-ray selected objects are presented in Giallongo et al. (2019, dark grey circles), and the galaxies are from Bouwens et al. (2015). Harikane et al. (2022) presents the broad-line AGN from CEERS (thick grey boxes). We see that the number densities of the compact red sources is comparable to what is seen from X-ray selection, and broad-line selection as well. Right : Same as left, but for lensing-corrected compact red sources between 5 < z < 7. The UV AGN luminosity function is presented by Matsuoka et al. (2018); all other sources are as at left. \n<!-- image --> \nM \n1450 \n<!-- image --> \n1450 \nsive, extremely compact, and at high redshift, a straightforward interpretation would be that we are witnessing the formation of the dense compact inner regions of the most massive ellipticals today (Baggen et al. 2023). \nWe note that the lack of a stacked ALMA 1.2 mm detection is unexpected if the reddest NIRCam wavelengths reflect typical dusty star-forming galaxies at these redshifts. Our complementary sample of dusty star-forming galaxies at z = 2 -6, selected with the same color selection as the compact red sources, have ALMA fluxes that are consistent with cool dust T ∼ 20 -40K (see Figure 8). Our sources are quite compact, and a relationship of increasing dust temperature with decreasing size has been found for SMGs (Hodge et al. 2016), which would imply hotter T d ∼ 60K for our sources. Dust temperatures based on Herschel+ALMA observations of z ∼ 5 -6 drop-out selected galaxies are found to be warmer T d = 40 -60K (B'ethermin et al. 2020; Sommovigo et al. 2022; Mauerhofer & Dayal 2023), and even higher T d ∼ 90K have been suggested at z ∼ 7 -8 (Laporte et al. 2017; Behrens et al. 2018; Bakx et al. 2020). Known examples of star forming galaxies with even hotter dust are rare. NGC1377 shows hot nuclear dust emission ( T d > 180K), but is likely powered by a buried AGN (Aalto et al. 2020). Luminous IR sources with hot dust at high-redshift (hot-DOGS) also typically \nharbor powerful AGN (Assef et al. 2018). Our observations do not preclude the existence of compact starbursts with high dust temperatures, although if they dominate our measured number densities then such systems are underrepresented in sub-mm/mm surveys of obscured star formation in the early universe (e.g., Zavala et al. 2021; Fujimoto et al. 2023).", '4.2. AGN interpretation': 'PreJWST , it was common to characterize AGN at z > 5 using the UV luminosity function, that being the only luminosity that we could measure. Our sources are very faint in the UV. Their magnification-corrected sources are fainter in the UV even than the lowestluminosity AGN discovered via deep X-ray data (Figure 11 left Giallongo et al. 2019). However, in our model the compact red sources are reddened, with A V ≈ 2 -6. Thus, the UV luminosity does not represent the intrinsic luminosity of the accretion disk, as in unreddened AGN. Instead, the inferred bolometric luminosities derived from the dereddened rest-frame optical flux may be more illustrative of the true AGN luminosity. \nWhat we find is that our targets span a wide range of bolometric luminosity, ranging from the Eddington limit for ∼ 10 6 M ⊙ black holes to ∼ 10 8 M ⊙ black holes. While we cannot infer black hole masses from the photometry, if we assume standard Eddington ratio \nM \nFigure 11. Left : UV luminosities of the compact red sources (Main sample, red circles; SED-selected sub-sample identified with black dots) compared with various comparison samples from the literature, including SDSS DR7 (Shen 2013), and the high-redshift and luminous tail of SDSS quasars (McGreer et al. 2013), the lower-luminosity quasars from the Hyper-Suprime Camera survey (Akiyama et al. 2018; Matsuoka et al. 2018), and the deep X-ray sample from GOODS-North (Giallongo et al. 2019). The UV luminosities for the compact red sources are corrected for lensing using the model from Furtak et al. (2023), and only the high-quality sample have the black circles around them. The compact red sources lie at the very faint end of these samples when compared in the rest-frame UV at λ = 1450 ˚ A. Right : Bolometric luminosity distribution of the same comparison samples, showing that the compact red sources span the full range of observed quasar bolometric luminosities at these redshifts; luminosities also corrected for magnification using the Furtak et al. (2023) model. We show the luminosity corresponding to a black hole at M BH = 10 6 , 10 8 M ⊙ when accreting at the Eddington limit. \n<!-- image --> \n<!-- image --> \nz \ndistributions for these sources, typically ∼ 10% of the Eddington limit (e.g., Aird et al. 2018), then the typical source has M BH ∼ 10 7 -10 9 M ⊙ , similar to what was inferred spectroscopically by Kocevski et al. (2023) and Harikane et al. (2023). \nIt it useful to compare our targets with known populations of reddened broad-line AGN at lower redshift (e.g., Glikman et al. 2011, 2012; Banerji et al. 2015; Assef et al. 2018; Hamann et al. 2017). Banerji et al. (2015) find that the reddened broad-line objects at z ≈ 2 . 5 have higher number density than UV-bright quasars at the very highest luminosities L bol > 10 47 erg/s, but then their number begins to flatten at lower L . In contrast we are finding that our sources could outnumber their unreddened counterparts by a factor of ten or more. \nWe can also compare with X-ray searches at high redshift. In the UNCOVER field specifically, the emission from the hot gas in the cluster also complicates the analysis in the X-ray (Bogdan et al. 2023), and will require more specialized analysis. However, the Little Red Dots presented in Matthee et al. are quite X-ray faint; we would likely need to reach ten-times deeper in the X-ray to find this population. \nGiven the very stringent limits on size measured for these sources, it is likely that the ratio of black hole to \ngalaxy mass will be quite high for these targets. However, we prefer to defer a discussion of this topic to the time when we spectroscopically confirm the nature of the sources.', '5. DISCUSSION AND SUMMARY': "Thanks to its remarkable red sensitivity, depth, and spatial resolution, JWST has unveiled an unexpected population of very red and compact sources with photometric redshifts z > 3 (Endsley et al. 2022; Labb'e et al. 2023; Furtak et al. 2022a; Akins et al. 2023). In a handful of cases, these compact red sources have been spectroscopically revealed to contain moderate-luminosity heavily reddened AGN (Kocevski et al. 2023; Harikane et al. 2023; Ubler et al. 2023; Oesch et al. 2023). Thus motivated, we searched systematically for such sources within the JWST Treasury Survey UNCOVER (Bezanson et al. 2022). \nWe found a sample of 26 sources with red colors (F277F444 > 1 or F200-F356 > 1). Their light profiles are PSF-dominated, leading to typical lensing-corrected size limits of < 100 pc at the presumed redshift range of 3 < z phot < 8, and a median size of ∼ 50pc. These sizes alone do not preclude that the presence of massive very compact dusty star-forming galaxies. However, fits to the broad-band SEDs including ALMA can rule out typ- \nal dusty star formation with cool/warm dust in more than half the cases (Fujimoto et al. 2023, in prep.). \nOn net, we argue that a natural explanation for the bulk of the compact red sources would be that they are powered by heavily reddened AGN. In that case, they have comparable number densities to X-ray selected samples from the deepest X-ray data (Vito et al. 2018; Giallongo et al. 2019) and they have implied bolometric luminosities spanning 10 44 < L bol < 10 46 erg/s. We find heavily reddened broad-line AGN at all redshifts, from the local Mrk 231 (Veilleux et al. 2016) and Mrk 1239 (e.g., Pan et al. 2021) to reddened AGN at cosmic noon (Glikman et al. 2012; Banerji et al. 2015; Hamann et al. 2017; Noboriguchi et al. 2019, 2022). However, at z < 3, highly reddened broad-line AGN with a UV component are rare. For instance, Noboriguchi et al. (2019) estimate that 1% of the HotDOGs that they select from the Hyper-Suprime Camera Survey (Akiyama et al. 2018) have a UV excess.", '5.1. Exciting prospects for selecting AGN with JWST': 'High-redshift quasars ( z > 5) have historically been selected via their blue continuum and Ly α break (e.g., Fan et al. 2022). The final SDSS quasar search over 11,000 deg 2 uncovered a sample of 52 AGN at z > 6. These luminous quasars at high redshift are very rare. Lower luminosity sources have been found using the deepest existing X-ray data, where near-infrared data are also required to determine galaxy counterparts and photometric redshifts for the X-ray sources. These lower-luminosity sources typically have higher number densities, but require very deep Chandra data (e.g., Aird et al. 2015; Vito et al. 2018; Giallongo et al. 2019) along with deep NIR data to determine photometric redshifts. \nEven just limiting attention to our high-quality sample, we find a substantial number of probable AGN with z > 5 in the ∼ 45 arcmin 2 field. Identifying these targets to requires the combination of moderately deep JWST data (2 -4 hr per filter) and HST data. It is clear that if these sources are confirmed, then there is a new and efficient way to identify AGN over a wide range of intrinsic luminosity through JWST /NIRCam imaging.', '5.2. Are most high-z AGN UV and X-ray faint?': 'Deep X-ray searches combined with JWST redshifts are finding new and very exciting sources even out to z ≈ 10 (Bogdan et al. 2023). However, the number densities of UV and X-ray selected AGN are rather low compared to the number densities of galaxies with z > 5. For instance, taking local relations between black hole mass and galaxy mass, and reasonable assumptions for Eddington ratio distributions, Volonteri & Reines \n(2016) predict a higher number density of AGN than have been found with UV or X-ray searches. Volonteri & Reines (2016) invoke a lower ratio of black hole to galaxy mass for the bulk of the population. On the other hand, luminous quasars at these epochs appear to have higher ratios of black hole mass to galaxy mass (see review in Fan et al. 2022). \nIt is possible that UV-luminous quasars are a highly biased population in which the black holes have outgrown their galaxies, while the bulk of the black hole population at these redshifts are under-massive (e.g., Zhang et al. 2023). Another possibility is that a large fraction of the AGN at these epochs are too red and obscured to be detected in the rest-frame UV by standard techniques, and too X-ray weak to be discovered in the X-ray. So far, neither the lensed Furtak et al. (2022a) object, nor the Kocevski et al. (2023) objects are X-ray detected. Many models predict that the fraction of accretion that is obscured will grow with redshift (e.g., Ni et al. 2020). So far, the empirical verdict on evolution of the obscuration fraction with redshift and luminosity has been a bit murky (e.g., Aird et al. 2015; Georgakakis et al. 2015; Vito et al. 2018; Giallongo et al. 2019). If we are discovering relatively obscured, moderate luminosity AGNs at high number density, our picture of the evolution in obscuration with redshift also may evolve. \nAlong similar lines, Harikane et al. (2023) use restframe optical spectroscopy to find that moderate luminosity AGNs account for ∼ 10% of the galaxy population at z > 5. Of these, 20% of the broad-line AGN are in very red sources. This fraction is hard to interpret at the moment given the complicated selection function for spectra. A recent spectroscopic search using JWST NIRCam grism data revealed a substantial number of broad-line AGN in highly obscured compact objects at z = 5 -6 (Matthee et al. 2023). Certainly, early JWST results suggest a higher number density of actively accreting black holes than was appreciated before. \nUNCOVER is a spectroscopic experiment. While the pre-imaging are in hand, the NIRSpec component is scheduled for July 2023. We hope to obtain spectroscopic identifications some of these targets at that time, to determine their nature as revealed by low-resolution spectroscopy in the UNCOVER region. Additional NIRCam grism spectroscopy and medium-band imaging of the UNCOVER field is scheduled in cycle 2, shedding additional light on the nature of these objects. Anything we find will be interesting in terms of understanding the nature of these targets.', 'ACKNOWLEDGMENTS': "J.E.G. and A.D.G acknowledge support from NSF/AAG grant# 1007094, and J.E.G. also acknowledges support from NSF/AAG grant # 1007052. L.J.F. and A.Z. acknowledge support by Grant No. 2020750 from the United States-Israel Binational Science Foundation (BSF) and Grant No. 2109066 from the United States National Science Foundation (NSF), and by the Ministry of Science & Technology of Israel. The Cosmic Dawn Center is funded by the Danish National Research Foundation (DNRF) under grant #140. This work has received funding from the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00072, as well as from the Swiss National Science Foundation (SNSF) through project \ngrant 200020 207349. PD acknowledges support from the NWO grant 016.VIDI.189.162 ('ODIN') and from the European Commission's and University of Groningen's CO-FUND Rosalind Franklin program. RPN acknowledges funding from JWST programs GO-1933 and GO-2279. Support for this work was provided by NASA through the NASA Hubble Fellowship grant HST-HF251515.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. The research of CCW is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.", 'A. SED FITTING': 'To perform SED fitting with multiple components, e.g., stellar populations with potentially different reddening or AGN contribution, custom fitting is needed. \nFor the AGN we use an empirical base model based on composite optical spectra from 2200 SDSS quasar spectra from (Vanden Berk et al. 2001) and 27 quasar spectra observed with IRTF in the near-infrared by (Glikman et al. 2006). This template is then reddened by A V =0.5 to 5 using a Calzetti (2001) attenuation law. When our fits include the ALMA 1.2 mm point, we model the mid-to-far infrared emission with the CLUMPY torus models (Nenkova et al. 2008a,b). These models are implemented within the Flexible Stellar Population Synthesis (FSPS) code (Conroy et al. 2009), with the torus thickness, number of clumps, power-law density of the clump distribution held constant at default values. The optical depth of a given clump is fixed at τ AGN = 10. The orientation of the torus is also fixed at 40 deg, which means that virtually no emission from the AGN accretion disk is transmitted, and there is no AGN emission in the optical from this component of the model. More thorough discussion can be found in Leja et al. (2018). We tie the UV/optical disk component to the mid-to-far infrared component by scaling the torus model by the luminosity of the obscured component, thus ensuring energy balance. In some cases we wish to model both the blue rest-UV and red rest-optical wavelengths of the sources by AGN, where the blue light could reflect scattered light. In that case we use two components, one dust-free and one reddened, each with a logarithmic prior in luminosity from 13 < L bol < 48. Additional freedom is allowed in the broad lines and narrow lines, where the equivalent widths are modified by a small factor with a Gaussian prior with a dispersion of 0 . 3 dex. \nFor the stellar populations we use three independent components based on FSPS: a dust-free star forming component, a dust obscured star forming component, and an old quiescent component. The star forming components assume constant-SFH and include nebular emission lines. The quiescent component is modeled as an exponentially declining SFH with τ = 10 Myr. The stellar population model assumes a Chabrier (2003) IMF and 0.2 Z ⊙ metallicity. The ages of all three components may vary independently, in logarithmic steps, from 10 Myr to a maximum age corresponding to a formation redshift of z = 12. The attenuation of the dusty component assumes a Calzetti (2001) attenuation law, with A V =0.5 to 5.5 and a uniform prior. The masses of each component are drawn from a uniform prior between 6 < log ( M/M ⊙ ) < 12 When modeling the ALMA data, we use dust emission models as implemented in FSPS (see description in Leja et al. 2017). These are based on Draine & Li (2007) dust models, which are parameterized with three parameters. As we cannot constrain the shape of the dust emission spectrum, the default settings for FSPS are adopted. The minimum starlight intensity is fixed to U min = 1 . 0, the relative fraction of the dust exposed to the minimum starlight is fixed to γ = 0 . 01, and the PAH mass fraction q PAH = 3 . 5. Together these three parameters establish the shape of the dust emission spectrum. The default settings correspond to an effective dust temperature of T ∼ 20K. We also ran the analysis with an elevated effective dust temperature, to account for dusty star formation with intrinsically higher temperatures or to take into account the effect the elevated CMB (daCunha et al. 2013). While there is no simple mapping from the parametrization of dust emission in FSPS to a typical modified black body at a single temperature, we increase U min = 20 . 0 finding it approximately corresponds to a dust peak emission of T ∼ 40K. The results do not change significantly with a higher dust temperature. \nThe model fitting is done by constructing composite models, drawing priors for each parameter, and evaluating the Bayesian evidence using the nested sampling algorithm nestle in Python . From the models, predicted photometry is generated using sedpy , the log likelihood is calculated by comparing to the HST , JWST , and ALMA observations, and credible intervals on the parameters are calculated from the posterior samples.', 'B. SAMPLE': '<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 12. Image stamps of high quality PSF + SED-selected sample \n<!-- image --> \nFigure 13. Image stamps of the rest of the Main sample (PSF selected) \nFigure 14. Several examples of sources that are indistinguishable from a PSF. As a PSF model, a nearby star was selected and fitted and subtracted. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 15. Image stamps of examples of extended dusty galaxies \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \ny \nµJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n0 \n1 \n2 \n3 \n0 \nID=13556 \nstars z=5.6 logM=9.5 Av=4.8 \nχ \nstars z=5.5 logM=9.0 Av=2.4 \nχ \nAGN z=5.2 Lbol=45.3 Av=5.1 \nχ \n1 \n2 \n3 \n4 \n5 \n6 \nobserved [ \nµm \n] \nID=16561 \nstars z=6.6 logM=10.3 Av=5.0 \nχ \n2 \n2 \n= \n27.5 \n2 \n2 \n2 \n= \n16.1 \n= \n127.6 \n= \n15.0 \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n0 \n1 \n2 \n3 \n4 \n5 \n6 \n0 \n10 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \n2 \n10 \n3 \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n- \n10 \n- \n10 \n- \n3 \n5 \n7 \n1 \n10 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \nID=16561 z=6.3 \njoint stars +AGN \n2 \n10 \n3 \ny \nµJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n10 \n- \n10 \n- \n10 \n- \n0 \n1 \n2 \n3 \nstars z=7.8 logM=11.3 Av=5.0 \nχ \nstars z=7.5 logM=9.1 Av=2.3 \nχ \nAGN z=6.8 Lbol=45.7 Av=5.1 \nχ \n1 \n2 \n3 \n4 \n5 \n6 \nobserved [ \nµm \n] \n10 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \n2 \n10 \n3 \njoint stars +AGN \n10 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \n2 \n10 \n3 \n<!-- image --> \nID=13556 z=5.2 \njoint stars +AGN \n10 -1 Figure 16. PSF + SEDs fits of SED-selected sample \ny \nmJ \ny \nt \n2 \n2 \n2 \n= \n48.4 \n= \n221.2 \n= \n20.7 \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n0 \n1 \n2 \n3 \n4 \n5 \n6 \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n1 \n3 \n5 \n7 \ny \nµJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \ny \nµJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \ny \nµJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \ny \nµJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n10 \n10 \n- \n10 \n- \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n10 \n- \n10 \n- \n10 \n- \n1 \n0 \n1 \n2 \n0 \n1 \n2 \n3 \n0 \n1 \n2 \n3 \n0 \n1 \n2 \n3 \n2 \n2 \nstars z=4.6 logM=10.1 Av=3.2 \nχ \nstars z=4.8 logM=10.2 Av=2.8 \nχ \nAGN z=4.8 Lbol=45.6 Av=2.1 \nχ \n1 \n2 \n3 \n4 \n5 \n6 \nobserved [ \nµm \n] \n10 \n1 \n10 \nobserved [ \nµm \n] \njoint stars +AGN \n10 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \nFigure 17. SEDs of the rest of the Main sample \n<!-- image --> \nID=31298 \nstars z=5.1 logM=10.9 Av=4.8 \nχ \nstars z=5.1 logM=11.0 Av=4.8 \nχ \nAGN z=4.8 Lbol=46.0 Av=4.7 \nχ \n1 \n2 \n3 \n4 \n5 \n6 \nobserved [ \nµm \n] \nID=35771 \nstars z=4.9 logM=10.2 Av=4.6 \nχ \nstars z=4.9 logM=10.1 Av=4.5 \nχ \nAGN z=4.3 Lbol=44.6 Av=2.6 \nχ \n1 \n2 \n3 \n4 \n5 \n6 \nobserved [ \nµm \n] \nID=35819 \nstars z=4.8 logM=8.9 Av=3.7 \nχ \nstars z=4.8 logM=8.9 Av=3.8 \nχ \nAGN z=4.6 Lbol=44.1 Av=2.5 \nχ \n1 \n2 \n3 \n4 \n5 \n6 \nobserved [ \nµm \n] \nID=37108 \nstars z=5.7 logM=9.2 Av=3.0 \nχ \nstars z=5.7 logM=9.0 Av=3.7 \nχ \nAGN z=4.5 Lbol=44.0 Av=3.2 \nχ \n1 \n2 \n3 \n4 \n5 \n6 \nobserved [ \nµm \n] \n2 \n2 \n2 \n= \n10.6 \n= \n14.1 \n= \n0.0 \n2 \n2 \n2 \n= \n10.4 \n= \n10.9 \n= \n15.9 \n2 \n2 \n2 \n= \n11.7 \n= \n12.0 \n= \n19.3 \n2 \nID=31298 z=4.8 \njoint stars +AGN \n10 \n0 \ny \nmJ \ny \nt \n2 \n2 \n2 \n= \n120.0 \n= \n119.8 \n= \n93.1 \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n1 \n0 \n1 \n2 \n3 \n4 \n5 \n6 \n1 \n0 \n1 \n2 \n3 \n4 \n5 \n6 \n0 \n1 \n2 \n3 \n4 \n5 \n6 \n7 \n0 \n1 \n2 \n3 \n4 \n5 \n6 \n7 \n10 \n10 \n10 \n10 \n0 \n0 \n0 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \n10 \n1 \n10 \nobserved [ \nµm \n] \n10 \n1 \n10 \nobserved [ \nµm \n] \n10 \n1 \n10 \nobserved [ \nµm \n] \n2 \n2 \n2 \n2 \n10 \n10 \n10 \n10 \n3 \n3 \n3 \n3 \ni \ns \nn \ne \nd \nx \nu \nfl \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n2 \n4 \n6 \n1 \n3 \n5 \n7 \n1 \n2 \n3 \n4 \n5 \n6 \n7 \n8 \n3 \n4 \n5 \n6 \n7 \n10 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \nID=35771 z=4.3 \njoint stars +AGN \n10 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \nID=35819 z=4.6 \njoint stars +AGN \n10 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \nID=37108 z=3.0 \njoint stars +AGN \n10 \n0 \n10 \n1 \n10 \nobserved [ \nµm \n] \n2 \n2 \n2 \n2 \n10 \n10 \n10 \n10 \n3 \n3 \n3 \n3 \ny \nµJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n10 \n10 \n- \n10 \n- \n1 \n0 \n1 \n2 \n= \n12.3 \n= \n12.8 \n= \n66.6 \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n10 \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n10 \n- \n1 \n0 \n1 \n2 \n3 \n4 \n5 \n10 \n0 \n2 \n10 \n3 \ny \nmJ \ny \nt \ni \ns \nn \ne \nd \nx \nu \nfl \n10 \n10 \n- \n10 \n- \n10 \n- \n0 \n2 \n4 \n6 \n2 \n10 \n3'}
2024arXiv240904650A	Type Ia supernovae SNe Ia constitute an historical probe to derive cosmological parameters through the fit of the HubbleLematre diagram i.e. SN Ia distance modulus versus their redshift. In the era of precision cosmology realistic simulation of SNe Ia for any survey entering in an HubbleLematre diagram is a key tool to address observational systematics like Malmquist bias. As the distance modulus of SNe Ia is derived from the fit of their lightcurves a robust simulation framework is required. In this paper we present the performances of the simulation framework skysurvey to reproduce the the Zwicky Transient Facility ZTF SN Ia DR2 covering the first phase of ZTF running from April 2018 up to December 2020. The ZTF SN Ia DR2 sample correspond to almost 3000 classified SNe Ia of cosmological quality. First a targeted simulation of the ZTF SN Ia DR2 was carried on to check the validity of the framework after some fine tuning of the observing conditions and instrument performance. Then a realistic simulation has been run using observing ZTF logs and ZTF SN Ia DR2 selection criteria on simulated lightcurves to demonstrate the ability of the simulation framework to match the ZTF SN Ia DR2 sample. Furthermore a redshift dependency of SALT2 lightcurve parameters stretch and colour was conducted to deduce a volume limited sample i.e. an unbiased SNe Ia sample characterized with zlim leq 0.06. This volume limited sample of about 1000 SNe Ia is unique to carry on new analysis on standardization procedure with a precision never reached those analysis are presented in companion papers.	2024-09-01T00:00:00Z	['arXiv:2409.04650', '10.48550/arXiv.2409.04650', '2024arXiv240904650A']	['Astrophysics - Cosmology and Nongalactic Astrophysics']	ZTF SN Ia DR2 Simulations and volume limited sample	2,024	192	0.51	['EPRINT_HTML', 'EPRINT_PDF']	2	https://arxiv.org/pdf/2409.04650.pdf	{'ZTF SN Ia DR2: Simulations and volume limited sample': "M. Amenouche 1 ⋆ , M. Smith 2 ⋆⋆ , P. Rosnet 3 ⋆⋆⋆ , M. Rigault 4 , M. Aubert 3 , C. Barjou-Delayre 3 , U. Burgaz 5 , B. Carreres 6 , 7 , G. Dimitriadis 5 , F. Feinstein 6 , L. Galbany 8 , 9 , M. Ginolin 4 , A. Goobar 10 , L. Harvey 5 , Y.-L. Kim 2 , K. Maguire 5 , T.E. Müller-Bravo 8 , 9 , J. Nordin 11 , P. Nugent 12 , 13 , B. Racine 6 , D. Rosselli 6 , N. Regnault 14 , J. Sollerman 12 , J.H. Terwel 5 , A. Townsend 11 , S.L. Groom 16 , S.R. Kulkarni 15 , M. Kasliwal 15 , R.R. Laher 16 , and J. Purdum 16 \n- 1 National Research Council of Canada, Herzberg Astronomy & Astrophysics Research Centre, 5071 West Saanich Road, Victoria, BC V9E 2E7, Canada\n- 2 Department of Physics, Lancaster University, Lancaster, UK, LA14YW\n- 3 Université Clermont Auvergne, CNRS / IN2P3, LPCA, F-63000 Clermont-Ferrand, France\n- 4 Univ Lyon, Univ Claude Bernard Lyon 1, CNRS, IP2I Lyon / IN2P3, UMR 5822, F-69622 Villeurbanne, France\n- 5 School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland\n- 6 Aix Marseille Université, CNRS / IN2P3, CPPM, Marseille, France\n- 7 Department of Physics, Duke University Durham, NC 27708, USA\n- 8 Institute of Space Sciences (ICE-CSIC), Campus UAB, Carrer de Can Magrans, s / n, E-08193 Barcelona, Spain\n- 9 Institut d'Estudis Espacials de Catalunya (IEEC), 08860 Castelldefels (Barcelona), Spain\n- 10 The Oskar Klein Centre, Department of Physics, AlbaNova, Stockholm University, SE-106 91 Stockholm, Sweden\n- 11 Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany\n- 12 The Oskar Klein Centre, Department of Astronomy, AlbaNova, Stockholm University, SE-106 91 Stockholm, Sweden Lawrence Berkeley National Laboratory, 1 Cyclotron Road MS 50B-4206, Berkeley, CA, 94720, USA\n- 13 Department of Astronomy, University of California, Berkeley, 501 Campbell Hall, Berkeley, CA 94720, USA\n- 14 LPNHE, CNRS / IN2P3, Sorbonne Université, Université Paris-Cité, Laboratoire de Physique Nucléaire et de Hautes Énergies, 75005 Paris, France\n- 15 Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA\n- 16 IPAC, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA Caltech Optical Observatories, California Institute of Technology, Pasadena, CA 91125, USA", 'ABSTRACT': 'Type Ia supernovae (SNe Ia) constitute an historical probe to derive cosmological parameters through the fit of the Hubble-Lemaître diagram, i.e. SN Ia distance modulus versus their redshift. In the era of precision cosmology, realistic simulation of SNe Ia for any survey entering in an Hubble-Lemaître diagram is a key tool to address observational systematics, like Malmquist bias. As the distance modulus of SNe Ia is derived from the fit of their light-curves, a robust simulation framework is required. In this paper, we present the performances of the simulation framework skysurvey to reproduce the the Zwicky Transient Facility (ZTF) SN Ia DR2 covering the first phase of ZTF running from April 2018 up to December 2020. The ZTF SN Ia DR2 sample correspond to almost 3000 classified SNe Ia of cosmological quality. First, a targeted simulation of the ZTF SN Ia DR2 was carried on to check the validity of the framework after some fine tuning of the observing conditions and instrument performance. Then, a realistic simulation has been run using observing ZTF logs and ZTF SN Ia DR2 selection criteria on simulated light-curves to demonstrate the ability of the simulation framework to match the ZTF SN Ia DR2 sample. Furthermore a redshift dependency of SALT2 light-curve parameters (stretch and colour) was conducted to deduce a volume limited sample, i.e. an unbiased SNe Ia sample, characterized with zlim ≤ 0 . 06. This volume limited sample of about 1000 SNe Ia is unique to carry on new analysis on standardization procedure with a precision never reached (those analysis are presented in companion papers). \nKey words. ZTF ; Cosmology ; Type Ia Supernovae ; Simulation', '1. Introduction': 'In the past two decades, Type Ia supernovae (SNe Ia) have been used to probe the Universe on cosmological scales, typically up to a redshift z ∼ 1. Early measurements discovered the acceleration of the expansion of the Universe, while the recent stateof-the-art analysis allows to measure the mass density parameter ( Ω m ) within the Λ CDM model with an uncertainty better than \ntwo percent (Brout et al. 2022). This kind of analysis is based on the cosmological fit of the Hubble-Lemaître diagram, the SN Ia luminosity distance as a function of their redshift. \nIn current analysis, the SN Ia luminosity distance is determined from the fit of their light-curves in di ff erent photometric bands with a phenomenological model tuned on SN Ia Spectral Energy Distributions (SED). The most common used model is SALT2 (Guy et al. 2007). But for precision cosmology, for instance to reach the two percent level precision in the cosmological parameters, the distance luminosity must be determined with \na typical accuracy of 0.15 mag, i.e. with a photometric precision of typically 1%. \nSuch precise measurements can be achieved only by correcting for any instrumental e ff ects, but also for any systematic effect like the Malmquist bias, but not only as discussed in DES cosmological analysis paper (Vincenzi et al. 2024). The way to quantify any bias is to use realistic simulation, i.e. able to reproduce typical light-curves observed by the survey. With a robust simulation framework, high-statistic simulations can be generated to estimate any systematic e ff ect starting from the SNe Ia light-curves that can a ff ect the cosmological results. \nAt low redshift ( z < 0 . 1) the DES Collaboration has shown that the bias on the SNe Ia distance modulus is limited to 0.05 mag, while at high redshift ( z > 0 . 1) it can reach 0.4 mag (Kessler et al. 2019). To overcome the right distance modulus, in their simulation framework (SNANA software package, Kessler et al. 2009), they include observational e ff ects (sky-noise, zeropoint...) and also detection e ffi ciency determined from fake SNe Ia introduced in real images. This example illustrates how critical it is to accurately determine such bias for cosmological analysis. \nThe Zwicky Transient Facility (ZTF) is a low redshift survey (Bellm et al. 2019). The necessity to measure with the best precision SN Ia peculiar velocity for bulk-flow analysis or for the measurement of f σ 8 is crucial (Carreres et al. 2023). Beyond pure low redshift analysis, the combination of the unique ZTF SNe Ia sample with high redshift surveys, especially the coming one like LSST (Gris et al. 2023), will also help in constraining better the cosmological parameters in the Hubble fit. \nThis paper highlights the tools developed by the ZTF Collaboration to simulate SN Ia light-curves in a realistic way by taking into account both the observing strategy and the instrumental limitations. In Sect. 2, the ZTF instrument is introduced as well as the ZTF Cosmology Data Release 2 of SNe Ia (ZTF SN Ia DR2) corresponding to the three first years of observations. Then the simulation framework is described in section 3, while section 4 outlined the methodology developed to test the simulation pipeline. The test of the simulation framework based on the ZTF SN Ia DR2 sample is discussed in section 5, before to conclude.', '2. Data: The Zwicky Transient Facility': 'The Zwicky Transient Facility (ZTF) is a wide field optical survey covering the visible northern sky (Bellm et al. 2019), using the Samuel Oschin 48-inch (1.2-m) Schmidt Telescope at the Palomar observatory (Graham et al. 2019). The ZTF camera, consisting in a mosaic of 16 CCDs, has a field-of-view of 47deg 2 (Dekany et al. 2020), 13.3% of which is lost due to chip gaps and vignetting. The first phase of ZTF, called ZTF-1, with an average nightly cadence of every 2.5 nights in g and r -band, and 5 σ limiting depths of 20.5 mag, started observations in March 2018. All data obtained are processed, using di ff erence imaging techniques by the Infrared Processing and Analysis Center (IPAC, Masci et al. 2019) to estimate observing conditions (e.g. limiting magnitude, atmospheric distortion and zero-point) and detect candidates. This information is then disseminated to the community through alert packets (Patterson et al. 2019). From March 2018 to December 2020, ZTF-1 discovered and classified 4156 supernovæ. A full description of the ZTF-1 observing pattern can be found in Rigault et al. (a).', '2.1. Spectroscopic Follow-up': 'Spectroscopic follow-up for ZTF-1 was lead by the Bright Transient Survey (BTS, Perley et al. 2020; Fremling et al. 2020). This project aimed to produce a magnitude-limited census of extra-galactic transients by classifying all ZTF discoveries with m peak < 18 . 5. Classifications were predominately obtained using the low resolution integral field unit, SEDm (Blagorodnova et al. 2018; Rigault et al. 2019; Kim et al. 2022) with additional time allocated on large facilities (see Fremling et al. 2020 for details) to classify missed or rapidly fading events. From ZTF-1, this project classified 3597 transients 1 , with completeness fractions estimated to 97%, 93% and 75% for objects brighter than 18 mag, 18.5 mag and 19 mag, respectively (see Perley et al. 2020 for selection requirements and details).', '2.2. ZTF SN Ia DR2': 'ZTF-1 has discovered and classified 3628 SNe Ia. This second SNe Ia data release (ZTF SN Ia DR2 detailed in Rigault et al. (a) and Smith et al., called DR2 Throughout in the text), following ZTF SN Ia DR1 presented in Dhawan et al. (2022), includes events discovered and classified from March 2018 to December 2020. About 80% of events in this sample were classified by the BTS survey, with the remaining classifications obtained from public surveys (e.g. Smartt et al. 2015) and individual spectra reported to the Transient Name Server (TNS 2 ). Photometry for this sample is determined using forced-photometry extracted from di ff erence images, created, as part of the IPAC pipeline, by subtracting each science image from a stack of ZTF images taken during survey operations. A full description of the sample, and its photometric characterisation is given in Smith et al., with companion papers on host galaxy properties and standardisation (Ginolin et al. 2024b,a; Popovic et al. 2024) with largescale structure correlations (Ruppin et al. 2024; Aubert et al. 2024), light-curves properties (Rigault et al. 2024; Deckers et al. 2024), spectral properties Johansson et al. and diversity (Burgaz et al. 2024), Dimitriadis et al. with astrophysics Burgaz et al. and cosmological (Carreres et al. 2024; Dhawan et al. 2024) impacts. \nIn this work, we aim to estimate the completeness for the DR2 in term of redshift defining a volume limited sample.', '3. Simulation Framework': 'To simulate the DR2 sample, we randomly generate samples of SNe Ia, drawn from a realistic model, each with a given redshift (drawn from a model describing the volumetric evolution of the rate of SNe Ia events), and sky position. Each simulated SN Ia is then matched to the true ZTF observing cadence and observing conditions to predict the flux and the associated uncertainty (from which we can deduce the signal-to-noise ratio) that ZTF would have observed at each epoch, and produce simulated light-curves. By applying the selection criteria (accounting for photometric detection, light-curve characterisation and spectroscopic selection) used to define the DR2 sample, we predict the distribution of events in the DR2 sample. \nTo do this we use a new python package skysurvey 3 (Rigault & Others 2024) developed for any transient survey, but described here for the purpose of ZTF. This \nues \nnew simulation framework is an improved (from computing point-of-view) version of simsurvey (Feindt et al. 2019) build in a modular way ( simsurvey has already been applied to model the ZTF survey De et al. 2020; Sagués Carracedo et al. 2021; Magee et al. 2022). It allows to simulate thousands of astronomical targets as they would be observed by a survey. The main scheme is the same as for simsurvey : a targeted transient described by its model convoluted with an instrument cadence characterised by its sky deepness and its transmission filters produce light-curves. A schematic representation of this workflow is given in Fig. 1. The main characteristics of skysurvey are: (i) the coding which fasten the transient generation by about a factor thousand compared to simsurvey and (ii) a modular setup allowing to build any transient model by taking into account all astrophysical inputs as the environmental e ff ects. This approach is consistent with the approach taken by SNANA (Kessler et al. 2009) framework commonly used in most SN cosmology analyses. \nThis new framework was used to study the DR2 selection and its potential biases. To this end skysurvey can be used to simulate the typical SNe Ia sample observed by ZTF during its first phase by using its observed log file, i.e. between 2018 1 st April to 2020 December 31 st in g, r and i-bands.', '3.1. SN Ia model': 'To simulate an individual object, we first draw a redshift and sky position. To do this, we randomly pick-up a redshift from a non-evolving volumetric rate in the range 0 < z < 0 . 2, based on Planck 2018 (Planck Collaboration et al. 2020a) fiducial cosmology (flat Λ CDM with H 0 = 67 . 66 kms -1 Mpc -1 and Ω m = 0 . 30966), and a sky position (RA, Dec) within the ZTF footprint (Dekany et al. 2020). Skysurvey can be run in two way: (i) by using a defined rate, by the default SN Ia rate is 2 . 35 × 10 -4 Gpc -3 yr -1 (Perley et al. 2020), or (ii) by specifying a number of SNe Ia to simulate. \nTo generate a SN Ia light-curve, we use the SALT2 timedependent spectral energy distribution model (Guy et al. 2007, 2010). This template parameterizes SNe Ia light curves by three parameters: t 0, the epoch of maximum brightness in B -band, x 1, the light-curve width, and c , the rest-frame colour. To ensure that our light-curves are representative of the overall SNe Ia population, for each event we draw x 1 from the distribution defined in Nicolas et al. (2021) using the parameters determined in Ginolin et al. (2024b) and c from that of Scolnic & Kessler (2016). t 0 is drawn randomly within the ZTF-1 observing time range. To set the apparent peak luminosity in B -band, mB , we generate a peak absolute magnitude of -19 . 3, scattered using a colour independent intrinsic dispersion of σ int = 0 . 1-mag and correct for the luminosity-decline rate and luminosity-colour relations with the nuisance parameters α = 0 . 14 and β = 3 . 15 from (Betoule et al. 2014), via a Tripp relation (Tripp 1998). For each simulated event, we use the sncosmo framework (Barbary et al. 2016), to generate observer-frame light-curves given the above information, and include the e ff ect of the Milky Way (MW) dust by using the extinction maps from Schlegel et al. (1998).', '3.2. Observing Logs': 'To match our generated model with the ZTF-1 survey, we compiled the observing conditions for every ZTF observation by \nquerying the IPAC database. In particular, for every observation, we retrieve 4 : \n- -date: mjdobs ,\n- -sky location (observed field within ZTF grid): fieldid ,\n- -ZTF-filter, CCD and readout-channel: fid , ccdid and qid ,\n- -processing status: infobits ,\n- -zero-point magnitude of the observation: magzp ,\n- -5 σ limiting magnitudes of the science image: scimaglim ,\n- -readout channel gain: gain . \nWe convert the measured limiting magnitude to an e ff ective sky brightness (hereafter referred to as skynoise) using \nsb = 10 0 . 4 × ( magzp -scimaglim ) 5 . (1) \nOur logs cover all observations from April, 2018 to December, 2020. A total of 431k exposures are retrieved, of which (37.1%, 55.5%, 7.4%) are in (g, r, i)-band, respectively.', '3.3. Signal-to-Noise Ratio': 'Flux uncertainties on each simulated event have three contributions: \nσ f = s s 2 b + \| f \| gain + ( f × σ calib ) 2 , (2) \nwhere sb is the sky brightness (defined in Eq. (1)), f is the simulated flux, gain ≈ 6 . 2 is the CCD gain and σ calib is the photometric calibration precision (Masci et al. 2019). \nBefore applying this framework to free simulation, it was first tested on the DR2 targeted SNe Ia to check its validity to reproduce measured light-curves. In order to estimate the accuracy of the simulations, we compare the simulated quantities of the DR2 light-curves to the measured ones. The key quantity used for this purpose is the Signal-to-Noise Ratio (SNR) defined as following: \nSNR = f σ f , (3) \nwhere f and σ f are respectively fluxes and their associated uncertainties from Eq. (2). We compute the SNR of every DR2 object at each epoch of its measured and simulated light-curve for each band ( g , r , i ).', '4. Testing the simulation framework': 'In this section, we describe our work to validate the accuracy of skysurvey in reproducing the DR2 light-curves. By taking into account the real-time observing conditions and cadence of ZTF and the observed properties of the objects gathered from the data, we aim to replicate individual objects from the DR2 sample and compare the simulated and measured light-curves. In 4.1, we describe the methodology we followed, in 4.2 we show the comparison of a simulated and measured light-curve for one object of the sample, then in 4.3 we discuss the comparison of the fluxes, their uncertainties and their SNR for the whole sample. \nFig. 1. Schematic representation of our simulation pipeline. Left: model light-curves are generated and combined with the true ZTF observing conditions to produce simulated light-curves. Selection criteria, matching those used to generate the ZTF SN Ia DR2 sample are then applied to produce a mock ZTF SN Ia DR2 dataset. Right: Comparing our predicted dataset to the ZTF SN Ia DR2 sample, we can estimate completeness and biases. \n<!-- image -->', '4.1. Methodology': 'To simulate the DR2 sample, we use the observed properties of the sample, i.e the SALT2 (Guy et al. 2007, 2010) fitted parameters of the measured light-curves as skysurvey inputs. The SALT2 parameters of the sample are presented in Rigault et al. (2024). We use t 0, x 1, c and z of every individual object in skysurvey combined to ZTF observing logs to simulate every SN Ia. In order to evaluate the accuracy of our simulations, we compare the fluxes, flux uncertainties and SNR of the simulated and measured light-curves at all epochs for all DR2 objects. Furthermore, to avoid statistical fluctuations, each SN Ia is simulated 10 times, meaning that for the whole DR2 sample, we generated about 30,000 SNe Ia.', '4.2. One example': 'As an example, Fig. 2 shows the measured and simulated lightcurve of ZTF19abgppki, in g (left), r (middle) and i -band (right). The top part of each plot shows a good agreement between the simulated and measured fluxes in every three bands at all epochs. We compute the SNR values of the simulated and measured light-curves and their ratios. The bottom part of each plot in Fig. 2 shows the evolution of the SNR ratios (SNR sim / SNR data ) as a function of time (in mjd ) for the three bands. The crosses represent the median SNR values per observation. We can observe that the SNR ratios stand above 1, especially for low-fluxes at the starting point and in the tail of the light-curves, indicating that the simulated light-curves display higher SNR than the DR2 ones. And as the fluxes seem to match in the upper part of the plots of Fig. 2, it indicates that the simulated uncertainties are not \nreproducing the measured ones. To get a quantitative idea of the discrepancy, and then the correction to apply in our simulations, a statistical analysis over the full DR2 objects is necessary.', '4.3. Statistical test of skysurvey': 'From the 3628 SNe Ia of the full DR2 sample, we apply the following set of basic cuts on the DR2 light-curves properties from Rigault et al. (a) \n- -\| x 1 \| < 3,\n- -c ∈ [ -0 . 2 , 0 . 8],\n- -δ t 0 ≤ 1,\n- -δ x 1 ≤ 1,\n- -δ c ≤ 0 . 1.\n- -probfit > 10 -7 ,\n- -we require data points with a phase ( Φ ) ∈ [ -10 , 40]. \nWe apply additional cuts of good sampling, they can be found in Rigault et al. (a). The phase cut is dictated by the study of DR2 light-curve residuals led by Rigault et al. (2024). The final sample counts 2662 SNe Ia. For the purpose of our study, we will use this sample as our baseline. \nSo, we simulate 10 times those 2662 SNe Ia with their SALT2 parameters and their redshift as inputs to skysurvey .', '4.3.1. Light-curves comparison': 'To ensure an accurate comparison between the measured and simulated quantities, for every SN Ia we match its measured and simulated light-curve along with the associated observing logs. \nFig. 2. Top: Simulated (10 times) and measured ZTF19abgppki ( z = 0 . 06096) light-curves in g (left), r (middle) and i (right) bands. Bottom: evolution of the SNR ratio of the simulated light-curves w.r.t. measured data points as a function of time (in mjd ). The crosses show the median of the 10 simulations for the SNR ratio for each observation date. \n<!-- image --> \nThe DR2 fluxes and associated uncertainties are extracted from the di ff erence images, based on reference images, as discussed in the Smith et al. companion paper. So, we computed the flux uncertainties following Eq. (2) using skynoise derived from the di ff erence image limiting magnitudes or from the science image limiting magnitudes. The ratio of the simulated uncertainties and the DR2 associated ones ( σ sim /σ data ) are shown in Fig. 3 for g (top), r (middle) and i (bottom) bands: as grey curves for the di ff erence image limiting magnitudes ( di f f maglim ) and as blue curves for the science image limiting magnitudes ( scimaglim ). \nAt low flux (typically f < 10 4 ), where the skynoise contribution (Eq. (2)) is dominant, we can see that the ratios of the simulated uncertainties computed with skynoise from the di ff erence image limiting magnitude and the DR2 one are systematically above 1, while the ratios of the uncertainties computed from the science limiting magnitude are systematically below 1. Furthermore, departure from 1 is more pronounced with the di ff erence image limiting magnitude, that the limiting magnitude associated to the di ff erence image is not correctly estimated. In the case where the skynoise is estimated using the science image limiting magnitude, the simulated flux uncertainties are under estimated compared to the measured ones, which is expected because the di ff erence image processing add some background noise, but the resulting values are closer. It is a clear indication for the need of a new estimated skynoise in the simulations, in order to accurately replicate the DR2 light-curves. To estimate the new skynoise, we use data points which are at low flux ( f < 5000). We compute the median of the measured and simulated flux uncertainties per flux ranges. By computing the median of the binned measured and simulated uncertainties we obtain the corrective factor for the skynoise. We find that a corrective factor of 1 . 23 for gband, 1 . 17 for r-band and 1 . 20 for i-band are needed to solve the discrepancy between the simulated and measured uncertainties. This means that we need to increase the skynoise level associated to science image to account for the forced photometry output which is applied to the di ff erence images. To obtain the values of the new skynoise, we multiply the skynoise obtained from the science limiting magnitude by the corrective factors we estimated for each of both bands. \nIn addition, an error-floor uncertainty of 2 . 5%, 3 . 5% and 6% flux level, in respectively g , r and i -bands, is added to the DR2 flux uncertainties Rigault et al. (2024). In order to accurately \ncompare the DR2 flux uncertainties to the simulated ones, we need to account for this error-floor in the simulation framework. \nWe simulate back our sample with the correct prescription for the skynoise and the error-floor. The ratio of the newly simulated uncertainties and the measured ones for the sample are shown in the same figure (Fig 3). We compute the median values for the ratios per flux range, they are represented in the same figure by the bigger points. We can notice the data points are scattered around 1, with a bigger scatter at low-flux due to the skynoise dominance in the flux uncertainties. We can also notice that the binned data points are between the ratio of uncertainties computed with the science and di ff erence images limiting magnitudes. The binned ones are compatible with 1 along with their evolution with the flux, indicating that the newly simulated flux uncertainties and the measured ones are in good agreement. \nAs discussed in Rigault et al. (2024), an error-floor is necessary to account for the Gaussian distribution of light-curve data points w.r.t. SALT2 model along the full fitted phase range [10, 40]. The simulated fluxes were compared to the measured ones by computing the pull = ( fsim -fdata ) /σ data for every points of the 10 simulated light-curves of the full DR2. Fig. 4 shows the pull distribution for the three bands. The full histograms correspond to the pull computed by adding the error-floor, while the open dashed histograms are the pull computed without the error-floor. As light-curve data points are distributed following a 1-sigma Gaussian centered on zero w.r.t. their SALT2 fitted model (see Fig. 1 of Rigault et al. (2024) where σ data -pull = 1) and as the simulated light-curves are generated following a random Gaussian distribution w.r.t. to the SALT2 model (i.e. such as σ sim -pull = 1), we expect for the pull of Fig. 4 a Gaussian distribution centered on zero with a sigma equal to σ pull = \nq σ 2 data -pull + σ 2 sim -pull = √ 2. This prediction is represented by the solid grey curves on Fig. 4. The comparison between the expected pull and the computed one are in good agreement when taking into account the error-floor (full histograms), while the computed pull distributions are wider when the error-floor is omitted. This result shows that the amount of error-floor of 2 . 5%, 3 . 5% and 6%, respectively in g , r , and i -bands, are necessary for a good matching between simulations and the DR2 data. \n<!-- image --> \nFig. 3. Ratio of simulated and DR2 flux uncertainties ( σ sim /σ data ) as a function of data flux in g ( top ), r ( middle ) and i ( bottom ) bands. Small points are comparison between every simulated and observed objects at every phase, while the bigger points are the associated median binned values, both after skynoise correction (see text for details). The blue curves ( scimaglim ) are the binned values obtained from the scienceimage, while the grey curves ( di f f maglim ) are the binned values obtained from the di ff erence-image limiting magnitudes. \n<!-- image -->', '4.3.2. Signal-to-Noise Ratio comparison': 'We use the newly simulated light-curves, with the prescriptions, from 4.3.1, for the skynoise computed from the science limiting magnitudes with the correcting factors and the error-floor needed in the flux uncertainties. We compute the SNR of the simulated and observed data points at all epochs, following Eq (3). We represent the ratio of the SNR from the simulation and the DR2 as a function of the DR2 one in Fig. 5 for g (top), r (middle) and i (bottom) bands. We compute the median of the SNR ratios per SNR data range, they are represented in the same figure with bigger points. One can notice that the data points display a wide scatter especially at low SNR, due to the skynoise fluctuations, knowing its dominance in the flux uncertainties budget at these SNR values. The scatter is reduced at higher SNR. The binned SNR ratios are consistent with 1 in the three bands for SNR > 5. For the low-SNR ( < 5), the data points are below the detection limit and the SNR ratio is a ff ected by negative fluxes, so the SNR ratio ( sim / data ) = 1 is no more valid. It shows that the simulations framework replicates the fluxes and associated uncertainties of the measured DR2 light-curves at all epochs, across ZTF redshift range.', '4.4. Summary': 'We aimed to simulate the most realistic DR2 SN Ia sample, with all our knowledge of ZTF observations and SN Ia modelling. We use the ZTF true observing strategy and state-of-the-art SN Ia models in skysurvey to replicate the whole DR2 sample. We simulated 2662 SNe Ia, passing the good sampling and the basic cuts and tested multiple configurations of the simulation framework. We replicate the DR2 fluxes, associated uncertainties and \nFig. 4. Light-curve pulls = (flux sim - flux data ) / σ data for g ( top ), r ( middle ) and i ( bottom ) bands. The full (open dashed) histograms show the pulls with (without) error-floor, while the grey curves correspond to a Gaussian centered on zero and with a sigma equal to √ 2.Fig. 5. SNR ratio of simulated and DR2 as a function of DR2 SNR in g ( top ), r ( middle ) and i ( bottom ) bands. Small points are comparison between every simulated and observed objects at every phase, the bigger points are the associated median binned values. \n<!-- image --> \nSNR by using the skynoise computed from the science limiting magnitudes with a corrective factor of 1 . 23 for g-band, 1 . 17 for r-band and 1 . 20 for the i-band. In addition, we account for an error-floor of 2 . 5%, 3 . 5%and 6% of the flux level in respectively g , r and i -band. With these prescriptions, we can accurately sim- \nulate the ZTF DR2 SNIa light-curves at all epoch for every band ( g , r , i ) up to the redshift limit of the survey z ≈ 0 . 15.', '5. Simulating the ZTF SN Ia DR2 sample': 'Simulations are crucial for correcting biases in the analysis of SNe Ia to infer cosmological quantities. In this section, we present our work on the DR2 selection sample with simulations.', '5.1. ZTF SN Ia DR2 simulation': 'To study selection e ff ects on DR2 sample a high-statistic simulation has been carried on. To avoid limited sample at very low redshift, this simulation was partitioned in redshift bins of 0.005 starting from 0 to 0.15, i.e. by generating 25k SNe Ia in 30 redshift bins. Then a weighting procedure is applied to retrieve the natural power law redshift evolution. SALT2 model is used for generation with a color parameters c drawn from an asymmetric distribution following the model from Scolnic & Kessler (2016) and a stretch parameters x 1 drawn from a bimodal distribution from Nicolas, N. et al. (2021). \nAll selection cuts applied to the DR2 sample are reproduced to the simulated light-curves. First, the BTS sampling cuts (see section 2.3 of Perley et al. 2020) with the selection function (see Fig. 4 of Perley et al. 2020) are applied to the generated lightcurve data points. Second, the basic DR2 cuts (see table 2 of Rigault et al. (a)) are applied to the fitted light-curves. \nAll together, the selections cuts allow to reproduce the DR2 redshift distributions as shown in Fig. 6. The natural redshift power law is well decreased by all selection cuts to be able to reproduce DR2 redshift distribution. The matching is not perfect but in very good agreement. Indeed the high-redshift part ( z > 0 . 07) reproduces the drastic reduction due mainly to the selection function up to a factor 100 at z = 0 . 15. In the lowredshift regime, a small bump is seen in DR2 data compared to the smooth simulation especially in the range 0 . 03 < z < 0 . 04. The origin of this bump has not been identified, but it might arise from the "local" large-scale structure of the Universe within the ZTF footprint or an unmodelled population of underluminous SNeIa. \nSimulated light-curves fitting is conducted as for real data using the SALT2 model. A comparison of the main outputs from SALT2 parameters is shown in Fig. 7: without redshift cut (grey data points and light-blue histogram for simulation) and for z ≤ 0 . 06 (smaller black data points and blue histogram for simulation). The three SALT2 output parameters are, at first order, well reproduced by the simulation, especially for the lowredshift selection z ≤ 0 . 06. We observed clearly the Malmquist bias for faint SNe Ia in the mb = -2 . 5 log 10( x 0) + 10 . 635 distribution (Fig. 7, upper plot): compared to the low-redshift sample ( z ≤ 0 . 06), the full sample shows a steeper distribution at high mb illustrating the selection e ff ect for higher redshift. On the other side, the shape of the stretch ( x 1, Fig. 7, middle plot) and the colour ( c , Fig. 7, lower plot) distributions look the same (at first order) for the full sample and the low-redshift selection. A deeper analysis of the redshift dependence of these distribution is discussed in the next section. \nTo quantify the ability of the simulation to reproduce DR2 distributions, we computed a sort of chi-square per degree-offreedom defined as \nχ 2 / nd f = 1 Nbin Nbin X i = 1 ( Ndata , i -Nsim , i ) 2 Ndata , i (4) \nFig. 6. Comparison of DR2 redshift distribution (black points) with skysurvey simulation in log scale (upper plot) and linear scale (lower plot) from generated SNe Ia distribution (black dotted line) to selected ones (full blue histograms). All simulated distributions (before and after selection cuts) are normalized to the data sample. \n<!-- image --> \nTable 1. χ 2 / nd f (with an estimation of its uncertainty) between DR2 and simulation distributions of SALT2 parameters ( mb , x 1 and c ) for the full sample (no z cut) and for the volume limited sample ( z ≤ 0 . 06). \nwhere Nbin is the number of bins in the compared histograms which plays the role of degree-of-freedom. In this definition, we assume a Gaussian uncertainty for the data counting per bin ( σ data , i = p Ndata , i ) and we neglect the statistical fluctuation of the simulation (the number of simulated SNe Ia passing all selection cuts is 47 times the DR2 dataset). The limits of the histograms are defined by cutting the 0 . 5%of the DR2 left and right tails using the quantile method, i.e. keeping 99% of the bulk of the distributions. For the robustness of the analysis, we varied Nbin from 80 to 100 by step of 1 bin (i.e. 20 χ 2 / nd f calculations for each distribution comparison) and then we computed the mean χ 2 / nd f and evaluated an uncertainty on the mean value. \nThe results are presented in Table 1 for the three SALT2 parameters ( mb , x 1 and c ) for the full DR2 sample (no z cut) and the lower redshifts part ( z ≤ 0 . 06). For the full data sample, the chi-square per degree-of-freedom ( χ 2 / nd f ) is between 1.4 and 1.7, depending on the SALT2 parameter, meaning that the simulations are not reproducing perfectly the full data-set. The worst agreement is observed for the B-band magnitude ( mB ) parameter. When comparing DR2 and simulated distributions for the lower redshift part ( z ≤ 0 . 06, which corresponds to what we call the volume limited sample, see Sect. 5.2) we observe a small improvement for the mB distribution, a much better matching for the stretch ( x 1) distributions, and a consequent improvement for the colour distribution ( c ) for which the χ 2 / nd f is decreased by 25%. This quantitative comparison shows that the simulation is close to reproduce the main properties of our SN Ia data sample when considering the volume limited sample ( z ≤ 0 . 06) which is described in more detail in the next section. \nFig. 7. Comparison of DR2 SALT2 parameters (points with error bars) -mb (upper plot), x 1 (middle plot) and c (lower plot) - with skysurvey SNe Ia simulation after light-curves fit (histograms): without redshift cut (grey points and light-blue histogram) and for z ≤ 0 . 06 (smaller black points and blue histogram). \n<!-- image -->', '5.2. Volume limited sample': 'An analysis of SALT2 parameters as a function of the redshift is necessary to estimate the limited redshift for which SNe Ia are not biased. Fig. 8 shows a comparison between the input stretch (left plot) and colour (right plot) distributions before (grey filled histograms) and after selection cuts (open histograms), as for the ZTF-DR2-Cosmos data, for cumulative redshift ranges sampling the bulk of the full ZTF redshift interval. From those plots it appears that input distributions are well reproduced at very low redshift and start to be distorted when redshift increased beyond 0.05. \nA volume limited sample can be defined as a data sample for which the input SALT2 distributions are not significantly distorted, meaning that there is no bias in the selected SNe Ia. To quantify the redshift limit of the volume limited sample a Kolmogorov-Smirnov (KS) test has been performed between the input distributions used for generation and the distributions retrieved after all selection cuts applied to simulated sample, as for real data, and as a function of redshift limit ( z < zlim ). The result \nFig. 8. Comparison of stretch (left plot) and colour (right plot) input distributions before (grey filled histograms) and after all selection cuts (open histogram) for lower redshifts ( z ≤ 0 . 06, open blue histogram) and higher redshifts ( z > 0 . 06, open red histogram). \n<!-- image --> \nof the KS test of both stretch and colour parameters is shown in Fig. 9 and compared to the probability limits at 1, 2 and 3 σ . This comparison allows to estimate the redshift limit zlim ≈ 0 . 07 (vertical black dashed line) of a SNe Ia sample with model parameters (here SALT2 stretch and colour) compatible with input model distributions within 3 σ . Indeed, up to this redshift limit the stretch distribution of selected SNe Ia matchs almost perfectly the input distribution, while the colour distribution starts to diverge from the input distribution for a redshift z > 0 . 05. \nFurthermore, we observe that the KS test behaviour for low redshift (typically z < 0 . 08) is similar when comparing the parameter distributions after selection to the generated input distributions, as well as for true parameter (full lines) than for fitted parameter with SALT2 (dotted lines). This result reinforces the confidence in the fact that the volume limited sample as define before is not biased. We can also note that for higher redshitfs ( z > 0 . 08) the KS test p-value increases for fitted parameters compared to generated ones. This e ff ect can be interpreted by a spreading of stretch and colour distributions with fitted parameters due to the fact that SNe Ia are fainter and then the uncertainty in the estimation of SALT2 parameters in the fitting of light-curves becomes more important. This spreading dilutes the distortion of the distributions after selection w.r.t. to the input ones. \nThis redshift dependency analysis of the input SALT2 parameters after selection explains the very small and the more consequent disagreement of the stretch and colour distributions, respectively, compared to the ZTF-DR2-Cosmo data integrated in the full redshift range ( z < 0 . 15) as seen in Fig. 7. \nIn conclusion, the redshift limit zlim ≤ 0 . 06 constitutes a robust estimation to define the volume limited sample of the DR2 dataset, i.e. a sample without any bias when the light-curves are fitted using the SALT2 model.', '6. Toward cosmological analysis': "The simulation framework was tested up to the Hubble-Lemaître diagram. To this end, the distance modulus of the fitted SNe Ia of the high-statistic simulation were computed by assuming the fiducial input parameters for the standardization: µ i = mb , i -( Mb -α x 1 , i + β ci ) for every ( i ) SN Ia with Mb = -19 . 3, α = 0 . 14 and β = 3 . 15. Then the residuals to the fiducial input cosmology (Planck 2018 Planck Collaboration et al. 2020b) were deduced: ∆ µ i = µ i -µ ( zi ), where µ ( zi ) is the theoretical result of the fiducial cosmology for every SN Ia redshift zi . \nFig. 9. KS test p-value for stretch (orange curve) and colour (red curve) as a function of the redshift limit: full lines show the comparison between generated distributions after w.r.t. before selection, while dashed lines compare distributions of fitted SALT2 parameters after selection w.r.t. to generated distributions before selection. \n<!-- image --> \nFig. 10. Hubble-Lemaître diagram residual distribution, assuming no environmental steps, for the DR2 (black points) and the simulation with fitted SALT2 parameters (full histogram) and the generated ones (open histogram). \n<!-- image --> \nThe residual distribution is presented in Fig. 10 with a comparison between DR2 data (2624 SNeIa) and simulation with fitted SALT2 parameters (full histogram) as well as generated parameters (open histogram). Firstly, the dispersion of the fitted residual distribution is close to the generated one, indicating that the residual is mainly driven by intrinsic dispersion ( σ int ). Secondly, the fitted simulation reproduces overall the distribution measured from the DR2 dataset ( χ 2 / nd f = 2 . 00), with small discrepancies that we will tackle in the upcoming DR2.5 cosmological data release. The robust agreement between our simulated samples and the DR2 dataset for the volume-limited sample, with χ 2 / nd f = 1 . 81 , 1 . 61 , 1 . 26 , 1 . 23 , 1 . 63 when considering the distribution of z , mB , x 1 , c , ∆ µ respectively, indicates that both the underlying distribution of parameters and selection of events are well understood, laying the foundations for a cosmological analysis. \nThe DR2 dataset (and other literature samples) have observed significant correlations between the residual luminosity, standardisation parameters and measures of the local environment (Ginolin et al. 2024b,a, e.g.). To ensure that our estimates of the cosmological parameters are unbiased, these must be included in our simulation framework, which must also be updated to match the reprocessed 'scene-model photometry'SMP Lacroix et al. and benchmarked against the widely used software, e.g. SNANA . This work will be released as part of the DR2.5 cosmological analysis, due for release in late 2025.", '7. Conclusion': "In this paper we presented the performance of the simulation framework skysurvey and its advanced version skysurvey - developed for the Zwicky Transient Facility (ZTF) survey. The targeted simulation of the first phase of ZTF, from April 2018 and December 2020, was used to validated the full pipeline. The confrontation to the ZTF SN Ia DR2 sample has shown that neither the science nor the di ff erence images magnitude limit are able to reproduce the measured flux uncertainties. A tuning of the sky-noise with an increasing factor of 1.23, 1.17 and 1.20 for g, r and i-bands, respectively, has been derived to match flux uncertainties when using science image magnitude limit with an additional calibration error of 2.5%, 3.5% and 6% for g, r and ibands. The simulation using realistic SALT2 stretch and colour distribution has shown the ability of the skysurvey package to reproduce the ZTF SN Ia DR2 sample. Finally, a redshift study of SALT2 parameters allowed to identify a volume limited sample, z ≤ 0 . 06, i.e. an unbiased sample of about 1000 SNe Ia. This volume limited sample is unique to perform new studies on SNe Ia and especially analysis to improve their standardization procedure for future cosmological analysis. \nAcknowledgements. Based on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grant No. AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann Institute of Science, the Oskar Klein Center at Stockholm University, the University of Maryland, the University of Washington, Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. The ZTF forced-photometry service was funded under the Heising-Simons Foundation grant #12540303 (PI: Graham). SED Machine is based upon work supported by the National Science Foundation under Grant No. 1106171 This work was supported by the GROWTH project (Kasliwal et al. 2019) funded by the National Science Foundation under Grant No 1545949. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement n 759194 - USNAC). P.R. acknowledges the support received from the Agence Nationale de la Recherche of the French government through the program ANR21-CE31-0016-03. UB is supported by the H2020 European Research Council grant no. 758638 GD is supported by the H2020 European Research Council grant no. 758638 L.G. acknowledges financial support from AGAUR, CSIC, MCIN and AEI 10.13039 / 501100011033 under projects PID2020-115253GAI00, PIE 20215AT016, CEX2020-001058-M, and 2021-SGR-01270. This work has been supported by the research project grant 'Understanding the Dynamic Universe' funded by the Knut and Alice Wallenberg Foundation under Dnr KAW 2018.0067 and the Vetenskapsrådet, the Swedish Research Council, project 2020-03444 LH is funded by the Irish Research Council under grant number GOIPG / 2020 / 1387 Y.-L.K. has received funding from the Science and Technology Facilities Council [grant number ST / V000713 / 1] KM is supported by the H2020 European Research Council grant no. 758638. T.E.M.B. acknowledges financial support from the Spanish Ministerio de Ciencia e Innovación (MCIN) and the Agencia Estatal de Investigación (AEI) 10.13039 / 501100011033 under the PID2020-115253GA-I00 HOSTFLOWS project, and from Centro Superior de Investigaciones Científicas (CSIC) under the PIE project 20215AT016 and the program Unidad de Excelencia María de Maeztu CEX2020-001058-M. JHT is supported by the H2020 European Research Council grant no. 758638.", 'References': 'Aubert, M., Rosnet, P., Popovic, B., et al. 2024, ZTF SN Ia DR2: Exploring SN Ia properties in the vicinity of under-dense environments \nBarbary, K., Barclay, T., Biswas, R., et al. 2016, SNCosmo: Python library for supernova cosmology, Astrophysics Source Code Library, record ascl:1611.017 \nBellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, PASP, 131, 018002 Betoule, M., Kessler, R., Guy, J., et al. 2014, Astronomy & Astrophysics, 568, A22 \nBlagorodnova, N., Neill, J. D., Walters, R., et al. 2018, PASP, 130, 035003 Brout, D. et al. 2022, Astrophys. J., 938, 110'}
2024arXiv240905801P	In this paper we investigate the properties of relativistic stars made of isotropic matter within the framework of the minimal Standard Model Extension where a bumblebee field coupled to spacetime induces spontaneous Lorentz symmetry breaking. We adopt analytic equationsofstate describing either condensate dark stars or strange quark stars. We solve the structure equations numerically and we compute the masstoradius relationships. The influence of the bumblebee parameter l is examined in detail and an upper bound is obtained using the massive pulsar PSR J07406620 and the strangely light HESS J1731347 compact object.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.05801', '2024arXiv240905801P', 'arXiv:2409.05801']	['General Relativity and Quantum Cosmology', 'Astrophysics - Solar and Stellar Astrophysics']	Strange Quark Stars and Condensate Dark Stars in Bumblebee Gravity	2,024	193	0.35	['EPRINT_HTML', 'EPRINT_PDF']	1	https://arxiv.org/pdf/2409.05801.pdf	{'Strange Quark Stars and Condensate Dark Stars in Bumblebee Gravity': "Grigoris Panotopoulos 1, ∗ and Ali Ovgun 2, † \n1 \nDepartamento de Ciencias F'ısicas, Universidad de la Frontera, Casilla 54-D, 4811186 Temuco, Chile. \n2 Physics Department, Eastern Mediterranean University, Famagusta, Cyprus.", 'Abstract': 'In this paper, we investigate the properties of relativistic stars made of isotropic matter within the framework of the minimal Standard Model Extension, where a bumblebee field coupled to spacetime induces spontaneous Lorentz symmetry breaking. We adopt analytic equations-of-state describing either condensate dark stars or strange quark stars. We solve the structure equations numerically, and we compute the mass-to-radius relationships. The influence of the bumblebee parameter l is examined in detail, and an upper bound is obtained using the massive pulsar PSR J0740+6620 and the strangely light HESS J1731-347 compact object. \nKeywords: Modified gravity; Relativistic stars; Stellar composition; Lorentz symmetry breaking; Bumblebee gravity.', 'I. INTRODUCTION': "General Relativity and the Standard Model respectively describe gravity classically and particles/interactions quantum mechanically. Unifying these theories remains a fundamental challenge. While proposed quantum gravity theories offer potential solutions, direct experimental verification is currently unattainable due to the requisite Planck scale ( 10 19 GeV) energies. Nonetheless, subtle quantum gravity effects might manifest at lower energy scales, opening avenues for experimental investigation. One such potential signal is the breaking of Lorentz symmetry, rooted in special relativity, ensures that physical laws are consistent for all observers in inertial frames. This symmetry, encompassing rotational and boost transformations, is fundamental to general relativity and the standard model of particle physics . In curved spacetimes, local Lorentz symmetry is maintained due to the Lorentzian nature of the background. However, violating Lorentz invariance introduces directional or velocity dependencies in physical variables, altering the dynamics of particles and waves. In 1989, Kosteleck'y and Samuel introduced bumblebee gravity [1], a model for spontaneous Lorentz violation. In this model, a bumblebee field with a vacuum expectation value (VEV) breaks Lorentz symmetry through the dynamics of a single vector field, B µ . This approach suggests that investigating Lorentz symmetry violation could provide insights into Planck-scale physics, combining aspects of both General Relativity and the Standard Model within the framework of effective field theories known as the Standard Model Extension (SME), which offers a framework that includes all possible coefficients for Lorentz/CPT violation, with its gravitational sector defined on a Riemann-Cartan manifold, treating torsion and the metric as dynamic geometric quantities. While the SME allows for non-Riemannian terms, research has primarily focused on the metric approach, using the metric as the main dynamic field. There are certain effects of Lorentz violation within the gravitational sector have been explored in [2-18]. \nNew discoveries of super-dense objects are pushing the boundaries of our understanding in astrophysics. While countless compact objects have been found, the exact laws governing their behavior remain a mystery. Recent observations have challenged the traditional theories, particularly general relativity and the standard model for neutron stars. One key example is the secondary object in the GW190814 merger [20-22]. Some scientists believe it might be a quark star, with its properties described by a single parameter ( λ ) within the framework of general relativity. This theory aligns with recent research suggesting non-strange quark matter as the most stable form of baryonic matter under extreme conditions [23]. Promising findings have also been obtained \nby similar investigations using the quark matter phase or using other models for matter behaviour (equation of state) [15-19]. \nStrange stars: Strange quark stars are hypothetical compact objects proposed as an alternative to neutron stars [24-30]. Composed of quark matter, which is theorized to be the absolutely stable ground state of hadrons [31, 32], these stars could explain the puzzling nature of super-luminous supernovae [33, 34]. Unlike ordinary supernovae, these exceptionally bright explosions are 100 times more luminous and occur in approximately one out of every thousand cases. \nWhile strange quark stars remain theoretical at present, their existence cannot be definitively excluded. Some observed compact objects exhibit anomalous properties, such as unusually small radii, that standard neutron star models cannot fully explain [35-37]. These discrepancies have fueled speculation about the potential presence of strange matter in these objects. While some researchers propose that strange matter could constitute the core of hybrid neutron stars [38-40], others argue that such stars would be virtually indistinguishable from ordinary neutron stars [41]. \nBoson stars: The standard cosmological model posits that dark matter consists of weakly interacting massive particles (WIMPs). While this assumption aligns well with large-scale cosmic observations (scales of megaparsecs and larger) ( ≥ Mpc ), it faces challenges at smaller galactic scales. Discrepancies such as the core-cusp problem, the diversity problem, the missing satellites problem, and the too-big-to-fail problem arise [42]. Self-interacting dark matter has been proposed as a potential solution [43, 44], as particle collisions within this model can mitigate the formation of sharp density cusps. Furthermore, if dark matter is composed of ultralight scalar particles with a mass below one electronvolt m ≤ eV and a weak repulsive self-interaction, it could form a BoseEinstein condensate (BEC) with long-range correlations. This BEC scenario has been proposed as a potential explanation for the aforementioned galactic scale discrepancies [45-47]. \nRegarding the composition and inner structure of compact objects, the most massive pulsars [48-50] observed over the last 15 years or so are putting constraints on different equations-ofstate, since any mass-to-radius relationship that predicts a highest mass lower than the observed ones must be ruled out. In order to model a certain star, it would be advantageous to know both its mass and its radius, which is not always the case as measuring the radius is way more difficult. There are some good strange quark star candidates, see e.g. Table 5 of [51] or Table 1 of [37], and also the recently discovered massive pulsar PSR J0740+6620 [52-54] and the strangely light object HESS [55], where both the stellar mass and radius are known observationally. \nOur current understanding of ultra-dense stars, particularly those exceeding the limitations of traditional neutron stars, remains a fascinating mystery in astrophysics. This work explores this field by examining the characteristics of isotropically-composed stars. \nOur key objective is to explore the impact of a phenomenon known as spontaneous Lorentz symmetry breaking within these exotic stars. This breaking is induced by a hypothetical field, the bumblebee field, coupled to spacetime in the framework of the minimal Standard Model Extension. \nIn the present article we propose to study non-rotating relativistic stars made of isotropic matter within bumblebee gravity adopting for the matter content two analytic equations-of-state. Our work is organized as follows: After this introductory section, we briefly review how to obtain interior solutions describing hydrostatic equilibrium within GR in the next section, while in section 3 we present the Bumblebee model as well as the corresponding modified structure equations. In the fourth section we comment on the equations-of-state assumed here, and we discuss our numerical results. Finally, we summarize and conclude our work in section 5. Throughout the manuscript we work in geometrical units setting the universal constants (Newton's constant and speed of light in vacuum) to unity, G = 1 = c , and we adopt the mostly positive metric signature {-, + , + , + } .", 'II. RELATIVISTIC STARS IN GENERAL RELATIVITY: HYDROSTATIC EQUILIBRIUM AND STRUCTURE EQUATIONS': "Due to their intense gravitational fields, the structure and dynamics of neutron stars (NSs) are governed by Einstein's equations of General Relativity (GR) [56], given by: \nG µν = R µν -1 2 Rg µν = 8 πT µν , (1) \nwhere R µν and R denote the Ricci tensor and Ricci scalar, respectively. The energy-momentum tensor for isotropic matter, T µν , is expressed as: \nT µν = Pg µν +( P + ρ ) u µ u ν , (2) \nwhere g µν is the metric tensor, P is the pressure, E is the energy density, and u µ is the fourvelocity. For static, spherically symmetric stars, the line element in Schwarzschild-like coordinates { t, r, θ, ϕ } is described as: \nds 2 = -e ν ( r ) dt 2 + e λ ( r ) dr 2 + r 2 ( dθ 2 + sin 2 θdϕ 2 ) , (3) \nwhere e ν ( r ) and e λ ( r ) are the metric functions. One obtains the Tolman-Oppenheimer-Volkoff (TOV) equations [57, 58] for the equilibrium structure of NSs by solving the Einstein field equation with the above-defined metric, \ndP ( r ) dr = -[ ρ ( r ) + P ( r )][ m ( r ) + 4 πr 3 P ( r )] r 2 (1 -2 m ( r ) /r ) , (4) \ndm ( r ) dr = 4 πr 2 ρ ( r ) . (5) \nThe metric functions become \ne λ ( r ) = (1 -2 m/r ) -1 , (6) \nν ( r ) = log (1 -2 M/R ) + 2 ∫ r R dr ' e λ ( r ' ) r ' 2 [ m ( r ' ) + 4 πr ' 3 P ( r ' )] . (7) \nCombined with the given EoS-i.e., P ( ρ ) of the matter-, TOV equations can be solved with the initial conditions at the center of the star, m ( r = 0) = 0 and ρ ( r = 0) = ρ c , where ρ c is the central energy density. The stellar mass and radius are determined using the matching conditions at the surface of the star upon comparison to the Schwarzschild exterior vacuum solution [59] \nds 2 = -(1 -2 M/r ) dt 2 + 1 1 -2 M/r dr 2 + r 2 d Ω 2 . (8) \nThus, the radius of the star is determined by requiring that the energy density vanishes at the surface, P ( R ) = 0, and the stellar mass is then given by M = m ( R ) .", 'III. BUMBLEBEE GRAVITY: MODIFIED TOV EQUATIONS': "In this section, we briefly review the bumblebee gravity, a theory that expands on General Relativity (GR). Inspired by the works of Kostelecky and collaborators [1], bumblebee gravity introduces a twist: it breaks a fundamental symmetry (Lorentz symmetry) within the realm of gravity. This twist manifests as a special value (nontrivial vacuum expectation value) that influences how other fields behave around a mysterious 'bumblebee field'. Interestingly, even with these extra interactions, bumblebee gravity preserves the geometrical framework and basic laws established by GR in curved spaces. \nAmong various models capable of breaking Lorentz symmetry, one of the simplest approaches involves a vector field B µ , known as the bumblebee field, within a torsion-free spacetime. This can be expressed as follows [15]: \nS = ∫ d 4 x √ \| g \| ( 1 16 π ( R + ξB µ B ν R µν ) -1 4 B µν B µν -V ( B µ B µ ± b 2 ) + L M ) , (9) \nwhich encapsulates the interaction between the bumblebee vector field ( B µ ) and gravity, represented by the Ricci tensor ( R µν ). The strength of this interaction is modulated by a coupling constant ( ξ ). \nThe bumblebee field B µν ≡ ∂ µ B ν -∂ ν B µ also interacts with matter, as outlined by the matter Lagrangian density ( L m ). The potential function V ( B µ B ν ± b 2 ) , where b 2 is a positive constant, dictates the behavior of the bumblebee field. Notably, this potential enables the field to achieve a non-zero vacuum expectation value, specifically B µ = b µ = (0 , b r ( r ) , 0 , 0) , where b r ( r ) is a function of the radial coordinate. \nTo break a specific symmetry, such as U (1) , the potential must reach a minimum at V = 0 , with its derivative V ' = 0 also being zero at that point. This condition allows the bumblebee field to acquire a constant, non-zero vacuum expectation value, denoted by b µ , which satisfies B µ ≡ b µ with b µ b µ = ∓ b 2 ≡ constant (note that b 2 is a real positive constant). \nBy varying the action, as given in Eq. (9), we derive the equations governing both the gravitational field and the dynamics of the bumblebee field, including the corresponding equations of motion. \nR µν -1 2 g µν R = 8 π ( T B µν + T M µν ) , (10) \n∇ µ B µν = J B ν + J M ν . (11) \nWithin the bumblebee gravity framework, two key terms come into play: the source term for the BF (represented by J M ν ) and the current due to the BF's self-interaction (denoted by J B ν = 2 V ' B ν -ϱB µ R µν / 8 π ). Note that the prime symbol ( ' ) denotes differentiation with respect to the potential argument, specifically V ' ≡ ∂V ( y ) /∂y , where y = B µ B ν ± b 2 . \nAdditionally, T B µν represents the energy-momentum tensor associated with the bumblebee field, which can be expressed as \nT B µν = B µα B α ν -1 4 g µν B αβ B αβ -g µν V +2 B µ B ν V ' + ϱ 8 π [ 1 2 g µν B α B β R αβ -B µ B α R αν -B ν B α R αµ + 1 2 ∇ α ∇ µ ( B α B ν ) + 1 2 ∇ α ∇ ν ( B α B µ ) -1 2 ∇ 2 ( B µ B ν ) -1 2 g µν ∇ α ∇ β ( B α B β ) ] . (12) \nRemember that T M µν represents the energy and momentum distribution of matter, and we have a constant value denoted by b µ b µ = ∓ b 2 ≡ constant related to the bumblebee field (where b µ is the VEVand b is its magnitude). These factors influence the radial component b r ( r ) of the BF when it reaches its minimum energy state (VEV). Interestingly, research by Casana et al. [15] shows that this radial component can be expressed as b r ( r ) = \| b \| e ζ ( r ) . \nThe next step is to understand how this VEV of the BF affects the geometry of spacetime itself. This is described by the metric, which will be expressed in the following spacetime \nds 2 = -e 2 χ ( r ) dt 2 + e 2 ζ ( r ) dr 2 + q 2 r 2 d Ω 2 , (13) \nwhere χ ( r ) and ζ ( r ) , valid in the region 0 ≤ r ≤ R , represent the sought metric functions. It is noteworthy that choosing e -2 ζ ( r ) = g ( r ) is always possible, resulting in: \ng ( r ) = 1 -2 M( r ) r , (14) \nin which the collective mass function denoted by M( r ) can be viewed as the combination of the BH mass. \nThus, for an interacting system of this nature, the energy-momentum tensor associated with the spacetime metric can be represented as T µ ν = diag [ -ρ ( r ) , P r ( r ) , P θ ( r ) , P ϕ ( r )] . Consequently, these considerations facilitate the derivation of the Einstein field equations G µν = 8 πT µν as \n8 πρ ( r ) = 1 -q 2 +2 q 2 M ' ( r ) 2 2 , \n8 πP θ ( r ) = 8 πP φ ( r ) = ( 1 r + χ ' ( r ) )( M ( r ) r 2 -M ( r ) r + ( 1 -2 M ( r ) r ) χ ' ( r ) ) + ( 1 -2 M ( r ) r ) χ '' ( r ) . \nq r (15a) 8 πP r ( r ) = 1 r 2 ( 1 -1 q 2 ) -2 M ( r ) r 3 + 2 r ( 1 -2 M ( r ) r ) χ ' ( r ) , (15b) ' (15c) \nWithin the bumblebee gravity framework, the parameter called q 2 = 1 / ( l +1) plays a crucial role. It depends on a constant known as parameter of Lorentz symmetry breaking l , which itself is linked to the coupling strength ξ of the BF and a constant related to the BF b as with l = ξb 2 . The prime symbol ( ' ) denotes differentiation with respect to the radial position (r). Interestingly, when we take the limit as l approaches zero, q approaches 1 , has a significant implication. In this specific limit ( l → 0 , q 2 → 1 ), the Einstein field equations (refer to Eq. (15) for reference) simplify back to their standard form. \nNow, let us bring in another piece of the puzzle: the equation of state P r ( r ) = ωρ ( r ) , which relates pseudo-pressure ( P r ( r ) ) and energy density ( ρ ( r ) ) with a constant factor ( ω ). By combining this equation of state with the modified Einstein field equations we obtained earlier (Eq. (15a)), and considering a well-known law in physics (conservation of energy-momentum) T ν µ ; ν = 0 , we can arrive at a modified version of the TOV equations. These modified TOV equations are crucial for understanding how bumblebee gravity affects the structure and stability of stars and other selfgravitating objects. \ndχ ( r ) dr = M ( r ) -( 1 2 -1 2 q 2 ) r +4 πr 3 P r ( r ) r ( r -2 M ( r )) , (16a) \ndP r ( r ) dr = -( ρ ( r ) + P r ( r )) dχ ( r ) dr , (16b) \nHence, in the exterior (vacuum) region where T µν = 0 , the solution simplifies to the bumblebee black hole (BH) solution [15] \nds 2 = -(1 -2 M/r ) dt 2 + l +1 1 -2 M/r dr 2 + r 2 d Ω 2 , (17) \nor, after a coordinate transformation and a mass redefinition as follows \nt → t, r → √ 1 + l r, M → √ 1 + l M, (18) \nit may be recast in the following form [17] \nds 2 = -(1 -2 M/r ) dt 2 + 1 1 -2 M/r dr 2 + q 2 r 2 d Ω 2 . (19) \n̸ \nWhen studying a star in bumblebee gravity, we need to ensure our solutions for the different regions (interior and exterior) seamlessly match at the star's surface, located at a specific radius ( R ). To achieve this perfect match, we impose three matching conditions when q = 1 :", '· Matching Mass:': 'M = M ( R ) (20) \nthe first condition states that the value of the mass function ( M ( r ) ) at the surface ( r = R ) must be equal to the total mass ( M ) of the star.', '· Zero Pressure at the Surface:': 'P r ( R ) = 0 , (21) \nthe second condition requires the pressure at the surface to be zero. This condition allows us to compute the radius of the star.', '· Setting the Spacetime Geometry:': "e 2 χ ( R ) = 1 -2 M/R, (22) \nthe third condition involves the exponential term in a metric function ( e 2 χ ( R ) ). Here, we stipulate that its value at the surface ( r = R ) is related to the star's mass ( M ) and radius ( R ) through a specific relationship ( 1 -2 M/R ). This essentially sets the initial condition for this metric function, which helps describe the geometry of spacetime around the star. \nIn summary, the aforementioned matching conditions act as bridge equations, ensuring that our solutions for the star's internal structure and the surrounding vacuum smoothly connect at the boundary, providing a physically consistent description of a bumblebee gravity star.", 'IV. PROPERTIES OF STARS: NUMERICAL ANALYSIS AND DISCUSSION OF MAIN RESULTS': 'In this section, we explore the properties of relativistic stars made of isotropic matter, P r = P θ = P ϕ = P , within the framework of Lorentz-violating gravity theories. Through numerical integration of the relevant structure equations, we subsequently present and analyze our key findings', 'A. Equation-of-state': 'In the discussion to follow we shall consider i) a polytropic EoS and ii) MIT bag model [60, 61] for quark matter. Polytropes of the form \nP = kρ γ , γ = 1 + 1 n (23) \nwith n being the polytropic index, are used to describe interior solutions of white dwarfs (index n = 3 or n = 3 / 2 ) as well as condensate dark stars (index n = 1 ). Here we shall consider the case \n2.5 \nFIG. 1: Mass-to-radius relationships for the two EoSs considered here. Upper panel corresponds to polytrope ( l = 0 , 0 . 15 , 0 . 30 from top to bottom), lower panel to the extreme MIT bag model ( l = 0 , 0 . 030 , 0 . 055 from left to right). The contours in magenta color indicate the allowed mass and radius range of the massive pulsar PSR J0740+6620 and the light HESS J1731-347 compact object. \n<!-- image --> \nn = 1 assuming k = 4 . 012 × 10 -4 fm 3 /MeV . On the other hand, the extreme SQSB40 MIT bag \nFIG. 2: Stellar mass versus normalized central energy density for the two EoSs considered here. Upper panel corresponds to polytrope, lower panel to the extreme MIT bag model. \n<!-- image --> \nmodel is given by a linear analytic function [62] \nP = a ( ρ -ρ 0 ) (24) \nwith a = 0 . 324 and ρ 0 = 3 . 0563 × 10 14 g/cm 3 [62]. The numerical values of the three parameters of the MIT bag model are as follows: The bag constant B = 40 MeV/fm 3 , while the mass of the \nFIG. 3: Factor of compactness versus stellar mass for the two EoSs considered here. Upper panel corresponds to polytrope, lower panel to the extreme MIT bag model. \n<!-- image -->', 'B. Numerical results': 'In Fig. 1, we show the mass-to-radius relationships for the two EoSs considered here, i.e. a polytrope with n = 1 (upper panel) and the extreme MIT bag model (lower panel). The dashed curves represent the standard mass-to-radius relationship within GR. Those curves serve as a reference for how quark stars behave under GR without modifications from the bumblebee gravity theory. The curves in blue color correspond to the bumblebee parameter l = 0 . 15 for the polytrope and l = 0 . 03 for the MIT bag model. In this case, the mass-to-radius relationship for quark stars shows slight deviations from the GR case. The introduction of a small bumblebee parameter indicates minor changes in the structure and stability of the quark star, reflecting how quark stars might behave under weak modifications to gravity. The curves in red color represent the bumblebee parameter l = 0 . 30 for the polytrope and l = 0 . 055 for the MIT bag model. With a larger bumblebee parameter, the mass-to-radius relationship exhibits more significant deviations from the GR case. This indicates that stronger modifications to gravity have a notable impact on the structure, stability, and maximum mass of quark stars. \nIn Fig. 2 the stellar mass versus central (normalized and dimensionless) energy density is shown. As in previous figure, the upper panel corresponds to the polytrope, while the lower panel to the extreme MIT bag model. The dashed curves correspond to GR, the blue curves to the lowest value of l , and the red curves to the highest value of l . The curves increase, reach a maximum value and then they decrease. The maximum stellar mass is the one that is shown in Fig. 1. The Harrison-Zeldovich-Novikov criterion [63, 64] for stability \ndM dρ c > 0 (25) \nstates that a stellar model is a stable configuration only if the mass of the star grows with the central energy density. Therefore according to the criterion, only the first part of the function M ( ρ c ) is physical, namely up to the maximum stellar mass. \nNext, in Fig. 3 we display the factor of compactness, M/R , as a function of the stellar mass for the two EoS assumed in this work. Panels and colors are the same as before. The factor of compactness increases with the stellar mass, which acquires a maximum value, and which is precisely the same that is shown in Figures 1 and 2. Notice that the factor of compactness satisfies the Buchdahl limit of GR, ( M/R ) ≤ 4 / 9 ≈ 0 . 444 [65].', 'V. CONCLUSION': 'To summarize our work, in the present article we investigated the properties of non-rotating relativistic stars made of isotropic matter within the bumblebee gravity in four-dimensional spacetime. For the matter content of the stars we considered analytic equations-of-state corresponding to quark matter (extreme MIT bag model) and condensate dark stars (polytropic EoS with index n = 1 ). In the first part of the article we presented the exterior vacuum solution, the structure equations describing hydrostatic equilibrium of interior solutions as well as the appropriate conditions both at the center and at the surface (matching conditions) of the stars. \nNext, we integrated numerically the generalized TOV equations imposing the initial conditions at the center of the star, and then making use of the matching conditions we computed the stellar mass and radius as well as the factor of compactness. The impact of the bumblebee parameter l on the properties of the stars was shown in three figures, where we displayed i) the stellar mass versus stellar radius, ii) the factor of compactness as a function of the stellar mass, and iii) stellar mass versus central energy density. \nOur results show that a) as we increase l the deviation from GR becomes more significant, as expected, b) the factor of compactness satisfies the Buchdahl limit of GR, ( M/R ) ≤ 4 / 9 ≈ 0 . 444 [65], c) regarding the MIT bag mode, increasing l shifts the M-R relationships to the left, which implies that at a certain point the EoS cannot model the HESS compact object, d) regarding the polytrope, increasing l shifts the M-R profiles downwards, which implies that at a certain point the EoS cannot support stars at two solar masses. Therefore, an upper limit on l was obtained, although the bound coming from the MIT bag model EoS is the strongest one.', 'VI. ACKNOWLEDGMENTS': "A. 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2024arXiv240618352H	We present the number densities and physical properties of the bright galaxies spectroscopically confirmed at zsim714. Our sample is composed of 60 galaxies at zmathrmspecsim714 including recentlyconfirmed galaxies at zmathrmspec12.3414.32 with JWST as well as new confirmations at zmathrmspec6.5837.643 with 24lt MmathrmUVlt 21 mag using ALMA and Keck. Our JWSTNIRSpec observations have also revealed that very bright galaxy candidates at zsim1013 identified from groundbased telescope images before JWST are passive galaxies at zsim34 emphasizing the necessity of strict screening and spectroscopy in the selection of the brightest galaxies at zgt10. The UV luminosity functions derived from these spectroscopic results are consistent with a double powerlaw function showing tensions with theoretical models at the bright end. To understand the origin of the overabundance of bright galaxies we investigate their morphologies using JWSTNIRCam highresolution images obtained in various surveys including PRIMER and COSMOSWeb. We find that sim70 of the bright galaxies at zsim7 exhibit clumpy morphologies with multiple subcomponents suggesting mergerinduced starburst activity which is consistent with SED fitting results showing bursty star formation histories. At zgtrsim10 bright galaxies are classified into two types of galaxies extended ones with weak highionization emission lines and compact ones with strong highionization lines including NIVlambda1486 indicating that at least two different processes e.g. mergerinduced starburst and compact star formationAGN are shaping the physical properties of the brightest galaxies at zgtrsim10 and are responsible for their overabundance.	2024-06-01T00:00:00Z	['arXiv:2406.18352', '10.48550/arXiv.2406.18352', '2024arXiv240618352H']	['Astrophysics - Astrophysics of Galaxies']	JWST ALMA and Keck Spectroscopic Constraints on the UV Luminosity Functions at z714 Clumpiness and Compactness of the Brightest Galaxies in the Early Universe	2,024	193	0.69	['EPRINT_HTML', 'EPRINT_PDF']	32	https://arxiv.org/pdf/2406.18352.pdf	{'JWST, ALMA, and Keck Spectroscopic Constraints on the UV Luminosity Functions at z ∼ 7 -14: Clumpiness and Compactness of the Brightest Galaxies in the Early Universe': "Yuichi Harikane, 1 Akio K. Inoue, 2, 3 Richard S. Ellis, 4 Masami Ouchi, 5, 1, 6 Yurina Nakazato, 7 Naoki Yoshida, 7 Yoshiaki Ono, 1 Fengwu Sun, 8, 9 Riku A. Sato, 2 Giovanni Ferrami, 10, 11 Seiji Fujimoto, 12, ∗ Nobunari Kashikawa, 13, 14 Derek J. McLeod, 15 Pablo G. P'erez-Gonz'alez, 16 Marcin Sawicki, 17 Yuma Sugahara, 3, 2 Yi Xu, 1, 13 Satoshi Yamanaka, 18 Adam C. Carnall, 15 Fergus Cullen, 15 James S. Dunlop, 15 Eiichi Egami, 8 Norman Grogin, 19 Yuki Isobe, 3 Anton M. Koekemoer, 19 Nicolas Laporte, 20 Chien-Hsiu Lee, 21 Dan Magee, 22 Hiroshi Matsuo, 5, 23 Yoshiki Matsuoka, 24 Ken Mawatari, 3 Kimihiko Nakajima, 5 Minami Nakane, 1, 7 Yoichi Tamura, 25 Hiroya Umeda, 1, 7 and Hiroto Yanagisawa 1, 7 \n- 1 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan 2 Department of Physics, School of Advanced Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan\n- 3 Waseda Research Institute for Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan\n- 4 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK 5 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan \n6 \nKavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan \n7 \nDepartment of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan \n8 \n9 \nSteward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA \nCenter for Astrophysics \n\| \nHarvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA \n10 \nSchool of Physics, University of Melbourne, Parkville, VIC 3010, Australia \n- 11 ARC Centre of Excellence for All-Sky Astrophysics in 3 Dimensions (ASTRO 3D)\n- 12 Department of Astronomy, The University of Texas at Austin, Austin, TX, USA\n- 13 Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan \n14 \n16 \nResearch Center for the Early Universe, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan \n15 \nSUPA, Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, UK \nCentro de Astrobiolog'ıa (CAB/CSIC-INTA), Ctra. de Ajalvir km 4, Torrej'on de Ardoz, E-28850, Madrid, Spain \n- 17 Department of Astronomy and Physics and the Institute for Computational Astrophysics, Saint Mary's University, 923 Robie Street, Halifax, NS B3H 3C3, Canada\n- 18 General Education Department, National Institute of Technology, Toba College, 1-1, Ikegami-cho, Toba, Mie 517-8501, Japan 19 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n- 20 Aix-Marseille Universit'e, CNRS, CNES, LAM (Laboratoire d'Astrophysique de Marseille), UMR 7326, 13388 Marseille, France 21 Hobby-Eberly Telescope, McDonald Observatory, 32 Mt. Fowlkes, Fort Davis, TX 79734, USA\n- 22 Department of Astronomy and Astrophysics, UCO/Lick Observatory, University of California, Santa Cruz, CA 95064, USA 23 The Graduate University for Advanced Studies, SOKENDAI, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 24 Research Center for Space and Cosmic Evolution, Ehime University, Matsuyama, Ehime 790-8577, Japan\n- 25 Division of Particle and Astrophysical Science, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan \n(Accepted for Publication in ApJ)", 'ABSTRACT': 'We present the number densities and physical properties of the bright galaxies spectroscopically confirmed at z ∼ 7 -14. Our sample is composed of 60 galaxies at z spec ∼ 7 -14, including recentlyconfirmed galaxies at z spec = 12 . 34 -14 . 32 with JWST, as well as new confirmations at z spec = 6 . 583 -7 . 643 with -24 < M UV < -21 mag using ALMA and Keck. Our JWST/NIRSpec observations have also revealed that very bright galaxy candidates at z ∼ 10 -13 identified from ground-based telescope images before JWST are passive galaxies at z ∼ 3 -4, emphasizing the necessity of strict screening and spectroscopy in the selection of the brightest galaxies at z > 10. The UV luminosity functions \nderived from these spectroscopic results are consistent with a double power-law function, showing tensions with theoretical models at the bright end. To understand the origin of the overabundance of bright galaxies, we investigate their morphologies using JWST/NIRCam high-resolution images obtained in various surveys including PRIMER and COSMOS-Web. We find that ∼ 70% of the bright galaxies at z ∼ 7 exhibit clumpy morphologies with multiple sub-components, suggesting mergerinduced starburst activity, which is consistent with SED fitting results showing bursty star formation histories. At z ≳ 10, bright galaxies are classified into two types of galaxies; extended ones with weak high-ionization emission lines, and compact ones with strong high-ionization lines including Niv] λ 1486, indicating that at least two different processes (e.g., merger-induced starburst and compact star formation/AGN) are shaping the physical properties of the brightest galaxies at z ≳ 10 and are responsible for their overabundance. \nKeywords: galaxies: formation - galaxies: evolution - galaxies: high-redshift', '1. INTRODUCTION': "Probing the properties of luminous galaxies in the early universe is key to understanding the physical process that governs galaxy formation. The luminosity function, representing the volume density of galaxies as a function of the luminosity, is one of the most important statistical measurements of galaxies. It is known that the luminosity functions at low redshifts are described by the Schechter function (Schechter 1976), which is derived from the shape of the halo mass function (Press & Schechter 1974). The exponential cutoff at the bright end of the Schechter function is thought to be caused by mass-quenching (Peng et al. 2010). However, previous wide-area imaging surveys using ground-based telescopes have reported that the bright end of the UV luminosity functions at z ∼ 4 -7 does not follow the Schechter function, but is well described by the doublepower-law luminosity function (e.g., Bowler et al. 2012, 2014, 2015, 2017; Stevans et al. 2018; Adams et al. 2020; Varadaraj et al. 2023). Even after subtracting the contributions from quasars, the galaxy luminosity function still shows a bright-end excess beyond the Schechter function at z ∼ 4 -7 (Ono et al. 2018; Harikane et al. 2022b). Studies using ground-based near-infrared imaging datasets have identified very bright galaxy candidates at z ∼ 10 -13 (Bowler et al. 2020; Harikane et al. 2022a), which also supports the double-power-law function rather than the Schechter function. However, these studies are based on samples of photometric galaxy candidates selected from imaging datasets, and spectroscopic observations are required to conclude the discussion about the shape of the bright end. \nSince its first operation in 2022, the James Webb Space Telescope (JWST) has sparked a revolution in high redshift galaxy studies. Various studies using \nJWST have reported that the abundance of bright galaxies at z ≳ 10 is higher than theoretical model predictions (e.g., Bouwens et al. 2023a,b; Castellano et al. 2023; Donnan et al. 2023b, 2024, 2023a; Finkelstein et al. 2022, 2023a,b; Harikane et al. 2023a, 2024; McLeod et al. 2024; Naidu et al. 2022b; Robertson et al. 2023; P'erezGonz'alez et al. 2023, but see also Willott et al. 2023 for a report of relatively low number densities). Several possibilities are raised and intensively discussed for the origin of this tension between the JWST observations and model predictions (see discussions in Harikane et al. 2023a, 2024 for a review), such as a high star formation efficiency (e.g., Dekel et al. 2023; Fukushima & Yajima 2021; Inayoshi et al. 2022), AGN activity (e.g., Harikane et al. 2023b; Hegde et al. 2024), a top-heavy IMF (e.g., Omukai et al. 2005; Chon et al. 2022; Steinhardt et al. 2023; Ventura et al. 2024, see also Rasmussen Cueto et al. 2023), bursty star formation (e.g., Pallottini & Ferrara 2023; Shen et al. 2023; Mu˜noz et al. 2023; Sun et al. 2023a,b), radiation-driven outflows (e.g., Ferrara et al. 2023; Ferrara 2024a), and a flaw in the current cosmology model (e.g., Parashari & Laha 2023; Hirano & Yoshida 2024). Although some studies have investigated the physical properties of galaxies (e.g., Cullen et al. 2024; Endsley et al. 2023; Langeroodi & Hjorth 2024; Roberts-Borsani et al. 2024; Topping et al. 2024b), including studies for individual bright galaxies such as GN-z11 at z = 10 . 60 (e.g., Bunker et al. 2023; Tacchella et al. 2023; Maiolino et al. 2024, 2023; Scholtz et al. 2023; Xu et al. 2024), so far the physical origin of this overabundance of z ≳ 10 galaxies is not clear. \nIn this study, we investigate the number density and the physical properties of spectroscopically confirmed bright galaxies at z ∼ 7 -14. We will discuss the shape of the bright end of the UV luminosity function and the morphologies and star formation histories of the bright galaxies using the JWST, ALMA, and Keck datasets. These results are useful to understand the physical ori- \nFigure 1. UV magnitudes as a function of the redshift. The circles show the spectroscopic redshifts of galaxies compiled in this study (see Table 1) and the literature (Harikane et al. 2024; D'Eugenio et al. 2024; Fujimoto et al. 2023c). The red, blue, green, and orange circles are galaxies in the JADES, lensing cluster (e.g., Abell2744), CEERS, and the other fields, respectively. The UV magnitudes of galaxies in D'Eugenio et al. (2024) are calculated from broad-band fluxes presented in Hainline et al. (2024). \n<!-- image --> \nn of the overabundance of z ≳ 10 galaxies and the process that governs galaxy formation in the early universe. Moreover, this study will be an important reference for future wide-area imaging surveys using Euclid (e.g., Weaver et al. 2024), Nance Grace Roman Space Telescope, and GREX-PLUS (Inoue et al. 2022), which will allow us to search for very bright galaxies at z ≳ 10. \nThis paper is organized as follows. We describe our galaxy sample and spectroscopic and photometric datasets in Section 2. Section 3 presents the calculation of the effective survey volume and the results of the UV luminosity functions based on the spectroscopically confirmed galaxies. In Sections 4 and 5, we show the morphologies of the bright galaxies and the results of the SED fitting. In Section 6, we discuss the physical origin of the overabundance of bright galaxies at z ∼ 7 and z ∼ 12 -14, and the impact of the low-redshift interlopers in the galaxy selection using the wide-area survey datasets. Section 7 summarizes our findings. Throughout this paper, we use the Planck cosmological parameter sets of the TT, TE, EE+lowP+lensing+BAO result (Planck Collaboration et al. 2020): Ω m = 0 . 3111, Ω Λ = 0 . 6899, Ω b = 0 . 0489, h = 0 . 6766, and σ 8 = 0 . 8102. With this cosmological parameter set, the angular size of 1 . '' 0 corresponds to \n5.338 kpc at z = 7 . 0. All magnitudes are in the AB system (Oke & Gunn 1983).", '2.1. Galaxy Sample': 'In this study, we use a sample of 60 galaxies spectroscopically confirmed at z spec = 6 . 538 -14 . 32. The sample is composed of 50 bright galaxies at z spec ∼ 7 -8 with UV magnitudes brighter than M UV < -21 . 0 mag including four new confirmations with Keck and ALMA, and 10 galaxies recently confirmed at z spec ∼ 10 -14 with JWST. Table 1 summarizes the properties of galaxies in our sample, and Figure 1 shows the spectroscopic redshifts of our sample as well as other studies including Harikane et al. (2024). In conjunction with the results in Harikane et al. (2024), we can investigate luminosity functions in a wide redshift range of 7 ≲ z ≲ 14. In addition to the confirmed galaxies at z spec ≳ 7, we use the results of JWST spectroscopic follow-ups targeting galaxy candidates at z ∼ 10 -13 that are found to be low-redshift interlopers. We describe the sample in detail below.', '2.1.1. Keck/LRIS Spectroscopy': 'We conducted Keck/Low Resolution Imaging Spectrometer (LRIS) spectroscopy targeting bright galaxy', 'HSC J084916+005311 ( z spec =6.606)': "<!-- image --> \nFigure 2. Keck/LRIS spectra of HSC J084916+005311 at z spec = 6 . 606 (top) and HSC J091519-012630 at z spec = 7 . 0 (bottom). For each object, the top panel shows the two-dimensional spectrum (yellow is positive), and the bottom panel shows the one-dimensional spectrum. For HSC J091519-012630, we plot the averaged spectra over 110 ˚ A bins with the red-filled circles to show the continuum. The Ly α line is clearly detected in HSC J084916+005311, and the continuum and a break around 9670 ˚ A are identified in HSC J091519-012630. \n<!-- image --> \ncandidates at z ∼ 7 identified in the HSC Wide field in Harikane et al. (2022b) from the Subaru/Hyper Suprime-Cam (HSC) survey datasets (Aihara et al. 2018a,b, 2019, 2022). Observations were conducted in the Multi-Object Spectroscopy (MOS) mode on 2023 April 14th and 2024 February 8th and 9th (S23A001N and S24A-001N, PI: Y. Harikane). We used the 600/10000 grating with the central wavelength of 9000 ˚ A and the D680 dichroic, resulting in the wavelength resolution of R ∼ 1500 at 9000 ˚ A. The slit width was 0. '' 8 and the seeing size was ∼ 1 '' in an FWHM. The ex- \n∼ 1 hour per each target. We reduced the data using PypeIt (Prochaska et al. 2020). \nIn the Keck/LRIS spectroscopy, we targeted five z ∼ 7 galaxy candidates, and determined spectroscopic redshifts of two bright galaxies, HSC J084916+005311 and HSC J091519 -012630, to be z spec = 6 . 606 and z spec = 7 . 0, respectively. Figure 2 presents the spectra of the two galaxies. The spectrum of HSC J084916+005311 shows a very bright and asymmetric emission line around 9250 ˚ A, consistent with the Ly α emission line at z = 6 . 606. The line width of the Ly α emission af- \n1.0 \nFigure 3. ALMA data of UVISTA-J-1212 at z spec = 7 . 642 (top) and XMM1-Z-151269 at z spec = 6 . 583 (bottom). The left panels show the [Cii] 158 µ m maps made with the CASA task immoments , by integrating over 140 and 220 km s -1 , comparable to the [Cii] line widths of UVISTA-J-1212 and XMM1-Z-151269, respectively. The red contours are drawn at 1 σ intervals from ± 2 σ . The backgrounds are rest-UV images (ground-based J -band images). The images are 4 '' × 4 '' , and the red ellipses at the lower left corner indicate the synthesized beam sizes of ALMA. The right panels show ALMA spectra around the [Cii] line after continuum subtraction. These spectra are extracted from a 1 . '' 4-diameter circular aperture. The [Cii] line is detected at the ∼ 6 σ significance level in both objects. It is conceivable that XMM1-Z-151269 is a merger given the possible double-peak [Cii] emission, but deeper and higher-resolution data is needed for a definitive conclusion. \n<!-- image --> \nter instrumental broadening correction is ∼ 500 km s -1 . The spectrum of HSC J091519 -012630 shows a continuum break around ∼ 9670 ˚ A, which is interpreted as the Lymanα break at z spec ∼ 7 . 0. The redshifts obtained here are consistent with those determined in ALMA [Cii] observations ( z [CII] = 6 . 600 and 6 . 955 for HSC J084916+005311 and HSC J091519 -012630, respectively, Sun et al. in prep.), which also supports our redshift determinations. The confirmed two galaxies are very bright with UV absolute magnitudes of -23 . 9 ≤ M UV ≤ -23 . 1 mag.", '2.1.2. ALMA Spectroscopy': "Two galaxies identified as z ≳ 7 galaxy candidates, UVISTA-1212 and XMM1-Z-151269, were observed in an ALMA large program Reionization Era Bright Emission Line Survey (REBELS; 2019.1.01634.L, PI: R. Bouwens; Bouwens et al. 2022) after the submission of the Bouwens et al's survey paper. The REBELS pro- \nam observed 40 UV-bright ( M UV ≲ -22 mag) galaxies at z > 6 . 5 with [Cii] 158 µ m or [Oiii ]88 µ m with a spatial resolution of ∼ 1 . '' 2 -1 . '' 6. We reduced and calibrated the archival data obtained in the REBELS program using the Common Astronomy Software (CASA; McMullin et al. 2007) pipeline version 6.4.1.12 in the standard manner with scripts provided by the ALMA observatory. Using the task tclean , we produced images and cubes with the natural weighting without taper to maximize point-source sensitivities. The beam sizes were ∼ 1 . '' 3 -1 . '' 6. The data analysis by the PI team will be presented in Schouws et al. in prep. (see also JWST GO-6480). \nThe right panels of Figure 3 display the obtained ALMA spectra of the two galaxies extracted with a 1. '' 4diameter circular aperture. The emission line is clearly detected around the frequencies of 219.9 and 250.6 GHz in UVISTA-1212 and XMM1-Z-151269 at the 6.0 and 5.9 σ significance levels, respectively. We calculate these \nFigure 4. JWST/NIRSpec spectra of bright galaxy candidates at z ∼ 10 -13, XMM3-3085 in Bowler et al. (2020) and HD1 and HD2 in Harikane et al. (2022a). The spectra show the Balmer breaks around 1 -2 µ m, indicating that these sources are low-redshift interlopers at z spec ∼ 3 -4. The blue circles represent photometric data points in Harikane et al. (2022a) for HD1 and HD2, and those measured using the final data release of the VIDEO survey (Varadaraj et al. 2023) for XMM3-3085. These sources were selected as z ∼ 10 -13 galaxies due to the photometric scatters seen in the discrepancies between the photometric data points and the spectrum. \n<!-- image --> \nsignal-to-noise ratios using 0. '' 6-diameter circular aperture in the same manner as Harikane et al. (2020). As shown in the left panels of Figure 3, these emission lines are cospatial with the rest-frame UV emission in the J -band images. These emission lines in UVISTA-1212 and XMM1-Z-151269 are interpreted as the [Cii] 158 µ mlines at z spec = 7 . 642 and 6 . 583, respectively, consistent with photometric redshift estimates in the literature (Bowler et al. 2020; Bouwens et al. 2022). The [Cii] line profile of XMM1-Z-151269 shows two peaks, suggesting the possibility of a merger, but deeper and higher-resolution data is needed for a definitive conclusions.", '2.1.3. JWST Spectroscopy': 'We conducted JWST/NIRSpec spectroscopy for very bright galaxy candidates at z ≳ 10 identified in \nthe ground-based images, XMM3-3085 at z phot ∼ 11 (Bowler et al. 2020) and HD1 and HD2 at z phot ∼ 12 -13 (Harikane et al. 2022a), whose best-fit photometric redshifts are z > 10 with ∆ χ 2 > 4. Observations for XMM3-3085 were conducted on 2024 January 8th with Prism using the S400A1 fixed slit (GO-2792; PI: Y. Harikane). The total integration time was 3545 seconds. Observation for HD1 and HD2 were conducted on 2023 January 6th and 2022 August 16th, respectively, with Prism using the S400A1 fixed slit (GO-1740; PI: Y. Harikane). The total integration times were 2873 and 1801 seconds for HD1 and HD2, respectively. We used the level-3 product obtained from the Mikulski Archive for Space Telescopes (MAST) for XMM3-3085, and reduced data in Sato et al. (2024) for HD1 and HD2. Note \nthat MAST level-3 products for HD1 and HD2 are almost identical to the reduced data. \nFigure 4 shows the obtained NIRSpec spectra of the three galaxies. We find that these three candidates are not z > 10 galaxies as suggested by previously-obtained best-fit photometric redshifts in Bowler et al. (2020) and Harikane et al. (2022a), but are passive galaxies at z ∼ 3 -4. The spectrum of XMM3-3085 does not display a clear continuum break like the Lyman break but shows a continuum detection below ∼ 1 . 4 µ m, not consistent with the z phot ∼ 11 solution. From spectral fitting using Prospector (Johnson et al. 2021), we estimate the spectroscopic redshift of XMM3-3085 to be z spec = 2 . 6. Similarly, HD1 and HD2 show continuum detections below ∼ 1 . 7 µ m, not consistent with the z phot ∼ 12 -13 solutions, and the spectroscopic redshifts are measured to be z spec = 4 . 0 and 3 . 2. These lower redshift solutions align with alternative solutions suggested in previous studies (Bowler et al. 2020; Harikane et al. 2022a; Kaasinen et al. 2023). In Figure 4, we also compare the obtained NIRSpec spectra with the photometric data points used to select these galaxy candidates. Since the measured photometric fluxes deviate from the spectrum in some bands, it is likely that these low-redshift passive galaxies are scattered into the galaxy selection at z ≳ 10 due to photometric errors, especially in the Spitzer bands where the background subtraction is not straightforward (see the data points at > 3 µ min HD1). Another bright galaxy candidate at z ≳ 12, HD3, was also observed in the program GO-1740 and turned out to be a low-redshift interloper. More detailed analyses of HD1, HD2, and HD3 using the NIRSpec spectra are presented in Sato et al. (2024). \nPrevious ALMA Band 6 spectroscopy for HD1 in Cycle 7 DDT showed a 3 . 8 σ line-like tentative signal around 237.8 GHz, which can be interpreted as the [Oiii] 88 µ m line at z = 13 . 3 (Harikane et al. 2022a). ALMA Band 4 data also showed a 4 σ tentative feature that can be consistent with the [Cii] 158 µ m line, but statistical tests demonstrated that these ∼ 4 σ -level signals are fully consistent with being random noise features (Kaasinen et al. 2023). To investigate the previously reported line-like signal, we conducted additional ALMA Band 6 observations covering 237.8 GHz in Cycle 8 (2021.1.00207.S; PI: Y. Harikane). Figure 5 shows the spectra obtained in Cycle 7 DDT and Cycle 8, resulting in no significant detection around 238 GHz in the Cycle 8 data, which is consistent with the lowredshift solution from the JWST/NIRSpec spectroscopy. Similarly, ALMA Band 7 observations for XMM3-3085 (2021.1.00341.S, 2022.1.00522.S; PI: Y. Harikane) do not \nFigure 5. ALMA spectrum of HD1. The green line shows the Cycle 7 data, where a 4 σ line-like signal was reported around the frequency of 237.8 GHz. No signal is identified in the Cycle 8 data (the blue line). \n<!-- image --> \nshow any significant emission line, which also agrees with the JWST/NIRSpec spectroscopic result. \nThese spectroscopic observations reveal that the three very bright galaxy candidates at z ≳ 10 selected from the ground-based images before JWST are low-redshift interlopers at z ∼ 3 -4. The possibilities of these lowredshift solutions were already discussed in the discovery papers (Bowler et al. 2020; Harikane et al. 2022a), but our observations highlight the importance of the spectroscopy when discussing very bright galaxies at z > 10. These results suggest that passive galaxies with Balmer breaks at intermediate redshifts can be selected as Lyman break galaxies at high redshifts due to photometric scatters, and are important contaminants that should be taken into account in the galaxy selection, in addition to galaxies with strong emission lines that boost the broad-band and medium-band fluxes and mimic a Lyman break-like SED (Arrabal Haro et al. 2023a; Naidu et al. 2022a; Zavala et al. 2023). The implications for the UV luminosity functions and future bright galaxy selections using wide-area survey datasets are discussed in Sections 3.2 and 6.2, respectively. As discussed in Section 6.2, these passive low-redshift interlopers are erroneously selected as high redshift galaxies because of 1) large photometric scatters originating from relatively shallow ground-based and Spitzer datasets, and 2) their very bright magnitudes. Note that these interlopers are not significant in JWST-selected photometric candidates because they are usually faint compared to the galaxies discussed here. Indeed, high spectroscopic success rates are reported in JWST-selected candidates (e.g., Arrabal Haro et al. 2023a,b; Fujimoto et al. 2023b,c). \nIn addition to the four galaxies spectroscopically confirmed in Sections 2.1.1 and 2.1.2, we have compiled bright galaxies with spectroscopic confirmations in the literature. We include 24 bright ( M UV < -21 . 5 mag) galaxies at z spec ∼ 7 -8 in the COSMOS and UDS fields from the REBELS program (Bouwens et al. 2022), and from Schouws et al. (2023). In the COSMOS field, we also take four galaxies at z spec ∼ 7 from Endsley et al. (2021a, 2022a). From the Subaru/HSC survey, we include one very bright ( M UV = -23 . 6 mag) galaxy, HSC J023526 -031737 at z spec = 6 . 913 (Ono et al. 2018; Harikane et al. 2022b, M. Sawicki et al. in prep.). In addition, we use six galaxies at z spec ∼ 7 -8 with M UV < -21 . 0 mag in the JWST CEERS and GLASS fields from Nakajima et al. (2023), and 11 galaxies from other spectroscopic studies (see Table 1 for their references).', '2.1.5. Literature at z ∼ 10 -14': 'To extend our analysis to higher redshifts, we include 10 galaxies at z spec ∼ 10 -14 recently confirmed with \nJWST. We use GHZ2, a bright galaxy initially identified in the JWST/NIRCam images (e.g., Naidu et al. 2022b; Castellano et al. 2022) and recently confirmed at z spec = 12 . 34 with NIRSpec and MIRI spectroscopy (Castellano et al. 2024; Zavala et al. 2024). We also include JADES-GS-z14-0 ( z spec = 14 . 32, hereafter GSz14-0) and JADES-GS-z14-1 ( z spec = 13 . 90, hereafter GS-z14-1), which are firstly identified in the JADES Origins Field (Eisenstein et al. 2023b) by Robertson et al. (2023) and recently confirmed with NIRSpec by Carniani et al. (2024a). Finally, we add seven galaxies at z ∼ 10 -11 recently confirmed in Napolitano et al. (2024). Although these galaxies are relatively faint ( -21 ≲ M UV ≲ -19 mag) compared to galaxies at z ∼ 7 -8 in this sample, they are useful to obtain meaningful constraints on the number densities of galaxies at z ∼ 10 -14. \nIn total, our sample consists of 60 galaxies at z spec = 6 . 538 -14 . 32 (Table 1). These galaxies are selected in multiple survey fields with various methods, and we carefully estimate the survey volume in Section 3.1. \nTable 1 . List of Spectroscopically-Confirmed Galaxies Compiled in This Study \nTable 1 continued', 'Bright-End of the UVLFs at z ∼ 7 -14 :': "Table 1 (continued) \nTable 1 continued \nTable 1 (continued) \nNote -(1) Name. (2) Right ascension. (3) Declination. (4) Spectroscopic redshift. (5) Absolute UV magnitude. (6,7) References for spectroscopic redshifts and photometry (At23: Atek et al. 2023, Bou22: Bouwens et al. 2022, Bou23: Bouwens et al. 2023a, Bow14: Bowler et al. 2014, Bow17a: Bowler et al. 2017, Bow20: Bowler et al. 2020, Cas22: Castellano et al. 2022, Cas24: Castellano et al. 2024, Car24a: Carniani et al. 2024a, Car24b: Carniani et al. 2024b, Do23: Donnan et al. 2023b, En21a: Endsley et al. 2021b, En21b: Endsley et al. 2021a, En22a: Endsley et al. 2022a, En22b: Endsley et al. 2022b, Fu16: Furusawa et al. 2016, Har22a: Harikane et al. 2022b, Har22b: Harikane et al. 2022a, Har23: Harikane et al. 2023a, He24: Helton et al. 2024 Hu16: Hu et al. 2016, Hu21: Hu et al. 2021, In16: Inoue et al. 2016, It18: Itoh et al. 2018, La17: Laporte et al. 2017, Ma17: Matthee et al. 2017, Ma18: Matthee et al. 2018, Nai22: Naidu et al. 2022b, Nak23: Nakajima et al. 2023, Nap24: Napolitano et al. 2024, Ou09: Ouchi et al. 2009, Ou13: Ouchi et al. 2013, Pe16: Pentericci et al. 2016, Pe18: Pentericci et al. 2018, RB16: Roberts-Borsani et al. 2016, Re23: Ren et al. 2023, Rob23: Robertson et al. 2023, Row24: Rowland et al. 2024, SaP: Sawicki et al. in prep., Sh12: Shibuya et al. 2012, Sc23: Schouws et al. 2023, Sc24: Schouws et al. 2024, Sm15: Smit et al. 2015, Sm18: Smit et al. 2018, So15: Sobral et al. 2015, Ste17: Stefanon et al. 2017, Ste19: Stefanon et al. 2019, Za24: Zavala et al. 2024, Zh20: Zhang et al. 2020). \n2.2. Imaging Dataset \n2.2.1. JWST/NIRCam \nWe will use JWST/NIRCam and HST images to investigate the photometric and morphological properties of bright galaxies at z ∼ 7 -14 compiled in this study and in Harikane et al. (2024). JWST/NIRCam images were taken in the COSMOS, UDS, CEERS, Abell2744, GOODS-North, and GOODS-South fields. Public Release IMaging for Extragalactic Research (PRIMER; Dunlop et al., in preparation) survey conducted imaging observations over a total of ∼ 400 arcmin 2 in the COSMOS and UDS fields taken with eight NIRCam filters, F090W, F115W, F150W, F200W, F277W, F356W, F410M, and F444W. The PRIMER data were reduced using the PRIMER Enhanced NIRCam Image Processing Library (PENCIL; Magee et al., in preparation) software. The PENCIL is built on top of STScI's JWST Calibration Pipeline (v1.12.5) but also includes additional processing steps not included in the standard calibration pipeline, such as the subtraction of 1 /f noise striping patterns and the subtraction of wisps artifacts in the short wavelength filters. Additionally, the COSMOS-Web survey (Casey et al. 2023) mapped a 0.6 deg 2 area in the COSMOS field with four filters, F115W, F150W, F277W, and F444W. The COSMOSWeb data were reduced using the JWST Calibration Pipeline (versions 1.12.5) and the Calibration Reference Data System context file of jwst 1193.pmap with custom modifications described in Harikane et al. (2023a). The CEERS field was observed in the CEERS survey (Finkelstein et al. 2023a) with seven NIRCam filters, F115W, F150W, F200W, F277W, F356W, F410M, and F444W. We use reduced images released by the CEERS \nteam (see Bagley et al. 2023, for the data reduction). 1 The NIRCam images of the Abell2744 cluster field were taken in two surveys, the GLASS survey (Treu et al. 2022) and the UNCOVER survey (Bezanson et al. 2022). Reduced images provided by the UNCOVER team are used in this study. 2 Finally, the JADES program (Eisenstein et al. 2023a) conducted NIRCam observations in the GOODS-North and GOODS-South fields. We use imagesthat were reduced with grizli (Brammer 2023) and are provided in the DAWN JWST Archive (versions 7.2 and 7.3 for GOODS-North and GOODS-South, respectively, see also Valentino et al. 2023).", '2.2.2. HST': 'We also use HST/ACS and WFC3 images in the COSMOS and UDS fields. The Cosmic Assembly Nearinfrared Deep Extragalactic Legacy Survey (CANDELS: Grogin et al. 2011, Koekemoer et al. 2011) obtained images over a total of ∼ 300 arcmin 2 in the COSMOS and UDS fields with V 606 , I 814 , J 125 , and H 160 filters. We use the reduced imaging data provided by the 3D-HST team (Brammer et al. 2012; Skelton et al. 2014). In addition, the COSMOS-Drift And SHift (COSMOS-DASH) survey (Mowla et al. 2019; Cutler et al. 2022) conducted H 160 imaging observations covering an area of 0.49 deg 2 . Data products provided in the MAST are used in this study.', '3.1. Effective Volume Estimate': 'Using the sample of spectroscopically confirmed galaxies constructed in Section 2.1, we calculate the UV luminosity functions at z ∼ 7 -14 in the bright magni- \nde range, which are not investigated in previous studies such as Harikane et al. (2024) and Fujimoto et al. (2023c). Because our samples is composed of 50 galaxies at z ∼ 7 -8 and 3 galaxies at z ∼ 12 -14, we divide our sample into the four redshift subsamples at z spec = 6 . 5 -7 . 5, 7 . 5 -8 . 5, 11 . 0 -13 . 5, and 13 . 5 -15 . 0 to calculate the number densities at z ∼ 7, 8, 12, and 14. Since the galaxies in our spectroscopic sample are confirmed with various instruments whose target selection and detection completeness are not well-known, we carefully estimate the effective volume for the luminosity functions. As detailed below, we use the two methods to estimate the luminosity functions. If the all photometric candidates in a magnitude bin are spectroscopically observed, we calculate the best estimate of the number density with errors using the effective volume published in the literature. If not all of the candidates are observed, we put a lower limit on the number density using the number of the confirmed sources and the survey area. \nIn the magnitude ranges of -23 . 5 < M UV < -22 . 0 mag at z spec = 6 . 5 -7 . 5 and -22 . 5 < M UV < -22 . 0 at z spec = 7 . 5 -8 . 5, we use galaxies in the COSMOS field where the spectroscopic completeness is high thanks to some intensive spectroscopic surveys (e.g., Endsley et al. 2021a; Bouwens et al. 2022). We count the number of galaxies spectroscopically confirmed in the area of 150 . 8 < R . A . < 149 . 3 deg and 1 . 70 < decl . < 2 . 75 deg, corresponding to the survey area of 1.5 deg 2 (comparable to one in Bowler et al. 2014, 2020), and calculate the survey volume in the redshift range of z = 6 . 5 -7 . 5 or z = 7 . 5 -8 . 5. At the magnitude bins of M UV = -23 . 2 mag ( -23 . 45 < M UV < -22 . 95 mag) and M UV = -22 . 7 mag ( -23 . 95 < M UV < -23 . 45 mag), we have found that all of the galaxy candidates reported in Bowler et al. (2017) are spectroscopically observed. Thus in these two bins, we calculate the number densities rather than lower limits, assuming a 100% completeness. In the other bins, we obtain the lower limits of the number densities, since there are remaining candidates that are not yet spectroscopically observed. \nIn the brightest magnitude bin at z ∼ 7, -24 . 5 < M UV < -23 . 5 mag, we use HSC J023526 -031737 at z spec = 6 . 913, which was first photometrically selected in Ono et al. (2018), and obtain a lower limit of the number density using the survey area of 102.7 deg 2 in Ono et al. (2018). Similarly, in the faint magnitude bins of -22 . 0 < M UV < -21 . 0 mag at z spec = 6 . 5 -7 . 5 and -21 . 5 < M UV < -21 . 0 mag at z spec = 7 . 5 -8 . 5, we calculate lower limits of the number densities with galaxies in the JWST CEERS and GLASS fields from Nakajima et al. (2023). We use the survey area of 72 \narcmin 2 , which corresponds to the effective coverage of NIRSpec pointings in CEERS and GLASS. Regarding the GLASS, we do not consider the gravitational lensing, resulting in a larger survey area and obtaining a conservative lower limit. \nWe also estimate the number densities of galaxies at z ∼ 12 and 14 using the recently spectroscopically confirmed galaxies at z = 12 -14. At z ∼ 12, since the confirmed galaxy, GHZ2 at z = 12 . 34 is the only galaxy in the brightest magnitude bin at z ∼ 12 in Harikane et al. (2023a), we use the survey area calculated therein to estimate the number density. At z ∼ 14, the two confirmed galaxies, GS-z14-0 ( M UV = -20 . 8 mag) and GS-14-1 ( M UV = -19 . 0 mag) are originally selected in Robertson et al. (2023). Since the brightest bin in Robertson et al. (2023) includes only GS-z14-0, we adopt the number density therein for the estimate at M UV = -20 . 8 mag. In the fainter magnitude bin ( M UV = 19 . 0 mag), we obtain a lower limit of the number density using the survey volume in Robertson et al. (2023). \nWe also obtain upper limits on the number densities of the brightest galaxies at z ∼ 10 and 12 based on the spectroscopic results of the z ∼ 10 -12 bright galaxy candidates, XMM3-3085, HD1, and HD2, which are found to be low-redshift passive galaxies (Section 2.1.3). At z ∼ 10, we estimate the survey volume from the inverse of the number density in Bowler et al. (2020). At z ∼ 12, we use the survey volume calculated in Harikane et al. (2022a). \nThe 1 σ uncertainty of the number density is calculated by taking the Poisson confidence limit (Gehrels 1986) and cosmic variance into account. We estimate the cosmic variance in the number densities following the procedures in Somerville et al. (2004). As the large-scale bias parameter needed for the cosmic variance calculation, we adopt b = 7 obtained by the clustering analysis of galaxies at z ∼ 7 in Harikane et al. (2016), which is broadly comparable with recent JWST estimates for higher redshift galaxies (Dalmasso et al. 2024). Note that the value of the bias parameter does not change our conclusion because the Poisson error is much larger than the cosmic variance due to the small number of galaxies. For example, if we adopt b = 10, the error for the number density at z ∼ 12 and M UV = -20 . 5 mag changes only by 3%. In this way, the 1 σ uncertainty presented in this study includes both the Poisson uncertainty and the cosmic variance.', '3.2. Results': "Figures 6 and 7 show our constraints on the number densities of galaxies at z ∼ 7, 8, 10, 12, and 14 and Table \nFigure 6. UV luminosity functions at z ∼ 7 (upper-left), z ∼ 8 (upper-right), z ∼ 10 (lower-left), and z ∼ 12 (lower-right). The red diamonds represent the number densities of galaxies with spectroscopic redshifts derived in this study (Table 2). The errors include the cosmic variance (see text). The red filled circles at z ∼ 10 and 12 are the spectroscopic constraints from Harikane et al. (2024), and the blue circles at z ∼ 7 are number densities of spectroscopically-confirmed quasars (QSOs) in Matsuoka et al. (2023). The other red symbols show spectroscopic constraints in the literature (Meyer et al. 2024; Rojas-Ruiz et al. 2024; Fujimoto et al. 2023c; Napolitano et al. 2024). The data point at z ∼ 12 and M UV = -20 . 5 mag in Napolitano et al. (2024) is shifted by +0 . 05 mag for clarity. The gray symbols are estimates based on photometric samples by previous studies (Adams et al. 2024; Bouwens et al. 2021, 2023a; Bowler et al. 2017, 2020; Castellano et al. 2023; Donnan et al. 2024; Finkelstein et al. 2015, 2023b; Harikane et al. 2022b, 2023a; Morishita & Stiavelli 2023; McLeod et al. 2024; P'erez-Gonz'alez et al. 2023; Robertson et al. 2023; Stefanon et al. 2019; Varadaraj et al. 2023; Willott et al. 2023). The red solid and dashed lines are our best-fit double power-law and Schechter functions, respectively, and the shaded region shows the 1 σ uncertainties for the double power-law fit (Table 3). At z ∼ 10 and z ∼ 12, although the bright-end slope is fixed to β = -4 . 60 in the fitting, the allowed β range ( z ∼ 10: β < -3 . 0, z ∼ 12: β < -2 . 4) is plotted to show the uncertainty of the constraint on the bright end. The red dotted lines are the lensed Schechter function calculated in Ferrami & Wyithe (2023) assuming the size-luminosity relation in Shibuya et al. (2015). At z ∼ 7, the spectroscopic constraints at the bright end prefer the double power-law or lensed Schechter function to the original Schechter function. \n<!-- image --> \n17 \nTable 2. Spectroscopic Constraints on the Luminosity Function at Each Redshift \nFigure 7. Same as Figure 6 but at z ∼ 14. The red solid and dashed lines are double power-law and Schechter functions, respectively, whose parameters are extrapolated from the z ∼ 10 -12 results (Table 3). \n<!-- image --> \n2 summarizes them. Our spectroscopic constraints are consistent with previous estimates of the number density in the literature based on photometric samples. Our results are also consistent with the number densities of spectroscopically confirmed [Oiii] λ 5007 emitters at z ∼ 7 -8 identified in the JWST FRESCO survey (Meyer et al. 2024), and spectroscopic lower limits obtained in Fujimoto et al. (2023c). \nWe fit our results at z ∼ 7 -12 with the double-powerlaw function, \nΦ( M UV ) = ln 10 2 . 5 ϕ ∗ × [ 10 0 . 4( α +1)( M UV -M ∗ UV ) +10 0 . 4( β +1)( M UV -M ∗ UV ) ] -1 , (1) \nand the Schechter function, \nΦ( M UV ) = ln 10 2 . 5 ϕ ∗ 10 -0 . 4( M UV -M ∗ UV )( α +1) × exp ( -10 -0 . 4( M UV -M ∗ UV ) ) , (2) \nwhere ϕ ∗ is the overall normalization, M ∗ UV is the characteristic magnitude, and α and β are the faint and bright-end slopes, respectively. In fitting, we use the results of this study and Bouwens et al. (2021) at z ∼ 7 -8, and this study and Harikane et al. (2024) at z ∼ 10 -12, whose samples are not overlapping each other. We also fix the faint end slope α = -2 . 10 and the bright end slope β = -4 . 60 in the fitting at z ∼ 10 and 12, and the characteristic UV magnitude to M ∗ UV = -20 . 60 mag in the fit at z ∼ 12, based on the lower redshift results. \nFigure 8. Comparison of the luminosity functions with theoretical predictions in the literature at z ∼ 7 (upper-left), z ∼ 8 (upper-right), z ∼ 10 (lower-left), and z ∼ 12 (lower-right). The red symbols show observational results based on the spectroscopically-confirmed galaxies obtained in this study (filled diamond), Harikane et al. (2024, filled circle), Meyer et al. (2024, open square), Rojas-Ruiz et al. (2024, open circle), Napolitano et al. (2024, cross), and Fujimoto et al. (2023c, open pentagon). The dashed lines and shaded region show predictions of theoretical and empirical models in Mauerhofer & Dayal (2023), Prada et al. (2023), Vogelsberger et al. (2020), Vijayan et al. (2021), Yung et al. (2019, 2024), Mason et al. (2015a, 2023, their model with dust extinction), Kannan et al. (2023), Wilkins et al. (2023), Ferrara (2024a), Li et al. (2023), and Shen et al. (2024). For models in Li et al. (2023), a range of a maximum efficiency parameter of ϵ max = 0 . 2 -1 . 0 is plotted as the grey shaded region. Spectroscopic constraints for bright galaxies with -24 < M UV < -23 mag at z ∼ 7 ( -21 < M UV < -20 mag at z ∼ 12) are higher than the number densities of some model predictions. \n<!-- image --> \n17 \nTo take the lower and upper limits into account in the fit, we follow a χ 2 minimization procedure presented in \nSawicki (2012) and define χ 2 as follows \nχ 2 = ∑ i ( Φ i -Φ model ,i σ Φ i ) 2 -2 ∑ j ln ∫ Φ upper ,j -∞ d Φ exp [ -1 2 ( Φ -Φ model ,j σ Φ j ) 2 ] -2 ∑ k ln ∫ ∞ Φ lower ,k d Φ exp [ -1 2 ( Φ -Φ model ,k σ Φ k ) 2 ] , (3) \nFigure 9. Same as Figure 8 but at z ∼ 14. Our constraints on the number density are higher than model predictions, especially at M UV = -21 mag. \n<!-- image --> \nwhere Φ, σ Φ , and Φ model are the observed number density, its uncertainty, and the number density from the fitted model, respectively. The indices i , j , and k correspond to the magnitudes bins with the best estimate, upper limit, and lower limit, and Φ upper and Φ lower are the 1 σ upper and lower limits, respectively. \nThe best-fit functions are plotted in Figure 6 and the estimated parameters are summarized in Table 3. At z ∼ 7, the bright end of the luminosity function is well described with the double-power-law function rather than the Schechter function with a 2 σ level. The bright end at z ∼ 7 is also consistent with the expectation of the lensed Schechter function in Ferrami & Wyithe (2023). In the other redshift bins, both the double-power-law and Schechter functions can reasonably reproduce our spectroscopic constraints of the number densities. Wide photometric and spectroscopic surveys are needed to constrain the shape of the bright-end luminosity functions at z ≳ 8. In Figure 7, we also plot the double-power-law and Schechter functions at z ∼ 14 whose parameters are estimated from the extrapolations using the best-fit results at z ∼ 10 -12 (see Table 3).", '3.3. Comparison with Model Predictions': "In Figures 8 and 9, we compare the spectroscopic constraints in this study and Harikane et al. (2024) with theoretical model predictions. At z ∼ 7, most of the models agree with the number densities of galaxies fainter than M UV ≃ -23 . 0 mag, but more than half of the models predict lower number densities than the observations at the magnitude brighter than M UV ≃ -23 . 0 mag. At z ∼ 8 and 10, the observed number densities can be \nreproduced by most of the models compared here. At z ∼ 12, the number density of GHZ2 at M UV = -20 . 5 mag is higher than most of the models except for Wilkins et al. (2023), Ferrara (2024a), and Li et al. (2023), similar to the lower limit at M UV = -20 . 1 mag obtained in Harikane et al. (2024). \nThe constraints at z ∼ 14 also show tension with the model predictions (Figure 9). Especially, the number density at M UV = -20 . 9 mag is more than 100 times higher than most of the predictions. This number density is based on the most distant galaxy recently confirmed, GS-z14-0 at z = 14 . 32. The redshift determination is considered to be reliable, given the unambiguous confirmation of the redshift via a clearly detected Lyman break in Carniani et al. (2024a). Although there is a lower redshift galaxy ( z spec = 3 . 475) that is close to GS-z14-0 with a separation of 0. '' 4, the lensing magnification factor is estimated to be small, less than µ = 1 . 2 (see Carniani et al. 2024a). Thus this number density estimate can be considered to be reliable, unless the cosmic variance is much stronger than our estimate and the JADES Origins Field is significantly biased. Larger area datasets are needed to understand the real effect of the cosmic variance. Physical origins for these discrepancies between the observations and model predictions at z ∼ 12 -14 are discussed in Section 6.1.", '3.4. SFR Density': "Based on the two recently-confirmed galaxies in Carniani et al. (2024a), we calculate the lower limit of the cosmic star formation rate (SFR) density at z ∼ 14. We use the survey volume discussed in Section 3.1, and convert the observed UV luminosities of the two galaxies to SFRs using the following equation assuming the Salpeter (1955) IMF, \nSFR ( M ⊙ yr -1 ) = 1 . 15 × 10 -28 L UV (erg s -1 Hz -1 ) . (4) \nWe assume the Salpeter (1955) IMF for comparison with the literature. Figure 10 shows our spectroscopic lower limit based on the two galaxies brighter than M UV = -18 . 0 mag, corresponding to the SFR of SFR UV = 0 . 8 M ⊙ yr -1 , and Table 4 summarizes the measurements in this study and in Harikane et al. (2024). We also plot estimates based on the photometric samples in the literature (Bouwens et al. 2020, 2023a,b; Donnan et al. 2024; Finkelstein et al. 2015, 2023b; McLeod et al. 2024; Harikane et al. 2023a; P'erez-Gonz'alez et al. 2023; Willott et al. 2023). Since some of these studies calculate the SFR densities with different integration limits from M UV = -18 . 0 mag, we have corrected their results based on the difference between the SFR density integrated down to their limit and that down \nFigure 10. Cosmic SFR density evolution. The red diamond represents a lower limit on the cosmic SFR density at z ∼ 14 obtained in this study integrated down to M UV = -18 . 0 mag (corresponding to SFR UV = 0 . 8 M ⊙ yr -1 , based on the Salpeter (1955) IMF with a conversion factor of SFR /L UV = 1 . 15 × 10 -28 M ⊙ yr -1 / (erg s -1 Hz -1 )). The error includes both the 1 σ Poisson error and the cosmic variance. The red circles are spectroscopic lower limits obtained in Harikane et al. (2024). The blue curves are predictions of the constant star formation (SF) efficiency models of Harikane et al. (2018, 2022b, solid), Mason et al. (2015a, 2023, dashed), and Sun & Furlanetto (2016, dotted). The obtained lower limits of the SFR densities at z ∼ 12 -14 are higher than the model predictions. The gray open symbols are estimates of previous studies using photometric samples (Bouwens et al. 2020, 2023a,b; Donnan et al. 2024; Finkelstein et al. 2015, 2023b; McLeod et al. 2024; Harikane et al. 2023a; P'erez-Gonz'alez et al. 2023; Willott et al. 2023). \n<!-- image --> \nto M UV = -18 . 0 mag using their fiducial luminosity function, in the same manner as Bouwens et al. (2023a). Our lower limit at z ∼ 14 is consistent with these photometric estimates and is more than 10 times higher than the model predictions assuming a constant star formation efficiency (Harikane et al. 2018, 2022b; Mason et al. 2015a, 2023; Sun & Furlanetto 2016).", '4.1. Multiple Sub-Components in Bright Galaxies at z ∼ 7': "We investigate morphologies of bright galaxies spectroscopically confirmed at z ≳ 7 using high-resolution HST and/or JWST images. Among the 50 galaxies at z ∼ 7 -8 used in this study, 23 galaxies are observed and clearly detected with either HST/WFC3 and/or JWST/NIRCam. From the 23 galaxies, we select 12 galaxies that are observed with both HST and JWST and are brighter than M UV = -21 . 5 mag. Figure 11 shows the selected 12 galaxies at z spec = 6 . 595 -7 . 154. \nWe find that some galaxies show clumpy structures spatially extended up to ∼ 1 '' ( ∼ 5 kpc). In addition, the JWST/NIRCam F115W image with a high spatial resolution of ∼ 0 . '' 07 allows us to identify multiple sub-components/clumps that are not identified with the HST images whose spatial resolution is ∼ 0 . '' 2 (e.g., COS-3018555981). Other rest-UV images (e.g., F150W) also show similar clumpy structures in these galaxies. \nTo quantitatively discuss the sub-components in the galaxies, we run SExtractor (Bertin & Arnouts 1996) on the JWST and HST images with a parameter set of DETECT MINAREA = 5 , DETECT THRESH=3 , ANALYSIS THRESH=3 , DEBLEND NTHRESH = 32 , and DEBLEND MINCOUNT = 0.01 , following Bowler et al. (2017). We then visually inspect the sub-components detected at > 5 σ significance levels to remove any spurious detections and foreground objects. In Figure 12, we show the surface brightness maps of the 12 galaxies in the JWST/NIRCam F115W images and the positions of the sub-components detected with SExtractor. We find that 8 of the 12 galaxies in Figure 12 have more than one component, indicating that 66 ± 14% of the bright galaxies exhibit multiple sub-components. Even if we include galaxies observed only with HST, the fraction of galaxies with multiple sub-components is still high, 60 ± 12%. Such a high fraction of galaxies with multiple sub-components at z ∼ 7 is comparable to or slightly higher than those of similarly bright galaxies ( M UV ∼ -22 mag) at z ∼ 6 -7 ( ∼ 40 -60% e.g., Jiang et al. 2013; Willott et al. 2013; Bowler et al. 2017; Shibuya et al. 2022, see also Asada et al. 2024 for fainter galaxies), but our estimate is the first result based on the spectroscopically confirmed galaxies at z ∼ 7. Our fraction at z ∼ 7 is higher than those at z ∼ 4 -5 ( ∼ 20 -30%) with similar luminosities (e.g., Shibuya et al. 2022), although this difference could be due to the difference in the spatial resolutions of the datasets used (e.g., JWST vs. other telescopes). \nTwo possible mechanisms are proposed to form these multiple sub-components in galaxies; galaxy mergers (e.g., Di Matteo et al. 2008) and the violent disk instability (e.g., Dekel et al. 2009, 2013). Recently, Nakazato et al. (2024) discuss that galaxies with multiple subcomponents extended up to 5 kpc can be made by a merger, where multiple clumps form in gas debris in tidal tails induced by the merger event. Since the merger fraction is expected to be high for the bright z ∼ 7 galaxies in this study which are hosted by massive dark matter halos, this merger-induced clump formation is a plausible scenario for the origin of the observed multiple sub-components (see also discussions in Asada et al. 2024). The possible increasing trend of the \nTable 3. Fit Parameters for Luminosity Functions \nNote -Errors are 1 σ . \n- † Extrapolated from the results at z ∼ 10 and 12.\n- ∗ Paremeters when multiple sub-components are split into separate 'galaxies' (see Section 4.2). \nTable 4. Spectroscopic Constraints on Cosmic UV Luminosity Density and SFR Density \nNote -Errors are 1 σ . ρ SFR , UV is the SFR density based on the Salpeter (1955) IMF without dust extinction correction. \n- † Taken from Harikane et al. (2024). \nclumpy galaxy fraction towards higher redshifts is also consistent with the redshift evolution of the halo merger rate predicted by simulations (e.g., Fakhouri et al. 2010; Rodriguez-Gomez et al. 2015). The other possibility is the violent disk instability, where clumps are predicted to form in unstable regions where the Toomre Q parameter (Toomre 1964) is below a critical value in thick and gas-rich galaxy disks (e.g., Noguchi 1998; Dekel et al. 2009; Ceverino et al. 2010; Tacconi et al. 2010; Romeo et al. 2010; Romeo & Agertz 2014; Inoue & Yoshida 2019). However, the scale of the observed clumpy structures (up to 5 kpc) is much larger than a typical scale of disks ( ∼ 1 kpc), where clumps are made via the violent disk instability (e.g., Fujimoto et al. 2024). IFU observations reveal complex velocity structures support- \ng the merger-induced clump formation in at least five galaxies studied here (e.g., Matthee et al. 2017; Carniani et al. 2018; Hashimoto et al. 2019; Matthee et al. 2020; Ren et al. 2023; Sugahara et al. 2024; Scholtz et al. 2024; Marconcini et al. 2024), rather than the violent disk instability, although recent ALMA [Cii] 158 µ m observations reveal a rotation of cold gas in one galaxy, UVISTA-Y-003 (Rowland et al. 2024). Further IFU observations for more galaxies are needed for a definitive conclusion, but the currently-available IFU data, the spatial extent of the clumpy structure, and theoretical simulations suggest that the majority of the observed multiple subcomponents are made by mergers.", '4.2. Luminosity Function at z ∼ 7 with Sub-Components': "In the luminosity function measurements in Section 3, we have classified each galaxy with multiple subcomponents as a single object due to the close separation of the clumps ( ≲ 5 kpc), in the same manner as Bowler et al. (2017). If we interpret the sub-components as clumps made by the violent disk instability, it is reasonable to treat these multiple sub-components as a single galaxy. If the sub-components are formed in tidal tails of a merger event, it is not clear whether we should regard these sub-components as a single object or not. \nIn either case, the measurements in Section 3 are useful to compare with previous observational and theoretical studies that have similarly classified each galaxy with multiple sub-components as a single object. Previous observational studies for bright galaxies rely on \nFigure 11. 2 '' × 2 '' cutout images of selected 12 galaxies used in this study with the UV magnitudes of M UV ≤ -21 . 5. From left to right we show the ground-based H -band, the HST/WFC3 J 125 , JH 140 , or H 160 , the JWST/NIRCam F115W, and the JWST/NIRCam false color images. The false color images are made from the F115W, F150W, and F277W data. The high-resolution JWST image with the PSF FWHM of ∼ 0 . '' 07 allows us to identify individual sub-components that are blended in the HST and ground-based images whose PSFs are ∼ 0 . '' 2 and ∼ 1 '' , respectively. Some objects (e.g., COS-788571 and COS-955126) are not clearly detected in the HST images because these images are taken in the COSMOS-DASH program whose survey depth is shallow, ∼ 25 mag. \n<!-- image --> \nCOS-zs7-1 ( z spec =7.154, M UV = -21.9) \nground-based imaging data whose spatial resolution is poor ( ∼ 1 '' ) and do not resolve these galaxies into multiple sub-components (e.g., Ono et al. 2018; Harikane et al. 2022b; Varadaraj et al. 2023). In most of the theoretical models (Mason et al. 2015b, 2023; Yung et al. 2024, 2019; Mauerhofer & Dayal 2023; Prada et al. 2023), star formation processes in galaxies are calculated based on halo merger trees, and a galaxy with multiple sub-components is treated as a single galaxy in a single \nhalo. Only cosmological simulations with high resolutions can detect sub-components in these galaxies. Thus the comparison conducted in Figure 8 is fair for most of the model predictions where these sub-components are treated as a single galaxy. \nNevertheless, it is useful to present the luminosity function measurements with multiple sub-components treated as individual galaxies. Using the subcomponents identified by the SExtractor in the JWST \nFigure 12. JWST/NIRCam F115W cutout images of the selected bright galaxies at z ∼ 7. The contours are drawn from 10% to 70% of the peak surface brightness with a 20% interval to highlight any extended emission. The black crosses show the positions of the sub-components (see Section 4.1). We find that more than half of the sources have multiple sub-components. The scale of 0 . '' 2, corresponding to 1.068 kpc at z = 7 . 0, is displayed as a scale bar in the lower right of each image. \n<!-- image --> \nand HST images (see Section 4.1) and assuming the MAG AUTO magnitude as a total magnitude for each component, we re-calculate the UV luminosity function of z ∼ 7 galaxies. Figure 13 presents the derived UV luminosity function. The luminosity function calculated from the individual sub-components shows a slightly steeper decline at the bright end than the previous measurement where the sub-components are treated as a single galaxy. Note that it is not clear whether the luminosity function calculated from the individual sub-components follows the Schechter function or not, because the brightest galaxies in our study with -24 < M UV < -23 mag, where the tension with the Schechter function can be seen, are not yet observed with either HST or JWST.", '4.3. Morphologies of Bright Galaxies at z ≳ 10': "We investigate morphologies of bright galaxies spectroscopically confirmed at z ≳ 10 using JWST/NIRCam images. Among the galaxies spectroscopically confirmed (Figure 1), we select five bright galaxies at z ≥ 10 . 6 with the UV magnitude brighter than M UV = -20 . 0 mag, where the tension between the observed number density and model predictions is seen (Figure 8). The five selected galaxies are GN-z11 (Bunker et al. 2023; Tacchella et al. 2023), CEERS2 588 (Finkelstein et al. 2023a; Harikane et al. 2024), Maisie's galaxy (Finkelstein et al. 2023a; Arrabal Haro et al. 2023a), GHZ2 (Ono et al. 2023; Castellano et al. 2024), GS-z14-0 (Robertson et al. 2023; Carniani et al. 2024a). Figures 14 and 15 show cutouts of these galaxies. We find that these galaxies are classified into two types of galaxies; extended ones with their effective radii in the rest-frame UVband of r e ∼ 200 -500 pc (Figures 14; CEERS2 588, \nFigure 13. Same as the upper-left panel of Figure 6, but with the measurements calculated from the individual subcomponents of each ground-based selected galaxy (the open red diamonds). The red-filled diamonds show the results when multi-component galaxies are instead plotted as single objects (as shown in Figure 6). \n<!-- image --> \nMaisie's galaxy, and GS-z14-0), and compact ones with r e ≲ 100 pc (Figure 15; GN-z11 and GHZ2). In addition to these five galaxies, Gz9p3 at z = 9 . 313 ( r e ∼ 270 pc Boyett et al. 2024) and GN-z9p4 at z = 9 . 380 ( r e ∼ 120 pc Schaerer et al. 2024) can be added into the extended and compact subsamples, respectively. \nIn Figure 16, we plot the rest-frame UV effective radii of the seven galaxies as a function of the redshift. We also plot the radius of a compact galaxy at z = 6 . 1057, RXCJ222024MNRAS.535..881J48-ID Top- \nFigure 14. Same as Figure 12 but for extended galaxies at z > 9. JWST/NIRCam F200W cutout images are used, and the displayed scale of 0 . '' 2 corresponds to 0.747 kpc at z = 12. \n<!-- image --> \nFigure 15. Same as Figure 12 but for compact galaxies at z > 9. \n<!-- image --> \nping et al. (2024a). These effective radii are taken from previous studies (Tacchella et al. 2023; Finkelstein et al. 2023a; Ono et al. 2023; Carniani et al. 2024a; Boyett et al. 2024; Schaerer et al. 2024; Topping et al. 2024a). We find that the sizes of the extended galaxies follow the redshift evolution measured at z ∼ 0 -8, while the compact galaxies have significantly smaller sizes than the redshift evolution. \nInterestingly, these two types of galaxies also have different emission line properties. Figure 17 shows restframe equivalent widths of Niv] λ 1486 and Civ λ 1549 as a function of the rest-UV effective radius. Rest-frame UV high ionization emission lines such as Niv] λ 1486 (47.5 eV), Civ λ 1549 (47.9 eV), and He ii λ 1640 (54.4 eV) are not significantly detected in NIRSpec spectra of the extended galaxies. In contrast, the compact galaxies exhibit prominent high ionization lines such as Civ λ 1549, He ii λ 1640, and Niv] λ 1486 (e.g., EW 0 NIV] ≃ 9 -30 ˚ A), indicating that the interstellar medium of compact galaxies are more nitrogen-enriched and highly ionized compared to that of the extended galaxies. A strong Niv] λ 1486 line with EW 0 NIV] ∼ 20 ˚ A is also reported in a compact ( r e ∼ 100 pc) and bright ( M UV ∼ -22 mag) galaxy with an AGN at z = 5 . 55, GS 3073 (Vanzella et al. 2010; Grazian et al. 2020; Ubler et al. 2023; Ji et al. 2024). CEERS 1019 at z = 8 . 68 also shows strong Niv] λ 1486 emission and a compact mor-ph \n023; Ono et al. 2024), following this anti-correlation trend. These two types are clearly different both in morphologies and emission line properties, suggesting that at least two different processes are shaping the physical properties of these bright galaxies at z ≳ 10, which will be further discussed in Section 6.1.", '5. SED FITTING': "To understand the physical properties of the bright galaxies at z ∼ 7 -12, we conduct SED fitting. The galaxies to be fitted here are limited to those with well-constrained rest-frame optical emission line fluxes. We select galaxies within the PRIMER footprint (COS2987030247, COS-3018555981, and Himiko), where the combination of the F410M medium-band filter and the spectroscopic redshift resolves the degeneracy between the Balmer break and rest-frame optical emission lines (e.g., Desprez et al. 2024), and GHZ2 at z = 12 . 34 whose [Oiii] λλ 4959,5007 and H α emission lines are detected with MIRI (Zavala et al. 2024). The photometric measurements and the [Oiii] λλ 4959,5007 and H α emission lines fluxes of GHZ2 are taken from Naidu et al. (2022b) and Zavala et al. (2024), respectively. For the other sources in the PRIMER footprint, we measure the fluxes of F444W-detected sub-components in the PSF-matched NIRCam images using SExtractor in the same manner as Harikane et al. (2023a). We calcu- \nFigure 16. Rest-frame UV effective radius of galaxies as a function of the redshift. The black symbols are size measurements for bright galaxies with M UV ∼ -21 mag at z = 0 -8 in Shibuya et al. (2015, square: star forming galaxies, circle: Lyman break galaxies), and the gray shaded region shows the size evolution for galaxies with M UV ∼ -21 mag and its 1 σ dispersion in Shibuya et al. (2015). The red circles are bright galaxies at z > 9 whose sizes follow the redshift evolution measured at z ∼ 0 -8 (Gz9p3: Boyett et al. 2024, CEERS2 588: Finkelstein et al. 2023a; Harikane et al. 2024, Maisie's galaxy: Finkelstein et al. 2023a; Arrabal Haro et al. 2023a, and GS-z14-0: Robertson et al. 2023; Carniani et al. 2024a), and the magenta diamonds are compact galaxies whose effective radii are smaller than ∼ 100 pc (RXCJ2248-ID: Topping et al. 2024a, GN-z9p4: Schaerer et al. 2024, GN-z11: Bunker et al. 2023; Tacchella et al. 2023, GHZ2: Ono et al. 2023; Castellano et al. 2024). Bright galaxies at z ≳ 10 can be classified into two types of galaxies; extended galaxies with weak high-ionization emission lines (the red circles), and compact galaxies with strong high-ionization lines such as Niv] λ 1486 (the magenta diamonds). The extended galaxies with sizes of r e ∼ 200 -500 pc do not exhibit prominent rest-UV high ionization emission lines and sometimes show a signature of merger activity (e.g., Gz9p3; Boyett et al. 2024). In contrast, the compact galaxies exhibit strong high ionization lines such as Civ λ 1549, He ii λ 1640, and Niv] λ 1486 (e.g., EW 0 NIV] ≃ 10 -30 ˚ A), suggesting compact and intense starburst or AGN activity. \n<!-- image --> \ntotal magnitude in each band from an aperture magnitude measured in a 0. '' 3-diameter circular aperture with an aperture correction. Since the PSF of the F444W image ( ∼ 0 . '' 16) is larger than that of the F115W image used in Section 4.1, some of the subcomponents detected in Section 4.1 are not identified here. In the SED fitting, we use prospector (Johnson et al. 2021), with changing the dust optical depth in the V -band, metallicity, star formation history, and total stellar mass as free parameters while fixing the redshift to the spectroscopically-determined value. Model spectra are derived from the Flexible Stellar Population Synthesis (FSPS; Conroy et al. 2009; Conroy & Gunn 2010) package with the modules for Experiments in Stellar Astrophysics Isochrones and Stellar Tracks (MIST; \nChoi et al. 2016). The boost of ionizing flux production of massive stars due to rotation is included in the MIST isochrones (Choi et al. 2017). Here we assume the stellar IMF determined by Chabrier (2003), the Calzetti et al. (2000) dust extinction law, and the intergalactic medium (IGM) attenuation model by Madau (1995). We adopt a flexible star formation history with five bins. The first bin is fixed at 0-10 Myr and the other bins are spaced equally in logarithmic times between 10 Myr and a lookback time that corresponds to z = 30, where the SFR within each bin is constant. We assume a continuity prior for the star formation history, and flat priors for other parameters in the range of 0 < τ V < 2, -2 . 0 < log( Z/Z ⊙ ) < 0 . 4, and 6 < log( M ∗ /M ⊙ ) < 12. We search for the best-fit model to the observed photo- \nFigure 18 shows the SED fitting result for GHZ2 at z = 12 . 34. With the blue rest-frame UV slope ( β UV = -2 . 4; Castellano et al. 2024) and the strong [Oiii] and H α emission lines detected with MIRI, the stellar age of GHZ2 is very young, about 10 Myr. The estimated star formation history exhibits a sharp increase \n<!-- image --> \nFigure 17. Rest-frame equivalent widths of Niv] λ 1486 (left) and Civ λ 1549 (right) as a function of an effective radius. The magenta diamonds and red circles are compact ( r e ≲ 100 pc) and extended ( r e ∼ 200 -300 pc) galaxies, respectively. The upper limit is 2 σ . The compact galaxies show strong Niv λ 1486 emission lines with EW 0 NIV] ≳ 10 ˚ A and sometimes strong Civ λ 1549 lines, while these lines are weak in the extended galaxies. The significance levels of these anti-correlations are 92% and 87% for Niv] λ 1486 and Civ λ 1549, respectively. GN-z11 shows a weak Civ λ 1549 emission line because of a strong absorption seen in the spectrum (Maiolino et al. 2024). \n<!-- image --> \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \n0.0 \nFigure 18. SED fitting results for GHZ2 at z spec = 12 . 34. The leftmost panel shows the 2 '' × 2 '' JWST/NIRCam false-color image made from F115W, F150W, and F277W. The position of GHZ2 is indicated with the white circle whose diameter is 0. '' 3. The second left panel is an SED of GHZ2. The red circles and arrows show the measured magnitudes and 2 σ upper limits, respectively, and the blue line and open circles denote the best-fit model SED with Prospector . The third left panel shows the comparison between observed and modeled line fluxes of H α and [Oiii] λλ 4959,5007. The rightmost panel shows the star formation history constrained with the SED fitting. GHZ2 exhibits a bursty star formation history with SFRs increasing by a factor of ≳ 10 within the last ∼ 100 Myr. \n<!-- image --> \nmetric data points (and the [Oiii] λλ 4959,5007 and H α fluxes for GHZ2) with the MCMC method by using emcee (Foreman-Mackey et al. 2013). Table 5 summarizes the results of the SED fitting. The values for the total component of COS-2987030247, COS-3018555981, and Himiko are the luminosity-weighted means of the measurements in individual sub-components ('Clumps'). \nby a factor of 10 in the recent ∼ 20 Myr, indicating that GHZ2 is in a starburst phase, although the uncertainty is large at > 20 Myr. Figures 19-21 present the results for COS-2987030247, COS-3018555981, and Himiko, respectively, at z ∼ 7. Although the sub-components of these galaxies have various rest-frame UV slopes, the strong [Oiii] and H α emission lines inferred from the broad- and medium-band fluxes suggest very young stellar ages of 10 -20 Myr and bursty star-formation histories with the SFR increasing by a factor of ∼ 10 -100 within the last 100 Myr, similar to GHZ2, albeit large uncertainties at > 100 Myr. Indeed, the observed increase of the SFR in the last ∼ 10 -100 Myr is more \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \n0.0 \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \n0.0 \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \nFigure 19. SED fitting results for sub-components (clumps A-C; from top to bottom) of COS-2987030247 at z spec = 6 . 808. The left panels show the JWST/NIRCam false-color image (same as Figure 11) and the position of each clump. The middle and right panels are SEDs and the star formation histories, respectively, in the same manner as Figure 18. All of the clumps exhibit bursty star formation histories with SFRs increasing by a factor of ∼ 10 -100 within the last 100 Myr. We do not include a northeast clump in our analysis because this clump is detected in the F090W image and is thus considered to be a foreground object. \n<!-- image --> \n0.0 \nrapid than an averaged dark matter halo growth history at the same redshifts (e.g., Fakhouri et al. 2010). Thus these starbursts may be responsible for the UV-bright nature of galaxies at z ∼ 7 -12, whose abundances show the tension with the theoretical predictions in the luminosity functions. More discussions on the physical origin of the overabundance are presented in Section 6.1. \nWe find that all of the clumps analyzed here show increasing star formation histories at the time of observations, which is in contrast to the results of Asada et al. (2024) for fainter galaxies at z ∼ 5 -7 showing both ris- \ng and declining star formation histories. This difference in the fraction of galaxies with rising star formation histories between this study and Asada et al. (2024) may be due to the difference in the UV luminosities of the samples used in these two studies. Given the bright UV magnitudes of our galaxies, it is possible that we are selectively observing the UV-bright phase of galaxies with rising star formation histories, while Asada et al. (2024) are looking at the UV-bright phase of low-mass galaxies and the UV-faint phase of massive galaxies. Indeed, Endsley et al. (2023) find a similar trend of a higher \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \n0.0 \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \nFigure 20. Same as Figure 19 but for COS-3018555981 at z spec = 6 . 854. Since we use the F444W image with a relatively large PSF of ∼ 0 . '' 16 in the FWHM as the detection image, a few neighboring clumps are identified and analyzed as a single clump (e.g., Clump A). \n<!-- image --> \n0.0 \nTable 5. Summary of the SED Fitting Results \nNote -Errors are 1 σ . The SFR presented here is the SFR averaged over the past 50 Myr, and the stellar age is the mass-weighted age calculated from the star formation history. Positions of the clumps in each galaxy (Clumps A, B, and C) can be found in Figures 19-21. \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \n0.0 \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \n0.0 \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \n0.0 \nFigure 21. Same as Figure 19 but for Himiko at z spec = 6 . 595. \n<!-- image --> \nfraction of galaxies with rising histories towards brighter magnitudes.", '6.1. Physical Origin of the Overabundance of Bright Galaxies': 'Various studies using JWST have reported that the abundance of bright galaxies at z ≳ 10 is higher than theoretical model predictions. The physical origin of this overabundance is not clear, but several possibilities are discussed, such as a high star formation efficiency, AGN activity, a top-heavy IMF, bursty star formation, radiation-driven outflows, and a flaw in the current cosmology model (see Section 1). As shown in Section 3.3, we have found that the number densities of bright galaxies with M UV ≲ -21 mag at z ∼ 12 -14 are higher than the model predictions (Figures 8 and 9), similar to previous studies using JWST. In addition, the measured number densities of bright galaxies with M UV ≲ -23 mag at z ∼ 7 are also higher than some model predictions.', '6.1.1. z ∼ 7 : Merger-Induced Starbursts': 'In Section 5, we find that the SFR of bright z ∼ 7 galaxies has increased by a factor of 10-100 within the last ∼ 100 Myr, indicating that the recent starburst is contributing to the UV-bright nature of these luminous galaxies. Since these UV-bright galaxies at z ∼ 7 exhibit clumpy morphologies with multiple subcomponents (Section 4.1, Figure 12), the starburst is thought to be triggered by a recent merger event. Indeed, previous studies have reported evidence of mergers in some galaxies at z ∼ 7 (Hashimoto et al. 2019; Sugahara et al. 2024). Numerical simulation results in Witten et al. (2024) also indicate a starburst during a merger phase at z > 7. These results suggest that the merger-induced starburst can explain the high number density of UV-bright galaxies at z ∼ 7. The tension between observations and theoretical predictions may be due to the SFRs of model galaxies in the merging phase not increasing sufficiently compared to what has been observed.', '6.1.2. z ≳ 10 : Merger-Induced Starburst and Compact Star Formation/AGN': "In contrast to the case at z ∼ 7, the situation for galaxies at z ≳ 10 is not simple. As described in Section 4.3, bright galaxies at z ≳ 10 can be classified into two types; extended galaxies with weak high ionization lines, and compact galaxies with strong high-ionization lines. For the extended galaxies, as discussed in Carniani et al. (2024a), the weak high ionization lines suggest no strong \nAGN activity, and the shape of the rest-UV continuum does not support a significant contribution from nebular continuum emission with a top-heavy IMF such as discussed in Cameron et al. (2023b). We discuss that a high star formation efficiency, possibly enhanced by a merger-induced starburst similar to the bright z ∼ 7 galaxies studied here, is responsible for their UV-bright nature. Such a merger-induced starburst is expected to be frequent because the merger rate of bright galaxies at z ≳ 10 is theoretically several times higher than those at z ∼ 7 (e.g., Fakhouri et al. 2010; RodriguezGomez et al. 2015). The major merger timescale, the inverse of the merger rate, is ∼ 200 Myr at z ∼ 12 for bright galaxies with M ∗ ∼ 10 9 M ⊙ (Rodriguez-Gomez et al. 2015), which is shorter than the age of the universe at z ∼ 12 (370 Myr), indicating that most of the bright galaxies at z ∼ 12 have experienced at least one major merger. Indeed, some bright galaxies at z ≳ 9 show merger signatures (Hsiao et al. 2023; Boyett et al. 2024). Theoretically, Ono et al. (2023) also discuss that simulated galaxies in Yajima et al. (2022) whose sizes are r e ∼ 200 -400 pc and follow the size evolution are experiencing major mergers or tidal interactions. Note that the bright galaxies at z ∼ 7 showing multiple subcomponents also follow the size evolution (Bowler et al. 2017). Thus the merger-induced starburst, similar to the bright galaxies at z ∼ 7, is a plausible scenario for the brightness of these extended galaxies at z ≳ 10. Weak emission lines in these extended galaxies are probably due to their low metallicites or high ionizing photon escape fractions (e.g., Carniani et al. 2024a; Ferrara 2024b). In contrast to the z ∼ 7 galaxies showing clumpy morphologies, some of these z ≳ 10 galaxies do not look clumpy. This is probably because the smaller angular diameter distance and galaxy's size at z ≳ 10 compared to z ∼ 7 make it difficult to observe clumps in z ≳ 10 galaxies. \nThe compact galaxies at z ≳ 10 such as GN-z11 and GHZ2 do not show significantly extended or clumpy structures unlike the bright z ∼ 7 galaxies and the extended z ≳ 10 galaxies. Given the high signal-to-noise ratios of the detection (e.g., > 30 σ ), if these galaxies exhibit multiple components or extended structures, these components/structures should be detected, indicating that the image depth is not the origin of their apparent compactness. Rather, their compact morphologies are made by either compact star formation or AGN activity, which enhances the UV luminosity and makes the tension with the model predictions. Compact star formation can easily happen at high redshifts (e.g., Zolotov et al. 2015; Tacchella et al. 2016; Fukushima & Yajima 2021). Ono et al. (2023) discuss that such a compact star \nformation phase with intense accretion of the material makes isolated galaxies whose sizes are less than ∼ 100 pc, similar to GN-z11 and GHZ2. Interestingly, these compact galaxies are very blue in the rest-UV continuum with slopes of β UV ≃ -2 . 4 (Bunker et al. 2023; Castellano et al. 2024), suggesting negligible/zero dust attenuation (Cullen et al. 2023, 2024; Topping et al. 2024b; Austin et al. 2024; Morales et al. 2024). Formation of such compact galaxies with weak dust attenuation is indeed predicted in the feedback-free starburst scenario proposed in Dekel et al. (2023). P'erez-Gonz'alez et al. (2023) discuss that such compact star formation are not predicted in some cosmological simulations, which may be the origin of the tension between the JWST observations and theoretical predictions. Such compact star formation may increase the density in a galaxy and the rate of runaway stellar collisions producing supermassive stars and/or tidal disruption events, enhancing the nitrogen production (e.g., Cameron et al. 2023a; Watanabe et al. 2024). \nAGN activity is also a plausible scenario for the compact galaxies at z ≳ 10. Several studies claim AGN activity in these compact galaxies such as GN-z11 and GHZ2, based on the detections of high ionization emission lines including Ne iv λ 2424 (63.5 eV; Maiolino et al. 2024) and a very strong Civ λ 1549 line with an equivalent width of EW 0 CIV ∼ 40 ˚ A (Castellano et al. 2024). AGN activity is also reported in similarly compact and bright galaxies, GS 3073 at z = 5 . 55 ( Ubler et al. 2023) and COS-zs-1 at z = 7 . 15 ( Ubler et al. 2024), with broad H β and/or H α emission lines. To make these galaxies UV-bright, their rest-frame UV continua should be dominated by emission from an accretion disk of the AGN, such as type-1 broad-line AGNs whose accretion disk emission can be directly observed. Alternatively, GNz11 and GHZ2 can be narrow-line type-2 quasars (e.g., Zakamska et al. 2003), and their UV continua are significantly contributed by scattered lights of emission from the accretion disk.", '6.2. Low-Redshift Interlopers at the Bright End': 'JWST/NIRSpec spectroscopy has revealed that very luminous ( M UV ≲ -23 mag) galaxy candidates at z ∼ 10 -12 identified in ground-based images before JWST (Bowler et al. 2020; Harikane et al. 2022a) are low-redshift passive galaxies at z ∼ 3 -4 whose Balmer break is redshifted to the wavelength of the Lyman break at z ∼ 10 -12 (Figures 4 and 5). The stellar masses of HD1 and HD2 are estimated to be ∼ 10 10 M ⊙ (Sato et al. 2024), lower than typical passive galaxies at similar redshifts (e.g., Carnall et al. 2022, 2024), indicating that HD1 and HD2 are relatively faint and easily affected by \nphotometric scatters. These results suggest that lowmass passive galaxies at z ∼ 3 -4 can be selected as bright Lyman break galaxies at z ∼ 10 -12 due to photometric scatters in relatively shallow ground-based and Spitzer images (see Figure 4). These passive galaxies at intermediate redshifts are important contaminants that should be taken into account in the high redshift galaxy selection, in addition to strong emission line galaxies seen in Arrabal Haro et al. (2023a). Even more careful galaxy selections are required to select very luminous galaxies at z ∼ 10 -12 and remove these low-redshift interlopers in relatively shallow datasets. \nTo understand how careful selection is needed, we calculate expected number densities of low-redshift interlopers. We convert the stellar mass functions of passive galaxies at z ∼ 2 . 5 -3 ( z ∼ 3 -4) in Davidzon et al. (2017) to the H -band ( K -band) luminosity function using the NIRSpec spectrum of XMM3-3085 (HD1) obtained in Section 2.1.3. Note that the conclusion does not change if we use the stellar mass functions in McLeod et al. (2021). We calculate the expected H -band ( K -band) luminosity function of low-redshift interlopers by assuming that a certain fraction of the passive galaxies are selected as z ∼ 10 ( z ∼ 12) galaxy candidates. We adopt a fraction of ≤ 0 . 2% to match the expected number density to the observed densities of the three low-redshift interlopers we have identified in this study. In Figure 22, we plot the calculated luminosity functions of low-redshift interlopers and the best-fit double-power-law functions of galaxies at z ∼ 10 -12 with errors constrained from the spectroscopic results in Section 3.2. We find that even if only 0 . 2% of lowredshift passive galaxies are erroneously selected as highredshift galaxy candidates, then the number density of low-redshift interlopers becomes higher than that of real high-redshift galaxies at z ∼ 10 -12. In future widearea bright ( M UV ≲ -22 mag) galaxy surveys at z ≳ 10 with Euclid, Roman, and GREX-PLUS, it is necessary to devise selection criteria, such as using a more strict dropout color criterion with deeper datasets at the wavelength shorter than the break, to limit the fraction of low-redshift passive galaxies entering into the selection to less than 0 . 2%. Note that these interlopers are much less of a concern for JWST-selected candidates because they are usually fainter than M UV ∼ -22 mag, where the effect of the contamination is not significant.', '7. SUMMARY': 'In this paper, we present the number densities and physical properties of galaxies at z ∼ 7 -14, based on the sample of 60 luminous galaxies spectroscopically \nFigure 22. Effect of low-redshift interlopers from passive galaxies at z ∼ 2 . 5 -4 (see also Fujimoto et al. 2023a). In the left (right) panel, the red solid line represents the best-fit double power-law function at z ∼ 10 ( z ∼ 12) obtained in this study, and the gray symbol is the number density of z ∼ 10 ( z ∼ 13) galaxy candidate from Bowler et al. (2020) (Harikane et al. (2022a)) that is identified as a low-redshift galaxy in this study. The gray dashed curve is the H -band ( K -band) luminosity function of passive galaxies at z ∼ 2 . 5 -3 ( z ∼ 3 -4), which are calculated based on the stellar mass functions of passive galaxies in Davidzon et al. (2017) and the NIRSpec spectrum of XMM3-3085 (HD1) obtained in Section 2.1.3. At these redshifts, passive galaxies contaminate the Lyman break galaxy selections at z ∼ 10 ( z ∼ 12) because the Balmer break is redshifted to ∼ 1 . 3 µ m ( ∼ 1 . 6 µ m), the wavelength of the Lyman break at z ∼ 10 ( z ∼ 12). The gray shaded region shows the expected number density of low-redshift interlopers assuming that ≤ 0 . 2% of the passive galaxies are selected as z ∼ 10 ( z ∼ 12) galaxies. The passive galaxies at intermediate redshifts can contaminate a sample of bright ( M UV ≲ -22 mag) galaxy candidates at z > 10, and strict screening is necessary to select real z > 10 bright galaxies. \n<!-- image --> \nm \n<!-- image --> \nm \nconfirmed at z spec = 6 . 538 -14 . 32. Our major findings are summarized below: \n- 1. We constrain the UV luminosity functions at z ∼ 7 -14. At z ∼ 7, the bright end of the luminosity function is well described by the double-powerlaw or lensed Schecher function rather than the original Schechter function (Figure 6). We find that the number densities of spectroscopicallyconfirmed bright galaxies at z ∼ 7 and 12 -14 are higher than theoretical model predictions (Figures 8 and 9).\n- 2. Using the high-resolution JWST/NIRCam images, we find that ∼ 70% of our bright ( M UV ≤ -21 . 5 mag) galaxy sample at z ∼ 7 exhibit clumpy morphologies with multiple sub-components, suggesting recent merger events (Figures 11 & 12).\n- 3. We conduct SED fitting for GHZ2 at z spec = 12 . 34 and three galaxies at z spec ∼ 7, whose [Oiii] λλ 4959,5007 and H α emission line fluxes are constrained with MIRI and the NIRCam F410M observations, respectively. We find that all of the clumps in the four galaxies show bursty starformation histories with the SFR increasing by a \nfactor of ∼ 10 -100 within the last 100 Myr (Figures 18-21). \n- 4. Based on the clumpy morphologies and the bursty star formation histories revealed in this study, we discuss that a recent merger event has triggered a starburst in the bright galaxies at z ∼ 7. Such a merger-induced starburst boosts the UV luminosity, resulting in the observed high number density of bright galaxies at z ∼ 7 showing the tension with theoretical models (Section 6.1).\n- 5. At z ≳ 10, bright galaxies are classified into two types of galaxies; extended ones with weak highionization emission lines and compact ones with strong high ionization lines including Niv ] λ 1486 (Figures 16 and 17). These two populations are different in both morphologies and emission line properties, suggesting that at least two different processes are contributing to the overabundance of bright galaxies at z ≳ 10. We discuss that a merger-induced starburst may be responsible for the UV-bright nature of the extended galaxies, similar to the bright z ∼ 7 galaxies studied here, while the UV luminosity of compact galaxies is enhanced by compact star formation or AGN activity (Section 6.1). \n- 6. Our JWST/NIRSpec observations have revealed that very bright galaxy candidates at z ∼ 10 -12 previously identified from ground-based images are low redshift passive galaxies at z ∼ 3 -4 (Figure 4). These passive low-redshift interlopers are erroneously selected as high redshift galaxies because of 1) large photometric scatters originating from relatively shallow datasets, and 2) their very bright magnitudes. This result indicates that strict selection criteria that keep the fraction of passive galaxies entering into the selection to less than 0 . 2% are required in the future wide-area bright galaxy surveys at z ≳ 10 with Euclid, Roman, and GREX-PLUS (Figure 22).', 'ACKNOWLEDGMENTS': 'We thank the anonymous referee for careful reading and valuable comments that improved the clarity of the paper. We are grateful to Rebecca Bowler and Rohan Varadaraj for providing the latest photometric measurements for XMM3-3085, and to Masayuki Akiyama, Yoshinobu Fudamoto, Takuya, Hashimoto, Taddy Kodama, Mariko Kubo, Masafusa Onoue, and Hannah Ubler for useful comments and discussions. This paper makes use of the ALMA data obtained in 2019.1.01634.L (REBELS), 2021.1.00207.S, 2021.1.00341.S, and 2022.1.00522.S. The authors acknowledge the REBELS team led by Richard J. Bouwens for developing their observing program. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. The JWST data presented in this paper were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with programs ERS-1324 (GLASS), ERS-1345 (CEERS), GO-1727 (COSMOS-Web), GO-1740, GO1837 (PRIMER), GO-2561 (UNCOVER), GO-2792, and GTOs-1180, 1181, 1210, and 1286 (JADES). The authors acknowledge the GLASS, CEERS, COSMOS-Web, UNCOVER, and JADES teams led by Tommaso Treu, Steven L. Finkelstein, Jeyhan Kartaltepe & Caitlin Casey, Ivo Labbe & Rachel Bezanson, and Daniel Eisenstein & Nora Luetzgendorf, respectively, for developing their observing programs. The JWST and HST data presented in this article were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed can be accessed via 10.17909/gbkc2b17. Some of the data products presented herein were retrieved from the Dawn JWST Archive (DJA). DJA is an initiative of the Cosmic Dawn Center (DAWN), which is funded by the Danish National Research Foundation under grant DNRF140. This publication is based upon work supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (20H00180, 21K13953, 21H04467, 22H04939, 23H00131, 23KJ0728, 24H00245), the JSPS Core-to-Core Program (JPJSCCA20210003), and the JSPS International Leading Research (22K21349). This work was supported by the joint research program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo. S.F. acknowledges the NASA Hubble Fellowship grant #HSTHF2-51505.001-A awarded by the Space Telescope Sci- \nSoftware: Prospector (Johnson et al. 2021), PypeIt (Prochaska et al. 2020; Prochaska et al. 2020), SExtractor (Bertin & Arnouts 1996)', 'REFERENCES': "Vijayan, A. P., Lovell, C. C., Wilkins, S. M., et al. 2021, MNRAS, 501, 3289 \nVogelsberger, M., Nelson, D., Pillepich, A., et al. 2020, MNRAS, 492, 5167 \n- Watanabe, K., Ouchi, M., Nakajima, K., et al. 2024, ApJ, 962, 50\n- Weaver, J. R., Taamoli, S., McPartland, C. J. R., et al. 2024, arXiv e-prints, arXiv:2405.13505\n- Wilkins, S. M., Vijayan, A. P., Lovell, C. C., et al. 2023, MNRAS, 519, 3118\n- Willott, C. J., McLure, R. J., Hibon, P., et al. 2013, AJ, 145, 4\n- Willott, C. 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2024arXiv240912336B	This work focuses on constructing electromagnetized black holes and vortexlike backgrounds within the framework of the ModMax theorythe unique nonlinear extension of Maxwells theory that preserves conformal symmetry and electromagnetic duality invariance. We begin by constructing the MelvinBonnor electromagnetic universe in ModMax through a limiting procedure that connects the spacetime of two charged accelerating black holes with that of a gravitating homogeneous electromagnetic field. Building on this result we proceed to construct the Schwarzschild and Cmetric MelvinBonnor black holes within the ModMax theory representing the first black hole solutions embedded in an electromagnetic universe in the context of nonlinear electrodynamics. While the characteristics of the MelvinBonnor spacetime and some of its black hole extensions have been widely examined we demonstrate for the first time that the SchwarzschildMelvinBonnor configuration exhibits an unusual KerrSchild representation. Following this direction we also unveil a novel KerrSchild construction for the spacetime of two accelerating black holes drawing on the intrinsic relationship between the MelvinBonnor spacetime and the Cmetric. Finally we expand the spectrum of exact gravitational solutions within EinsteinModMax theory by constructing a vortexlike background that coexists with the MelvinBonnor universe. In this process the TaubNUT spacetime in ModMax has played a crucial role. We also present an extended TaubNUT solution that incorporates the contribution of a monopolelike magnetic component in the gauge field.	2024-09-01T00:00:00Z	['2024arXiv240912336B', 'arXiv:2409.12336', '10.48550/arXiv.2409.12336']	['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']	Electromagnetized Black Holes and Swirling Backgrounds in Nonlinear Electrodynamics The ModMax case	2,024	193	0.14	['EPRINT_HTML', 'EPRINT_PDF']	5	https://arxiv.org/pdf/2409.12336.pdf	{'Electromagnetized Black Holes and Swirling Backgrounds in Nonlinear Electrodynamics: The ModMax case': "José Barrientos, 1, 2, ∗ Adolfo Cisterna, 1, 3, † Mokhtar Hassaine, 4, ‡ and Konstantinos Pallikaris 5, § \n1 Sede Esmeralda, Universidad de Tarapacá, Avenida Luis Emilio Recabarren 2477, Iquique, Chile 2 Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic 3 Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 180 00 Praha 8, Czech Republic 4 Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile 5 Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia \nThis work focuses on constructing electromagnetized black holes and vortex-like backgrounds within the framework of the ModMax theory-the unique nonlinear extension of Maxwell's theory that preserves conformal symmetry and electromagnetic duality invariance. We begin by constructing the Melvin-Bonnor electromagnetic universe in ModMax through a limiting procedure that connects the spacetime of two charged accelerating black holes with that of a gravitating homogeneous electromagnetic field. Building on this result, we proceed to construct the Schwarzschild and C-metric Melvin-Bonnor black holes within the ModMax theory, representing the first black hole solutions embedded in an electromagnetic universe in the context of nonlinear electrodynamics. While the characteristics of the Melvin-Bonnor spacetime and some of its black hole extensions have been widely examined, we demonstrate for the first time that the Schwarzschild-Melvin-Bonnor configuration exhibits an unusual Kerr-Schild representation. Following this direction, we also unveil a novel Kerr-Schild construction for the spacetime of two accelerating black holes, drawing on the intrinsic relationship between the Melvin-Bonnor spacetime and the C-metric. Finally, we expand the spectrum of exact gravitational solutions within Einstein-ModMax theory by constructing a vortex-like background that coexists with the Melvin-Bonnor universe. In this process, the TaubNUT spacetime in ModMax has played a crucial role. We also present an extended Taub-NUT solution that incorporates the contribution of a monopole-like magnetic component in the gauge field.", 'I. INTRODUCTION': "Classical modifications to Maxwell's theory that become significant in strong field regimes, while naturally reducing to Maxwell's electrodynamics in the weak field approximation, are known as nonlinear theories of electrodynamics (NLE) [1]. Historically, they have been inspired by the development of Born-Infeld [2] and EulerHeisenberg [3] theories. Born-Infeld theory aimed to address the infinite self-energy in the electron's electric field, whereas Euler-Heisenberg theory provided a complete nonperturbative one-loop effective action for Quantum Electrodynamics, accounting for vacuum polarization effects due to virtual electrons and positrons. Being the electromagnetic duality of Maxwell field equations and the conformal invariance of Maxwell action some of the most remarkable features of Maxwell theory in 4 d , NLE have ultimately evolved to produce a nonlinear generalization of Maxwell's theory that preserves both of these symmetries. Dubbed ModMax theory [1, 4, 5], the recently proposed model represents a one-parameter extension of Maxwell's theory in four dimensions. Several \n‡ \nElectronic address: [email protected] \naspects of this model have come under scrutiny in the last years, see e.g. [6-17]. Naturally, investigating the black hole spectrum of the theory, in essence, exact solutions to the Einstein-ModMax field equations, serves as a guide to better understand the effects that arise from nonlinearities of the ModMax theory. \nIn electrovacuum, black hole spacetimes are characterized by seven parameters being the mass m , the Kerr-like rotation parameter a , the electric and magnetic charges q e and q m , the cosmological constant Λ , and the more exotic NUT and acceleration parameters l and A , respectively. The largest spacetimes involving all these parameters are the Plebanśki-Demiański configuration [18-20] which is the most general algebraically special solution of type D of the Einstein-Maxwell equations, and its recent generalization of algebraically general nature (type I), dubbed as Enhanced Plebanśki-Demiański [21]. Additionally, two specific scenarios that fall outside these constructions involve the consideration of external electromagnetic and vortex-like fields. These scenarios correspond to Melvin-Bonnor geometries [22, 23] and swirling geometries [24, 25], both of which can accommodate the embedding of the families of solutions mentioned above. Their construction generally requires the use of a certain group of Lie point symmetries of the electrovacuum, specifically, Harrison and Ehlers symmetries of the magnetic type [26, 27]. \nUp to now, the study of black hole spacetimes within ModMax theory has primarily focused on spherically \nsymmetric configurations, see e.g. [28, 29]. However, there are two notable exceptions: a stationary solution characterized by Taub-NUT geometry [30, 31] and an accelerating black hole solution [32], with the latter being the first C-metric spacetime identified in a NLE model. Conversely, rotating solutions appear to fully engage the nonlinear aspects of the theory, as no such geometries have been constructed to date, even within the slowly rotating approximation [33]. \nThe main objective in this work is to expand the spectrum of black hole solutions within the ModMax theory and to gain deeper insight into the technical challenges that prevent the construction of spacetimes with certain characteristics. Considering the current catalog of solutions in Einstein-ModMax theory and the persistent challenge of constructing rotating configurations, we focus on the remaining set of spacetimes characterized by background electromagnetic and vortex-like fields, specifically the Melvin-Bonnor and swirling geometries. Although Melvin-Bonnor spacetimes have been previously explored within the context of NLE, these constructions have faced limitations, particularly when a nontrivial cosmological constant is involved [34]. Moreover, no black hole immersed in a Melvin-Bonnor spacetime has yet been realized in any NLE theory whatsoever. On the other hand, swirling geometries have not been investigated in the context of NLE, even at the background level without an immersed black hole. The study of such a geometry is especially intriguing, as even if the swirling rotation is not of the Kerr-type, massive bodies do not drag the spacetime via its rotation but an intrinsic stationarity of the spacetime itself drags massive bodies instead. These geometries may provide valuable insights for the future development of standard rotating spacetimes within the theory. \nTo construct Melvin-Bonnor geometries, we employ the innovative procedure proposed in [35, 36]. This latter represents a self-gravitating extension of the well-known concept that a homogeneous electric field in Minkowski spacetime is easily retrieved by taking Born's solutions describing two charged accelerating test particles and enlarging the distance between them while properly increasing the value of the charges. Therefore, following this approach, we first construct the Melvin-Bonnor-ModMax universe by starting from the spacetime of two accelerated charged black holes, specifically the C-metric spacetime discovered in [32]. By examining the near-horizon geometry around the Rindler horizon and sending both black holes to infinity, we obtain the homogeneous, selfgravitating electromagnetic background characteristic of the Melvin-Bonnor-ModMax spacetime. Notably, unlike the analogous procedure in flat spacetime, the charge of the resulting configuration remains finite. Furthermore, this method naturally incorporates a nontrivial cosmological constant without requiring fine-tuning of the external electromagnetic field. Building on this foundation, an informed conjecture allows us to construct the Schwarzschild-Melvin and C-metric-Melvin black holes \nwithin ModMax theory, a result that could potentially indicate the existence of an underlying electromagnetization symmetry in the Einstein-ModMax framework. In the specific case of these configurations, such symmetry can be attributed to the electromagnetization symmetries inherent to the Einstein-Maxwell model. As we will show for the Melvin-Bonnor, Schwarzschild-Melvin, and C-metric-Melvin spacetimes, the electromagnetic invariants that define the ModMax action simplify, allowing the action to ultimately reduce to that of Maxwell's theory. \nFurthermore, we begin the exploration of charged spacetimes with a background vortex-like field, focusing on a swirling background geometry for the ModMax theory. While this configuration does not exhibit a Kerr-type rotation, it represents the first example of a highly nontrivial stationary spacetime in ModMax theory. A key aspect of its construction is the identification of the associated magnetic field. Similar to the TaubNUT case [30, 31], the nonlinearity of the theory becomes pronounced when the magnetic field includes a monopole contribution. Consequently, the field equations can be directly integrated by considering the magnetic field generated by the interaction between the spacetime's stationarity and a monopole electric charge. In our pursuit of constructing a vortex-like background solution, we first revisit the Taub-NUT-ModMax spacetimes [30, 31]. This examination provides us with the necessary insights to achieve our construction, while also extending prior results by incorporating a monopole-like magnetic component in the gauge field. For the swirling spacetime, we utilize an external electromagnetic field of the MelvinBonnor type. The stationarity of this spacetime allows the magnetic field component to generate an external electric field, thereby simplifying the integration of the field equations in analogy with the Taub-NUT scenario. A swirling Melvin-Bonnor background spacetime is thus obtained. While an exact solution for the black hole case has not yet been found, we suggest that the main challenge is computational in nature. This hints at the potential existence of another underlying Lie point symmetry, this time of the Ehlers type, in Einstein-ModMax theory. This symmetry, unlike the electromagnetic one, should be entirely nonlinear. The stationarity of the spacetime prevents the ModMax action from resembling Maxwell's theory, as there is no particular simplification of the electromagnetic invariants in these configurations. \nThe plan of the paper is divided as follows. In Section II, we provide a concise introduction to ModMax theory and its field equations, together with a review of the accelerating ModMax black hole solutions discovered in [32]. In Section III, we describe and apply the limiting procedure used to derive the Melvin-ModMax spacetime from the ModMax-C-metric, with a brief discussion of the key properties of this electromagnetic background. We also present a generalization that incorporates a nontrivial cosmological constant. In Section IV, we extend this background solution to include cases where a static \nor accelerating black hole is immersed in the MelvinModMax background. While these configurations in Einstein-Maxwell theory have been extensively studied, we present two new results: a novel Kerr-Schild representation for the Schwarzschild-Melvin-Bonnor spacetime and the first Kerr-Schild representation for the spacetime of two charged accelerating black holes, specifically the charged C-metric. These two findings pertain to the ModMax case, but they are equally applicable to the Einstein-Maxwell framework in a straightforward manner. In Section V, we focus on constructing a swirling background geometry. We begin by highlighting its similarities to the Taub-NUT case and introducing a new Taub-NUT-ModMax solution where the gauge field also includes a monopole contribution. This sets the stage for the presentation of a novel electromagnetic MelvinBonnor-Swirling background configuration. Finally, in Section VI, we conclude by summarizing our findings and outlining potential directions for future research on these geometries.", 'II. MODMAX THEORY AND ACCELERATING BLACK HOLES': "We consider the Einstein-ModMax action principle given by \nI = 1 16 π ∫ d 4 x √ -g ( R -4 L ) , (1) \nwhere R is the Ricci scalar and L stands for the ModMax Lagrangian density whose expression reads \nL = 1 2 ( S cosh γ -√ S 2 + P 2 sinh γ ) . (2) \nHere, S and P represent the simplest scalar and pseudo scalar electromagnetic invariants that can be constructed from the Maxwell-Faraday tensor F µν = ∂ µ A ν -∂ ν A µ and its dual ∗F µν = 1 2 /epsilon1 µνλρ F λρ , namely, \nS = 1 2 F µν F µν , P = 1 2 F µν ∗ F µν . (3) \nThe Lagrangian density depends on P 2 ; therefore parity invariance is not compromised. The parameter γ is a dimensionless coupling constant restricted to γ ≥ 0 , thus well-posedness is ensured being causality and unitarity achieved [4]. In addition, since L is a convex function of the electric field, its energy-momentum tensor respects weak, strong, and dominant energy conditions [1]. \nVariations with respect to the metric and gauge fields yield the Einstein-ModMax field equations \nG µν = 8 πT µν , d ∗ E = 0 , d F = 0 , (4) \nwhere E = E ( F , ∗F ) represents a nonlinear function of the Maxwell-Faraday tensor and its dual \nE µν = ∂ L ∂ F µν = 2( L S F µν + L P ∗ F µν ) , (5) \nand T µν being the following the energy-momentum tensor \n8 πT µν = 4 F µσ F σ ν L S +2( PL P -L ) g µν . (6) \nWe make use of the shorthand notation L S = ∂ L /∂ S and L P = ∂ L /∂ P . The function E ( F , ∗F ) is manifestly nonanalytic, as can be observed from the explicit expression \nE µν = ( cosh γ -S sinh γ √ S 2 + P 2 ) F µν -P sinh γ √ S 2 + P 2 ∗ F µν , (7) \nand hence is ill-defined for null electromagnetic configurations. Notwithstanding, the Hamiltonian formulation of ModMax does support null configurations, showing at the same time how flat spacetime naturally belongs to the spectrum solution of the theory. Due to the conformal invariance g → Ω 2 g , being Ω an arbitrary function of the spacetime coordinates, Maxwell's theory does not form part of the weak field limit of ModMax theory; however, it is recovered in the γ → 0 limit [1]. This, up to some extend, is implied by the nonexistence of null field solutions in the Lagrangian formulation of the theory. Finally, the field equations are easily proven to be invariant under the electromagnetic duality transformation \n( E ' µν ∗F ' µν ) = ( cos θ sin θ -sin θ cos θ )( E µν ∗F µν ) , (8) \nhenceforth showing that the ModMax theory is the unique NLE model sharing both conformal and electromagnetic duality invariance. \nIt is direct to recognize that purely electric or magnetic solutions of Maxwell theory, namely, those for which P = 0 , are going to be solutions of ModMax as well. However, the situation is different for P /negationslash = 0 . Already, for simple static and spherically symmetric spacetimes a deviation, although small, is retrieved. This is the case for Reissner-Nordström black holes in ModMax [28, 29]. Pertinent to the first part of this work is the C-metric spacetime constructed in [32]. Having an evident Maxwell-like gauge field profile, its main features and causal structure coincide with those of the standard geometry of a pair of charged accelerating black holes in electrovacuum. In spherical-like coordinates, and noticing the introduction of a cosmological constant Λ = -3 l 2 , the solution is given by the following spacetime line element \nds 2 = 1 Ω 2 ( -fdt 2 + dr 2 f + r 2 [ dθ 2 h + h sin 2 θ dϕ 2 K 2 ]) , (9) \nwhere \nf = (1 A 2 r 2 ) f 0 + r 2 2 , \n-/lscript h = 1 + 2 Am cos θ + A 2 w 2 cos 2 θ , Ω = 1 + Ar cos θ, w 2 = e -γ ( q 2 e + q 2 m ) , \nbeing f 0 the static metric function characterizing the (asymptotically flat) static solution \nf 0 = 1 -2 m r + w 2 r 2 , (10) \ntogether with the corresponding ModMax gauge field \nA = -e -γ q e r dt + q m cos θ dϕ K , F = d A . (11) \nThe solution is characterized by the five parameters A, m, q e , q m and K , being respectively the acceleration, the mass, the electric and magnetic charges, and the conical deficits. An appropriate choice of K can remove one of the spacetime's conical singularities; we will return to this point later. Note that the effect of the nonlinearity of ModMax consists of 'screening' the charges by the factor of e -γ . \nIn the next section, we will make use of the original method proposed by Havrdova and Krtouš [35, 36] to yield in the corresponding ModMax electromagnetic universe via the C-metric configuration (9). In order to achieve this task, it is more convenient to express the above solution (without regard to the cosmological constant for now) in prolate coordinates ( x, y ) . Indeed, performing a change of coordinates of the form \nt = ¯ t A , r = 1 Ay , cos θ = x, (12) \nthe line element reads \nds 2 = 1 A 2 ( x + y ) 2 [ -Fd ¯ t 2 + dy 2 F + dx 2 G + G dϕ 2 K 2 ] , (13) \nwhere \nF ( y ) = -(1 -y 2 )(1 -2 Amy + A 2 w 2 y 2 ) , (14) \nG ( x ) = (1 -x 2 )(1 + 2 Amx + A 2 w 2 x 2 ) , (15) \nwhile the electromagnetic tensor solution is given by \nF = -e -γ q e dy ∧ d ¯ t + q m dx ∧ dϕ K . (16) \nThe spacetime coordinates are restricted as t ∈ R , x ∈ ( -1 , 1) , y < -x and ϕ ∈ ( -π, π ) . In this convenient set of coordinates originally presented in [37], it is easy to observe that F ( z ) = -G ( -z ) , which means that these two functions will have the same number of roots. In fact, the roots of F ( y ) are explicitly given by \ny c = -1 , y A = 1 , y i = 1 Aw 2 ( m -√ m 2 -w 2 ) , y o = 1 Aw 2 ( m + √ m 2 -w 2 ) , (17) \nwhere y A represents the accelerating horizon, while y i and y o are the inner and outer black hole horizons naturally contained in a charged configuration. Notice that y c = -1 lies outside the physical causal structure of the solution. As usual in a C-metric configuration with no external background fields, the acceleration mechanism is provided by the occurrence of conical singularities along each of the semi-axis \nδ x = ± 1 = 2 π ( 1 -\| 1 ± 2 Am + A 2 w 2 \| \| K \| ) . (18) \nIn what follows, we fix the conicity parameter to be K = 1+2 Am + A 2 w 2 , in such a manner that the conical deficit along the positive semi-axis is removed. The whole conicity is translated to the negative semi-axis x = -1 .", 'III. THE MELVIN-BONNOR-MODMAX SPACETIME': "After presenting the C-metric configuration in ModMax (9), we proceed with the construction of the MelvinBonnor-ModMax spacetime via the limiting procedure exposed in [35, 36]. In essence, this limit corresponds to a self-gravitating extension of what is known for the case of a homogeneous test electric field in Minkowski spacetime. In such situation, the homogeneous electric field is achieved starting from the configuration of two charged accelerating particles by enlarging the distance between them, while properly increasing the value of their charge. In the backreacting extension, it is then direct to consider the utilization of the spacetime of two charged accelerating black holes, namely the C-metric configuration [32]. The limiting procedure described in [35, 36] consists in two main ingredients: a wisely defined coordinate transformation that allows performing a near horizon approximation around the Rindler horizon of the accelerating geometry, and a proper redefinition of the black hole inner and outer horizons, redefinition that ultimately prescribed the behavior of the physical parameters m and w such that the limit as a solution of the Einstein-ModMax field equations is guaranteed. The parameter A is kept fixed in the limiting process. \nIn mathematical terms, the inner and outer horizons are reparametrized as follows \ny i = 1 + ˜ y i /epsilon1, y o = 1 + ˜ y o /epsilon1, (19) \nbeing ˜ y i and ˜ y 0 two constants related to m , w and A and satisfying ˜ y i < ˜ y o ; this for the order of the horizons to be maintained. The parameter /epsilon1 is a small parameter to be sent to zero when performing the limit. For the limit to be carried around the Rindler horizon, the following change of coordinates has to be considered \n¯ t = 1 ˜ y i ˜ y o 1 /epsilon1 2 τ, y = 1 + ˜ y i ˜ y o /epsilon1 2 v. (20) \nThe coordinates x and ϕ remain unchanged. Notice that this implies that y i and y o scale as /epsilon1 causing the new horizon locations v i and v o to scale as /epsilon1 -1 . As a consequence, the horizons are pushed away in the limit /epsilon1 → 0 , confirming the analogy with the flat spacetime case in which the charges are pushed away from each other leaving a remnant electric field between them [35, 36]. In addition, the relevant causal structure condition y < -x becomes v < ∞ , while the three roots of the function G degenerate to x = -1 in the limit. As a result of the simultaneous consideration of (19) and (20), in the limit \n/epsilon1 → 0 , the full configuration (9) takes the form \nds 2 = 1 A 2 (1 + x ) 2 [ -2 vdτ 2 + dv 2 2 v + dx 2 (1 -x )(1 + x ) 3 + (1 -x 2 ) dϕ 2 16 A 2 ] , (21) = e -γ q e dv dτ + q m dx dϕ. \nF -∧ 4 ∧ \nNotice that (19) implies that ( m,w ) → 1 /A in the limit of vanishing /epsilon1 , a necessary condition for (21) to solve the Einstein-ModMax field equations. The spacetime configuration (21) already represents the line element and gauge field profile of the Melvin-Bonnor-ModMax configuration; however, in Rindler-like coordinates, the coordinates adapted to an accelerating observer. To properly identify the Melvin-Bonnor-ModMax universe in an intuitive form, we move to the coordinates of a static global observer as suggested in Refs. [35, 36]. We start with a suitable redefinition of the acceleration parameter \nA = e -γ/ 2 √ E 2 + B 2 4 , (22) \nand the following change of coordinates \nv = e -γ ( E 2 + B 2 ) 8 ( -t 2 + z 2 ) , τ = tanh -1 ( t z ) , x = 1 -e -γ ( E 2 + B 2 ) 4 ρ 2 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 . (23) \nAs a result, the line element of the Melvin-BonnorModMax spacetime, in cylindrical coordinates, takes the recognizable form \nds 2 = [ 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ] 2 ( -dt 2 + dρ 2 + dz 2 ) + ρ 2 [ 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ] 2 dϕ 2 , (24) \nwhile its gauge field, via the ulterior redefinition \nq e = 4 E e -γ ( E 2 + B 2 ) , q m = 4 B e -γ ( E 2 + B 2 ) , (25) \nbecomes \nF = e -γ Edz ∧ dt + Bρ [ 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ] 2 dρ ∧ dϕ. (26) \nNote that, here, the time coordinate t must not be confused with the original time coordinate used in (9). \nThis configuration describes the backreaction of a parallel bundle of electromagnetic flux in ModMax. As we can observe, the electric and magnetic fields are screened by a factor e -γ , however, the causal structure of the solution is practically unchanged with respect to the MelvinBonnor spacetime in Einstein-Maxwell. It belongs to the Kundt class of type D spacetimes and asymptotically approaches the Levi-Civita spacetime, modulo a reparametrization of the noncompact coordinates [38]. The main features of this spacetime have been extensively studied in the literature [38-40], and hence it is unnecessary to reproduce them here. Notwithstanding, it is important to stress that the solution remains nonsingular, as the electromagnetic theory under consideration is not Maxwell's theory anymore. For a generic NLE the occurrence of curvature singularities will depend on the form of the NLE Lagrangian [34], usually imposing bounds on E and B for the solution to be curvature singularity free. This is not the case of ModMax, where a straightforward exploration of the curvature scalars, the Kretschmann invariant K := R µνρσ R µνρσ , reveals the singularity free nature of the spacetime \nK = [ 3( E 2 + B 2 ) 2 ρ 4 e 2 γ -24( E 2 + B 2 ) ρ 2 e γ +80 ] ( E 2 + B 2 ) 2 4 e 2 γ ( 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ) 8 . (27) \nIn addition, the charges of the solution are proven to be finite, just as in the case of Einstein-Maxwell theory \nQ e = 1 4 π ∫ H F = 2 E E 2 + B 2 , Q m = 1 4 π ∫ H ∗F = 2 e γ B E 2 + B 2 . (28) \nThis agrees with the gravitational bounded nature of the homogeneous electromagnetic field of the MelvinBonnor-ModMax configuration, and once again contrasts what occurs in the flat spacetime case. \nIn Einstein-Maxwell the limit proposed in [35, 36] has been applied to the case in which a cosmological constant is included [41]. The procedure applies in direct analogy with respect to the asymptotically flat case starting from the factorized form of the charged (anti)-de Sitter Cmetric proposed in [42]. Here, we extend those findings to the case of Einstein-ModMax and deliver the (A)dS Melvin-Bonnor-ModMax spacetime in a suitable set of coordinates in which the Λ = 0 case is easily retrieved [43]. The configuration solution in this case reads \nds 2 = [ 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ] 2 -dt 2 + ρ 2 dρ 2 ρ 2 -4 3 Λ e -2 γ ( E 2 + B 2 ) 2 [ 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ] 4 + dz 2 + ρ 2 -4 3 Λ e -2 γ ( E 2 + B 2 ) 2 [ 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ] 4 [ 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ] 2 dϕ 2 , A = e -γ Ezdt + Bρ 2 2 [ 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ] dϕ. (29) \nAs noted in [43], the spacetime in question involves the presence of a spinning string. At a fixed z , we have \nlim ρ → 0 g ϕϕ = -4 3 e 2 γ Λ ( E 2 + B 2 ) 2 . (30) \nWhen the cosmological constant is positive, removing the string is impossible; the parameter γ can not be used to tune the constants to any particular value such that the string is removed. Consequently, the de-Sitter electromagnetic spacetime also inherits the cosmic string from the C-metric [41]. However, if Λ < 0 , we can reparametrize the coordinates as \n( t, ϕ ) = ( ˜ t + 2 √ -Λ √ 3 e -γ ( E 2 + B 2 ) ˜ ϕ, ˜ ϕ ) , (31) \nand re-glue the spacetime to eliminate the string. The resulting spacetime is then free of conical singularities. Choosing a negative cosmological constant also ensures the metric retains a Lorentzian signature. By analyzing the Kretschmann scalar, we can further confirm that the spacetime is free of curvature singularities for any value of the cosmological constant \nR µνρσ R µνρσ = 8Λ 3 + K 0 , (32) \nwhere K 0 corresponds to the value of the Kretschmann scalar given in equation (27).", 'IV. STATIC BLACK HOLES WITHIN THE MELVIN-BONNOR-MODMAX UNIVERSE': "It is well-known that the construction of magnetized black hole solutions [44, 45] of the Einstein-Maxwell equations has been made possible by exploiting the Lie point symmetries of the field equations [46, 47]. This approach leverages the symmetries inherent in the equations in order to generate new solutions from known ones. However, this method is highly dependent on the specific form of the underlying theory and cannot be easily \nadapted to accommodate even small modifications of the action. To illustrate this, even the simple addition of a cosmological constant to the action complicates the generalization of the method. This constraint is the reason why we use limiting procedures to construct the Melvin(A)dS solution [34, 41, 43]. The challenges are even more pronounced when considering nonlinear electrodynamics. In these cases, the nonlinearity of the field equations breaks the symmetries that the Ernst mechanism relies on, rendering this method completely inapplicable. Consequently, constructing (electro)magnetized black holes within NLE models becomes an highly nontrivial task. Surprisingly, due to the 'Maxwell-like' character of static solutions in ModMax, we can embed static black hole solutions within the Melvin-Bonnor-ModMax universe. Specifically, we construct Schwarzschild and accelerating black holes embedded in the ModMax electromagnetic geometry. \nThe key insights for the existence of such a configuration are twofold. First, it is observed in Einstein-Maxwell theory that the presence of a Schwarzschild black hole onto the Melvin-Bonnor spacetime modifies the form of the electromagnetic field in a very precise manner, that it does not alter the magnetic field while introducing the lapse function of the line element into the electric component. Second, the Melvin-Bonnor-ModMax spacetime, just as the C-metric solution [32] and all spherically symmetric configurations in ModMax satisfy the key condition S ∝ P , in fact, for the Melvin-Bonnor-ModMax configuration we have \nS = B 2 -e -2 γ E 2 ( 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ) 4 , P = -2 e -γ EB ( 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 ) 4 . (33) \nThis enormously reduces the field equations overlapping the spectrum of solutions in Einstein-Maxwell and Einstein-ModMax theories modulo a constant factor to be redefined. \nHence, the Schwarzschild-Melvin-Bonnor-ModMax black hole takes the simple form \nds 2 = [ 1 + e -γ ( E 2 + B 2 ) 4 r 2 sin 2 θ ] 2 ( -fdt 2 + dr 2 f + r 2 dθ 2 ) + r 2 sin 2 θ [ 1 + e -γ ( E 2 + B 2 ) 4 r 2 sin 2 θ ] 2 dϕ 2 , A = e -γ Erf cos θdt + Br 2 sin 2 θ 2 [ 1 + e -γ ( E 2 + B 2 ) 4 r 2 sin 2 θ ] dϕ, f = 1 -2 m r . (34) \nNotice that we have written the metric in spherical-like coordinates to favor comparison with the Schwarzschild solution. In this case, we explicitly obtain \nS = ( f + 2 m r cos 2 θ ) ( B 2 -e -2 γ E 2 ) ( 1 + e -γ ( E 2 + B 2 ) 4 r 2 sin 2 θ ) 4 , (35) \nP = -2 ( f + 2 m r cos 2 θ ) e -γ EB ( 1 + e -γ ( E 2 + B 2 ) 4 r 2 sin 2 θ ) 4 . (36) \nds 2 = 1 Ω 2 [ 1 + e -γ ( E 2 + B 2 ) 4 Pr 2 sin 2 θ Ω 2 ] 2 ( -Qdt 2 + dr 2 Q + r 2 dθ 2 P ) + Pr 2 sin 2 θ [ 1 + e -γ ( E 2 + B 2 ) 4 Pr 2 sin 2 θ Ω 2 ] 2 dϕ 2 , A = e -γ Erf 2 cos θ + Ar (1 + cos 2 θ ) 2Ω 2 dt + BPr 2 sin 2 θ 2Ω 2 [ 1 + e -γ ( E 2 + B 2 ) 4 Pr 2 sin 2 θ Ω 2 ] dϕ, (37) \nwhere for simplicity we have defined Q = ( 1 -A 2 r 2 ) f , P = 1 + 2 Am cos θ and Ω = 1 + Ar cos θ , with f defined in (34). The invariants explicitly read \nS = ( ( P 2 + A 2 m 2 sin 4 θ ) r -2 Pm sin 2 θ ) ( B 2 -e -2 γ E 2 ) r ( 1 + e -γ ( E 2 + B 2 ) 4 Pr 2 sin 2 θ Ω 2 ) 4 , \n(38) \nP = -2 ( ( P 2 + A 2 m 2 sin 4 θ ) r -2 Pm sin 2 θ ) e -γ EB r ( 1 + e -γ ( E 2 + B 2 ) 4 Pr 2 sin 2 θ Ω 2 ) 4 . (39) \nNotice that in all the cases we have exposed, namely, the Melvin-Bonnor-ModMax and its Schwarzschild and Cmetric generalizations the ratio S / P acquires the same value. These black hole spacetimes represent the first examples of a black hole ever embedded in an NLE model.", 'A. Novel Kerr-Schild representations': "As discussed earlier, the geometric properties and causal structures of the spacetimes considered in this work have been extensively examined in the literature [38, 43-45, 48]. For this reason, we refrain from delving into those aspects further. However, we offer two novel and, to the best of our knowledge, intriguing observations concerning unknown Kerr-Schild representations of the Schwarzschild-Melvin-Bonnor and C-metric spacetimes. While our constructions are presented explicitly for these geometries within the framework of EinsteinModMax theory, they remain equally applicable in the context of electrovacuum. \nWe begin by showing that the Schwarzschild-MelvinBonnor-ModMax black hole solution (34) can be cast in a Kerr-Schild form. As is well known, the Kerr-Schild construction allows mass to be introduced from an initially massless seed configuration. Typically, for charged solutions such as Reissner-Nordström, Kerr-Newman, or even those of dilatonic origin, the potential vector does not depend on the mass. Consequently, the Kerr-Schild transformation only acts on the metric and not on the electro- \nThe condition S ∝ P can be further explored to integrate even more intricate black hole spacetimes within the Melvin-Bonnor-ModMax background, in fact, it is possible to construct the accelerating extension of (34), the C-metric-Melvin-Bonnor-ModMax black hole configuration \nmagnetic field. However, in the solution (34), the gauge field explicitly depends on the black hole's mass, necessitating a generalization of the Kerr-Schild transformation to account for this dependence. Here, we present for the first time a Kerr-Schild construction of this black hole solution. As the construction is novel, we will present it step by step for clarity. Our starting point is the seed solution, hereafter denoted with a 0 subscript, \nds 2 0 = H ( r, θ ) 2 ( -dt 2 + dr 2 + r 2 dθ 2 ) + r 2 sin 2 θ H ( r, θ ) 2 dϕ 2 , A 0 = e -γ Er cos θdt + Br 2 sin 2 θ 2 H ( r, θ ) dϕ, (40) \nwhere for simplicity we have defined the function \nH ( r, θ ) = 1 + e -γ ( E 2 + B 2 ) 4 r 2 sin 2 θ. \nIt is simple to see that a null, geodesic, and shear-free one-form field for this metric is given by l = dt -dr . Hence, let us consider the following Kerr-Schild ansatz \nds 2 = ds 2 0 + Z ( r, θ ) ( dt -dr ) 2 , (41a) \nA = A 0 + A 1 ( r, θ ) ( dt -dr ) . (41b) \nThe novelty compared to known charged solutions in the literature is that the null and geodesic vector also serves to modify the Abelian gauge field since the new potential vector A is also shifted along l . Starting from such new ansatz, and denoting the Einstein-ModMax equations by E µν and M µ , one can note that the equation E θφ = 0 imposes \nZ ( r, θ ) = -∂ θ A 1 ( r, θ ) Er sin θ H ( r, θ ) 2 , (42) \nand that the Maxwell equation M ϕ = 0 imposes A 1 ( r, θ ) to be as a separable sum, i.e., A 1 ( r, θ ) = F 1 ( r ) + F 2 ( θ ) . Additionally, due to the equation E tϕ = 0 , one ends up with \nds 2 = ds 2 0 + 2 m H ( r, θ ) 2 ( dt dr ) 2 , \nA = A 0 +2 mE cos θ ( dt -dr ) = Er cos θ ( 1 -2 m r ) dt -2 mE cos θdr + Br 2 sin 2 θ dϕ. \nr -(43a) (43b) (43c) \n2 H ( r, θ ) \nAs usual, the off-diagonal terms of the Kerr-Schild metric (43a) can be canceled by a change of variable, and hence one yields precisely configuration (34). Additionally, it is also interesting to note that the Kerr-Schild transformation (43) has the following remarkable representation \ng µν = g (0) µν -( 2 m r ) [ g (0) αβ ξ α (0) ξ β (0) ] l µ ⊗ l ν , A µ = A (0) µ -( 2 m r ) [ A (0) α ξ α (0) ] l µ , (44) \nwhere ξ α (0) ∂ α = -∂ t is a timelike Killing vector field. \nContinuing with our new Kerr-Schild approach, one can show that the C-metric solution (9), as well as the standard C-metric of Einstein-Maxwell, has also a generalized Kerr-Schild representation, which, to our knowledge, has never been put in light before. As in the previous case, since this derivation is original, we intend to detail each step. Nevertheless, let us re-derive the solution (9) only in the purely electric case and without a cosmological constant. In fact, the cosmological constant can be easily incorporated into the seed metric, while the magnetic contribution of the gauge field can be added later by applying duality, as is typically done. \nWe start from the following seed configuration, \nds 2 0 = 1 Ω 2 ( -f 0 dt 2 + dr 2 f 0 + r 2 [ dθ 2 h 0 -h 0 sin 2 θ dϕ 2 K 2 ]) , A = -e -γ q e r dt. (45) \nwhere we have defined \nf 0 = (1 -A 2 r 2 )(1 + w 2 r 2 ) , h 0 = 1 + A 2 w 2 cos 2 θ , Ω = 1 + Ar cos θ, w 2 = e -γ q 2 e . \nThis definition of f 0 should not be confused with the one provided in equation (10). It is important to note that the seed metric is a solution with signature sgn (2 , 2) since the metric component g (0) ϕϕ < 0 . Hence, let us operate a double Kerr-Schild transformation on the sgn (2 , 2) seed metric given by \nds 2 = ds 2 0 + H 1 ( r ) Ω 2 l ⊗ l + r 2 H 2 ( θ ) Ω 2 k ⊗ k, (46) \nwhere we have defined \nH 1 ( r ) = 2 m r (1 -A 2 r 2 ) , H 2 ( θ ) = -2 mA cos θ h 2 0 , (47) \nand, where the null vectors l and k along which the KerrSchild transformations are operated are given by \nl = dt -dr f 0 , k = dθ + h 0 sin θ K dϕ. (48) \n/negationslash \nNote that l , as usual, is geodesic and shear-free. The novelty here is that, although k is not geodesic, k a ( ∇ a k b ) = 0 , the norm of the geodesic vector is unexpectedly zero, \nk a ( ∇ a k b ) k c ( ∇ c k b ) = 0 . (49) \nFinally, by means of the coordinate transformations \ndt → dt + H 1 ( r ) f 0 ( r ) ( H 1 ( r ) -f 0 ( r )) dr, dϕ → dϕ -KH 2 ( θ ) ( H 2 ( θ ) h 0 ( θ ) -1) sin θ dθ, \nthat eliminate off-diagonal terms, and after the Wick rotation ϕ → iϕ in (46), the metric (9) is recovered. \nIn summarize, we have highlighted unexpected aspects of the Schwarzschild-Melvin-Bonnor-ModMax solutions (34), as well as the C-metric (9), by showing that each of these solutions admits a novel Kerr-Schild type representation.", 'V. A VORTEX-LIKE BACKGROUND IN MODMAX': "To construct our swirling spacetime in ModMax, it is instructive, as a prelude, to revisit the case of the TaubNUTgeometries already existing in the literature [30, 31]. Contrary to the accelerating solutions of the previous sections, the Taub-NUT spacetimes, as we will also see for the swirling case, the condition S ∝ P is not respected, and thus we can expect a more significant deviation from Einstein-Maxwell configurations. However, S and P behave in such a way that the integrability of the ModMax field equations is not compromised. \nThe Taub-NUT solutions are described by the line element \nds 2 = -f ( r )( dt -2 n cos θdϕ ) 2 + dr 2 f ( r ) +( r 2 + n 2 ) d Ω 2 , (50) \nwhere, as usual, d Ω 2 = dθ 2 +sin 2 θdϕ 2 represents the line element of the 2-sphere, and n denotes the NUT parameter. Solutions have been found for the simplest choice of gauge field, specifically one compatible with the underlying U (1) Hopf fibration of the spacetime, in which the standard monopole contribution proportional to q m cos θ has been omitted. \nHere, we generalize this solution by including a monopole magnetic charge that is not determined by the stationarity of the spacetime. Thus, we consider \nA = A t ( r )( dt -2 n cos θdϕ ) + q m cos θdϕ. (51) \nThe most general Taub-NUT spacetime in ModMax then takes the form \nds 2 = -( r 2 -2 mr -n 2 + e -γ ( q 2 e + q 2 m ) r 2 + n 2 ) ( dt -2 n cos θdϕ ) 2 + dr 2 ( r 2 -2 mr -n 2 + e -γ ( q 2 e + q 2 m ) r 2 + n 2 ) +( r 2 + n 2 ) d Ω 2 , A t = -q e 2 n sin ( e -γ [ π -2 arctan ( r n )]) -q m 2 n cos ( e -γ [ π -2 arctan ( r n )]) + q m 2 n . (52) \nFollowing the same procedures outlined in [30, 31], it is straightforward to verify the limits of (52) to its standard Einstein-Maxwell form [43] and to the dyonic ReissnerNordström solution in ModMax [28, 29]. The existence of these solutions hinges on how the ModMax interaction specifically modifies the Maxwell equations. Having the following electromagnetic invariants \nS = -( A ' t ) 2 + ( 2 nA t -q m r 2 + n 2 ) 2 , P = 2 A ' t (2 nA t -q m ) r 2 + n 2 (53) \nthe ModMax equation modifies the standard Maxwell equation by introducing different constant pre-factors in front of each term of the Maxwell differential equation to be solved. However, the fundamental structure of the equation remains unchanged \ne γ A '' t + e γ 2 rA ' t r 2 + n 2 + e -γ 2 n (2 nA t -q m ) ( r 2 + n 2 ) 2 = 0 , (54) \nup to the exponentials of the γ parameter. The overall integration of the ModMax equation is therefore not compromised, but the form of the gauge potential is slightly modified, yielding solution (52). Due to the presence of e -γ inside the trigonometric sine and cosine functions, this solution does not reduce to the usual form \nin Einstein-Maxwell theory. The key point behind this simplification is the form of the gauge potential, particularly the term proportional to ωA t , the induced magnetic term due to the stationarity of the spacetime. \nTo broaden the spectrum of solutions in EinsteinModMax theory, it is wise to adopt a similar approach and seek an electromagnetic configuration where an induced field of the previous form is achieved. We can draw on insights from the Ernst approach to Einstein-Maxwell theory [46, 47]. It is known that the Reissner-NordströmNUT spacetime can be obtained, in addition to direct integration, through the successive application of an electric Harrison transformation [26], which transforms a Schwarzschild spacetime into a Reissner-Nordström configuration, followed by an electric Ehlers transformation that ultimately introduces the NUT parameter [27, 49]. Due to the structure of these electric transformations, the so-called twisted Ernst equations-integrability conditions of the Einstein-Maxwell system expressed in terms of complex Ernst potentials-naturally induce a magnetic gauge field component of the desired form, specifically proportional to ωA t . The most promising approach to introduce a new stationary spacetime in EinsteinModMax with the desired electromagnetic configuration \nis to consider a Melvin-Bonnor spacetime endowed with a vortex-like rotation [43]. A spacetime closely related to the Taub-NUT spacetime is the swirling or vortex-like spacetime [24, 25]. Unlike the standard Kerr-like rotation, where the spacetime itself does not rotate unless a rotating source induces nontrivial dragging, the swirling geometry features a background that rotates and drags bodies along with it, rather than being dragged by a source. Given that mixing electric and magnetic Ehlers or Harrison transformations disrupts the commutativity of the process, and after analyzing these backgrounds in [43], an educated guess guides us to consider a MelvinBonnor-Swirling spacetime, which in the Ernst scheme is constructed by composing a Harrison transformation with an Ehlers transformation of the magnetic type, thereby providing a gauge field with an induced electric field of the desired form via the corresponding twisted equations, ∼ ωA ϕ . With this approach, the integration of the Einstein-ModMax equations remains intact, just as with the Taub-NUT case. The relevant Maxwell equation, specifically the magnetic component (noting that the swirling spacetime can be viewed as a variant of the \nTaub-NUT geometry achieved through a magnetic Ehlers transformation), differs from the Einstein-Maxwell case only in the pre-factors in front of each term of the differential equation. In this case, the equation reads \ne -γ A '' ϕ + ( 2 V V ' -V 2 -3 j 2 ρ 4 ρ ) e -γ A ' ϕ ( V 2 + j 2 ρ 4 ) + 4 j ( E +4 je γ A ϕ ) ρ 2 ( V 2 + j 2 ρ 4 ) 2 = 0 . (55) \nSee below for the ansatz of the line element and gauge field, Eq. (57). Notice that, as well as in the TaubNUT case, the electromagnetic invariants S and P are not proportional to each other \nS = A ' 2 ϕ ρ 2 -(4 jA ϕ + e -γ E ) 2 ( V 2 + j 2 ρ 4 ) 2 , P = -2 A ' ϕ (4 jA ϕ + e -γ E ) ( V 2 + j 2 ρ 4 ) ρ . (56) \nThe Melvin-Bonnor-Swirling-ModMax spacetime configuration, here presented for the first time, reads \nds 2 = ρ 2 V 2 + j 2 ρ 4 ( dϕ +4 jzdt ) 2 + ( V 2 + j 2 ρ 4 ) ( -dt 2 + dρ 2 + dz 2 ) , A = ( e -γ Ez +4 jzA ϕ ) dt + A ϕ dϕ, (57) \nV = 1 + e -γ ( E 2 + B 2 ) 4 ρ 2 , A ϕ = e -γ B 4 j sin ( e γ [ π -2 arctan ( V jρ 2 )]) + e -γ E 4 j cos ( e γ [ π -2 arctan ( V jρ 2 )]) -e -γ E 4 j . (58) \nRecall that, in this case, the integrated function is the magnetic function A ϕ , rather than A t . This magnetic function induces an electric field of the desired form ∼ ωA ϕ . Additionally, a background electric field of the form e -γ Ez serves the same role as the q m cos θ term in the Taub-NUT analog (52). \nJust as in the Taub-NUT case, the line element is only slightly modified by the inclusion of the screening factor e -γ , which means a detailed analysis of the causal structure is not necessary, as it has been addressed in [43]. However, the electromagnetic configuration does undergo significant modification. Nevertheless, the corresponding Einstein-Maxwell and Melvin-Bonnor-ModMax limits are properly recovered for γ → 0 and j → 0 . In fact, the Einstein-Maxwell Melvin-Swirling spacetime of [43] is recovered, as seen from the expansion \nA = A (0) + γ A (1) + O ( γ 2 ) , (59) \nwith \nA (0) = [ E ( ¯ V 2 -j 2 ρ 4 ) z ¯ V 2 + j 2 ρ 4 + 2 jB ¯ V ρ 2 z ¯ V 2 + j 2 ρ 4 ] dt + [ B ¯ V ρ 2 2( ¯ V 2 + j 2 ρ 4 ) -jEρ 4 2( ¯ V 2 + j 2 ρ 4 ) ] dϕ, (60) \nA (1) = -A (0) + ( B ( ¯ V 2 -j 2 ρ 4 ) ¯ V 2 + j 2 ρ 4 -2 jE ¯ V ρ 2 ¯ V 2 + j 2 ρ 4 ) × ( 8 j ( E 2 + B 2 ) ρ 4 ¯ V 2 + j 2 ρ 4 + π -2 arctan ( ¯ V jρ 2 )) × ( zdt + dϕ 4 j ) , (61) \nand where ¯ V corresponds to the polynomial V evaluated at γ = 0 . Furthermore, the Melvin-Bonnor-ModMax configuration constructed in Section II is easily retrieved \nwhere \nin the j = 0 expansion \nA ∼ e -γ Ezdt + Bρ 2 2 V dϕ + O ( j ) , (62) \nthe leading order corresponding to the desired gauge field producing the expression (26). Indeed, the swirling geometry of pure vacuum is obtained via ( E,B ) → 0 .", 'VI. CONCLUSIONS AND FURTHER PROSPECTS': "In this paper, we have extended the study of black hole solutions within the framework of Einstein-ModMax theory. Building on the existing catalog of exact solutions and considering the prevalent limitations in the characterization of rotating solutions, we have constructed Melvin-Bonnor and vortex-like geometries. Specifically, we have derived a Melvin-Bonnor-ModMax background, along with its (A)dS extension, and used these insights to construct, for the first time, both Schwarzschild and Cmetric Melvin-Bonnor black holes immersed in the ModMax theory. Although the properties of these spacetimes closely resemble their Einstein-Maxwell counterparts, and therefore a detailed re-examination of their causal structures is not required, we present two novel contributions. We identify a previously unnoticed aspect of the Schwarzschild-Melvin-Bonnor-ModMax solution (34) and the C-metric solution (9). In particular, we show that both solutions can be reformulated through two distinct novel Kerr-Schild-type representations. These representations not only offer a fresh perspective on these familiar spacetimes but also unveil structural features that were previously hidden in their traditional forms. By introducing these innovative Kerr-Schild frameworks, we provide new insights that may have significant implications for the study of charged black hole solutions. It is important to note that the significance of these results is partly due to the fact that these constructions remain equally valid in the standard electrovacuum case by simply taking the limit γ → 0 . \nIn the context of stationary solutions, we also present, for the first time, a vortex-like solution, referred to as the Melvin-Bonnor-Swirling background in ModMax. The necessary intuition for constructing this solution was \n- [1] D. P. Sorokin, Fortsch. Phys. 70 , 2200092 (2022), 2112.12118.\n- [2] M. Born and L. Infeld, Proc. Roy. Soc. Lond. A 144 , 425 (1934).\n- [3] W. Heisenberg and H. Euler, Z. Phys. 98 , 714 (1936), physics/0605038.\n- [4] I. Bandos, K. Lechner, D. Sorokin, and P. K. Townsend, Phys. Rev. D 102 , 121703 (2020), 2007.09092.\n- [5] B. P. Kosyakov, Phys. Lett. B 810 , 135840 (2020), \ndrawn from the previously known Taub-NUT solutions in ModMax [30, 31], which we have also slightly generalized by enhancing their gauge field's magnetic component. \nThere are several promising directions for further exploration of the Einstein-ModMax theory, building upon the work we have presented here. One interesting avenue is embedding static and accelerating black holes within the Melvin-Bonnor-Swirling-ModMax background. Based on our understanding, the challenges we have faced are primarily due to computational limitations rather than conceptual ones, leading us to believe that these black hole solutions belong to the spectrum of exact solutions in Einstein-ModMax theory. Additionally, following the approach in [50], the solutions presented here can be generalized to include a scalar field with conformal coupling. Moreover, these solutions can be further extended by introducing a (super-)renormalizable potential that breaks the conformal symmetry of the scalar field action, as discussed in [51]. \nAnother valuable direction would be the full generalization of the Ernst scheme [46, 47] for ModMax theory. Although achieving a comprehensive formulation of Einstein-ModMax theory using complex Ernst potentials will likely require significant effort, investigating potential Lie point symmetries, such as those identified in more complex systems [52, 53], may yield valuable insights. \nLastly, a major and highly nontrivial challenge lies in constructing rotating solutions within ModMax theory. We aim to report progress on this in future work.", 'Acknowledgments': "J.B. and A.C. appreciate interesting discussions with Cristóbal Corral and Prof. Pavel Krtouš. 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2024ApJ...974...70P	We have used the Dark Energy Camera DECam on the CTIO Blanco 4 m telescope to perform a new emissionline survey of the Large Magellanic Cloud LMC using narrowband H and S II filters in addition to a continuum band to create pure emissionline images. We refer to this new survey as DeMCELS to distinguish it from the earlier Magellanic Cloud Emissionline Survey MCELS. DeMCELS covers 54 degSUP2SUP encompassing most of the bright optical disk of the LMC. With DECams pixel size of only 0.27 our DeMCELS survey provides a seeinglimited improvement of 35 times over MCELS and is comparable in depth with surface brightness limits of inlineformula mmlmath overflowscrollmmlmn3.3mmlmnmmlmommlmommlmspace width0.25emmmlmspacemmlmsupmmlmn10mmlmnmmlmrowmmlmommlmommlmn17mmlmnmmlmrowmmlmsupmmlmiergmmlmimmlmspace width0.33emmmlmspacemmlmsupmmlmicmmmlmimmlmrowmmlmommlmommlmn2mmlmnmmlmrowmmlmsupmmlmspace width0.33emmmlmspacemmlmsupmmlmi mathvariantnormalsmmlmimmlmrowmmlmommlmommlmn1mmlmnmmlmrowmmlmsupmmlmspace width0.50emmmlmspacemmlmsupmmlmiarcsecmmlmimmlmrowmmlmommlmommlmn2mmlmnmmlmrowmmlmsupmmlmath inlineformula and inlineformula mmlmath overflowscrollmmlmn2.9mmlmnmmlmommlmommlmspace width0.25emmmlmspacemmlmsupmmlmn10mmlmnmmlmrowmmlmommlmommlmn17mmlmnmmlmrowmmlmsupmmlmiergmmlmimmlmspace width0.33emmmlmspacemmlmsupmmlmicmmmlmimmlmrowmmlmommlmommlmn2mmlmnmmlmrowmmlmsupmmlmspace width0.33emmmlmspacemmlmsupmmlmi mathvariantnormalsmmlmimmlmrowmmlmommlmommlmn1mmlmnmmlmrowmmlmsupmmlmspace width0.50emmmlmspacemmlmsupmmlmiarcsecmmlmimmlmrowmmlmommlmommlmn2mmlmnmmlmrowmmlmsupmmlmath inlineformula in H and S II respectively. DeMCELS provides detailed morphological information on nebulae of all scales from the largest supershells to individual H II regions and supernova remnants to bubbles of emission surrounding individual stars and even to faint structures in the diffuse ionized gas of the LMC. Many complex regions of emission show significant variations in the ratio of S II to Ha sign of a mixture of shocks from stellar winds andor supernovae with photoionization by embedded hot young stars. We present the details of the observing strategy and data processing for this survey and show selected results in comparison with previous data. A companion project for the Small Magellanic Cloud is in progress and will be reported separately. We are making these new data available to the community at large via NOIRLabs Data Lab site.	2024-10-01T00:00:00Z	['10.48550/arXiv.2409.04846', 'arXiv:2409.04846', '2024ApJ...974...70P', '10.3847/1538-4357/ad6766', '2024arXiv240904846P']	['Magellanic Clouds', 'Large Magellanic Cloud', 'Interstellar medium', 'Supernova remnants', 'Nebulae', '990', '903', '847', '1667', '1095', 'Astrophysics - Astrophysics of Galaxies']	The Dark Energy Camera Magellanic Clouds Emissionline Survey	2,024	193	0.53	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	1	https://arxiv.org/pdf/2409.04846.pdf	{'No Header': 'Draft version September 10, 2024 \nTypeset using L A T E X preprint style in AASTeX631', 'The Dark Energy Camera Magellanic Clouds Emission-Line Survey': "Sean D. Points, 1 Knox S. Long, 2, 3 William P. Blair, 4 Rosa Williams, 5 You-Hua Chu ( 朱有花 ) , 6, 7 P. Frank Winkler, 8 Richard L. White, 2 Armin Rest, 2, 4 Chuan-Jui Li ( 李傳睿 ) , 7 and Francisco Valdes 9 \n1 NSF's NOIRLab/CTIO \nCasilla 603 \nLa Serena, Chile \n2 Space Telescope Science Institute, 3700 San Martin Dr, Baltimore MD 21218, USA \n3 Eureka Scientific, Inc. 2452 Delmer Street, Suite 100, Oakland, CA 94602-3017, USA \n4 The William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD, 21218, USA \n- 5 Department of Earth and Space Sciences, Columbus State University, 4225 University Avenue, Columbus, Georgia 31907, USA\n- 6 Department of Physics, National Sun Yet-Sen University, No. 70, Lienhai Rd., Kaohsiung 80424, Taiwan 7 Institute of Astronomy and Astrophysics, Academia Sinica, No.1, Sec. 4, Roosevelt Rd., Taipei 106216, Taiwan 8 Department of Physics, Middlebury College, Middlebury, VT, 05753, USA \n9 NSF's NOIRLab, 950 Cherry Ave, Tucson, AZ 85719", 'ABSTRACT': "We have used the Dark Energy Camera (DECam) on the CTIO Blanco 4-m telescope to perform a new emission-line survey of the Large Magellanic Cloud (LMC) using narrow-band H α and [S II ] filters in addition to a continuum band for use in creating pure emission-line images. We refer to this new survey as DeMCELS, to distinguish it from the earlier Magellanic Cloud Emission Line Survey (MCELS) that has been in service for nearly 30 years. DeMCELS covers ∼ 54 degrees 2 , encompassing most of the bright optical disk of the LMC. With DECam's pixel size of only 0 . '' 27, our DeMCELS survey provides a seeing-limited improvement of 3 - 5 × over MCELS and is comparable in depth, with surface brightness limits of 3 . 3 × 10 -17 ergs cm -2 s -1 arcsec -2 and 2 . 9 × 10 -17 ergs cm -2 s -1 arcsec -2 in H α and [S II ], respectively. DeMCELS provides detailed morphological information on nebulae of all scales, from the largest supershells to individual H II regions and supernova remnants, to bubbles of emission surrounding individual stars, and even to faint structures in the diffuse ionized gas of the LMC. Many complex regions of emission show significant variations in the ratio of [S II ] to H α , a sign of the mixture of shocks from stellar winds and/or supernovae with photoionization by embedded hot, young stars. We present the details of the observing strategy and data processing for this survey, and show selected results in comparison with previous data. A companion project for the Small Magellanic Cloud is in progress and will be reported \nCorresponding author: Sean D. Points \[email protected] \nseparately. We are making these new data available to the community at large via the NOIRLab's Data Lab site. \nKeywords: Galaxies: LMC, Supernova Remnants", '1. INTRODUCTION': "The Large Magellanic Cloud (LMC) is the largest and most massive satellite galaxy of the Milky Way. At a distance of 49 . 59 ± 0 . 09 kpc (Pietrzy'nski et al. 2019), and seen along a line of sight with low Galactic foreground absorption, the LMC is the best laboratory for studying a wide variety of astrophysical phenomena, including the life cycle of stars (star formation and stellar death in all forms), the interstellar medium (ISM), and the interplay between them. With a mass of 1 . 8 × 10 11 M ⊙ (Shipp et al. 2021), the LMC has a star formation rate of 0 . 2 M ⊙ yr -1 (Harris & Zaritsky 2009), dominated by the spectacular 30 Dor complex, and has a metallicity of [Fe / H] = -0 . 42 dex (Choudhury et al. 2021), similar to that of M33. The LMC contains complexes of emission-line gas, excited by both photoionization from young, hot stars, and indirectly by shocks from stellar winds and the supernovae that have exploded over time. Even away from regions of active star formation, little-studied faint diffuse gas fills the interstellar regions of the galaxy, ionized by starlight leaking out of the star forming regions. \nAt optical wavelengths, the ISM is observed primarily in the Balmer lines of hydrogen, and various forbidden lines of intermediate mass elements like O, N, and S. The principal method of distinguishing shocked from photoionized gas has traditionally been the ratio of [S II ] λλ 6716,6731 to H α (Mathewson et al. 1983, and references therein). Images of portions of the LMC have been accumulated by many observers over the decades, but in the CCD era, the most widely used global emission-line survey of both the Large and Small Magellanic Clouds has been the Magellanic Clouds Emission Line Survey (MCELS, Smith & MCELS Team 1999). From 1997-2001, MCELS produced data covering the central 8 · × 8 · of the LMC using the University of Michigan/CTIO Curtis Schmidt telescope with the Newtonian focus CCD and five filters: three centered on emission lines ([O III ], H α , and [S II ]) and two continuum bands (green and red) for stars and continuum subtraction. The individual MCELS observations covered a 1 . · 4 × 1 . · 4 field of view with a pixel scale of 2 . '' 3 pixel -1 and an effective spatial resolution of ∼ 5 '' . \nThe MCELS data have been an extremely powerful tool for investigating many aspects of the interstellar component of the LMC for more than 20 years. Because the ratio of [S II ] to H α is, to first order, a diagnostic of shocks vs. photoionization 1 , MCELS has provided critical data in studies of supernova remnants (SNRs, Kavanagh et al. 2015a,b, 2022; Yew et al. 2021; de Horta et al. 2012; Crawford et al. 2010), superbubbles, and supergiant shells (Dunne et al. 2001; Warth et al. 2014; Kavanagh 2020; Collischon et al. 2021; Sasaki et al. 2022). MCELS data have also been used in a variety of investigations such as: (1) determining the optical depth of H II regions (Pellegrini et al. 2012); (2) searching for planetary nebulae (Reid & Parker 2013); (3) investigating the physical conditions of Wolf-Rayet nebulae (Hung et al. 2021); (4) helping to separate the thermal and nonthermal radio emission in the LMC (Hassani et al. 2022); and many other multiwavelength campaigns \nin the LMC (Kim et al. 2003; Blair et al. 2006; Maggi et al. 2016; Bozzetto et al. 2017, to mention a few). \nAlthough MCELS was revolutionary for its time, advances in instrument technology and computing power are now allowing larger aperture telescopes the ability to image wide fields of view at significantly higher angular resolution. In this paper, we have undertaken the next generation CCD emission-line survey of the LMC using the Dark Energy Camera (DECam; Honscheid & DePoy 2008; Flaugher et al. 2015) on the CTIO Blanco 4-m telescope, starting with imagery at H α and [S II ], and a red continuum band. Our DECam narrow-band imaging survey provides a significant improvement in spatial resolution ( ≥ 3 × , depending on seeing) and reaches comparable depth to the MCELS data, thus greatly improving investigations of the detailed morphology of both bright and faint, diffuse ISM structures in the MCs on a global scale. A companion survey covering the Small Magellanic Cloud with five DECam fields is in progress and will be reported separately. \nThe H α and [S II ] observations described below were not obtained at the same time. In this paper, we combine the DECam N662 (H α ) images of the LMC from a survey led by Puzia (PropID: 2018A0909; 2018B-0908) and a separate N673 ([S II ]) and DES r ' surveyled by Points (PropID: 2021B-0060) into a single unified analysis. We describe the observations in § 2 and the data reduction and quality assessments in § 3. Additional data processing beyond the initial pipeline reductions is discussed in § 4. In § 5, we provide some examples and comparisons for various types of objects between our survey and MCELS to provide a sense of the power of this new higher resolution emission-line resource. We are making the processed data available to the community at large via the NOIRLab DataLab site, as described in § 6.", '2. OBSERVATIONS': "Our DECam Magellanic Cloud Emission-Line Survey (hereinafter DeMCELS) consists of 20 overlapping and dithered DECam pointings in several filters covering the entire bright optical disk of the LMC, approximately 54 deg 2 . Each DECam image covers a roughly circular region of 1 . · 8 diameter with an array of 60 2048 × 4096 CCDs and a pixel size of 0 . '' 27. Hence, depending on seeing conditions, a spatial resolution improvement of a factor of 3 to 5 or more can be expected over MCELS. \nOur observations used the following filters: the narrow-band N662 (H α + [N II ] λλ 6548 , 6583) and N673 ([S II ] λλ 6716 , 6731) to image optical emission lines, and the DES r ' filter which was used for continuum subtraction. The properties of these filters are listed in Table 1 and shown in Figure 1. We note that even though the N662 filter contains emission from both H α and [N II ] λλ 6548 , 6583, we will refer to it as the H α filter throughout this work. This is because the [N II ] emission lines are generally weak due to the lower N abundance in the LMC and because the spatial distribution of both H α and [N II ] is expected to be quite similar. 2 The N673 filter will be referred to as the [S II ] filter. \nThe total desired exposure time per field in H α was 4980 s, chosen to reach a surface brightness sensitivity limit of ∼ 3 × 10 -18 erg s -1 cm -2 arcsec -2 . This was normally divided into six short exposures of 30 s each and six long exposures of 800 s for each field. The telescope was dithered between each of the individual exposures for each field, allowing the gaps between the DECam \nTable 1. DECam Filter Properties \nFigure 1. DECam filter curves used in this investigation with DES r ' (black), N662 (red), and N673 (blue). \n<!-- image --> \nindividual detectors to be filled in. The total exposure time per field for the [S II ] observations was double the exposure time of the H α observations, in order to reach a similar surface brightness limit because of the fainter S emission lines. These observations were normally divided into 12 short exposures of 60 s each and 12 long exposures of 800 s, although some fields with bright emission and/or heavy crowding were instead observed with 24 long exposures of 400 s. The exposure times of the r ' observations were selected to reach the expected photometric depth of point sources in the narrow-band observations in order to facilitate continuum subtraction. This corresponds to short and long exposure times of 8 s and 60 s, respectively. \nA summary of all of the DECam observations for each field per filter is presented in Table 2. Inspection of this Table shows that, given observing conditions and the time available, we were not able to observe all of the fields to the desired depth, with some fields lacking some or all observations in either H α or [S II ]. We present the MCELS H α image of the LMC with the DECam fields overlaid in Figure 2. The color coding in this Figure shows where data shortcomings are present. \nFigure 2. MCELS H α image of the LMC with 2 · diameter circles representing the approximate DECam footprint overlaid for all narrow-band data of the LMC found in the NOIRLab Astro Data Archive. As discussed in more detail in § 3, colors of the circles represent the completeness of the long observations compared to a desired uniform depth: ≥ 2 / 3 complete in H α and [S II] (green); ≥ 2 / 3 complete in H α and ≤ 2 / 3 complete in [S II] (blue); ≤ 2 / 3 complete in H α and ≥ 2 / 3 complete in [S II] (cyan); and ≤ 2 / 3 complete in both H α and [S II] (red). \n<!-- image --> \nTable 2. DECam Observations \nTable 3. DECam Data Quality", '3. DATA REDUCTION AND QUALITY ASSESSMENT': 'We begin with the object images, processed with the DECam Community Pipeline (hereafter the DCP, Valdes et al. 2014) for bias subtraction and flat-fielding (i.e., ooi images). The DCP produces flux-calibrated images with accurate astrometric solutions based on Gaia data. One portion of the Pipeline process is to remove sky background and a pupil ghost from the data. The standard process is to fit the sky background with a high-order polynomial across several of the individual CCDs of a DECam exposure, but with our narrow-band images this had the effect of removing a portion of the diffuse emission as well, leaving negative backgrounds in some subtracted data. As a result, for our analysis we requested special processing of the data with options set to fit the sky background with a 2nd-order polynomial across all detectors in addition to removing the pupil ghost.', '3.1. Data Quality Assessment': 'To assess the quality of the data listed in Table 2, we examined the distribution of the image quality (PSF) and the photometric depth of the observations, as given by the FITS header keywords SEEING and MAGZERO, respectively. These values are determined by the DCP as part of the standard processing. Because the narrow-band DECam surveys of the LMC were performed to surpass the existing MCELS data in terms of angular resolution, the DCP values for SEEING and MAGZERO are the fundamental measures needed to determine if that goal is met. We present histograms of the seeing and photometric depth measurements of the observations listed in Table 2 in Figures 3 & 4, and summarize the results in Table 3.', '3.1.1. Seeing': "As seen in the top-left panel of Fig. 3, the seeing histogram for all exposures and all filters has the most counts in the bin centered on 1 . '' 15, but amplitude is similar to the bins centered on 0 . '' 95 and 1 . '' 05. The overall seeing distribution has a median value of 1 . '' 13. Furthermore, the distribution is not Gaussian in shape; it has a sharp lower cutoff at 0 . '' 8 and a tail that extends to 2 . '' 0. The seeing distributions for the short and long exposures in all filters (top-center and top-right panels, \nrespectively) follow the same general trend. We note that the histogram of the short exposures peaks in the bin centered on 1 . '' 05 and that the long exposures have relatively more instances of measured seeing > 1 . '' 7 than the short exposures. \nIn order to investigate the delivered image quality as revealed by seeing histograms in more detail, we also examine the seeing on a per filter basis as shown in Fig. 3. In general, the seeing distribution of the [S II ] and r ' observations follows the distribution seen for all filters. This is not surprising because, as shown in Table 3, the vast majority of individual exposures ( ∼ 85%) were taken with the [S II ] and r ' filters contemporaneously in the 2021B semester. The H α observations, taken during the 2018A (LMC 30Dor field) and 2018B (all other LMC fields), however, have a distinct bimodal seeing distribution (Fig. 3, second row). Given that this bi-modality is apparent in both the short and the long H α observations, it seems that the seeing was generally poor and unstable during the nights when the H α data were obtained. The short H α observations have a median seeing 0 . '' 25 greater than the median seeing for all short exposures and the long H α exposures have a median seeing 0 . '' 13 greater than the median for all long exposures (see Table 3).", '3.1.2. Photometric Depth': "Similar to our analysis of the delivered image quality of the DECam data using the seeing histograms, we also investigate the photometric depth of our observations. As seen in the left-hand side panels in Fig. 4, the distribution of the photometric depth for all exposure times is bimodal and independent of the filter used. This is expected as the short exposures are not as deep as the long exposures. When we separate the data into short and long exposures and plot histograms of the photometric depth, we find that the short and long observations have a median depth of 27.10 mag and 29.70 mag, respectively. For the short exposures, we go deepest with the r ' observations. For the long exposures, we are deepest with the H α images, as measured by the DCP. \nWe also see from Fig. 4 that there is more spread in the photometric depth values for the short exposures in general and for the H α data in particular. Again, this is indicative of the more unstable observing conditions for the observing run when the H α data were obtained.", '3.2. Final Data Set': "As discussed below in § 4, continuum-subtraction is necessary to detect the faintest emission from the ISM, but can be difficult in practice. In order to minimize the effects of poor seeing and poor weather (i.e., clouds) on the data presented here, we use the plots shown above in Figs 3 & 4 to flag low-quality data and remove it from our reduction pipeline. In general, observations with a SEEING value > 1 . '' 45 and observations with MAGZERO values inconsistent with the median values determined for the short and long exposures in each filter were not processed further 3 . \nAfter removing the data with a seeing value > 1 . '' 45, we re-examined the distribution of the seeing and photometric depth and present those revised results in Fig. 5 and Fig. 6 and summarize them in Table 4. A comparison between Table 3 and Table 4 shows that, in general, removal of data based on the seeing value did not drastically improve the median seeing and median photometric depth of the [S II ] and r ' observations. We do, however, see a significant improvement of the measured median seeing in the H α images. Before removal of poor seeing data, the median seeing of the H α observations \nFigure 3. Histogram of the seeing value for the data as measured by the DCP. The top row plots the measured seeing for all filters and all exposure times ( left ), all filters for short exposure times ( center ), and all filters for long exposure times ( right ). The second, third, and fourth rows plot the seeing for all exposure times ( left ), short exposure times ( center ), and long exposure times ( right ), for the H α , [S II], and r ' filters, respectively. The histograms have been divided into 0 . '' 1 bins, with the first bin centered at 0 . '' 75 and the last bin centered on 1 . '' 95. Note that the y-axis for these histograms is not constant. \n<!-- image --> \nFigure 4. Histogram of the photometric depth measured by the DCP. The top row plots the depth for all filters and all exposure times ( left ), all filters for short exposure times ( center ), and all filters for long exposure times ( right ). The second, third, and fourth rows plot the photometric depth for all exposure times ( left ), short exposure times ( center ), and long exposure times ( right ), for the H α , [S II], and r ' filters, respectively. The histograms have been divided into 0.5 mag bins with the first bin centered on 23.25 mag and the last bin centered on 30.75 mag. Note that the y-limit for these histograms is not constant. \n<!-- image --> \nTable 4. Revised Data Quality \nwas 1 . '' 37 and after removal the median seeing value is 1 . '' 04. Likewise, the median photometric depth of all of the H α images, after removal of low-quality data, improves by 0.45 mag. \nAs seen in Figure 6, the histograms of the photometric depth of the long H α and short r ' observations have counts in bins centered at 29.25 mag and 28.75 mag, respectively. The data in these bins come from the LMC 30Dor field (see Table 2) where the long H α and short r ' observations had an individual exposure times of 300 s and 20 s. Therefore, in comparison to our more typical long H α and short r ' of 800 s and 8 s, one expects these fields to have a lower than average depth in the long H α observations and greater than average depth in the short r ' observations. \nFinally, for all observations, we did not use any data from the 'S7' detector in DECam. As mentioned on the DECam web page (https://noirlab.edu/science/programs/ctio/instruments/DarkEnergy-Camera/Status-DECam-CCDs), amplifier B on this detector has an unstable gain. As a result, the background across this detector has a discontinuity that affects image combination when matching the sky background across multiple detectors for image combination. Because we are performing sky- and continuum-subtraction to investigate faint and large-scale nebular emission, we removed this detector from all data processed by our custom pipeline. \nIn Table 5, we present the data for the long exposures we process with our custom software, described in § 4. As seen in the table, we did not end up achieving complete and uniform coverage of the LMC with the current data.", '4. DATA PROCESSING': "The survey data were obtained over a number of nights and under varying conditions with an instrument containing 60 CCDs with significant gaps between individual detectors. Our goal in processing these data was to produce mosaicked versions of these images and to accurately represent diffuse H α and [S II ] emission (on large and small scales) at brightness levels that are significantly lower than that observed from the night sky. This presents significant challenges since on the one hand (a) all of the images contain significant contribution from stars, which range from isolated \nFigure 5. Same as Fig. 3 except data with a seeing value > 1 . '' 45 have been removed. \n<!-- image --> \nobjects to clusters, and (especially in the bar of the LMC) to the diffuse light which is brighter than the emission line gas, and (b) the band pass of the r ' -band filter is sufficiently broad that these images contain contributions from both H α and [S II ], albeit at low levels due to the shorter exposure times (see Fig 1). Our goals and the associated challenges are very different from those who wish to measure transient phenomena in sets of images, or to study starlight. \nFigure 6. Same as Fig 4 except data with seeing value > 1 . '' 45 have been removed. \n<!-- image --> \nTo address these goals, and after significant experimentation with alternatives, we have developed a processing pipeline that is designed to be optimized for accurately assessing the diffuse emission that pervades the LMC. The pipeline is Python-based, but makes extensive use of the SWARP software package (Bertin et al. 2002) to re-project and mosaic images. The principal steps of this procedure are outlined below: \nTable 5. Revised Observation Summary \na Completeness of the observations for the final dataset as shown in Figure 2. 'Y' indicates that at least 2/3 of the observations for a field were completed and passed the quality assessment. 'N' indicates that fewer than 2/3 of the observations for a field were completed and/or failed the quality assessment. The incomplete observations for the individual filters are labeled as 'H' (H α ), 'S' ([S II]), and 'r' (r ' ). \n- · We begin with the data (ooi images) reduced by the DCP that has passed the data quality checks discussed in sections § 3.1 and § 3.2.\n- · We re-scale all of the data to a common (stellar) magnitude scale where 1 count (DN) corresponds to that expected for a 27th magnitude star. The re-scaling is based on the flux conversion to magnitudes provided by the DCP which for the declinations appropriate to the Magellanic Clouds is based on the Gaia (3rd early release) G band catalog. To zeroth-order, this conversion means that one can simply subtract any two images taken at the same position from one another to produce a continuum-free difference image. We also remove a single background from all the CCD images that comprise each exposure, providing a zeroth-order subtraction of the sky background.\n- · We create a grid of 4 × 4 overlapping tiles for each field, each 0 . · 67 across. This is basically a convenience that allows for producing a set of uniform data products. The remaining analysis is carried on the individual CCD images that are part of each tile.\n- · For each tile and for each set of exposures with filter and exposure time, we measure the difference in flux levels in the overlap regions, and use this to adjust the background levels \non each CCD. Specifically, we use SWARP to project the individual CCDs onto a common astrometric frame. We then estimate the difference in (sky) background in each of the two overlapping CCD images (from the mode of the difference between the two images in the overlap region). Because the typical number of overlaps greatly exceeds the number of images to be combined, we fit the overlaps by assigning weights to each measurement and use a leastsquares procedure to minimize the overlap differences assuming a single background be added or subtracted from each exposure, namely \nΞ = Σ ij w ij (∆ ij -( b i -b j )) 2 (1) \nwhere w ij is a weight based on the size of the overlap, ∆ ij is the calculated difference in the fluxes, and b i is the extra background to be added or subtracted. \n- · We then use SWARP to create tile images on the same world coordinate system for each exposure and filter. All of the images at this point in the processing contain a combination of continuum (from stars) and line emission (from gas). The stars are of roughly equal brightness in all of the images, but the relative amount of line emission depends upon the filter bandpass.\n- · To create 'pure' emission-line images, we first create 'emission-line free continuum images' from the r ' -band images and subtract these images from the H α and [S II ] images. \nThe basic process is straightforward. The r ' -band image contains continuum emission from stars as well as line emission from H α (and [N II ]) and from [S II ], e.g., \nr ' = r ' cont + r ' Hα + r ' [ S II ] (2) \nAll of the images have been scaled so that stars are equally bright in the images. Consequently, the counts (DN) from H α emission in the r ' band images is only a fraction of the H α counts (DN) in the H α images, that is \nr ' Hα = ∆ λ Hα ∆ λ r ' Hα (3) \nwhere ∆ λ is the effective band pass, approximately the FWHM of each filter. Similarly for [S II ], \nr ' [ SII ] = ∆ λ [ S II ] ∆ λ r ' [ SII ] (4) \nSo one can now create a line free r ' -band image by subtraction, e.g., \nr ' cont ∝ ( r ' -∆ λ Hα Hα +∆ λ [ S II ] [ SII ] ∆ λ r ' ) (5) \nHowever, in the process of subtracting the line emission from the r ' -band image, we have also removed some of the continuum emission. In order to produce a final emission-line subtracted image with stars of the same apparent brightness as in the original image, one must renormalize: \nr ' cont = ∆ λ r ' r ' -(∆ λ Hα Hα +∆ λ [ S II ] [ SII ]) ∆ λ r ' -(∆ λ Hα +∆ λ [ S II ] ) (6) \nThus, given an r cont image, one can in principle produce a pure H α or [S II ] image by simple subtraction. This process is not perfect for a variety of reasons, which include that (a) there are color corrections associated with the differences in central wavelengths, and these can vary from field to field, and (b) the r ' -band images were typically taken at different times from the narrow-band images and so the stellar PSFs differ between the narrow and r ' -band images. Nevertheless, this process produces superior continuum images for using to subtract from the emission-line images. \nIn principle, further improvements could be made, especially by convolving the images to a common PSF prior to image subtraction, but the current version of the processing pipeline does not include this. 4 As described in § 3.2, we attempt to at least partially ameliorate this problem by limiting the data we use to have a seeing value < 1 . '' 45. \nAs noted earlier, as part of the processing the images, we rescaled them all based on G-band magnitudes in the Gaia catalog. Given this scaling, the narrow-band flux in the final H α and [S II ] images should be given by \nF λ o = DN 1 A ( λ o ) λ o ∫ A ( λ ) λF λ, 27 dλ (7) \nwhere A represents the effective area of the DECam system, allowing for the transmission of the atmosphere, the reflectivity of the mirror system of the telescope and any filters used in the observation, and F λ, 27 is the flux per unit wavelength above the atmosphere for a 27th magnitude star. For the narrow-band H α and [S II ], and for a wavelengths in region where the transmission through the narrow-band filter is near the peak, the flux conversion reduces to \nF λ o = DN F λ,o, 27 ∆ λ (8) \nwhere ∆ λ is the filter bandpass, 160 and 100 ˚ A in the cases of the H α and [S II ] filters, respectively. The 3rd data release of Gaia data includes spectra of a large number of stars that appear in the images, and based on the spectra, estimates of the effective temperatures and gravities of many of these stars. To improve on the flux calibration provided by DCP, we have selected stars with temperatures near 10,000 K and g-band magnitudes of order 16 and matched them to objects for which we have measured net counts using aperture photometry. Scaled to 27th magnitude, we find that the emission line flux corresponding 1 DN in the H α and [S II ] images is 2 . 4 × 10 -18 ergs cm -2 s -1 and 2 . 1 × 10 -18 ergs cm -2 s -1 , respectively. Given the pixel size of 0 . '' 27, these fluxes correspond to surface brightnesses of 3 . 3 × 10 -17 ergs cm -2 s -1 arcsec -2 and 2 . 9 × 10 -17 ergs cm -2 s -1 arcsec -2 , respectively. The H α surface brightness limit corresponds to an emission measure of 16.5 cm -6 pc. Extended objects with excess of 1 DN are readily visible in the final images.", '5. A SAMPLING OF INITIAL RESULTS': "We consider the currently available processing of the data to be preliminary, but the superior resolution and overall depth and quality of the DECam survey data are already apparent. In this section, we provide some examples of the new data for various types of nebulae and compare to the previous MCELS survey data. Of course, these examples barely scratch the surface; the ultimate \ngoal is to provide the full data set to the community, allowing larger, global studies to be performed, taking advantage of the improved spatial resolution. \nFigure 7. A comparison of MCELS and DeMCELS data for the N70, a superbubble ∼ 7 . ' 8 in diameter, and located in an isolated region in the eastern LMC. Top row shows MCELS H α and [S II] images in black and white (linear stretch), with a color combination at right (red: H α ; green: [S II], log scaling). Yellow indicates both ions are strong. The bottom row shows the same sequence, but for our DECam data. The scale is shown in the lower right color panel. The inset at right shows detail of the northern rim. \n<!-- image -->", '5.1. Bubbles and Superbubbles': "Fig. 7 shows the region surrounding N70 (Henize 1956), an isolated nebular superbubble in the eastern LMC, also known as DEM L301 (Davies et al. 1976). N70 has a diameter of 7 . ' 8 ( ∼ 115 pc) and surrounds the stellar association LH114 (Lucke & Hodge 1970). N70 is an X-ray emitter (Chu & Mac Low 1990; Wang & Helfand 1991; Zhang et al. 2014) but is relatively weak. \nOptical spectroscopic observations show the nebula to be expanding at ∼ 40 kms -1 (Chu & Kennicutt 1988, and references therein). Optical spectra also show [S II ] λλ 6716,6731 to H α ratios to be variable but averaging ∼ 0.3, somewhat enhanced above the value expected for bright photoionized nebulae. Shock heating from strong stellar winds and/or previous supernovae within the shell are possible sources, but early modeling by Dopita et al. (1981) did not conclude shock heating was required. It was surprising that far ultraviolet observations of stars within the superbubble showed an excess of O VI λ 1032 absorption compared with non-superbubble sight lines (Danforth & Blair 2006). Since shock velocities in excess of 150 kms -1 are needed to produce O VI , a model involving thermal conduction from nebular interfaces with the hot interior gas seems favored. \nThe stellar content exciting the superbubble emission has been analyzed extensively and compared with other superbubbles and comparable H II regions (Oey 1996a,b). The photoionizing input for N70 is apparently dominated by a few very early type O stars (Oey 1996b) and calculations indicate there should be more than sufficient ionizing radiation to photoionize the entire nebula (Oey 1996c; Skelton et al. 1999). In fact, an embarrassingly little of the available energy is needed, leading to the suggestion that much of the ionizing radiation must escape the nebula. The differing optical nebular morphology between classical H II regions and superbubbles like N70 does not appear to be driven by the stellar content, as color magnitude diagrams (CMDs) show very similar behavior (Oey 1996b). Fitting the CMD for N70 with a Salpeter IMF implies that one or more very massive association members should have already exploded in the region, raising the likelihood that a hybrid model involving both photoionization and shocks play a role in exciting the nebula, a concept that was well-studied and modeled by Oey et al. (2000). \nSkelton et al. (1999) used the Rutgers Fabry-Perot on the CTIO 1.5m to obtain CCD imagery of N70 with very narrow effective filter bandpasses to sample a wider range of emission lines, including [O III ] λ 5007, and cleanly separating H α from [N II ] λλ 6548,6583, but at a seeing-limited 2 . '' 2 resolution. 5 Those observations show the nebular ionization structure (including [O III ]) to good advantage, and indicate a mix of radiative and collisional (shock) processes is likely responsible for ionizing and exciting the nebular shell, with the fraction of each changing outward from the center (more photoionization) to the outer rim (more evidence of shocks). This highlights the advantage of eventually adding [O III ] to our own survey as an additional nebular ionization diagnostic. \nThe comparison in Fig. 7 shows MCELS data in the top three panels and our new DECam survey images below. DECam resolves the filamentary structure of N70 at higher angular resolution, showing the nebular structure to be even more filamentary than seen previously. This may help explain why so little of the available ionizing flux is captured in the observed nebula. The brightening of the nebula on the west side likely indicates the expansion is encountering denser material on that side. The color variations in the lower right panel show an even finer separation of regions with enhanced [S II ] compared with H α , highlighting more dramatically those filaments dominated by shock heating, including a cluster of filaments just east of north on the north rim (see inset), faint filaments to the southeast, and an apparent and impending blow-out from the shell in the south. Targeting specific regions for further spectroscopic studies will allow the full extent of shock heating versus photoionization to be investigated, providing more detailed constraints to modeling such nebular systems. Of course, the power of the new survey is that such studies can be performed on numerous nebulae, sampling a wide range of parameter space.", '5.2. Supernova Remnants': "One of our team's primary goals is to improve upon observations of LMC SNRs, particularly the population of larger, fainter SNRs that are associated with later stages of SNR evolution leading to merging with the ISM. This survey allows us to use [S II ]/H α ratios to verify candidate SNRs suggested in various wavelength regimes, and to uncover new SNRs. \nA handful of the brightest LMC SNRs have been observed individually in depth and at many wavelengths, including some of the most famous SNRs observed with HST. However, many of the fainter or lesser known objects have received little attention and have relatively poor quality imagery \nFigure 8. A comparison of MCELS and DeMCELS data for the faint SNR 0527-6549, aka DEM L204. Presentation is the same as in Fig. 7. This is a low surface brightness SNR in field c42 (tile 12). Note the subtle color (ratio) variations in the DECam image at lower right. \n<!-- image --> \navailable. The broad spatial coverage of DeMCELS addresses this set of SNRs by providing improved imagery in H α and [S II ]. \nFig. 8 shows one such object, DEM L204, where previous MCELS and DeMCELS data are compared. This faint SNR lies in an isolated region in the northern LMC. The DECam images resolve the nebular structure in significant detail, showing a partial shell of emission brightest in the south and open to the west, where faint filaments extend beyond the main shell. At high resolution, variations in ratio for individual filaments are apparent as color variations in the panel at lower right. \nAnother type of SNR where DeMCELS data are superior are in regions of complex emission. SNR 0523-6753 is a little-studied SNR on the NE edge of the complex N44 emission region (Chu et al. 1993). Fig. 9 shows the comparison of MCELS and DeMCELS data for this region. Despite a fair amount of adjacent complexity in emission, the [S II ] image shows details of the filamentary structure of this object clearly. \nEven in very complex regions of emission, DeMCELS makes it possible to discern embedded SNR emission. The Honeycomb SNR (aka SNR 0535-6918) is a region of [S II ] bright loops first identified by Wang (1992) in a complex field not far from SN 1987A (also contained in this tile). This region contains ten or more loops of emission with sizes of ∼ 12 '' , corresponding to 2.8 pc at the distance of the LMC. Chu et al. (1995) used bright X-ray emission, non-thermal radio emission, and high [S II ] to H α ratios to identify this structure as resulting from a supernova shock front, perhaps propagating through a porous region of dense interstellar gas. Meaburn et al. (2010) modeled the spatial and kinematic structure of this object and suggested two possible scenarios: one in which the supernova \nFigure 9. A comparison of MCELS and DeMCELS data for the faint SNR 0523-6753. Presentation is the same as in Fig. 7. The SNR stands out by way of it's relatively brighter [S II] emission despite the complexity of the emission in this region. \n<!-- image --> \noccurred within the edge of a giant LMC shell, and one in which the loops are produced by a precessing jet from a binary microquasar, as in SS 433. \nThe large area covered by the DECam pointing in this region (Fig. 10) allows us to examine both the larger context of the SNR's surroundings, and to examine the loops or 'bubbles' themselves in detail. If the Honeycomb is part of a larger structure with varying density, as suggested by Chu et al. (1995), it would be useful to examine both the loops themselves and other [S II ] bright features in the vicinity. By using the [S II ] to H α ratios to identify possible features associated with the Honeycomb structure, one could perform follow-up high-resolution velocity studies of these features, building on the work of Meaburn et al. (1993), to see whether, e.g. , their line profiles are consistent with a common physical origin. Such work may help to clarify whether the Honeycomb can be associated with other nearby shocked gas, indicating its status as part of a larger structure. In particular, it would be useful if fainter filaments associated with the parts of the SNR expanding into lower-density surroundings can be identified. SNRs are thought to lose many of their prominent observational characteristics when expanding into rarefied ISM such as the interior of a superbubble. Identifying fainter, more extended structure in this instance may help to quantify the contributions of such otherwise invisible SNRs to the energy and hot gas component in active stellar regions that produce similar low-density conditions. \nFinally, the DECam survey also allows detailed follow-up examination of faint and little-studied SNR candidates, many suggested by observers at radio or X-ray wavelengths. The MCELS dataset, \nFigure 10. A comparison of MCELS and DeMCELS data for the SNR 0535-6918, aka the Honeycomb, within the outer regions of 30 Dor. Presentation is the same as in Fig. 7. DECam resolution is particularly effective in separating different kinds of emission in complex regions like this. \n<!-- image --> \ncombined with optical spectroscopy, was used by Yew et al. (2021) to study three SNRs and 16 SNR candidates in the LMC. Fig. 11 shows three of these objects in the DeMCELS data. Yew 3 (SNR 0454-7003) lies at the edge of superbubble DEM L25 (aka N185), which shows some indications of shock activity overall (Oey et al. 2002). Echelle studies of the DEM L25 superbubble by Zhang et al. (2014) show expansion velocities of up to 200 km/s in part of this object, consistent with the presence of SNR shocks, as well as diffuse X-ray emission. In DeMCELS, the SNR candidate shows up with relatively bright [S II ] emission, as with SNR 0523-6753 mentioned above; but the increased resolution of the DECam survey shows additional structure within the larger superbubble, including other [S II ]-enhanced regions. Using DeMCELS data to produce a high-resolution [S II ]/H α ratio map, in combination with existing optical spectra and/or detailed X-ray imaging spectroscopy, could help to more narrowly constrain the contribution of shock structures to the overall superbubble emission. Yew 6 (SNR 0502-6739) is an SNR candidate with a comparatively low [S II ]/H α ratio of 0.55 (Yew et al. 2021), which the authors point out may be due to dilution from the overlapping H II region. In the higher resolution DeMCELS data, we can see that the brightest and best defined [S II ] features correspond well with the boundaries estimated from MCELS images, but it also displays additional faint filamentary structure. In addition, the well-defined filaments may allow better measurement of shock-heated emission above that of the H II region. Similarly, Yew 12 (SNR 0528-7017) shows fine filaments in the DECam data that were unresolved in MCELS. \nFigure 11. DeMCELS data for three SNRs studied by Yew et al. (2021). Shown here are (a) Yew 3 (SNR 0454-7003), (b) Yew 6 (SNR 0502-6739), and (c) Yew 12 (SNR 0528-7017). Ellipses indicate the listed sizes of the SNRs from Table 1 of that paper: for Yew 3 the longest axis is 129 '' , for Yew 6, 190 '' , and for Yew 12, 380 '' . H α emission is shown in red and [S II] in green. \n<!-- image --> \nSome objects suggested to be SNRs by multiwavelength data may not pan out under closer inspection at optical wavelengths. Fig. 12 shows a moderately wide view of region of complex emission in the northern LMC. SNR J0521-6543 (DEM L142) was noted as a possible SNR candidate from MCELS data (e.g., Williams 2009) and was listed by Maggi et al. (2016) as a confirmed SNR. In the DECam images, the [S II ] bright circular shell stands out nicely from the background H α emission. Not previously noted, however, are the complex loops seen in both emission lines along and below the circular rim of the SNR, reminiscent of the structure in the Honeycomb SNR. This raises the possibility, as suggested for the Honeycomb SNR, that SNR shocks are traveling through a larger structure of porous gas. \nA circular region of about the same size as SNR J0521 -6543 appears at left in Fig. 12, and shows a circular rim of enhanced [S II ] emission. This feature aligns with a shell-like radio feature noted by Bozzetto et al. (2023) in ASKAP and ATCA images. Its shell-like radio morphology, and its radio spectral index of -0 . 51 ± 0 . 05, typical of SNRs, led the authors to suggest this region as a confirmed SNR J0522-6543. They also note an enhanced [S II ] to H α ratio of 0.4 from MCELS images. However, the DECam images of this region show bright central H α emission with a diffuse surrounding ring of enhanced [S II ]; as viewed at DECam resolution, the ring does not show the sharp filamentary structure seen in most LMC SNRs (including J0521-6543 at right). This diffuse morphology more strongly suggests that the [S II ] emission marks the outer boundary of photoionized gas in an H II region. Although continuum has been subtracted in Fig. 12, residuals from a central star cluster can be readily seen. \nAdding to the interest of this optical complex is the slightly enhanced [S II ] in an irregular filamentary feature that lies between the SNR and SNR candidate. \nFigure 12. A comparison of MCELS and DeMCELS data for the region near SNR J0521-6543, aka DEM L142, which is the well-defined filamentary shell at right. Presentation is the same as in Fig. 7. SNR J0521-6543 is ∼ 170 '' in diameter Maggi et al. (2016). The similar sized shell at left is J0522-6543, an SNR according to Bozzetto et al. (2023). However, this shell is center-filled with H α and does not resolve into crisp filaments at DECam resolution. The centrally-placed star cluster makes it likely that this nebula is a stellar bubble. See text for details. \n<!-- image --> \nFigure 13. LMC planetary nebula SMP83 from field c48 (tile 2). The MCELS data are shown at left, and DeMCELS at right, with H α in red and [S II] in green. LMC PNe have very little [S II] emission and typically appear stellar or nearly stellar. Shaw et al. (2006) list this PN as being 3 . '' 98 by 3 . '' 63 based on HST imagery, and is one of the largest PNe in their survey. \n<!-- image -->", '5.3. PNe and other Small Scale Nebulae': "The LMC contains hundreds of identified planetary nebulae (PNe) (Reid & Parker 2006, 2013), which appear as stellar or nearly stellar sources of H α (and/or [N II ]) emission since PNe emit at very low levels in [S II ]. As a practical matter, many of what appear to be stellar residuals in the subtracted H α images in MCELS and DeMCELS surveys are actually unresolved PNe or one of several possible types of emission-line stars in the LMC (W-R stars, X-ray binaries, cataclysmic variables, and the \nlike). Shaw et al. (2006) obtained images and spectroscopy of a number of LMC PNe using STIS on HST in slitless spectroscopy mode. In Fig. 13 we show the PN SMP83 as seen in both MCELS and DeMCELS. SMP83 is one of the largest PNe listed by Shaw et al. (2006) with angular size of 3 . '' 98 × 3 . '' 63 (0.96 pc × 0.87 pc). Many of the LMC PNe listed by Shaw et al. (2006) are below 1 '' in diameter, and hence only marginally-resolved or unresolved even by DeMCELS. \nFigure 14. DeMCELS data for a small angular size bi-lobed emission nebula from field c42 (tile 7). The MCELS data only show an unresolved point-like source (primarily from the brighter central edge-on disk) and is not shown here. Otherwise the presentation is the same as in Fig. 7. The bi-lobed structure is ∼ 18 '' long, much too large for a PN. \n<!-- image --> \nIt is somewhat surprising then that the object shown in Fig. 14 has been classified as a PN (PN3464 Reid & Parker 2013). The MCELS data show only the bright central region as an unresolved small emission region and is not shown in the figure. The identification was based on line ratios, since the nebula itself was not resolved in the UK Schmidt survey data used by Reid & Parker (2013). Our survey data are apparently the first to resolve the structure of this nebula, showing a bright central region which appears to be an edge-on disk plus a much fainter bi-lobed outer structure. The extent of the bi-lobed structure is ∼ 18 '' ( ∼ 4.4 pc), which would be exceedingly large for a PN. Reid & Parker (2013) show a spectrum of this source, no doubt dominated by the bright central disk, and unlike most of the H α emission sources, this object shows very strong [N II ] emission, which is indicative of enhanced N abundance. So it is likely that this material is circumstellar medium that was ejected by the central evolved star or interacting binary, a conclusion supported by the morphology of the nebula. \nAlthough it is beyond the scope of this survey overview paper to catalog the numerous small H α nebulae that are unresolved in MCELS but are at least marginally resolved with DECam, we show some examples of such nebulae in Fig. 15. This figure highlights a small region north of the N44 complex that contains several small star forming regions of various sizes. The larger nebula above and left of center is clearly being excited by a small handful of hot interior stars. On the southern edge of this larger nebula, a small 'single star' nebula can be seen. The enlargement at right shows a 30 '' region with three other very small (individual star) nebulae. Faint structured diffuse emission from the outskirts of the N44 region is also visible in the lower part of the frame. \nFigure 15. A comparison of MCELS and DeMCELS for a small region from field c39 that contains several small to very small star forming regions. Presentation is the same as in Fig. 7 except for MCELS, we have used the emission line files prior to continuum subtraction so stars remain visible. For scale, the box at right is 30 '' . These smallest emission nebulae are totally unresolved in MCELS data. \n<!-- image --> \nIf one ignores the bright regions of emission and looks in the background, the LMC is filled with extensive diffuse emission, some of which has been noted above in earlier figures. With its larger pixel size, MCELS actually shows just how extensive this emission is, extending outward from and between many of the bright H II region complexes. Much of this emission is akin to what has been called either the warm ionized medium (WIM) or diffuse ionized gas (DIG) in other nearby galaxies (see Haffner et al. 2009, for a review), but it has received relatively little attention in the LMC. Generally speaking, the DIG shows relatively strong [S II ] emission in comparison to normal photoionized gas, but it is nonetheless photoionized by radiation leaking out from active star forming regions. Some of this faint background emission has a fluffy morphology, and for this component, MCELS actually shows it as effectively or more so than DeMCELS, owing to the larger pixel scale in the former. However, to the extent that some of this background emission is structured, DeMCELs is effective at showing it. This is an example of where using the two surveys together can be effective. \nFig. 16 shows one example of this comparison, for a background region ∼ 0 . 5 · north of the N44 emission complex. The scaling has been adjusted to enhance what is generally quite faint emission compared with what has been shown in earlier figures. A region of structured emission is seen in projection against more diffuse emission, particularly visible along the left side of the Figure. Not only does the morphology change between these two components of the background, but the line ratio of [S II ] to H α also changes, being somewhat higher in the structured emission. As no star cluster or ionizing source is seen within the structured emission and since the [S II ]/H α ratio is somewhat \nenhanced compared with other local emission, it is conceivable that this emission represents an old SNR that has nearly faded into the background, but it has not been catalogued as such previously. \nFigure 16. A comparison of MCELS and DeMCELS for a region of structured background from field c42 (tile 5). Presentation is the same as in Fig. 7. Although the overall depth of the two surveys is about the same, truly diffuse emission shows up somewhat better in the original MCELS data, owing to its larger pixel size. However, for any of these background regions that show fine-scale structure, such as the arc of emission shown here, the higher resolution of DeMCELS shows those details better. \n<!-- image -->", '6. DATA AVAILABILITY': "Our interests in obtaining an improved version of the MCELS survey are primarily related to our interests in the study of shocked gas in SNRs and other settings. However we also felt that a new survey would be of interest to others for a variety of projects. As a result, and working with scientists and engineers at NOIRLab, we have made our initial reduction of the DeMCELS survey data available at NOIRLab's Data Lab survey page. There one can find mechanisms to obtain all or portions of the data. We expect to acquire additional images of the LMC in the near future and have a companion study of the SMC underway. We expect future releases of the survey data (including any improvements in the data processing) to be posted on this site as well.", '7. SUMMARY': "We performed a new multiband (H α and [S II ]) emission-line survey of the LMC using DECam on the Blanco 4-m telescope at CTIO, using images with the DES r ' filter for continuum-subtraction. Twenty overlapping fields were required to cover the visible extent of the LMC, leading to huge and complex data sets to process into large-area mosaic images. We have described the data and processing, and then have presented representative examples of various kinds of emission nebulae \nin the LMC, highlighting the improvements in spatial resolution and diagnostic power provided by the new survey in comparison with the previously available MCELS survey (Smith & MCELS Team 1999). \nHowever, all of the above are simply representative examples. The power of the DECam survey presented here is that it allows more global studies of a variety of emission nebulae and objects across the entire LMC at a spatial resolution some 3-5 × higher than the MCELS survey. These data make an ideal complement to other multiwavelength surveys of the LMC (cf. Kim et al. 2003; Meixner et al. 2006; Maggi et al. 2016). \nAs a service to the community, we are making the processed data available to enable a much broader range of science than our team's particular scientific interests. Work on a companion survey of the Small Magellanic Cloud is ongoing. As these data are processed, we expect to make them available in the same manner. \nThis project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the US Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute for Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Funda¸c˜ao Carlos Chagas Filho de Amparo 'a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico and the Minist'erio da Ciˆencia, Tecnologia e Inova¸c˜ao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. \nThe Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones En'ergeticas, Medioambientales y Tecnol'ogicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci'encies de l'Espai (IEEC/CSIC), the Institut de F'ısica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF NOIRLab, the University of Nottingham, the Ohio State University, the OzDES Membership Consortium, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. \nBased on observations at NSF Cerro Tololo Inter-American Observatory, NSF NOIRLab (NOIRLab Prop. ID 2018A-0909; PI: T. Puzia; NOIRLab Prop. ID 2018B-0908; PI: T. Puzia; and NOIRLab PropID 2021B-0060; PI: S. Points), which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the U.S. National Science Foundation. \nThis work has made use of data from the European Space Agency (ESA) mission Gaia (https: //www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. \nSDP would like to thank Thomas Puzia and Eric Peng for the purchase of the N662 filter and starting this work with their H α survey of the LMC. SDP would also like to thank Kyoungsoo Lee and the ODIN survey for permitting us to use the N673 filter in this investigation. WPB acknowledges support from the Johns Hopkins Center for Astronomical Sciences during this work. PFW acknowledges the support of the NSF through grant AST-1714281. YHC acknowledges the support of the grants NSTC 112-2112-M-001-065 and NSTC 111-2112-M-001-063 from the National Science and Technology Council of Taiwan. RMW would like to acknowledge the contributions of the following Columbus State University undergraduate students for data review and object searches: Kayleen Linge, Devin Janeway, Sharmaine Motin, A'naja Houston, Delta Flowers, Griffin McLeroy, Cory Mitchell, Trinity Smith, Samuel Kimball, and William Morgan. \nFacility: Blanco \nSoftware: SWarp (Bertin et al. 2002), Astropy (Astropy Collaboration et al. 2013, 2018, 2022), Matplotlib (Hunter 2007), NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), and SAOImage DS9 (Joye & Mandel 2003)"}
2024arXiv240912754O	We investigate the discovery potential at the MATHUSLA experiment of a longlived particle LLP which is the heavier state of inelastic scalar dark matter DM in third generationphilic U1 U1X3 extension of the standard model. Since the heavier state and DM state form the complex scalar charged under the U1X3 it is natural that the heavier state P is almost degenerate with the DM state and hence longlived. We find that third generationphilic righthanded U1 U1R3 model is the most interesting because third generationphilic models are less constrained by the current experimental results and righthanded U1 interactions leave visible final decay products without produing neutrinos. For a benchmark of the model parameters consistent with the current phenomenological constraints we find that the travel distance of the LLP can be mathcalO100 m and the LLP production cross section at the 14 TeV LHC can be mathcalO10 fb. Thus we conclude that the LLP can be discovered at the MATHUSLA with a sufficiently large number of LLP decay events inside the MATHUSLA detector.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.12754', 'arXiv:2409.12754', '2024arXiv240912754O']	['High Energy Physics - Phenomenology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	Discovery potential of a longlived partner of inelastic dark matter at MATHUSLA in U1X3 extension of the standard model	2,024	193	0.22	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.12754.pdf	{'Discovery potential of a long-lived partner of inelastic dark matter at MATHUSLA in U (1) X 3 extension of the standard model': 'Nobuchika Okada ∗ \nDepartment of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35487, USA \nOsamu Seto † \nDepartment of Physics, Hokkaido University, Sapporo 060-0810, Japan', 'Abstract': 'We investigate the discovery potential at the MATHUSLA experiment of a long-lived particle (LLP), which is the heavier state of inelastic scalar dark matter (DM) in third generation-philic U (1) ( U (1) X 3 ) extension of the standard model. Since the heavier state and DM state form the complex scalar charged under the U (1) X 3 , it is natural that the heavier state P is almost degenerate with the DM state and hence long-lived. We find that third generation-philic right-handed U (1), U (1) R 3 , model is the most interesting, because third generation-philic models are less constrained by the current experimental results and right-handed U (1) interactions leave visible final decay products without produing neutrinos. For a benchmark of the model parameters consistent with the current phenomenological constraints, we find that the travel distance of the LLP can be O (100) m and the LLP production cross section at the 14 TeV LHC can be O (10) fb. Thus, we conclude that the LLP can be discovered at the MATHUSLA with a sufficiently large number of LLP decay events inside the MATHUSLA detector.', 'I. INTRODUCTION': "Explanation of the neutrino oscillation phenomena between neutrino flavor species due to non-zero neutrino mass and existence of dark matter (DM) in the Universe require undiscovered particles and interactions of beyond the standard model (BSM) of particle physics. Nevertheless, any evidence of new particles in BSM have not yet been reported in any ongoing experiments. A reason of non-discovery of a new particle might be due to not its large mass but its extremely weak interaction strength, in other words, a relevant coupling constant is very small. Even if coupling constants and hence the resultant production cross sections of new particles are very small, we may expect certain amount of production of new particles if experiments are conducted with huge luminosity. The small coupling constant makes the new particle long-lived, and signals of such long-lived particles (LLPs), if they are electrically neutral, would be discovered through their 'displaced vertex' signatures. In fact, displaced vertex signatures have been searched for at, for instance, ATLAS [1] and FASER [2] in Large Hadron Collider (LHC). Some near-future experiments such as FASER2 [3] will start soon, and other several future experiments have been proposed, targeting various range of LLP masses and decay lengths. Among such proposed future experiments for the displaced vertex search, MAssive Timing Hodoscope for Ultra Stable neutraL pArticles (MATHUSLA) experiment is striking, because it will be able to search LLPs with the decay length of the order of 100 m [4]. Theoretical studies on search sensitivity for various BSM models have been examined in Refs. [5-40]. \nThe null results in dark matter direct detection experiments constrain models of Weakly Interacting Massive Particles (WIMPs) DM [41]. The null results in indirect searches of DM also constrain the present DM annihilation cross section and exclude WIMPs annihilating into b ¯ b quarks in s-wave processes with the cross section of about 3 × 10 -26 cm 3 / s, which is the typical value for thermal WIMPs, with mass up to a few hundreds of GeV [42, 43]. Despite such stringent constraints, there are still many viable WIMP models. One of those model proposed in Ref. [44] is an extra U (1) interacting scalar inelastic dark matter, which is consistent with null results of direct DM searches due to its inelastic nature [45, 46] and of indirect DM search because thermal abundance in the early Universe is fixed by coannihilation cross section [47, 48] rather than the self-annihilation cross section [49]. \nIn the models of Ref. [44], both DM state and the heavier state relevant for both co- \nannihilation and inelastic scattering of DM state originate from one complex scalar field. Since they belong to the single complex scalar field, those two states are naturally well degenerate. The decay rate of the heavier state into the lighter one (DM state) would be very small due to its very small phase space volume, so that the heavier state could be a natural candidate for the LLP that we are interested in. To examine the MATHUSLA prospects, inelastic DM models have been proposed [12, 23, 27] 1 . While the longevity of LLP in fermionic inelastic DM model proposed in Ref. [12] is due to the mass degeneracy as well as its tiny kinetic mixing of dark photon, the LLP in our scalar inelastic DM model is long-lived due to only about 1% mass degeneracy. \nThis paper is organized as follows: In Sec. II, we describe the extra gauge U (1) interacting inelastic model, which is a variant of the original models in Ref. [44]. We identify a parameter set to reproduce thermal DM abundance and a benchmark point in Sec. III. In Sec. IV, for the benchmark point, we present a search prospect of LLPs by the MATHUSLA experiment. Section V is devoted to our summary.", 'II. THE MODEL': "There are several possibilities of anomaly free gauged U (1) extension of the standard model. Among them, the best studied model is the flavor universal U (1) B -L [50-53]. However, because the new neutral gauge boson Z ' in the U (1) B -L model interacts with the first and the second generation of SM fermions, the experimental constraints on the couplings are very stringent [54-56]. Thus, we may not expect a sizable production cross section for the LLP production from the Z ' boson decay. Note that anomaly cancellation of U (1) B -L model is realized in each generation of fermions, hence it is generally possible for only a particular generation to be charged under the U (1). The model where only the third generation fermions are charged, U (1) ( B -L ) 3 , is such a choice [44, 57-60] and well studied. As expected, the experimental constraints on the U (1) ( B -L ) 3 model are much weaker compared with the universal U (1) B -L model [61-65]. It is easy to generalize the U (1) B -L model to the U (1) X model by assigning the U (1) charge for a field as a linear combination of its U (1) Y and U (1) B -L charge. Since U (1) Y and U (1) B -L are independently anomaly free, the U (1) X \nTABLE I: The particle contents of our U (1) X 3 model. In addition to the SM particle content ( i = 1 , 2 , 3), three RH neutrinos ( N i R ( i = 1 , 2 , 3)) and two U (1) X 3 scalar fields ( φ 1 and φ 2 ) are introduced. The scalar φ 2 is the Higgs field and develops its VEV to break the U (1) X 3 symmetry, while φ 1 has no VEV and includes the inelastic DM and its partner LLP as its components. \nis automatically anomaly free [66-68]. A special case is so-called U (1) R model, where only right-handed (RH) SM fermions and RH neutrinos are charged under the U (1) [69]. In this paper, we generalize U (1) ( B -L ) 3 to U (1) X 3 , where X 3 denotes a linear combination of U (1) Y 3 and U (1) ( B -L ) 3 . We introduce an U (1) X 3 charged scalar field φ 1 with its charge +1 in addition to the minimal particle contents. The total particle contents are listed in Tab. I. The field φ 2 is responsible to break U (1) X 3 gauge symmetry and x H is a real parameter which parameterize a combination weight of ( B -L ) 3 and Y 3 . \nThe U (1) X 3 gauge interaction for an SM chiral fermion and RH neutrinos ( f L/R ) can be read from the usual covariant derivative, \nL int = ∑ f j f j γ µ X µ g X 3 ( q f jL P L + q f jR P R ) f j , (1) \nwhere X µ is the U (1) X 3 gauge field, g X 3 is the gauge coupling constant, and q f jL/R is a U (1) X 3 charge of f jL/R (see Tab. I). \nThe scalar potential, which is gauge-invariant and renormalizable, is expressed as [70, 71] \nV (Φ , φ 1 , φ 2 ) = -M 2 Φ \| Φ \| 2 + λ 2 \| Φ \| 4 + M 2 φ 1 φ 1 φ † 1 -M 2 φ 2 φ 2 φ † 2 + 1 2 λ 1 ( φ 1 φ † 1 ) 2 + 1 2 λ 2 ( φ 2 φ † 2 ) 2 + λ 3 φ 1 φ † 1 ( φ 2 φ † 2 ) +( λ 4 φ 1 φ † 1 + λ 5 φ 2 φ † 2 ) \| Φ \| 2 -A ( φ 1 φ 1 φ † 2 + φ † 1 φ † 1 φ 2 ) , (2) \nwith Φ being the SM Higgs doublet field. All parameters, M 2 i , λ i and A , in the potential (2) are taken to be real and positive.", 'A. Dark matter mass and interactions': "At the U (1) X 3 and the electroweak (EW) symmetry breaking vacuum, the SM Higgs field and the U (1) Higgs field are expanded around those VEVs, v and v 2 , as (in the unitary gauge) \nΦ = 0 v + ϕ √ 2 , (3) \nφ 1 = S + iP √ 2 , (4) \nφ 2 = v 2 + ϕ 2 √ 2 . (5) \nThe physical states ( ϕ and ϕ 2 ) are diagonalized to the mass eigenstates ( h and H ) with masses m h and m H as \n ϕ ϕ 2 = cos α sin α -sin α cos α h H . (6) \nFor a small mixing angle α , h is identified with the SM-like Higgs boson. Hereafter, we set α negligibly small. With the U (1) X 3 and the EW symmetry breaking, the Z ' boson which is the mass eigenstate after the X becomes massive, S and P acquire their masses, respectively, as \nm 2 Z ' = g 2 X 3 4 v 2 2 , (7) \nm 2 S = M 2 φ 1 + 1 2 λ 3 v 2 2 + 1 2 λ 4 v 2 -√ 2 Av 2 , (8) \nm 2 P = M 2 φ 1 + 1 2 λ 3 v 2 2 + 1 2 λ 4 v 2 + √ 2 Av 2 . (9) \nNote that the parameter A controls the mass splitting between S and P . Since we take A positive, S is lighter than P and becomes the DM candidate. The other choice of A < 0 causes no essential difference in our final results, except that P is the DM particle in the case. \nGauge interaction of the DM particle is expressed as \nL int = g X 3 Z ' µ (( ∂ µ S ) P -S∂ µ P ) , (10) \nand similarly 3rd generation quarks and leptons also interact with Z ' boson with corresponding charges. The absence of Z ' -DM-DM coupling indicates that the Z ' -mediating DM scattering off with a nucleon is inelastic and ineffective for a mass splitting larger than the maximal energy transfer in the scatterings [45, 46]. Elastic scattering through Higgs bosons exchange [72] can be neglected since we have set a very small Higgs mixing α . We also set the coupling constant A very small, so that S and P are well degenerate in mass. This degeneracy is crucial not only for the scalar P being long-lived, but also for reproducing the observed DM relic density through S and P coannihilation process.", "B. The decay width of Z ' boson": "The partial decay width of Z ' → f ¯ f is given by \nΓ( Z ' → f ¯ f ) = 1 48 πm 2 Z ' N c \|M\| 2 √ m 2 Z ' -4 m 2 f , (11) \n\|M\| 2 =2 g 2 X 3 ( q 2 fL ( m 2 Z ' -m 2 f ) +6 q fL q fR m 2 f + q 2 fR ( m 2 Z ' -m 2 f )) , (12) \nwhere the number of color N c is 3 for quarks and 1 for leptons. If m f /lessmuch m Z ' , then Eq. (11) is reduced to \nΓ( Z ' → f ¯ f ) /similarequal g 2 X 3 N c m Z ' 24 π ( q 2 fL + q 2 fR ) . (13) \nWe obtain the partial decay width of Z ' into S and P as \nΓ( Z ' → SP ) = g 2 X 3 48 π ( m 2 Z ' -( m P -m S ) 2 ) 3 / 2 ( m 2 Z ' -( m P + m S ) 2 ) 3 / 2 m 5 Z ' , (14) \nfrom the vertex (10). The decay branching ratios are shown in Fig. 1. Note that Br ( Z ' → ν τ ν τ ) vanishes at x H = -2, which corresponds to U (1) right-handed, U (1) R . This fact plays an important role in the following discussion. \nFIG. 1: Branching ratios of the Z ' decay for g X = 0 . 1 , m S = 130 GeV and m Z ' = 350 GeV. \n<!-- image -->", 'C. Decay of P': "If S and P are strongly degenerate so that the mass difference is much smaller than the mass of Z ' , the two body decay of P is kinematically forbidden. Thus, for the main decay mode, P → SZ '∗ → Sf ¯ f , the total decay width (Γ), or equivalently the inverse of the lifetime ( τ ), \nΓ = 1 τ = 1 2 p 0 ∫ \|M ( P → Sf ¯ f ) \| 2 dQ, (15) \nis suppressed by the phase space volume [73]. As a result, P is short-lived in cosmology but can be long-lived in collider experiments. The expected travel distance cτ for x H = -2, in other words U (1) R 3 , is shown as the function of its mass m P in Fig. 2. We note that the mass difference, \nm 2 P -m 2 S = 2 √ 2 Av 2 , (16) \n/negationslash \nfrom Eqs. (8) and (9), is always adjustable by suitably choosing the free parameter A such that cτ becomes about 100 m, which is ideal for the detection of P decay inside the MATHUSLA detector. Here and hereafter, we have forcused on the U (1) R 3 case. For x H = -2, the decay mode of P → SZ '∗ → Sν τ ν τ exits and becomes the dominant mode for M P -m S ≤ 2 m τ . In this case, the LPP P provides only invisible decay products. \nFIG. 2: cτ for g R 3 = 0 . 1 , m S = 150 GeV and m Z ' = 350 GeV. Near the threshold m P -m S /similarequal 2 m τ with m τ being the tau-lepton mass, cτ is very long. \n<!-- image -->", 'III. DARK MATTER AND LHC CONSTRAINTS': '/negationslash \nFor evaluation of prospect at MATHUSLA in the next section, in this section, we will find a benchmark point which satisfies the thermal DM abundance and the latest LHC constraints. As metioned above, we concentrate on the U (1) R 3 model. The LLP P in other general U (1) X 3 ( x H = -2) may decay into neutrinos which are invisible for the MATHUSLA detector. Thus, the U (1) R 3 model is the best choice from the viewpoint of the detection with a large decay length of O (100) m.', 'A. Thermal relic abundance': "We estimate the thermal relic abundance of the real scalar DM, S , by solving the Boltzmann equation, \ndn dt +3 Hn = -〈 σ eff v 〉 ( n 2 -n 2 EQ ) , (17) \nwhere H and n EQ are the Hubble parameter and the DM number density at thermal equilibrium, respectively [49]. In our model, the main annihilation mode is coannihilation SP → f ¯ f through s -channel Z ' exchange for m Z ' > m S and the annihilation mode SS → Z ' Z ' by u ( t )-channel P exchange for m Z ' < m S [44]. We use the effective thermal averaged annihilation \ncross section \n〈 σ eff v 〉 = ∑ i,j = S,P 〈 σ ij v ij 〉 n i n EQ n j n EQ , (18) \nto include the coannihilation effects properly and n in Eq. (17) should be understood as n = ∑ i n i for i = S, P [47, 48]. The contours of thermal DM abundance Ω h 2 /similarequal 0 . 1 [74] (blue curve) and Br( Z ' → SP ) (green curves) for the U (1) R 3 model are shown in Fig. 3. \nFIG. 3: The contour (in blue) along which the observed DM relic abundance Ω h 2 /similarequal 0 . 1 is reproduced for g R 3 = 0 . 1 and x H = -2. \n<!-- image -->", 'B. Benchmark points': "To find a viable benchmark point, we also take the Z ' boson search at the LHC with ditau final state [75, 76] into account. We evaluate the production cross section for the process pp → τ + τ -given by \nσ ( pp → Z ' )BR( Z ' → τ + τ -) , (19) \nwhere the branching ratio is calculated from Eq. (11). As can be seen in Eq. (12), BR( Z ' → τ + τ -) depends on the value of x H , and we forcus on x H = -2 as stated above. \nWith the decay rates of X estimated in Sec. II B, as in Ref. [56], since the total X boson decay width is very narrow, we use the narrow width approximation to evaluate the X boson production cross section \nσ ( pp → X ) = 2 ∑ q, ¯ q ∫ dx ∫ dyf q ( x, Q ) f ¯ q ( x, Q )ˆ σ (ˆ s ) , (20) \nˆ σ (ˆ s ) = 4 π 2 3 Γ X ( X → q ¯ q ) m X δ (ˆ s -m 2 X ) , (21) \nwhere f q and f ¯ q are the parton distribution function (PDF) for a quark and antiquark ( b and ¯ b in our model), ˆ s = xys is the invariant mass squared of colliding quarks for the center of mass energy s . The factor 2 in Eq. (20) counts two ways of q coming from which proton out of two colliding protons. Since the most severe bound is from the dilepton channel ( /lscript = τ ), we calculate σ ( pp → X )Br( X → τ + τ -) and compare it with the CMS results [75] 2 . We employ PDFs of CTEQ6L [77] with a factorization scale Q = m Z ' for simplicity. \nThe resultant production cross section is shown in the second column in Tab. III. The production cross section is consistent with and smaller than the latest most stringent bound from the CMS [75] at the LHC. We summarize parameters and masses of relevant particles in Tab. II. \nTABLE II:", 'IV. PROSPECT FOR MATHUSLA': "The travel distance cτ /similarequal 100 m is the most sensitive range at MATHUSLA [4, 78, 79]. The pair production cross section of such a LLP χ for its 5 σ discovery can be read as σ χχ > 0 . 3 fb from Fig. 1 in Ref. [9]. On the other hand, in our scenario, the LLP P is not pair produced but singly produced through pp → Z ' → SP . Thus, the required production \nTABLE III: \ncross section for the LLP P to be discovered at the MATHUSLA would be set to be larger than 0 . 6 fb. In Tab. III, we list the SP production cross section at 13 TeV LHC and 14 TeV LHC for our benchmark on Tab. II satisfying the DM abundance and the LHC bounds. We find that the MATHUSLA will be able to discover those LLPs because the production cross section σ SP is sufficiently larger than 0 . 6 fb.", 'V. SUMMARY': "We have proposed a simple extension of the SM with an extra U (1) R 3 gauge interaction with scalar particle of DM candidate with the charge 1, third generation of SM fermions and third generation of right-handed neutrinos. After the SM singlet U (1) R 3 breaking scalar with the charge 2 develops the VEV, the gauge boson acquires the mass, and the tiny mass splitting between real and imaginary component of the charge 1 scalar appears through the scalar tri-linear interaction. The lighter scalar S is inelastic DM candidate with the heavier state P . Due to the mass degeneracy, the slightly heavier state P is long-lived and will be able to be discovered as LLPs. \nWe have calculated the production cross section of LLP P through the process pp → Z ' → SP at the MATHUSLA for a benchmark point which satisfies stringent LHC bounds and thermal DM abundance. The production cross section of benchmark point turns out to be about 10 fb, which is about an order of magnitude larger than the cross section required by 5 σ discovery as a LLP at the MATHUSLA. Thus, we conclude that the heavier state of inelastic DM in our model can be discovered at the MATHUSLA.", 'Acknowledgments': 'This work is supported in part by the U.S. DOE Grant No. DE-SC0012447 and DESC0023713 (N.O.) and KAKENHI Grants No. JP23K03402 (O.S.).', 'Appendix A: The decay rate of P': "The spin averaged squared amplitude for P ( p ) → f ( q 1 ) ¯ f ( q 2 ) S ( q 3 ) is given by \n\|M\| 2 = 2 g 4 R 3 m 4 Z ' ( -2 ( p · q 3 ) + m 2 P + m 2 S -m 2 Z ' ) 2 ( -2 ( -m 2 P + m 2 S + m 2 Z ' ) 2 ( p · q 1 ) 2 -2 ( m 2 P -m 2 S + m 2 Z ' ) 2 ( q 1 · q 3 ) 2 +4 ( ( m 2 P -m 2 S ) 2 -m 4 Z ' ) ( p · q 1 ) ( q 1 · q 3 ) -2 ( -m 2 P + m 2 S + m 2 Z ' ) 2 ( p · q 1 ) ( p · q 3 ) + 2 ( m 2 P -m 2 S + m 2 Z ' ) 2 ( q 1 · q 3 ) ( p · q 3 ) + m 2 f ( -2 ( ( m 2 P -m 2 S ) 2 -m 4 Z ' ) ( p · q 3 ) + m 4 Z ' ( m 2 P + m 2 S ) + ( m 2 P -m 2 S ) 2 ( m 2 P + m 2 S -2 m 2 Z ' ) ) +( p · q 1 ) ( m 4 Z ' ( m 2 P -3 m 2 S ) +2 m 2 Z ' ( m 4 S -m 4 P ) + ( m 2 P -m 2 S ) 2 ( m 2 P + m 2 S ) ) +( q 1 · q 3 ) ( m 4 P ( m 2 S -2 m 2 Z ' ) + m 2 P ( m 4 S +3 m 4 Z ' ) -m 6 P -( m 3 S -m S m 2 Z ' ) 2 )) . (A1) \nThe integration of phase space volume is reduced to \ndQ = d 3 q 1 (2 π ) 3 2 q 0 1 d 3 q 2 (2 π ) 3 2 q 0 2 d 3 q 3 (2 π ) 3 2 q 0 3 (2 π ) 4 δ (4) ( p -q 1 -q 2 -q 3 ) = 1 (2 π ) 5 d 3 q 1 2 q 0 1 d 3 q 3 2 q 0 3 δ ( ( m P -q 0 1 -q 0 3 ) 2 -( \| q 1 \| 2 + \| q 3 \| 2 ) -2 \| q 1 \|\| q 3 \| cos θ 13 -m 2 f ) = 1 8(2 π ) 5 dq 0 1 dq 0 3 d Ω 1 dϕ 13 d cos θ 13 δ ( ( m P -q 0 1 -q 0 3 ) 2 -\| q 1 \| 2 -\| q 3 \| 2 -m 2 f 2 \| q 1 \|\| q 3 \| -cos θ 13 ) . 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2024arXiv240908674S	Observations of quasars show that the polarization position angle of the emission coming from them varies greatly over time including periods called rotations during which the angle changes in an orderly manner. The study proposes a method for identifying such events and assessing their statistical significance. The operation of the method is demonstrated using the example of longterm polarimetric observations of the blazars CTA 102 3C 454.3 and OT 081. During the analysis of light curves 51 rotations of the polarization position angle were found and it was shown that for CTA 102 and 3C 454.3 the rotations are predominantly oriented in one direction.	2024-09-01T00:00:00Z	['2024arXiv240908674S', '10.48550/arXiv.2409.08674', 'arXiv:2409.08674']	['Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	The Method of Searching for Rotations of the Polarization Position Angle of Quasars	2,024	193	0.49	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.08674.pdf	{'The Method of Searching for Rotations of the Polarization Position Angle of Quasars': 'S. S. Savchenko, 1, 2, 3, ∗ D. A. Morozova, 1, † S. G. Jorstad, 1, 4 D. A. Blinov, 5, 6 G. A. Borman, 7 A. A. Vasilyev, 1 T. S. Grishina, 1 A. V. Zhovtan, 7 E. N. Kopatskaya, 1 E. G. Larionova, 1 1 1 1 1 \nI. S. Troitskiy, Yu. V. Troitskaya, E. V. Shishkina, and E. A. Shkodkina 1 St. Petersburg University, St. Petersburg, 199034 Russia 2 Central (Pulkovo) Astronomical Observatory, Russian Academy of Sciences, St. Petersburg, 196140 Russia 3 Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnii Arkhyz, 369167 Russia 4 Institute for Astrophysical Research, Boston University, Boston, MA 02215 USA 5 Institute of Astrophysics, Foundation Research and Technology-Hellas, Heraklion, GR-71110 Greece \n6 Department of Physics and Institute for Theoretical and Computational Physics, University of Crete, Heraklion, GR-70013 Greece \n7 Crimean Astrophysical Observatory, Russian Academy of Sciences, Nauchny, 298409 Russia \nObservations of quasars show that the polarization position angle of the emission coming from them varies greatly over time, including periods called rotations during which the angle changes in an orderly manner. The study proposes a method for identifying such events and assessing their statistical significance. The operation of the method is demonstrated using the example of long-term polarimetric observations of the blazars CTA 102, 3C 454.3, and OT 081. During the analysis of light curves, 51 rotations of the polarization position angle were found and it was shown that for CTA 102 and 3C 454.3 the rotations are predominantly oriented in one direction. \nKeywords: Methods: data analysis /emdash.cyr techniques: polarimetric /emdash.cyr quasars: general', 'I. INTRODUCTION': "Active galactic nuclei (AGN), constituting less than 7% (Roy, 1995) of the total number of galaxies in the Universe, have been studied with increasing interest for more than half a century. Studies of these objects, which began in the optical range, have now spread to all ranges available for observation: from radio to ultra-high energies. The properties of active galaxies are most clearly manifested in the subclass of blazars. The reason for the extreme properties of blazars is that their jet is oriented almost directly at the observer, and the jet's relativistically amplified emission dominates the entire wavelength range. A non-thermal spectrum, high variability in flux density, and high and variable polarization are the distinctive characteristics of blazars in the optical range. \nVariability of brightness and polarization can occur on long time scales of the order of weeks, months and years, and short scales of days or even within one day. For the first time, such ultra-fast (intra-day) polarization variability was discovered in 1972 by V. A. Hagen-Thorn for an extragalactic object, the blazar OJ 287, when within one hour a change in the degree of polarization by 2.5% and a change in the polarization position angle χ of about 10 · were observed (Hagen-Thorn, 1972). \nThe polarization of optical emission is explained by its synchrotron nature, and the direction of the electric vector position angle (EVPA) is perpendicular to the projection of the magnetic field onto the celestial plane. The specific values of the degree and EVPA depend on the structure of the magnetic field in the emitting region and the number of emitting regions along the observer's line of sight. Typically, flux density and polarization change in a chaotic manner, which is consistent with the random walk model (Kiehlmann et al., 2016; Marscher, 2014; Moore et al., 1982). \nHowever, in some cases, rotations of the EVPA are smooth, long-lasting, and have a large amplitude, which is most often observed during flare activity in a wide range of wavelengths. For the first time, the relationship between the EVPA rotations in the optical and radio ranges was discovered for the object OJ 287 in the work of Kikuchi et al. (1988). Later, in the work of Marscher et al. (2008) a similar behavior for one of the BL Lac blazar flares was observed using VLBI observations to show that the rotation is associated with the appearance of a new superluminal component passing through the jet core at millimeter wavelengths. Later, in a number of works (for example, Marscher et al. 2010) a similar behavior was discovered in other blazars during individual flares. However, not every rotation is accompanied by the appearance of a new component from the jet core at millimeter wavelengths (Jorstad and Marscher, 2016). \nCurrently, among the attempts made to search and analyze a large number of the EVPA rotations, the RoboPol project (Blinov and Pavlidou, 2019) stands out; this is an instrument and observation program that was designed to systematically study the optical polarization of blazars. As part of the program, regular observations of a selection of blazars were carried out from 2013 to 2017 and 40 rotations were found in 24 objects. \nSince the direction of the EVPA is related to the magnetic field, a detailed study of the variability of the angle will provide information about the fine structure of blazars jets. Rotations can be generated by both deterministic and stochastic processes. Deterministic processes are associated with ordered magnetic fields, for example, shock waves traveling along a jet (Marscher et al., 2008, 2010), jet curvature (Nalewajko, 2010), and a two-component model (Cohen and Savolainen, 2020). Stochastic processes are characterized by entangled magnetic fields and turbulent plasma motion (Kiehlmann et al., 2017; Marscher, 2014). Obtaining as large a sample of EVPA rotations as possible will clarify which rotations can be explained by deterministic processes and which are associated with chaotic changes. In addition, since many rotations are observed during flare activity in the gamma-ray range (Blinov et al., 2018; Marscher et al., 2010), their study will help provide a better understanding of the physical relationship between optical synchrotron and high-energy emission, and determine the structure of the magnetic field in the emitting areas. \nMulti-wavelength data analysis during rotation periods became of great interest after the launch of the IXPE (Imaging X-ray Polarimetry Explorer) instrument and the first measurements of X-ray polarization of blazars (Di Gesu et al., 2022). For example, Di Gesu et al. (2023) found a very rapid rotation of the polarization angle in the X-ray range for Mrk 421 (about 85 · per day for five days), while the optical EVPA remained constant. Such comparisons provide very important information about the location of emitting regions and the structure of the magnetic field (Di Gesu et al., 2023). The possible connection of astrophysical neutrinos with blazars is currently being actively discussed. At least some of the rotations associated with the repeating structure of gamma-ray bursts (Blinov et al., 2021) may in turn be associated with neutrinos (Novikova et al., 2023), so creating a large sample of rotations in the future is important when searching for correlations with high-energy neutrino detection events. \nThe study of rotations is a challenging task. First, rotations are relatively rare events, therefore, a long series of observations is necessary. Secondly, the measured angle contains ambiguity ± πn , the resolution of which imposes even more stringent requirements on the density of the series. Thus, the task of purposefully searching for rotations and identifying them in the light curve is difficult, primarily from an observational point of view. \nThis work proposes a new method for detecting polarization angle rotations and assessing their reliability, taking into account the experience of similar studies in previous works (for example, Blinov and Pavlidou (2019); Blinov et al. (2015, 2016a,b); Kiehlmann et al. (2016), etc.). \nThe structure of the article is as follows. In Section II we describe the acquisition of observational data. Section III describes the method for finding rotations. The results of applying this method to observations of three blazars are given in Section IV. Section V contains conclusions.", 'II. OPTICAL OBSERVATIONS': "The data used in this work was obtained by the authors as part of a monitoring program for a sample of bright gamma-ray blazars, carried out at Saint Petersburg State University 1 . Optical photometric and polarimetric data were obtained in the R band on the following telescopes: LX200 (40 cm, Saint Petersburg State University, Peterhof), AZT-8 (70 cm, Crimean Astrophysical Observatory, Nauchny), Perkins (1.83 m, Lowell Observatory, Flagstaff, Arizona, USA). The telescopes LX-200 (CCD camera FLI MicroLine ML4710) and AZT-8 (CCD camera SBIG ST-7) are equipped with almost identical polarimeters from Saint Petersburg State University. Polarimetric observations were carried out using two Savart plates rotated against each other by 45 · . The relative Stokes parameters q and u can be obtained from two separate images of each source in the field by observing from each plate in turn. The Perkins telescope is equipped with a PRISM 2 instrument with a CCD camera and a rotating half-wave plate polarimeter. To determine the polarization, four measurements are made at position angles of 0 · , 45 · , 90 · and 135 · . \nThe polarization measurements were carried out in the R filter on the AZT-8 and Perkins telescopes; on the LX-200 telescope, the measurements were carried out in 'white light' (without a filter) with a central wavelength λ eff = 670 nm; starting from the fall of 2018 the R filter was used. Instrumental polarization was determined from stars located near the object, under the assumption that their emission was unpolarized. As a rule, errors do not exceed 1% for the degree of polarization and 10 · for the EVPA for objects with a stellar magnitude of about 17 m . \nDetails of data acquisition and processing for LX-200 and AZT-8 are given in Larionov et al. (2008), and for the Perkins telescope in the paper by Jorstad et al. (2010). \nThe data for three objects, 3C 454.3 (2005/emdash.cyr2021), CTA 102 (2005/emdash.cyr2022) and OT 081 (2009/emdash.cyr 2021), was used in this work. Figure 1 shows the behavior of the degree and angle of polarization as a function of time for the objects mentioned above.", 'III. METHOD FOR ISOLATING ROTATIONS OF POLARIZATION POSITION ANGLE': "In this section, we will describe a technique for searching for significant rotations of the polarization position angle in the observational data we obtained. Generally, such events occur suddenly and are unevenly distributed in the curve of the EVPA. The presence of such events in short observation sessions is often determined by eye, by the presence or absence of large-scale trends in the curve of EVPA after which a section of the curve containing rotation is isolated and the angle its are determined. A systematic search program for rotations based on long-term observations must operate with a more stringent criterion that allows such a search to be performed uniformly across the entire light curve to obtain the most complete sample of rotations possible. \nFor example, in the work of Blinov et al. (2015) the criterion that a section of the EVPA curve contains a rotation is a monotonic and significant (exceeding measurement errors) change in the EVPA consistently in at least four observations in a row and with a total amplitude of more than 90 · . This approach made it possible to detect 14 rotations in the light curves of 12 blazars obtained in the observing season of 2013. A similar approach is used in a number of other works (for example, Blinov et al., 2016a,b; Liodakis et al., 2017). The undoubted advantage of this method of searching for rotations is its simplicity and extreme transparency of the results, however, it is not without a very significant drawback: the strict requirement of monotonic changes in the EVPA leads to the fact that individual points deviating from monotonic behavior either break one long rotation into several separate episodes, or rotation is not detected at all. Such points deviating from \n4 \n3C 454.3 \nFigure 1. The observational data used in the work are presented from top to bottom: the dependence of the degree and angle of polarization on time for 3C 454.3 (a), CTA 102 (b) and OT 081 (c), respectively. Due to the large range of EVPAs, the error bars for them are smaller than the icon size. \n<!-- image --> \nMJD \nmonotonicity can be both a manifestation of measurement noise and a consequence of the presence of several sources of polarized optical emission in the active nucleus: even if one source leads to a long-term and smooth rotation of the polarization vector, short-term bursts of other sources with different polarization parameters can lead to the fact that the observed total Stokes parameters exhibit complex behavior and the polarization angle during such flares deviates from monotonic behavior while maintaining the general trend. The authors of the method note this problem and \nmanually combine individual parts of large rotations, broken up by periods of non-monotonicity (for example, Fig. 2 in Blinov et al., 2015). \nIn this work, we propose an alternative approach to the criterion for identifying the rotation periods of the observed polarization vector. The new criterion should have the following properties: \n- 1. Allow individual episodes of non-monotonicity to exist within the rotation to solve the problem of breaking the rotation into separate parts.\n- 2. Not be tied to the full range of a rotation. Observations show that relatively long but slow rotations can exist so that the total amplitude is low. Imposing a strict limit on the rotation amplitude can lead to the loss of a certain proportion of such events. Moreover, it introduces a subjective parameter into the measurement process: a restriction on the minimum value of the rotation amplitude.\n- 3. It should produce a certain value characterizing the reliability of rotation detection, that is, allowing one to estimate the probability that such a rotation occurred randomly. Even in the absence of a source producing emission with a truly monotonically rotating polarization vector, the variable emission of individual jet cells turbulently varying in time can randomly add up in such a way that the total polarization exhibits a monotonically rotating angle over a fairly long period of time (Kiehlmann et al., 2016; Marscher, 2014). In long-term observational data, such random rotations can be present in considerable quantities, and there must be a criterion that allows one to cut off insignificant rotations. \nBefore proceeding directly to the description of the criteria, the need for general preprocessing of the EVPA curve should be noted. First, the ambiguity of the position angle χ must be resolved: since its value is determined with an ambiguity of ± πn , the rotations will inevitably contain discontinuities when the angle passes through 0/180 degrees. In practice, the standard approach to solving this problem (for example, Abdo et al., 2010; Blinov et al., 2015; Ikejiri et al., 2011) is the assumption of smooth behavior of the EVPA. In this case, if two neighboring points differ by more than 90 · , then the value ± πn is added to the second one, where n is selected in such a way as to minimize the difference. This approach makes it possible to reconstruct long-term rotations with an amplitude of hundreds of degrees, but can only be applied if there is a sufficiently dense observational series. \nIf there is a large gap in the observations, then it can no longer be assumed that there is a correlation between the values at two neighboring points. In this work, we use the approach described above to resolve the ambiguity of the EVPA; however, with large gaps in observations, it cannot be guaranteed that neighboring points are connected, and the EVPA must be counted from zero, and the possible rotation is broken into separate events. The specific period of time after which EVPAs cannot be considered related depends on the object and its local behavior. The upper limit of this interval can be obtained from the autocorrelation function of the EVPA (it cannot be larger than the interval over which a high correlation between the χ values remains). Figure 2 shows the autocorrelation function calculated using the LDCF method (Welsh, 1999) for the three sources studied in this work. The values of χ are strongly correlated even over a fairly long period of time, but it is obvious that the main contribution here comes from states without rotations. In this case, the limitation on the maximum gap in observations will be given by the average rate of an EVPA rotation: if during the observational gap the angle of polarization has time to rotate by more than 180 · , then it will be impossible to resolve the ambiguity ± πn . The characteristic values of the rotation rate of the polarization vector published in the literature are about ten or slightly more degrees per day (Blinov et al., 2015; Kiehlmann et al., 2016). Therefore, if there is a gap in observations of about 10-15 days, a complete uncertainty in the angle value will arise. Faster rotations will require denser observations (Kiehlmann et al., 2021). \nAnother important point is the smoothing of observational data. The relative accuracy of de- \nFigure 2. Autocorrelation function for the EVPA of objects 3C 454.3, CTA 102 and OT 081. \n<!-- image --> \ntermining polarization parameters is lower than the accuracy of photometric measurements. As a result, the measurement error of the EVPA can significantly exceed the measured change of this value between adjacent observations. In this case, only significant changes in χ are important for searching and analyzing rotations. Recently, a data smoothing algorithm based on Bayesian blocks has become widespread (Scargle et al., 2013). The main idea of the method is to replace the observed time series with a piecewise constant function; the time series is divided into non-overlapping intervals (blocks), in which the dependent variable (the position angle χ ) is described by a constant value. That is, the partition into blocks is performed in such a way that the measured value within the blocks does not change strongly enough to recognize these changes as significant. Individual blocks represent periods between which the value changed significantly. Among all possible partitions, one that minimizes the discrepancy between real observations and the approximating piecewise constant function is searched for. The best result will be obtained by a partition in which each observation point is a separate block, but such a partition is meaningless since it does not smooth the data. To solve this problem, an a priori number of blocks that is less than the number of observation points is set; as a result, neighboring points with close χ values are forced to combine into a single block. The number of partition blocks established a priori is selected from the following considerations: in the absence of a signal, that is, when the EVPA does not change, and the observed changes arise only due to errors, the probability of identifying a random change in a separate false block should be less than 0.05. Note that the chosen a priori number of partition blocks is not the final number of blocks into which the time series will be split: it is the expected value of the number of blocks before the splitting begins; in the case of a highly variable signal, the final number of blocks will be larger, and in the absence of variability, all points can be combined into one block (this is the Bayesian part of the method: as in any Bayesian modeling, an a priori model is first specified, which changes during the modeling process according to the available observational data). A direct algorithm for finding the optimal partition is presented in the article by Scargle et al. (2013). After the partition is completed, the points that fall into the common block are averaged by calculating the weighted average (that is, the errors of individual measurements are taken into account). The advantage of this approach in comparison with other smoothing methods, for example, running average, is the absence of 'smearing' of sharp changes in value: sharp changes are always divided into a separate block, which is important in the context of our task. \nIn the rest of our work, we use the approach described above to smooth the observed EVPA curves. All calculations are performed with smoothed curves, however, for illustrative purposes, \nwe plot both the smoothed curves (in the form of piecewise constant curves) and the original observations (in the form of points) in the figures containing the dependence of the EVPA on time.", 'A. Binomial Test': "As a basis for a criterion that satisfies the stated requirements, we propose to use a one-sided binomial test. In the absence of an ordered direction of rotation of the polarization vector χ (i.e. when χ experiences only chaotic changes), both directions of change of the angle between two adjacent measurements are equally probable. In any part of the curve with such a chaotic change in the polarization angle, the number of clockwise changes in the angle should be balanced by the number of counterclockwise changes. If the polarization vector has a dominant direction of rotation, then the number of angle changes in this direction will prevail. \nNaturally, measurement errors, randomness, as well as stochastic processes in the jet lead to the fact that in the first case (no ordered rotation) the number of changes of χ in both directions will not be strictly equal whereas in the second case (with rotation) not all the changes will be one-sided. The binomial test will help to distinguish these cases. Let's assume that N obs changes of the value of χ are detected during the observation process. Of these, N cw occurred clockwise and N ccw counterclockwise ( N obs = N cw + N ccw ). Also, let's assume that N ccw > N cw . The probability that such an imbalance could occur by chance is determined through binomial coefficients by the following equation: \np binom = 0 . 5 N obs N obs ∑ i = N ccw ( N obs i ) . (1) \nThe null hypothesis in this case is the assumption that both directions of change of the angle χ are equally probable (there is no ordered rotation). If the resulting value of p is not close to zero (not lower than a certain significance level chosen in advance), then the null hypothesis cannot be rejected and, thus, the presence of a rotation cannot be confirmed. If the value of p turns out to be below a given significance level, then in this area there is likely to be an ordered direction of the EVPA rotation.", 'B. T-Test': "A weakness of the proposed criterion based on the binomial test is its abstraction from the rate of change of the EVPA and from how the average rotation rate relates to the variation of rates between individual measurements. To solve this problem, we propose a second test, based on the requirement of a significant average rotation rate. \nLet us assume there are N changes in the EVPA between neighboring observations. If in a given section of the light curve variations of χ are produced only by stochastic processes, without stable rotation, then the average rotation rate will be close to zero. Otherwise, if in addition to stochastic changes, there is also a stable rotation, then the rate averaged over individual measurements will differ from zero, revealing a constant component in the change of the EVPA. Statistical tests (Shapiro - Wilk and Q - Q) showed that in certain sections of the EVPA curve, the distribution of the angle variation rates is close to normal, so checking for the significance of the difference of the average rates from zero can be performed using Student's T-test. \nLet us denote by χ i and χ j the EVPA values measured at times t i and t j , respectively. Then the EVPA variation rate in this area can be taken as follows: \nr ij = χ i -χ j t i -t j . \nIf the measurement errors of the polarization angle σ χ i and σ χ j are known, then the rate error will be: \nσ r ij = √ σ 2 χ i + σ 2 χ j t i -t j . (2) \nIn this case, the weighted average EVPA variation rate in a certain area will be equal to: \n¯ r = ∑ N k =1 r k σ 2 r k ∑ N k =1 1 σ 2 r k , (3) \nσ r = √ √ √ √ √ √ √ √ ∑ N k =1 ( r k -¯ r ) 2 σ 2 r k N -1 N ∑ N k =1 1 σ 2 r k . (4) \nand its standard deviation: \nHaving these values, we can calculate the Student's t -statistic to test the null hypothesis that the average EVPA variation rate is equal to zero: \nt = ¯ r σ r / √ N , (5) \nfrom where the p -value is calculated in a standard way using the Student distribution for a given number of degrees of freedom ν = N -1 : \np t -test ( t ) = Γ ( ν +1 2 ) √ νπ Γ ( ν 2 ) ( 1 + t 2 ν ) -ν +1 2 . (6) \nValues of p close to zero indicate a low probability that the stochastic process will generate such a consistent change in the EVPA and lead to the emergence of an average rate significantly exceeding the observed scatter. Otherwise, the average rate of the EVPA in this section does not differ significantly from zero, and it cannot be said that a rotation takes place.", 'C. Discussion of Criteria': 'Note that the described criteria meet all the requirements mentioned above: \nS4 1749+70 \n<!-- image --> \nFigure 3. Examples of rotations identified using the proposed criteria (according to Robopol, Blinov et al., 2020). Circles with error bars are observational data, lines are data smoothed using Bayesian blocks. Filled circles and a dark continuous line show the area of the selected rotation, open circles and a dotted line show areas outside the rotations. The numbers are the p -values of rotations determined by the binomial test ( p binom ) and the T-test ( p t -test ). For comparison, crosses indicate points assigned to rotations based on criterion of the strict monotonicity of changes of the EVPA in the work of Blinov et al. (2016a). \n<!-- image --> \n- 1) there is no requirement for a strict monotonic change in the polarization angle, as long as the predominance of some specific direction over the noise component is observed;\n- 2) the full amplitude of a rotation and the average rate do not play any role; if the accuracy of the measurements allows, then with a sfficient number of observations, arbitrarily small and a slow rotation can be detected;\n- 3) there is a numerical characteristic of a rotation reliability. \nAnother important point is that the number of observation points inside the rotation naturally affects its significance, determined by both criteria. Since the direction of χ can exhibit a random walk, short periods when EVPA rotates monotonically several times in a row can occur spontaneously, without the presence of mechanisms generating a true rotation. The criterion used in previous works, which requires four or more consecutive one-sided changes in the EVPA to detect a rotation, does not possess this property; as a result, for short segments of the observed curve the probability of false positives increases, since a randomly changing EVPA with a probability of 0 . 0625 will rotate four times in a row in one direction (the restriction on the minimum rotation amplitude used in these works only partly solves this problem, since at a low degree of polarization, random walks of the EVPA can be very large (Larionov et al., 2016)). \nA comparison of the results of identifying rotations using the proposed criteria with a method based on searching for a strictly monotonic EVPA variation is presented in Fig. 3 via two objects: S41749+70 (left) and 3C 454.3 (right) according to Robopol data (Blinov et al., 2020). Since in that work a minimum of four one-sided χ changes in a row was required to detect rotations, which corresponds to a p -value of 0.0625 in the binomial test, we used this limit as the signficance level of our tests to detect rotations. The crosses mark the points that were included in the rotations found using the criterion of monotonicity of χ variation in the work of Blinov et al. (2016a). Also, since in that work, there was no limit on the maximum gap in observations that would break the rotation, we also did not impose any restrictions for a correct comparison. Our proposed criteria make it possible to identify longer rotations by including new points, separated from the rest by a short period of non-monotonicity. For example, in the case of object S4 1749+70 (Fig. 3a), only a \nFigure 4 shows examples of reliable rotations with low amplitude (less than 90 · ) found in observations of objects 3C 454.3 (Fig. 4a) and CTA 102 (Fig. 4b) in our observational data. Although in both cases the EVPA varies monotonically over a significant number of observations, the lower bound on the rotation amplitude used in past work to guard against false positives would not allow these rotations to be detected. The calculated p -values of both criteria (see Fig. 4) show that these rotations are statistically significant. \n<!-- image --> \nCTA 102 \n5 \n3 \n. \nMJD \n+5 \n. \n769 \n5 \n4 \n× \n. \n0 \n10 \nFigure 4. Examples of detected rotations with low (less than 90 · ) amplitude based on our observational data. The structure of the figure is similar to Fig. 3. \nsingle point (near MJD 58852) deviates from monotonicity, as a result all previous points are lost and are not included in the rotation. \nThe ability to detect low-amplitude rotations is especially important because the observed rotation amplitude may be small if there are multiple sources of polarized (and non-polarized) emission in the object. In this case, the observed Stokes vector is the sum of the vectors of individual sources, and the observed direction of the polarization vector will depend not only on the directions of the polarization vectors in these sources but also on their relative intensity and degree of polarization. Thus, if there is a bright constant source of polarized emission in an object and a relatively weak source demonstrating rotation of the angle χ , then the observer, instead of rotating the polarization vector, will observe its oscillations around the preferential direction corresponding to the direction of polarization in the constant source. Moreover, the lower the luminosity and degree of polarization of the variable source, the smaller the oscillation amplitude will be. Our proposed method makes it possible to isolate half-periods of such small oscillations if the measured EVPA error is much smaller than the amplitude of the oscillations; however, an additional check can be attained by the study of structures outlined by the Stokes vector on the Q -U plane (see Shablovinskaya and Afanasiev 2019; Uemura et al. 2016)).', 'D. Numerical Experiments': "To test the performance of the proposed criteria, we conducted two numerical experiments, the main goal of which was to demonstrate, on the one hand, the low probability of detecting rotations arising due to random walk of the polarization vector within the limits of measurement errors, and on the other hand, the high probability of detecting a true rotation in noisy data. \nFor the first experiment, we created artificial curves of the EVPA that do not contain rotations: \n] \ng \ne \nd \n[ \nχ \n140 \n130 \n120 \n110 \n100 \nbinom \np \nt \np \n1 \n- \n5 \n2 \n. \ntest \n= 0 \n= 0 \n. \n. \n016 \n036 \n0 \n2 \n. \n0 \n3 \n. \n. \n4 \nthe true value of χ is constant for them, but the measurements contain an error, so that the observed values of χ have a scatter. In practice, the EVPA error increases as the degree of polarization decreases, so that for weakly polarized objects it can be quite large, and random deviations of the measured χ values can line up in a consistent rotation. If the measurement errors of the EVPA are known, then our proposed method is resistant to such errors, since the calculation of p binom and p t -test is preceded by smoothing using Bayesian blocks: random changes in χ within the errors do not give a significant change between neighboring measurements, so such a walk will be averaged within a single block even if the errors are very large. \nHowever, in practice it may turn out that EVPA errors are underestimated (for example, due to some unaccounted factor). In this case, a random change in χ can be taken as significant and this measurement will be allocated as a separate block. If the EVPA measurement errors are greatly underestimated, this can lead to the identification of several such blocks, which with some probability can demonstrate a smooth variation of χ . We tested this possibility by introducing an additional factor f σ into the simulation, which determines how much the EVPA error is underestimated: \nσ χ, used = f σ σ χ, true , \nwhere σ χ, true , true is the true EVPA error utilized to create the artificial curves, and σ χ, used , used is the value used in searching for rotations. \nTherefore, the numerical experiment looks like this: we created 1000 random EVPA curves, each consisting of 1000 points with a constant value of χ , to which random errors distributed according to the normal law are added. For each curve, a search for rotations was performed, varying the value of f σ in the range from 0.1 to 1.0 (that is, in the extreme case, the measurement error of χ is underestimated by a factor of 10). It is important to note here that the absolute value of the added error does not matter since the probability that the measured χ will randomly change in the same direction several times in a row does not depend on the magnitude of the error. Different values of the added error can only change the average rate of such false rotation, but our proposed method is not sensitive to the rate. \nFigure 5 shows the average number of detected random rotations in an EVPA curve consisting of 1000 measurements, depending on how much the errors are underestimated. The figure shows that smoothing observations using Bayesian blocks protects against the detection of random walks of the EVPA within errors (even if the errors are underestimated by half, such false detections are practically excluded). If observational errors are greatly underestimated, then the number of false detections increases. \nThe second experiment aims to determine the ability to detect true rotations in the presence of large measurement errors. If the errors are large and the rotation rate is low, the systematic change in the EVPA may be less than the scatter of measurements and the rotation may not be detected. It is natural to expect that in our proposed approach this problem can be solved by increasing the number of observations since both proposed criteria are statistical. For example, a binomial test requires significantly more changes of χ in one direction than in the other. As the measurement error increases, the statistics will deteriorate, since random changes in the EVPA that are opposite to the dominant direction will occur. In any case, on average, there will be more changes in χ in the dominant direction, and with a sufficient number of observations, it will be possible to accumulate a significant signal. Similarly with the T-test: a sufficiently large number of measurements will allow one to determine that the average change of EVPA is significantly different from zero, even in the presence of errors that exceed the change of EVPA between successive observations. \nTo demonstrate this, we ran the following simulation. An EVPA curve containing a monotonic variation (i.e. a true rotation) is created; then normally distributed measurement errors are added to the curve. The first parameter of the simulation is the ratio of the average rotation rate to the magnitude of the measurement error (the smaller this ratio, the more difficult it is to detect the \nFigure 5. Average number of detected random rotations per thousand measurements as a function of underestimation of EVPA measurement error. \n<!-- image --> \nFigure 6. The probability of detecting a rotation depending on its duration (number of observations) and the ratio of its average rate to the average observation error. \n<!-- image --> \nrotation since changes in χ between individual observations begin to 'drown' in the measurement errors). The second parameter is the duration of rotation (that is, the number of observations assuming a uniform observational series). Next, a search for rotation occurs, and the results are averaged over 1000 implementations for each pair of simulation parameter values. Figure 6 shows the results of this experiment, demonstrating a limitation of our method: the short, slow rotation is the hardest to detect. For example, if the average daily change in the EVPA is approximately equal to the measurement error, even 14 daily observations in a row will detect such a rotation with only a 50% probability. To detect a slow rotation reliably, it must be long-lasting so that a significant change in EVPA occurs. If the daily variations of the EVPA are three time larger than the measurement errors, then six observations are almost guaranteed to detect such a rotation. \nFigure 7. All EVPA rotations detected for the object 3C 454.3. \n<!-- image -->", 'IV. RESULTS': "The criteria described above were applied to search for rotations in observations of three objects: 3C 454.3 ( z = 0 . 859 , Jackson and Browne, 1991), CTA 102 ( z = 1 . 037 , Schmidt, 1965) and OT081 ( z = 0 . 320 , Stickel et al., 1993). These objects, included in the monitoring program of Saint Petersburg State University, were selected based on the presence of long and dense series of observations: the objects are included in the subsample with the highest priority of observations and have periods of regular (almost daily) observations. \nA total of 51 rotations were identified: 17 rotations for 3C 454.3, 23 rotations for CTA 102, and 11 rotations for OT 081. Currently, this is the most complete sample of known rotations for these objects, obtained as a result of a systematic search. The values of the EVPA for all detected rotations are shown in Fig. 7(3C 454.3), Fig. 8 (CTA 102) and Fig. 9 (OT 081). The rotation parameters (dates, amplitudes, values of average rates) are given in Tables I, II and III. The sign of the amplitude determines the direction of rotation (positive is for counterclockwise rotation). For clarity, the amplitudes and average rotation rates are also shown in histograms in Figs. 10, 11 and \nTable I. Rotations parameters of the object 3C 454.3: date of rotation, ∆MJD is the rotation duration, A is the total rotation amplitude (in degrees), ¯ r is the average rotation rate (in degrees per day), p binom is the p -value by binomial test, and p t -test is the p -value from T-test \n× \nTable II. Rotations parameters of the object CTA 102, the columns are similar to Table I \n× \n12. \nThe average frequency of observation of rotations for these objects in the observer's system is 1.05 (3C 454.3), 1.31 (CTA 102) and 0.93 (OT 081) events per year. Time in the jet system is related to time in the observer system as follows: \n∆ T jet = ∆ T obs δ 1 + z , \nwhere δ is the jet Doppler factor, and z is the redshift. Using Doppler factor estimates from Weaver et al. (2022), the following average rotation frequencies in the jet system are calculated:0.068 \nTable III. Rotations parameters of the object OT 081, the columns are similar to Table I \n- \n- \n(3C454.3), 0.065 (CTA 102) and 0.047 (OT 081) events per year. \nWhen compared with data from other researchers using other instruments (for example, Blinov et al., 2015; Itoh et al., 2016), it can be noted that our rotations parameters are consistent with those previously published. Thus, the duration of the shortest rotations in the Robopol program is 5-7 days (Blinov et al., 2015), while the minimum amplitude is at least 90 · due to criteria limitations. The duration of some of the rotations we discovered exceeded 100 days, while, according to RoboPol, the maximum duration is 90 days (Blinov et al., 2016a), and according to the KANATA telescope, long-term rotations of more than 100 days are also visible (Itoh et al., 2016). Some models predict faster and shorter rotations (Hosking and Sironi, 2020), and such rotations are indeed detected (Ahnen et al., 2018), but this occurs sporadically when such an event coincides with a dense observation campaign. When systematically searching for rotations in long-term observational programs, such a density of observations cannot be achieved, and the upper limit of the observed rotation rate is determined by the average frequency of observations (Kiehlmann et al., 2021). \nThe significant number of rotations detected in this work allows us to carry out a statistical analysis. In particular, the histograms for objects 3C 454.3 and CTA 102 show a clear asymmetry in the distribution of rotation amplitudes relative to zero. For object 3C 454.3, 14 of the 17 detected rotations occur counterclockwise, and for CTA 102, 17 of the 23 detected rotations occur clockwise. The probabilities of such (or greater) asymmetry with a random distribution of rotation directions are 0.025 and 0.017, respectively. At the same time, although the object OT 081 has asymmetry in the direction of rotation (8 out of 11 rotations occur clockwise), its significance is lower: the probability of such or greater asymmetry is 0.113. \nThe presence of the EVPA rotations can be explained by the spiral structure of the magnetic field in the jet (for example, Marscher et al. (2008), model of a shock wave traveling along the jet). In this case, the observed dominant direction of rotations reflects the global structure of the magnetic field, which is related to the direction of rotation of the black hole or accretion disk (Semenov et al., 2004). \nIt is assumed that in the acceleration and collimation zone closer to the black hole, the magnetic field should be twisted into a tighter spiral (Vlahakis, 2006), and with distance, on parsec scales, the degree of twist decreases. Thus, the different rotation rates we observe in the same object may be an indication of the position of the emission region in the jet. On the other hand, observations show (Weaver et al., 2022) that for an individual object, the apparent velocity of the radio components in the jet can differ significantly. This will also be reflected in the rotation rate if the rotations are associated with the movement of superluminal components in the jet. In addition, geometric effects associated with different viewing angles of individual sections of the curved jet may have an influence, through a change in the Doppler factor (Raiteri et al., 2017). \nTable IV. Average rotation amplitudes in the dominant direction (2nd column) and the opposite direction (3rd column) \n± \n± \nThe existence of rotations occured in the opposite direction from the dominant may indicate the simultaneous action of the mechanism of random walk of the polarization vector as a result of turbulent movements in the jet. In this model, the emission from individual turbulent cells of the jet, in which the field is assumed to be uniform, adds up to the total emission. The observed polarization is determined by the sum of the Stokes parameters from individual cells (Marscher, 2014). If a stochastic mechanism takes place, then it should produce rotations in both directions with equal probability. Therefore, one cannot expect all rotations in the dominant direction to reflect the global structure of the jet's magnetic field, since the set of such rotations also contains stochastic rotations. However, one can expect differences in the statistical characteristics of rotations occured in different directions. For example, the probability of stochastic rotation decreases with increasing amplitude, since it requires long-term random alignment of the magnetic field in disconnected turbulent cells. For the objects studied in this work, the average amplitude of rotations in the dominant direction ( 〈 ∆ χ rot 〉 ) and in the opposite direction ( 〈 ∆ χ counter 〉 ) is given in Table 4. The table shows that for two out of three objects, the average amplitudes of rotations in the dominant direction significantly exceed the aver age amplitudes of rotations in the opposite direction, which may indicate that the contribution of random walks is insignificant. However, to confirm this, a study of a larger sample of objects is required.", 'V. CONCLUSIONS': 'In this work, a new approach to identifying rotations of the EVPA was proposed and implemented. The method is based on two statistical criteria that allow assessing the reliability of the found rotations, that is the likelihood of their random occurrence. Compared to previous work, the new method has greater flexibility, allowing us to find not only rotations with a strictly monotonic variation of the EVPA but also rotations in which the EVPA briefly deviates from monotonicity while maintaining the average direction of rotation. In addition, the method does not have a restriction on the minimum amplitude of rotation; the statistical significance of rotations is determined by the number and accuracy of observations. \nThe proposed method was tested in numerical experiments on artificial data and demonstrated robustness to observational errors, both in terms of detecting false rotations due to χ random walk within the errors, and in terms of detecting true noisy rotations. Application of the method to three blazars (3C 454.3, CTA 102, OT 081) made it possible to detect 51 events of significant EVPA rotations: this is the largest sample of such events currently published in the literature. In the future, we plan to apply the described method to a larger sample of galaxies with active nuclei to systematically study the rotations parameters and compare them to the behavior of objects in the optical and other regions of the spectrum', 'FUNDING': 'The study was supported by a grant from the Russian Science Foundation № 23-22-00121, https://rscf.ru/project/23-22-00121/ .', 'ACKNOWLEDGMENTS': 'The authors thank the anonymous reviewers for their comments, which helped to significantly improve this work.', 'CONFLICT OF INTEREST': "The authors of this work declare that they have no conflicts of interest. \n- A. A. Abdo, M. Ackermann, M. Ajello, et al., Nature 463 (7283), 919 (2010). DOI:10.1038/nature08841\n- M. L. Ahnen et al. (MAGIC Collab.), Astron. and Astrophys. 619 , id. A45 (2018). DOI:10.1051/00046361/201832677\n- D. Blinov, S. G. Jorstad, V. M. Larionov, et al., Monthly Notices Royal Astron. Soc. 505 (3), 4616 (2021). DOI:10.1093/mnras/stab1484\n- D. Blinov, S. 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Suppl. 98 , 393 (1993).\n- M. Uemura, R. Itoh, L. Xu, et al., Galaxies 4 (3), id. 23 (2016). DOI:10.3390/galaxies4030023\n- N. Vlahakis, ASP Conf. Ser. 350 , 169 (2006).\n- Z. R. Weaver, S. G. Jorstad, A. P. Marscher, et al., Astrophys. J. Suppl. 260 (1), id. 12 (2022). DOI:10.3847/1538-4365/ac589c\n- W. F. Welsh, Publ. Astron. Soc. Pacific 111 (765), 1347 (1999).\n- V. A. Hagen-Thorn, Astronomicheskii Tsirkulyar 714 , 5 (1972). \nFigure 8. All EVPA rotations detected for the object CTA 102. \n<!-- image --> \nFigure 9. All EVPA rotations detected for the object OT 081. \n<!-- image --> \nFigure 10. Distribution of (a) amplitudes and (b) rotation rates of the EVPA for object 3C 454.3. \n<!-- image --> \n\| \n\| \nFigure 12. Distribution of (a) amplitudes and (b) rotation rates of the EVPA for object OT 081. \n<!-- image --> \n\| \n\| \nFigure 11. Distribution of (a) amplitudes and (b) rotation rates of the EVPA for object CTA 102. \n<!-- image --> \n\| \n\|"}
2024arXiv240907274R	Energetic particles in the form of stellar energetic particles and cosmic rays can lead to disequilibrium chemical effects in exoplanetary atmospheres. In Earthlike atmospheres energetic particles can drive the formation of prebiotic molecules the building blocks of life. Here instead I study the transport of energetic particles through a hydrogendominated exoplanet atmosphere and calculate the resulting ionisation rate of molecular hydrogen using a Monte Carlo energetic particle transport model. I focus on a GJ436 blike atmosphere at orbital distances between 0.010.2 au which includes the orbital distance of the exoplanet GJ436 b 0.028 au. I found that stellar energetic particles lead to high ionisation rates in a GJ436 blike atmosphere between 0.010.2 au. These results motivate the use of chemical models of gas giant atmospheres including energetic particle ionisation to ultimately produce synthetic James Webb Space Telescope JWST and Ariel transmission spectra in the future.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.07274', '2024arXiv240907274R', 'arXiv:2409.07274']	['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics']	Stellar energetic particle and cosmic ray effects in exoplanetary atmospheres	2,024	193	0.41	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.07274.pdf	{'No Header': 'Solar and Stellar Coronal Mass Ejections Proceedings of the IAU Symposium No. 388, 2024 N. Gopalswamy, O. Malandraki, A. Vidotto & W. Manchester, eds. doi:10.1017/xxxxx', 'D. Rodgers-Lee': 'Dublin Institute for Advanced Studies \nAbstract. Energetic particles, in the form of stellar energetic particles and cosmic rays, can lead to disequilibrium chemical effects in exoplanetary atmospheres. In Earth-like atmospheres, energetic particles can drive the formation of prebiotic molecules, the building blocks of life. Here instead, I study the transport of energetic particles through a hydrogen-dominated exoplanet atmosphere and calculate the resulting ionisation rate of molecular hydrogen using a Monte Carlo energetic particle transport model. I focus on a GJ436 b-like atmosphere at orbital distances between 0.01-0.2 au which includes the orbital distance of the exoplanet GJ436 b (0.028 au). I found that stellar energetic particles lead to high ionisation rates in a GJ436 b-like atmosphere between 0.01-0.2 au. These results motivate the use of chemical models of gas giant atmospheres including energetic particle ionisation to ultimately produce synthetic James Webb Space Telescope (JWST) and Ariel transmission spectra in the future. \nKeywords. Stellar energetic particles, Galactic cosmic rays, exoplanet atmospheres', '1. Introduction': "There are two types of energetic particles that are important for exoplanet atmospheres. These are stellar energetic particles from the coronal mass ejections and flares of low-mass stars like the Sun and Galactic cosmic rays that enter an astrosphere† from the interstellar medium (ISM). These two types of energetic particles then reach the location of an exoplanet. Here, they encounter any existing exoplanetary magnetosphere and interact with the atmosphere of the exoplanet. \nEnergetic particles are of interest in the context of exoplanet atmospheres because they can cause a number of interesting effects. They can drive the formation of prebiotic molecules in Earth-like atmospheres (Airapetian et al. 2016; Dong et al. 2019) which are the molecules thought to be important for the origin of life on Earth. On the other hand, they can also cause fake biosignatures in Earth-like atmospheres (Grenfell et al. 2012). A biosignature is a chemical signature in an exoplanet atmosphere thought to be indicative of life. Thus, since energetic particles can also contribute to fake biosignatures it is important to constrain their effect so that it can be removed when searching for real biosignatures. Last, to isolate the chemical effect of energetic particles, Helling & Rimmer (2019) and Barth et al. (2021) found that energetic particles can lead to the formation of exotic molecules in hydrogen-dominated atmospheres that are not expected to form otherwise which are known as fingerprint ions. \nMore broadly, studying the effect of energetic particles is important because high fluxes of energetic particles affect life-forms by damaging DNA (Herbst et al. 2019; Atri 2020). It has even been suggested that energetic particles (in this case Galactic cosmic rays but the same logic applies to stellar energetic particles) indirectly left an imprint on the helicity of DNA (Globus & Blandford 2020). \n† An astrosphere is the stellar equivalent of the Sun's heliosphere.", '2. Methodology': 'Here, I will focus on the impact of energetic particles in the GJ436 system. GJ436 is a wellstudied M dwarf star with a known close-in planet, GJ436 b orbiting at 0.028au (Butler et al. 2004). GJ436 b is a warm mini-Neptune (with a calculated effective temperature of T = 880K, Rodgers-Lee et al. 2023) and is an interesting exoplanet because it has a cometary-like outflow detected via a Lyα transit with the Hubble Space Telescope (Ehrenreich et al. 2015). While these hydrogen-dominated gas giant atmospheres are less likely to be hospitable to any life-as-we-know-it in comparison to an Earth-like atmosphere, they do have an advantage: gas giant atmospheres constitute the majority of JWST and Ariel exoplanet targets. Their low mean molecular weight results in a large scale height, making them easier to observe with transmission spectroscopy than an Earth-like atmosphere with a higher mean molecular weight. \nIn order to model the transport of energetic particles in the GJ436 system a stellar wind model and a model for the exoplanetary atmosphere are required. In Rodgers-Lee et al. (2023) we use a stellar wind model from Mesquita et al. (2020). The stellar wind properties necessary to model Galactic cosmic ray transport from the ISM into an astrosphere are the stellar wind velocity, magnetic field strength and mass loss rate. The level of turbulence present in the magnetic field, rather than the field strength itself, dictates the energetic particle transport through the magnetised stellar wind. The turbulence properties for winds other than the solar wind are unknown and we simply adopt similar properties as for the solar wind. See RodgersLee et al. (2023) for more details.', '2.1. Energetic particle spectra at the top of the exoplanet atmosphere': "The spectrum for Galactic cosmic rays in the local ISM (known as the Local Interstellar Spectrum), outside of the solar system, has been measured by Voyager 1 and 2 (e.g. Cummings et al. 2016). The Galactic cosmic ray spectrum is then suppressed by the solar wind, mainly at cosmic ray energies below ∼ 30 GeV so that a somewhat reduced spectrum is detected at Earth with instruments such as PAMELA (e.g. Potgieter & Vos 2017). Thus, we calibrated our 1D energetic particle transport model (Rodgers-Lee et al. 2020) using the Voyager and PAMELA detections. The LIS is used as the input spectrum outside the astrosphere of another stellar system. The energetic particle transport model from Rodgers-Lee et al. (2020), based on the diffusive-advection transport equation from Parker (1965), is used in combination with the stellar wind properties (Case 'A' presented in Mesquita et al. 2020) to calculate the Galactic cosmic ray fluxes at different orbital distances within the GJ436 b system. These fluxes at the different orbital distances are the values we adopt as the values at the top of the hypothetical exoplanet atmospheres. \nThe Galactic cosmic ray fluxes are suppressed as they travel through the magnetised stellar wind because they are charged particles. The solar wind, and therefore we assume stellar winds from other low-mass stars, are turbulent. This means that Galactic cosmic rays diffuse into the solar, or stellar, system as they are deflected and bounce off perturbations in the magnetic field rather than travelling ballistically. The stellar wind is expanding, relating to its velocity, which advects the Galactic cosmic rays out of the solar system. The Galactic cosmic rays suffer adiabatic losses due to the divergence of the stellar wind's velocity field, which also can be thought of as advection in momentum space. The overall level of suppression that the Galactic cosmic rays experience in any given stellar system is a balance between these diffusive and advective processes. \nFor GJ436b, the resulting Galactic cosmic ray differential intensities at orbital distances between 0.01 - 0.2 au are shown in the left panel of Fig. 1. The Galactic cosmic ray fluxes increase with increasing orbital distance. The differential intensity at the orbital distance of GJ436b, shown by the black line, is lower than the differential intensities observed at Earth. \nFigure 1. Differential intensities of (left) Galactic cosmic rays and (right) stellar energetic particles as a function of cosmic ray kinetic energy at orbital distances between 0.01 - 0.2 au for the GJ436 system (from Rodgers-Lee et al. 2023). \n<!-- image --> \nThe right panel of Fig. 1 shows the stellar energetic particle differential intensities for the same orbital distances. Again, the black line represents the differential intensities at the orbital distance of GJ436 b. It is important to note that the scale on the left and right panels is significantly different. An interesting feature of these spectra is that the Galactic cosmic ray spectra peak at ∼ GeV energies. In comparison for the stellar energetic particles, the intensities are much higher generally but they begin to decrease rapidly at ∼ Gev energies. The stellar energetic particle spectra are produced based on 3 assumptions. The first is that the acceleration mechanism produces a power-law index of -2 for the low-energy part of the spectrum (indicative of diffusive shock acceleration). There is a maximum momentum that the star can accelerate stellar energetic particles to, beyond this momentum the differential intensities decrease exponentially. The final assumption is that the stellar energetic particles are constant in time. \nThe exoplanet atmosphere properties (e.g. T , p ) in Rodgers-Lee et al. (2023) were obtained using the HELIOS model (Malik et al. 2017, 2019). The temperature-pressure profiles are shown in Fig. 3 of Rodgers-Lee et al. (2023). The density is the most important atmospheric property to model energetic particle transport in exoplanet atmospheres which was calculated using the ideal gas law. In Rodgers-Lee et al. (2023), for the energetic particle transport we assume a hydrogen-dominated atmosphere.", '3. Analysis and results': 'Here, I show our results for the penetration of cosmic rays in a GJ436 b-like atmosphere. Each of the spectra shown in Fig. 1 is propagated down through an exoplanet atmosphere using a Monte Carlo cosmic ray transport code (Rimmer & Helling 2013). For example, the solid black line in Fig. 2 shows the initial Galactic cosmic ray spectrum at the top of the atmosphere for a = 0 . 2au. The dashed coloured lines show the subsequent spectra for increasing pressures in the atmosphere as the Galactic cosmic rays lose energy by interacting with the atmosphere. The resulting spectra for the Galactic cosmic rays and the stellar energetic particles is then used to calculate the ionisation rate of molecular hydrogen at each height in the atmosphere. \nFigure 2. Differential intensities of Galactic cosmic rays as a function of cosmic ray kinetic energy at different heights in an exoplanet atmosphere for a GJ436 b-like exoplanet at a = 0 . 2 au in the GJ436 system (from Rodgers-Lee et al. 2023). \n<!-- image -->', '3.1. Ionisation rate of molecular hydrogen': 'The ionisation rate of molecular hydrogen, ζ , due to energetic particles gives an indication of how important, and where, they might be at driving chemistry in the exoplanet atmosphere. The H2 ionisation rate is calculated from the differential intensities and the H2 ionisation crosssection (Eq.4 from Rodgers-Lee et al. 2023). \nFig. 3 shows the ionisation rate of molecular hydrogen, ζ , as a function of pressure for a GJ436b-like planet for a = 0 . 01 -0 . 2 au (adapted from Rodgers-Lee et al. 2023). The left and right panels plot ζ as a result of ionisation due to Galactic cosmic rays and stellar energetic particles, respectively. The black solid line represents ζ in an exoplanet atmosphere at 0.028 au, the orbital distance of GJ436 b. The dotted grey line denotes P = 1 bar for reference. For comparison, the LIS results in ζ ∼ 10 -17 s -1 . \nAdditionally, for both Galactic cosmic rays and stellar energetic particles, the value of ζ remains constant with increasing pressure up to a certain point. In this region of the atmosphere the energetic particles are not losing a significant amount of their energy. For Galactic cosmic rays, ζ begins to decrease between p ∼ 10 1 -10 -1 bar for a = 0 . 01 -0 . 2au, respectively. The stellar energetic particles are absorbed at much higher heights in the atmosphere, corresponding to p ∼ 10 -2 bar. Thus, stellar energetic particles dominate as a source of ionisation over Galactic cosmic rays in the upper atmosphere. At high pressures, p > ∼ 1bar, although overall the ionisation rate is low Galactic cosmic rays dominate instead. Transmission spectroscopy with JWST and Ariel will probe between 10 -4 -10 -1 bar in hydrogen dominated atmospheres (Welbanks & Madhusudhan 2019). Thus, transmission spectroscopy with JWST and Ariel is more likely to be sensitive to chemistry driven by stellar energetic particles for the atmospheres and orbital distances studied here. \nFigure 3. The ionisation rate of molecular hydrogen due to (left) Galactic cosmic rays and (right) stellar energetic particles as a function of atmospheric pressure for a GJ436 b-like planet at orbital distances between 0.01 - 0.2 au for the GJ436 system (adapted from Rodgers-Lee et al. 2023). \n<!-- image -->', '4. Discussion and Summary': "I presented our results of the ionising effect of stellar energetic particle and Galactic cosmic rays on a hydrogen-dominated exoplanet atmosphere for a = 0 . 01 -0 . 2au in the M dwarf system, GJ436. This includes the orbital distance of the known mini-Neptune, GJ436 b. I discussed how the stellar wind properties, along with the composition and density of an exoplanet atmosphere determine the energetic particle fluxes in the exoplanet atmosphere. Stellar energetic particles lead to high ionisation rates in a GJ436 b-like atmosphere at the orbital distances discussed here, namely between 0.01-0.2 au. The Galactic cosmic ray ionisation rates for a hypothetical exoplanet at these orbital distances were generally much lower but dominate over stellar energetic particles deep in the exoplanet atmosphere. \nThe results presented here did not include the effect of a planetary magnetic field. A planetary magnetic field would shield the atmosphere by deflecting both stellar energetic particles and Galactic cosmic rays, up to a certain energy, towards the poles (see e.g. Herbst et al. 2019). Investigating the effect of a planetary magnetic is of great interest. As mentioned above, the stellar energetic particle fluxes are assumed to be constant here. In reality, stellar flares and CMEs are unlikely to produce constant fluxes of associated stellar energetic particles that impact an exoplanet between a = 0 . 01 -0 . 2au for this system, given that it is not an active star. It is not clear what impact this would have on chemistry in the atmosphere at this point. Chemical modelling of an exoplanet atmosphere including energetic particle ionisation are needed to address this time dependent effect. \nUsing chemical models, Helling & Rimmer (2019) identified H + 3 and H3O + as ions indicative of Galactic cosmic ray ionisation in a brown dwarf/gas giant atmosphere. The important pathway for creating these ions is: \nH2 + CR → H + 2 + e -+ CR (1) \nH + 2 + H2 → H + 3 + H (2) \nH + 3 + H2O → H3O + + H2 (3) \nwhere 'CR' is short for cosmic ray. Stellar energetic particles will produce similar effects (e.g. Barth et al. 2021). Note, the presence of water is important. Scaling the abundances of H + 3 and H3O + from Helling & Rimmer (2019), Bourgalais et al. (2020) produced synthetic JWST and Ariel transmission spectra for a GJ1214 b-like atmosphere, showing that absorption features from H + 3 and H3O + in the near-infrared would be strong enough to be observed with Ariel and NIRSpec on JWST (see their Fig. 6). This remains to date the only synthetic JWST and Ariel transmission spectrum for a hydrogen-dominated atmosphere that includes H + 3 and H3O + , formed as a result of energetic particle ionisation. \nThe stellar energetic particle and Galactic cosmic ray spectra shown here can be used in the future with chemical and radiative transfer models to produce a suite of synthetic transmission spectra, building on the results presented in Bourgalais et al. (2020). This would identify the energetic particle ionisation rate needed to result in sufficient abundances of H + 3 and H3O + to be detected with transmission spectroscopy. In turn this would help us identify the types of exoplanets to target, in terms of orbital distance and stellar activity, to detect these features. Alongside this more direct connection with observations, more sophisticated 2D/3D energetic particle transport models can be employed in the future (see e.g. Engelbrecht et al. 2024).", '5. Acknowledgements': 'I would like to thank the Scientific Organising Committee for the invitation to take part in the IAU symposium 388. DRL would like to acknowledge that this publication has emanated from research conducted with the financial support of Science Foundation Ireland under Grant number 21/PATH-S/9339.', 'References': 'Airapetian V. S., Glocer A., Gronoff G., et al. 2016, Nature Geoscience, 9, 452 \nAtri D., 2020, Scientific Reports , 1011646 \nBarth P., Helling C. St¨ueken E. E., et al. 2021, MNRAS , 502, 6201 \nBourgalais J., et al., 2020, \nApJ \n, 895, 77 \nButler R. P., Vogt S. S., Marcy G. W., et al. 2004, ApJ , 617, 580 \nCummings A. C. et al. 2016, ApJ , 831, 18 \nDong, C. F., Lingam, M., Fang, X. H. et al. 2019, The First Billion Years: Habitability , Conference \nProceedings, 2134, 1046 \nEngelbrecht, N. Eugene, Herbst, K., Strauss, R. Du Toit et al. 2024, ApJ , 964, 89 \nEhrenreich, D., Bourrier, V., Wheatley, P. J. et al. 2015, Nature , 522, 459 Globus N., Blandford R. D., 2020, ApJ , 895, L11 \nGrenfell, J. L., Grießmeier, J.-M., von Paris, P., et al. 2012, Astrobiology , 12, 1109 \nHelling C., Rimmer P. B., 2019, Philosophical Transactions of the Royal Society of London Series A , 377, 20180398 \nHerbst, K., Grenfell, J. L., Sinnhuber, M. et al. 2019, A&A , 631, 101 \nMalik M., Grosheintz L., Mendonc¸a, J. M., et al. 2017, AJ , 153, 56 \nMalik M., Kitzmann D., Mendonc¸a J. M., et al. 2019, AJ , 157, 170 \nMesquita A. L., Rodgers-Lee D., Vidotto A. A., 2021, MNRAS , 505, 1817 \nParker E. N., 1965, Planetary and Space Science , 13, 9 \nPotgieter, M. S., Vos, E. E., A&A , 601, 23 \nRimmer P. 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2024arXiv240903822A	Semianalytic methods can generate baryoncorrected fields from Nbody simulations baryonification and are rapidly becoming a ubiquitous tool in modeling structure formation on nonlinear scales. We extend this formalism to consistently model the weak lensing and thermal SunyaevZeldovich tSZ fields directly on the fullsky with an emphasis on higherorder correlations. We use the auto and cross Nthorder moments with N in 2 3 4 as a summary statistic of the lensing and tSZ fields and show that our model can jointly fit these statistics measured in IllustrisTNG to within measurement uncertainties for scales above gtrsim 1 rm Mpc and across multiple redshifts. The model predictions change only minimally when including additional information from secondary halo properties such as halo concentration and ellipticity. Each individual moment is dependent on halos of different mass ranges and has different sensitivities to the model parameters. A simulationbased forecast on the ULAGAM simulation suite shows that the combination of all moments measured from current and upcoming lensing and tSZ surveys can jointly constrain cosmology and baryons to high precision. The lensing and tSZ field are sensitive to different combinations of the baryonification parameters with degeneracy directions that are often orthogonal and the combination of the two fields leads to significantly better constraints on both cosmology and astrophysics. Our pipeline for maplevel baryonification is publicly available at httpsgithub.comDhayaaAnbajaganeBaryonForge.	2024-09-01T00:00:00Z	['arXiv:2409.03822', '10.48550/arXiv.2409.03822', '2024arXiv240903822A']	['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies']	Maplevel baryonification Efficient modelling of higherorder correlations in the weak lensing and thermal SunyaevZeldovich fields	2,024	193	0.5	['EPRINT_HTML', 'EPRINT_PDF']	3	https://arxiv.org/pdf/2409.03822.pdf	{'No Header': 'Preprint typeset using L A T E X style openjournal v. 09/06/15', 'Map-level baryonification: Efficient modelling of higher-order correlations in the weak lensing and thermal Sunyaev-Zeldovich fields': 'Dhayaa Anbajagane ( ) 1 , 2 , Shivam Pandey 3 , 4 , and Chihway Chang 1 , 2 \n- 1 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA\n- 2 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA\n- 3 Department of Physics, Columbia University, 538 West 120th Street, New York, NY, USA 10027, USA and\n- 4 Columbia Astrophysics Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027, USA \nVersion December 11, 2024', 'ABSTRACT': "Semi-analytic methods can generate baryon-corrected fields from N-body simulations ('baryonification') and are rapidly becoming a ubiquitous tool in modeling structure formation on non-linear scales. We extend this formalism to consistently model the weak lensing and thermal Sunyaev-Zeldovich (tSZ) fields directly on the full-sky, with an emphasis on higher-order correlations. We use the auto- and cross𝑁 th-order moments, with 𝑁 ∈ { 2 , 3 , 4 } , as a summary statistic of the lensing and tSZ fields, and show that our model can jointly fit these statistics measured in I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG to within measurement uncertainties, for scales above ≳ 1 Mpc and across multiple redshifts. The model predictions change only minimally when including additional information from secondary halo properties, such as halo concentration and ellipticity. Each individual moment is dependent on halos of different mass ranges and has different sensitivities to the model parameters. A simulationbased forecast on the U/l.pc/a.pc/g.pc/a.pc/m.pc suite shows that the combination of all moments, measured from current and upcoming lensing and tSZ surveys, can jointly constrain cosmology and baryons to high precision. The lensing and tSZ field are sensitive to different combinations of the baryonification parameters, with degeneracy directions that are often orthogonal, and the combination of the two fields leads to significantly better constraints on both cosmology and astrophysics. Our pipeline for map-level baryonification is publicly available at https://github.com/DhayaaAnbajagane/BaryonForge .", '1 INTRODUCTION': "Measurements of weak lensing, which is a direct probe of the cosmic density field, have provided some of the best constraints on the properties of the Universe, both on the initial conditions and on their subsequent evolution (Abbott et al. 2022; Asgari et al. 2021; Secco et al. 2022a; Amon et al. 2022; More et al. 2023; Omori et al. 2023; Chang et al. 2023; Madhavacheril et al. 2024). These constraints are expected to significantly improve with the advent of surveys like the Dark Energy Spectroscopic Instrument DESI Collaboration et al. (2016), Vera C. Rubin Observatory (The LSST Dark Energy Science Collaboration et al. 2018), the Euclid mission (Racca et al. 2016), the Roman space telescope (Spergel et al. 2015) and more, all of which will provide vastly richer datasets and therefore enable measurements of the small-scale density field \nwith improved statistical precision. \nHowever, an ongoing limitation in using such small-scale measurements is modeling the impact of baryons ('baryonic imprints') on the density field. Baryons, which constitute ≈ 15% of the total matter in our Universe, with the rest being accounted for by dark matter (DM), experience a vast set of astrophysical process across a range of scales. Such processes alter the distribution and thermodynamics of these baryons, and also the DM phase space via the gravitational coupling between the two components ( e.g. , Gnedin et al. 2004; Abadi et al. 2010; Duffy et al. 2010; Anbajagane et al. 2022a; Shao & Anbajagane 2024). The signatures of these processes can be prevalent even on quasi-linear scales (roughly a few times a halo radius) and will be a significant component of any smallscale signal (i.e. the density near/within a halo). Their presence can bias any cosmological analyses that do not account for this effect ( e.g. , Secco et al. 2022a; Amon & Efstathiou \n2022; Gatti et al. 2022; Anbajagane et al. 2023). This limitation in scales is further accentuated by the fact that many novel cosmological signatures are imprinted onto these small scales. Thus, efficient models of these baryonic imprints have the potential to not only improve the constraining power for current models of interest, such as Λ CDM and 𝑤 CDM, but also open the opportunity to constrain extended models like those of primordial non-gaussianities (Coulton et al. 2022; Anbajagane et al. 2024b; Jung et al. 2023; Goldstein et al. 2024). \nThe most precise model of baryonic imprints comes from studying the formation of structure in various hydrodynamical simulations which model the evolution of gas, stars, black holes etc. in addition to the DM ( e.g. , McCarthy et al. 2017; Springel et al. 2018). The simulations have a finite resolution scale that is larger than many relevant astrophysical processes (such as supernovae explosions, or supermassive black hole accretion, to name a few) and must therefore implement these processes as 'sub-grid' physics that approximate their effects (see Vogelsberger et al. 2020, for a review). The subgrid models of each simulation vary in the equations that are solved, and in how the equations are parameterized. Studies of these simulations, however, have showcased the variety of predictions manifesting from these different, but often equally realistic, choices in the models of galaxy formation ( e.g. , Anbajagane et al. 2020; Lim et al. 2021; Lee et al. 2022; Cui et al. 2022; Stiskalek et al. 2022; Anbajagane et al. 2022a,b; Shao et al. 2023; Gebhardt et al. 2023). The type and amplitude of these differences vary significantly depending on the exact properties - such as the gas mass/temperature/pressure, the stellar mass/age/metallicity etc. - being studied and the regime in mass, redshift, environment etc. being explored. Thus, while the simulations are valuable in capturing the range of possible baryonic imprints, one cannot use them to robustly model these imprints in the density field. Furthermore, studies on the thermodynamic properties of gas also find differences between the measurements from data and the predictions from these hydrodynamic simulations ( e.g. , Hill et al. 2018; Amodeo et al. 2021; Pandey et al. 2022; Anbajagane et al. 2022c). \nGiven the current challenges in modelling baryons in simulations, it is advantageous to instead use a phenomenological model. Such a model does not require specifying any sub-grid physics and instead requires inputs that are observables, or are quantities directly obtained from observables, of the different baryonic matter components. Baryonification is a method through which N-body simulations - which evolve collisionless matter (dark matter) under gravity and accurately capture all non-linear evolution - are modified through phenomonological models such that the new density field mimics one that has baryonic imprints (Schneider & Teyssier 2015; Schneider et al. 2019). This method has been used extensively to model the matter power spectrum with baryonic imprints, and has \nbeen shown to have the flexibility needed to capture the behaviors ranging across many different simulations (Schneider et al. 2019; Giri & Schneider 2021; Aric'o et al. 2021b). It has already been used to analyse 2-point correlation functions measured in widefield surveys (Chen et al. 2023; Aric'o et al. 2023; Bigwood et al. 2024). \nThe fiducial baryonification method is performed by displacing the positions of particles in the N-body simulations to alter the overall density distribution in the volume. This procedure has been computationally tractable for building emulators that translate the model parameters to prediction a given statistic in real space, e.g. , the 3D power spectrum such as done in Giri & Schneider (2021); Aric'o et al. (2021b). These emulators can then be used in standard analysis pipelines to predict, for example, the weak lensing correlation functions ( e.g. , Aric'o et al. 2023). However, surveys are rapidly adopting simulation-based analyses as a way to accurately forward model systematic effects, and also to incorporate more information from the field ( e.g. , Gatti et al. 2023, 2024b; Jeffrey et al. 2024; Harnois-Deraps et al. 2024; Cheng et al. 2024). Some work has been done to show the baryonification model generates fields that accurately model baryonic imprints on higher-order statistics, such as the density field bispectrum (Aric'o et al. 2021a) and the peaks of the weak lensing field (Lee et al. 2023). These higher-order statistics extend upon the simpler, two-point statistics - which describe the correlation between two points in space, e.g. , the power spectrum - by also extracting correlations between three or more points ( e.g. , Fluri et al. 2019; Gatti et al. 2020; Zurcher et al. 2021; Fluri et al. 2022; Euclid Collaboration et al. 2023; Anbajagane et al. 2023; Gatti et al. 2024a; Jeffrey et al. 2024). \nThe use of baryonification in simulation-based analyses of surveys is limited by the computational expense of the method. Survey datasets frequently require lightcone maps covering a large sky fraction for their predictions (upcoming surveys will need ≈ 15 , 000 deg 2 coverage) and these maps are constructed from O( 10 2 ) 3D snapshots/boxes in a simulation ( e.g. , Kacprzak et al. 2023; Anbajagane et al. 2024b; Jeffrey et al. 2024). Applying baryonification on each box is an computationally expensive procedure - and also a memory intensive one, since it requires storing the particles of each snapshot - given analyses often require O( 10 3 -10 4 ) such simulations. An attractive alternative then is to perform baryonification directly on pixels of 2D maps. This has been explored by Fluri et al. (2019) for performing simulation-based inference of weak lensing data from the Kilo-Degree Survey (Kuijken et al. 2015). An aspect of our work focuses on updating this method, by robustly accounting for the pixel window functions and line-of-sight projections and then validating its performance on simulations. \nWe have so far only discussed the total density field, \nwhereas other probes - often more closely linked to the baryonic matter components - are actively used to constrain astrophysics and cosmology. Of these, one of the most commonly used and actively studied ones is the thermal SunyaevZeldovich (tSZ) field (Sunyaev & Zeldovich 1972), observed using millimeter surveys such as the South Pole Telescope (Carlstrom et al. 2011; Benson et al. 2014), the Atacama Cosmology Telescope (Fowler et al. 2007; Thornton et al. 2016), and the Planck satellite (Planck Collaboration et al. 2020). The tSZ effect is the inverse Compton scattering of CMB photons with energetic electrons along the line of sight. It probes the integrated gas pressure and therefore the thermodynamics of baryons in and around the halo (see Carlstrom et al. 2002; Mroczkowski et al. 2019, for reviews). As a result, it is advantageous to include the tSZ probe in analyses, both for its cosmological information - as it, like the total matter field, is sensitive to the initial conditions and their subsequent evolution - and even more so for its information on the gas thermodynamics. However, pursuing this direction requires a consistent, common model that predicts the impact of astrophysical processes on both the distribution of matter and the thermodynamics of baryons. Studies of baryonification have thus far been exclusively on the density field (Schneider et al. 2019; Giri & Schneider 2021; Aric'o et al. 2021b; Lee et al. 2023; Aric'o et al. 2023), with the inclusion of tSZ only recently being explored (Aric'o & Angulo 2024). /one.sup \nPandey et al. (2024, henceforth, P24) show that the halo profiles used in the existing baryonification model can be extended, with simple modifications, to also predict the gas pressure profiles (and gas number density profile) in addition to the total matter density profiles. To et al. (2024, see their Figure 1) have further demonstrated that this model can consistently predict the tSZ-halo mass scaling relation alongside the density two-point correlations. Thus, the original baryonification model of Schneider et al. (2019, henceforth, S19) can be built on to now also predict the tSZ observables (as well as other observables that trace the gas number density, such as the kinematic SZ effect or the X-ray flux). While P24 focus on using the halo model approach to predict two-point measurements of weak lensing and the tSZ, the model remains to be explored and validated at the map-level for different summary statistics that combine information from the lensing and tSZ fields. \nThis work explores exactly this aspect, by both building and validating a map-level baryonification model that can be used to predict higher-order correlations. We do this in three ways: (i) we first validate the map-level baryonification method in modelling the density and tSZ fields, using the moments of the fields up to fourth order as our summary statistic, (ii) then we test the sensitivity of our predictions to the in- \nclusion of secondary halo properties such as concentration and ellipticity, as well as additional features in the input halo profiles such as cosmological shocks and non-thermal pressure, and (iii) finally, we showcase the ability of existing and upcoming surveys to constrain these baryonic effects, either separately or in a joint analysis with cosmological parameters varied as well. \nOf particular note is that our modelling pipeline is built atop the open-source C/o.pc/r.pc/e.pc C/o.pc/s.pc/m.pc/o.pc/l.pc/o.pc/g.pc/y.pc L/i.pc/b.pc/r.pc/a.pc/r.pc/y.pc suite (CCL, Chisari et al. 2019) developed for the Rubin LSST Dark Energy Science Collaboration (DESC). This choice both allows our methods to be easily accessible/usable by the community, by borrowing the many user accesibility features built for the suite, and more importantly allows the halo profiles used in this method to also be propagated through other halo model tools already built with CCL. \nWe organize this paper as follows: Section 2 describes the baryonification model used in this work. Section 3 describes the simulations, the summary statistics used to model and validate our method, and the quantification of measurement uncertainty and model best-fit. Section 4 describes a subset of the tests we perform to validate our modelling pipeline and characterize its behavior. Finally, section 5 details the power of current and upcoming surveys in constraining the parameters associated with these baryonic effects. We conclude in Section 6. \nA number of appendices provide further details on the methods we used: the lensing and tSZ forward modelling approach in Appendix A, the sensitivity of the predictions to additional model and methodology choices in Appendix B and C, the redshift evolution of the baryonification predictions in Appendix D, the dependence between Fisher information and baryonification model-complexity in Appendix E, and finally, some salient computation details on the method, include the runtimes, in Appendix F.", '2 BARYONIFICATION': 'We first describe the halo model used in this work in Section 2.1. Then, we detail the procedures used to include baryon signatures into the different large-scale structure fields we consider: the matter density field (Section 2.2) and the tSZ field (Section 2.3). Our model follows from that presented in P24, which builds on the density field-only baryonification model of S19 and Giri & Schneider (2021). Following the CCL convention, all distance scales used in defining the profiles are written in comoving Mpc units, and all masses as in M ⊙ . We will frequently use 𝑀 200c, which is a spherical overdensity mass defined as the mass contained within a halo-centric sphere of average density ⟨ 𝜌 ⟩ = 200 𝜌 𝑐 ( 𝑧 ) , with 𝜌 𝑐 ( 𝑧 ) being the critical density of the universe at redshift 𝑧 . The radius of \nthe sphere is 𝑅 200c. \nA number of salient computational details about the method are described in Appendix F, including the characteristic runtimes of the pipeline. The baryonified lensing and tSZ maps for a simulation /two.sup can be generated on the order of minutes on an Intel broadwell chip with 40 cores; see Appendix F for more details.', '2.1 Dark matter baryon (DMB) halo model': 'The DMBhalomodelisasumovermultiple different components. Each component is parameterized in a different way, with the parameterization informed by prior work in simulations and/or observations. We now describe each component below: \nThe dark matter profile is modelled by a simple NFW form (Navarro et al. 1997) with an additional truncation term that scales as 𝑟 -4 , \n𝜌 NFW ( 𝑟 ) = 𝜌 0 GLYPH<18> 𝑟 𝑟 𝑠 GLYPH<19> -1 GLYPH<18> 1 + 𝑟 𝑟 𝑠 GLYPH<19> -2 GLYPH<18> 1 + 𝑟 2 𝑟 2 𝑡 GLYPH<19> -2 , (1) \nwhere 𝜌 0 is a normalization coefficient set by requiring that the mass enclosed within 𝑅 200c is 𝑀 200c, \n𝜌 0 = 𝑀 200c GLYPH<20> ∫ 𝑅 200c 0 4 𝜋𝑟 2 𝜌 NFW ( 𝑟 ) 𝑑𝑟 GLYPH<21> -1 . (2) \nNote that this profile is implicitly a function of 𝑟 𝑠 , or alternatively the halo concentration 𝑐 200c = 𝑅 200c / 𝑟 𝑠 . The truncation term is necessary as the baryonification model requires as estimate of the total enclosed mass at 𝑟 → ∞ and this is a divergent quantity for the standard NFW profile as the profile scales as 𝑟 -3 at 𝑟 ≫ 𝑟 𝑠 . We follow S19 and set 𝑟 𝑡 = 4 𝑅 200c (see Table 1). \nThetwohalo profile , which describes the extended matter distribution around a halo contributed to by its neighboring structures, is defined similar to S19, as \n𝜌 2h = 𝜌 𝑚 ( 𝑧 ) GLYPH<20> 1 + 𝜉 𝑚𝑚 𝑏 ( 𝜈 200c ) GLYPH<21> , (3) \nwith 𝜌 𝑚 ( 𝑧 ) being the comoving matter density at redshift 𝑧 , 𝜉 𝑚𝑚 the linear matter correlation function, and 𝑏 ( 𝜈 200c ) is the halo bias given by \n𝑏 ( 𝜈 200c ) = 1 + 𝑞𝜈 2 200c -1 𝛿 𝑐 + 2 𝑝 𝛿 𝑐 ( 1 + 𝑞𝜈 2 200c ) 𝑝 . (4) \nHere, 𝛿 𝑐 = 1 . 686 / 𝐷 ( 𝑧 ) , with 𝐷 being the growth factor normalized to 𝐷 ( 𝑧 = 0 ) = 1, and 𝜈 200c is the peak height of \nthe halo, and is computed as, \n𝜈 200c = 𝛿 𝑐 / 𝜎 ( 𝑅 200c ) , (5) \nwith 𝜎 ( 𝑅 200c ) being the root-mean square of the density field smoothed with a tophat of radius 𝑅 200c. We fix 𝑞 = 0 . 707 and 𝑝 = 0 . 3, following S19 which in turn is based on the findings of Sheth & Tormen (1999). \nThe NFW profile and the two halo profile can be summed to predict the total matter distribution in the dark matter-only (DMO) case. Now, we describe the profiles of the baryonic components. \nThe stellar profile model follows Mohammed et al. (2014); Schneider & Teyssier (2015) and is a simple powerlaw with an exponential cutoff, \n𝜌 star = 𝑀 star 4 𝜋 3 / 2 𝑅 ℎ 1 𝑟 2 exp GLYPH<20> -GLYPH<18> 𝑟 2 𝑅 ℎ GLYPH<19> 2 GLYPH<21> , (6) \nwhere 𝑅 ℎ = 𝜖 ℎ 𝑅 200c is the half-light radius of the stellar halo, and 𝑀 star is the normalization of the profile obtained as, \n𝑀 star = 𝑓 cga 𝑀 tot , (7) \nwhere we have defined a theoretical total halo mass, 𝑀 tot ( 𝑟 → ∞) , through the expression \n𝑀 𝑋 ( 𝑟 ) = ∫ 𝑟 0 d 𝑟 4 𝜋𝑟 2 𝜌 𝑋 ( 𝑟 ) . (8) \nThe central galaxy fraction, 𝑓 cga, is then given by \n𝑓 cga = 2 𝐴 GLYPH<20> GLYPH<18> 𝑀 200c 𝑀 1 GLYPH<19> 𝜏 cga + GLYPH<18> 𝑀 200c 𝑀 1 GLYPH<19> 𝜂 cga GLYPH<21> -1 . (9) \nHere, the two power-law exponents are given by 𝜏 cga = 𝜏 + 𝜏 𝛿 and 𝜂 cga = 𝜂 + 𝜂 𝛿 , where 𝜏 and 𝜂 correspond to the scaling of the total stellar fraction, and 𝜏 𝛿 and 𝜂 𝛿 are a correction to convert the total stellar fraction scaling to the central galaxy stellar fraction scaling. The factor of 2 ensures that for 𝑀 200c = 𝑀 1, we have 𝑓 cga = 𝐴 . The original baryonification model of S19 used a different version of Equation (9), which included 𝜂 and 𝜂 𝛿 but did not have 𝜏 and 𝜏 𝛿 . These latter parameters control the slope of the stellar-mass halo-mass relation for low-mass halos and is included in this work. /three.sup In practice, the 𝜏 and 𝜏 𝛿 parameters have no impact in our work as the signal for both weak lensing and tSZ is generated by halos that are above the 𝑀 1 = 3 × 10 11 M ⊙ mass scale (see Table 1 for the fiducial values of different parameters), and such halos are sensitive only to 𝜂, 𝜂 𝛿 and not 𝜏, 𝜏 𝛿 . However, we include this parameter here as it is a simple extension that generalizes the model for smaller mass halos. \n/three.sup S19 note that analyses of weak lensing only use halos above 𝑀 200c > 10 12 M ⊙ , for which their single power-law model is accurate and adequate. We extend this to the double power-law model traditionally used in observational and theoretical analyses of galaxies ( e.g. , Behroozi et al. 2019, see their Figure 9). This extension allows the baryonification model to be used consistently with halos of lower masses. \nAs discussed before, the quantity 𝑀 tot does not diverge now due to our addition of a truncation radius, 𝑟 𝑡 , in Equation (1). Without this factor, 𝑀 tot = 𝑀 NFW ( 𝑟 → ∞) → ∞ . This is relevant as we use this total mass, 𝑀 tot, alongside the relevant mass fractions discussed above, to define the total mass of stars/gas mass in a halo. The baryonification model assumes that 𝑀 tot for a given halo is the same in Universes with and without the presence of baryons. This assumption is valid as the redistribution of matter is mostly localized around the halo, and for sufficiently large radii (we use 𝑟 → ∞ in definition 𝑀 tot) the enclosed mass is constant even after this redistribution process ( e.g. , Ayromlou et al. 2022; Gebhardt et al. 2023). \nThe gas profile is a modified version of the generalized NFW profile (Nagai et al. 2007, see their Equation A1), with two distinct length scales, /four.sup \n𝜌 gas = 𝜌 gas , 0 GLYPH<18> 1 + 𝑟 𝑅 co GLYPH<19> -𝛽 GLYPH<18> 1 + GLYPH<18> 𝑟 𝑅 ej GLYPH<19> 𝛾 GLYPH<19> -𝛿 -𝛽 𝛾 . (10) \nHere, 𝑅 co = 𝜃 co 𝑅 200c is the length scale of the gas core, and 𝑅 ej = 𝜃 ej 𝑅 200c is that of the ejected gas. The scaling 𝛽 , which controls the slope between 𝑅 co < 𝑟 < 𝑅 ej, has an additional mass dependence as, \n𝛽 = 3 ( 𝑀 200c / 𝑀 𝑐 ) 𝜇 𝛽 1 + ( 𝑀 200c / 𝑀 𝑐 ) 𝜇 𝛽 , (11) \nwhich asymptotes to 3 for 𝑀 ≫ 𝑀 𝑐 , and 0 for 𝑀 ≪ 𝑀 𝑐 . Equation (11) is the updated parameterization from Giri & Schneider (2021) and guarantees positivitity of 𝛽 for all masses. The other power-law indices, 𝛾 and 𝛿 , control the slopes at 𝑟 ∼ 𝑅 ej and 𝑟 ≫ 𝑅 ej, respectively. \nThe normalization, 𝜌 gas , 0 is set in an analagous manner to the stellar profile normalization of Equation (7), and is given as \n𝜌 gas , 0 = 𝑓 gas 𝑀 tot GLYPH<20> ∫ ∞ 0 4 𝜋𝑟 2 𝜌 gas ( 𝑟 ) 𝑑𝑟 GLYPH<21> -1 , (12) \nwith 𝑓 gas derived from the baryon and stellar mass fraction, \n𝑓 gas = 𝑓 b -𝑓 star , (13) \n𝑓 b = Ω b / Ω m , (14) \n𝑓 star = 2 𝐴 GLYPH<20> GLYPH<18> 𝑀 200c 𝑀 1 GLYPH<19> 𝜏 + GLYPH<18> 𝑀 200c 𝑀 1 GLYPH<19> 𝜂 GLYPH<21> -1 , (15) \nThe baryon fraction within an isolated, toy-model halo is simply the cosmological baryon fraction, and once we specify the stellar fraction - which contains in it the rich physics of star formation - we can obtain the gas fraction of the halo. Note that 𝑓 star denotes the fraction of all stellar matter, including those in satellite galaxies, and we will use this fact in Equation \n(20). Next, following P24, we add additional mass and redshift dependence to multiple gas profile parameters through the following promotion for each parameter, \n𝑋 → 𝑋 GLYPH<18> 𝑀 200c 𝑀 𝑋 GLYPH<19> 𝜇 𝑋 ( 1 + 𝑧 ) 𝜈 𝑋 ( 𝑐 200c ) 𝛼 𝑋 . (16) \nwhere 𝑀 𝑋 is a pivot mass, 𝜇 𝑋 , 𝜈 𝑋 , 𝛼 𝑋 control the scaling with mass, redshift, and halo concentration. Note that in practice, most of these additional scaling parameters are not included in the final model; see Section 2.5 and Table 1 for more details. \nThe collisionless matter profile is the final component of the baryonification density profile model. This profile constitutes both the DM as well as the subhalos/galaxies in the halo, as both components are collisionless and interact only gravitationally (and not hydrodynamically) with the gas distribution. We follow previous baryonification models in ignoring any explicit modelling of gas within individual subhalos/galaxies. For the observables we wish to model - the tSZ field, and the baryonic imprints in the density field - the gas distribution in substructure is irrelevant compared to the gas of the host halo. The collisionless matter profile is modelled by accounting for the adiabatic, gravitational contraction/expansion of the original matter distribution (Blumenthal et al. 1986; Gnedin et al. 2004) - where this original distribution is taken to be an NFW profile; see Equation (1) - due to the presence of baryonic components, namely gas and stars. \nThis adiabatic relaxation is performed via the transformation 𝑟 → 𝑟 / 𝜁 , where 𝜁 is obtained by solving \n𝜁 ( 𝑟 ) -1 = 𝑎 GLYPH<20> GLYPH<18> 𝑀 𝑖 ( 𝑟 ) 𝑀 𝑓 ( 𝑟 ) GLYPH<19> 𝑛 -1 GLYPH<21> , (17) \nwhere 𝑎 and 𝑛 are phenomenological parameters. For 𝑎 = 𝑛 = 1, the expression corresponds to exact angular momentum conservation. We instead take 𝑎 = 0 . 3 and 𝑛 = 0 . 2 (see Table 1), which provides a better match to simulations (Abadi et al. 2010). The deviation from 𝑎 = 𝑛 = 1 arises because galaxy growth is not an instantaneous process (Gnedin et al. 2004; Abadi et al. 2010). \nThe initial (final) mass distribution 𝑀 𝑖 ( 𝑀 𝑓 ) is given by, \n𝑀 𝑖 = 𝑀 nfw ( 𝑟 ) (18) \n𝑀 𝑓 = 𝑓 CLM 𝑀 nfw ( 𝑟 ) + 𝑀 gas ( 𝑟𝜁 ) + 𝑀 star ( 𝑟𝜁 ) . (19) \nWesolve for 𝜁 iteratively, and find that in practice it converges ( < 1% differences) in under 10 iterations. Upon obtaining a solution, we compute the final 𝜌 CLM profile as, \n𝜌 CLM ( 𝑟 ) = 𝑓 CLM 4 𝜋𝑟 2 𝑑 𝑑𝑟 𝑀 nfw ( 𝑟 ) , (20) \nwith the definition 𝑓 CLM = ( Ω m -Ω 𝑏 )/ Ω m +( 𝑓 star -𝑓 cga ) . The latter two fractions are defined in Equation (9) and (15), and their difference 𝑓 star -𝑓 cga corresponds to the mass fraction of \nstars in satellite galaxies. This quantity, 𝑓 CLM, corresponds to the fraction of collisionless matter in the total matter distribution. \nWe have now described all the profile components needed for modelling the baryonic imprints. We can write the total density distribution in the DMO and DMB halo models as \n𝜌 DMO = 𝜌 NFW + 𝜌 2h (21) \n𝜌 DMB = 𝜌 CLM + 𝜌 gas + 𝜌 star + 𝜌 2h (22) \nFigure 1 shows examples of the different profiles we use in this work. We see that the collisionless matter profile (CLM) is amplified at small scales and suppressed at larger scales, when compared to the dark matter NFW profile, as is expected from the process of adiabatic relaxation. The DMB model is further amplified at small scales due to the presence of the stellar component. We have verified that our pipeline reproduces the profiles shown in Figure 1 of S19.', '2.2 Matter density field': "Thenovelty of the baryonification model is in taking results of an N-body simulation - which includes accurate predictions for the gravitationally induced correlations of non-linear structure - and perturbing it to induce baryon-like features in the density field. Such an approach requires a displacement function, Δ 𝑑 ( 𝑟 ) , which specifies the distance a particle must be perturbed from its current location. \nTo define this function, we first compute the enclosed mass corresponding to the density distributions in Equations (22) and (22). Then, the displacement function is given by \nΔ 𝑑 ( 𝑟 ) = 𝑀 -1 DMB ( 𝑀 DMO ( 𝑟 )) -𝑟, (23) \nwhere 𝑀 -1 DMB ( 𝑀 ) is the functional inversion of enclosed massradius function /five.sup and outputs a radius given a mass, and 𝑀 DMO ( 𝑟 ) is the enclosed mass in the DMO case. For a given halo, we evaluate this function at the location of every particle within some maximum halo-centric distance (nominally set to 20 𝑅 200c), and perform a radial offset to the particle position with respect to the halo. In practice, we do not immediately offset the particle locations. Instead, we loop over all halos in the volume, and accumulate the offsets per particle. Once the loop is finished, we now shift the particles by the total, accumulated offset. In this way, the algorithm and its predictions are invariant to changes in the ordering of halos in the loop. The offset positions obey periodic boundary conditions of the simulation box; for example, any pixel that is offset beyond the left edge of the simulation volume will reappear on the right edge of the volume. \nThus far, we have used the formalism of S19. However, as discussed previously, we require a faster method for applying this displacement function to large suites of full-sky simulations. Fluri et al. (2019) introduced the shell baryonification method, where the displacement is performed on individual map pixels rather than on individual matter particles. In this case, the displacement is written as, \nΔ 𝑑 ( 𝑟 p ) = 𝑀 -1 DMB , p ( 𝑀 DMO , p ( 𝑟 p )) -𝑟 p , (24) \nwhere we have substituted the 3D distance 𝑟 with the projected distance 𝑟 p. The enclosed masses are all now masses enclosed within the projected radius, \n𝑀 𝑋,𝑝 ( 𝑟 p ) = ∫ 𝐿 p / 2 0 2 𝑑𝑙 ∫ 𝑟 p 0 𝑑𝑟 p2 𝜋𝑟𝜌 𝑋 GLYPH<18> √︃ 𝑙 2 + 𝑟 2 p GLYPH<19> , (25) \nwhere we now integrate over the line-of-sight distance 𝑙 . The factor of 2 in the integral over 𝑑𝑙 arises from the symmetry in 𝑙 used to set integration limits to 0 < 𝑙 < 𝐿 p / 2 instead of -𝐿 p / 2 < 𝑙 < 𝐿 p / 2. The choice of 𝐿 p / 2 as the limit, where 𝐿 p is the thickness of the shell (or simulation box, when considering a 3D snapshot), is different from Fluri et al. (2019), where the integration limits were set to 50 𝑟 p instead. We motivate the former choice by recognizing that the mass in a given map pixel is a line-of-sight integration overing the distance -𝐿 p / 2 < 𝑙 < 𝐿 p / 2. Thus, a similar choice must be made in the baryonification model. \nThis choice of projection scale can also be motivated by taking the limit 𝐿 p →∞ , where 𝑀 DMB , p ≈ 𝑀 DMO , p since the large-scale matter distribution (which is much less affected by baryons than the small-scale distribution, and is modelled by the two-halo term in our work) dominates the mass in the pixel. In this limit, the displacement function is simply Δ 𝑑 ( 𝑟 p ) ≈ 0. Bysetting 𝐿 p as the integration distance, our model predictions are formally consistent in these extreme limits. Appendix B shows the changes in the predicted displacement due to variations in the assumed projection scale. We note that the projection integral in Equation (25) assumes the halo is spherically symmetric. We test the anisotropic baryonification corrections in Section 4.1.3, but the baryonification model in those cases will still use the integral in Equation (25). \nSimilar to the case with the particle baryonification, we loop over all halos in a given shell and accumulate offsets to the pixel locations. Once the loop is done, we shift the pixel locations by the sum of the accumulated offsets. For each offset parent pixel, we use the H/e.pc/a.pc/l.pc/p.pc/y.pc interpolation routine get interp weights to (i) determine the nearest four H/e.pc/a.pc/l.pcP/i.pc/x.pc child pixels that the parent pixel contributes to, and (ii) determine the contribution the parent pixel makes to each child pixel, which is determined by the weights provided by the function. Thus, we regrid the offset pixel back onto the H/e.pc/a.pc/l.pcP/i.pc/x.pc grid using the neighboring pixels and corresponding \nFigure 1. An illustration of the different profiles used in the baryonification pipeline. The top left shows the dark-matter only (DMO) and dark matter baryon (DMB) models, including all the components that constitute each profile. The top right, bottom left, and bottom right, show the fiducial gas, star, and pressure profile, respectively. In addition to the fiducial parameter choice, listed in Table 1, we show a handful of variations (see legends) to illustrate the impact of certain parameters on the profiles. \n<!-- image --> \ncontributions as determined by the H/e.pc/a.pc/l.pcP/i.pc/x.pc routines. This procedure is guaranteed to preserve the total mass/density in the shell. \nIn Section 4, we perform a number of tests to evaluate the model's accuracy and these tests use 2D rectilinear simulation maps rather than curved sky maps. In this case, the baryonification-based mass reassignment is done similar to the H/e.pc/a.pc/l.pcP/i.pc/x.pc case: for each offset parent pixel, we find its nearest four child pixels, calculate the area overlap with each child pixel, and re-assign the parent pixel's values to the four child pixels. \nWehave verified that our particle baryonification pipeline, operating on the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG DMO simulation, can reproduce the 3D power spectrum results from Figure 2 of S19. We have also checked - using the shell baryonification pipeline - that the angular power spectrum of the baryon-corrected density fields shows the same scale-dependent variation with model parameters as in that Figure; for example, using 𝜃 ej = 2 leads to an excess of power on intermediate scales. /six.sup", '2.3 Thermal Sunyaev-Zeldovich (tSZ) effect': "The tSZ effect is sourced by the thermal pressure of the electrons in the cosmic distribution of gas. Given we already have a model for the total density profile, in Equation (22), and for the gas density profile, in Equation (10), we can now assume the halo is in hydrostatic equilibrium and trivially obtain an estimate of the gas pressure, \n𝑑𝑃 𝑑𝑟 = -𝐺𝑀 1 -halo DMB ( 𝑟 ) 𝑟 2 𝜌 gas ( 𝑟 ) . (26) \nThis expression can be integrated to obtain the pressure, 𝑃 = -∫ 𝑟 ∞ 𝑑𝑃 𝑑𝑟 𝑑𝑟 , using the boundary condition of 𝑃 ( 𝑟 →∞) = 0. Note that Equation (26) uses the enclosed mass corresponding to only the one-halo term, denoted above as 𝑀 1 -halo DMB ( 𝑟 ) , and ignores the two-halo contribution. This is so our pressure profile corresponds to only the one-halo component. The tSZ field is generated by pasting tSZ profiles around all halos in the simulation. Under such a method, the two-halo term of the tSZ field is naturally modelled as the clustering of halo positions includes this two-halo component. Thus, we avoid double-counting the two-halo term by explicitly ignoring it in Equation (26). \nEquation (26) is derived using hydrostatic equilibrium, which assumes that the gravitational pull on the gas is balanced entirely by the thermal pressure - random motions of the ions \n-of the gas. While this is generally true for the gas closer to the core of the halo, it is not accurate for the gas at the outskirts ( e.g. , Nelson et al. 2014). Thus, Equation (26) only represents the total pressure required to support the gas, and we must estimate the fraction contributed separately by the thermal and the 'non-thermal' pressure, where the latter primarily constitutes turbulent motions of the gas (i.e. the gas' velocity dispersion). \nWe follow P24 by including in our predictions a parameteric model, based on Shaw et al. (2010), of the non-thermal pressure, \n𝑓 nt = 𝛼 nt 𝑓 𝑧 GLYPH<18> 𝑟 𝑅 200c GLYPH<19> 𝛾 nt , (27) \n𝑓 𝑧 = min [( 1 + 𝑧 ) 𝜈 nt , ( 𝑓 max -1 ) tanh ( 𝜈 nt 𝑧 ) + 1 ] , (28) \n𝑓 max = 6 -𝛾 nt / 𝛼 nt , (29) \nwhere 𝛼 nt is the amplitude of non-thermal pressure, 𝛾 nt its scaling with radius, and 𝜈 nt its scaling with redshift. We set 𝛾 nt = 0 . 5 and 𝜈 nt = 0 . 3 following P24. The quantities 𝑓 𝑧 and 𝑓 max are defined so 𝑓 nt is between 0 and 1 for 𝑟 < 6 𝑅 200c. We also explicitly clip its value between 0 < 𝑓 nt < 1. The non-thermal pressure is varied in all our predictions. We also show in Section C2 the predictions of this model compared to others in the literature and show the model is adequately flexible. \nAric'o & Angulo (2024) also introduce an extension of their baryonification method (Aric'o et al. 2021b) to include a tSZ field consistent with the baryon-corrected density field. They first assign 'synthetic' gas and star particles to their simulations using the DM particle dataset. The pressure profile is then modelled using the prescription of Komatsu & Seljak (2002). That prescription connects the gas pressure and density by assuming the gas is a polytrope ( 𝑃 ∝ 𝜌 Γ ), which is allowed under the condition of hydrostatic equilibrium, and then by assuming the total matter distribution follows an NFW-like profile (Komatsu & Seljak 2001, see their Equation 7). Note that Γ is furthermore assumed to be independent of scale, whereas it has been found to be a function of radius in both simulations ( e.g. , Battaglia et al. 2012, see their Figure 4) and observations (Capelo et al. 2012); it does, however, depend explicitly on halo concentration and therefore halo mass. Finally, a particle at a given radial distance, 𝑟 , is assigned a value given by the pressure profile evaluated at that distance. \nIn comparison, our work predicts the pressure field directly on the sky, avoiding the use of particles to reduce computational cost and relying instead on the halo positions. Our pressure profile is obtained using the assumptions of hydrostatic equilibrium as well, but we do not assume an NFW profile for the total matter distribution. Instead, we directly use 𝑀 dmb ( 𝑟 ) predicted by the baryonification model; see Equation (22). Our model also does not assume the gas is a polytrope or that \nit has a scale-independent polytropic index. This allows our predictions to be more easily generalized to both the outskirts of halos and to lower-mass halos. Our profile painting method can capture anisotropic features in the field via the use of elliptical profiles (see Section 4.1.3), but it does not capture more fine-grained morphological features, as would be possible in the particle-based method of Aric'o & Angulo (2024).", '2.4 Pixelization': "Our predictions are pixelized maps rather than a discrete, particle/point. There is therefore a window function needed due to this the pixelization, and must be accounted for during the theory modelling. We do so by modifying the profiles as, \n𝑋 ( 𝑟 ) → GLYPH<20> 𝑊 𝑝 ★ 𝑋 GLYPH<21> ( 𝑟 ) , (30) \nwhere [ 𝑊 pix ★ 𝑋 ] represents a convolution between the pixel window function, 𝑊 pix, and the radial profile, 𝑋 . This convolution is performed entirely in Fourier space. In the case of 2D maps, which have square pixels, we assume a circular tophat with an area equivalent to that of the square pixel. /seven.sup We have tested that this approximation makes a negligible difference in the convolved profiles: the results are identical for scales smaller/larger than the pixel-scale, and differ at the percent level around the pixel-scale. For the full-sky maps, we approximate the Healpix pixel window function by a Gaussian window function which we verifiy is accurate to better than 1%, \n𝑊 pix ( ℓ ) = exp GLYPH<20> -𝜎 2 2 ℓ ( 1 + ℓ ) GLYPH<21> , (31) \nwith 𝜎 = 𝜃 pix /( 4 √ ln 2 ) , 𝜃 pix the pixel resolution scale obtained from the nside2resol function of H/e.pc/a.pc/l.pcP/y.pc, and ℓ is the angular multipole. This analytic function is advantageous as it can be evaluated at any ℓ -which is necessary for performing FFTlog-based Fourier transforms to high numerical precision - whereas the default Healpix window function is only computed/provided up to an ℓ max set by the Healpix map's resolution. /eight.sup \nFigure 2 shows an example of the pixelization's impact on the density profiles and therefore the displacement function. For larger pixelizations, we smooth over baryonic imprints and so the displacement function approaches zero. It is at zero for 𝐿 = 10 Mpc, which is about 20 times larger than the radius 𝑅 200c = 0 . 45 Mpc for the halo mass we show. There is a pileup of mass (identified as the 'bump' in the profiles) at the scale of pixel, where this scale is shown in vertical dotted lines. The \nFigure 2. Thedisplacement function (top) and the DMB profile (bottom) corresponding to different pixel/map resolutions (pixel length scale denoted in the legend). The black dotted line is the result from no convolutions, and it is consistent with the result from convolving the predictions with a vanishingly small pixel. The pixel window function smooths the mass on small scales, thereby leading to (i) nearly zero displacement at small scales, and (ii) a pileup of mass at the scale of the pixel (denoted by vertical lines). \n<!-- image --> \nblack dotted line in Figure 2 shows the unsmoothed result, and there is numerical consistency between it and the result from using a very small pixel window function. We have verified that the FFTlog calculations, /nine.sup which are used to convert the profiles to Fourier space and perform the convolution, preserve the profiles down to the 10 -5 % level, which is adequate for our modelling. We use these pixel convolutions for all steps in our work. The pixel scales vary depending on the exact simulation/analysis being performed, and we will denote the exact values where relevant.", '2.5 Model parameterization': 'The profile models in Section 2.1 and Section 2.3 has a number of free parameters, but not all of them need to be varied. Table 1 summarizes the parameters of the model, and then the priors we use when performing any model comparisons/fits in the discussions to follow. The key parameters to vary are those associated with the gas profile. Note that this includes 𝜂 , given the stellar fraction directly controls the gas fraction as shown in Equation (13). The rest of the parameters are fixed to values discussed in S19. The parameter 𝜏 , which we introduce in this work in Equation (15), is set to 𝜏 = -1 . 5 which is the 1 𝜎 upper bound from the findings of Moster et al. (2013, see their Table 1). \nIn Equation (16) we modified the gas profile parameters to include additional dependence on mass, redshift, and halo concentration. We set these additional scalings to zero for most parameters; the notable exceptions are the scale radii, 𝜃 ej and 𝑀 𝑐 , where we follow P24 in varying the relevant mass- and redshift-dependence parameters associated these quantities. We also follow P24 in setting the concentrationdependence parameter 𝛼 𝑋 = 0. The concentration dependence of baryonic feedback has been recently measured in many works ( e.g. , Shao et al. 2023; Shao & Anbajagane 2024; Gebhardt et al. 2023; Pandey et al. 2024) but given our model parameterization is already flexible we do not include this additional dependence. \nThe halo concentration is formally a free parameter in the model. In practice, for a given halo, we use the halo concentration measured directly in the simulation when available and otherwise assume the simulation-calibrated relation of Diemer & Joyce (2019) - alongside the halo mass, redshift, and assumed cosmology of the simulation - to assign the halo a 𝑐 200c value.', '3 SIMULATIONS & SUMMARY STATISTICS': 'Weuse three distinct simulation suites and one set of summary statistics to validate our model predictions against the simulation measurements. The I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG suite (Section 3.1) and the Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc suite (Section 3.2) are used to validate the baryonification model. The U/l.pc/a.pc/g.pc/a.pc/m.pc suite (Section 3.3) is used to simulate surveys as part of our Fisher forecast. All analyses of the fields are done after summarizing the fields into a data vector containing moments (from 2nd to 4th order) of the lensing and tSZ field (Section 3.4). All maps made from the simulations are either a 2D rectilinear grid of 512 2 pixels or a H/e.pc/a.pc/l.pcP/i.pc/x.pc curved-sky map of NSIDE = 1024.', '3.1 I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG simulations': "The I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG /one.sup/zero.sup suite (Nelson et al. 2018; Pillepich et al. 2018b; Springel et al. 2018; Naiman et al. 2018; Marinacci et al. 2018) is a set of high-resolution hydrodynamical simulations that self-consistently evolve a wide set of astrophysical processes (Weinberger et al. 2017; Pillepich et al. 2018a). The simulation volume spans a box with side 𝐿 box = 205 Mpc / ℎ . At such volumes, the particle mass resolution is high but at the cost of poor number statistics for the larger halos. The latter limits our accuracy in validating the pressure field model, as the pressure field depends more heavily on the most massive halos. The suite contains simulations of three resolution levels, and we use the third, TNG300-3, variant. This run has 625 3 particles each of dark matter and \nTable 1. The input parameters to the profiles of the baryonification model, described in Section 2.1 and 2.3. For each parameter, we list its fiducial value, the prior used when fitting it (left blank if the parameter is fixed in our analysis) and a description of the parameter's physical meaning. The prior ranges are similar to, but generally broader than, the ones defined in P24, see their Table 1. Any mass, redshift, and concentration-dependent parameters - as defined by Equation (16) - are fixed to 0 if they are not shown above. \ngas, with a DM particle mass resolution of 4 . 5 × 10 9 M ⊙ . While higher resolution runs are also available, we utilize only this low-resolution run as its resolution more closely mimics that of the larger simulations used for modelling full-sky lensing and tSZ fields. The pixel-scale of the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG 2D maps is 0 . 6 Mpc. \nEach simulation in the suite also has a DMO counterpart, that shares the exact same initial conditions. The pair of simulations can be used to make measurements whose uncertainties are cosmic variance suppressed (see Section 3.5 for details) and therefore allow more precise validations than otherwise possible. We use the public data release, as introduced in Nelson et al. (2019). There also exist other, public simulation suites that span much large volumes than I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG -such as M/a.pc/g.pc/n.pc/e.pc/t.pc/i.pc/c.pc/u.pc/m.pc (Hirschmann et al. 2014) and B/a.pc/h.pc/a.pc/m.pc/a.pc/s.pc (McCarthy et al. 2017) - and would therefore be better suited for our tests of the tSZ field and lensing field cross-correlations. These simulations also span a wide range of baryonic feedback prescriptions, some of which are stronger compared to the relatively weaker feedback in I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG. However, unlike the case with I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG, the quantities \nwe require from these simulations (the particle snapshots) are not publicly accessible. For these reasons, we use I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG for our initial validation here, but highlight the need for a more precise validation - using a larger simulation before employing this model on survey data.", '3.2 Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc simulations': 'Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc /one.sup/one.sup is a large suite of DMO simulations spanning a wide range of cosmological parameters (Villaescusa-Navarro et al. 2020). We use the high-resolution runs generated the fiducial cosmology which is based on Planck Collaboration et al. (2016). These run have 1024 3 particles spanning a large volume ( 𝐿 box = 1000 Mpc / ℎ ), and therefore has much poorer mass/spatial resolution than I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG with each DM particle of mass ≈ 10 11 M ⊙ , but also have a statistical sample of halos of higher masses. In our work, we use these simulations for tests that (i) use the halo concentration, as we can utilize the 𝑐 200c estimates from the R/o.pc/c.pc/k.pc/s.pc/t.pc/a.pc/r.pc catalogs (Behroozi et al. 2013) provided with these simulations, and \n(ii) for checking sensitivities to higher masses that are not testable in I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG. The pixel-scale of the 2D maps is 3 Mpc.', '3.3 U/l.pc/a.pc/g.pc/a.pc/m.pc simulations': "The U/l.pc/a.pc/g.pc/a.pc/m.pc /one.sup/two.sup suite is a set of full-sky N-body simulations designed for widefield survey analyses, and was introduced in Anbajagane et al. (2024b). The simulations are based on the Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc and Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc-P/n.pc/g.pc suites (Villaescusa-Navarro et al. 2020; Coulton et al. 2022) and complement them by enabling Fisher forecasts of observables from wide-field surveys. In particular, they vary four cosmology parameters, and four inflationary parameters as well. In this work, we use these simulations to perform forecasts for weak lensing and tSZ surveys (Section 5). \nThe simulations resolve 512 3 particles in a volume with 𝐿 box = 1 Gpc / ℎ , following the same configuration as the fiducial (not high-resolution) Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc run. However, the halo catalogs provide friends-of-friends (FoF) mass, estimated with a linking length of 𝑏 = 0 . 2 in units of the mean interparticle separation, instead of 𝑀 200c. This catalog is used only in our forecast analysis (Section 5), and we make the approximation of treating 𝑀 fof masses as though they were 𝑀 200c ones. FoF linking lengths of 𝑏 = 0 . 2 will generally correspond to halo overdensities of 200 𝜌 𝑐 , but this statement depends on the concentration of the specific halo (More et al. 2011, see their Figure 4). Even with this caveat, we can still meaningfully study the degeneracy directions of the different baryonification parameters, as well as the change in relative constraining power across different analysis configurations. \nHalos of 𝑀 fof > 10 14 M ⊙ are adequately resolved by the simulation as these halos contain at least 100 particles. /one.sup/three.sup Anbajagane et al. (2024b, see their Figure 12 and 13) show that the halo mass function and halo bias of this catalog agreed with those from higher-resolution simulations. This mass limit is also adequate for studying tSZ auto-correlations: we show in Section 4.3, using the high-resolution Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc suite, the sensitivity of our observable (the moments of the fields) to different mass scales. For the scales of interest to us, the tSZ-only measurements are sensitive to halos above 𝑀 200c > 10 14 . 5 M ⊙ . For the lensing-only measurements, we have already shown in Anbajagane et al. (2024b, see their Figure 10 and 11) that the density and lensing field are accurately predicted in these simulations. \nModelling the cross-correlation of the lensing and tSZ field through these simulations incurs one caveat. Pandey et al. \n(2022, see their Figure 2) compute the mass-dependence of the lensing and tSZ real-space, two-point cross-correlation function. For the smallest scales we measure ( 𝜃 ≈ 3 ' ) they show the peak contributing halo mass is 𝑀 200c ≈ 10 14 M ⊙ but halos below that mass-scale also contribute a significant amount to the total signal. Our map will therefore underestimate this signal since we only paint pressure profiles around halos with a minimum mass of 10 14 M ⊙ . Higher-order correlations push the peak contributing halo mass to higher masses, therefore making any missing contribution from lower mass halos less relevant. Regardless, we stress that the absence of lower mass halos in our analysis will reduce the impact of baryons on our simulated signal (relative to their impact when using all halos) and therefore will weaken our Fisher information on baryonic processes. We do not risk quoting artificially tighter bounds due to this caveat. In summary, the U/l.pc/a.pc/g.pc/a.pc/m.pc suite still has sufficient, though not ideal, characteristics for performing our joint analysis of lensing and tSZ fields.", '3.4 Field moments statistic': "The baryonification model has been validated for the density field on both the power spectrum (Schneider et al. 2019; Aric'o et al. 2023), as well as the bispectrum (Aric'o et al. 2021a) and the weak lensing peaks (Lee et al. 2023). It has also been validated for the tSZ power spectrum (Aric'o & Angulo 2024). In this work, we focus on the moments of the field which have been used extensively on weak lensing data ( e.g. , Van Waerbeke et al. 2013; Chang et al. 2018; Peel et al. 2018; Petri et al. 2015; Gatti et al. 2022, 2023, 2024b,a), and are a computationally efficient estimators of higher-order information in the fields. \nThese moments are computed as, \n⟨ 𝐴 ( 1 ) 𝐴 ( 2 ) . . . 𝐴 ( 𝑁 ) ⟩( 𝜃 ) = 1 𝑁 pix -1 𝑁 pix ∑︁ 𝑖 = 1 𝐴 ( 1 ) 𝑖 𝐴 ( 2 ) 𝑖 . . . 𝐴 ( 𝑁 ) 𝑖 , (32) \nfor some set of fields 𝐴 ( 1 ) , 𝐴 ( 2 ) , . . . , 𝐴 ( 𝑁 ) , where all fields are smoothed on some scale, 𝜃 . We follow previous works ( e.g. , Gatti et al. 2020; Anbajagane et al. 2023) in performing the smoothing using a tophat-filter, which is defined in harmonic space as \n𝐵 ( ℓ ) = 2 𝐽 1 ( ℓ𝜃 ) ℓ𝜃 (33) \nwhere 𝐽 1 ( 𝑥 ) is a Bessel function of the first order. In this work, we will consider two fields: the density field and the tSZ field. For 𝑁 = 2, the moments capture the same information as a power spectrum, and this has been checked extensively in many recent works ( e.g. , Anbajagane et al. 2023; Gatti et al. 2024b). \nAn ideal validation of our model would also use other summary statistics - such as the poly-spectra, cumulative \ndistribution functions, the wavelet harmonics, etc. - but we choose a single statistics, the moments, for simplicity in presenting and interpreting our results. Different statistics will probe somewhat similar types of information, even if the exact definitions vary significantly. Thus, validating our approach for one set of statistics serves as a positive sign for other statistics as well, though an explicit validation must always be performed before employing the baryonification method, with that given statistic, on data. One particular note is that the isotropic shape of the filter in Equation (32) means the moments are an angle-averaged quantity. Correlation functions for 𝑁 > 2 points depend on the exact configuration (or shape) of the points being correlated; for correlations of three points, the shapes are different triangles. Statistics computed on isotropically smoothed fields, such as the moments defined here, average over all these configurations/shapes. Thus, the validation of baryonification for these moments does not guarantee the validation of shape-specific correlations.", '3.5 Estimation of measurement uncertainty and model best fit': "One key validation done in this work is to compare the predictions of the baryonification model to the fields from I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG. The robustness of the match is quantified in relation to the uncertainty on the simulation measurement. There are two different contributions to this uncertainty: (i) cosmic/sample variance, with quantifies the intrinsic uncertainty of our measurement due to using only a finite volume/sample of the Universe, and (ii) shot noise, which is the poission uncertainty from using a finite number of discrete objects. \nIn the case of I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG, our key measurement is the fractional difference between the moments measured on the true fields and those measured on the predicted 'baryonification-based' fields. The latter fields are derived using products from the dark matter-only simulation in I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG. The DMO and hydrodynamical simulations share the same initial conditions, and therefore our predicted fields have the same initial conditions of the hydrodynamical simulations. In this case, the uncertainty on the fractional difference is cosmic variance-suppressed. Note that it is suppressed , not cancelled , as the late-time density distributions in the two simulations still vary slightly due to differences in the dynamical equations being solved in each (the DMO simulation does not solve any hydrodynamical equations). \nThe uncertainty is estimated as follows: we define the ratio 𝑅 ≡ 𝑋 dmo / 𝑋 hydro, where 𝑋 is the 𝑁 th -order moment of the density field and/or tSZ field. The uncertainties on 𝑅 are computed through a leave-one-out jackknife resampling of the 2D fields, with 64 patches. In each sampling, we remove a contiguous, square patch of the field and compute 𝑅 using the remaining area. In the case of the density field, this \nis straightforward as both DMO and hydrodynamical simulations produce density fields. However, this process cannot be repeated as is for the tSZ field as the DMO simulation does not have an associated tSZ field. We instead create a mock field through the following transformation, \n𝑦 dmo = iFFT GLYPH<18> FFT ( 𝑦 hydro ) × FFT ( 𝛿 dmo ) FFT ( 𝛿 hydro ) GLYPH<19> . (34) \nEquation (34) uses the tSZ field of the hydrodynamical simulation and modifies it in Fourier space to now correspond to the density field of the DMO simulation. This field is then used in estimating 𝑅 for moments of the tSZ field. This new 'artificial' tSZ field accounts for differences in the exact realizations of the hydrodynamical and DMO simulations, which requires information about fourier-mode phases, and also the differences in the clustering of matter on small scales, which requires information about the fourier-mode amplitudes. Note that there are many other reasonable ways to estimate a 'artificial' tSZ field, such as the combination 𝑦 dmo = iFFT ( FFT ( 𝛿 dmo ) × √︁ 𝑃 𝑦𝑦, hydro / 𝑃 𝛿𝛿, hydro ) . We find these alternatives result in larger covariances for the ratio 𝑅 , andwetherefore use the method described above since a tighter (or smaller) covariance estimate results in a more stringent test of our method. \nDue to the small volume probed by I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG, not all jackknife patches are representative volumes of the Universe (for example, only two out of 64 patches contains a halo with 𝑀 200c > 10 15 M ⊙ ), and therefore the jackknife provides a biased estimate of the covariance. However, this estimate is still useful in interpreting the amplitude of deviations between the baryonification model and I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG. \nWe find the best fit baryonification model for the simulation measurement by generating predictions for a large number of points in parameter space and performing a simple minimum𝜒 2 search using the fractional residuals: given as 𝜒 2 = ( 𝑀 / 𝑠 -1 ) 𝑇 C -1 ( 𝑀 / 𝑠 -1 ) , where we compare simulations, 𝑠 , with the model, 𝑀 . During this procedure, we set the off-diagonal terms of the covariance matrix, C , to zero. Our numerical estimate of this matrix is noisy, and given the comparatively long datavectors (120 to 240 datapoints) we can incur numerical artifacts/errors during matrix inversion. Increasing the number of jackknife resamplings will alleviate this issue at the cost of further biasing our covariance estimates. /one.sup/four.sup For the purpose of checking that the baryonification model jointly fits the simulation measurements, this choice still enables a meaningful analysis while avoiding numerical artifacts. However, this highlights that a more precise validation on larger simulations - for example, by comparing the \nbaryonification predictions to full-sky simulations like M/i.pc/l.pc/l.pc/e.pc/n.pc/i.pc/u.pc/m.pcTNG (Pakmor et al. 2023) or the F/l.pc/a.pc/m.pc/i.pc/n.pc/g.pc/o.pc project (Schaye et al. 2023) - will be needed to more robustly characterize this model.", '4 MODEL VALIDATION': 'A primary focus of this work is validating the baryonification model for the density and tSZ fields up to 4th order, by using auto/cross moments of the field as our summary statistic. Here, we will use the density field as an analog of the lensing field as our validation is done with the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG simulation for which we cannot create full-sky maps (such maps require simulations, like those of U/l.pc/a.pc/g.pc/a.pc/m.pc suite, which have at least twenty to thirty times larger volumes). Validating the model for the density field will automatically validate it for the lensing field, as the latter derives directly from the former: see Equation (A1). /one.sup/five.sup Wewill use the full-sky approach for our forecasting results in Section 5. \nWe split our validation steps into two categories: (i) methodology and (ii) model. The former encapsulates all decisions related to how we move the particles or paint the tSZ field (including how we use the simulation data as inputs). The latter includes changes in the profile model that can then be propogated into the rest of the pipeline. For brevity, we discuss only the former case here. The latter analysis, which is detailed in Appendix C, shows that our profile models are adequate in their flexibility/parameterization.', '4.1.1 Comparing Particle and Shell baryonification': "An critical choice in this work is modelling the signatures of baryon astrophysics using just 2D fields and halo catalogs. As mentioned earlier, the original baryonification methods focused explicitly on 3D particle snapshots (S19, Aric'o et al. 2023) though some work has employed a 2D approach for weak lensing analyses (Fluri et al. 2019). In the 3D approach, denoted as 'particle' baryonifiction, one applies the displacement function to the particles in the 3D snapshot. This snapshot can then be compressed into a lightcone, which is the relevant dataproduct for lensing analyses. In the 2D approach, denoted 'shell' baryonification, we apply the displacement function to the 2D pixels. The latter is computationally cheaper and the discretization step in converting particles to maps significantly reduces the memory footprint. \nThis is an increasingly important consideration given the community is already producing simulation suites nearing O( 10 4 ) individual simulations ( e.g. , Villaescusa-Navarro et al. 2020; Coulton et al. 2022; Kacprzak et al. 2023; Anbajagane et al. 2024b). \nFor our tSZ profile painting method, we can interchangeably use the 3D and 2D approach as the difference is simply in what step the line-of-sight integration is done. /one.sup/six.sup In the first method, we paint profiles on a 3D grid (we do not use particle information when painting profiles) and then integrate along line-of-sight by collapsing the map along one direction. In the second, the integration is done within the definition of the projected, 2D profile, before any painting happens. Thus, the tSZ field does not suffer accuracy drops between using 3D fields and 2D fields. \nOn the other hand, the density field predication can vary between the particle and shell baryonification methods. We test this explicitly by performing the two on the 𝑧 = 0 snapshot from the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG simulation; in this case 'shell' baryonification is done on the 2D rectilinear map and not the full-sky maps. We first perform both types of baryonification for 32 points in the parameter prior space (defined in Table 1), drawn using a Sobol sequence (Sobol 1967). We use only 32 models given the computational expense of the particle baryonification procedure, which takes O( 10 3 ) longer than the shell-based method. We will use a significantly finer sampling (1024 samples) in Section 8 when comparing our shell baryonification predictions to I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG measurements. \nOnce we have made the map for a given model, we take the fractional difference of the moments measured on the two maps from particle/shell baryonification, and then compute the average absolute ratio over all 32 models. Figure 3 shows this average difference for the moments of the density field. In practice, the offsets are within the 1-2% level, which is a negligible error. The error in the third moments increases at around the pixel-scale of ≈ 1 Mpc, and the same for the fourth moments increases below ≲ 2 Mpc. These differences are still within the cosmic variance-suppressed uncertainties of the measurement (see gray bands in Figure 8). We also show the 1 𝜎 spread of these absolute, fractional differences and find them to also be around the 1 -2% level for the scales mentioned above. \nIn general, a difference between the two methods can occur in two different ways: (i) the geometry of the offsets are fundamentally different in the particle and shell baryonification methods, as in the former some particles will be displaced just along the line-of-sight and have no impact on the final, 2D \nFigure 3. The absolute fractional difference in the baryonification predictions, averaged over 32 parameter choices, between the 2D shell method and the 3D particle method (see Section 2.2 for details on both methods). The dashed lines are the standard deviation of the absolute fractional differences. The 32 models are a Sobol sequence drawn from a prior defined in Table 1. The bias is within 2% across all scales and increases for higher-order statistics for scales below ≲ 2 Mpc. We deem this a negligible systematic; see text for more details. \n<!-- image --> \nfield, whereas the projected displacements are always perpendicular to the line-of-sight; and (ii) the presence of interlopers, which are halos that overlap in 2D space but are distant in 3D space, as the offsets predicted by the smaller halo will impact the matter distribution of the larger halo given both distributions will contribute to the pixel being displaced. Formally, this latter effect is indeed accounted for through the two halo profile in Equation (3). Both effects are alleviated when the projection is done over smaller distances, i.e. if the shells are thin. The tests above are done with the projection scale set to 𝐿 = ≈ 300 Mpc whereas in full-sky simulations - such as the U/l.pc/a.pc/g.pc/a.pc/m.pc we perform shell baryonification on in Section 5 the shell thickness is ≈ 60 -100 Mpc. Therefore, Figure 3 is a more strict test of the method. \nFollowing the discussions in Section 2.2, the model for the displacement function and projected displacement function should be equivalent as both are derived from the same input profile and are physical models of the 3D and projected density distribution, respectively. Figure 3 shows our predictions obey this condition down to at least ≈ 1 Mpc (2 Mpc) for up to thirdorder (fourth-order) in the density field.", '4.1.2 Dependence on concentration': "To first order, halos can be defined using just their mass and redshift. Once these two properties are given, many other halo properties can be approximately derived through simple scaling arguments. However, there are a number of secondary halo properties that add significant information about the halo. Of these, the halo concentration plays a key role and has been studied extensively as a result. \nIn our current baryonification model, which does re- \nFigure 4. The impact of concentration on the displacement function for halos of two different masses. There is a strongly scale-dependent effect from varying 𝑐 200c . Note that we use a large range of 𝑐 200c values (beyond our physical expectations for halos of these masses) for illustrative purposes. In general, lower 𝑐 200c halos require stronger contraction (more negative displacements) on small scales, and the physical scale of maximum displacement is also shifted to slightly larger scales. This prediction is made using the fiducial values of the baryonification parameters (see Table 1). \n<!-- image --> \nquire an input 𝑐 200c per halo, we simply use a precomputed concentration-mass relation - in our case the model of Diemer & Joyce (2019) - and assign each halo a 𝑐 200c. This is then used in the rest of the pipeline. This is effectively a mass-only model as the halo is still defined using just its mass; a halo of a given mass will also have a fixed concentration value. However, at fixed halo mass there is a ∼ 40% scatter in the concentration values ( e.g. , Wechsler et al. 2002; Diemer & Kravtsov 2015; Diemer & Joyce 2019; Ragagnin et al. 2019; Anbajagane et al. 2022a). Furthermore, the concentration value of a halo depends on the halo's cosmological context, i.e. its entire accretion history, its environment and more ( e.g. , Wechsler et al. 2002; Mansfield & Kravtsov 2020). Thus, the baryonification model's predictions for the cosmological correlations can be more accurate if we self-consistently include the concentration of each halo as measured in the simulation. \nFirst, we show in Figure 4 the impact of 𝑐 200c on the displacement function for halos of the Milky Way mass-scale and the cluster mass-scale. In both cases, lower 𝑐 200c values lead to a more negative displacement at small scales. A lower 𝑐 200c implies a more diffuse halo and therby requires a stronger displacement value to transform the mass distribution into the more concentrated form that is generated by the presence of baryonic matter. The second effect of a lower 𝑐 200c is \nFigure 5. The fractional difference in the baryonification predictions, for the 2nd, 3rd and 4th moments of the density and tSZ field, when including/ignoring the concentration information. The difference is below 1% for almost all correlations, and is most amplified only for the cross-correlations where varying the concentration will generate more spatially correlated features across the density and tSZ field. \n<!-- image --> \nincreasing the radial scale corresponding to the maximum displacement. This is because for a more diffuse halo, the gas distribution starts dominating the dark matter profile at larger radii. This analysis uses the fiducial baryonification parameters as defined in Table 1. \nWe then test the impact of this difference on the statistics of interest. Note that this test is done on Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc given those simulations have a readily available concentration measurement, and therefore the range of scales we probe here is larger than that of the other tests. We generate 32 models across the parameter prior and show the median, fractional deviation in the statistics. Figure 5 shows there is a ≈ 2%offset in the measured moments down to 5 Mpc, which is the smallest scale we can probe in Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc. In summary, while the 𝑐 200c variation does have an impact on the statistics, it is not significant above > 5 Mpc and is completely negligible above > 10 Mpc. It is likely the effect is more pronounced towards smaller scales and for higher orders of the fields. We find it is best to use the simulation-measured concentration when available. \nFigure 4 also shows that the difference in the displacement function is minor when varying the concentration by nearly two orders of magnitude. This implies the model is insensitive to using other halo concentration-mass relations ( e.g. , Child et al. 2018; Ishiyama et al. 2021; Anbajagane et al. 2022a; Shao et al. 2023; Sorini et al. 2024) which deviate, to varying \nlevels, from the predictions of Diemer & Joyce (2019). \nOne can also trivially amplify the impact of concentration on the measurements by setting 𝛼 𝑋 ≠ 0 (see Equation (16) and Section 2.1) to enforce a relationship between 𝑐 200c and the gas profile parameters. Thus our test above is primarily showing that the default, fiducial model is not sensitive to the choice of concentration. While P24 find values of 𝛼 𝜃 ej that allow the model predictions to match the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG simulations, more detailed simulation studies are required to understand the correlation between the gas parameters and the halo concentration. Such a correlation could be important for the baryonification technique as the halos' concentration values are correlated with their spatial distribution, an effect commonly referred to as 'assembly bias' ( e.g. , Zentner et al. 2014). Note that some aspect of this phenomenon can already be included in the current baryonification method, through using the concentration measured for each individual halo in the simulation (as opposed to assuming a concentration-mass relation, which will not include the effects of assembly bias).", '4.1.3 Dependence on ellipticity': "Another critical secondary property of the halo is its shape. To good approximation, halos are quasi-spherical objects. However, they are not exactly spherical, and their ellipticities/orientations are sourced by the cosmological process of structure formation; meaning the shape and orientation of halos exhibit cosmologically sourced, spatial correlations. Such correlations are closely connected to the 'intrinsic alignments' effect studied in the weak lensing community (see Troxel & Ishak 2015, for a review). \nIt is therefore important to determine the impact of the ellipticity correlations in the baryonification pipeline, for modelling the statistics of interest to us. The implementation of ellipticities in our model is trivial. The halo-centric distance to a pixel is now scaled as 𝑥 2 + 𝑦 2 → 𝑥 2 + 𝑦 2 / 𝑞 2 where 𝑞 ≤ 1 is the ratio of the minor axis to the major axis. /one.sup/seven.sup Under this formalism, the major axis is assumed to be of length 𝑅 200c. In practice, this scaling is done by transforming the coordinate system - rotating the pixel grid so the major/minor axis of the halo is along the x-axis and y-axis of the grid - and then rescaling the 𝑦 distance by 𝑞 as denoted above. The relevant quantity, a displacement or gas pressure, is computed at the location given by this rescaled distance. We perform this test with I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG, where we computed the orientations of halos using the same pipeline as Anbajagane et al. (2022a); this work computes the mass inertia tensor in 3D to compute the 3D ellipticity. In our case, we take only the 2 × 2 subset of the tensor corresponding to the x and y directions, and \nFigure 6. Different maps of the gas pressure: from the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG hydrodynamic run (top left), from the baryonification model of the DMO run assuming circular profiles (top right), elliptical profiles (bottom left), and 'extreme' elliptical profiles (bottom right). The former elliptical model uses ellipticies computed directly in the simulation, whereas the latter is simply a scaled, more-elliptical version of the former, 𝑞 ex = 0 . 6 𝑞 TNG . We can clearly see the correlation between the halo orientations and the large-scale density field: the alignments are along filaments or point in the direction of the nodes. \n<!-- image --> \ncompute a 2D ellipticity using the new tensor. \nFigure 6 shows a visual comparison of theoretical maps made with and without the orientation information. One can clearly see the correlation of the ellipticities with the largescale density field. The alignments are often in the direction of filaments or are pointing towards nodes in the field. This is most prevalent in 'Ellip. Ex.' map, which is an extreme version of the ellipticity, given as 𝑞 ex = 0 . 6 𝑞 TNG, and shown purely for illustrative purposes. \nFigure 7 then shows the change in the moments of the fields due to the inclusion/exclusion of the halo ellipticity in our modelling. The change in the statistics is below percentlevel in almost all cases. In summary, the ellipticity of halos is not necessary for modelling the moments of the field. Other statistics - particularly those with anisotropic information, such as the bispectrum which is a function of angle, and not just isotropic information like the moments - may \nFigure 7. The fractional difference in the baryonification model predictions when excluding/including ellipticity in the model. We show all moments of the density and tSZ field, up to the 4th moment. The impact is sub-percent for all moments, and generally grows towards small-scales. \n<!-- image --> \nhave stronger sensitivity to this choice and therefore must be separately tested. Similarly, methods that involve the explicit detection/characterization of filamentary structures, such as topological measures ( e.g. , Heydenreich et al. 2021, 2022; Euclid Collaboration et al. 2023), could have a stronger sensitivity to the choice of including/ignoring ellipticity.", '4.2 Comparison to hydro simulation': "Wenowcheckthatour baryonification model can jointly fit measurements from hydrodynamical simulations. This is done by computing the fractional difference between the simulation measurements and our best-fit baryonification predictions for all moments of the density and tSZ field from 2nd to 4th order - and comparing the difference against the measurement uncertainty described in Section 3.5. \nFigure 8 compares our predictions with measurements made on I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG. Our predictions use the DMO simulation whereas the measurements are made on the full hydrodynamical simulation. Each subplot shows the fractional difference for different moments. The gray-bands denote the 1 , 2 , 3 𝜎 uncertainty bounds computed using the method in Section 3.5. The baryonification model is accurate to within the cosmic variance-suppressed uncertainties of the moments; the fraction differences are always within the 1-2 𝜎 bounds (gray bands). Note that the percent-level agreement for the large-scales of the density auto-correlations (at any order) is expected as the baryonic imprints are negligible on such scales. \nFigure 8. The fractional difference between the moments of the baryonified DMO simulations and of the TNG hydrodynamical simulation. All moments are computed on 2D maps at 𝑧 = 0. The gray bands show the 1, 2, 3 𝜎 uncertainties. The dotted black lines denote 10% fractional errors. Cosmic variance is highly suppressed given both maps derive from the same initial conditions. The joint fit (yellow line) is within 1-2 𝜎 for all different moments. We a-priori expect some irreducible deviations due to the low statistics of massive halos in I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG; see Section 4.2 for details. For the panel with ⟨ 𝑦𝑦 ⟩ , the 'Individual Fit' line is underneath the 'Joint Fit' line. \n<!-- image --> \nWe a-priori expect to poorly match some features on small-scales, and particularly for any moments that involves the tSZ field. The signal for the tSZ field comes from the most massive clusters, but the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG simulation has only a handful of such clusters ( e.g. , only two clusters above 𝑀 200c > 10 15 M ⊙ ). For a larger simulation, with a larger sample of clusters contributing to the signal, any individual characteristics of the cluster would average out over the population, and the baryonification prediction - which by construction represents only the average cluster - would match the simulation measurement better. When the simulation contains only a few clusters, these individual characteristics can have a more dominant role in the emergent signal. For example, the most massive cluster in I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG - which will dominate the total tSZ signal, especially for the higher-order moments (see Figure 9) - has undergone a recent merger. Our model will not account for this characteristic. However, a significantly larger simulation would have enough massive systems such that the individual merger history of a single object would not have a notable impact on the observed field. \nFigure 8 also shows the results from fitting only the density \nfield, or only the tSZ field. The density field is not a sensitive probe of all the different baryonic physics, and therefore a fit to the density alone does not correctly estimate the amplitude of the tSZ field. We also show the result from fitting a single moment at a time. Unsurprisingly, this fit is the best of all, given the large degrees of freedom of the model compared to number of datapoints used to constrain it. Note that in some moments, even the invidiual fit is in the 2 𝜎 regime, which indicates the presence of the individualistic features in the few most massive clusters (which are not captured in the model). We have separately used the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pc simulations (Vogelsberger et al. 2014), a predecessor to the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG suite we use here, and verified that our model predictions are not systematically low on small-scales in those analyses. This further indicates that the offsets seen in Figure 8 is a stochastic fluctuation, induced by the individual histories of the handful of the largest clusters in the simulation. \nUnder the current statistical uncertainties, we find that our model is able to jointly fit the simulation data at a fractional accuracy that is at worst between 5% to ≈ 50%, depending on the exact moment being considered, with the fractional \ndifference increasing for moments with larger measurement uncertainties (for example, ⟨ 𝑦 3 ⟩ or ⟨ 𝑦 4 ⟩ , whose signal is dominated by just a few halos in the TNG volume). Given the caveats related to TNG that we have discussed throughout the text above, a more robust analysis is needed to calibrate the baryonification model accuracy to much better precision. Such an analysis would require a much larger simulation; for example B/a.pc/h.pc/a.pc/m.pc/a.pc/s.pc (McCarthy et al. 2017) or M/i.pc/l.pc/l.pc/e.pc/n.pc/i.pc/u.pc/m.pc-T/n.pc/g.pc (Pakmor et al. 2023). We leave these efforts to future work. \nWhile the validation of the baryonification method for the density two-point statistics (or 2nd moments) has been done to the percent-level ( e.g. , Giri & Schneider 2021; Aric'o et al. 2021b), the model for the other eleven measurements shown in Figure 8 can still be useful even if the model accuracy is larger than 1%: the noise level in the higher-order moments increases significantly compared to the 2nd moments ( e.g. , Anbajagane et al. 2023, see their Figure 6, for the lensing-only moments), while the tSZ field is generally more noise-dominated than the lensing field. These two factors add leniency to the validation requirements of these other eleven moments. The actual requirements for using baryonifcation is survey-specific and must be determined explicitly for each survey and choice of statistic. \nAppendix D also shows that the baryonification model can jointly fit all moments measured across multiple redshifts in the I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG simulations.", '4.3 Mass dependence': 'In Section 4.2 we show that our model is flexible enough to jointly fit the simulation measurements within their statistical uncertainties. We now use this model to check the massdependence of the predicted signal for the different moments. This is analogous to similar plots made for the two-point function in Pandey et al. (2022, see their Figure 2) or To et al. (2024, see their Figure 3), and lend insight into the origin of the signal in different measurements. In general, we expect higher-order moments to be more sensitive to more massive halos than the 2nd moments, and similarly, for the tSZ field to be more sensitive to more massive halos than the lensing field. The former is because taking higher powers of a variable amplify the tails of its distribution, and the density field has a strongly positive-skewed tail - occupied by massive halos -whose presence/importance is amplified by taking higher powers of the field. The latter is because weak lensing depends linearly on the density field and therefore mass, whereas the tSZ signal follows ∝ 𝑀 5 / 3 ( e.g. , Lee et al. 2022, see their Figure 1 for a comparison across simulations) and therefore is sensitive to higher masses than lensing. \nWe estimate the mass-dependence of the moments using the Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc simulations, and in particular using their high- \nresolution runs, where halos of 10 13 M ⊙ are resolved by 100 particles. The contributions from halos of a given mass, 𝑀 1, is computed through the difference 𝑋 ( > log 10 𝑀 1 ) -𝑋 ( > log 10 𝑀 2 ≡ log 10 𝑀 1 -𝛿 ) , where 𝛿 = 0 . 25 and 𝑋 is a Nthorder moment measured on a density and/or tSZ field, with the field generated using only halos of mass greater than 𝑀 1 or 𝑀 2. We use the baryonification model to make fields for different choices of 𝑀 and take simple numerical differences to estimate the derivative of the measurement with mass, d ln 𝑋 / d ln 𝑀 . We approximate d ln 𝑋 ≈ Δ 𝑋 / 𝑋 fid where 𝑋 fid is the fiducial signal from using all halos. Thus d ln 𝑋 / d ln 𝑀 represents the fractional change - compared to the fiducial case of using all available halos - in the moments of the fields due to halos of a given mass. \nFor the tSZ field, the change d ln 𝑋 / d ln 𝑀 represents the contribution of a given halo mass to the total tSZ signal, but for the density field it represents the contribution to the baryonic imprints in the field and not the density field as a whole. Note that for the density field, the 𝑋 fid used in the expression Δ 𝑋 / 𝑋 fid is the total density field signal (and not just the total baryonic imprints signal). For this reason, the fractional differences for the density field will have a lower amplitude than those for the tSZ field. For all estimates, we have used 10 high-resolution simulations and averaged the derivatives across realizations. The mass-dependence will change based on the chosen baryonification parameters; for example, changing the 𝜇 𝑋 parameters in Table 1 can modify these derivatives. For our estimate of the derivative, we will use the fiducial parameter values listed in Table 1. \nFigure 9 shows these derivatives across different halo masses, for five different apertures used in computing the moments. As expected, the ⟨ 𝛿𝛿 ⟩ correlation has the lowest mass-dependence, as all other correlations have derivatives that peak towards higher masses. The derivatives of the density auto-correlations are also negative as the presence of halos (and namely, the astrophysical processes within them) generally suppresses power on these scales, while for the tSZ field the derivative will be positive as the presence of these halos increases the signal. The derivatives of moments that include the tSZ field are orders-of-magnitude larger than those that include the density field alone, showcasing the dramatically higher sensitivity of the tSZ field to baryonic physics. As discussed previously, the mass-dependence of a measurement peaks at higher masses when we increase the order of the moment or if we include the tSZ field. \nThe mass-dependence in Figure 9 highlights the difficulty in validating our method using simulations that span only sub- Gpc volumes. If we focus on the 2nd moments of the tSZ field, the minimum contributing mass is 𝑀 200c ≈ 10 14 . 5 M ⊙ , whereas for the 4th moments this mass scale is 𝑀 200c ≈ 10 15 . 2 M ⊙ . The I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG simulations have \nFigure 9. The derivative of the baryonification signal, with respect to the halo mass, for the moments of the density field and tSZ field (different panels, 𝑋 = ⟨ 𝛿𝛿 ⟩ , ⟨ 𝛿𝑦 ⟩ , . . . ) measured at different smoothing scales (colors). The y-axis represents the fractional contribution, from halos of a given mass 𝑀 , to the total signal. For the density field, the signal is the baryonic imprints in the field rather than the density field itself. See text for details on the measurement. The mass-dependence of higher-order moments, or any moments including the tSZ field, is increased to higher masses when compared to the 2nd moments of the density field. The exact values of the derivatives depends on the baryonification parameters used, and we use the fiducial values shown in Table 1. \n<!-- image --> \n/circledot \n/circledot \n/circledot \n/circledot \naround O( 100 ) halos at the former mass scale, but only a single halo at the latter mass scale. We require hydrodynamical simulations of Gpc-scale volumes in order to validate these higher-order moments to good accuracy. However, we reiterate that in practical use-cases, the accuracy requirements for these higher-order moments - especially those of the tSZ field - will be more lenient given the larger measurement uncertainties of such moments.', '5 FORECASTING': "Wenowforecast the power of current and future surveys in constraining these baryonification parameters, and study the parameters' different degeneracy directions. This is done by forward modelling the observables from surveys; namely, the lensing convergence field and the thermal Sunyaev-Zeldovich field. For brevity, we only briefly describe the procedure below and provide a detailed discussion in Appendix A. \nThe lensing field is modelled using the same pipeline from Anbajagane et al. (2023, 2024b). The maps are generated from the density fields of the U/l.pc/a.pc/g.pc/a.pc/m.pc simulations (Section 3.3), and then processed to contain all the observational effects found in the data, including the relevant survey mask/area and source galaxy redshift distribution. The two lensing surveys we focus on are: (i) the Dark Energy Survey (DES, The Dark Energy Survey Collaboration 2005), which is an optical imaging survey of 5,000 deg 2 of the southern sky, and is cur- \nrently the largest precision photometric dataset for cosmology. The Year 3 data products and cosmology results are available (Sevilla-Noarbe et al. 2021; Abbott et al. 2022), while the legacy Year 6 dataset is not yet available at the time of publishing this work. We focus on this Year 6 dataset. We then consider (ii) the Rubin Observatory Legacy Survey of Space and Time (LSST), which is a 14,000 deg 2 survey that probes higher redshifts, and is the successor to current weak lensing surveys. We forecast only for the final, year 10 dataset. The characteristics of the surveys (shape noise, and 𝑛 ( 𝑧 ) distribution parameters) are given in Table A1. More details can be found in Appendix A1 \nNext, the tSZ field is modelled by following the approach of Raghunathan et al. (2022a,b). We include simple Gaussian models for a number of 'foreground' componenets, such as the cosmic infrared background, radio background etc., as well as the thermal and atmospheric noise. We focus on two CMB surveys: (i) SPT-3G (Benson et al. 2014) is a survey of the southern sky with considerably deeper (lower noise) observations than other CMB surveys. The entire footprint covers O( 10 4 ) deg 2 of the sky with different noise levels. We focus on the main 1500 deg 2 footprint (i.e. excluding any wide/summer fields) that also overlaps completely with DES Y6. (ii) The Simons Observatory (SO, Ade et al. 2019) is an upcoming survey of ≈ 20 , 000 deg 2 of the sky, and has nearly complete overlap with the LSST Y10 footprint. For both surveys, we assume the analysis is supplemented by Planck data like in ex- \nting surveys (Madhavacheril et al. 2020; Bleem et al. 2022). The characteristics of the surveys (noise properties, beam, frequency bands etc.) are given in Table A2. Further details are provided in Appendix A2. \nWe note that all our analyses use relatively 'medium'resolution data-products: the U/l.pc/a.pc/g.pc/a.pc/m.pc simulations are run with 512 3 particles, and produce maps of NSIDE = 1024. These choices set a limit on the scales we can analyse. Simulations of higher resolution enable access to even smaller scales, which would place tighter constraints on the baryonic processes discussed in this work. Therefore all our constraints below could be improved further by pushing the analysis to even smaller scales. In this work, we are limited by the resolution of the simulations available for a Fisher forecast. \nThese results should also not be over-interpreted or generalized since our analysis explicitly limits the tSZ measurements to 𝜃 > 3 ' , whereas the tSZ power spectrum has the highest signal-to-noise at even smaller scales, ℓ ∼ 3000 (Raghunathan et al. 2022b, see their Figure 4). Our tSZ maps are generated at NSIDE = 1024 - the same choice as that of the lensing field - whereas the maps from current SPT datasets are made at NSIDE = 8192 (Bleem et al. 2022). Thus, there is more usable information in the tSZ field that we have not accessed in our datavectors given we limit our maps (and therefore our analysis) to a coarser resolution. This is done primarily in the interest of minimizing the computation cost, as generating a tSZ map of NSIDE = 8192 would take around 64 × longer than generating one of NSIDE = 1024. Note that for the lensing analysis, the choice of NSIDE = 1024 is realistic, though it may be possible to access smaller scales ( NSIDE = 2048 , 4096) - if the lensing systematics are better controlled/modelled - which would improve our constraints. Theconstraints from both fields could also improve/deteriorate depending on the summary statistics being considered; for examples in the lensing field, see Euclid Collaboration et al. (2023); Anbajagane et al. (2023); Gatti et al. (2024a). We have not explored this direction, and have limited ourselves to the moments in this work, as they have already been used to constrain cosmology (Gatti et al. 2022, 2024a). \nWe now present the Fisher information for a number of cases, each changing the statistics being measured, the fields being used, and/or the parameters being varied. The key goal of these estimates is to understand the degeneracy directions of the different parameters under different analysis choices, and also the relative constraining power of surveys. The Fisher information is estimated as, \n𝑭 𝑖 𝑗 = ∑︁ 𝑚,𝑛 𝑑 e 𝑋 𝑚 𝑑𝜃 𝑖 GLYPH<0> C -1 GLYPH<1> 𝑚𝑛 𝑑 e 𝑋 𝑛 𝑑𝜃 𝑗 , (35) \nwhere 𝑑 e 𝑋 𝑚 𝑑𝜃 𝑖 is the mean derivative of point 𝑚 in data vector 𝑋 with respect to parameter 𝜃 𝑖 , where the mean is computed us- \nns of the surveys for the analysis of DES and SPT, and 300 realizations for that of LSST and SO. We compute this derivative around the fiducial parameter values, as detailed in Table 1. Then, C -1 is the inverse of the numerically estimated covariance matrix and includes the Hartlap correction factor (Hartlap et al. 2007), \nC -1 → 𝑁 sims -𝑁 data -2 𝑁 sims -1 C -1 . (36) \nThe Hartlap factor for all analyses in this work is ≳ 0 . 9. The datavector, 𝑋 , consists of the moments of the field, computed on 10 scales ranging from 3 . 2 ' < 𝜃 < 200 ' . These are the same choices as previous works on the moments of the lensing field (Gatti et al. 2020; Anbajagane et al. 2023; Gatti et al. 2024a) and we adopt it for the tSZ field as well. The covariance matrix is estimated by generating 6000 to 8000 realizations of the surveys, with the number varying depending on the survey being analysed. The O( 10 4 ) realizations are enabled by the 2000 independent simulations available in the U/l.pc/a.pc/g.pc/a.pc/m.pc simulation suite. These simulations are run at the fiducial cosmology, chosen to be the best fit values of Planck Collaboration et al. (2016), and the derived baryonification products use the fiducial values of the model parameters (see Table 1). \nThe Fisher information estimated in Equation (35) can be artificially increased by numerical noise as this noise can break parameter degeneracies; see Coulton et al. (2023, their Appendix A) for a discussion of this. We quantify this amplification in our estimates by varying the number of realizations used to compute the derivatives and the covariance. The Fisher information changes by ≲ 2% if we estimate the covariance using half the number of realizations. Doing the same exercise on the derivative estimates change the Fisher information by ≲ 10%. Our discussion below focus on constraints improving by more than factors of 2, and are therefore unaffected by any such numerical uncertainties at the 10% level. \nFigure 10 shows the constraints for different combinations of moments, as measured on mock maps of DES Y6 and SPT 3G; the numbers are listed in Table 2 as well. In addition to the baryonification parameters, we also include the cosmological parameters Ω m, which is the matter energy fraction at 𝑧 = 0, and 𝜎 8, which is the root-mean squared fluctuations of the density field at 𝑧 = 0, smoothed on a 8 Mpc / ℎ scale. We include, and therefore marginalize over, all baryonification parameters that are listed with a prior in Table 1. We however only show a subset of them in this plot for visibility. \nThe inclusion of higher-order moments (blue and yellow contours) greatly improves the constraints over using the 2nd momentsalone(purple contours). This is expected as baryonic imprints are tied to the astrophysical processes within halos and are therefore stronger on non-linear scales. These scales are also where the field is more non-Gaussian, and parameter constraints improve from including measurements of any \nFigure 10. Fisher constraints on the baryonification/cosmology parameters (see Table 1 and Section 2) from moments of the lensing and tSZ fields constructed with DES Y6 and SPT 3G specifications, respectively. The contours show 1 𝜎 and 2 𝜎 constraints from using the 2nd/3rd/4th moments. The dashed green/peach lines along the diagonal panels show the 1D posteriors from using only the 3rd/4th moments, respectively. We show only a subset of parameters for the sake of brevity; the full constraints are listed in Table 1. The inclusion of higher-order information significantly increases constraining power. \n<!-- image --> \nhigher-order correlations at these scales. Aric'o et al. (2021a) already show that a range of baryonification parameter choices can accurately fit a matter power spectrum from simulations but only a subset of those can also jointly fit the matter bispectrum, indicating that the addition of higher-order information constrains the parameter space further. We expand on this by using up to 4th order in the fields, and also by using both the lensing and tSZ fields. Figure 10 also shows the marginalized constraints from using only the 3rd moments (4th moments), presented as a green (peach) dashed line in the 1D posteriors \nalong the diagonal panels of the triangle plot. /one.sup/eight.sup In many cases, the 'only 3rd moments' and 'only 4th moments' constraints are more constraining than the 2nd moments alone (though this is not the case for the cosmology parameters). \nFor most parameters, the largest relative improvement is between using 2nd moments alone and using the combination of 2nd and 3rd moments. Including the 4th moments to this \nTable 2. The constraints presented in Figure 10, 11, and 12 for (i) different statistics measured on both the lensing and tSZ field (left columns), (ii) for different combinations of fields using the 2nd and 3rd moments (middle columns), and (iii) for different combinations of surveys using the 2nd and 3rd moments of the lensing and tSZ fields (right columns), respectively. The inclusion of higher-order statistics improves all constraints by factors of 3 to 4. Including the tSZ field in the analysis similarly improves constraints by up to factors of 10. The constraints from LSST Y10 and SO are factors of 4-5 better than that of DES Y6 and SPT-3G. The parameters are partitioned visually into cosmological parameters, baryonification parameters shown in the figures of this work, and remaining parameters that are not presented but are marginalized over. In analysis (ii), we do not use 𝛼 nth due to it having no impact on the 𝜅 field; see text in Section 5 for details. \ncombination improves constraints as well but the relative gain is smaller. This is similar to the lensing-only analysis of Anbajagane et al. (2023), where the 4th and 5th moments did not improve the constraining power. The one exception we find is 𝜃 ej, which improves by nearly a factor of three when including the 4th moment. \nFigure 10 indicates that a number of degeneracy directions are broken by including higher-order information. It also shows that the cosmological parameters, Ω m and 𝜎 8, can be constrained even when marginalizing over astrophysical systematics. Note that these analyses use both the lensing field and the tSZ field. Limiting the analysis to the lensing field alone deteriorates the constraining power significantly (see Figure 11 below). In general, this result highlights the utility in jointly constraining cosmology and astrophysics using a combination of the lensing field and tSZ field. P24 have already shown this to be the case when using only two-point statistics in the data vector (see their Figure 8). \nWhile the above analysis decomposes the information into different orders, we now decompose it by field. In Figure 11 we split the Fisher information by convergence field and tSZ field. In the former case, we use all auto/cross-correlations between all tomographic bins of the lensing data, while in the latter we use only auto-correlations of the singular tSZ field. We then show the constraints from moments that have at least one power of the lensing field and one of the tSZ field, and finally the constraints from all moments of the two fields. While the above analysis also used the 4th order moments, we will now use the combination of the 2nd and 3rd as our fiducial datavector, given this measurement has already been used (for the lensing field) to constrain cosmology ( e.g. , Gatti et al. 2022, 2024a). This analysis also does not vary the \nFigure 11. Fisher constraints on the baryonification and cosmological parameters from different combinations of the convergence and tSZ field as measured by DES Y6 and SPT 3G. The datavector uses the 2nd and 3rd moments. The convergence field and tSZ field have different sensitivities to the parameters, and the degeneracy directions are often orthogonal. As a result the combination of fields is significantly more constraining. The 1D marginal distributions of Ω 𝑚 and 𝜎 8 also show a dotted blue line, which is the lensing-only constraints without marginalizing over any baryonification parameters. \n<!-- image --> \nµ \nnon-thermal pressure parameter 𝛼 nt, see Equation (27), since this parameter has no impact on the 𝜅 field, leading to zero Fisher information and therefore a singular Fisher matrix. We \nchoose to remove this parameter for three variants (lensingonly, tSZ-only, cross correlations-only) in this analysis - and not just for the 𝜅 -only analysis where the issue of a singular Fisher matrix arises - to preserve consistency across analysis setups and therefore simplify the comparisons of the different constraints. \nFigure 11 shows that the lensing and tSZ field have different sensitivities to combinations of parameters; in many panels, the degeneracy directions of the convergence-only and tSZonly case are nearly orthogonal. As a result, any analysis using combinations of the convergence and tSZ field is far more constraining than the individual fields. Another interesting note is that the lensing field is able to better constrain combinations of the redshift parameters, 𝜈 𝜃 ej and 𝜈 𝑀 c , as shown in Table 1. This is because the lensing survey has more redshift information via the availability of tomographic bins, whereas the tSZ field is a singular field integrated over all redshifts. This result highlights the highly complementary nature of combining the lensing and tSZ fields when constraining baryonification models; P24 (see their Figure 8) have shown the same for the two-point functions. It is also interesting that the crosscorrelations alone can provide a relatively good constraint on the baryonification parameters (as well as cosmology). Such cross-correlations are more robust measurements than their auto-correlation counterparts, as additive systematics in each field (that are uncorrelated across the different fields) will not be present in the measurement. \nThe field-by-field comparison of Figure 11 also highlights the improvement in cosmology constraints ( Ω m and 𝜎 8) due to the addition of the tSZ field. The cross-correlation of the lensing and tSZ fields is a significant factor in this improvement. The results also indicate that the tSZ-only analysis is unable to jointly constrain baryonification models alongside cosmology. However, this result should not be over-interpreted or generalized since our analysis explicitly limits the tSZ measurements to 𝜃 > 3 ' , whereas the tSZ power spectrum has the highest signal-to-noise at even smaller scales, ℓ ∼ 3000 (Raghunathan et al. 2022b, see their Figure 4). Our tSZ maps are generated at NSIDE = 1024 - the same choice as that of the lensing field - whereas the maps from current SPT datasets are made at NSIDE = 8192 (Bleem et al. 2022). Thus, there is more usable information in the tSZ field that we have not accessed in our datavectors given we limit our maps (and therefore our analysis) to a coarser resolution. This is done primarily in the interest of minimizing the computation cost, as generating a tSZ map of NSIDE = 8192 would take around 64 × longer than generating one of NSIDE = 1024. For the lensing analysis, the choice of NSIDE = 1024 is realistic, with some potential to extend to higher resolution if the systematics can be controlled robustly. \nThe 1D marginal distributions in Figure 11 also shows a \nµ \nFigure 12. Fisher constraints of the baryonification parameters from different survey datasets. The datavector includes only the 2nd and 3rd moments, which contain the bulk of the signal-to-noise. The constraints from LSST Y10 and SO is between 3 to 5 times better than that from DES Y6 and SPT-3G. We have marginalized over all other parameters in the analysis. The constraints are also listed in Table 2. \n<!-- image --> \ndotted blue line, which are the results from a lensing-only analysis (using 2nd and 3rd moments) but with all baryonification parameters fixed. The constraints on cosmology 𝜎 ( Ω 𝑚 ) = 0 . 008 and 𝜎 ( 𝜎 8 ) = 0 . 013 - are comparable to those from using all 2nd and 3rd auto/cross-moments of the lensing and tSZ field after marginalizing over all baryonification parameters. Note that this compares lensing-only with a lensing plus tSZ analysis. In Appendix E, we consistently use a lensing plus tSZ analysis, with all baryonification parameters either fixed or varied in different stages, and find the constraints on cosmology degrades by ≈ 60% if we vary all baryonification parameters. \nFinally, Figure 12 compares constraints from current surveys (DES Y6 and SPT-3G) with upcoming ones (LSST Y10 and SO). We find the parameter degeneracies in the upcoming surveys are broadly similar to those in the current surveys. The extended redshift range probed by LSST Y10 (compared to DES Y6) has not changed the degeneracy directions. The parameter constraints themselves improve by factors of 3 to 5 depending on the specific parameter. While these results are simplistic in their accounting of systematic uncertainties - we have not marginalized over intrinsic alignments for the lensing field, and we include no systematic model to be marginalized over for the tSZ field - they show the gain that could be realized from the final survey datasets of the next decade, relative \nto constraints from ongoing surveys. \nAll analyses above, summarised in Table 2, show that a numberofanalysis setups result in parameter uncertainties that are of the same order as the fiducial parameter values listed in Table 1: (i) the 𝜈 𝑋 parameters, which characterize the redshift evolution of 𝜃 ej and 𝑀 c, (ii) 𝛾 which is the gas slope at 𝑅 ∼ 𝑅 ej, and (iii) the mass-dependence parameters 𝜇 𝑋 . This may indicate that robust predictions of the lensing and tSZ moments can be obtained while fixing these parameters. However, a robust validation of this statement requires fitting this model (with and without varying the parameters mentioned here) to mock surveys derived from different hydrodynamical simulations. Appendix D also presents results that indicate the 𝜈 𝑋 parameters may not be necessary for jointly modelling the moments measurements across redshift. Bigwood et al. (2024), who use the baryonification model of Giri & Schneider (2021) to analyse data from DES and the Atacama Cosmology Telescope (ACT), also find qualitatively similar behavior to us for constraints on their mass-dependence parameter 𝜇 (see their Table B2).", '6 CONCLUSIONS': 'Cosmology is in a golden age of data-driven science, and this is set to only improve over the upcoming decade. For example, measurements of weak lensing will extend to larger fractions of the sky, and to much higher statistical precision (Spergel et al. 2015; The LSST Dark Energy Science Collaboration et al. 2018; Racca et al. 2016). Such measurements will access a broader range of scales than was previously available. However, the actual use of these measurements - namely those on smaller, more non-linear scales - for obtaining constraints is a non-trivial task due to uncertainties in modelling the physics of baryons on these scales. Addressing this issue is tantamount to better constraining Λ CDM cosmology ( e.g. , Secco et al. 2022b; Amon et al. 2022; Amon & Efstathiou 2022) and also to using lensing to extract information about other cosmological phenomenon, such as inflation (Anbajagane et al. 2024b; Goldstein et al. 2024). \nWe introduce here a map-level baryonification technique that jointly generates baryon-corrected density fields (and thereby lensing fields) as well as tSZ fields, using N-body simulations as an input. This is done in a computationally efficient manner by working directly with full-sky maps from simulations rather than with three-dimensional particle snapshots. We perform various validations on this technique and characterize its performance on 2D simulation fields, as well as in mock analyses of wide-sky datasets. We perform all validations by using the moments of the fields, from 2nd to 4th order, as our summary statistics. We include all autoand cross-correlations at these orders. Our main results are as follows: \n- · The baryonification method can be applied directly to 2D fields (rather than 3D snapshots) in a robust manner. After accounting for the pixel window function and the line-of-sight projection scale, the residuals between the 2D and 3D version is under 2% for most physical scales (Figure 3).\n- · Themodelpredictions are fairly insensitive to secondary halo properties, such as halo concentration and ellipticity. (Figure 5 and 7).\n- · The model is adequately flexible and can jointly fit all moments (up to 4th order) of the total matter density and gas pressure fields, as measured in I/l.pc/l.pc/u.pc/s.pc/t.pc/r.pc/i.pc/s.pcTNG. The residuals are within the uncertainties in all cases (Figure 8). The model can also jointly fit across multiple redshifts (Figure D1).\n- · The moments have significant differences in the halo mass their signal is most sensitive to; the total mass range spans nearly two orders of magnitude across the different moments, highlighting their complementary nature (Figure 9).\n- · A Fisher forecast of current/upcoming surveys shows the inclusion of higher-order information dramatically improves the constraining power on the baryonification model (Figure 10).\n- · The tSZ and lensing fields are more constraining for specific subsets of parameters, and the parameter degeneracy directions in the different fields are nearly orthogonal, leading to the combination of fields providing significantly better constraints (Figure 11). The lensing data is also more informative on redshift-dependent parameters due to the tomographic redshift information of the dataset. \nThis work provides a first, initial confirmation that the baryonification method is viable for jointly modelling the lensing and tSZ fields directly on the sky, across multiple orders in the fields. We stress that this is not a final and conclusive validation of the map-level baryonification technique. As was pointed out before, a more thorough validation will require full-sky hydrodynamical simulations, such as M/i.pc/l.pc/l.pc/e.pc/n.pc/i.pc/u.pc/m.pc-TNG (Pakmor et al. 2023) and/or the F/l.pc/a.pc/m.pc/i.pc/n.pc/g.pc/o.pc project (Schaye et al. 2023), and a focus on other popular summary statistics used for analyses of weak lensing and tSZ fields. \nOnce the baryonification method has been more precisely validated, a limiting step in applying it to data will be the forward model of the tSZ field and its foregrounds, done consistently across multiple orders . Omori (2022) have provided a precisely validated prescription for converting an N-body \nsimulation into realistic tSZ fields (with all foregrounds consistently modelled, and then also verified at the two-point level), but this requires high fidelity simulation products which are often not available for the large simulation suites used for forward modelling the lensing field (Kacprzak et al. 2023; Anbajagane et al. 2024b; Jeffrey et al. 2024; Gatti et al. 2024a). Explorations of more approximate tSZ forward models will be needed in order to analyse the tSZ field with the same simulation-based forward modelling approach used for the lensing field. \nThe baryonification method has rapidly become a important tool in characterizing the two-point correlation functions measured by lensing surveys. The method, however, is a general prescription on how to add baryonic fields to N-body simulations through an approximate, phenomological approach. In this work, we use this formalism to model the lensing and tSZ field. One could similarly use this prescription to model other fields that are observables of gas physics, such as the kinematic SZ effect, which has already been measured using different combinations of CMB and galaxy surveys ( e.g. , Mallaby-Kay et al. 2023; Hadzhiyska et al. 2024), and the X-ray luminosity, measured for a large fraction of the sky by the eROSITA survey (Bulbul et al. 2024). In fact, some of these observables have already been used to constrain baryonification models (Schneider et al. 2019, 2022; Grandis et al. 2024; Bigwood et al. 2024). Thus, we have only probed some aspects of the full potential provided by the baryonification formalism. \nThe current state of widefield survey science is rapidly advancing towards surveys that cover O( 1 ) fractions of the sky, and have overlapping areas of O( 10 4 ) deg 2 . In this datarich age, the advent of multi-survey, multi-wavelength science provides powerful ways to extract robust physical inferences at ever-increasing precision. The baryonification method provides one way of performing joint-analyses across different surveys in a rapid, flexible way, and is complementary to other approaches such as dedicated hydrodynamical simulations. Our prescription for the map-level baryonification pipeline is documented and made publicly available. We hope this enables easy explorations/extensions of the baryonification pipeline that build up the synergies of the growing multisurvey landscape.', 'ACKNOWLEDGEMENTS': 'DAissupported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1746045. SP is supported by the Simons Collaboration on Learning the Universe. CC is supported by NSF grant AST-2306166. \nWe thank Marco Gatti and Yuuki Omori for discussions during early stages of this work, Srini Raghunathan for guid- \nance on the tSZ forward models used in Raghunathan et al. (2022a,b), and Kayla Kornoelje for details on the linear combination method used by SPT. We also thank the referee for their many helpful comments that greatly enhanced our discussion and results. The conception and pursuit of this work arose from many conversations between DA and Eric Baxter. The authors are grateful to Eric for his collaboration and friendship over the many years, and for his efforts in making our research communities vibrant avenues for discussion and discovery. He will be missed. \nAll analysis in this work was enabled greatly by the following software: P/a.pc/n.pc/d.pc/a.pc/s.pc (McKinney 2011), N/u.pc/m.pcP/y.pc (Van der Walt et al. 2011), S/c.pc/i.pcP/y.pc (Virtanen et al. 2020), and M/a.pc/t.pc/p.pc/l.pc/o.pc/t.pc/l.pc/i.pc/b.pc (Hunter 2007). We have also used the Astrophysics Data Service (ADS) and arXiv preprint repository extensively during this project and the writing of the paper.', 'DATA AVAILABILITY': 'All simulation datasets used in this work are publicly available, and can be found at the links provided in the relevant subsections of Section 3. \nThe baryonification pipeline is publicly available at https://github.com/DhayaaAnbajagane/ BaryonForge .', 'REFERENCES': "Abadi M. G., Navarro J. F., Fardal M., Babul A., Steinmetz M., 2010, MNRAS, 407, 435 \nAbbott T. M. C., et al., 2022, Phys. Rev. D, 105, 023520 \nAde P., et al., 2019, J. Cosmology Astropart. Phys., 2019, 056 \nAmodeo S., et al., 2021, Phys. Rev. D, 103, 063514 \nAmon A., Efstathiou G., 2022, MNRAS, 516, 5355 \nAmon A., et al., 2022, Phys. Rev. D, 105, 023514 \nAnbajagane D., Evrard A. E., Farahi A., Barnes D. J., Dolag K., McCarthy I. G., Nelson D., Pillepich A., 2020, MNRAS, 495, 686 \nAnbajagane D., Evrard A. 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V., Dekel A., 2002, \nApJ, 568, 52 Weinberger R., et al., 2017, MNRAS, 465, 3291 Zentner A. R., Hearin A. P., van den Bosch F. C., 2014, MNRAS, 443, 3044 Zhang Z. J., Chang C., Larsen P., Secco L. F., Zuntz J., LSST Dark Energy Science Collaboration 2022, MNRAS, 514, 2181 Zhu Z., et al., 2021, in Galaxy Cluster Formation II (GCF 2021) - Virtual \nWorkshop. p. 56 ( arXiv:2107.01663 ), doi:10.5281/zenodo.5013757 \nZurcher D., Fluri J., Sgier R., Kacprzak T., Refregier A., 2021, J. Cosmology Astropart. Phys., 2021, 028 \nZurcher D., et al., 2022, MNRAS, 511, 2075", 'A FORWARDMODELLING PIPELINE': "Akeyresult of our work is the simulation-based forecast of current and future surveys (Section 5). The forecast is done by forward modeling the observables from these surveys; namely, the lensing convergence field (Section A1) and the thermal Sunyaev-Zeldovich field (Section A2). This appendix details the exact procedures used to create our forward model for each field. \nThroughout, we use maps with NSIDE = 1024. The angular scale of the pixels (3 . 4 ' ) correspond to physical scales of 0 . 4 < 𝑟 < 7 Mpc for the redshift range spanned by the simulation, 0 . 1 < 𝑧 < 3 . 5.", 'A1 Weak lensing': "The lensing convergence field, 𝜅 , is a line-of-sight integral of the density field \n𝜅 ( ˆ n , 𝑧 𝑠 ) = 3 2 𝐻 2 0 Ω m 𝑐 2 ∫ 𝑧 𝑠 0 𝛿 ( ˆ n , 𝑧 𝑗 ) 𝜒 𝑗 ( 𝜒 𝑠 -𝜒 𝑗 ) 𝑎 ( 𝑧 𝑗 ) 𝜒 𝑠 𝑑𝑧 𝑗 𝑑𝜒 𝑑𝑧 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> 𝑧 𝑗 , (A1) \nwhere 𝑧 𝑠 is the redshift of the 'source' plane/galaxies being lensed, ˆ n is the pointing direction on the sky, 𝛿 is the overdensity field, 𝜒 is the comoving distance from an observer to a given redshift, 𝑎 is the scale factor, 𝐻 0 is the Hubble constant, Ω m is the matter energy density fraction at 𝑧 = 0, and 𝑐 is the speed of light. Here, we have used the shorthand 𝜒 ( 𝑧 𝑠 ) ≡ 𝜒 𝑠 and 𝜒 ( 𝑧 𝑗 ) ≡ 𝜒 𝑗 . \nWe use the same forward-modelling pipeline from Anbajagane et al. (2023, 2024b), whose description we reproduce below for completeness. The lensing maps are generated from the density fields of the U/l.pc/a.pc/g.pc/a.pc/m.pc simulations, and then processed to include all the observational effects found in the data. These procedures have been utilized in many analyses/forecasts of weak lensing data (Fluri et al. 2019; Zurcher \net al. 2021; Fluri et al. 2022; Gatti et al. 2022; Zurcher et al. 2022; Gatti et al. 2023; Anbajagane et al. 2023, 2024b; Gatti et al. 2024b,a; Jeffrey et al. 2024). As mentioned in Section 5, we focus on two surveys: (i) the Dark Energy Survey (DES, The Dark Energy Survey Collaboration 2005), Year 6 dataset, and the (ii) the Rubin Observatory Legacy Survey of Space and Time (LSST), year 10 dataset. We detail below our forward modeling procedure: \nConstructing lensing convergence shells. The model starts with lightcone shells of the particle counts, which are two-dimensional, HEALP/i.pc/x.pc maps of the (projected) particle counts at different redshifts. We convert the particle count map to an overdensity map as \n𝛿 𝑖 = 𝑁 𝑖 p /⟨ 𝑁 p ⟩ -1 , (A2) \nwhere 𝑁 𝑖 p is the number of particles in pixel 𝑖 , and the average is computed over all pixels in the shell. The density shells can then be converted into the convergence 𝜅 using Equation A1, after converting the integral over redshift into a discrete sum over lightcone shells. Note that our baryonification step is performed separately on each individual density shell, using the halos in just that shell. The baryonified density shells are then used in the summation of Equation (A1). \nSource galaxy redshift distributions. We perform a weighted average of the convergence shells to construct the convergence within the different tomographic bins of each survey. The weights used in this averaging are the source galaxy redshift distribution, 𝑛 ( 𝑧 ) , of the chosen bin and survey, \n𝜅 𝐴 ( ˆ n ) = 𝑁 steps ∑︁ 𝑗 = 1 𝑛 𝐴 ( 𝑧 𝑗 ) 𝜅 ( ˆ n , 𝑧 𝑗 ) Δ 𝑧, (A3) \nwhere 𝜅 𝐴 is the true convergence of a tomographic bin, 𝐴 . The 𝑛 ( 𝑧 ) is obtained from the following: for DES Year 6 and LSST Year 10 we use the same 𝑛 ( 𝑧 ) as that used in Zhang et al. (2022, see their Table 1). The LSST modeling in that work follows the baseline analysis choices of The LSST Dark Energy Science Collaboration et al. (2018, see their Appendix D2.1). The redshift distributions for DES Y6, and LSST Y1 and Y10 are parameterized as, \n𝑑𝑁 𝑑𝑧 ∝ 𝑧 2 exp GLYPH<20> -GLYPH<18> 𝑧 𝑧 0 GLYPH<19> 𝛼 GLYPH<21> , (A4) \nwith parameters given in Table A1. Once the 𝑛 ( 𝑧 ) of the full survey is defined, we split it into 4 (5) tomographic bins for DES Y6 (Y10) of equal number density. Each bin is then convolved with a Gaussian of width given by the photometric redshift uncertainty, also quoted in Table A1. The DES Y6 distribution is non-zero only between 0 . 2 < 𝑧 < 1 . 3, following Zhang et al. (2022, see their Table 1). The LSST distributions are cut at 𝑧 < 3 . 5. See Figure 1 of Anbajagane et al. (2024b) for the exact distributions we use in this work. \nTableA1. Theredshift distribution and source galaxy number density assumed for the upcoming surveys. All numbers are taken from Zhang et al. (2022, see their Table 1). \nConstructing lensing shear shells. Weak lensing surveys measure galaxy shapes, which primarily trace the shear field, 𝛾 , and not the convergence field, 𝜅 . However, the shear and convergence field can be transformed into each other using the Kaiser-Squires (KS) transform (Kaiser & Squires 1993), implemented in harmonic space as \n𝛾 ℓ𝑚 𝐸 + 𝑖𝛾 ℓ𝑚 𝐵 = -√︄ ( ℓ + 2 )( ℓ -1 ) ℓ ( ℓ + 1 ) GLYPH<18> 𝜅 ℓ𝑚 𝐸 + 𝑖𝜅 ℓ𝑚 𝐵 GLYPH<19> , (A5) \nwhere 𝑋 { 𝐸,𝐵 } are the E-mode and B-mode (or Q and U polarizations, in H/e.pc/a.pc/l.pcP/i.pc/x.pc notation) of the field. \nShape noise. Upon generating the two shear fields, we add shape noise in real space. The forward-modelled field includes Gaussian shape noise with a standard deviation given as \n𝜎 𝛾 = 𝜎 𝑒 √︁ 𝑛 gal 𝐴 pix , (A6) \nwhere 𝑛 gal is the source galaxy number density, and 𝐴 pix is the pixel area for a given map resolution. All maps in this work use NSIDE = 1024, corresponding to a pixel resolution of 3 . 2 ' . The per-galaxy shape noise is taken to be 𝜎 𝑒 = 0 . 26. \nSurvey mask & mass map construction. The noisy shear field - which is the sum of the true shear fields and the shape noise fields - is then masked according to the survey footprint. For DES Y6 we use the provided survey mask in the year 3 data release (Sevilla-Noarbe et al. 2021). For LSST Y10, we divide the full-sky into three equal-area cutouts of roughly 14,000 deg 2 each, which is the expected area coverage (The LSST Dark Energy Science Collaboration et al. 2018, see Appendix C1). The noisy shear maps are converted to convergence using the relation in Equation A5. We only use the resulting E-mode field, 𝜅 𝐸 , for our analyses. This follows the same procedures used in Chang et al. (2018); Jeffrey et al. (2021). \nTo summarize, lensing convergence maps are constructed from the raw particle number count maps. The 𝑛 ( 𝑧 ) distributions for a given survey are used to obtain the convergence map in a given tomographic bin. This convergence map is converted to shear maps, the relevant shape noise is added, the relevant survey mask is applied, and then the noisy shear maps are converted back to a noisy convergence map. The set of procedures listed above is the standard approach for forward modeling the lensing field ( e.g. , Zurcher et al. 2021, 2022; Gatti et al. 2022; Anbajagane et al. 2023, 2024b). Thus, our final convergence maps will be an accurate representation of \nthe survey data. We have intentionally ignored the modelling of intrinsic alignments (Troxel & Ishak 2015; Lamman et al. 2023) - which is a systematic effect considered in all the models cited above - as this work focuses on a pure test of the degeneracy directions between cosmology and the baryonification paraemeters. We will explore, in future works, the relationship between the baryonification model and other systematics.", 'A2 Thermal Sunayev-Zeldovich effect': "Therawmeasurements from a CMB survey are the temperature or power measured at different frequencies. At a given frequency, the observations contain contributions from a wide variety of astrophysical and cosmological sources, of which one is the tSZ field. This tSZ field can be filtered out of these maps through a 'Linear Combination' (LC) procedure, which is a weighted sum of frequency maps performed in harmonic space ( e.g. , Madhavacheril et al. 2020; Bleem et al. 2022). The exact method used to calculate the weights results in a variety of LC methods, such as the internal Linear combination (ILC), Needlet ILC etc. We use the LC algorithm of Bleem et al. (2022), which uses a theoretically estimated covariance matrix for the angular power spectra of the frequency maps. This estimate requires models for the other, 'foreground' fields that contribute power to the different frequency maps. For this, we follow the approach of Raghunathan et al. (2022a,b), who base their model on the measurements of George et al. (2015) from the South Pole Telescope (SPT) data. \nAs mentioned above in Section 5, we consider two CMB surveys: SPT-3G (Benson et al. 2014) and the Simons Observatory (SO, Ade et al. 2019). For both surveys, we assume their analyses are supplemented by Planck data, following the procedures used in existing surveys (Madhavacheril et al. 2020; Bleem et al. 2022). As a result, we forward model the Planck maps in this work. We follow Bleem et al. (2022) in using data from only the high-frequency instrument (HFI) of Planck . We also additionally use the 545 GHz band (which is not used in Bleem et al. (2022)), but as we will show later in Figure A1, the weights for the map is such that it provides no information to the final data product. The characteristics of the surveys (noise properties, beam, frequency bands etc.) are given in Table A2. We now detail our forward modelling procedure for the tSZ field: \nTrue tSZ field. Our simulated tSZ emission is obtained by pasting pressure profiles onto a H/e.pc/a.pc/l.pcP/i.pc/x.pc map. We use all halos in the lightcone to do so. The pasted profiles are already integrated over the line of sight. We therefore only need to change the units of the map through the simple rescaling, \n𝑦 ( ˆ n ) = 𝜎 𝑇 𝑘 𝐵 𝑚 𝑒 𝑐 2 𝑃 𝑒 ( ˆ n ) (A7) \nTable A2. The noise and beam properties of the different surveys in different frequency bands (in units of GHz). For Planck , we list the 100, 143, and 217 properties under the 90, 150, 220 columns for brevity; our actual analysis makes Planck frequency maps at 100, 143, and 217 GHz. The 𝛼 knee and ℓ knee of SO are constant across all bands. For SPT-3G, 𝑁 atm = 𝑁 white , and for Planck 𝑁 atm = 0. All noise estimates are in micro-kelvin-arcminute units. The beam full-width half-max, 𝜃 FWHM , is given in arcminutes. The values for SPT-3G and SO are taken from Raghunathan et al. (2022b, see their Table 1 and 2) while the values for Planck are from Madhavacheril et al. (2020, see their Table 1). See references therein for how these values are derived. \nwhere 𝑃 𝑒 ( ˆ n ) is the projected electron pressure in direction ˆ n , 𝜎 𝑇 is the Thomson cross-section, 𝑘 𝐵 is the Boltzmann constant, and 𝑚 𝑒 𝑐 2 is the rest energy of an electron. As mentioned above, the raw dataproducts of a CMB survey are the temperature maps at different frequencies. We can convert the tSZ signal into these frequencies via, \nΔ 𝑇 ( 𝜈, ˆ n ) = 𝑓 SZ ( 𝜈 ) 𝑇 cmb 𝑦 ( ˆ n ) (A8) \nwhere the frequency response of the tSZ signal, 𝑓 SZ, is given as, \n𝑓 SZ GLYPH<18> 𝑥 ≡ ℎ𝜈 𝑘 𝐵 𝑇 cmb GLYPH<19> = 𝑥 coth ( 𝑥 / 2 ) -4 . (A9) \nHere, ℎ is the Planck constant and 𝑇 cmb = 2 . 7255 𝐾 is the temperature of the CMB. \nForegrounds. As we discussed previously, the temperature map at a given frequency has contributions from multiple fields, normally called 'foregrounds'. We follow Raghunathan et al. (2022b) in modelling the CMB, radio, and infrared foregrounds. The kinematic SZ emission is ignored as it is subdominant, relative to these foregrounds, by more than an order of magnitude and is thus negligible for our work (George et al. 2015, see their Figure 3). We generate Gaussian realizations of all these foregrounds, by modelling their harmonic power spectra and using the synfast routine in H/e.pc/a.pc/l.pcP/y.pc to make full-sky realizations. The lensed 𝐶 CMB ℓ is modelled using C/a.pc/m.pc/b.pc run at the best-fit cosmology from Planck (Planck Collaboration et al. 2016). \nIn differential temperature units ( Δ 𝑇 ), the CMB template has no frequency dependence. The radio and infrared foregrounds follow George et al. (2015), and can be written as \n𝐶 Radio ℓ,𝜈 1 ,𝜈 2 = 𝐴 Radio 𝐺 15 𝐹 Radio 𝜈 1 ,𝜈 2 GLYPH<18> ℓ 3000 GLYPH<19> 2 GLYPH<18> 2 𝜋 ℓ ( ℓ + 1 ) GLYPH<19> (A10) \n𝐶 CIB , poiss . ℓ,𝜈 1 ,𝜈 2 = 𝐴 CIB , poiss . 𝐺 15 𝐹 CIB , poiss . 𝜈 1 ,𝜈 2 GLYPH<18> ℓ 3000 GLYPH<19> 2 GLYPH<18> 2 𝜋 ℓ ( ℓ + 1 ) GLYPH<19> (A11) \n𝐶 CIB , clus . ℓ,𝜈 1 ,𝜈 2 = 𝐴 CIB , clus . 𝐺 15 𝐹 CIB , clus . 𝜈 1 ,𝜈 2 GLYPH<18> ℓ 3000 GLYPH<19> 0 . 8 GLYPH<18> 2 𝜋 ℓ ( ℓ + 1 ) GLYPH<19> (A12) \nThe radio background is modelled as having a poisson spatial distribution (and is therefore essentially constant in ℓ ), while the CIB is modelled as a combination of a poisson-distributed component and a clustered component. The frequency response function, 𝐹 𝜈 1 ,𝜈 2 is defined as, \n𝜖 𝜈 1 ,𝜈 2 = GLYPH<18> 𝐵 ' ( 𝜈 0 ) 𝐵 ' ( 𝜈 0 ) 𝐵 ' ( 𝜈 1 ) 𝐵 ' ( 𝜈 2 ) GLYPH<19> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> T = Tcmb (A13) \n𝐹 Radio 𝜈 1 ,𝜈 2 = 𝜖 𝜈 1 ,𝜈 2 GLYPH<18> 𝜈 1 𝜈 2 𝜈 2 0 GLYPH<19> -𝛼 (A14) \n𝐹 CIB , x 𝜈 1 ,𝜈 2 = 𝜖 𝜈 1 ,𝜈 2 GLYPH<18> 𝜈 1 𝜈 2 𝜈 2 0 GLYPH<19> 𝛽 𝑥 GLYPH<18> 𝐵 ( 𝜈 1 ) 𝐵 ( 𝜈 2 ) 𝐵 ( 𝜈 0 ) 2 GLYPH<19> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> T = Tcib (A15) \nwhere 𝐵 is the blackbody function, 𝐵 ' ≡ 𝑑𝐵 / 𝑑𝑇 its derivative with temperature, and 𝑥 ∈ { poiss . , clus . } . The function 𝜖 𝜈 1 ,𝜈 2 converts spectral energies to differential temperature ( Δ 𝑇 ) units. The radio component is modelled with a power-law frequency dependence, while the CIB is a modified blackbody where the modification is to the low-frequency, Rayleigh-Jeans regime of the distribution. For the CIB model in Equation (A15), we use 𝛽 poiss . = 1 . 505, 𝛽 clus . = 2 . 510, 𝑇 cib = 20 𝐾 (George et al. 2015). For the radio foreground, we use 𝛼 = -0 . 6, following Raghunathan et al. (2022b). \nNote that our goal in including foregrounds is only to obtain a reasonably accurate noise term for the tSZ maps. As a result, we follow Raghunathan et al. (2022a,b) in modelling these foregrounds as uncorrelated Gaussian fields, whereas in actuality these fields will be correlated. Such correlations were found to be negligible for analyses of clusters (Raghunathan et al. 2022a, see their Section 3.6), which dominate the information in the tSZ field. However, a more realistic forward model of the tSZ field will need to consistently connect the foregrounds to the density field and halo catalog in a given simulation realization; see Omori (2022), for an example of a robust, self-consistent model. \nInstrumental and Atmospheric noise. The primary noise contributor to the tSZ maps are the thermal noise of the detector and atmospheric noise (for ground-based surveys). This is modelled through a two-term noise component, \n𝑁 ℓ = 𝑁 white + 𝑁 atm GLYPH<18> ℓ ℓ knee GLYPH<19> -𝛼 knee . (A16) \nThe first term is the frequency-independent, 'white noise' component arising from detector noise. The second term is a 1 / 𝑓 -noise that increases towards large-scales. The values we use for the different experiments are given in Table A2. For Planck , we set 𝑁 atm = 0. /one.sup/nine.sup Equation (A16) is the beamconvolved noise estimate, and to get the true noise model we deconvolve Equation (A16) as 𝑁 ℓ → 𝑁 ℓ / 𝐵 2 ℓ , where the beam 𝐵 ℓ is defined below. \nInstrument Beam. The point-spread function (or beam) varies by experiment and detector. The values for the experiments we consider are given in Table A2. We use Gaussian beams, \n𝐵 ℓ = exp GLYPH<18> 𝜎 2 FWHM 2 ℓ ( ℓ + 1 ) GLYPH<19> (A17) \nwith 𝜎 FWHM = 𝜃 FWHM / √︁ 8 ln ( 2 ) . Here 𝜃 FWHM is the beam full-width half-max in radians. \nLinear Combination (LC): Finally, the (beamdeconvolved) frequency maps are summed together through an LC, similar to that of Bleem et al. (2022). In this formalism, the final map is, /two.sup/zero.sup \n𝑦 LC ℓ,𝑚 = ∑︁ 𝑖 𝑤 ℓ ( 𝜈 𝑖 ) 𝑇 ℓ,𝑚 ( 𝜈 𝑖 ) (A18) \nwhere 𝑇 ℓ ( 𝜈 𝑖 ) are (harmonic space) temperature maps at different frequencies and 𝑤 ℓ ( 𝜈 𝑖 ) are the LC weights. These weights are the optimal (minimum-variance, or MV) weights that maximimize the tSZ signal, defined as \n𝑤 ℓ ( 𝜈 𝑖 ) = 𝑓 𝜈 𝑖 SZ 𝐶 -1 ℓ 𝒇 𝑇 SZ 𝐶 -1 ℓ 𝒇 SZ . (A19) \nwhere f SZ = { 𝑓 𝜈 0 , 𝑓 𝜈 1 , . . . } specifies that the tSZ is our target signal, and 𝐶 -1 ℓ is the inverse of the covariance matrix between the different frequency maps at a given ℓ . Following Bleem et al. (2022); Raghunathan et al. (2022a) we model the covariance matrix theoretically as, \n𝐶 𝜈 𝑖 ,𝜈 𝑗 ℓ = 𝑁 ℓ,𝜈 𝑖 ,𝜈 𝑗 𝛿 𝑖 𝑗 + ∑︁ 𝑎 ∈ 𝑋 𝐶 𝑎 ℓ,𝜈 𝑖 ,𝜈 𝑗 (A20) \nwhere 𝑁 ℓ is already beam-deconvolved, and 𝛿 𝑖 𝑗 ensures that instrument and atmospheric noise only affects the autocorrelation of frequency maps. /two.sup/one.sup The frequency and ℓ -dependence of the foregrounds, 𝑋 ∈ { CMB , Radio , CIB } are defined above. \n/one.sup/nine.sup Note that Planck still exhibits some 1 / 𝑓 -noise (particularly on ℓ < 100) arising from other, non-atmospheric components. For the purposes of this work, we ignore this term. \n/two.sup/zero.sup Under the assumption of uncorrelated Gaussian foregrounds and instrumental/atmospheric noise, the residual noise in the tSZ field is given by the spectra 𝑁 𝑦𝑦 ℓ = 𝒇 𝑇 SZ 𝐶 -1 ℓ 𝒇 SZ , which is the denominator in Equation (A19). One can obtain a realistic, noisy tSZ field by simply generating a Gaussian field with 𝐶 ℓ = 𝑁 𝑦𝑦 ℓ and adding it to the true tSZ field. However, our forward model still adopts the approach of decomposing the signal into different frequency bins and performing the LC, as this approach is more generalized. \n/two.sup/one.sup Nominally, atmospheric noise should be correlated across frequency bands, however we follow Raghunathan et al. (2022a,b) in ignoring this. \nFigure A1. The harmonic-space weights used in the linear combination procedure - see Equation (A19) - to combine the different frequency maps into a minimum-variance tSZ map, shown for both SPT-3G (top) and Simons Observatory (bottom). The weights for the five Planck maps are shown as dashed lines. A black dotted line shows 𝑤 = 0 for reference. \n<!-- image --> \nThe emission from these foregrounds is completely correlated across bands. Once this final map, 𝑦 LC ℓ,𝑚 , is made, we convolve it with a Gaussian beam of 1 . 7 ' , which is the final resolution of the map. Note that in practice, we make maps at NSIDE = 1024, which already has a larger pixel scale (3 . 4 ' ) than the beam. \nIn summary, we generate true tSZ maps from our baryonification pipeline and decompose the signal into different frequency channels. We then add multiple other foregrounds to these frequency channels, including the CMB, Radio emission, and CIB. We also add the thermal noise from detectors and then the noise due to the atmospheric. The maps are linearly combined using weights that result in the minimum variance of the final tSZ map. The weights we use in this work are shown in Figure A1. We have verified our weights estimation procedure reproduces the results of Raghunathan et al. (2022b, their Figures 3 and 4), and we also find our dominant frequency band at a given ℓ broadly follows that of Bleem et al. (2022); /two.sup/two.sup for example the 100 GHz Planck data dominates at low ℓ while the 90 GHz SPT data dominates at high ℓ . \nOmori (2022) provide a detailed, and precisely validated, prescription for consistently producing a realistic tSZ field \n/two.sup/two.sup We do not expect precise agreement given our approach is still a significantly simpler version of the analysis done in Bleem et al. (2022), and given that analysis is for a previous release of the SPT data. \nFigure B1. The impact of the chosen projection scale on the projected displacement function for halos of two different masses; see Equation (25) and Section (2.2). Increasing the limits of the lineof-sight projection integral washes out the variations in the density profile arising from baryonic imprints and lowers the amplitude of the displacements. \n<!-- image --> \n- including all relevant contaminant foregrounds, lensing effects, cross-correlations etc. - through the use of highresolution simulation products and data-driven models. The U/l.pc/a.pc/g.pc/a.pc/m.pc suite does not have the required resolution or fidelity of simulation products (for example, we have no merger trees or particle shells) to employ their approach. Instead, our forward model focuses solely on capturing the noise level of the tSZmaps, which is adequate for understanding the different degeneracy directions (for the different moments measurements) of the different baryonification and cosmology parameters.", 'B IMPACT OF PROJECTION SCALE': 'We extend on the discussions of methodology validation in Section 4 by checking the sensitivity of our predictions to choices in the projection operation. As noted in Section 2.2, we set the limits of the projection integral according to the simulation box (or more generally, according to the thickness of the volume that we are applying baryonification on). This is in contrast to some previous works that set the the projection scale based on the projected separation 𝑟 p (see Equation (25) for definitions of the quantity). Figure B1 presents the impact of the projection choices on the estimated projected displacement function. In the limit of an infinite projection length, the impact of baryons will be completely washed out and there is no baryonification required. Figure B1 shows that increasing the projection length does indeed reduce the predicted dis- \nFigure C1. The dependence of the displacement function on the including/excluding the DM adiabatic relaxation. The effect changes the displacement by only 10%, and thus, this displacement is fairly insensitive to the exact modelling choices of the DM relaxation and is instead highly sensitive to the gas distribution. \n<!-- image --> \ncements from the model. We use a minimum projection scale of 60 Mpc, which is the width of density shells used in full-sky mocks of the lensing field (Kacprzak et al. 2023; Anbajagane et al. 2024b), and a maximum of 1000 Mpc, which is twice the projection scale we use for the Q/u.pc/i.pc/j.pc/o.pc/t.pc/e.pc. \nThe displacements on the largest scales are affected most by varying the projection scale ( e.g. , see bottom panel of Figure B1) as at these radii the one-halo density profile becomes comparable/subdominant to the large-scale two-halo term. On smaller scales the impact is less prominent. Though, for adequately small halos ( e.g. , , 𝑀 = 10 12 M ⊙ , see top panel of Figure B1), where the two-halo term can dominate over the one-halo term even at smaller radii, the suppression in amplitude is still seen as we increase the projection scale. This result highlights the importance in choosing a consistent, physically meaningful projection scale.', 'C ALTERNATIVE CHOICES IN HALO PROFILE MODEL': 'The halo model described in Section 2 is not a unique parameterization of the different matter components in the halo; there are several modifications to the model that either extend it to enhance accuracy or simplify it to enhance computational cost. We test a few of these modifications below. In all cases, we show the differences at the profile level for brevity and do not present the summary statistics-level consistency checks.', 'C1 Adiabatic contraction of dark matter': "The collisionless matter component of the baryonification model includes galaxies as well as DM. The distribution of \nFigure C2. The profile of total gas pressure compared with only the thermal pressure, assuming some model for the non-thermal component. We show predictions from Green et al. (2020), and for the fiducial baryonification parameters (Table 1) following the model of Shaw et al. (2010) for the non-thermal component as described in Section 2.3. \n<!-- image --> \n200c \nthe latter does not follow the same NFW profile as it did in the DMO case; this has been highlighted by many works on a multitude of simulations ( e.g. , Gnedin et al. 2004; Duffy et al. 2008; Ragagnin et al. 2019; Beltz-Mohrmann & Berlind 2021; Forouhar Moreno et al. 2021; Anbajagane et al. 2022a; Shao et al. 2023; Shao & Anbajagane 2024; Sorini et al. 2024). The gravitational response of the DM to the presence of baryons has been studied by many works, and approximated through various (simple) relations. Our baryonification model adopts one such relation from Gnedin et al. (2004); Abadi et al. (2010); Teyssier et al. (2011). In Figure C1 we show the change in the displacement function if we ignore the adiabatic relaxation of the DM. This provides an estimate of the model's sensitivity to inaccuracies in the modelling of this relaxation process. The effect is at most 15% of the maximum displacement value. Thus, the displacement function is dominated by the form of the gas distribution rather than by the redistribution of the DM due to adiabatic relaxation. This is consistent with findings in hydrodynamical simulations ( e.g. , Springel et al. 2018, see their Figure 9).", 'C2 Sensitivity to non-thermal pressure models': 'The tSZ modelling described in Section 2.3 starts by assuming halos are in hydrostatic equilibrium, which means the gas distribution is supported against gravitational collapse solely due to its thermal pressure. However, simulations have shown that order ∼ 10% of the pressure support comes from non-thermal pressure, primarily constituted by turbulent motions of the gas, i.e. a velocity dispersion (Nelson et al. 2014; Green et al. 2020). We have corrected for this in our predictions by explicitly modelling the contribution of nonthermal pressure and varying/marginalizing the parameters of \nFigure C3. The dependence of the gas density profile (top) and displacement function (middle, bottom) when including features from a large-scale cosmological shock; see Equation (C1). We vary the shock location, 𝑅 sh , and the shock width Δ ln 𝑅 sh . In the top panel, the profiles varying the shock width are offset vertically for visual purposes. The bottom two panels show the shock feature changes the displacement quite significantly. However, these effects can be captured through the existing, flexible parameterization of the gas profile. See Section C3 for more details. The results from profiles with no shock features is shown in black. \n<!-- image --> \nthis model. Our parameterization follows Shaw et al. (2010), which is the same choice as P24. \nFigure C2 compares this model to other alternatives, particularly the predictions of Green et al. (2020). The latter model is an analytic prediction (see their Figure 1 for a summary of the model) that depends only on the peak height of the halo, and is validated using a set of hydrodynamic simulations from Nelson et al. (2014). For comparison, we plot thermal gas pressure profiles, using either Green et al. (2020) or our prescription in Section 2.3 to estimate the non-thermal pressure. In the latter approach, predictions are made for fiducial parameter values listed in Table 1. The phenomenological model, based on Shaw et al. (2010), qualitatively matches the form predicted by Green et al. (2020), which in turn is a advanced version of previous non-thermal models, such as that of Nelson et al. (2014).', 'C3 Sensitivity to shock features': "The gas distribution around massive halos is expected to undergo shock heating due to the generic process of gravitational infall; cold matter from nearby large-scale structure (eg. filaments) infalls into the massive halo. Cold gas has a \nrelatively low sound speed, but the infall velocity can exceed the sound speed by factors of O( 100 ) for gas around massive halos. This causes the gas to shock heat and thereby imprint features into the gas distribution on large scales. Such features have been predicted by simulations of gas ( e.g. , Quilis et al. 1998; Miniati et al. 2000; Molnar et al. 2009; Baxter et al. 2021; Aung et al. 2020), and there are indications of these features from measurements at different wavelengths ( e.g. , Hurier et al. 2019; Pratt et al. 2021; Zhu et al. 2021; Anbajagane et al. 2022c, 2024a; Hou et al. 2023). \nThe change in density due to a cosmological shock can be found following the Rankine-Hugonoit jump conditions (Rankine 1870; Hugoniot 1887). For these shocks, which have a very high mach number 𝑀 ≡ 𝑣 / 𝑐 𝑠 ≫ 1, where 𝑣 is the infall velocity and 𝑐 𝑠 is the sound speed of the gas - the density profile post-shock is transformed as 𝜌 → 𝜌 / 4. This estimate corresponds to a gas adiabatic index of 𝛾 = 5 / 3, which is valid under the assumption that the gas is monoatomic. We then introduce this shock feature through a modified version of the gas profile from Section 2.1, \n𝜌 gas , sh / 𝜌 gas = GLYPH<18> 1 -0 . 25 1 + exp GLYPH<2> 1 Δ ln 𝑅 shock ln ( 𝑟 / 𝑅 shock ) GLYPH<3> GLYPH<19> + 0 . 25 . (C1) \nThe ratio is a simple sigmoid function that asymptotes to 1 / 4 for 𝑟 ≫ 𝑅 shock and 1 for 𝑟 ≪ 𝑅 shock. Note that in Equation (10), which defines the gas profile, the normalization of the profile is set by integrating the profile to 𝑟 → ∞ . Given the shock feature changes the shape of the profile, it would nominally affect this normalization. However, we keep the normalization fixed and only change the large-scale behavior of the profile. This is because the shock feature only affects the physics of the large scales and is largely decoupled from the gas physics internal to the halo. Thus, it is unrealistic to couple large-scale features like shocks to changes in the density throughout the halo (the normalization is a scaling factor for the entire profile and thus changes the density at all scales). \nFigure C3 shows the profiles with shocks, and the displacement functions corresponding to such profiles. There is a significant effect both at the profile level and thereby at the displacement level. As discussed before, the feature is prominent at large-scales and so the displacement is affected only on these scales. The small-scale displacement is left largely unaltered. However, the baryonification model is flexible and therefore the large-scale changes to the gas profile can be captured through the gas profile slope parameters, 𝛾 and/or 𝛿 . The impact of such shocks on the pressure profile can also be captured by the non-thermal pressure model, as that model fundamentally captures a 'lack of thermal pressure' in the halo, which is the signature that will be generated by the shocks (Anbajagane et al. 2022c, see their introduction for a \ndescription of this process). We leave a dedicated study of shocks in the baryonification model to a future study, and note that Figure C3 highlights the potential impact it can have on the total density distribution within halos.", 'D VALIDATION AT HIGHER REDSHIFTS': 'WhileSection 4.2 shows that the baryonification model can fit the simulation measurements at 𝑧 = 0, we must also verify that the model can jointly fit measurements across redshifts. Figure D1 shows such a joint fit across all moments, up to 4th order, from 𝑧 = 0 and 𝑧 = 0 . 5. \nThe fit is done using three models: (i) the fiducial model shown in Figure 8, with 𝜈 𝑋 = 0 (no redshift dependence in the gas profile parameters) and fit to the 𝑧 = 0 data alone, then (ii) the same fiducial model now jointly fit to 𝑧 = 0and 𝑧 = 0 . 5, and finally (iii) the same model but with those redshift-dependence parameters varied, and fit to both 𝑧 = 0 and 𝑧 = 0 . 5 data. The first model is still a decent match to the simulations, though a few predictions at 𝑧 = 0 . 5 are discrepant with the simulation beyond the 3 𝜎 bound. The second model, which is a joint fit, is a better match across both redshifts. The predictions are almost always within 1 -2 𝜎 of the datapoints. Finally, the third model, which now varies two redshift-dependence parameters, 𝜈 𝜃 ej and 𝜈 𝑀 c , provides a similar/better match depending on the subset of measurements we focus on. There are some potential discrepancies in the higher order moments of the tSZ field, but the discrepancies are still within the 2 𝜎 uncertainties. The fractional uncertainties of the higher redshift measurements -particularly those involving the tSZ field, or higher-order combinations of any fields - can be larger due to the weaker signal at higher redshift. The statistical limitations of small simulation volumes, which we had discussed earlier in Section 4.2, are only exacerbated at higher redshift as there are now even fewer massive halos generating the signals. \nIt is interesting to note that an adequate fit is achieved even when fixing the redshift-dependence parameters, 𝜈 𝑋 . This complements the results of Section 5, and Table 2 in particular, which showed that all versions of our analysis were poorly constraining these parameters. We have thus seen two different analyses show that the 𝜈 𝑋 parameters may not be a necessary feature of the model when applying baryonification to survey data. P24 found these parameters necessary for fitting the profiles of halos across multiple redshifts. However, it may be that for summaries of the whole lensing/tSZ field measured in surveys, the mass-dependence is adequate enough without the need for any additional redshift-dependence as well. However, one limitation of our analysis here is it spans a relatively narrow range in redshift, 0 < 𝑧 < 0 . 5 (though our Fisher forecast in Section 5 uses 0 < 𝑧 < 3 . 5) and fitting simulation measurements across a broader range could once again necessitate \nTable E1. Constraints on two cosmology parameters, and the two key baryonification parameters (for lensing) as a function of the number of baryonifciation parameters varied in the analysis (also denoted with the brackets). We forecast for a DES Y6 x SPT 3G analysis, using the 2nd and 3rd moments of the fields. \nthe redshift-dependence parameters in this particular analyses. Further analyses, using other simulations, is needed to better confirm the necessity of these parameters for analyses of survey data. \nIn conclusion, we find the model specified in Section 2 adequately captures the moments of the lensing and tSZ fields, from 2nd to 4th order, and across multiple redshifts. As mentioned before, more precise validation will be enabled through the use of larger simulations.', 'E IMPACT OF PARAMETER SET OF FISHER INFORMATION': "Anumber of previous works have constrained the baryonification parameters using data ( e.g. , Chen et al. 2023; Aric'o et al. 2023; Grandis et al. 2023; Bigwood et al. 2024), and they span a variety of choices in the parameters being varied. For example Chen et al. (2023) vary a single parameter, 𝑀 𝑐 , whereas Aric'o et al. (2023) varies six parameters. Additionally, Giri & Schneider (2021) explored the minimal set of parameters needed for baryonification to still predict the matter power spectrum from different simulations. \nHere, we show our Fisher constraints on cosmology using different sets of baryonification parameters. Ordered by increasing complexity and number of parameters, we consider: varying no baryonification parameters, varying only 𝑀 𝑐 , 𝜃 ej (base), varying the two mass-dependence parameters 𝜇 𝑋 , the two redshift dependence parameters 𝜈 𝑋 , and varying the four remaining parameters. See Table 1 for a complete list of all parameters we vary. Figure E1 shows the results of this lensing plus tSZ analysis. As expected, varying all parameters degrades cosmology constraints by roughly around 60%. Table E1 also lists the constraints for cosmology, as well as 𝑀 𝑐 , 𝜃 ej, alongside the number of baryonification parameters varied per analysis.", 'F COMPUTATIONAL DETAILS': "There are a number of salient computational choices and details in the baryonification model of Section 2 that we list \nOnly z = 0 fit \nz = 0, 0.5 fit \nz = 0, 0.5 fit (z-dep. param.) \nFigure D1. Similar to Figure 8, but now showing a variety of a different fitting cases and their predictions for 𝑧 = 0 . 5 measurements (top three rows) and for the 𝑧 = 0 measurements previously shown in Figure 8 (bottom). The different curves show fits made to different combinations of redshifts. The purple line ('fixed') is from a joint fit of 𝑧 = { 0 , 0 . 5 } with all redshift dependence of the baryonification model set to 𝜈 𝑋 = 0; see Equation (16). The yellow line ('varied') is the same joint fit, but after allowing these values to be non-zero. Both the fixed and varied predictions are a decent fit to the 𝑧 = 0 . 5 measurements, with the varied one providing the best fit of the two. The fit from using only the 𝑧 = 0 measurements (blue) is shown for comparison and is the worst performing. In the bottom three rows, the blue line overlaps exactly with the yellow and so we show it as a thicker/translucent line to aid visibility. \n<!-- image -->", 'here for completeness.': "Projection: It is computationally efficient to perform the projection integral in Fourier space, which is the choice made in the CCL code base. However, this increases the likelihood of aliasing, or 'ringing' effects in the resulting profile, particularly for profiles computed in the extreme regions of parameter space. Such profiles can deviate very significantly from simple \npower-laws, and in these cases the F/f.pc/t.pcL/o.pc/g.pc algorithm can fail. To make our pipeline more robust, we perform all projection integrals in real-space, \n𝜌 𝑝 ( 𝑟 𝑝 ) = ∫ 𝐿 / 2 0 2 𝑑𝑙𝜌 GLYPH<18> √︃ 𝑙 2 + 𝑟 2 𝑝 GLYPH<19> . (F1) \nWe find that using the real-space calculation, instead of the Fourier-space one, does not slow the pipeline a noticeable \nFigure E1. Constraints on two cosmology parameters as a function of the number of baryonifciation parameters varied in the analysis. We forecast for a DES Y6 x SPT 3G analysis, using the 2nd and 3rd moments of the fields. The constraints, and the number of baryonification parameters varied in each, are listed in Table E1. The inclusion of all 11 baryonification parameters leads to a ≈ 60% degradation in cosmology constraints. \n<!-- image --> \namount. Note that in practice we only compute the profiles once, at the start of the baryonification pipeline, to create a table. This lookup table is then used for generating profiles of all the halos in the simulations (see the 'Tabulation' discussion below). \nProfile interpolation: We use tabulation and interpolation for all integrals/derivatives of the profiles appearing in this work; for example, the volume integral to get the normalization in Equation (2) and (12). For almost all interpolation steps (the one exception is described below), we use the PchipInterpolator (Fritsch & Butland 1984; Moler 2004) in S/c.pc/i.pc/p.pc/y.pc as it preserves the monotonicity of the function under the presence of outliers. The CubicSpline interpolator, which is a popular choice and also implemented in S/c.pc/i.pc/p.pc/y.pc, can break monotonicity if the input function has any rapid (but still monotonic) variation and this will then lead to ringing in the interpolated result. While our profiles are all well-behaved and do not contain sharp changes over radius, their pixel-convolved versions may present such rapid deviations if the halo radius is far smaller than the pixel-scale. The PchipInterpolator enables the pipeline to adequately handle such extreme edgecases as well. We use this interpolator to accurately compute integrals of functions, for example when converting density profiles to enclosed mass profiles. \nWe also require interpolation for computing the derivatives - primarily the operation in Equation (20) to compute the collisionless matter profile - and here we do use the \nCubicSpline . This is because this spline guarantees continuous first and second derivatives in the interpolated function (whereas PchipInterpolator only guarantees continuous first derivatives). In Equation (20), we take the derivative of a enclosed mass function in order to obtain a density profile, and if the second derivative of the former function is discontinuous that will cause (unphysical) jumps in the resulting density profile. The CubicSpline prevents this by enforcing smooth second derivatives in the interpolator of the enclosed mass. \nTabulation: For all of our work here, we use a tabulated profile to speed up our calculations. These profiles (or the displacement function) are computed over some specified range in log 10 𝑟 , log 10 𝑀 200c and log 10 ( 1 + 𝑧 ) . When computing the dependence of the profiles on external parameters, such as concentration 𝑐 200c or the projection distance 𝐿 proj (see Figures 4 or B1), we extend the tabulation to include this parameter. These tables are defined on regular grids, and we use simple linear interpolation of the table through the RegularGridInterpolator routine in S/c.pc/i.pc/p.pc/y.pc. This approach greatly increases the computational speed of the model predictions, and leads to negligible drops in accuracy given all quantities (profiles, displacement functions etc.) are smoothly varying across radius, mass, and redshift. Grids of O( 10 5 ) points (1000 points in log 10 𝑟 , 10 in log 10 𝑀 200c, and 10 in log 10 ( 1 + 𝑧 ) ) take less than five minutes to build, on a singlethreaded process of an Intel Broadwell CPU, and result in interpolators that are accurate to within < 0 . 1%. \nComputing runtimes: The baryonification process is 'embarassingly parallel' given the contributions from halos are accumulated through a loop, and each iteration in the loop can be evaluated completely disjoint from the rest. Our pipeline is parallelized across all cores in a node using the J/o.pc/b.pc/l.pc/i.pc/b.pc routines, i.e. we have implemented shared-memory parallelization. For the standard full-sky simulation used in this work - 100 density shells of NSIDE = 1024 and halo catalogs with 𝑀 > 10 14 M ⊙ ; see Section 3.3 - baryonification of the density field is done in under 2 minutes on an Intel Broadwell chip with 40 cores. The profile pasting step, which uses approximately two million halos across 0 < 𝑧 < 3 . 5, generates the tSZ field in under 2 minutes as well. The runtime scales linearly with the number of halos in the simulation volume as well as the number of shells and number of pixels. If we used halo catalogs of 𝑀 > 10 13 M ⊙ , the run time would increase by a factor of two to three; a halo catalog with 𝑀 > 10 13 M ⊙ has 10 × more halos than one with 𝑀 > 10 14 M ⊙ but these additional, less-massive halos will span a smaller area on the sky which in turn will reduce the computational load. \nThis paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and \neasy peer review for new papers in the astro-ph section of the \narXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theoj.org ."}
2024arXiv240905505P	This contribution summarizes the main activities and objectives of the outreach project Astroaccesible whose main aim is to carry the teaching and diffusion of astronomy among all kinds of collectives focusing on blind and visually impaired BVI people. This project is led by a blind astronomer and aims to use a variety of resources based on different sensory channels avoiding limiting the transmission of concepts to visual perception. This principle favors inclusion and benefits everyone as the information is not presented using just one channel. This strategy is especially convenient for the nowadays typical data acquisition where a variety of sources of information not solely based on the collection of different spectral domains of electromagnetic radiation is used. Moreover the study of new multimessenger astronomy could be much better understood using a multimessenger teaching approach favoring inclusion motivation and creativity.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.05505', 'arXiv:2409.05505', '2024arXiv240905505P']	['Astrophysics - Instrumentation and Methods for Astrophysics']	Astroaccesible A multimessenger outreach for a multimessenger science	2,024	193	0.36	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.05505.pdf	{"Enrique P'erez-Montero 1": "1 Instituto de Astrof'ısica de Andaluc'ıa - CSIC, Apdo. 3004, Granada, Spain E-mail: [email protected] \nAbstract. This contribution summarizes the main activities and objectives of the outreach project Astroaccesible , whose main aim is to carry the teaching and diffusion of astronomy among all kinds of collectives, focusing on blind and visually impaired (BVI) people. This project is led by a blind astronomer and aims to use a variety of resources based on different sensory channels, avoiding limiting the transmission of concepts to visual perception. This principle favors inclusion and benefits everyone, as the information is not presented using just one channel. This strategy is especially convenient for the nowadays typical data acquisition, where a variety of sources of information, not solely based on the collection of different spectral domains of electromagnetic radiation, is used. Moreover, the study of new multi-messenger astronomy could be much better understood using a multi-messenger teaching approach, favoring inclusion, motivation, and creativity.", '1 Introduction': 'The teaching and outreach of science for every public are essential for the formation and establishment of a general culture for any individual. The constant scientific and technological advancements being made nowadays must be accompanied by appropriate dissemination and explanation for the entire society. This fundamental objective, explicitly recognized in several declarations such as the Universal Declaration of Human Rights (1948) and the Convention on the Rights of Persons with Disabilities (2006), cannot be satisfactorily achieved without including all impaired people and those who, due to their characteristics, find barriers to accessing information [1]. \nAstronomy, which includes astrophysics and cosmology, is one of the fastest-growing branches of science and is providing numerous new scientific results. This growth is partly driven by investments in new facilities for observing the Universe, such as large telescopes, space observatories, and missions to objects in our Solar System. The new data and findings from these facilities always have a significant impact on the media. \nGiven that a large portion of the dissemination of these results is predominantly done through visual media (e.g., pictures, videos, animations, graphics), blind and visually impaired (BVI) people are often excluded from having full and satisfactory access to all this content. Although there is a growing awareness of the need to adapt scientific information for teaching or outreach, we are still far from achieving the necessary objectives to make all content fully accessible for everyone. \nMoreover, in the last decades, we have witnessed how the acquisition of observational information coming from the huge heterogeneity of sources and ranges present in the Universe has widened, opening the gate to a new multi-messenger astronomy, which includes data from gravitational waves, cosmic rays, and neutrinos, presenting both a challenge and an opportunity for inclusion. Adapting these new types of data for a wider audience requires innovations in the presentation and teaching of science, fostering inclusion and benefiting everyone by presenting information through multiple sensory channels. This is crucial not only for BVI individuals but also for enhancing understanding and motivation in the study of multi-messenger astronomy. \nFigure 1: Several sketches representing the concepts (from left to right) of exclusion, segregation, integration, and inclusion. \n<!-- image -->', '2 The inclusive aspect of the project Astroaccesible': "The outreach project Astroaccesible 1 [2] started as an initiative to teach basic concepts of astronomy among the collective of BVI. Although these first activities had a character mainly restricted to impaired people, the project has incorporated an inclusive aspect [3] whose benefits have been shared and transmitted to other educators or professionals of outreach. \nInclusion is not a concept easy to understand for those who are not familiar with the teaching or outreach for collectives of people with any disability. In Figure 1, several basic sketches are used to illustrate the different steps that are necessary to follow before reaching total inclusion in the design of an outreach activity 2 . The first necessary step is the recognition of exclusion, including the methods and procedures that can exclude a part of the target public. A first treatment of this exclusion is partially adopted by means of segregation, or the organization of specific and separated actions. A more appropriate strategy can be followed by means of integration, in which people with or without any special need are treated in the same activity, although using different resources. The desirable and ideal scenario is inclusion, in which everyone receives the same information using the same resources. This last is an ideal situation that can be partially reached through a pathway covering the previous stages and adopting an inclusive philosophy in the design of our outreach programs, which enormously helps to improve the outreach to everyone. \nTherefore, Astroaccesible has not just the objective of connecting the BVI colective to science, but also of convincing other scientists and teaching professionals to incorporate an inclusive aspect to their projects, as this largely benefit the collective of impaired people, helping at the same time to improve the quality of their contents for everyone. This strategy can be considered as a facet of the well-known Universal Design for Learning (UDL) method, already put in practice for other outreach and learning projects related with astronomy and space sciences, like AstroAcess [5]. \nThis not only makes all transmitted concepts easier to understand for everyone but also helps to sensitize the whole population about the importance of inclusion of people with disabilities, letting this collective be more interested in science, and promoting the idea of the incorporation of this collective to begin a professional scientific career, enhancing diversity in research groups, which is demonstrated to increase the productivity and impact factors of the published work [6]. \nAmong the various activities undertaken by Astroaccesible in recent years are sessions specifically designed for BVI individuals, often in collaboration with the Organizaci'on Nacional de Ciegos de Espa˜na (ONCE). These sessions, however, also cater to a wide range of audiences at different educational levels, from primary school to college, always employing an inclusive methodology. In all these activities, it is common to provide as a least inclusive level, not necessarily based on the use of specific resources based on a multisensory approach, comprehensive oral descriptions of the presented material, supplemented by appropriate comparisons, emphasizing the significant role of sight in acquiring the explained information.", '3 A multi-messenger approach': "The outreach and teaching of the new multi-messenger astronomy, involving a wide range of electromagnetic ranges, going from gamma rays down to radio sources, and also involving detection of cosmic rays, neutrinos, or gravitational waves, can be thus done by multiple sensory channels, not just \nFigure 2: Distorted images of the spiral galaxy M51 to show how an individual with a visual impairment can perceive it. From left to right and from up to down: a) loss of visual field, b) patched vision, c) loss of central vision, d) night blindness, e) photophobia, and f) loss of visual acuity. These plots were done thanks to the application VR Tengo baja visi'o developed by the association Begisare . \n<!-- image --> \nrestricted to the use of visual resources, helping the inclusion of BVI. In addition, this strategy also takes advantage of a richness of resources to reinforce the idea of a richness of sources and, as many of these means provide information about the same objects as seen from different energies and perspectives, to provide a more complete view of the different aspects coming from the variety of properties that can be explored. \nThe adoption of a multi-messenger strategy implies thus the use of a diversity of resources and channels for the transmission of the information not exclusively based on the sense of sight, but extended to other senses. To address this, Astroaccesible has implemented the use of all kind of alternative resources,including maquettes and models to represent astronomical objects and phenomena, allowing BVI individuals to physically explore the shapes and structures of celestial bodies. Additionally, the project incorporates soundscapes and audio descriptions to convey data and observations in an auditory format. For example, the sonification of data from gravitational waves or the translation of visual graphs into sound patterns can provide an alternative means for BVI individuals to perceive and understand complex astronomical information [7]. In the below subsections, more details are given around each one of these adaptations.", '3.1 An inclusive use of images': "One of the most common approaches when dealing with an audience belonging to the BVI colective is a total renunce to the use of images, accepting the prior assumption that all assistants are totally blind. Instead, the preparation of every documention with the Braille language is used. This consideration can be wrong as most BVI individuals are not totally blind and, in fact, do not dominate the Braille system. On the contrary the preferred system to access towritten information is much more efficiently done by means of screen readers in mobile phones. Therefore, it is usually a much more accesible system to provide all documentation in an electronic format that can be accesed by every assistant. For the use of images, there is a similar problem. Apart from the fact that BVI are usually accompanyed by other people without any sight problem that are also participants in the activities, the heterogeneity of visual impairments is not usually considered; most visually impaired people have residual vision or visual memory, so the needs and approaches to make the disseminated information accessible vary greatly ([8]). \nIn Figure 2 it is represented the spiral galaxy M51 as observed through several optical filters, as seen by \n<!-- image --> \nFigure 3: Two examples of models used in the activities of Astroaccesible . Left: A model of the night sky as seen from the northern hemisphere. Rigt: A model of the Sun, with its different inner layers. \n<!-- image --> \npeople with different types of visual affections, including loss of peripherical vision, patched vision, loss of central vision, night blindness, photophobia, or loss of visual acuity. These images were obtained using the application VR TEngo Baja Visi'on , developed by the Spanish association Begisare . The mobile phone can be also used by any user to try to overcome these difficulties if the images are previously provided in an electronic form. This also includes some solutions based on artificial intelligence able to provide very accurate descriptions. In any case, it is preferable to adopt a preventive strategy trying to show the images under different configurations, contrasts, sizes, and colors to try to let most people access to the visual information, even if a part of them cannot correctly see it., Thus, one must never renonce to the use of images, given the inclusive strategy taht it is prferable to follow in any activity, giving information in as many different channels as possible.", '3.2 Use of tactile material': "Another important resource used in all the activities of Astroaccesible is the tactile material, both in the form of relief sheets and 3D models. Some of them can be seen in Figure 3. \nAmong them, are those developed by the project A Touch of the Universe [9], representing various rocky planets and moons of the Solar System. Our models of asteroids and comets, featuring detailed surfaces, help users understand these celestial bodies' shapes and structures. These are not only useful for a complete recognition of the characteristics of the bodies of the Solar System for BVI, but also for a much more complete identification of their geological features for everyone, overcoming the limitation of a much more simplistic exploration based on a projected image. \nAdditionally, 3D models of the night sky, printed on the outer surface of a hemisphere and featuring stars in relief connected by constellation lines, have been extensively utilized in our activities. These models assist BVI individuals in forming a spatial mental image of the sky's layout, and they also aid sighted individuals by projecting the same figures onto a screen, facilitating identification and contextual understanding of the night sky. Over the past few years, we have employed 3D models depicting the night sky visible from the Northern Hemisphere during autumn and winter, from Orion to Gemini constellations. These models are particularly useful when public observations are infrequent due to adverse weather conditions. Moreover, we have developed 3D models representing the summer night sky, including constellations such as Scorpius, Sagittarius, and the Summer Triangle. These models can be used during both outdoor and indoor inclusive activities, providing a comprehensive view of the sky during warmer months. \nTactile models of the night sky offer BVI individuals the chance to explore the stars' arrangement and to locate deep space objects, while sighted participants can compare these tactile impressions with visual representations. This approach not only enhances spatial awareness but also enriches the educational experience for all participants. \nThere is a pletora of other projects whose tactile resources have been widely used by Astroaccesible , \nFigure 4: Two snapshots of animations developed within the project Cosmonic , sonifying plots. Left: LIght curve of the transit of the exoplanet HATP7. Right: Predicted and observed radial stellar velocity curve around the disk of the spiral disk M33 to explore the effect of the presence of a dark matter halo. \n<!-- image --> \nincluding the kit Astro TES , comprising different elements like a model of the Sun 3 ; the Astro BVI project 4 , that includes low relief representations of different galaxy types; or a model of a protoplanetary disk 5 . All this material, used in combination of images, appropriate descriptions and, as we will see in the next subsection, sonifications, offer a incredible diversity of resources that are not only inclusive, but also expands in a unexpected way the motivation, compromise and inspiration of every participant for outreach and educational activities.,", '3.3 Using Sonifications': 'Sounds constitute an invaluable resource that can significantly enhance the multisensory aspect of the activities carried out by Astroaccesible , adding an extra dimension to all presented material. The use of sound, combined with images and tactile models, creates a comprehensive multisensory learning environment. This approach not only facilitates the inclusion of BVI individuals but also enhances the educational experience for all, fostering a deeper understanding of the universe through a multi-messenger perspective [10]. There is a growing interest from researchers and sound designers who are increasingly committed to the task of converting astronomical data into sound [11]. \nThe variety of techniques available for converting astronomical data into sound, such as audification, sonification, and musification, allows for their incorporation into different adapted activities, deepening their inclusive aspect. These techniques enable direct interaction of BVI astronomers with astronomical data, facilitating their integration into research groups in various contexts [12]. Sonification presents several advantages when used in conjunction with images, helping to present a more complete vision. For instance, the Cosmonic project [13], whose various products can be accessed on its website \nhttp://rgb.iaa.es/es/cosmonic/ , offers different animations with accompanying sounds. These resources enable BVI individuals to access the data, while also helping sighted individuals interpret the graphical content more effectively. The integration of sound not only aids in data comprehension but also enhances the overall sensory experience for all participants. Two examples of the graphics sonified by Cosmonic are shown in Figure 4, including the light curve of the star HATP7 with a transit of an exoplanet, and the radial stellar velocity around the spiral galaxy M33 to illustrate the effect of a dark matter halo. \nBeyond the inclusive aspect of sound, explaining the meaning of each property of the sound, including the chosen timbre, loudness, and tone, highlights the arbitrary nature of every sensory channel we use \nfor astronomical data. This feature of sound makes it especially convenient to raise awareness about the role of data processing both for analysis and education. Unlike images, which most people consider the natural way to represent astronomical data, sound can emphasize the conversion factor between a digital signal and its representation, making the process more transparent and understandable. Additionally, the ability of sound to be perceived at a much higher time resolution by the human ear, as well as the broader frequency range that can be detected, can be particularly useful for specific time-dependent data series [14]. In this context, it is a misconception to associate sounds exclusively with gravitational waves. The quality of sound can represent any type of astronomical data, enriching the multisensory outreach. Sound can convey information about the structure of galaxies, the dynamics of star formation, or the motion of celestial bodies, thus providing a versatile tool for education and inclusion.', '3.4 Audio descriptions: Using language as an additional window': "Given that the basis of a multi-messenger transmission of the astronomical information must reside on the principle of supplying alternative sources to the information, not exclusively based on the use of images to favour inclusion, other sources of information not based on senses can be envisaged. This is the case of language, as a tol to provide information at different levels. In this case, Astroaccesible tries to foster its use above all in Spanish promoting the diffusion of written texts, easily converted into a spoken speech, covering different matters related with astronomy and science. \nIn addition, another useful adaptation that Astroaccesible has implemented is the use of audio description (AD) [15]. AD is a valuable access service that translates visual information into words, enhancing the multisensory aspect of activities carried out by Astroaccesible . This intersemiotic translation practice bridges visual and linguistic signs, providing a comprehensive understanding of astronomical data through an inclusive, multi-messenger perspective. The primary goal of AD is to compensate for the lack of visual information, ensuring that people with sight loss can comprehend the described source material (e.g., films or paintings) similarly to sighted individuals. \nAD practice, traditionally focused on films and audiovisual media, is expanding into new areas such as museums and live events and its potential application in science, particularly astronomy, for both research and educational purposes, is immense. By incorporating AD, Astroaccesible not only supports BVI individuals but also enhances the educational experience for all participants, promoting a deeper understanding of the universe. \nIn science communication, AD scripts need to be meticulously crafted to convey essential visual details, aiding BVI individuals in forming mental images of the described objects. This guided viewing experience benefits sighted people as well, enriching their sensory experience and enhancing information retention . The El Universo en palabras 6 ( The Universe in words ), exemplifies this approach, where final-year Translation and Interpreting students at the University of Granada created audio-described videos under the supervision of both AD trainers and astronomers. \nThese videos, focusing on very popular astronomical objects, including objects from the Messier catalogue,, provide detailed visual and scientific information, helping both blind and sighted people understand complex astronomical concepts. The project's success underscores the potential of AD as a truly inclusive resource, enhancing the multisensory learning environment by incorporating elements like tactile aids and sonifications. \nAD's evolving role in science education highlights its broader application. It helps non-experts and experts alike by focusing attention on critical visual features, thus improving understanding and engagement. This approach is particularly relevant in astronomy, where expert knowledge significantly influences meaning-making processes.", '4 Summary and conclusions': 'Astroaccesible is an astronomy outreach project developed at IAA-CSIC, led by a blind astronomer, Over ten years of activities, it has proven that total inclusion is possible by extending strategies and resources originally designed for BVI people to all individuals, regardless of any disability. \nAdopting a multi-messenger philosophy for transmitting information that is collected through ground radiation or detecting particles-has proven effective. The use of different types of images combined \nwhich began as an initiative to demonstrate that astronomy can be made accessible to BVI individuals. observatories, space telescopes, and underground experiments covering various ranges of electromagnetic \nwith tactile materials, sounds useful for representing time variations, and audio descriptions that help the general public understand the context of these representations creates an excellent, inclusive set of tools. This approach makes astronomy, astrophysics, and astroparticle physics more accessible and engaging, opening inspiring new ways to handle the diverse ways the Universe sends us information. Implementing a teaching, outreach, and scientific content dissemination strategy based on Universal Design for Learning (UDL) criteria is essential for including disabled people in the scientific community. This approach encourages them to take an interest in science, potentially leading to professional involvement, and promotes the formation of more diverse research groups capable of finding creative and innovative solutions to scientific problems.', 'References': "- [1] A. Ortiz-Gil, M. Gomez, S. Martinez, P. Blay, J. C. Girado, A. T. Gallego, and M. Lanzara. Communicating Astronomy to Special Needs Audiences. Communicating Astronomy with the Public Journal , 11:12, July 2011.\n- [2] E. P'erez-Montero, E. Garc'ıa G'omez-Caro, Y. S'anchez Molina, A. Ortiz-Gil, S. L'opez de Lacalle, and A. Tamayo. Astroaccesible: Bringing the study of the Universe to the visually impaired. In S. Arribas, A. Alonso-Herrero, F. Figueras, C. Hern'andez-Monteagudo, A. S'anchez-Lavega, and S. P'erez-Hoyos, editors, Highlights on Spanish Astrophysics IX , pages 742-747, March 2017.\n- [3] Enrique P'erez-Montero. Towards a more inclusive outreach. Nature Astronomy , 3:114-115, February 2019.\n- [4] James W. Trayford and Chris M. Harrison. Introducing STRAUSS: A flexible sonification Python package. arXiv e-prints , page arXiv:2311.16847, November 2023.\n- [5] Caitlin O'Brien and Anna Voelker. AstroAccess: Improving Human Spaceflight through Universal Design. In American Astronomical Society Meeting Abstracts , volume 242 of American Astronomical Society Meeting Abstracts , page 116.06, June 2023.\n- [6] Nature Editorial. Science benefits from diversity. Nature , 558:5, 2018.\n- [7] E. P'erez-Montero, M. Lanzara, A. Ortiz-Gil, M. Villaverde, R. Garc'ıa-Benito, T. Gallego-Calvente, and E. Garc'ıa G'omez-Caro. A tactile model of the night summer northern sky for the teaching of astronomy to the BVI. In M. Manteiga, L. Bellot, P. Benavidez, A. de Lorenzo-C'aceres, M. A. Fuente, M. J. Mart'ınez, M. V'azquez Acosta, and C. Dafonte, editors, Highlights on Spanish Astrophysics XI , page 468, May 2023.\n- [8] Leopoldo Benacchio. Catch the Stars in the Net: astronomy and new technologies in an Italian education and outreach project. In Robert I. Kibrick and Anders Wallander, editors, Advanced Global Communications Technologies for Astronomy , volume 4011 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series , pages 124-132, June 2000.\n- [9] A. Ortiz Gil. A Touch of the Universe. In XIV.0 Scientific Meeting (virtual) of the Spanish Astronomical Society , page 265, July 2020.\n- [10] A. Zanella, C. M. Harrison, S. Lenzi, J. Cooke, P. Damsma, and S. W. Fleming. Sonification and sound design for astronomy research, education and public engagement. Nature Astronomy , 6:1241-1248, November 2022.\n- [11] Rub'en Garc'ıa-Benito. Astronomy & Astrophysics in ICAD History. arXiv e-prints , page arXiv:2311.12101, November 2023.\n- [12] A. Deandra, M. Putra, M. I. Mandasari, C. Kunjaya, D. Herdiwijaya, and Aprilia. An Application of Sonification as an Alternative for the Accessibility of Astronomical Images to the Visually Impaired. In Revista Mexicana de Astronomia y Astrofisica Conference Series , volume 54 of Revista Mexicana de Astronomia y Astrofisica Conference Series , pages 118-121, July 2022.\n- [13] R. Garc'ıa-Benito and E. P'erez-Montero. Painting graphs with sounds: CosMonic sonification project. In Revista Mexicana de Astronomia y Astrofisica Conference Series , volume 54 of Revista Mexicana de Astronomia y Astrofisica Conference Series , pages 28-33, July 2022.\n- [14] J. Tucker Brown, C. M. Harrison, A. Zanella, and J. Trayford. Evaluating the efficacy of sonification for signal detection in univariate, evenly sampled light curves using ASTRONIFY. Monthly Notices of the Royal Astronomical Society , 516(4):5674-5683, November 2022.\n- [15] E. P'erez-Montero, C. Barn'es-Casta˜no, and E. J. Garc'ıa G'omez-Caro. The Universe in Words: Astronomy for all through audio description within the outreach project Astroaccesible. In Revista Mexicana de Astronomia y Astrofisica Conference Series , volume 54 of Revista Mexicana de Astronomia y Astrofisica Conference Series , pages 111-114, July 2022."}
2024PhRvD.110b3018G	Highfrequency gravitational waves HFGWs are predicted in various exotic scenarios involving both cosmological and astrophysical sources. These elusive signals have recently sparked the interest of a diverse community of researchers due to the possibility of HFGW detection in the laboratory through gravitonphoton conversion in strong magnetic fields. Notable examples include the redesign of the resonant cavities currently under development to detect the cosmic axion. In this work we derive the sensitivities of some existing and planned resonant cavities to detect a HFGW background. As a concrete scenario we consider the collective signals that originate from the merging of compact objects such as two primordial black holes PBHs in the asteroid mass window. Our findings improve over existing work by explicitly discussing and quantifying the loss in the experimental reach due to the actual coherence of the source. We elucidate on the approach we adopt in relation with recent literature on the topic. Most notably we give a recipe for the estimate of the stochastic background that focuses on the presence of the signal in the cavity at all times and showing that in the relevant PBH mass region the signal is dominated by coherent binary mergers.	2024-07-01T00:00:00Z	['2024PhRvD.110b3018G', '10.1103/PhysRevD.110.023018', 'arXiv:2403.18610', '10.48550/arXiv.2403.18610', '2024arXiv240318610G']	['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Experiment', 'High Energy Physics - Phenomenology']	Cavity detection of gravitational waves Where do we stand	2,024	193	0.33	['EPRINT_HTML', 'EPRINT_PDF']	8	https://arxiv.org/pdf/2403.18610.pdf	{'Cavity Detection of Gravitational Waves: Where Do We Stand?': 'Claudio Gatti, 1, ∗ Luca Visinelli, 2, 3, † and Michael Zantedeschi 2, 3, ‡ \n1 INFN Laboratori Nazionali di Frascati, via Enrico Fermi 54, 00044 Frascati (Roma), Italy 2 Tsung-Dao Lee Institute (TDLI), No. 1 Lisuo Road, 201210 Shanghai, China \n3 School of Physics and Astronomy, Shanghai Jiao Tong University, \nDongchuan Road 800, 201240 Shanghai, China \n(Dated: September 17, 2024) \nHigh frequency gravitational waves (HFGWs) are predicted in various exotic scenarios involving both cosmological and astrophysical sources. These elusive signals have recently sparked the interest of a diverse community of researchers, due to the possibility of HFGW detection in the laboratory through graviton-photon conversion in strong magnetic fields. Notable examples include the redesign of the resonant cavities currently under development to detect the cosmic axion. In this work, we derive the sensitivities of some existing and planned resonant cavities to detect a HFGW background. As a concrete scenario, we consider the collective signals that originate from the merging of compact objects, such as two primordial black holes (PBHs) in the asteroid mass window. Our findings improve over existing work by explicitly discussing and quantifying the loss in the experimental reach due to the actual coherence of the source. We elucidate on the approach we adopt in relation with recent literature on the topic. Most notably, we give a recipe for the estimate of the stochastic background that focuses on the presence of the signal in the cavity at all times and showing that, in the relevant PBH mass region, the signal is dominated by coherent binary mergers.', 'I. INTRODUCTION': "The detection of gravitational waves (GWs) from compact mergers, made possible via a network of groundbased interferometers, has marked the dawn of GW astronomy [1, 2]. At present, these efforts focus on the sub-kHz GW frequency band, corresponding to the range in reach of the Laser Interferometer Gravitational-Wave Observatory, the Virgo interferometer, and the Kamioka Gravitational Wave Detector. \nNevertheless, the GW spectrum is expected to extend over various decades in frequencies. In fact, the detection in the nHz region of a GW background (GWB) has been recently confirmed by several consortia operating at the global scale, through pulsar timing array techniques [36]. Moreover, the cosmic microwave background (CMB) provides with an indirect constrains on the primordial GWB spectrum at frequencies below the pHz [7-9]. \nNear-future experiments plan to scan different bands encompassing different methods including the forthcoming ground-based [10, 11] and space-based [12, 13] laser interferometers, atom interferometers [14-16], and probes of the CMB [9, 17]. These endeavors cover the GW bands at the kHz frequency and below, where astrophysical and cosmological sources from the merger of known compact objects are expected to provide with a GWB besides other possible sources of unknown origin. \nA parallel search can be designed to involve high frequency gravitational waves (HFGWs), spanning fre- \nquency ranges well above the kHz. A GWB at high frequencies could potentially be sourced by a collection of exotic physical phenomena originating both in the early and late Universe. Examples include the merging of primordial black holes (PBHs) [18-25] or boson stars [26, 27], black hole (BH) superradiance [28], models of modified gravity [29, 30], the primordial thermal plasma [31], phase transitions in the early Universe [3234], the 'slingshot' mechanism, taking place upon the coexistence of confined and unconfined vacua in the presence of heavy quarks [35], a network of cosmic strings [3640], post-inflation (p)reheating [41, 42], and inflationary mechanisms [43, 44]. \nA novel avenue for detecting GWs at such high frequencies in the GW spectrum has gained momentum in recent years, with various proposal being explored including interferometers [45-48], microwave and optical cavities [49-54], mechanical resonators [55-60], superconducting rings [61], and superconducting resonant cavities [62]. Insightful reviews are found in Refs. [63, 64]. \nOne promising line that has recently been proposed involves the conversion of gravitons into photons via the inverse Gertsenshtein effect [65-67]. This intriguing concept has positioned resonant cavities as promising candidates for detecting gravitational wave signals within the MHz-GHz HFGW band [63]. Previous experiments such as Explorer at CERN, Nautilus at INFN-LNF, and Auriga at Legnaro predominantly targeted gravitational waves in the kHz range, emanating from the mergers of compact objects or comprising a GWB [68, 69]. Cavity experiments extend the detection capability to higher frequencies, potentially spanning the range (0.1-10) GHz. This groundbreaking approach not only widens the spectrum of detectable gravitational waves but also paves the way for exploring phenomena such as the existence and \ndistribution of PBHs within this yet uncharted frequency domain. \nVarious cavity searches are currently undergoing, including the Axion Dark Matter eXperiment (ADMX)G2 [70-72], ADMX EFR [73], the Oscillating Resonant Group AxioN Experiment (ORGAN) [58, 74, 75], the Haloscope At Yale Sensitive To Axion CDM (HAYSTACK) [76], the Center for Axion and Precision Physics Research (CAPP)-8T [77, 78], CAPP-9T [79], CAPP-PACE [80], CAPP-18T [81], CAST-CAPP [82], GrAHal [83], RADES [84-86], TASEH [87], QUAX [8893], and Dark SRF [62]. Other searches planned to be operational in coming years are the FINUDA magnet for Light Axion SearcH (FLASH) [94, 95], the International Axion Observatory (IAXO) in its intermediate stage BabyIAXO [96, 97], the Axion Longitudinal Plasma HAloscope (ALPHA) [98, 99], A Broadband/Resonant Approach to Cosmic Axion Detection with an Amplifying B-field Ring Apparatus (ABRACADABRA) [100], DM-Radio [101, 102], the Canfranc Axion Detection Experiment (CADEx) [103], and the magnetized disk and mirror axion experiment (MADMAX) [104]. A proposed network of cavities that employs 'quantum squeezing' would lower the noise and boost the efficiency of the search [105-109]. These endeavors collectively represent a concerted effort to advance our understanding of highfrequency gravitational waves and their potential implications in fundamental physics and cosmology. \nIn this work, we derive the sensitivities for some of the cavities above. We comment on the detection techniques for either a GW signals or a stochastic background. As a practical example, we consider the GW signal released from BH mergers. This system is characterized by its potentially fast frequency swiping which, contrary to the axion case, can lead to a significant loss of the cavity reach [23, 110]. Our goal is to quantify explicitly such a suppression, detailing the procedures outlined in a previous experimental report [95]. \nWhile our approaches may seem straightforward, we believe this to be worth explicitly commenting on, given the above mentioned advancement and the excitement in the realm of HFGWs. As it turns out, the actual reach given by physical sources such as PBHs mergers is significantly worse than what previously discussed in the literature, see also Refs. [111-113]. Even in the best case scenario, the discrepancy between current experimental reaches and the physical GW signal amounts to about eight orders of magnitudes, for the case of BH mergers. Rather than interpreting our results as a negative outcome for the detection of HFGWs, we envision them as a catalyst for inspiring novel experimental setups and the study of GW sources. \nOur paper is organized as follows. In Sec. II we outline the details of the expression used in the analyses. The results of our approach are presented in Sec. III and a discussion is developed in Sec. IV. Conclusions are drawn in Sec. V. We set ℏ = c = 1 unless otherwise stated.", 'A. Sensitivity forecast': 'The coupling of the photon with gravity is described by the Maxwell-Einstein action, \nS = ∫ d 4 x √ -g ( -1 4 g µα g νβ F µν F αβ ) , (1) \nwhere g µν is the space-time metric with determinant g and F µν is the electromagnetic field strength. \nThe term in Eq. (1) modifies the usual expressions of electrodynamics by introducing a new source, see Eq. (A1) in Appendix A, and leads to a deposition of energy in the resonant cavity. To see this, the metric tensor is expanded to first order around a flat background as \ng µν = η µν + h µν , (2) \nwhere \| h µν \| ≪ 1 describes the perturbations. The action in Eq. (1) predicts a coupling between the GW signal and the electromagnetic (EM) energy tensor T µν EM with the Lagrangian L = (1 / 2) h µν T µν EM . This leads to the effective coupling L ∝ h 0 B 0 δB z for a GW strain h 0 in an external magnetic field B 0 ˆ z , where the magnetic field variation coupled within the cavity δB z can be picked up with a magnetometer. \nTo address the detection of a GW source by a resonating cavity we follow the discussion in Ref. [114], adapting the treatment originally expressed in terms of the search through interferometers to the case of a haloscope. Given the energy stored in the cavity U , the signal-to-noise ratio (SNR) is obtained as \nSNR = 2 πU T sys √ ∆ t ∆ f , (3) \nwhere T sys is the coldest effective temperature of the instrumentation and noise amplifier, ∆ f is the frequency bandwidth, and ∆ t is the time under which the signal is collected. The expression for the energy stored in the cavity is detailed in Eq. (A20). \nTo begin with, we assume that we observe a sufficiently stationary source of GWs for a number of cycles N cycles > ∼ 1. The source emits within the cavity frequency bandwidth ∆ f = f/Q , where Q is the quality factor of the cavity, so that the reach of the strain h 0 is obtained by setting SNR = 1. Once the expression for the power of the signal in Eq. (A24) is considered, this leads to an estimate for the strain at resonance as \nh 0 ≈ 6 . 0 × 10 -23 ( 1 T B 0 ) ( 0 . 14 η ) ( m 3 V ) 5 / 6 ( 10 6 Q eff ) 1 / 2 × ( T sys K ) 1 / 2 ( ∆ f kHz 1 min t int eff ) 1 / 4 ( GHz ω n / 2 π ) 3 / 2 , (4) \nwhere Q eff = min( N cycles , Q ) is the effective quality factor, η the coupling of the cavity with the gravitational signal defined in Eq. (A21), V the effective volume of the cavity, T sys the system temperature, ω n = 2 πf the resonant pulsation, and t int eff = min( N cycles ω -1 n , ∆ t ) the minimum value between the experimental integration time and the source duration. For a detailed derivation and definition of all the above quantities we refer the reader to Appendix A. The expression in Eq. (3) can also be used to estimate the sensitivity over a stochastic GW background. A GWB spectrum is generally assumed to be nearly isotropic, unpolarized, stationary, and characterized by a Gaussian distribution with zero mean. The fractional contribution to the density parameter Ω GW can be alternatively expressed in terms of a dimensionless characteristic strain h c as [115, 116] \nΩ GW ( f ) = (2 π ) 2 3 H 2 0 f 2 h 2 c , (5) \nwhere H 0 is the Hubble constant. \nAs we discuss in Appendix A, provided Q eff ∼ Q -whose validity depends on the properties of the sources discussed below - the characteristic strain reach reads \nh c ≈ 6 . 0 × 10 -20 ( 1 T B 0 ) ( 0 . 14 η ) ( m 3 V ) 5 / 6 × ( T sys K ) 1 / 2 ( ∆ f kHz 1 min t obs ) 1 / 4 ( GHz ω n / 2 π ) 3 / 2 , (6) \nwhere the total observation time of the experiment t obs can strategically largely exceed the value chosen for the search for a coherent source. \nNote, that the studies in Refs. [20, 21] find a much smaller sensitivity for the GW strain h ∼ 10 -30 , which would be ideal to probe the floor of GW background from coalescing compact objects of asteroidal mass. It is unclear to us how their analysis in time-domain can increase the experimental reach in such a non-trivial manner, given the analysis leading to Eq. (6). However, as stated in their conclusions, several assumptions are made regarding the coherence and duration of the stochastic background. Furthermore, the problem of frequency width of a stochastic GW background, as compared to the narrow width of the cavity, remains unaddressed. In this work, we characterize some of these effects. \nSo far, we have discussed the response of the cavity to a stationary signal. We now briefly comment on the coherence of the source. For the dark matter axion, the quality of the source is limited by thermal and quantum fluctuations which impact over the maximal capacity of the cavity to resonate with the signal. For the case of a GW signal from PBH mergers or other sources of HFGWs, the actual coherence of the GW signal at a given frequency is granted by the large number of gravitons in the parameter space of interest. This implies that the same approach as for the axion can be used to parametrize the non-stationarity of a HFGW source. \nIn fact, we can describe the gravitational wave signal as a coherent state of highly occupied gravitons of energy ω , with the occupation number N g ∼ ρ/ω 4 ∼ m 2 Pl h 2 0 /ω 2 corresponding to the number of gravitons per de Broglie volume ω -3 . Here, we have used the fact that the energy density of the GW signal is given by ρ ∼ h 2 0 ω 2 m 2 Pl . For a coherent signal, typical fluctuations are of the order of δN g ∼ √ N g , leading to a quality of the signal Q h = N g /δN g ∼ √ N g . This quality factor is potentially degrading the quality of the source when it is smaller than the quality of the cavity Q . Therefore, the quality of the source is determined by the quantum origin of the signal. \nFor example, consider a GW signal with the frequency f ∼ 0 . 1 GHz and a typical physical strain of the reach h 0 ∼ 10 -22 , corresponding to the typical current reach of the cavity given in Eq. (11) below. In this setup, the number of gravitons is N g ∼ 10 32 , which implies a source quality Q h ∼ 10 16 that is much larger than the quality factor of the cavity. Even in the desirable scenario where the cavity would reach a sensitivity comparable to the actual physical signal of strain h 0 ∼ 10 -30 , the source would possess the quality Q h ∼ 10 6 > ∼ Q . The consideration above therefore justifies the dropping of such a contribution from the treatment. Note, however, that if the sensitivity of the experiment could reach an even smaller sensitivity for this range of frequencies, part of the signal could be degraded.', 'B. Gravitational wave sources': "Potential sources of GWs are generally divided into two categories, namely sources of cosmological origin produced before recombination and sources of astrophysical origin. A cosmological GWB at high frequencies, expected from exotic sources, can be constrained by BBN considerations and CMB data as [33] \nΩ GW h 2 H < ∼ 5 . 6 × 10 -6 ∆ N eff , (7) \nwhere h H = H 0 / (100 km s -1 Mpc -1 ) is the reduced Hubble constant. The excess in the number of relativistic active neutrinos is constrained as ∆ N eff < ∼ 0 . 3 by various considerations on big-bang nucleosynthesis (BBN) in combinations with CMB data. Using Eq. (5), the bound above reads [110, 117] \nh c < ∼ 2 × 10 -30 (GHz /f ) ∆ N 1 / 2 eff , (8) \nwhich is several orders of magnitude below the expected reach of resonant cavities expressed in Eq. (6). Therefore, a potential detection of HFGWs can only be due to astrophysical sources in the late Universe. \nHowever, since there are no known astrophysical sources releasing GWs at such high frequencies, new physics is likely required to motivate the search in the \nHFGW band. Possibilities include the decay of an unstable axion star after a binary merging [118] and the stimulated decay of dark matter (DM) in theories of ChernSimons gravity [29]. We comment below on the merging of compact objects, with particular focus on the case of PBHs. We consider BHs that are too light to be explained by known stellar dynamics, so that their indirect detection through mergers would require new physics to explain their origins. A possibility is that these PBHs form in the early Universe, hence the name primordial. The exact details of the formation scenario are currently unknown. For example, they could result from inflationary overdensities [119], from the collapse of bubbles in supercooled phase transitions [120, 121], from the confinement of heavy quarks [122], or many other methods [119]. \nIn Table I we report the main setups of some experiments that employ a resonant cavity, see the caption for the specific references. For each experiment, the noise temperature T sys is obtained through the formula T sys = ω c [1 / (exp( ω c /T phys ) -1) + N A +0 . 5], where ω c / (2 π ) is the central frequency in the band covered by the experiment and the number of states is N A = 1 / 2 everywhere except for FLASH and BabyIAXO, for which N A = 10. In this context, 'FLASH HighT' refers to the initial phase of the experiment [95]. This phase will undergo an upgrade with an enhanced cryostat to achieve the performance levels of 'FLASH LowT', which serve as benchmarks for the results discussed below. 1 \nA fundamental question arises regarding whether these objects, regardless of how they formed, can constitute a substantial portion of DM. Cavity searches are sensitive to mergers involving sub-solar PBHs with masses ranging below about 10 -8 M ⊙ . This result, as we outline below, derives from requiring that at least one complete revolution of the binary system appears in the cavity tuned at the frequency around the GHz. This PBH mass range partially overlaps with the region of masses heavier than 10 -11 M ⊙ , in which stringent lensing constraints have already discounted PBHs as the primary constituents of DM [124]. To sum up, in the region of masses [10 -11 -10 -8 ] M ⊙ a minor contribution from PBHs to the DM abundance is still plausible [119], while the region of masses below 10 -11 M ⊙ is not currently constrained by microlensing results. \nAlthough cavity experiments do not reach the sensitivity required to probe the type of signal from these sources, several uncertainties in cosmological history could substantially amplify the signal. One explicit possibility under discussion is the incorporation of significant non-Gaussianity from the inflationary period, which, in turn, could lead to an escalation in the merger rate [23]. Note, that even in the best scenario a maximal increase \nof about two orders of magnitude is feasible in the merger rate [23]. \nCompact objects such as BH and neutron stars (NS) form in astrophysical environment. Possible and more exotic configurations such as PBHs or boson stars could have already been present in the earliest stages of the Universe [26, 27]. Binaries of compact objects could fall in the frequency range and with a GW strain that is sufficiently strong to be detectable with present or nearfuture technologies. Consider two compact objects of similar mass M and size R , and each of compactness C ≡ GM/R , forming a system of total mass M TOT ≈ 2 M . The frequency of the emitted GW spectrum at the end of the inspiral phase, when the stars occupy the innermost stable circular orbit, is [125] \nf = C 3 / 2 3 √ 3 πGM TOT . (9) \nFor example, a signal in the bandwidth O (100 MHz) gives \nM TOT ≈ 4 × 10 -6 M ⊙ ( C 0 . 1 ) 3 / 2 , (10) \nwhich, for the compactness of a BH C BH = 0 . 5, corresponds to the frequency of PBH binaries with mass M BH ∼ 10 -5 M ⊙ .", 'III. RESULTS': 'We focus on the detection of light PBHs in the asteroid mass window, whose merging would lead to the release of a HFGW signal with the GW strain [23] \nh 0 ≈ 10 -22 ( 10 kpc d )( M TOT 2 . 2 × 10 -5 M ⊙ ) 5 3 ( f 200 MHz ) 2 3 . (11) \nThe distance d is fixed by requiring that at least one PBH merger per year occurs in the Galaxy and assuming f PBH ≡ ρ PBH /ρ DM = 1, where ρ DM is the DM density today. In the derivation of the merger rate, a further enhancement coming from the galactic overdensity has been accounted for [126]. See also Fig. 3 in Ref. [23]. \nAn event within a distance d < ∼ 1 kpc could therefore be within the reach of the cavity experiments. Unfortunately, the source of GW, namely the in-spiraling binary, cannot be treated as a coherent source. The system emits at a given frequency for a number of cycles given by [127] \nN na¨ıve cycles = f 2 ˙ f ≃ ( M 2 . 2 × 10 -5 M ⊙ ) -5 / 3 ( f 200 MHz ) -5 / 3 . (12) \nThis can result in an effective limitation of the source at resonating with the detector. Eq. (12) describes the number of oscillations performed by the source at a given frequency, in a frequency width of order f during the swiping. In the literature, this is assumed to be the number \nTABLE I. Parameters defining the resonant cavity experiments considered in this work. Specifically, ADMX-G2 [70-72], ADMX EFR [123], FLASH low-frequency (LF) and high-frequency (HF) [95], ALPHA [98, 99], HAYSTACK [76], and BabyIAXO [96, 97]. \nof cycles for which the merger is a proper source inside of the cavity [23]. However, a merger can only resonate in a cavity as long as its frequency lies within the frequency width of the cavity ∆ f ≪ f itself. This leads to a further loss of reach, as the effective number of cycles within the cavity is expressed as \nN cycles = ∆ f f N na¨ıve cycles ≃ 1 Q N na¨ıve cycles . (13) \nAs we discuss below, the expression in Eq. (13) forces the reach of the detector described in Eq. (4) to be valid upon replacing the quality factor with the effective quantity Q eff ≡ min( Q,N cycles ). Similar comments regarding the swiping time t eff int and the value of Q eff have been pointed out in Ref. [24]. Our work extends the discussion by pointing out that the optimal mass range for the detection of GWs from PBH mergers relies on maximizing the cavity resonance. \nFig. 1 shows the effective quality factor Q eff (left panel) as a function of the PBH binary merger mass, at fixed frequencies for the experiments in Table I. 2 A maximal resonance is possible in this class of experiments only for PBH mergers lighter than about 10 -11 M ⊙ . For heavier mergers, the number of cycles scales as M -5 / 3 , leading to the complete absence of resonance Q eff ∼ 1 for masses around 10 -9 M ⊙ . No detection is possible for heavier PBH mergers with current strategies, perhaps suggesting the adoption of broadband type of experiments in those region, as discussed e.g. in Ref. [23, 110, 129]. \nTherefore, two competing effects come into play when the detection of PBH mergers is considered. On one hand, at a fixed frequency one would like to consider heavier binaries, as the signal expected in Eq. (11) would be more prominent for at higher masses. On the other hand, it would be optimal to exploit the cavity resonance with the highest possible Q eff , thus requiring the search \nfor light PBH binaries. As discussed below, it is indeed the former effect determining the optimal reach to be around 10 -11 M ⊙ given the physical signal of PBH mergers. \nAnother factor impacting the experimental reach for an individual source is expressed by the effective integration time t eff int , which is here defined as the minimum value between the source duration and the experimental observation time t obs . Here, we set t obs = 120s as a reference. The effective integration time is typically of the order of a few minutes at a given frequency and is limited by the swiping time of the PBH merger within the frequency width of the cavity. This is shown in the right panel of Fig. 1. The effect is in place for basically all PBH mergers with a total mass heavier than 10 -14 M ⊙ . Adirect inspection of Eq. (11) shows that the effect is not as impactful as the decrease in sensitivity coming from the scaling of Q eff with the binary mass. \nCompact objects that come close to each other in unbound orbits perform a hyperbolic motion, leading to a single scattering event that manifests itself through a burst of GWs [130-133]. The potential reach due to hyperbolic encounter does not greatly differ from the case of bound orbits, since the duration of the signal resonating in the cavity is extremely short. When considered, this additional contribution would affect the detectable strain by a factor of the order of O (1), so that the conclusions drawn above would not be altered considerably. \nA significant enhancement of Eq. (11) would be brought in by the possible presence of a non-Gaussianity component in the density perturbations [23]. In the region of optimal reach around 10 -11 M ⊙ , the maximal enhancement is similar in amplitude, to the decrease in signal due to the typical bounds on lensing, requiring f PBH ∼ O (10 -2 -10 -3 ) in the PBH population. This justifies the adoption of f PBH = 1 as an optimistic case scenario. \n<!-- image --> \nFIG. 1. The effective quality factor Q eff (left panel) and the minimum value between the source duration and the experimental observation time t obs , here t eff int (right panel) for the experiments listed in Table I. As a reference, the value t obs = 120s is chosen. \n<!-- image -->', 'A. Coherent sources': 'We first discuss the phenomenology related to the detection of a coherent source such as a binary merger. To showcase the effects affecting the cavity reach due to the non-stationary behaviour of the source, we first consider the setup of the resonant cavity employed in the FLASH experiment, 3 as considered in Table I. The reach for the experiment according to Eq. (4) is shown in Fig. 2 (red solid line) in comparison with the signal generated by mergers with f PBH = 1 (black solid line). Note, that the expected signal scales linearly with f PBH . In Fig. 2, the frequency is fixed at the cavity value f c = 200MHz, which is within reach for the FLASH setup. The drop in reach for M < ∼ 10 -11 M ⊙ is due to the non-stationary behaviour of the source, resulting in a lower effective quality factor Q eff as discussed below Eq. (12) and explicitly shown in Fig. 1. \nAnother loss in sensitivity takes place for PBHs lighter than about 10 -15 M ⊙ . In this mass region, the limit factor originates from the effective time required by the source to swipe over a frequency width of the order of f/Q = ∆ f , as shown in the right panel of Fig. 1. In fact, the optimal reach region balancing between the physical signal and the cavity reach is around the M BH ≈ 10 -11 M ⊙ mass window, for which the optimal reach is achieved when \nQ eff ≃ N cycle ≃ Q. (14) \nFIG. 2. Comparison of the forecast reach for FLASH for different PBH masses. The red line correspond to the reach according to the values in Table I. The black line corresponds to the theoretical prediction range for a monochromatic PBH distribution of values f PBH = 1. The cavity frequency is fixed at f c = 200MHz. \n<!-- image --> \nSimilar results as in Fig. 2 qualitatively hold true for the other cavity setups expressed in Table I. \nWe now consider the forecast reach in the cavity experiments given in Table I. Results for the reach as a function of the cavity frequency are shown in Fig. 3 for the PBH mass M BH = 10 -11 M ⊙ (left) and M BH = 10 -9 M ⊙ (right). For each panel, the expected reach (dark shaded area) is compared with the ideal reach of a perfectly coherent source (lightly shaded area). For the binary masses considered, most of the actual reach loss accounts for the frequency swiping time being shorter than the typical integration time. In fact, according to Eq. (14) the effective quality factor Q eff is maximized in this mass region. This corresponds to an ideal source that allows \nfor the cavity to fully resonate without any loss due to a quality factor at source. This is the reach usually showed in the literature, see e.g. Refs. [23, 129, 134]. For each panel, the black solid line marks the physical signal expected by a population of PBH binary mergers obtained using the results in Ref. [23]. Although cavity experiments swiping higher frequency ranges have a higher chance of working in the regime closer to the actual potential physical signal, there exists a discrepancy by many orders of magnitude from a potential detection of this source. \nThe right panel in Fig. 3 shows the dramatic improvement in considering a heavier mass range M BH ≈ 10 -9 M ⊙ . While the physical signal increases in strain, the actual reach (dark shaded areas) significantly decreases. This is due to the quality of the source N eff being placed away from the condition in Eq. (14) for the mass range considered. In this scenario, the physical cavity reach effectively places even further away from the physical signal when compared with the optimal case scenario M BH ≈ 10 -11 M ⊙ previously discussed and shown in the left panel.', 'B. Stochastic source': 'We now turn to the discussion over the stochastic signal sourced by merging PBH binaries, characterized by a superposition of weak, incoherent, and unresolved GW sources [135-137]. For this signal, the loss in coherence described in the previous section does not affect the reach for the mass region considered. However, the typical strain signal turns out to be significantly lower even for an optimistic scenario where the strain h 0 < ∼ 10 -26 is expected [23]. \nThe analysis of a stochastic signal demands the coincident detection by two correlated and co-aligned GW detectors, each picking up a GW strain h i ( f ) with i = 1 , 2 labeling the detector. 4 The signal-to-noise ratio can be expressed as [114] \nSNR ≈ 3 H 2 0 10 π 2 √ t obs (∫ + ∞ -∞ d f γ 2 ( f ) Ω 2 GW ( f ) f 6 P 1 ( f ) P 2 ( f ) ) 1 / 2 . (15) \nHere, P i ( f ) is the noise power spectrum of the i -th detector, which is related to the variance of the crosscorrelation signal as \nσ 2 i = ∫ + ∞ 0 d f P i ( f ) . (16) \nThe quantity γ ( f ), known as the overlap reduction function, quantifies the reduction in sensitivity due to the \ntime delay between the two detectors and the non-parallel alignment of the cavity axes [137]. For coincident and co-aligned detectors it is γ ( f ) = 1, while it is expected γ ( f ) < 1 when the detectors are shifted apart or rotated. \nTo claim the successful detection of a GWB, we assume that such a signal is indeed present at the frequencies considered, with a mean value that allows for its correct identification for a fraction δ of the times. In this context, the SNR should satisfy [114] \nSNR ≥ [erfc -1 (2 α ) -erfc -1 (2 δ )] / √ 2 , (17) \nwhere α quantifies the false alarm rate. Setting a false alarm rate α = 0 . 05 and a detection rate δ = 0 . 95 leads to the requirement SNR > ∼ 1.64. Note, that the presence of t obs appearing under the square root in Eq. (15) allows to consider much longer integration times when searching for the stochastic GWB compared with the coherent signal. Moreover, increasing t obs in a search of a coherent, but transient, signal would not lead to an improved sensitivity because of the shortness of the signal duration. \nWe assess the sensitivity of the strain h c in Eq. (6) assuming a period of observation t obs = 6months, leading to the results in Fig. 4 for the different experiments in Table I. To estimate the expected physical signal we adopt the derivation of Refs. [138, 139], see also Ref. [23] for a detailed discussion. We assume that the PBH masses are distributed according to a log-normal distribution centered at M c = 10 -11 M ⊙ and of width σ = 0 . 26 as [140, 141] \nψ ( M BH ) = f PBH √ 2 πσM BH exp [ -log ( M BH /M c ) 2 2 σ 2 ] . (18) \nNote, that an optimal reach is guaranteed for the range of masses considered since Q eff ∼ Q . The distribution in Eq. (18) is normalized so that the fraction of PBHs that make up the DM is \nf PBH = ∫ d M BH ψ ( M BH ) . (19) \nThe result for the expected signal shown in Fig. 4 (black solid line) scales linearly with the PBH abundance f PBH , which is here set as f PBH = 1. Even in this optimistic scenario, the values lie several orders of magnitude below the bound from BBN considerations in Eq. (7) (red-dashed line) and the actual cavity sensitivity of the various experiments.', 'C. Can we expect a continuous signal?': 'To address the stochastic GWB from PBH mergers, a monochromatic mass distribution with f PBH = 1 is now assumed for simplicity, in analogy with the treatment for a coherent signal. Our findings thus allow for a direct comparison between the coherent and the stochastic \n<!-- image --> \nFIG. 3. Cavity reach for the experiments listed in Table I as a function of frequency. For illustration purposes, we report the reach for the central value of the frequency range of each experiment. The lightly-shaded area denotes the expected reach for an ideal source, not limiting the capability of the cavity to fully resonate. These results are consistent with the sensitivity reach found, e.g., in Refs. [23, 110, 134] which, indeed, relied on the same assumption. \n<!-- image --> \nThe shaded area denotes the actual reach as compared to the actual source of a binary with mass 10 -11 M ⊙ (left panel) and 10 -9 M ⊙ (right panel), whose physical expected amplitude is marked by a black solid line. The PBH population is assumed to satisfy f PBH = 1. \nFIG. 4. Stochastic GW background reach. The black line denotes the actual physical signal for f PBH = 1 and the central value of the log-normal distribution M c = 10 -11 M ⊙ , see Eq. (18). The BBN bound is reported for comparison (dashed red line). \n<!-- image --> \ncases. We generally obtain that for large PBH masses it is more convenient to search for a coherent signal, while the stochastic signal is stronger for light PBH mergers. Note, that for this type of analysis it is required to coordinate at least two detectors [63]. \nA necessary requirement for a stochastic noise to be detectable is that, at any time, at least O (1) mergers are sourcing within the cavity. As discussed above, the swiping time t swiping across frequencies of the order of the frequency width of the cavity ∆ f depends on the masses of the BH pair, with the typical t swiping as given in Fig. 1. Therefore, at least one merger per swiping time should take place at every moment. This requirement provides the typical distance d swiping at which a merger takes place, therefore allowing for an optimistic characterisation of the amplitude of the signal. \nGiven a rate per unit volume of PBH mergers R PBH , see e.g. Ref. [23], the typical distances at which one \nmerger event is constantly present in the cavity is found from integrating over a volume of radius d swiping = (3 V swiping / 4 π ) 1 / 3 , to obtain \n1 < ∼ t swiping ∫ d swiping 0 d r 4 πr 2 R PBH . (20) \nWhile the above condition might not be sufficient to provide a stochastic signal, it surely suffices. More pragmatically, we address the question whether a continuous signal offers better detection prospects rather than fewer and rarer events of larger amplitude. As a byproduct, we determine the best search strategy for detecting PBH mergers in the mass window considered. \nIn previous literature the distance has been fixed through a different requirement, more precisely by looking for the volume within which one coherent merger per year is realized [23]. Under the condition in Eq. (20), the typical distance of a merger found by our requirement to realize at least one merger at each moment of time is significantly larger than the case of one coherent merger per year. \nIn Fig. 5, we show the comparison between the typical distances of black hole mergers sourcing a stochastic background (continuous line) with the coherent case of one merger per year f PBH (dashed line) for different values of f PBH . In the latter case, the resulting curve is found to be consistent with the one in Ref. [23]. In the former case, the distance now also depends on the typical swiping time. For PBH masses above ∼ 10 -14 M ⊙ , the typical distances increase significantly as they become larger than the galactic size, therefore resulting in fewer merging events for a given volume due to the absence the typical Galactic overdensity enhancement [126]. A similar effect is observed in the coherent case at heavier black hole masses [23]. The distances obtained for the \nstochastic case are significantly larger than in the coherent case and extend to extragalactic regions for heavier PBHs. For clarity, the plot in Fig. 5 focuses on the mass range for which the typical merger distances are larger than 10 5 kpc. For even larger distances local galactic enhancements of nearby galaxies become relevant and, eventually, the redshift of the signal should also be included for z > ∼ 0 . 1. \nFIG. 5. Distance for mergers sourcing a gravitational wave stochastic background under the condition in Eq. (20) (solid line). The coherent case of one merger here is shown with dashed lines. \n<!-- image --> \nOne might expect that the requirement adopted here would lead to a significant loss in the potential reach of the experiment as compared to the true physical signal. However, this is partially balanced by the simple fact that the actual reach of the experiment is now integrated over the full experiment time and not just over the swiping time, see Eq. (4), therefore partially making up for such a loss. For simplicity, in the following we assume at total observation time of 1 year. Note, that the following approach allows for a direct utilisation of the axion search data since, due to the small frequency range explored by cavity experiment, changing slightly frequency every few minutes does not alter the magnitude of our estimates. \nWe now compare the potential reach for both the stochastic and the coherent cases. In this comparison we directly use Eq. (4), replacing the integration time with the running time of the cavity experiment which, for simplicity, we take to be 1 year. Since we expect O (1) mergers to resonate within the cavity at every time, for a fixed BH mass we must use the effective quality factor Q eff in Eq. (4), analogously to the coherent case. Part of the resonance is suppressed by the presence of multiple sources, potentially resulting in a lower Q eff which is compensated by the presence of multiple detectors. \nRegarding the physical source, the difference between the stochastic and the coherent cases is given by the ratio of the distances obtained by adopting Eq. (20) and shown in Fig. 5. Therefore, it is useful to define the potential for a discovery as S i = h i physical /h i reach , with i =coherent, stochastic. This quantity should be larger than one in \norder to ensure detection. For simplicity, in the following we focus on the FLASH experiment in Table I provided that other experiments lead to similar results. \nThe ratio between the stochastic and coherent potentials for a detection is given by \nS coherent S stochastic ≃ d (1 year) d ( t swiping ) ( 1 year t int eff ) 1 / 4 . (21) \nThis ratio is shown in Fig. 6 as a function of the PBH merger masses and for a population with f PBH = 1. Similarly to the distance plot comparison in Eq. (5), there is a dip in the figure coming from the missing galactic enhancement in the PBHs density. At smaller masses the stochastic signal is comparable to the coherent one, therefore suggesting the possibility of a joint search. Note, that the signal distances between the stochastic and coherent case become comparable due to the high density of light PBHs in the galaxy. In this rage of masses, the swiping times are macroscopically large and reach up to O (0 . 1 year), therefore the condition in Eq. (20) is likely not leading to a clean stochastic background. Likely, a different condition is necessary, possibly leading to potentially larger distances and to smaller values of Q eff due to interference. Consequently, the coherent signal would likely dominate even for light mergers. In any case, in the light mass range window, close to asteroid masses where all of the dark matter could come in the form of primordial black holes, the actual signal is extremely far from the current reach due to the lightness of the primordial masses, as expressed in Fig. 2. \nFIG. 6. Ratio of potential reach for discovery between stochastic and direct merger detection. See Eq. (21) \n<!-- image --> \n. \nFor higher masses, the stochastic signal gets a suppression due to the large volumes necessary to attain a stochastic background. Therefore, for heavy PBHs it is more feasible to detect a coherent signal from one merger, suggesting the need to develop accurate templates for describing the signal from such events. Our estimate of the stochastic signal is clearly approximate and somewhat optimistic, so we therefore expect these results to remain valid in light of a potential and more accurate analysis.', 'V. CONCLUSION: IS THERE A DETECTION OUTLOOK?': 'We have discussed the usage of resonant cavities, generally employed as haloscopes to search for the cosmic axion, for the search of gravitational wave signals of astrophysical origin at frequencies (0.1-10) GHz. This highfrequency gravitational wave (HFGW) domain spans a significantly broader spectrum compared to what is currently accessible by both ground-based and planned space-based interferometers. Various proposals for these investigations have been advanced in the scientific literature, including one outlined in the FLASH experiment [95]. \nIf the HFGW signal originates from coalescing primordial black holes (PBHs), cavity experiments can probe the existence of PBH mergers in the mass window M BH < ∼ 10 -9 M ⊙ . Adopting the setup of the FLASH cavity and considering simultaneous observations of both axions and gravitational waves (GWs) from compact mergers yields the forecast reach as a function of the binary mass in Fig. 2. Other experiments employing similar setups lead to comparable results. \nThe signal expected from individual coherent sources in the Galaxy is too faint to be observed with current setups, based on conservative models for the distribution and merging details of the compact objects. This stems from quantifying the loss in the experimental reach due to the actual coherence of the source as discussed in Sec. III. The results do not improve when the collective stochastic signal that originates from the merging of multiple compact objects is considered. In this view, we have given a recipe for the estimate of the stochastic background that differs from previous literature and focuses on the presence of the signal in the cavity at all times, see Sec. IV C. This allows us to predict the region of the PBH masses where the signal is dominated by coherent binary mergers, see Fig. 6. Our method relies on the evaluation of the distance that assures to attain at least one event in the cavity at all times, which is obtained from the condition in Eq. (20) and shown in Fig. 5 in comparison with the previous literature. Contrary to the coherent case, the distance thus obtained depends on the typical frequency swiping time. Given the vast range of masses in which the detection of a coherent signal would be facilitated over the stochastic background, our findings suggest to push for a broadband search for HFGWs with ∆ f ∼ f . \nDespite this result does not push towards the search of a HFGW signal through resonant cavities, we remind that i) such a search comes at practical no expenses, given that the cavity searches for the cosmic axion are already in place. For this, the detection apparatus should be equipped with an appropriate pick up and acquisition system of the signal generated in cavity modes not excited by the axion, as discussed elsewhere [95]. Moreover, ii) the frequency band covered has not been explored by any dedicated search to date, so that unexpected sources \ncould be present and add up to the GWB as a result of exotic new physics. In view of these considerations, the search for HFGWs is bright.', 'ACKNOWLEDGMENTS': "We thank Diego Blas, Gabriele Franciolini, Gianluca Lamanna, and Francesco Enrico Teofilo for the useful discussions that led to the present work and for reviewing the draft. This research was supported by the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP), which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2094 - 390783311. M.Z. and L.V. acknowledge support by the National Natural Science Foundation of China (NSFC) through the grant No. 12350610240 'Astrophysical Axion Laboratories', as well as hospitality by the Istituto Nazionale di Fisica Nucleare (INFN) Frascati National Laboratories and the Galileo Galilei Institute for Theoretical Physics in Firenze (Italy) during the completion of this work. L.V. thanks for the hospitality received by the INFN section of Napoli (Italy), the INFN section of Ferrara (Italy), and the University of Texas at Austin (USA) throughout the completion of this work. This publication is based upon work from the COST Actions 'COSMIC WISPers' (CA21106) and 'Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse)' (CA21136), both supported by COST (European Cooperation in Science and Technology).", 'Appendix A: Detailed resolution of the Einstein-Maxwell equations': "Adapting the derivation in Ref. [99] for the search of axions in a resonant cavity, we consider Maxwell's equations in a dielectric medium sourced by a GW strain, \n∇· D = ρ, ∇× H -∂ t D = J , ∇· B = 0 , ∇× E + ∂ t B = 0 . (A1) \nHere, the displacement and the electric fields are related by D = ϵ E , where ϵ is the permittivity tensor, while the magnetic flux density and field strength are assumed to coincide, B = H . We solve the set of equations for an infinitely extended cylindrical cavity filled with a plasma, assuming that the axis is aligned along ˆz where the permittivity ϵ z differs from unity. \nUnder axial symmetry the fields are decomposed as \nB = B t + B z ˆz ; E = E t + ϵ z E z ˆz , (A2) \nwhere the subscripts ' t ' and ' z ' refer to the orientations transverse and parallel to the z-axis, respectively. Taking the Fourier transform of the fields and sources and assuming that the fields oscillate with frequency ω gives \n( ∇ t + ∂ ∂ z ˆz ) × ( B t + B z ˆz )= -iω ( E t + ϵ z E z ˆz )+( J t + J z ˆz ) , ( ∇ t + ∂ ∂ z ˆz ) × ( E t + E z ˆz )= -iω ( B t + ϵ z B z ˆz ) , \n(A3) \nwhere ∇ t is the gradient in the direction transverse to the z-axis. Decomposing Eq. (A3) into the directions transverse and parallel to ˆz yields \nˆz · ∇ t × B t = -iωϵ z E z + J z , ˆz · ∇ t × E t = iωB z , ˆz × ∂ B t ∂ z + ∇ t B z × ˆz = -iω E t + J t , ˆz × ∂ E t ∂ z + ∇ t E z × ˆz = iω B t . (A4) \nThe last two expressions in Eq. (A4) lead to a relation for the transverse fields, once a Fourier transform along the z-axis with momentum k z is applied, to give \nE t = 1 ω 2 -k 2 z ( ∇ t ∂E z ∂z + iω ∇ t B z × ˆz -iω J t ) , B t = 1 ω 2 -k 2 z ( ∇ t ∂B z ∂z -iω ∇ t E z × ˆz -∂ J t ∂ z ) . (A5) \nContrary to the axion case discussed in Ref. [99], in which the transverse components of the fields are sourced by z components of the current only, in this more general case these components are sourced by the transverse current as well. In fact, in the case of axion J t = 0 and the B z component can be neglected. Inserting Eq. (A5) in the first two expressions of Eq. (A4) gives \nω 2 ω 2 -k 2 z ∇ t 2 E z + ω 2 ϵ z E z + iωJ z = iω ω 2 -k 2 z ˆz · ∇ t × ∂ J t ∂z , ω 2 ω 2 -k 2 z ∇ t 2 B z -ω 2 B z = ω 2 ω 2 -k 2 z ˆz · ∇ t × J t . (A6) \nNote, that the negative sign in front of the second term of B z equation can lead to tachyonic instability, which are avoided when the dissipation in the cavity walls are considered. \nWe now specialize these results to a resonant cavity by expanding the electric field in modes of pulsation ω n labeled by an integer n , as \nE z = ∑ n e n E n ( x ) , (A7) \nwhere e n is the coefficient of the expansion and the eigenmodes satisfy [99] 5 \n∇ t 2 E n = ϵ ' z ( k 2 z -ω 2 n ) E n , ∫ d 3 x E n E m = δ nm . (A8) \nHere, the permeability has been decomposed as ϵ z = ϵ ' z -iϵ '' z following standard notation [142]. Since the induced current has frequency ω n , ϵ '' = ω n / ( ωQ ), therefore leading, from Eq. (A6), to \nω 2 ϵ ' z ω 2 -ω 2 n ω 2 -k 2 z e n -i ωω n Q e n = -iω ∫ d 3 x E n ˜ J t , (A9) \nwhere we defined \n˜ J t = J z -1 ω 2 -k 2 z ˆz · ∇ t × ∂ J t ∂z . (A10) \nSolving for e n we finally obtain \ne n = -iω ( ω 2 -ω 2 p ) ω 2 -ω 2 n ω 2 -k 2 z -i ωω n Q ∫ d 3 x E n ˜ J t . (A11) \nFollowing Eq. (20) in Ref. [134], we decompose the current as \n˜ J t = B 0 ω 2 V 1 / 3 ∑ a h a ( ω ) ˆ j a , (A12) \nso that \ne n = -iB 0 ω 3 V 1 / 3 ( ω 2 -ω 2 p ) ω 2 -ω 2 n ω 2 -k 2 z -i ωω n Q ∑ A h A ( ω ) (∫ d 3 x E n ˆ j A ) . (A13) \nThis leads to the signal \ne n = -iω B 0 V 5 / 6 T ( ω ) ∑ A h A ( ω ) η A , (A14) \nwhere we introduced the cavity-GW coupling coefficient \nη A = 1 V 1 / 2 ∫ d 3 x E n ˆ j A . (A15) \nIn Eq. (A14), the function controlling the resonance is given by \nT ( ω ) = [( 1 -ω 2 p ω 2 ) ω 2 -ω 2 n ω 2 -k 2 z -i Q ω n ω ] -1 . (A16) \nThe energy stored in the n -th cavity mode is \nU = ∫ d f d f ' ⟨ e n ( ω ) e n ( ω ' ) ⟩ , (A17) \nor, using the expression above, \nU = ∫ d f d f ' ω ( ω ' ) B 2 0 V 5 / 3 T ( ω ) T ∗ ( ω ' ) × ∑ AA ' ⟨ h A ( ω ) h A ' ( ω ' ) ⟩ η A η A ' . (A18) \nFor a stochastic signal, we write \n⟨ h A ( ω ) h A ' ( ω ' ) ⟩ = 3 H 2 0 32 π 3 δ AA ' δ ( f -f ' ) f -3 Ω GW ( ω ) , (A19) \nso that when k z = 0 we find \nU = ∫ d ω 2 π (4 π ) (2 π ) 3 ωB 2 0 V 5 / 3 η 2 Ω GW ( ω ) ω 2 [( 1 -ω 2 p ω 2 ) ( 1 -ω 2 n ω 2 )] 2 + ( ω n Q ) 2 3 H 2 0 32 π 3 , (A20) \nwhere an extra (4 π ) comes from the angular integration and an extra (2 π ) 3 from converting f into ω . We have also introduced the coupling \nη = ( ∑ A η 2 A ) 1 / 2 ≈ 0 . 14 , (A21) \nwhere the last expression corresponds to the cavity mode TM 012 . Inserting the expression for Ω GW in Eq. (5) into Eq. (A20) gives \nU = 1 2 B 2 0 V 5 / 3 η 2 ∫ + ∞ 0 d ω ω 3 h 2 c ω 2 ( 1 -ω 2 p ω 2 ) 2 ( 1 -ω 2 n ω 2 ) 2 + ( ω n Q ) 2 . (A22) \nWe first consider a coherent source with N cycle cycles that can be observed inside the cavity, so that the strain \n- [1] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116 , 061102 (2016), arXiv:1602.03837 [gr-qc].\n- [2] B. P. 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2024arXiv240909559E	We present a new family of regular black holes RBH in Pure Lovelock gravity where the energy density is determined by the gravitational vacuum tension which varies for each value of n in each Lovelock case. Speculatively our model may capture quantum effects through gravitational tension. In this way a hypothetical analogy is drawn between the pair production ratio in the Schwinger effect and our energy density. A notable feature of our model is that the regular solution closely resembles the vacuum solution before reaching the event horizon. For odd n the transverse geometry is spherical with phase transitions occurring during evaporation and the final state of this process is a remnant. For even n the transverse geometry is non trivial and corresponds to a hyperboloid. In the case of d2n1 with even n we find an RBH without a dS core and no inner horizon whose presence has been recently debated in the literature due to the question of whether its presence is unstable or not and no phase transitions. For d gt 2n 1 with even n the RBH possesses both an event horizon and a cosmological horizon also with no inner horizon present. The existence of the cosmological horizon arises without the usual requirement of a positive cosmological constant. From both numerical and analytical analysis we deduce that as the event horizon expands and the cosmological horizon contracts thermodynamic equilibrium is achieved in a remnant when the two horizons coincide.	2024-09-01T00:00:00Z	['arXiv:2409.09559', '2024arXiv240909559E', '10.48550/arXiv.2409.09559']	['General Relativity and Quantum Cosmology']	Pure Lovelock Gravity regular black holes	2,024	194	0.17	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.09559.pdf	{'Pure Lovelock Gravity regular black holes': 'Milko Estrada 1, ∗ and Rodrigo Aros 2, † \n1 (Dated: December 13, 2024) \nFacultad de Ingenier´ıa y Empresa, Universidad Cat´olica Silva Henr´ıquez, Chile \n2 \nDepartamento de Ciencias Fisicas, Universidad Andres Bello, Av. Republica 252, Santiago,Chile', 'Abstract': 'We present a new family of regular black holes (RBH) in Pure Lovelock gravity, where the energy density is determined by the gravitational vacuum tension, which varies for each value of n in each Lovelock case. Speculatively, our model may capture quantum effects through gravitational tension. In this way, a hypothetical analogy is drawn between the pair production ratio in the Schwinger effect and our energy density. A notable feature of our model is that the regular solution closely resembles the vacuum solution before reaching the event horizon. For odd n , the transverse geometry is spherical, with phase transitions occurring during evaporation, and the final state of this process is a remnant. For even n , the transverse geometry is non trivial and corresponds to a hyperboloid. In the case of d = 2 n + 1 with even n , we find an RBH without a dS core and no inner horizon (whose presence has been recently debated in the literature due to the question of whether its presence is unstable or not), and no phase transitions. For d > 2 n +1 with even n , the RBH possesses both an event horizon and a cosmological horizon, also with no inner horizon present. The existence of the cosmological horizon arises without the usual requirement of a positive cosmological constant. From both numerical and analytical analysis, we deduce that as the event horizon expands and the cosmological horizon contracts, thermodynamic equilibrium is achieved in a remnant when the two horizons coincide.', 'I. INTRODUCTION': "In recent years, several events, such as the discovery of gravitational waves from the collision of two rotating black holes [1], have positioned General Relativity as an effective theory for describing gravitational phenomena. \nNowadays, it is well-accepted that black hole physics and gravity are far from being finished and several core aspects are still unclear. For instance, since the original works by Penrose and Hawking, it is well-accepted that, considering only the known quantum phenomena, gravitational collapse could lead to black hole solutions containing singularities. In this connection, it is well established that the infinite tidal forces near a black hole's singularity can lead to the infinite stretching of an object, a phenomenon known as spaghettification. See for example the introduction of [2]. However, this prediction now is sometimes disregarded as the introduction of potential new quantum (gravity) effects could change the scenario drastically, avoiding the formation of singularities. See also [3]. \nIt is well known that an invariant associated with the measurement of tidal forces is the Kretschmann scalar [4]. In a four dimensional vacuum spherically symmetric space, this scalar is proportional to K ∼ ¯ M 2 /r 6 , leading to infinite tidal forces at the origin. Gravitational field tension in the spherically symmetric case is characterized by the curvature term given by the square root of the Kretschmann scalar of the vacuum theory, F ∼ √ K Schw ∼ ¯ M r 3 in 4 D . This correlation is logical, as the spacetime tension should increase with the mass of the vacuum source [5]. In this sense, Dymnikova's energy density [6] can be interpreted as: \nρ -Dymnikova ∼ exp ( -F c F ) (1) \nbeing F c , a constants. It is worth mentioning that in the four-dimensional case, this energy density behaves as follows: \nρ -Dymnikova ∼ exp ( -r 3 a 3 ) (2) \nDymnikova proposes a model that encodes the gravitational information of the spherically symmetric vacuum solution in such a way that near the central singularity-where the gravitational tension and tidal forces of the vacuum black hole become infinite-the energy density in the Dymnikova model (DyM) remains finite. This premise will be crucial for the model we examine in this work. This finite density in the DyM behaves as a positive cosmological constant near the origin, resulting in a regular black hole solution with a de Sitter core. It is also worth noting that \nthe DyM has garnered significant attention in recent years for addressing various issues in physics [5, 7-11]. \nReference [12] suggests an interesting speculative analogy between the DyM and the quantum Schwinger effect: the density of the DyM attained during collapse could approach the Planck scale or possibly the GUT scale, depending on the nature of the fields in the energy-momentum tensor. Thus, the energy-momentum tensor encapsulates the effects of these fields, relating them to the gravitational field tension. In this framework, reference [13] considers the case where the gravitational source is of electric origin, E ∼ F . Recently, references [5, 10] explored this analogy for regular black holes and wormholes under the influence of quantum GUP. However, despite what has been discussed in this paragraph, a deeper model for this quantum analogy would require a more thorough investigation, which is beyond the scope of this work. It is worth mentioning that in a very different framework is also associated the value of an electric field in the Schwinger effect with the value of the Kretschmann scalar in references [14, 15]. \nOn the other hand, it is well-known that several branches of theoretical physics predict the existence of extra dimensions. Even though several experiments have tried to test this idea, this is yet to be observed. Consequently, any theory incorporating extra dimensions must align with General Relativity in four dimensions or with one of its generalizations. Among these theories is Lovelock gravity. The Lagrangian of Lovelock gravity includes higher curvature terms as corrections to the Einstein-Hilbert action [16]. Furthermore, Lovelock's theories adhere to the fundamental principles of General Relativity; for example, their equations of motion are of second order. It is important to mention that the specific case of Lovelock gravity, known as Einstein-Gauss-Bonnet theory, has garnered attention in recent years for its applications in inflationary theories and has been compared with the results from GW170817 [17, 18]. \nA special case of Lovelock theories is Pure Lovelock theory (PL) [19]. As will be discussed below, this theory considers a single term in the Lagrangian. Pure Lovelock theory has drawn attention in recent years for various problems in physics [20-22]. See also [23, 24] . As we will see below, the value of the Kretschmann scalar for the vacuum solution depends on the power n of the curvature. This makes it interesting to study the behavior of a non-vacuum model that encodes the gravitational information of the empty geometry in an analogous way to the Dymnikova model. \nPure Lovelock is a theory characterized by specific properties that distinguish it from general Lovelock theory and all other higher derivative theories: vacuum solutions of the motion equations in Pure Lovelock theory are doubly degenerate for even n , a situation that does not occur in General \nRelativity where n = 1. This also raises an intriguing question about what these degenerate solutions represent in the non-vacuum case and whether they allow for transverse geometries that differ from spherical symmetry. Additionally, it is well known that General Relativity has a non-trivial vacuum solutions, where the gravitational potential of the form -¯ M/r d -2 n -1 does not depend radially on its denominator when d = 3 (i.e., d = 2 n + 1 where n = 1). An interesting feature is that pure Lovelock theory retains this property for d = 2 n + 1 with n > 1. Related to this, references [25-27] discuss a universal property of pure Lovelock theory, highlighting its kinematic nature in critical odd dimensions d = 2 N +1. This property raises the question of what the physical interpretation would be in the non-vacuum case, i.e., when d = 2 n +1 in the presence of high curvature terms. Another remarkable property of Lovelock theory is the existence of bound orbits in higher dimensions [28]. \nIn this work, we present a regular black hole model for Pure Lovelock gravity, based on the previously described idea: a model of energy density that encodes the gravitational information of the vacuum case, ensuring that tidal forces and energy density remain finite at the radial origin. We will analyze what happens for each value of the power of the Riemann tensor in each theory with n > 1. In the case where degenerate solutions exist, we will investigate whether there are black hole solutions with transverse geometries different from the usual spherically symmetric ones. We will also examine the physical interpretation of the solutions obtained for the cases d = 2 n +1 and d > 2 n + 1. As we will discuss later, we will interpret the horizon structure, identifying scenarios with the presence and absence of inner and cosmological horizons, in addition to the event horizon. For all the cases studied, we will investigate the temperature and radial evolution, which will provide insights into the evaporation process and help determine under what conditions the final stage of this process would correspond to a remnant. \nOn the other hand, in recent years, there has been growing interest in determining whether the presence of an inner horizon is inherently unstable. For example, reference [29] asserts that instability in the cores of regular black holes is inevitable because mass inflation instability is crucial for regular black holes with astrophysical significance. However, this remains an unresolved issue from a theoretical standpoint. Conversely, reference [30] argues that semiclassical effects due to Hawking radiation might mitigate the instability associated with the inner horizon, potentially stabilizing the existence of a de Sitter core. Therefore, in this work, we will also test whether black holes with only an event horizon and no inner horizon can exist in non-spherical transverse geometries for d = 2 n +1 and d > 2 n +1.", 'II. A BRIEF REVIEW ABOUT LOVELOCK GRAVITY AND THE PURE LOVELOCK CASE': 'The Lovelock Lagrangian [16] is : \nL = √ -g N ∑ n =0 α n L n , (3) \nwhere N = d 2 -1 for d even and N = d -1 2 for d odd, and α n are arbitrary coupling constants. L n is a topological density defined as: \nL n = 1 2 n δ µ 1 ν 1 ...µ n ν n α 1 β 1 ...α n β n Π n r =1 R α r β r µ r ν r , (4) \nwhere R αβ µν is an n -order generalization of the Riemann tensor for the Lovelock theory, and: \nδ µ 1 ν 1 ...µ n ν n α 1 β 1 ...α n β n = 1 n ! δ µ 1 [ α 1 δ ν 1 β 1 ...δ µ n α n δ ν n β n ] (5) \nis the generalized Kronecker delta. \nIt is important to emphasize that the terms L 0 , L 1 , and L 2 are proportional to the cosmological constant, the Ricci scalar, and the Gauss-Bonnet Lagrangian, respectively. The corresponding equation of motion is given by: \nN ∑ n =0 α n G ( n ) AB = T AB , (6) \nwhere G ( n ) AB represents an n -order generalization of the Einstein tensor, influenced by the topological density L n . For example, G (1) AB corresponds to the Einstein tensor related to the Ricci scalar (with Einstein-Hilbert theory as a specific case of Lovelock theory), and G (2) AB corresponds to the Lanczos tensor associated with the Gauss-Bonnet Lagrangian.', 'A. Pure Lovelock case': 'Pure Lovelock theory involves only a single fixed value of n (with n ≥ 1), without summing over lower orders. In some cases, it is considered as a single value of n ≥ 1 plus the n = 0 term, i.e., L = L 0 + L n , as shown in references [31-33]. For simplicity, in this work, we consider only the L n term without the L 0 term, as illustrated in references [34, 35]. Thus, the Lagrangian is: \nL = √ -gαL n = √ -gα n 1 2 n δ µ 1 ν 1 ...µ n ν n α 1 β 1 ...α n β n Π n r =1 R α r β r µ r ν r , (7) \nThe equations of motion are given by: \nwhere \n( G ( n ) ) A B = -1 2 n +1 δ AB 1 ...B 2 n BA 1 ...A 2 n R A 1 A 2 B 1 B 2 · · · R A 2 n -1 A 2 n B 2 n -1 B 2 n . (9) \nAnd the coupling constants were set to unity, consistent with references [34, 35].', 'III. OUR HIGHER DIMENSIONAL MODEL': "The vacuum solution in Pure Lovelock gravity can be found in reference [19], and is given by: \nf ( r ) = γ -( 2 ¯ M r d -2 n -1 ) 1 /n (10) \nwhere, as we will see below, γ is an integration constant related to the geometry of the nontransversal section [36]. On the other hand, the Kretschmann invariant is given by \nK = ( f '' ( r )) 2 + 2( d -2) r 2 ( f ' ( r )) 2 + 2( d -2)( d -3) r 4 ( γ -f ( r )) 2 (11) \nEvaluating solution (10) in the previous equation: \nK = C ¯ M 2 /n r (2 /n )( d -1) (12) \nwhere \nC = 4 1 n n 4 ( 2 n 4 d ( d -3) + (8 d -16) n 4 +( -8 d 2 +12 d -4) n 3 + (2 d 3 +5 d 2 -16 d +9) n 2 -6( d -1) 3 n +( d -1) 4 ) \nFollowing the previously described idea and motivated by Dymnikova's model, we propose a model where the energy density encodes the gravitational information of the vacuum solution with constant transversal curvature. In our model, near the central singularity-where the gravitational tension and tidal forces of the vacuum black hole become infinite-the energy density assumes a finite value. \nAs previously mentioned, the gravitational tension is associated with the square root of the Kretschmann scalar of the vacuum solution, which, in this case, corresponds to pure Lovelock gravity: \nF ∼ √ K ∼ ¯ M 1 /n r ( d -1) /n (13) \nG ( n ) AB = T AB , (8) \nThus, analogous to equation (1), we will define the energy density as: \nρ = A exp ( -r ( d -1) /n a ( d -1) /n ) (14) \nwhere a is a constant and where, for simplicity, the constant A has been adjusted to: \nA = d -2 n M a d -1 / ( d -1) (15)", 'IV. OUR REGULAR BLACK HOLE SOLUTION IN PURE LOVELOCK GRAVITY': "In this work, we study the static d -dimensional metric, which is given by: \nds 2 = -µ ( r ) dt 2 + µ ( r ) -1 dr 2 + r 2 d Σ γ . (16) \nThe constant γ can be normalized to ± 1, 0 by appropriately rescaling. Thus, the local geometry of Σ γ is a sphere, a plane, or a hyperboloid [36]: \nΣ γ locally = S d -2 for γ = 1 T d -2 for γ = 0 H d -2 /G H for γ = -1 . \nHere γ stands for the value of the local (constant Riemannian) curvature. G H could be any element of SO (1 , d -3) which acts freely on H d -2 and such that the quotient be compact. In the same fashion for R d -2 , T d -2 is the d -2-torus, i.e., ( S 1 ) d -2 . \nThe energy-momentum tensor corresponds to a neutral perfect fluid: \nT A B = diag( -ρ, p r , p θ , p θ , ... ) , (17) \nOn the one hand, it is well known that this form of the metric imposes the condition ρ = -p r , which implies that the ( t, t ) and ( r, r ) components of the equations of motion have the same structure, given by: \nd dr ( r d -2 n -1 ( γ -µ ) n ) = 2 d -2 r d -2 ρ, (18) \nOn the other hand, due to transversal symmetry, we have p θ = p ϕ = p t = . . . for all the ( d -2) angular coordinates. Thus, using the aforementioned condition ρ = -p r , the conservation law T AB ; B = 0 gives: \np t = -r d -2 ρ ' -ρ (19) \nDefining the mass function as: \nm ( r ) = 1 d -2 ∫ ρr d -2 dr (20) \nBy choosing M · ( n -1)! as integration constant, the mass function is given by \nm ( r ) = M ( ( n -1)! -Γ [ n, r ( d -1) /n a ( d -1) /n ]) = M n 1 F 1 ( n, n +1 , -( r a ) d -1 n ) ( r a ) d -1 (21) \nwhere Γ is the Gamma function and where 1 F 1 is hyperbolic confluent function. As expected, provided r →∞ describe a proper region of the space, \nlim r →∞ m ( r ) = ( n -1)! M (22) \nOn the other hand, for small values of r , such that r ≪ a , it is satisfied \nm ( r ) \| r ≪ a ≈ M a d -1 · n r d -1 , (23) \nThe solution of equation (18) for n =odd is given by \nµ ( r ) = γ -( 2 m ( r ) r d -2 n -1 ) 1 /n (24) \nThus, for odd n , we will focus only on the solution where γ = 1, whose transversal section corresponds to a sphere, since, as we will see below, this is the only case that has horizons. \nµ ( r ) γ =1 = 1 -( 2 m ( r ) r d -2 n -1 ) 1 /n (25) \nHowever, for even n , the equation (18) has the following solutions: \nµ ( r ) = γ ± ( 2 m ( r ) r d -2 n -1 ) 1 /n (26) \nFor even n , we will be interested in the following cases, which, as we will see below, possess horizons: \n- · One of them corresponds to the negative branch of the previous equation with γ = 1, which corresponds to Equation (25).\n- · The other corresponds to the positive branch of the previous equation with γ = -1, which corresponds to: \nµ ( r ) γ = -1 = -1 + ( 2 m ( r ) r d -2 n -1 ) 1 /n (27) \nFrom the above, we can note the following characteristics: \n- · Solution (25) represents a black hole solution with a cross-section that corresponds to an S d -2 sphere.\n- · Solution (27) represents a black hole solution with a cross-section that corresponds to an H d -2 hyperboloid.\n- · Both solutions are asymptotically flat \nlim r ≫ a µ ( r ) γ =1 = 1 -( 2( n -1)! M r d -2 n -1 ) 1 n , (28) \nlim r ≫ a µ ( r ) γ = -1 = -1 + ( 2( n -1)! M r d -2 n -1 ) 1 n (29) \nIt can be noticed that any information of a becomes undetectable in the asymptotic flat region. \n- · It is worth mentioning that the solution with γ = 1 behaves near the origin as \nλ ( r ) = µ ( r ) γ =1 ∣ ∣ ∣ r ≪ a 1 -( 2 M a d -1 · n ) 1 n r 2 . (30) \nTherefore, this later solution possesses a de Sitter core, representing a regular black hole. Thus, at small scales, the behavior of the energy density results in a de Sitter core instead of a central singularity. \n- · Also we can check that the solution with γ = -1 and even n behaves near the origin as \nµ ( r ) γ = -1 ∣ ∣ ∣ r ≪ a -1 + ( 2 M a d -1 · n ) 1 n r 2 . (31) \nTherefore, this later solution possesses an Anti de Sitter core, representing a regular black hole. \nOn the other hand, it is worth noting that both cases, with γ = 1 and γ = -1, have a finite value of the Kretschmann scalar, so both solutions correspond to regular black holes. \nK ≈ d ( d -1) ( 24 1 / 2 M na d -1 ) 2 /n (32)", 'V. HORIZON STRUCTURE AND PHYSICAL INTERPRETATION': 'Given from of the metric (16), (21), for any value r = r s such that µ ( r s ) = 0 the space presents a Killing horizon. Now, for both cases, equations (25) and (27), the zeros of µ ( r ) can be obtained from the condition \nM ( r s ) = r ( -1+ d -2 n ) s / ( 2 ( ( -1 + n )! -Γ [ n, a 1 -d n r -1+ d n s ])) (33) \nwhere M ( r s ) corresponds to the mass parameter. In what follows, we will analyze separately the branch with γ = 1 (with n even and odd) and the branch with γ = -1 (with n even). \nFor the analysis below, it will be useful to express this equation in terms of r s /a with r = r s : \nM [ r s /a ] a d -2 n -1 = ˜ M = n 2 ( a r s ) 2 n 1 1 F 1 ( n, n +1 , -( r s a ) d -1 n ) . (34) \nwhich resembles a parabolic curve for d -2 n -1 > 0. As we will see below, for d -2 n -1 > 0 this implies that the presence of a minimal mass at a value r s = r ∗ = r + = r -(or r s = r ∗ = r + = r ++ ) with M ( r ∗ ) = M 0 > 0, where r -, r + , and r ++ represent the inner, event, and cosmological horizons, respectively. This is the case where the temperature of the system vanishes. See below.', '1. d = 2 n +1': 'In this case, as mentioned earlier, since r + represents the only horizon corresponding to the event horizon, we can check the left side of Figure 1, for n = 2 , 4, that the derivative dM dr + < 0. On the other hand, we can see on the left side of Figure 9 that the sign of the derivative dT dr + ≤ 0. Thus, the specific heat is always positive. Since interpreting the behavior solely from the graphs of the parameter M and temperature does not seem straightforward, we outline the behavior of the specific heat, given by Equation (46), in Figure 10. \nMoving from left to right in figure 10, we observe that there is a value of the event horizon where an asymptote of zero is reached. In other words, this point, where the specific heat approaches zero, can be associated with a remnant. Thus, speculating, as the black hole horizon expands and cools, it transfers energy to the environment, halting evaporation once the mentioned asymptote is reached. \nFIG. 10. Left side: Specific heat for n = 2, d = 6, a = 1. Right side: Specific heat for n = 4, d = 9, a = 1 \n<!-- image -->', '2. d > 2 n +1': 'In the literature, it has been theoretically explored that the cosmological horizon could also have a temperature [10, 40, 50]. This is beyond the scope of our work. However, this fact might provide some insights into the radial evolution of the system. From Equation (40): \ndM = -( 1 4 π ∂f ∂r + ) dr + = -( 1 4 π ∂f ∂r ++ ) dr ++ (49) \nFirst, we can check that the quantity -∂f ∂M < 0 and ∂M ∂r ++ > 0 at the cosmological horizon (see Figure 3). Thus, following the triple chain rule, analogous to Equation (40), it is customary to define the temperature at the cosmological horizon as T ++ ∼ -∂f ∂r ++ ∼ ∂f ∂M ∂M ∂r ++ > 0. Thus, we can rewrite the last equation as: \ndM = -T + dr + = T ++ dr ++ (50) \nFirst, we can observe in Figure 3 that ∂M ∂r + < 0, noting that in this case, the event horizon corresponds to the smallest of the horizons, r + < r ++ . On the other hand, in the right side of Figure 9, we can see that ∂T ∂r + < 0. Therefore, the specific heat at the event horizon is always positive. \nAssuming that the event horizon expands from left to right while the temperature decreases until it reaches T = 0 on the right side of Figure 9, our interpretation is as follows: From the latter equation (50), we can deduce that since the specific heat at the event horizon is positive, if the object emits energy to its surroundings ( dM < 0), the event horizon expands ( dr + > 0), and the cosmological horizon contracts ( dr ++ < 0). Once both horizons reach the point M ( r ∗ ) where the temperature becomes zero, the evaporation process slows down. In other words, the physical system reaches equilibrium when both horizons coincide. It is worth mentioning that this process is analogous to the evaporation of a black hole with a positive cosmological constant and a \nspherically symmetric cross section. However, in our case, this process occurs without the presence of a cosmological constant, and the transversal section represents a hyperboloid.', 'C. A brief discussion about the topology': "Given that in our solutions the mass function is such that the spacetime behaves differently both near and far from the origin, it is of physical interest to test the topology of the spacetime at different locations. In this regard, initially, in reference [43], using the analogy with the ReissnerNordstrom spacetime, conditions for the so-called Borde's theorem were established, which leads to the fact that regular black holes with spherical symmetry (and consequently compact on the event horizon) in their cross-section are associated with a change in topology. Subsequently, in the \nreferences [44-46] the topology of four-dimensional regular black holes with (A)dS cores and other types of cores and with spherical symmetry in their cross-section was studied. In reference [44], it was found that the geometry of spacelike slices of spherically symmetric and (locally) static regular solutions is S 3 for a de Sitter core and H 3 for an AdS core. In that reference, it was pointed out that the regular black holes with dS core satisfying Borde's theorem [43] has the topology of S 3 in their cores, assuming the compact slice is smooth and simply connected. On the other hand, there is a transition between the topology of the core and the topology far from it, from S 3 to R× S 2 . Related to the latter, in reference [45], a way of representing the spacetime of a regular black hole was addressed, providing an identification of the regions corresponding to the origin in the usual Penrose diagram of Reissner-Nordstrom. This allows the spacetime to be described as causally simple towards the future and geodesically complete towards the future. It is also shown that all the requirements of Borde's theorem are satisfied by imposing the null energy condition. Therefore, there must be some compact slice in the causal future of the eventually trapped surface. This spacelike identification implies that the topology of the slices in the inner region of the regular black hole is S 3 , where the metric ds 2 = r -dχ 2 + r 2 ( χ ) d Ω 2 is such that r ( χ ) fluctuates between 0 and r -. This type of identification is also consistent with the aforementioned transition in topology. \nIn this subsection, we will have a brief discussion on the topology associated with the cases described in the previous subsections.", '1. Branch with γ = 1 for both even and odd n for d > 2 n +1': 'As mentioned before, the case where d = 2 n +1 does not represent a black hole; therefore, the case of interest corresponds to d > 2 n +1. As we can observe in equation (30), the core corresponds to a dS space. In order to test the topology, we will follow a strategy analogous to that in reference [44]. Since we are interested in testing whether the topology of the core corresponds to a sphere S d -1 , we will assume that the cross section of a coordinate system in any spacelike slice is constant and positively normalized to unity, γ = 1. However, a more detailed study, such as the one in reference [44], may be required, which goes beyond the scope of this work. Such a coordinate system can be written as \nds 2 = dr 2 λ ( r ) + r 2 d Ω d -2 . (35) \nwhere λ ( r ) is given by (30). This last equation can be rewritten as: \nλ ( r ) = 1 -r 2 l eff = 1 -R dS d ( d -1) r 2 = 1 -2 ( d -2) Λ eff ( d -1) r 2 (36) \nwhere l eff , R dS , and Λ eff represent the effective radius, Ricci scalar, and effective cosmological constant for the de Sitter spacetime, respectively. Intuitively motivated by reference [44], the following coordinate transformation is proposed: \n1 -R dS d ( d -1) r 2 = cos 2 ( χ ) ⇒ r = ( d ( d -1) R dS ) 1 / 2 sin( χ ) (37) \nwith this change of coordinate, the line element (35) is: \nds 2 = d ( d -1) R dS ( dχ 2 +sin 2 ( χ ) d Ω d -2 ) (38) \nAnd therefore, the topology of the spacelike slices is described by either S d -1 (corresponding to a de Sitter core with R dS > 0). On the other hand, we can check in equation (28) and figure 4 that, given the form of our mass function, the space time far from the origin behaves as the vacuum solution whose topology corresponds to R× S d -2 . Thus, we can infer that in our case, there is a transition in topology from an S d -1 near the core to a topology of R× S d -2 far from it. This result in 4 D reduces to that in reference [44]', '2. Branch with γ = -1 for even n': 'First of all, as pointed out previously, the case d = 2 n + 1 does not have an inner horizon associated with a potentially unstable core. On the other hand, for d > 2 n +1, there is a black hole with both an event horizon and a cosmological horizon. As we can observe in equation (31), the solution has an AdS core. This feature makes it intriguing to test whether or not there is a topological transition analogous to the previous case, where now the cross-section corresponds to a hyperboloid spacetime. However, the fact that in our case the cross-section corresponds to a compact hyperboloid, together with the existence of a cosmological horizon, implies that the analogous identification with the Reissner-Nordstrom spacetime is not direct and requires a deep study, which is beyond the scope of this work and could be addressed in future studies.', 'VI. TEMPERATURE': "As mentioned above, the nonvanishing temperature black hole solutions can be computed at r = r + . In this case, the thermodynamics can be analyzed considering the space between r + < \nr < ∞ . Evaluating the equations of motion (18) at the horizon yields the following expression for the temperature \nT = 1 4 πnγ n -1 ( d -2 n -1 r + -2 d -2 ρ ( r + ) r 2 n -1 + ) (39) \nwhere the mass parameter present in the energy density, equations (14) and (15), corresponds to that in equation (33). \nSince the mass parameter M can be expressed as a function of r + , r ++ or r -(as shown in Equation (33) and illustrated in Figures 1, 2, and 3), this implies the existence of a minimal mass parameter, denoted M ( r ∗ ), where r + = r -= r ∗ (or where r + = r ++ = r ∗ ) , with r ∗ being the radius at which the system's temperature vanishes. This can also be observed in the parameter space using the triple product chain rule: \nδf = 0 = ∂f ∂r + dr + + ∂f ∂M dM → ∂f ∂r + ∂r + ∂M ∂M ∂f = -1 → T ∼ ∂f ∂r + ∼ -∂f ∂M ∂M ∂r + (40) \nThus, since there is a minimum at ∂M ∂r + , where the plots show that the inner and outer horizons coincide, at that point T = 0. We can also note that: For the branch with γ = 1, -∂f ∂M > 0 and ∂M ∂r + ≥ 0, so the sign of the temperature is always positive. For the branch with γ = -1, -∂f ∂M < 0 and ∂M ∂r + ≤ 0, so the sign of the temperature is also always positive. \nTo find an expression for the minimum value of M where T = 0, we can notice that the temperature can also be expressed as an equation for r s a \nT + = 1 4 π dµ dr ∣ ∣ ∣ ∣ r = r + = 1 γ -1 2 πr + + 1 F 1 ( [ n +1] , [ n +2] , -( r + a ) d -1 n ) ( r + a ) d -1 n +2 n ( d -1) 2 πr + n 2 ( n +1) (41) \nwhere the value of r + corresponds to the largest solution of µ ( r ) = 0 from Equation (25) for the branch γ = 1, and to the smallest solution of µ ( r ) = 0 from Equation (27) for the branch γ = -1. \nEq.(41) shows that the temperature T + can vanish, i.e. , there are values of M/a d -2 n -1 for which T + = 0 = T -(or T + = 0 = T ++ ). Because the relation between M/a d -2 n -1 and r + is not analytic, it is much simpler to determine T + = 0, or equivalently r + = r -= r ∗ (or r + = r ++ = r ∗ ), using the condition \n1 F 1 ( [ n +1] , [ n +2] , -( r ∗ a ) d -1 n ) ( r ∗ a ) d -1 n +2 n ( d -1) n 2 ( n +1) = 1 . (42) \nOne can notice that this is an equation for r ∗ a and thus given d and n its solution can be computed at least numerically. This implies that the minimum value of the (normalized) mass, says M ∗ , is \ngiven by \nM min a d -2 n -1 = M ∗ = d -1 2 n ( n +1) ( r ∗ a ) d -1 n (43)", 'A. branch with γ = 1': 'For d > 2 n +1, as mentioned earlier, since r + represents the outermost horizon, r + > r -. In Figures 2 and 3, we can observe that the derivative dM dr + ≥ 0. Thus, the sign of the specific heat depends solely on the sign of the derivative dT dr + . \nFollowing Figure 8, we propose the following interpretation from right to left: Starting from the right of r max , the specific heat is negative, so energy is emitted into the environment while the temperature increases and the horizon radius contracts. Subsequently, at the point r max , the specific heat diverges and a phase transition occurs from an unstable black hole to a stable black hole. Later, to the left of r max , the specific heat is positive. Thus, while the horizon radius continues to contract, the black hole continues to emit energy and the temperature decreases. Finally, at the point where the temperature goes to zero, the specific heat also goes to zero, and the event horizon halts its contraction. Thus, once the contraction stops at the zero-temperature point, a black remnant could form, referred to as what is left behind once evaporation ceases [48]. Some references have explored the possibility that evaporation would stop once the horizon radius contracts to a value close to the Planck length, a phenomenon that might be linked to the emergence of quantum effects at this scale [49].', '1. d > 2 n +1': 'We can observe the behavior in Figure 8. It is straightforward to verify that this behavior is generic for other values of n and d . \nWe can verify that the point r ∗ , where the mass parameter reaches a minimum (i.e., the internal and black hole horizons coincide), is the same point where the temperature vanishes. Hence, the temperature reaches to zero at the extremal black hole. As we will discuss later, in this case, the zero-temperature point is associated with a black remnant, which refers to what remains after evaporation has ceased. \nFrom right to left in the figure 8, we observe that: first, being the derivative dT/dr + < 0, the temperature increases as the horizon decreases, reaching a maximum at r = r max . At this maximum point, dT/dr + = 0. Then, with dT/dr + > 0, the temperature decreases along with the horizon until T = 0 at r ∗ . Below, we will discuss the physical implications of this temperature behavior in the context of radial evolution and evaporation. \nFIG. 8. Left side: Temperature for n = 2, d = 6, a = 1. Right side: Temperature for n = 3, d = 8, a = 1. \n<!-- image -->', 'VII. ENTROPY AND THE FIRST LAW': "Given the form of the Lagrangian and the family of solutions considered it is straightforward to define an entropy at the event horizon. For this one can use the methodology showed in reference [47] \nS + = ∫ H ∂L ∂R 01 01 ≈ ( γ ) n -2 r d -2 n Σ γ (44) \nwhich coincides with the known expression for the vacuum non-regular black holes [19]. One can notice that this is an always increasing function of r + . Two important caveats exist. First, while the expression is the same, even though the values are extremely close, the values of r + differ between the solution above and the solution in [19]. The second, as the solution exists for γ = 1 for any n but for γ = -1 only if n is even, the expression of the entropy is the same in both cases. \nFollowing Wald's construction it is straightforward to confirm that, under an evolution of the parameters of the solution, \nT + δS + = δM (45) \nis satisfied. Even though this is the standard result, it must be considered that in this case, T + can vanish. Moreover, it seems like the parameter a is completely absent, but this is only apparent as the value of r + depends on a . \nIt must be emphasized in this point that Eq.(45) does not provide the whole scenario to understand the evolution of the system.", 'VIII. RADIAL EVOLUTION AND HEAT CAPACITY': 'It is important to emphasize that a discussion of thermodynamics is only possible in the presence of an event horizon. In this case, the mass M can be expressed as a function of r s = r + , as given by Equation (34). \nAfterwards, the heat capacity of these solutions can be computed as \nC ( r + ) = ∂M ∂T = dM dr + ( dT dr + ) -1 (46) \nFirst, we will outline an analytical expression and then analyze the radial evolution and evap- \noration for each of the cases of interest. From Eq.(41), \nd dr + T ( r + ) = 1 γ ( 1 2 πa ( r + a ) 2 -1 F 1 ( [ n +2] , [ n +3] , -( r + a ) d -1 n ) ( r + a ) d -1 n ( d -1) 2 ( r + a ) d -1 n +2 n 2 ( n +2) n 3 ( r + a ) 2 πa + 1 F 1 ( [ n +1] , [ n +2] , -( r + a ) d -1 n ) ( r + a ) d -1 n +2 n ( d -1 n +2 n ) ( d -1) 2 ( r + a ) 2 πan 2 ( n +1) (47) -1 F 1 ( [ n +1] , [ n +2] , -( r + a ) d -1 n ) ( r + a ) d -1 n +2 n ( d -1) 2 πa ( r + a ) 2 n 2 ( n +1) ) \nand \ndM dr + = a d -2 n -1 n 1 F 1 ( [ n +1] , [ n +2] , -( r + a ) d -1 n ) ( r + a ) d -1 n ( d -1) 2 1 F 1 ( [ n ] , [ n +1] , -( r + a ) d -1 n ) 2 ( r + a ) d -1 ( n +1) ( r + a ) -a d -2 n -1 n ( d -1) 2 1 F 1 ( [ n ] , [ n +1] , -( r + a ) d -1 n ) ( r + a ) d -1 ( r + a ) (48) \nwhere again the value of r + corresponds to the largest solution of µ ( r ) = 0 from Equation (25) for the branch γ = 1, and to the smallest solution of µ ( r ) = 0 from Equation (27) for the branch γ = -1. \nThere are two important features to notice. First dT/dr + can vanish pointing out potential phase transitions. Second, the values of r + where this occurs are not obviously connected with r ∗ , the value that defines the minimal mass M ∗ and T + = 0. Moreover, dM/dr + can vanish as well, in another seemly independent value of r + /a , pointing out regions where the heat capacity goes smoothly from C > 0 to C < 0. This usually implies a rich adiabatic evolution of the solutions. \nOf lesser relevance but still worthwhile to mention is that, even though equations (47) and (48) may look quite cumbersome, in fact once expressed for a given d and n they simplify greatly. \nFor our analysis, we will consider the following: The heat capacity will be utilized to study the thermodynamic evolution of the black hole. In this work, a positive heat capacity indicates that when the temperature decreases (increases), the black hole emits (absorbs) thermal energy, thus dM < 0 ( dM > 0) in the black hole, in order to reach thermodynamic equilibrium with the external environment, i.e., the black hole is stable. Conversely, a negative heat capacity represents that, if the temperature increases (decreases), the black hole also emits (absorbs) thermal energy toward the external environment, i.e., the black hole is unstable. A second-order phase transition is characterized by a change in the sign of the heat capacity.', 'IX. DISCUSSION AND SUMMARIZE': "In this work, we have constructed a new family of regular black hole solutions for pure Lovelock gravity. The energy density was developed using arguments analogous to those of the Dymnikova model: the energy density encodes the gravitational information of the vacuum case through the Kretschmann scalar. Near the radial origin, where tidal forces and the gravitational tension in the vacuum case diverge, the tidal forces in our model become finite due to the specific form of the energy density. This can be verified as both the geometry and the curvature invariants in our model remain finite. Unlike the Dymnikova model, our energy density varies with the power n of the Riemann tensor in the Lovelock theory and with the number of dimensions. Additionally, we have considered cases where the cross-section in the vacuum case corresponds to a hyperboloid. \nSpeculatively, our model might capture quantum effects through gravitational tension, which is proportional to the square root of the Kretschmann scalar in the vacuum case. In this context, some references, which we mention further below, suggest that, under these assumptions, vacuum polarization could occur within the resulting gravitational field. Consequently, these reference proposes an analogy between the ratio of pair production in the Schwinger effect and the energy density of the Dymnikova model, denoted as Γ ∼ ρ Dymnikova . This idea has been proposed in works [12, 13], and more recently in references [5, 9] for general relativity in 4 D . In our model, the vacuum corresponds to pure Lovelock gravity, characterized by either spherical or hyperbolic transverse geometry. However, a more in-depth study is required, which is beyond the scope of this work. \nWe have found two cases of interest: the branch with γ = 1, which has a spherically symmetric cross-section for both even and odd n ; and the branch with γ = -1, which has a hyperbolic cross-section for even n . \nFor the branch γ = 1, whose transversal section corresponds to a sphere, we can highlight the following: \n- · For d = 2 n +1, if the mass parameter satisfies M > M ∗ , the solution can be interpreted as a static cosmology, since in this case there is a cosmological horizon present.\n- · For d > 2 n +1, when M = M ∗ , the geometry represents a regular extremal black hole where \nthe inner and event horizons coincide and where the temperature is zero. \n- · For d > 2 n + 1, when M > M ∗ , the geometry corresponds to a regular black hole with both an external and an internal horizon. This solution has a dS core. One very interesting feature of our solution is how quickly it converges to the non-regular vacuum solution [19]. In fact, it seems that the two functions match within the region r -< r < r + . In reality, the separation between the two functions becomes very small rapidly, making it nearly impossible to distinguish between the solutions outside the outermost horizon.\n- · We have also analyzed the radial evolution of the temperature, which provides insights into the evaporation process. The temperature reaches a maximum at r max . Starting from the right of r max , the specific heat is negative, so energy is emitted into the environment while the temperature increases and the horizon radius contracts. At r max , the specific heat diverges, and a phase transition occurs from an unstable black hole to a stable one. Moving to the left of r max , the specific heat becomes positive. Thus, while the horizon radius continues to contract, the black hole continues to emit energy, causing the temperature to decrease. Finally, at the point where the temperature approaches zero, the specific heat also drops to zero, and the event horizon halts its contraction. This occurs in the extremal case. Thus, when the contraction stops at the zero-temperature point, a black hole remnant is formed, which is what remains once evaporation ceases [48]. Some references have explored the possibility that evaporation would stop once the horizon radius contracts to a value close to the Planck length, a phenomenon that might be linked to the emergence of quantum effects at this scale [49]. \nFor the branch γ = -1 with even n and with d = 2 n + 1, whose transverse section corresponds to a hyperboloid, we can highlight the following: \n- · For M > M ∗ , the solution represents a regular black hole with an event horizon and no inner horizon. This feature is particularly interesting because, in recent years, the literature has discussed both the instability of an inner horizon with a dS core due to mass inflation [29], as well as the possibility that Hawking radiation might mitigate the instability associated with the inner horizon and dS core [30].\n- · As we move radially outward from r + → 0, the derivative dT dr + is negative, approaching T = 0 asymptotically. The specific heat is positive and also approaches zero asymptotically. \nThe point where the temperature and specific heat approach zero can be associated with a remnant. Thus, speculating, as the black hole expands and cools, it transfers energy to the environment, halting evaporation once the aforementioned asymptote is reached. \nFor the branch γ = -1 with even n and with d > 2 n + 1, whose transverse section corresponds to a hyperboloid, we can highlight the following: \n- · For M < M ( r ∗ ), the solution has no horizons and seems to lack physical interest .\n- · For M > M ( r ∗ ), there is an event horizon and a cosmological horizon. The presence of the cosmological horizon prevents the computation of energy using Noether's charge, meaning the parameter M is not directly associated with the mass. However, through thermodynamic arguments, a local definition of mass variation can be established, which depends on M [40]. A remarkable feature of this solution is the presence of both an event horizon and a cosmological horizon without the need for a positive cosmological constant, as seen in the Schwarzschild-de Sitter case.\n- · For M = M ( r ∗ ), this represents the extremal case where the event horizon and the cosmological horizon coincide. This geometry is analogous to the Nariai solution [42], which arises from the evolution of the Schwarzschild-de Sitter black hole. At this point, the temperature becomes zero.\n- · As we move radially outward from r + → 0, the derivative dT dr + is negative, reaching T = 0 in the extremal case where r + = r ++ . From both analytical and graphical analysis, we can deduce that since the specific heat at the event horizon is positive, if the object emits energy to its surroundings ( dM < 0), the event horizon expands ( dr + > 0), and the cosmological horizon contracts ( dr ++ < 0). Once both horizons reach the point M ( r ∗ ), where the temperature becomes zero, the evaporation process slows down. In other words, the physical system reaches equilibrium when the horizons coincide. It is worth noting that this process is analogous to the evaporation of a black hole with a positive cosmological constant and a spherically symmetric cross-section. However, in our case, this process occurs without the presence of a cosmological constant, and the transverse section corresponds to a hyperboloid. \nMoreover, it is worth mentioning that, following references [43-46], we have established that for the branch with a spherically symmetric cross-section for d > 2 n +1, near the de Sitter core, the \ntopology of the spacelike slices is described by an S d -1 sphere (corresponding to a de Sitter core with R dS > 0). On the other hand, given the form of the mass function, the spacetime far from the origin behaves as the vacuum solution, whose topology corresponds to R× S d -2 . 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2024arXiv240904564R	Context. On 2022 January 20 the Energetic Particle Detector EPD on board Solar Orbiter measured a solar energetic particle SEP event showing unusual first arriving particles from the antiSun direction. NearEarth spacecraft separated 17 in longitude to the west from Solar Orbiter measured classic antisunwarddirected fluxes. STEREOA and MAVEN separated 18 to the east and 143 to the west from Solar Orbiter respectively also observed the event suggesting that particles spread over at least 160 in the heliosphere. Results. Solar Orbiter was embedded in a MC erupting on 16 January from the same active region as the one related to the SEP event on 20 January. The SEP event is related to a M5.5 flare and a fast CMEdriven shock of 1433 kms which injected particles within and outside the MC. The hard SEP spectra the presence of a Type II radio burst and the cotemporal Type III radio bursts being observed from 80 MHz that seems to emanate from the Type II points to the shock as the relevant accelerator of the particles. Conclusions. The detailed analysis of the SEP event strongly suggest that the energetic particles are injected mainly by a CMEdriven shock into and outside of a previous MC present in the heliosphere at the time of the particle onset. The sunward propagating SEPs measured by Solar Orbiter are produced by the injection of particles along the longer western leg of the MC still connected to the Sun at the time of the release of the particles. The determined electron propagation path length inside the MC is around 30 longer than the estimated length of the loop leg of the MC itself based on the graduated cylindrical shell model consistent with a low number of field line rotations.	2024-09-01T00:00:00Z	['arXiv:2409.04564', '2024arXiv240904564R', '10.48550/arXiv.2409.04564']	['Astrophysics - Solar and Stellar Astrophysics']	Solar energetic particles injected inside and outside a magnetic cloud The widespread solar energetic particle event on 2022 January 20	2,024	194	0.51	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.04564.pdf	{'The widespread solar energetic particle event on 2022 January 20': 'L. Rodríguez-García 1 , 2 , R. Gómez-Herrero 2 , N. Dresing 3 , L. A. Balmaceda 4 , 5 , E. Palmerio 6 , A. Kouloumvakos 7 , I. C. Jebaraj 3 , F. Espinosa Lara 2 , M. Roco 2 , C. Palmroos 3 , A. Warmuth 8 , G. Nicolaou 9 , G. M. Mason 7 , J. Guo 10 , T. Laitinen 11 , I. Cernuda 2 , T. Nieves-Chinchilla 4 , A. Fedeli 3 , C. O. Lee 12 , C. M. S. Cohen 13 , C. J. Owen 9 , G. C. Ho 14 , O. Malandraki 15 , R. Vainio 3 , and J. Rodríguez-Pacheco 2 \n- 1 European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s / n, 28692 Villanueva de la Cañada, Madrid, Spain\n- e-mail: [email protected]\n- 2 Universidad de Alcalá, Space Research Group (SRG-UAH), Plaza de San Diego s / n, 28801 Alcalá de Henares, Madrid, Spain\n- 3 Department of Physics and Astronomy, University of Turku, FI-20014 Turku, Finland\n- 4 Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA\n- 5 Physics and Astronomy Department, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA\n- 6 Predictive Science Inc., San Diego, CA 92121, USA\n- 7 The Johns Hopkins University Applied Physics Laboratory, 11101 Johns Hopkins Road, Laurel, MD 20723, USA\n- 8 Leibniz-Institut für Astrophysik Potsdam (AIP), D-14482 Potsdam, Germany\n- 9 Mullard Space Science Laboratory, University College London, Dorking RH5 6NT, UK\n- 10 Deep Space Exploration Laboratory / School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China\n- 11 Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, UK\n- 12 Space Sciences Laboratory, University of California, Berkeley, CA 94720, USA\n- 13 California Institute of Technology, Pasadena, CA 91125, USA\n- 14 Southwest Research Institute, San Antonio, TX 78238, USA\n- 15 National Observatory of Athens / IAASARS, I. Metaxa & Vas. Pavlou, GR-15236 Penteli, Greece \nReceived xxx, 2024; Accepted xxx, 2024', 'ABSTRACT': 'Context. On 2022 January 20, the Energetic Particle Detector (EPD) on board Solar Orbiter measured a solar energetic particle (SEP) event showing unusual first arriving particles from the anti-Sun direction. Near-Earth spacecraft separated 17 · in longitude to the west from Solar Orbiter measured classic antisunward-directed fluxes. STEREO-A and MAVEN, separated 18 · to the east and 143 · to the west from Solar Orbiter respectively, also observed the event, suggesting that particles spread over at least 160 · in the heliosphere. Aims. The aim of this study is to investigate how SEPs accelerated and transported to Solar Orbiter and near-Earth spacecraft, as well as to examine the influence of a magnetic cloud (MC) present in the heliosphere at the time of the event onset on the propagation of the energetic particles. \nMethods. We analysed remote-sensing data, including flare, coronal mass ejection (CME), and radio emission to identify the parent solar source of the event. We investigated energetic particles, solar wind plasma, and magnetic field data from multiple spacecraft. Results. Solar Orbiter was embedded in a MC erupting on 16 January from the same active region as the one related to the SEP event on 20 January. The SEP event is related to a M5.5 flare and a fast CME-driven shock of ∼ 1433 km s -1 , which injected particles within and outside the MC. The hard SEP spectra, the presence of a Type II radio burst, and the co-temporal Type III radio bursts being observed from 80 MHz that seems to emanate from the Type II, points to the shock as the relevant accelerator of the particles. Conclusions. The detailed analysis of the SEP event strongly suggest that the energetic particles are injected mainly by a CME-driven shock into and outside of a previous MC present in the heliosphere at the time of the particle onset. The sunward propagating SEPs measured by Solar Orbiter are produced by the injection of particles along the longer (western) leg of the MC still connected to the Sun at the time of the release of the particles. The determined electron propagation path length inside the MC is around 30% longer than the estimated length of the loop leg of the MC itself (based on the graduated cylindrical shell model) consistent with a low number of field line rotations. \nKey words. Sun: particle emission- Sun: coronal mass ejections (CMEs) -Sun: flares - Sun: corona - Sun: heliosphere', '1. Introduction': "Solar energetic particle (SEP) events are sporadic enhancements of particle intensities associated with solar transient activity. In \nthe inner heliosphere, these intensity enhancements are usually measured in situ at energy ranges spanning many orders of magnitude, from a few keV to hundreds of MeV or even above 1 GeV. For the most energetic events near-relativistic and rel- \nFig. 1. Longitudinal spacecraft constellation and magnetic connectivity at 05:58 UT on 20 January 2022 (left) along with multi-spacecraft SEP measurements (right). Left : Spacecraft configuration using the Solar-MACH tool (Gieseler et al. 2023), available online at https://doi.org/ 10.5281/zenodo.7100482 . Right: The upper panel shows near-relativistic electron intensities and the lower panel ∼ 5 MeV proton intensities observed by the spacecraft indicated with the same color code shown on the left panel. The blue vertical line indicates the time of the soft-X ray peak of the flare ( ∼ 05:58 UT) associated with the SEP event. \n<!-- image --> \nativistic electrons and protons are observed. The mechanisms proposed to explain the origin of large SEP events include: (1) acceleration during magnetic reconnection processes associated with solar jets (Krucker et al. 2011) and flares (Kahler 2007); (2) acceleration at shocks driven by fast CMEs (e.g. Simnett et al. 2002; Desai et al. 2016; Kouloumvakos et al. 2019; Jebaraj et al. 2024); and / or (3) acceleration during magnetic restructuring in the aftermath of coronal mass ejections (CMEs) and in the current sheets formed at the wake of CMEs (e.g. Kahler & Hundhausen 1992; Maia & Pick 2004; Klein et al. 2005). \nSEP events are often classified into two categories, impulsive and gradual (Cane et al. 1986; Reames 1999), on account of their observed properties, such as timescales, spectra, composition and charge states, and the associated radio bursts. During most gradual events, SEPs are detected over a very wide range of heliolongitudes. These widespread SEP events have been extensively researched (e.g. Reames et al. 1996; Lario et al. 2006, 2013, 2016; Wibberenz & Cane 2006; Dresing et al. 2012, 2014, 2023; Papaioannou et al. 2014; Richardson et al. 2014; Gómez-Herrero et al. 2015; Paassilta et al. 2018; Guo et al. 2018, 2023; Xie et al. 2019; Rodríguez-García et al. 2021; Kouloumvakos et al. 2022; Dresing et al. 2023; Khoo et al. 2024) thanks to constellations of spacecraft widely distributed throughout the heliosphere, such as Helios (Porsche 1981), Ulysses (Wenzel et al. 1992), the SOlar and Heliographic Observatory (SOHO; Domingo et al. 1995), the Solar TErrestrial RElations Observatory (STEREO; Kaiser et al. 2008), MErcury Surface Space ENvironment GEochemistry and Ranging (MESSENGER; Solomon et al. 2007), and more recently Solar Orbiter (Müller et al. 2020; Zouganelis et al. 2020), Parker Solar Probe (PSP; Fox et al. 2016), BepiColombo (Benkho ff et al. 2021), Mars Atmosphere and Volatile EvolutioN (MAVEN; Jakosky et al. 2015), and even Mars Science Laboratory (MSL; Grotzinger et al. 2012) on the surface of Mars. \nCMEs are large eruptions of magnetized plasma that are ejected from the Sun into the heliosphere as a result of the release of the huge energy stored in the coronal magnetic field. \nRemote-sensing observations of CMEs close to the Sun provide evidence for the existence of magnetic flux-rope (MFR) structures within CMEs (Vourlidas 2014). They consist of confined plasma within a helically organized magnetic structure. In interplanetary (IP) space, evidence of MFRs is found in structures known as magnetic clouds (MCs; Burlaga et al. 1981). In the best examples, the in situ MFR signatures show a monotonic rotation of the magnetic field direction through a large angle along with a low plasma temperature and low plasma β . \nSEPs can be injected inside IP CMEs (hereafter ICMEs) either due to impulsive acceleration at flares occurring at the footpoints of the parent ICME and / or when a new CME-driven shock intercepts one or both legs of other ICMEs (Richardson et al. 1991; Masson et al. 2012; Dresing et al. 2016; Palmerio et al. 2021; Wimmer-Schweingruber et al. 2023). Kahler et al. (2011a) found some electron events inside ICMEs in which the active regions responsible for the accelerated particles were di ff erent from the CME source region, suggesting an interconnection with adjacent loops. Independently of the source, SEPs propagating inside ICMEs provide a valuable tool to study their magnetic topology (e.g. Richardson & Cane 1996; Larson et al. 1997; Malandraki et al. 2001; Kahler et al. 2011a,c; Dresing et al. 2016; Gómez-Herrero et al. 2017). In particular, near-relativistic electrons may be used as probes of the magnetic structure inside the MC, as they only require a few minutes to travel between the solar source and 1 astronomical unit (au). \nOn 2022 January 20, a widespread SEP event was observed by di ff erent spacecraft located in the inner heliosphere, namely near-Earth probes, Solar Orbiter, STEREO-A, and MAVEN, spanning a longitudinal range of ∼ 160 · in the ecliptic plane (assuming that PSP did not observe the SEP event). The SEP origin was associated with an M-class flare and a wide and fast CME, erupting near the west limb from Earth's perspective. Figure 1 (left) illustrates the spacecraft locations in the heliographic equatorial plane along with nominal Parker field lines connecting each spacecraft with the Sun in the center of the plot, using measured solar wind speeds when available. The black arrow marks \nthe longitude of the associated flare (W76), and the dashed black spiral depicts the nominal magnetic field line connecting to this location. Near-Earth spacecraft (1, green) show the best nominal connection to the flare site. Solar Orbiter (2, blue) -separated 17 · eastwards from Earth- and STEREO-A (3, red) -separated 18 · eastwards from Solar Orbiter- were also well connected to the flaring region. PSP (4, purple)- separated 147 · eastwards from Earth- and MAVEN · (5, brown) -separated 126 · westwards from Earth- presented large longitudinal separation between the solar source and the footpoint of the respective nominal field lines. \nThe top panel in Fig. 1 (right) shows near-relativistic ( ∼ 20150 keV ) electron intensities and the bottom panel ∼ 5 MeV proton intensities di ff erent spacecraft following the color code of the left plot. The legend show the telescopes' looking directions (when available) used for computing the intensities of the first arriving particles for each spacecraft. As expected due to their closest magnetic connection, near-Earth spacecraft observed the highest intensities, measuring usual antisunward-directed particles. However Solar Orbiter, close to Earth's location, measured unusual sunward-directed fluxes for the first arriving particles. \nThe detection of predominantly sunward-propagating beams is quite uncommon. This sunward particle flux might be related to a source located beyond the observer (i.e., a connection to an IP shock) or to a particular interplanetary magnetic field (IMF) configuration (i.e., folded magnetic lines or a close structure such as an IP flux-rope). For example, over the STEREO mission until 2017, only six SEP events were found with dominant sunward particle fluxes (Gómez-Herrero et al. 2017). Over the Solar Orbiter mission, from a survey of ∼ 300 solar energetic electron (SEE) events observed from November 2020 until December 2022 by EPD, as listed by Warmuth et al. (2024), only this SEP event on 2022 January 20 presents clear sunward electron fluxes. We note however that Warmuth et al. (2024) survey is based on an unambiguous association between flare and SEE events, which might not favor events injected into IP clouds. When including energetic protons, the SEP event on 2022 February 15, analysed by Wei et al. (2024), also shows clear sunward fluxes, but not related to a solar origin. \nTo shed some light on which physical mechanisms are behind the unusual sunward-directed particles observed by Solar Orbiter, there are two main objectives in this study: (1) to identify the solar source of this widespread SEP event, and (2) to investigate the acceleration and propagation conditions that could a ff ect the observed SEP properties at the di ff erent but closelyspaced observers, in particular at Solar Orbiter location. The paper is structured as follows. The instrumentation used in this study is introduced in Sect. 2. A summary of the SEP event observations and analysis is shown in Sect. 3. We include the remote-sensing observations and data analysis of the SEP parent solar source in Sect. 4. A detailed analysis of the ICME present in the heliosphere at the time of the particle release is shown in Sect. 5. Section 6 traces the interplanetary propagation of the particles within the ICME. In Sect. 7, we summarise and discuss the main findings of the study and in Sect. 8 we outline the main conclusions.", '2. Instrumentation': 'The study of the wide spread of particles and the relation with the parent solar source requires the analysis of both remotesensing and in situ data from a wide range of instrumentation on board di ff erent spacecraft. We used data from Solar Orbiter, PSP, MAVEN, STEREO, SOHO, Wind (Ogilvie & Desch 1997), the Advanced Composition Explorer (ACE; Stone et al. 1998), \nthe Solar Dynamics Observatory (SDO; Pesnell et al. 2012), the Geostationary Operational Environmental Satellites (GOES), and the Fermi spacecraft. \nRemote-sensing observations of CMEs and related solar activity phenomena were provided by the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on board SDO, the C2 and C3 coronagraphs of the Large Angle and Spectrometric COronagraph (LASCO; Brueckner et al. 1995) instrument on board SOHO, and the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI; Howard et al. 2008) instrument suite on board STEREO-A. In particular, we used imaging data from the COR1 and COR2 coronagraphs and the Extreme Ultraviolet Imager (EUVI; Wuelser et al. 2004), which are part of the SECCHI suite. We used STEREO-Heliospheric Imager (HI; Eyles et al. 2009) data to track the evolution of the CME in the heliosphere. Radio observations were provided by the Radio and Plasma Wave Investigation (SWAVES; Bougeret et al. 2008) instrument on board the STEREO mission, the YAMAGAWA solar radio spectrograh (Iwai et al. 2017), and the e-Callisto network, in particular data from the Astronomical Society of South Australia (ASSA; Benz et al. 2009). The solar flare is primarily studied with X-ray observations provided by the Spectrometer / Telescope for Imaging X-rays (STIX; Krucker et al. 2020) on board Solar Orbiter, the Gamma-ray Burst Monitor (GBM; Meegan et al. 2009) on board the Fermi spacecraft, and the soft X-ray Sensor (XRS; García 1994) on board GOES 1 . \nThe properties of energetic particles near 1 au were measured by the SupraThermal Electrons and Protons (STEP) instrument, the Electron Proton Telescope (EPT), the High Energy Telescope (HET), and Suprathermal Ion Spectrograph (SIS) of the Energetic Particle Detector (EPD; Rodríguez-Pacheco et al. 2020) instrument suite on board Solar Orbiter. We also used the Solar Electron and Proton Telescope (SEPT, Müller-Mellin et al. 2008), the Low-Energy Telescope (LET, Mewaldt et al. 2008), and the High-Energy Telescope (HET, von Rosenvinge et al. 2008), and the Suprathermal Ion Telescope (SIT, Mason et al. 1998) on board STEREO (all of them part of the IMPACT instrument suite, Luhmann et al. 2008); the Electron Proton and Alpha Monitor (EPAM, Gold et al. 1998), and the Ultra-Low Energy Isotope spectrometer (ULEIS Mason et al. 2008) on board ACE; the Electron Proton Helium INstrument (EPHIN), part of the Comprehensive Suprathermal and Energetic Particle Analyzer (COSTEP, Müller-Mellin et al. 1995) and the Energetic Relativistic Nuclei and Electron Instrument (ERNE, Torsti et al. 1995) on board SOHO; and the 3D Plasma and Energetic Particle Investigation (3DP; Lin et al. 1995) on board Wind. SEP observations within 1 au were provided by the Integrated Science Investigation of the Sun (IS ⊙ IS; McComas et al. 2016) suite on board PSP. Low-energy electrons and ions are covered by the Energetic Particle Instrument-Low (EPI-Lo; Hill et al. 2017), while high-energy particles are measured by the Energetic Particle Instrument-High (EPI-Hi; Wiedenbeck et al. 2017). SEP data beyond 1 au were measured by the Solar Energetic Particle (SEP; Larson et al. 2015) instrument on board MAVEN. \nSolar wind plasma and magnetic field observations were provided by the Magnetometer (MAG; Horbury et al. 2020) and the Solar Wind Analyzer (SWA; Owen et al. 2020) on board Solar Orbiter. We used the Electron Analyser System (EAS), part of the SWA instrument, to measure the pitch-angle distribution of the suprathermal electrons. We also used the Plasma and Suprathermal Ion Composition (PLASTIC; Galvin et al. 2008) investigation and the Magnetic Field Experiment (MAG; Acuña \nTable 1. Magnetic connectivity between spacecraft and the Sun at 05:58 UT on 2022 January 20. Columns 1-4 present the respective spacecraft and its location in Carrington coordinates (the first row provides the flare location). Column 5 lists the measured solar wind speed (one-houraveraged at the SEP onset), 6-7 and 8-9 respectively provide the backmapped magnetic footpoints of the observer at 2.5 R ⊙ and at the solar surface using the PFSS model. Column 10 gives the observed (O) and modelled (M) polarity.Notes. ( a ) PFSS footpoints at 1 R ⊙ ; ( b ) Longitude and latitude values are given in the Carrington coordinate system. \net al. 2008) on board STEREO; and the Magnetic Fields Investigation (MFI; Lepping et al. 1995) as well as the Solar Wind Experiment (SWE; Ogilvie et al. 1995) on board Wind. Magnetograms from the Global Oscillations Network Group (GONG; Harvey et al. 1996) are available from the National Solar Observatory website 2 . \nFig. 2. Semi-logarithmic representation of the spacecraft constellation in the Carrington coordinate system at 05:58 UT on 2022 January 20. The orange circle at the centre indicates the Sun and the black arrow corresponds to the flare location. Color-coded solid circles mark the various spacecraft of the constellation, and the lines connected to them represent the nominal Parker spiral solutions calculated using their heliocentric distances and the observed solar wind speeds. The potential field source surface (at 2.5 R ⊙ in this case), which is the outer boundary of the potential-field model, is shown with the dashed circle. Below the source surface the magnetic field lines are extrapolated using a PFSS model, where the color of the lines corresponds to heliospheric latitude. The reddish closed lines around the flare location are also given by the PFSS model. Below the source surface the plot is linear and above it is logarithmic in distance. \n<!-- image -->', '3. SEP event on 2022 January 20: In situ observations and analysis near 1 au': 'We summarise here the particle observations and analysis of the SEP event on 2022 January 20. As shown in Figure 1 right, Solar Orbiter, STEREO-A, near-Earth spacecraft, and MAVEN (only electrons) observed the SEP event. The periodic decrease observed in the MAVEN electron data is due to its elliptical orbit. The dips occur when going into and out of the periapsis. PSP, regardless of the data gaps during the observing time, shows a late increase observed in the middle of January 21. However, the enhancement at PSP may be related to an eruption that originated in the vicinity of AR 12934 (close to the southeastern limb as seen from Earth) around ∼ 08:30 UT on January 21. Thus, based on the available observations, it can be argued that the SEPs accelerated by this solar event resulted in a particle spread over at least 160 · around the Sun, from STEREO-A to MAVEN. \nThe right panel of Fig. 1 also shows how the event features, such as intensity-time profiles, onset times, and peak intensities vary across the di ff erent observers. MAVEN observed very gradually growing electron fluxes with a small increase, compatible with its large connection angle to the source region. We focus in this study on the near-1 au observations of the SEP event, namely Solar Orbiter, near-Earth spacecraft, and STEREO-A, which are well connected to the solar source. Therefore PSP and MAVEN data are not included in the detailed study shown below.', '3.1. Magnetic connectivity': 'A fundamental parameter for interpreting the SEP event profiles at di ff erent locations is the longitudinal separation between the solar source and the footpoint of the IMF lines connecting to the respective observer. The location and magnetic connectivity of the di ff erent spacecraft around the estimated time of the soft-X ray peak of the flare ( ∼ 05:58 UT) is shown in Fig. 2 and detailed in Table 1, where Cols. 2-4 present the spacecraft locations at the time of the soft-X ray peak of the flare. \nWe note that the determination of the magnetic connectivity presented here is based on the assumption of nominal IP magnetic field lines following the shape of a Parker spiral, from which magnetic field lines are tracked downwards to the photosphere using the PFSS model. As explained later this assumption is likely not valid for Solar Orbiter during the SEP event. \nFigure 2 shows the instantaneous connectivity derived with the Potential Field Source Surface (PFSS; Schatten et al. 1969; Altschuler & Newkirk 1969; Wang & Sheeley 1992) coronal field solution. The corresponding footpoint connectivity is listed in Cols. 6-7 of Table 1 and the observed solar wind speed that is used to calculate the Parker spiral is shown in Col. 5. Columns 8-9 of Table 1 shows the magnetic connection points from the various spacecraft to the photosphere. Based on Cols. 8-9, the connectivity of near-Earth, Solar Orbiter, and STEREO-A to the solar surface is very close, ∼ 316 · longitude and ∼-16 · latitude. Then, the di ff erence with the solar flare region is of ∼ 9 · in longitude and ∼ 24 · in latitude for the three aforementioned spacecraft. Column 10 shows the magnetic field polarity observed (O) and modelled (M), indicating a good agreement between the Parker spiral-PFSS model and observations except for Solar Orbiter. This discrepancy might be related to the IP structure present at Solar Orbiter at the time of the SEP event, which is not considered in the whole back-mapping process described above. We note that the observed magnetic polarity is derived from the magnetic field vector observed in situ by the corresponding spacecraft, being positive ( + 1) for outward IMF and negative (-1) for inward IMF. \nFig. 3. Radially-scaled solar wind density snapshot from the ENLIL simulation in the ecliptic plane at 06:00 UT on 2022 January 20. The black and white dashed lines represent the IMF lines and the black contours track the ICMEs. The white lines correspond to the HCS, which separates the regions with opposite magnetic polarity, shown in blue (negative) or red (positive) on the outer edge of the simulation region. The yellow circle indicates the flank arrival of an ICME to Solar Orbiter (details given in the text). Credit: CCMC \n<!-- image -->', '3.1.1. ENLIL simulation': 'To further investigate the impact of previous CMEs in the magnetic connectivity of the di ff erent observers we performed a detailed simulation of the IP conditions during the event. Figure 3 shows a snapshot of the solar wind density in the WSAENLIL + Cone model (hereafter ENLIL model; Odstrcil et al. 2004) simulation around the SEP onset time on 2022 January 20 at 06:00 UT. We describe in detail the input parameters for the ENLIL model and the link to the online simulation in Appendix A. The black contours track the ejecta of the ICME. They are manifested in the simulation as coherent and outward propagating high-density regions. The black and white dashed lines represent the IMF lines connecting the Sun with the various observer positions. The simulation shows several stream interaction regions present near Solar Orbiter, Earth, STEREO-A, and Mars at the time of the onset of the particle event that might modify the magnetic connectivity and SEP propagation conditions. However the connectivity given by the ENLIL model is similar to the one given by the nominal Parker Spiral. According to ENLIL, there is a wide ICME just leaving behind the Earth environment but reaching Solar Orbiter through its eastern flank during the SEP event on January 20 (indicated with a yellow circle in Fig. 3). This ICME, which is studied in detail in Sect. 5, was ejected on 2022 January 16 from the same active region as the one related to the SEP event on January 20. Results of the ENLIL simulation are also presented in the six bottom panels of Figs. 4 and 5 that show the in-situ plasma and magnetic field data (discussed in the following sections) over-plotted with the pink line showing the result of the ENLIL simulation from mid 18 January to 22 January. The pink dashed lines represent the ENLIL simulation results without including the CMEs. As discussed in the following sections, ENLIL follows the general trend of the measured solar wind speed at the locations of Solar Orbiter and near-Earth locations.', '3.2. SEP observations and interplanetary context': 'The heliospheric conditions in which particles propagate at the time of release may a ff ect the SEP timing and intensity profiles (e.g. Laitinen et al. 2013; Dalla et al. 2020; Lario et al. 2022). We used both multi-point solar wind and IMF observations as well as the results of the ENLIL model presented above to provide a comprehensive understanding of the interplanetary structures and their possible influence on the propagation of the SEPs. In this section, we discuss multi-spacecraft SEP observations as well as in situ plasma and magnetic field. To classify the di ff erent in situ signatures within an ICME, we considered the following criteria. We defined the ICME start with the IP shock, followed by the sheath region and by the magnetic obstacle (MO). Within the MO, the core of the structure-that is, the MC-is restricted to periods where the following features are shown: (1) an increase in the magnetic field strength, (2) a monotonic magnetic field rotation (flux rope) resulting in large net rotation of at least one of the magnetic field components, (3) low proton temperature, and (4) plasma β below 1 (Burlaga et al. 1981).', '3.2.1. SEP observations and IP context: Earth': 'Figure 4a shows the particle, plasma, and magnetic field observations by near-Earth spacecraft. The first two panels show the SEP event on January 20 using the Sun-directed telescopes of the different instruments. The onset of the solar flare is indicated with an arrow head at the top of the panel 1, which shows a fast rise of \nFig. 4. In situ SEP time profiles and plasma and magnetic field observations by near-Earth spacecraft (a), and Solar Orbiter (b). Top: Energetic electron (1) and proton (2) temporal profiles. The arrow in the top panels indicates the flare peak time. Bottom: In situ plasma and magnetic field observations. The panels present, from top to bottom, the magnetic field magnitude (3), the magnetic field components (4), the magnetic field latitudinal (5) and azimuthal (6) angles, θ B-RTN and ϕ B-RTN, the solar wind speed (7), and the proton density (8), where RTN stands for radialtangential-normal coordinates (e.g. Hapgood 1992). Solid vertical lines indicate IP shocks, blue shaded areas indicate MC. The horizontal line on the top panel indicates a period of proton contamination. The pink lines represent the ENLIL simulation results. \n<!-- image --> \nenergetic electrons that reach energies of at least 0.7 MeV. The proton intensity-time profile (panel 2) also shows a fast intensity increase for ion energies above 4 MeV up to 100 MeV, showing a gradual increase for the lower ion energies. Electrons and protons arrived to the spacecraft from the Sun at pitch-angle 180 (inwards polarity), as discussed in Sect. 3.3.1. Near-Earth spacecraft also observed a prior SEP event on January 18 followed by a shock-driving ICME (indicated by the first vertical line and blue shading) arriving just before the onset of the January 20 SEP event, as described below. A second IP shock indicated by a second vertical line at 12:56 UT on January 21 locally accelerated low-energy protons ( ≲ 4 MeV). There are periods when the protons of ∼ 500 keV (blue line in panel 2) contaminate the electron channels ( ∼ 200-500 keV), as observed before and during the passage of the ICME and before and after the second IP shock (second vertical line). The possible contaminated periods are indicated with a horizontal line in panel 1. \nThe solar wind speed at the onset of the particle event is ∼ 480 km s -1 , as shown in panel 7. At the time of the SEP event, an ICME (blue shading) had recently left the near-Earth environ- \nnt and did not appear to a ff ect the particle propagation at this location. This ICME is likely the same structure arriving later at Solar Orbiter, as discussed in Sect. 3.2.2. The ICME starts with the arrival of an IP shock (first vertical line) at 22:57 UT on January 18. At this time we observe a simultaneous increase in the magnetic field magnitude (panel 3) and solar wind speed (panel 7), followed by a region of increased magnetic field and large fluctuations in the orientation of the magnetic field, which corresponds to the sheath region. After the sheath, we observe a region of coherent magnetic field rotation indicated with the blue shaded area, starting at 12:40 UT on January 19 lasting until 00:11 UT on January 20. The general trend of the solar wind speed is well simulated by ENLIL, which predicts the IP shock arrival within the model time uncertainties (Wold et al. 2018).', '3.2.2. SEP observations and IP context: Solar Orbiter': 'Panels 1 and 2 of Fig. 4b show the SEP event observed by Solar Orbiter. The data correspond to the particles measured by the anti-sunward looking telescopes, which measured the earli- \nFig. 5. In situ SEP time profiles and plasma and magnetic field observations by STEREO-A. Panels and interplanetary structures are as in Fig. 4. The salmon-shaded area indicate an SIR. The vertical dashed line indicates the SI within the SIR. The grey shaded area shows a flux-rope embedded in an SIR. \n<!-- image --> \nest onsets and highest intensities, as shown in Figs. 6 and 7 2 and discussed in Sect. 3.3.2. The electron event (panel 1) is observed to reach energies above ∼ 1 MeV, showing a fast intensity increase in both EPT and HET measurements. The energetic ion observations (panel 2) by Solar Orbiter / EPD / HET show a clear energy dispersion, reaching energies up to ∼ 80 MeV, used for the velocity dispersion analysis (VDA) discussed in Sect. 3.4.2. The intervening MC present at the time of the SEP onset (presented below) likely played a major role in the observed electron and proton anisotropies, as described in Sect. 3.3.2. Solar Orbiter also measured the previous SEP event on January 18, as can be seen in panel 1 of Fig. 4b. The lower energy EPT protons ( ≲ 7 MeV) showed an increase prior to the SEP event on January 20, probably related to the arrival of an ICME, as discussed below. This increase might also a ff ect the onset time determination and therefore these channels were not included in the VDA analysis, as discussed in Sect. 3.4.2. The ion contamination is also present in the decay phase of both events (January 18 and 20), indicated with the horizontal lines in panel 1, clearly visible in the high energy electrons ( ≳ 218 keV). \nThe solar wind speed at the time of the SEP event onset is ∼ 510 km s -1 , as shown in panel 7, measured by the SWA in- \nrument, which is fairly reproduced by ENLIL (pink line). As shown by the magnetic field and plasma data in panels 3-8, using MAG and SWA, the SEP event onset takes place during the passage of an ICME at Solar Orbiter. An IP shock is impacting the spacecraft at 08:02 UT on January 19, which is indicated by the vertical line in Fig. 4b. An MC arrives at 03:28 UT on 20 January, just before the particle onset, being observed until 17:52 UT (blue shaded area). The energetic electrons and the higher energetic ion particles ( ≳ 7 MeV) are propagating inside the ICME, which, however, seems to have a modulation e ff ect on the flux of ions ≲ 5 MeV.', '3.2.3. SEP observations and IP context: STEREO-A': 'Observations of the SEP event at STEREO-A are shown in Fig. 5. STEREO-A observed the earliest increases in the north telescope and the highest intensities in the south telescope, as presented in Sect. 3.3.3. However, after the solar superior conjunction of the STEREO spacecraft (from January to August 2015) until the approach to the Earth in August 2023, the STEREO-A spacecraft was rolled 180 degrees about the spacecraft-Sun line in order to allow the high-gain antenna to remain pointing at Earth. Consequently, the nominal pointing directions of the SEP suite of instruments were di ff erent from what was originally intended, and therefore we used omnidirectional fluxes in the plot. \nWith a fast intensity increase, a clear electron and proton event is observed up to ∼ 3 MeV energies (panel 1) and ∼ 60 MeV (panel 2), respectively, where clear velocity dispersion is also observed. The prior SEP event that occurred on 2022 January 18 is also measured by STEREO-A, whose background might affect the determination of the onset times, discussed in the VDA in Sect. 3.4.3. Just before the January 20 SEP event, the lowenergy ions ( ∼ 500 keV) show a small increase that coincides with the arrival of an MO (grey shaded area) and a stream interaction region (SIR, shaded in salmon colour) as detailed below. Both structures are present at the time of the flare peak time (arrow in top panel). The ion contamination is also present in the decay phase of both events (January 18 and 20), indicated with the horizontal lines in panel 1, clearly visible in the high energy electrons ( ≳ 165 keV). \nAs shown by the magnetic field and plasma data in panels 3-8 in Fig. 5, the SEP event onset at STEREO-A takes place during the passage of an MO from 03:45 UT to 07:00 UT on January 20 (grey shaded area) embedded in an SIR (salmon area). From 03:54 UT to 10:38 UT on January 20, the speed rises from ∼ 400 to ∼ 500 km s -1 ; sudden changes of the magnetic field polarity close to the stream interface (SI; dashed vertical line), and drops in the magnetic field strength together with temperature increases (not shown), which suggests that local reconnections are occurring. The SI is indicated with the dashed line, which coincides with the maximum total pressure (not shown). The solar wind speed at the time of the SEP event onset is ∼ 357 km s -1 , as shown in panel 7, fairly simulated by ENLIL.', '3.3. SEP pitch-angle distributions and first arriving particles near 1 au': 'In this section we study the pitch-angle distribution (PAD) of the three spacecraft, namely Solar Orbiter, STEREO-A, and Wind, which all have energetic particle anisotropy information. We used the four apertures of the three-axis stabilized Solar Orbiter and STEREO-A spacecraft, namely EPD / EPT, EPD / HET, \nFig. 6. Pitch-angle distribution of electrons measured by Wind / 3DP / 75-140 keV ( 1 ), Solar Orbiter / EPT / 86-102 keV ( 2 ), and STEREO-A / SEPT / 85125 KeV ( 3 ). Panels description: (a) : Pitch-angle coverage of the each of the eight pitch-angle bins of the Wind / 3DP (1); and of the centre of the four telescopes of Solar Orbiter / EPT (2) and STEREO / SEPT (3) (Sun in red, anti-Sun in orange, north in blue, and south in green); (b) : Intensities observed by each field of view. (c) : Pitch-angle distribution with color-coded intensities. (d) : first-order anisotropy values, in the range [-3, 3] (e.g. Dresing et al. 2014). The top left panel shows the longitudinal spacecraft constellation and nominal connectivity at 05:58 UT on 20 January 2022. \n<!-- image --> \nSTEREO-A / SEPT, and STEREO-A / LET (e.g. Dresing et al. 2014; Gómez-Herrero et al. 2021). The coverage of pitch angles by the four apertures of EPD and STEREO-A depends on the orientation of the magnetic field with respect to the telescopes. However, Wind is a spin-stabilized spacecraft, which allows the 3DP instrument to scan di ff erent regions of the sky and thus infer a more complete estimate of the 3D particle distribution.', '3.3.1. SEP pitch-angle distributions: Earth': 'Particle intensities measured by Wind / 3DP are stored into eight pitch-angle bins (sector 0 to sector 7). The panel 1 of Fig. 6 shows the electron PAD observed by Wind / 3DP at ∼ 75-140 keV, which shows clear anisotropies during the onset of the SEP event for about two hours until ∼ 07:30 UT. Panel a shows the intensities measured by the eight sectors of the instrument. During the onset of the SEP event the sectors measuring particles coming from the Sun presented higher intensities, covering pitch-angles from ∼ 135 · to ∼ 180 · , as shown in panel b, which shows the pitch-angles of each of the center of the sectors. Panel c shows the color-coded PAD intensity. The plot shows a discontinuity \nat pitch-angle ∼ 90 · , where the two hemispheres of pitch-angle seem to be separated. This pitch-angle discontinuity is discussed further in Appendix B. The first-order anisotropy A (e.g. Dresing et al. 2014) shown in panel d is negative, corresponding to particles propagating from the Sun, as the magnetic field polarity is negative during the period (Br negative, as shown in panel 4 of Fig. 4a). We note that large values of A (i.e. \| A \| ≳ 2) indicate highly anisotropic flows of particles, whereas small values (i.e. \| A \| ≲ 0.2) indicate nearly isotropic flows (Dresing et al. 2014). \nThe panel 1 of Fig. 7 shows that the early phase of the ∼ 3 . 15.7 MeV proton event is anisotropic for more than twelve hours (whole interval not shown), with higher fluxes in the sunwardlooking telescope that corresponds to pitch angles near 180 · , consistent with the inward magnetic polarity, showing much longer lasting anisotropies than for electrons.', '3.3.2. SEP pitch-angle distributions: Solar Orbiter': 'The four apertures of EPD / EPT and EPD / HET cover four viewing directions that are oriented along the nominal Parker spiral to the Sun and away from the Sun, to the north and to the south with \nFig. 7. Pitch-angle distribution of protons measured by Wind / 3DP / 3MeV ( 1 ), Solar Orbiter / HET / 7 MeV ( 2 ), and STEREO-A / LET / 6 MeV ( 3 ). Panels information as in Fig. 6. Panel 3 shows the 16 sectors of STEREO-A / LET, eight front-side (reddish colours) and eight back-side sectors (bluish colours). \n<!-- image --> \nsome inclination (Fig. 4 in Rodríguez-Pacheco et al. 2020). The panel 2 of Fig. 6 shows the electron PAD observed by Solar Orbiter / EPT at ∼ 87 -102 keV, which shows clear anisotropies during the onset of the SEP event at ∼ 06:30 until ∼ 09:00 UT. Panel a shows the intensities measured by the sun (red), asun (orange), north (blue), and south (green) telescopes. During the onset of the SEP event the asun and north telescopes measured slightly higher intensities, covering pitch-angles from ∼ 100 · to ∼ 140 · , as shown in panel b, which shows the pitch-angles of the center of the telescopes. Panel c shows the color-coded intensity PAD. The plot shows a discontinuity at pitch-angle ∼ 60 · , better seen in Fig. B.1. However, we note that the coverage around pitchangle 50 · -100 · is not ideal during the early phase of the event (from ∼ 06:30 to ∼ 08:00 UT). We present in more detail this discontinuity in Appendix B, including pitch-angle data from the STEP instrument. The anisotropy index shown in panel d is negative, corresponding to particles propagating towards the Sun, as the local magnetic vector is pointing outwards during the period (shown in panel 4 of Fig. 4b). While the maximum anisotropy value at Solar Orbiter is lower than at Wind, the duration of significant electron anisotropies is about three hours, and therefore slightly longer compared to Wind observations. \nThe panel 2 of Fig. 7 shows the ∼ 7 MeV proton intensities observed in the four telescopes of Solar Orbiter / HET. HET shows a one-and-a-half-hour anisotropic period starting shortly \nafter 08:30 UT on January 20 (panel d). The pitch-angle coverage is similar during this period (panels b and c). The asun telescope was measuring the higher intensities, covering pitch-angle ∼ 140 · (panel c). This means that particles propagated towards the Sun, as discussed above.', '3.3.3. SEP pitch-angle distributions: STEREO-A': 'SEPT apertures on board STEREO-A have a similar configuration to Solar Orbiter / EPT. However, since the spacecraft was put upside-down after the superior solar conjunction in 2015 until August 2023 as discussed above, the sun and asun telescopes pointed perpendicular to the nominal Parker Spiral within the ecliptic plane. The sun telescope pointed in the [ -R, -T] quadrant, whereas the asun aperture pointed in the [ + R, + T] quadrant. The north and south telescopes pointed opposite to the nominal configuration. \nThe panel 3 of Fig. 6 shows the electron PAD observed by STEREO-A / SEPT at ∼ 85 -105 keV, showing a data gap in the anisotropy panel during the onset of the SEP event, as shown in panel d. Due to the peculiar configuration of STEREO-A and the orientation of the magnetic field vector during this period, the pitch-angle coverage is not appropriate to detect field-aligned particles, as seen in panel b. The anisotropy can therefore not \nTable 2. Timing of the main solar phenomena and inferred SEP injection times t inj . All times shifted to 1 au on 2022 January 20. \nbe determined. Coinciding with an increase in the pitch-angle coverage, we observed some electron anisotropy after the onset, from ∼ 07:20 UT until ∼ 07:50 UT. During this time the asun and south telescopes measured slightly higher intensities, covering pitch-angles from ∼ 90 · to ∼ 180 · . The anisotropy index shown in panel d turns from negative to positive at ∼ 07:40 UT, when Br changed from negative to positive (panel 4 in Fig. 5). \nThe panel 3 of Fig. 7 shows the 6-10 MeV proton intensities observed in the 16 sectors of STEREO-A / LET, eight front-side (reddish colours) and eight back-side sectors (bluish colours). LET measured a one-and-a-half-hour anisotropic period starting shortly after 08:00 UT on 20 January, where most of the particles are observed in the sunward-facing sectors. The pitch-angle coverage is stable during this period, shown in panel b, which shows the pitch-angles of the sector centres. As for the electrons, the coverage is not ideal during the onset but su ffi cient to see the period where the beam has a discontinuity at pitch-angle 90 · (panel c).', '3.4. SEP timing': 'We analysed the timing of the SEP event by using the socalled velocity dispersion analysis (VDA) method, which is based on the assumption that first-observed SEPs of each energy have been injected simultaneously and propagate scatter-free and without adiabatic cooling which may cause energy changes. We include details about the VDA method in Appendix C. For this event, we focused our timing analysis in the three spacecraft located near 1 au, namely Solar Orbiter, near-Earth probes, and STEREO-A, which are all well-connected to the source and show clear energy dispersion to perform VDA. The results presented below are included in Table 2, which shows the timing of the inferred SEP injection times and of the solar phenomena (discussed in the following sections).', '3.4.1. SEP timing: near Earth': 'Fig. 8. VDA of the onset of the SEP event at near-Earth spacecraft. The horizontal and vertical axes correspond to the reciprocal of the particle velocities (1 /β = c / v ) and onset times, respectively. The green and blue data points respectively identify the onsets of the 3DP electron and ERNE proton at the corresponding velocities (energies), with the respective errors indicated. The dashed line is the linear regression to fit all points. The legend gives the e ff ective path length (L) and the estimated release time (t\\_inj) discussed in the text. \n<!-- image --> \nTo estimate the path length and infer the injection time of the particles for the near-Earth spacecraft (Wind and SOHO), we used a modified Poisson-CUSUM method that employs statistical bootstrapping (e.g. Huttunen-Heikinmaa et al. 2005; Palmroos et al. 2022). The method and the background windows used for the fitting are explained in detail in Appendix C.1. To estimate the onset times we used sector 5 of the Wind / 3DP instrument, which covers pitch-angles of anti-sunward propagating particles, as this sector observed the first arriving particles. The electron channels selected for the fitting are 27.84-401.3 keV, where the increase of the peak intensity over the background \nwas at least × 4400. For protons we used the ERNE energy channels between 13 and 50 MeV, where velocity dispersion was observed in the onset times and the peak-to-background intensity ratios were × 860-3890. \nThe VDA results are presented in Fig. 8. Two sets of data points represent the onset times as observed by SOHO / ERNE (protons, blue) and Wind / 3DP (electrons, green). The horizontal error bars represent the width of the energy channels, and the vertical error bars represent the 95% confidence interval of the onset times as provided by the Poisson-CUSUM-bootstrap hybrid method. A first-order polynomial is fitted to the data points with orthogonal distance regression (ODR) algorithm, and it is shown as the orange line over the points. The slope of this line (L = 1.4 ± 0.1 au) represents the e ff ective path length travelled by the particles, which is close to the nominal Parker spiral length for near Earth ( ∼ 1.08 au) using the measured solar wind speed. The intersection with the vertical axis represents the time of the particle injection tin j = 05:54 UT ± 4 min, or 06:02 UT ± 4 min shifted ∼ 8.2 min to compare with electromagnetic observations from 1 au. To compare, the VDA performed on ERNE (13-64 MeV) protons and the lowest electron energy channel (0.25-0.7 MeV) of EPHIN (not shown) yielded a path length of L = 1 . 3 ± 0 . 3 au and an injection time of 06:00 UT ± 9 min or 06:08 UT ± 9 min shifted ∼ 8.2 min, which is consistent with results given by SOHO / ERNE + Wind / 3DP.', '3.4.2. SEP timing: Solar Orbiter': 'In the case of Solar Orbiter, we utilised the anti-sunward measurements of the EPT electrons (33.37-218.18 keV) and HET protons (7.045-89.46 MeV), which observed the first arriving particles. These channels were not a ff ected by the enhanced levels of protons related to the arrival of the ICME to Solar Orbiter, as shown in Fig. 4b (blue shaded area). The method used for fitting and the background window is detailed in Appendix C.2. \nFig. 9. VDA of the onset of the SEP event at Solar Orbiter. Electron and proton intensities (colour-coded) from EPT and HET sensors, respectively, as function of time and inverse speed ( c / v which is 1 /β as used in Fig. 8). The colour-coded intensities are multiplied by the cubed energy to enhance the contrast. Over-plotted in black are the onsets of electrons (triangles) and protons (squares), and the velocity dispersion fitted line. The path length and injection time values shown in the legend are the result from bootstrapping (details given in the text). \n<!-- image --> \nFigure 9 shows the onsets for each energy channel indicated with a triangle (rectangle) for EPT electrons (HET protons), plotted on the particle spectrogram, with the corresponding uncertainties. We considered the bins of the channels as the uncertainties for the y-axis (c / v). In Appendix C.2 we detail how we', 'VDA STA SEPT(north) electrons, 2022-01-20': 'Fig. 10. Electron VDA of the SEP event at STEREO-A in the same format as Fig. 8. \n<!-- image --> \nestimated the uncertainties for the x-axis. Then we used an orthogonal distance regression (ODR) method to fit c / v against the onset times, to calculate the path length and the injection time. The linear fit is shown in Fig. 9. \nThe final values of path length and injection time, using a bootstrapping method detailed in Appendix C.2 to estimate the uncertainties, are given in the legend of Fig. 9. It shows an effective propagation path length of L = 2.6 ± 0.1 au, much longer than the length of ∼ 0.99 au expected for a nominal Parker spiral field with the measured solar wind and scatter free propagation. It might indicate a relatively poor pitch-angle coverage or a non-standard interplanetary magnetic field topology. The injection time given is 05:48 UT ± 4 min on January 20 (time at the Sun). Using the light-travel time to Earth, the injection time is 05:56 UT ± 4 min. Within uncertainties, the injected times derived from Solar Orbiter and near Earth are in agreement.', '3.4.3. SEP timing: STEREO-A': "To estimate the path length and infer the injection time of the particles for STEREO-A spacecraft, we followed the same process as described in Sect. 3.4.1 for near-Earth spacecraft. We note that due to the presence of the MO at the time of the SEP onset at STEREO-A spacecraft (discussed in Sect. 3.2.3), only electrons could be used for VDA since the elevated proton levels associated with the MO masked the proton onsets. For the electrons, we used energies from 45-145 keV (SEPT) measured by the north telescope, which registered the first arriving particles. The background time was chosen from to 02:00 to 05:20 UT, being short due to a previous event masking the background intensity. \nThe results of the VDA using SEPT electrons are shown in Fig. 10. The results of the fitting show an e ff ective path length travelled by the electrons of L = 2.3 ± 0.5 au, being much longer than the nominal Parker spiral length for STEREO-A ( ∼ 1.15 au) using the measured solar wind speed. It might indicate a relatively poor pitch-angle coverage or a non-standard interplanetary magnetic field topology, although we note that the uncertainty is large. The injection time is 05:44 UT ± 8 min, or 05:52 UT ± 8 min shifted ∼ 8.2 min to compare with electromagnetic observations from 1 au. This timing is in agreement within uncertainties with the injection time derived from near-Earth and Solar Orbiter data. \nSOIEPDISIS & HET Sunward Spectra (0.92 au, -1.5 deg): \nThe elemental composition of this event was measured by EPD / SIS and EPD / HET on board Solar Orbiter, by SIT on board STEREO-A, and by ULEIS on board ACE. The di ff erential energy spectral fluences measured by SIS and HET are shown in the top panel of Fig. 11. The H and 4 He spectra flatten at low energies, then steepen above a break at a few MeV / nucleon. The O and Fe spectra are similar but less certain due to the smaller energy range covered. These features are typical of large gradual SEP events (e.g., Desai & Giacalone 2016; Cohen et al. 2021). For 1 MeV / nucleon the 20 January event fluence for O was ∼ 4 × 10 3 particles / (cm 2 sr MeV / nucleon), roughly a factor of 25 below the fluences in the large October-November ('Halloween') 2003 events (Cohen et al. 2005), which are among the most intense events observed at 1 au in recent solar cycles. The top panel of Fig. 11 shows fluence spectra for major elements. Dashed lines are Band functions fits (Band et al. 1993) to H, 4 He , O, and Fe. The resulting spectral fitting coe ffi cients are listed in Table D.1. They fall within the distribution of results from the survey by (Desai et al. 2016), which is based on large gradual SEP events. \nThe bottom panel of Fig. 11 shows the average elemental abundances measured between 0.32-0.45 MeV / nucleon for the 2022 January 20 SEP event measured at Solar Orbiter, ACE, and STEREO-A. The average abundances from the three spacecraft show a very similar pattern. Comparing this event with the average from the 64-event survey of Desai et al. (2016) measured at the same energy, it is clear that the composition of the 2022 January 20 event is typical for gradual SEP events. The measured 3 He abundance was below 1%. The maximum Fe / O abundance ratio at Solar Orbiter is around 0.64 at an energy of 0.19 MeV / nucleon, placing this event close to the average ratio found in the Desai et al. (2006) survey of gradual SEP events.", '3.6. Electron peak spectra': 'Following the method described by Dresing et al. (2020) and Strauss et al. (2020), we determined the electron peak spectra, as observed by Wind, Solar Orbiter, and STEREO-A, with results shown in Fig. 12 and summarized in Table 3. In the case of Wind (panel 1 in Fig. 12) and using sector 6 of the 3DP instrument, no spectral transition was found, representing a single power law shape according to \nI ( E ) = I 0 E E 0 ! δ 1 , (1) \nwhere δ 1 represents the spectral index and I 0 is the intensity at E 0 = 0 . 1 MeV. We used sector 6 of the 3DP sensor instead of sector 7, which would cover a more field-aligned pitch-angle sector, because sector 7 was full of data gaps. \nFor STEREO-A (panel 3 in Fig. 12) we used the south telescope of SEPT, which presented the highest intensity peak. We found a broken power law to best describe the data represented by \nI ( E ) = I 0 E E 0 ! δ 1 E α + E α b E α 0 + E α b ! δ 2 -δ 1 α . (2) \nThis model yields a spectral transition at the energy Eb and a second spectral index δ 2 at energies above Eb . The parameter α describes the sharpness of the spectral transition. We note that there is a sudden drop around 07:30 UT in the intensity-time series caused by magnetic field changes (cf. Fig. 6 3). This drop \n<!-- image --> \nFig. 11. SEP fluences and relative abundances. Top panel: Fluence spectra from SIS (filled circles) and HET (circles) summed over the event, and fitted Band function spectra (dotted lines). Bottom panel: Abundances from 0.32-0.45 MeV / nucleon for the 2022 January 20 event compared with averages at the same energy from the survey of 64 large SEP events by Desai et al. (2006). Blue half-filled squares are from Solar Orbiter SIS, filled red circles from ACE / ULEIS, and orange diamonds from STEREO-A / SIT. \n<!-- image --> \npotentially a ff ects the peak intensities of the lower half of (or even all) the energy channels as they had not yet reached the peak. This means that especially the low energy channels, which usually reach their peak later, could in reality have higher peak intensities, which in turn could potentially lead to a steeper spectrum in that energy range. However, we are confident that the spectral break is not caused by this e ff ect as the intensity drop rather a ff ects also higher energies. We note however, that the break energy might be a ff ected by this issue. \nIn the case of Solar Orbiter (panel 2 in Fig. 12), we used the EPT and HET anti-Sun telescopes, showing the highest intensity peak. Potentially due to the much higher energy resolution of the Solar Orbiter data we found a triple power law to best represent \nFig. 12. Electron peak intensity spectra measured by Wind (1), Solar Orbiter (2), and STEREO-A (3). The legend shows the fit values: the spectral index ( δ 1, δ 2 δ 3) observed in between the spectral transitions: Eb; and α ( β ), which determines the sharpness of the break(s) (Strauss et al. 2020). The lower and fainter set of points correspond to the pre-event background level. Details given in the main text. \n<!-- image --> \nTable 3. Summary of the electron peak spectra results. Parameters based on the method described by Dresing et al. (2020) and Strauss et al. (2020). Details given in the main text. \nthe observations, which is described by \nI ( E ) = I 0 E E 0 ! δ 1 E α + E α bl E α 0 + E α bl ! δ 2 -δ 1 α E β + E β bh E β 0 + E β bh δ 3 -δ 2 β . (3) \nThis model yields two spectral transitions Ebl at lower energies and Ebh at higher energies and correspondingly three spectral indices δ 1, δ 2, and δ 3. We note the high uncertainty of the second spectral transition. The parameters α and β describe the sharpnesses of the two breaks, respectively. The spectral transition and indices below and above the spectral break are also summarized in Table 3. \nFor comparison, we selected the spectral index near 200 keV, namely δ 200. The spectral indices based respectively on Wind, \nSolar Orbiter, and STEREO-A data are similar within uncertainties and are summarized in the second column of Table 3. The spectral indices observed in this event are clearly harder than a large sample of events (781 near-relativistic electron events measured by both STEREO) studied by Dresing et al. (2020), who find ⟨ δ 200 ⟩ = -3.5 ± 1.4. Moreover, Dresing et al. (2022) analysed 33 energetic electron events that were related to coronal pressure waves. They derived a mean spectral index of ⟨ δ 200 ⟩ = -2.5 ± 0.3, similar to the indices found in this study ( δ 200 ≈ -2.6).', '4.1. Flare observation and analysis': 'Fig. 13. Sketch showing interplanetary configuration of the 2022 January 20 SEP event. The Sun (not to scale) is shown at the center indicated by the yellow circle. The grey circles represent, from the innermost and going outwards, the orbits of Mercury, Venus, and Earth. Earth, Solar Orbiter, and STEREO-A are shown by the green circle, blue and red squares, respectively. The ICME corresponding to the CME erupting on 2022 January 16 is shown in blue. The CME and CMEdriven shock associated with the SEP event on January 20 are indicated by the red shading and red curve, respectively. The dashed coloured lines indicate the nominal Parker spirals using measured solar wind speed. The rightmost dark-red dashed lines connects to the flare site using a nominal Parker spiral and 400 km s -1 . \n<!-- image -->', '4. SEP parent solar source: remote-sensing observations and data analysis': "The National Oceanic and Atmospheric Administration (NOAA) active region (AR) number 12929 produced a series of eruptions around the time of the study. A first detected CME was released from N08W30 (in Stonyhurst coordinates) at 20:48 UT on January 16, which arrived at Earth at 23:40 UT on January 18 and at Solar Orbiter at 17:10 UT on January 19, as discussed in Sect. 3.2. This CME and its corresponding ICME are studied in more detail in Sect. 5, as it is a ff ecting the particle propagation as observed by Solar Orbiter. A second CME was launched at 17:00 UT on January 18 from N07W53 (in Stonyhurst coordinates), related to the SEP event on January 18. The particle increase related to this event is shown in Figs. 4 and 5, whose background is a ff ecting the onset of the SEP event on January 20. A third eruption was observed to be released from N08W76 (in Stonyhurst coordinates; 325 · in Carrington longitude) at 06:12 UT on January 20, related to the SEP event under study. This CME is also represented in the sketch of Fig. 13 as a red shading area. In the following, we present the remote-sensing observations and analysis of this third eruption. \nAn M5.5 flare was observed on 2022 January 20 at N08W76 (in Stonyhurst coordinates) by near-Earth assets such as SDO and by Solar Orbiter. Since we are mainly interested in energetic particles, we focus on hard X-ray (HXR) observations that constrain nonthermal electrons in solar flares. In Fig. 14a, we plot X-ray count rates in five energy bands as recorded by STIX on Solar Orbiter. At lower energies (4-25 keV), the flare shows a smooth time profile, which is typical for thermal emission. It peaks at 05:58 UT (all STIX times have been shifted by 30 s to be consistent with observations from 1 au) and shows an extended gradual decay lasting more than 1.5 hours. Between 05:54 and 06:00 UT, three more impulsive peaks can be discerned at energies above 25 keV, which is consistent with nonthermal bremsstrahlung emission generated by accelerated electrons. However, the nonthermal emission is very weak. \nSTIX provides Fourier-synthesis imaging capabilities (cf. Massa et al. 2023), so we reconstructed the HXR sources at thermal and nonthermal energies. We found a single coronal source above the solar limb, also at higher energies, where usually the emission is predominantly emitted by chromospheric footpoints. Since the flare is observed right at the solar limb as seen from Solar Orbiter, we conclude that the footpoints are actually occulted. This is corroborated by data from Fermi-GBM, where the count rates show a much more pronounced nonthermal component above 25 keV (as shown in Fig. 14c). While GBM has no imaging capability, we know from SDO / AIA that the flare was fully visible from Earth, and we are thus confident that GBM has complete coverage of the X-ray emission of this flare. The emission above 25 keV shows multiple impulsive peaks, including three major ones that extended to at least 300 keV. \nDue to the full coverage of the flare from Earth's perspective, we used GBM data to get quantitative constraints on the electrons accelerated in the flare. We performed a series of spectral fits using the OSPEX (Object Spectral Executive) package 3 , which is part of the SolarSoft IDL software library. We forwardfitted the background-subtracted GBM count spectra with a combination of an isothermal component and a nonthermal thicktarget model assuming a power-law spectrum for the injected electrons (Brown 1971). As GBM is not optimized for solar observations, the spectra su ff er from pulse pileup, particularly during times of high count rates during solar flares. This mostly a ff ects the thermal component and the transition to the nonthermal range. We therefore do not consider the thermal fits here. Concerning the transition to the high-energy power-law, we determined the e ff ective low-energy cuto ff in the early phase of the impulsive phase when pileup is still comparatively small. We found low-energy cuto ff s around 22 keV, and then adopted this value as a constant parameter for all nonthermal fits. It should be stressed that this is the lowest cuto ff energy that is consistent with the data, because the true cuto ff is usually masked by the thermal component (e. g. Warmuth & Mann 2020). \nFigure 15 shows the spectral fit results for the thick-target electron component together with the GBM count rates in the nonthermal energy range. We focus here on the impulsive phase of the flare. The top panel shows the GBM count rates in three broad energy bands that are dominated by nonthermal emission, as shown by the multiple impulsive peaks with typical duration of ≈ 1-2 min. It is thus clear that this flare was characterized by multiple discrete episodes of energy release and particle acceleration. The middle panel shows the power-law index δ of the electron flux spectrum. Note that when the count rates are high, \nFig. 14. Inferred SEP injection times shifted to 1 au (vertical lines with temporal error bars on top) overplotted on the radio spectrogram as observed from STEREO-A / WAVES and Earth (ASSA, and YAMAGAWA) and the X-ray count rates from Solar Orbiter / STIX. The zoom-in on the right corresponds to the dashed line square indicated on the left. It shows Fermi-GBM X-ray count rates against the same radio spectrogram. The STIX times have been shifted by 30 s for comparison with electromagnetic observations from 1 au. Legend on the top right refers to lines in panels (a), (b), and (c). The observed radio structures are indicated in panels b and c. Details given in the main text. \n<!-- image --> \nthe spectral index becomes lower, i.e. the spectrum hardens. This anti-correlation is known as the soft-hard-soft evolution (e.g. Grigis & Benz 2004). The hardest spectra are characterized by an index of δ ≈ 4. Finally, the bottom panel of Fig. 15 shows the total injected electron flux above the low-energy cuto ff of 22 keV. Again, this is anti-correlated with the spectral index. During the impulsive phase, a total of 4 . 9 ± 0 . 1 × 10 37 electrons were accelerated, which contained an energy of 2 . 5 ± 0 . 1 × 10 30 erg. These values are typical for mid-M-class flares (Warmuth & Mann 2020). \nWe note that the spectrum of the injected electrons deduced from the HXR observations is softer than the in-situ spectra discussed in Sect. 3.6, namely δ HXR ≥ 4 as opposed to δ in -situ ≈ 2 . 5-2.8. This is consistent with what has been found by statistical studies of impulsive solar energetic electron events, which all show that the spectra of electrons precipitating on the Sun (assuming thick-target emission) are apparently softer than the spectra of the electrons injected into space (cf. Krucker et al. 2007; Dresing et al. 2021). It is not yet clear whether this truly means that the injection spectrum is di ff erent for the downwardand upward-moving electrons, or whether this di ff erence rather results from propagation e ff ects, di ff erent acceleration mecha- \nthat might be involved, or modeling assumptions that are made for inferring the electron spectrum from the measured photon spectrum.", '4.2. Radio observations and analysis': 'In Fig. 14b, we present a composite dynamic radio spectrum constructed using observations from several ground-based and space-based instruments. This provides an uninterrupted coverage of processes from the low corona to interplanetary space (8 GHz-30 kHz). The part of the spectrum from centimetric to metric wavelengths (in frequency, this corresponds to 8 GHz to 70 MHz) was constructed using data from the YAMAGAWA solar radio spectrograph, supplemented with data from the e-Callisto network of radio telescopes, and in particular with data from ASSA. For the part from decametric to hectometric wavelengths (corresponding to 16 MHz to 30 kHz in frequency) we used data from SWAVES on board STEREO-A. Such a spectrum can be leveraged to distinguish between the nuances of particle acceleration and transport from the corona to the inner heliosphere \nFig. 15. Results of the spectral analysis of the Fermi-GBM data. Top : GBM HXR count rates in three broad energy bands. Middle : spectral index of injected electron flux. Bottom : injected electron flux above the low-energy cuto ff of 22 keV. \n<!-- image -->', '(e.g. Ergun et al. 1998; Voshchepynets et al. 2015; Badman et al. 2022).': 'The solar radio event presented here is rich with a number of di ff erent emission types such as type II (TII), type III (TIII), and type IV (TIV), marked in Fig. 14 b. In the low-decimetric to metric wavelengths ( ≪ 1 GHz to 30 MHz) the observed radio emissions are mostly dominated by di ff erent types of plasma emission such as type II, III, and IV radio emission. These are associated with non-thermal electrons accelerated by propagating shock waves (TII), electron beams propagating along open and quasi-open magnetic field lines (TIII), and electrons trapped within rising post-flare loops or within CME flux ropes (TIV). The time evolution of the radio event, that is the starting times of TII and TIII are directly compared with the results from the VDA analysis (Sect. 3.4) and discussed here.', '4.2.1. Type II radio bursts': "Two di ff erent TII lanes may be distinguished, Type IIa (TIIa) and Type IIb (TIIb) indicated in Fig. 14d, both with their own complexities. Multiple TII emissions from the same shock may have their sources at di ff erent regions of the shock, therefore investigating them allows us to constrain regions where electrons are accelerated (Jebaraj et al. 2020, 2021). TIIa starts from 250 MHzpromptly at 05:55 UT suggesting shock formation early on during the event, as discussed in Sect. 4.3.2. TIIa drifts to lower frequencies at a rate of approximately d f / dt ∼ 10 MHz per minute between 05:55 (1.09 R ⊙ ) and 06:11 UT (1.5 R ⊙ ). Using this drift rate, and the approximate coronal heights at which they are formed, which was obtained from a commonly used Newkirk (1961) coronal electron density model, we calculated the speed of the emitting source to be ∼ 340 km s -1 . Previous research has often associated such slow propagation of the source with the shock's propagation within a streamer or at sector boundaries \n(Kouloumvakos et al. 2021; Morosan et al. 2024). It is noteworthy to mention the close correspondence between the speed of the EUV wave, discussed in Sect. 4.3.2, and the speed deduced from the TIIa drift. Moreover, the derived height at which TIIa was formed (1.09 R ⊙ ) is low in the corona where the EUV wave propagates (Warmuth 2015). TIIa continues up to the hectometer wavelengths, 3 MHz in the frequency spectra where it stops at 06:40 UT. \nFig. 16. Zoom-in of Fig. 14b. It shows in detail the Type IIIs and HBs radio structures, indicated with the blue and black arrows, respectively. \n<!-- image --> \nTIIb exhibits a far more complex structure. It is first characterized by a spectral kink-like morphology, previously linked to source propagation through regions of varying density (Kouloumvakos et al. 2021; Koval et al. 2023). It features distinct herringbone (HB) structures which are observed for a brief period between 05:56:30 and 06:00:00 UT within the 80 to 50 MHz range (corresponding to ∼ 1 . 5 R ⊙ ). They are indicated with red arrows in Fig. 14d and with the black arrows in the zoomedin plot in Fig. 16. HBs are known to be electron beams accelerated by a nearly perpendicular shock wave, emanating from a backbone that represents the width of the shock's nearly perpendicular region (Mann et al. 2018; Morosan et al. 2022). TIIb is observed for about ten minutes and drifts to 16 MHz (decameter wavelengths) by 06:05 UT. By applying a Newkirk density model to estimate the speed of the source linked with these emissions, we find it to be approximately 1400 km s -1 . This estimated shock speed is in agreement with that derived from the spheroid 3D reconstruction in Sect. 4.3.2, which was 1433 km s -1 .", '4.2.2. Type III radio bursts': 'In the meter to kilometer wavelengths (corresponding to 80 MHz to 30 kHz), we identified one group of TIII emissions close to the flare time, as indicated with the blue arrows in Figs. 14 and 16. This indicates electrons streaming away from the corona during this time period. They are observed to start around 80 MHz ( ∼ 1 . 45 R ⊙ based on the Newkirk (1961) density model) at 05:55:40 UT and are seen across the deca-hecto-kilometric wavelengths as observed from space-based receivers. STEREOAand L1 spacecraft did not measure any Langmuir waves at the time of the energetic particle event and the type III radio burst. The Radio and Plasma Waves (RPW; Maksimovic et al. 2020, \n<!-- image --> \nFig. 17. EUV and coronagraph images and GCS 3D reconstruction of the CME (green mesh) and associated driven shock (red mesh) as seen by two di ff erent points of view: STEREO / EUVI (b) and STEREO / COR2-A (d); SDO / AIA (a) and SOHO / LASCO-C2 (c), at two di ff erent times (left and right). \n<!-- image --> \n2021; Vecchio et al. 2021) on Solar Orbiter was not observing during this time period and therefore we cannot be conclusive about the lack of Langmuir waves at Solar Orbiter. However, given that Solar Orbiter was located between L1 and STEREO during the time of the SEP event, it is highly unlikely that it would have observed local wave, which suggests that the TIII emitting electron beams never traverse the vicinity of any spacecraft. This group of TIII are observed until 05:58 UT at metric wavelengths ( ∼ 60-30 MHz) where most individual TIII within the group seem to originate.', '4.2.3. Co-temporal Type III bursts and HBs': 'The co-occurrence of TIII and TIIB suggests that some of the electron beams accelerated by the shock (manifesting as HB) may contribute to the group of TIII bursts. Given the morphological similarities between HBs and TIII, it may be speculated that some TIII observed in low frequency spacecraft observation ( < 15 MHz) may have also been continuations of the HBs. Correlation between near-relativistic electrons observed in situ and HBs emitted by the coronal shock have been qualitatively discussed in prior studies, such as Jebaraj et al. (2023b). Following their conclusions, we may suggest that the shock strongly interacted with the field lines where the flare-accelerated electrons propagated. Since a near-perpendicular geometry is required to generate the HB structures, the lateral regions of the shock are the most-likely locations for such interactions. It is also worth noting that the HB and TIII bursts occur co-temporally with the HXR peak observed by Fermi-GBM (Fig. 14c). While, qualitatively this lends credibility to a shock re-acceleration phenomena, it is impossible to quantify these correlations due to the lack of precise X-ray and radio imaging. \nWe provide below a scenario where the above qualitative result is self-consistent. The acceleration mechanism which is invoked is the fast-Fermi process, particularly in its relativistic form (Jebaraj et al. 2023a). If the electrons accelerated during reconnection at the flaring site interact with a near-perpendicular shock ( θ Bn > 85 · ), they may be re-accelerated resulting in beams. Such beams can simultaneously emit plasma radiation which manifests in the radio spectrogram as HB or TIII bursts. \nThe process is highly e ffi cient and would result in a significantly changed electron spectra than the ones accelerated by flares. This is corroborated by the fact that the electron spectra discussed in Sect. 3.6 deviates from the photon spectra, likely due to shock modification.', '4.2.4. Solar phenomena-SEP timing comparison': 'In Fig. 14 we include the VDA timing results from Sect. 3.4 to compare with the HXR and radio signatures. The red, blue, and green vertical lines represent the injection times derived for STEREO-A, Solar Orbiter, and near-Earth spacecraft, respectively. The uncertainties of these onset times are represented by the arrows in the top panel a and bottom panel d. We also present in Table 2 a summary of this inferred SEP injection times and the timing of the solar phenomena discussed above (HXR peaks, TIIs, HBs, and TIIIs). For the spacecraft with less uncertainty in the VDA analysis (Solar Orbiter, t inj = 05:56 ± 4 min; Near-Earth, t inj = 06:02 ± 4 min), the injection times are co-temporal with the emission of TIIa, TIIb, HBs, TIIIs (partly co-temporal with the HBs), and the nonthermal HXR peaks. In the case of STEREOA, the inferred SEP injection time shows a larger uncertainty (t inj = 05:52 ± 8 min), however, this time is still co-temporal with the solar phenomena aforementioned.', '4.3. EUV and coronagraph observations and analysis': 'The extreme ultraviolet (EUV) observations of the solar eruption associated to the SEP event under study have been examined in detail by Zhang et al. (2022). We include here a summary of the most relevant information from that study and further observations and analysis using the EUV and coronagraph imagery as presented below.', '4.3.1. CME observation and analysis': "The early phase of the eruption started before 05:51:30 UT on 2022 January 20, as shown in Figure 1 of Zhang et al. (2022) in the hot channels of AIA at 131Å and 94Å. The overlying \nloop is tardy during the slow rise of the flux rope observed at the hot channels. It is pushed upward to form the leading front of a CME as the hot flux rope accelerates (Cheng et al. 2013). The final speed of the flux rope and the overlying loops are close to each other ( ∼ 830 km s -1 ). At 05:55:04 UT the eruption is clearly observed by SDO / AIA at the west solar limb, as shown in the left part of Fig. 17 (top panel a). About 30 minutes later, at 06:24:05 UT, the flux rope has evolved high enough in the corona to be fully observed by both SOHO / LASCO (panel c) and STEREO / COR2-A (panel d) coronagraphs. \nTo characterize the CME associated with the SEP event, mainly in terms of final coronal CME speed, width, and location, we took advantage of the multi-view spacecraft observations and reconstructed the 3D CME to minimize projection e ff ects using the graduated cylindrical shell (GCS; Thernisien et al. 2006; Thernisien 2011) model. The GCS model uses the geometry of what looks like a hollow croissant to fit a flux-rope structure using coronagraph images from multiple viewpoints. The sensitivity (deviations) in the parameters of the GCS analysis is given in Table 2 of Thernisien et al. (2009). The COR1 / 2-A and C2 and C3 quasi-simultaneous images were used to fit the flux-rope shape of CME at di ff erent times. The routine used for the reconstruction is rtcloudwidget.pro , available as part of the scraytrace package in the SolarSoft IDL library 4 . The main CME reconstruction period, using two vantage points of view, covered from 05:50 UT to 07:54 UT on 2022 January 20. \nThe lower panels of Fig. 17 show on top of the coronagraph images, the GCS fit analysis for the CME (green mesh) and the spheroid model fit for the CME-driven shock (red mesh), discussed below. The 3D reconstruction shows that the CME follows a radial path with a Stonyhurst (Carrington) latitude and longitude of 10 · and 74 · (323 · ), respectively. The tilt angle ( γ ), namely the inclination of the flux rope with respect to the ecliptic plane, does not show deviations, staying at a fixed value of 40 · . The CME speed at the leading edge estimated from the linear fit to the height-time measurements is 1410 km s -1 . The uncertainty of the CME speed is considered to be 7% of the value based on Kwon et al. (2014). The width or total angular extent of the CME is 51 · , based on Dumbovi'c et al. (2019), where the semi-angular extent in the equatorial plane is expressed by R maj -( R maj -R min) × \| γ \| / 90. The value of R maj (face-on CME half-width) is calculated by adding R min (edge-on CME half-width) to the half-angle, and R min was calculated as the arcsin( aspect ratio ). The CME width deviation was derived from the mean half-angle error, estimated by Thernisien et al. (2009) as + 13 · / -7 · . Thus, at the latest time of the 3D reconstruction at 07:54 UT, corresponding to a CME height of 16.10 R ⊙ , the narrow CME ( ∼ 51 · ) is propagating in the direction W74N10 with a relatively high speed ( ∼ 1410 km s -1 ). Figure 13, which depicts a sketch showing the interplanetary configuration of the 2022 January 20 SEP event, shows this CME represented by the red shading.", '4.3.2. CME-driven shock observation and analysis': 'The CME eruption leads to the formation and propagation of an EUV wave (shown in figure 1 of Zhang et al. 2022). The signatures of the EUV wave propagating on the solar surface, are clearly visible from 05:52 to 06:09 UT on 2022 January 20 in AIA images, as shown in panel a of Fig. 17. The EUV wave on the solar disk extends to about 365 Mm from the source region with a speed of 373 km s -1 . This value is in agreement with the \nTIIa drift of 340 km s -1 derived from radio observations in Sect. 4.2. The successful eruptions of the flux rope and the overlying system of loops evolve higher in the corona into a fast and wide CME, which drives a shock wave (shown in figure 4 of Zhang et al. 2022). This shock wave is observed as a well-formed bubble over the west solar limb by SDO / AIA at 05:55:04 UT, shown in the top panel a of Fig. 17. \nIn order to gain a detailed understanding of the magnetic connectivity to the CME-driven shock associated with the SEP event, the coronal shock 3D reconstruction, shown as the red mesh in lower panels of Fig. 17, was performed using the model developed by Olmedo et al. (2013). The model uses a spheroid shape to fit the CME-driven shock using quasi-simultaneous images from COR1 and COR2, and from C2 and C3. The images underwent a basic process for calibration, and base-di ff erence or running-di ff erence procedure was used to highlight the front of the shock better. The process of the fitting are explained in detail by Rodríguez-García et al. (2021). The main shock reconstruction period, using two vantage points of view, covered from 05:55 UT to 07:54 UT on 2022 January 20, when the shock height changed from ∼ 1.27 R ⊙ to ∼ 16.10 R ⊙ . \nThe parameters of the 3D reconstructed shock (red mesh in Fig. 17) are consistent during the main reconstruction period. The resultant spheroid is oblate ( e = 0.28) and the self-similarity coe ffi cient ( κ ) is ∼ 0.52. The longitude and latitude values show that the origin at the Sun of the coronal shock is located at W73N10. Lastly, the coronal shock speed, estimated as the linear fit of the evolution of the shock height, is 1433 km s -1 . The uncertainty of the CME-driven shock speed is considered to be 8% of the value, following Kwon et al. (2014). We note that the shock speed deduced from the spheroid reconstruction is in good agreement with the estimated type II drift ( ∼ 1400 km s -1 ) deduced in Sect. 4.2. \nWe used the 3D reconstruction of the CME-driven shock to estimate the first time the shock wave intersects the magnetic field lines connecting to the near-Earth, Solar Orbiter, and STEREO-A spacecraft. For this purpose and to have a reliable value of the uncertainty of the crossing time, we utilized the magnetic field lines given by the Magnetic Connectivity Tool 5 , which uses the measured solar wind and both a fixed value of high (800 km s -1 ) and low (300 km s -1 ) speed of the solar wind to estimate a set of magnetic field lines connecting the spacecraft to the solar surface (2.5 R ⊙ ). These sets of lines are modelled back to the solar surface using the PFSS model, with a bundle of 100 magnetic field lines for each solar wind speed value. We note that the assumption of nominal IP magnetic field lines is likely not valid for Solar Orbiter. \nThe first time the shock intersects more than 50% of the number of lines in the bundle is 05:58 UT ± 1 min for Solar Orbiter and 06:00 UT ± 1 min for both near-Earth spacecraft and STEREO-A. This timing is summarized in Table 2 and indicated in the right panel of Fig. 14 with the horizontal dashed segments on the bottom axis, following the colour code of the spacecraft, namely green, blue, and red for near-Earth probes, Solar Orbiter, and STEREO-A, respectively. These shock-linesintersection times can be compared with the results given by the VDA analysis from Sect. 3.4, already indicated in the same Fig. 14 as vertical lines discussed above. In the case of near-Earth spacecraft, the injection of the particles is estimated to happen two minutes after the first connection to the shock. For Solar Orbiter and STEREO-A, the injection time is respectively estimated three and eight minutes earlier than the first time of con- \nSDO AIA\\_2 211 16-Jan-2022 19.59.11 UT \nSDO AIA 2 193 20-Jan-2022 05.55.05 UT \n<!-- image --> \nFigure 18 shows the evolution of AR 12929 during 2022 January 16-20. Panel a presents an image taken by SDO / AIA / 211Å on January 16 at 19:59 UT, which shows the accumulated pixels of the dimming areas within the period of observation of the CME erupting on January 16. Panel b presents a base-di ff erence image taken by SDO / AIA / 193Å on 2022 January 20 at 05:55:05 UT, showing the EUV wave, as discussed in Sect. 4.3.2. We overplotted in red the position of the dimming lobes shown in panel a as they have would rotated in time from January 16 to January 20. To better visualise the location of the footpoints of the CME on January 16 relative to the centre of the source region of the CMEonJanuary 20, the image is now derotated to four days earlier using the solar di ff erential rotation formula of Howard et al. (1990), as shown in panel c. \n<!-- image --> \nFig. 18. Evolution of AR 12929 from 16 to 20 of January 2022. (a) Image taken by SDO / AIA\\_2 211Å on 2022 January 16 at 19:59:11 UT, showing the accumulated pixels of the dimming areas in the period of time of observation. (b) It shows the CME-driven shock and the EUV wave on January 20 along with the dimming lobes shown in panel a overplotted in red in the position they would have been four days later. (c) the image shown in panel b is derotated to the time of panel a using the solar di ff erential rotation formula by Howard et al. (1990). Details given in the main text. \n<!-- image --> \nnection to the shock. However, the timing is co-temporal for the three spacecraft if we consider the time uncertainties. \nWe also estimated for the three spacecraft the angle between the 3D shock normal and the magnetic field lines at the intersection θ Bn, being θ Bn = 82 · , 71 · , and 84 · , for near-Earth, STEREO-A, and Solar Orbiter, respectively. The heights at the time of first connection between the shock wave and the magnetic field lines linking to the near-Earth, Solar Orbiter, and STEREO-A spacecraft are 1 . 38 ± 0 . 18 R ⊙ , 1 . 47 ± 0 . 15 R ⊙ , and 1 . 39 ± 0 . 20 R ⊙ , respectively. These heights are consistent with our estimated formation heights of HBs, which is approximately 1 . 5 R ⊙ . In the presence of open magnetic fields in the laterally expanding shock regions, this configuration meets the steep geometry requirement for generating HB radio bursts. This alignment further supports the electron acceleration scenario proposed in Sect. 4.2 and the early connection to the escaping electron beams from the shock.', '5. CME/ICME on 2022 January 16 observations and analysis': 'At the time of the onset of the SEP event under study, Solar Orbiter was embedded in an ICME, namely from January 19 at 08:02 UT to January 20 at 17:52 UT. This ICME passed near Earth on January 18 at 22:57 UT and left the near-Earth environment on January 20 at 00:11 UT, a few hours before the SEP onset, as discussed in Sect. 3.2. An inspection of the STEREO and SOHO coronagraph images, and the CDAW SOHO LASCO CME catalogue 6 (Yashiro et al. 2004) together with the near-Earth ICME list provided by I. Richardson and H. Cane 7 (Richardson & Cane 2010), revealed that this ICME is most likely associated with a CME that appeared in LASCO C2 field of view at 20:48 UT on 2022 January 16. As this ICME might influence the transport of solar energetic particles to the Solar Orbiter location, we present below the CME evolution in the corona and in the heliosphere with some detail.', '5.1. January 16 CME observation and analysis': "The CME detected by SOHO / LASCO C2 at 20:48 UT on 2022 January 16 is associated with a flare erupting at 17:42 UT from AR12929, the same region as for the event on January 20. Based on GOES observations, the flare being classified as C1.1 level, peaked at 17:48 UT and was located on W27N07. We used the GCSmodel described in Sect. 4.3.1 to derive the 3D morphology and average speed of the CME close to the Sun. \nThe 3D fitting of the CME shows a tilted flux-rope ( γ = -45 · ) with a speed of ∼ 773 km s -1 . The ecliptic CME width based on Dumbovi'c et al. (2019) is estimated as 38 · . Based on the reconstructed CME nose longitude and latitude, the CME left the near-Sun environment propagating towards the Stonyhurst direction W25N17 (Carrington 315) at 15 R ⊙ . However, to fit the in situ observations of the ICME, based on arrival time at the locations being encountered, namely Earth and Solar Orbiter as presented in Sects. 3.2.1 and 3.2.2 respectively, the CME should have being oriented towards the south and east, specifically towards Stonyhurst W08N04 (Carrington 298). This is consistent with previous studies that fast CMEs turn to be blocked by the background solar wind ahead and deflected to the east (Wang et al. 2004). The additional observations from the STEREOA / HI cannot confirm if the CME rotated in the interplanetary space after leaving the solar corona or if the di ff erence in the CME nose position is due to inherent uncertainties associated with the GCS fitting (e.g. Verbeke et al. 2023; Kay & Palmerio 2024). \nFig. 19. Pitch-angle distribution function of solar wind electrons. (a) Pitch angle distribution of solar wind electrons with energies between 69 eV and 5 keV, for the time interval from 09:00 on 2022 January 19 to 18:00 on January 20. (b), (c), and (d) are 2D speed, pitch-angle distributions averaged over the three selected 70 minutes intervals, marked by the magenta shadowed regions in panel a. \n<!-- image --> \nUsing the EUV wave velocity of 373 km s -1 , as discussed in Sect. 4.3.2, we estimated that the EUV wave reaches the footpoints of the CME that erupted on 2022 January 16, within 3 ± 1 (western footpoint) and 5 ± 1 (eastern footpoint) minutes respectively after being firstly observed at 05:52 UT. Thus, the first time that the EUV wave related to the SEP event on January 20 intersects the centroid of the west dimming lobe is at 05:55 UT ± 1 min. This time is indicated in Table 2 and in Fig. 14 as a purple vertical dashed line in panels (c-d). We note that the inferred injection time for the particles observed by Solar Orbiter is co-temporal within uncertainties with the intersection time of the EUV wave associated to the January 20 SEP event with the western leg of the January 16 ICME, represented by the centroid of the dimming lobes.", '5.2. January 16 ICME observation and analysis': "The details about the solar wind and plasma data related to the ICME arriving at Earth and Solar Orbiter near the SEP event onset were presented in Fig. 4 and discussed in Sect. 3.2. It is unusual that an ICME directed towards the Earth and Solar Orbiter locations arrived ∼ 9 hours earlier at 1 au, as Solar Orbiter was located near 0.96 au. This might be related to the ICME propagating along a high-speed stream observed at near-Earth spacecraft before the arrival of the ICME, as shown in panel 7 of Fig. 4a. A coronal hole (not shown) was located to the southwest of AR 12929, a ff ecting the plasma conditions in which the ICME propagates, which may have caused distortion in the ICME shape, as discussed by Rodríguez-García et al. (2022). \nThe parameters of the CME in terms of speed (773 km s -1 ) and orientation (W08N04) inserted in the ENLIL model fit well the observed arrival time at near-Earth and Solar Orbiter loca- \ntions, presented in Figs. 4 and 5. Figure 3 shows how ENLIL simulates an earlier arrival of the ICME at Earth than at Solar Orbiter, being a flank arrival for both locations, as indicated with the yellow circle in the figure. This relative configuration of the ICME is represented in Fig. 13 by the blue shading. The ICME passed left the near-Earth (green circle) environment a few hours before arriving to Solar Orbiter (blue square) located at 0.96 au. \nFigure 19 shows the pitch-angle distribution function of solar wind electrons with energies between 69 eV to 5 keV, built from Solar Orbiter SWA-EAS and MAG observations between 09:00 on 2022 January 19 and 18:00 on January 20. From the start of the shown interval, at 09:00 on January 19, to 18:00 on the same day, there is a clear beam flowing anti-parallel to the magnetic field (peak at pitch angle 180) and a faint beam flowing parallel to the magnetic field (secondary peak at pitch-angle 0). From 18:00 on January 19 until 11:00 on January 20, we observe only the anti-parallel beam signature, and then, we have a signature of a bi-directional beam, lasting for 4 hours. After 15:00 on January 20, the pitch-angle distributions do not exhibit clear beam signatures. In panels b, c, and d, we show three 2D speed, pitch-angle distribution functions, averaged over 70 minutes intervals within the three time periods characterized by di ff erent electron distribution function properties. The distribution in Fig. 19b, is stretched towards the antiparallel direction, as there is a clear antiparallel beam, dominating over a much more 'faint', and broad parallel beam. The distribution in 19c , which is cotemporal with the SEP onset, shows only the antiparallel beam. The distribution in 19d has a signature of a bi-directional beam, resulting in a highly anisotropic distribution function. \nTherefore, the PAD function of the solar wind electrons is in agreement with the following. (1) A flank arrival of an ICME to Solar Orbiter from the beginning of the shown interval ( ∼ 9 \nUT on January 19) until 15 UT on January 20, with the presence of a flux-rope structure with both legs still connected to the Sun between 11:50 and 16:00 UT on 2022 January 20; (2) At the time of the SEP onset the solar wind particles were propagating towards the Sun, as the pitch-angle is 180 and the local IMF vector was pointing outwards (Fig. 4b). This is congruent with the anti-sunward flux observed in energetic particles measured by Solar Orbiter; (3) At the time of the SEP onset, the eastern leg of the ICME passing through Solar Orbiter is disconnected from the Sun, as we observe only the anti-parallel beam signature. \nThe ICME reconstruction using the EC analytical model (Nieves-Chinchilla et al. 2018) is shown in Appendix E. The results are not coherent, as the central magnetic fields are pointing to opposite directions at Solar Orbiter and near Earth. This result could be related to the flank nature of the encounter at both locations, and / or the potential deformation of the shape of the ICME in the heliosphere during propagation.", '6. Tracing the interplanetary propagation of the energetic particles': "Assuming that the energetic particles observed by Solar Orbiter are injected inside the western leg of the ICME on 2022 January 16, as discussed below, we could estimate the field line twisting taking into account the energetic particle timing. The level of magnetic fluctuations inside MCs is generally lower than in the solar wind (Dasso et al. 2005) and therefore energetic particle propagating inside MCs tend to have long mean-free paths. Based on Kahler et al. (2011b), using a simple cylindrical fluxrope approximation, the number of field rotations N from the Sun to 1 au is given by \nN = 1 2 π X R r L 2 X 2 -1 , (4) \nwhere L is the total field line length, R is the radius of the flux rope and X is the axial field line length. The VDA in Sect. 3.4.1 provides an estimation of the field line length of L = 2.6 ± 0.1 au. Using the GCS analysis (Sect. 5.1) extrapolated to Solar Orbiter's location and taking into account that the CME did not centrally sweep over the spacecraft, the axial field line length of the western (longer) leg when it reaches Solar Orbiter can be estimated as 2.07 au, with an estimation of the total loop length at 0.92 au of 3.21 au. Assuming R / X ∼ 0.05-0.3 (Kahler et al. 2011b), Eq. 4 gives an estimation of N ∼ 0.4-2.74 turns along the longer leg of the ICME. Hereafter we use 'longer leg' to denote the most distant ICME leg relative to Solar Orbiter, which connects the spacecraft to the Sun from the anti-sunward direction. This results is in agreement with the values found by Kahler et al. (2011b), namely 1-10 turns along the full MC length. These low number of field line rotations is represented in Fig. 13 by the purple line winding around the main axis depicted in black. The orientation of the magnetic field line is indicated in agreement with the in situ observations (local IMF vector pointing outwards at Solar Orbiter location), as presented in Table 1, and discussed in Sect. 3.3.2.", '7. Summary and discussion': 'On 2022 January 20, Solar Orbiter observed a SEP event showing strong sunward-directed beams for the first arriving particles, as presented in Figs. 6 2 and 7 2. The presence of velocity dispersion evidenced a solar origin, confirmed by radio and \nremote-sensing observations. Solar Orbiter was located at 0.92 au and 18 · eastwards of near-Earth spacecraft, which measured usual antisunward-directed particles. At the time of the SEP onset, based on solar wind and magnetic field signatures discussed in Sect. 3.2, Solar Orbiter was crossing the eastern flank of an ICME present in the heliosphere that erupted from the Sun four days earlier on 16 January from the same active region as the one related to the SEP event. This ICME is well simulated by ENLIL model, as shown in Fig. 3. An IP shock is impacting the spacecraft at 08:02 UT on January 19 and an MC arrives at 03:28 UT on 20 January, just before the particle onset, being observed until 17:52 UT. This ICME had passed the near-Earth environment a few hours before the SEP onset (shown in Fig. 13) and did not appear to a ff ect the particle propagation. However, this ICME could still have played a role in forming the overall SEP pitch-angle distributions, which showed distinct discontinuities between the two pitch-angle hemispheres as discussed in Sect. 3.3 and Appendix B. The SEP event was widespread in the heliosphere, as it was observed by Solar Orbiter, near-Earth spacecraft, STEREO-A, and MAVEN, namely spanning at least a longitude of ∼ 160 · , as presented in Fig. 1. The solar source related to the SEP event was located at AR 12929, close to the west limb as observed by Earth at the time of the particle onset. \nAn M5.5 flare was observed erupting from AR 12929 located at Stonyhurst N08W76 and peaking at 05:58 UT on 2022 January 20. The flare was characterized by multiple discrete episodes of energy release and particle acceleration. Several nonthermal HXR peaks were observed being co-temporal with TIIs and TIIIs radio bursts, the latter starting at 80 MHz, as discussed in Sect. 4.2. The spectrum of the injected electrons deduced from the HXR observations in Sect. 4.1 is softer than the insitu spectra discussed in Sect. 3.6, namely δ ≥ 4 as opposed to δ ≈ 2 . 5 -2 . 8, summarized in the second column of Table 3. We note that in itself, this mismatch does not rule out a flare-related origin of the interplanetary electrons, as similar relations were found to be typical for impulsive electron events where no CME nor CME-driven shock was present. \nThe overlying loop of the eruption is pushed upward to form the leading front of a CME at 05:50 UT on 2022 January 20. The CME eruption leads to the formation and propagation of an EUVwave on the solar surface (shown in figure 1 of Zhang et al. 2022), clearly visible from 05:52 to 06:09 UT on 2022 January 20 in AIA images with a speed of 373 km s -1 . The EUV wave intersects at 05:55 UT ± 1 min for the first time the centroid of the west dimming lobe from the CME erupting on January 16, passing Solar Orbiter at the time of the SEP event. This time is cotemporal with the inferred injection time of the particles (05:56 UT ± 4 min) observed by Solar Orbiter based on the VDA analysis presented in Sect. 3.4. A CME-driven shock was observed early at 05:55 UT. As discussed in Sect. 4.3.2, the first time the 3D shock intersects the magnetic field lines based on PFSS model connecting to both near-Earth spacecraft and STEREOA is 06:00 UT ± 1 min. This timing is also in agreement with the injections times derived for STEREO-A (05:52 UT ± 8 min) and near-Earth spacecraft (06:02 UT ± 4 min). Regarding connectivity, there is a good agreement between the PFSS model and observations except for Solar Orbiter. The polarity of the source region from where the particles were ejected is negative based on near-Earth and STEREO-A in situ data, but positive based on Solar Orbiter in situ observations. This is in agreement with the particles observed by Solar Orbiter propagated inside the previous ICME erupting on January 16. \nRadio observations of the SEP event shown in Sect. 4.3.2 suggest that a shock wave formed in the low corona since TIIa \nstarted at 05:55 UT. The shock speed estimated from TIIa ( ∼ 340 km s -1 ) is in agreement with the speed of the EUV wave (373 km s -1 ) observed starting at 05:52 UT, discussed above. A second Type II, TIIb was also identified and featured herringbone (HB) structures which are observed for a brief period of time between 05:57 and 06:00 UT. The co-temporal occurrence of HB and TIII radio bursts seem to suggest that they may be physically related such that some TIII may emanate from TIIb. That is, the electron beams generating HB may also generate TIII radio bursts. The TIIb is observed for about ten minutes and approached the decameter wavelengths ( ∼ 16 MHz) at 06:05 UT. The shock speed estimated from TIIb (1400 km s -1 ) is also in agreement with the 3D CME-driven shock speed (1433 km s -1 ) estimated from coronagraph data (Sect. 4.3.2). The coronal TII radio bursts provide evidence that the shock was a significant particle accelerator, further supported by the co-temporal occurrence of TIII bursts and the most intense part of TIIb, known as HB bursts, which also align with the solar release time of the energetic particles. \nThe PAD in panel 2 in Figs. 6 and 7 shows that particles arriving to Solar Orbiter propagated mostly anti-parallel to the magnetic field direction (distribution peaking at pitch-angle 180 · ). The local magnetic field vector at the time of the SEP onset is pointing outwards at Solar Orbiter, indicating that the energetic particles propagated towards the Sun. This is in agreement with the PAD of the solar wind electrons. Fig. 19 shows that at the time of the SEP onset only the anti-parallel beam is observed. This agrees with the solar wind electrons propagating towards the Sun inside the western (longer) leg of an ICME connected to the Sun and the eastern (shorter) leg being disconnected. In the case of STEREO-A and near-Earth spacecraft, the analysis of the anisotropies in Sect. 3.3 correspond to particles propagating from the Sun. \nWe also used the VDA analysis to estimate an e ff ective particle propagation length of L = 1.4 ± 0.1 au for near-Earth observers, which is close to the nominal Parker spiral length for near Earth ( ∼ 1.08 au) using the measured solar wind speed. However, in case of Solar Orbiter the e ff ective length travelled by the particles is estimated to be L = 2.6 ± 0.1 au, much longer than the length of ∼ 0.99 au expected for a nominal Parker spiral field with the measured solar wind and scatter free propagation. It might indicate a non-standard interplanetary magnetic field topology. This long e ff ective path agrees with the particles propagating inside an ICME to arrive to the Solar Orbiter location. We note that STEREO-A also presents a long e ff ective path length of L = 2.3 ± 0.5 au. We speculate that the presence of the previous ICME could pushed the field line leading to STEREOA out of the ecliptic making it longer. However, we note the relatively poor pitch-angle coverage and the high uncertainty of the path length. Further analysis might be needed to discuss the apparently long path followed by the particles to reach STEREOA, which is out of the goal of this study. \nFrom this VDA analysis we derived also the estimated injection time of the particles, as discussed above, being similar within uncertainties for the three spacecraft near Earth, Solar Orbiter, and STEREO-A. This timing is summarized in Fig. 14 and Table 2 to compare with radio and HXR signatures presented above. For the three spacecraft, the estimated injected times are co-temporal with the presence of nonthermal HXR peaks, Type IIs, HBs, and Type IIIs starting at 80 MHz. \nWe determined the electron peak spectra, as observed by Wind, Solar Orbiter, and STEREO-A, as summarized in Table 3. For comparison, we selected the spectral index near 200 keV, namely δ 200. The δ 200 indices found in this study -2.8 < δ 200 \n< -2.5 are similar to the study by Dresing et al. (2022). They derived a mean spectral index of ⟨ δ 200 ⟩ = -2.5 ± 0.3 analysing 33 large gradual SEP events that were related to coronal pressure waves. Moreover, the elemental composition measured by EPD / SIS and EPD / HET are typical of large gradual SEP events (Desai & Giacalone 2016; Cohen et al. 2021). \nAt Solar Orbiter, based on several indicators, namely (1) the energetic and solar wind particles propagating towards the Sun, (2) the solar origin of the energetic particles based on VDA, (3) the anisotropy pattern, (4) the long e ff ective path, (5) the presence of the ICME that erupted on January 16 at the location of Solar Orbiter, (6) and the early connection of the EUV wave with the west lobe of the ICME, we argue that the energetic particles of the SEP event on 2022 January 20 propagated inside the ICME that erupted on January 16 and arrived to Solar Orbiter, travelling along the longer (western) leg of the ICME. \nThis configuration is shown in Fig. 13 that shows the sketch of the interplanetary configuration of the 2022 January SEP event. STEREO-A and Earth are connected to the solar source through the nominal Parker spirals indicated with the dashed coloured lines using the measured solar wind speeds. We note that STEREO-A, being located to the east of Solar Orbiter, is estimated to be magnetically connected to the same region as near-Earth spacecraft, based on the PFSS model (not shown in Fig. 13). The injection times of both spacecraft are in agreement with the intersection times of the respective magnetic field lines, based on the PFSS model, and the reconstructed CME-driven shock, as discussed above. \nSolar Orbiter is embedded in an ICME, shown with a blue shading, with the axial magnetic field line in black. The longer leg of the ICME was still anchored to the solar surface, based on the solar wind electron PAD. The solar source identification in Sect. 4 indicates that such leg is connecting to AR12929, and connected to the EUV wave and CME-driven shock related to the particle event, indicated as a red curve. The winding of the magnetic field lines of the magnetic flux rope is found to be of moderate size. The number of magnetic field turns in the MC structure inferred using the particle timing is below 6. The calculated particle path is around 30% longer than the modelled lengths of the loop legs. This result is in agreement with previous observations of energetic particles inside ICMEs (e.g. Kahler et al. 2011a; Dresing et al. 2016; Palmerio et al. 2021). We note that the ICME shows evidence of deformation of the front, since the ICME arrived at 1 au (Earth) before arriving at Solar Orbiter, located at 0.92 au. These observations support the importance of considering ejecta as irregular or deformable structures rather than "rigid" bodies and their propagation direction can be significantly influenced by the ambient solar wind (Wang et al. 2004; Rodríguez-García et al. 2022). \nThe injection into both inside the western loopleg -Solar Orbiter-, and outside -near-Earth and STEREO-A- of the magnetic cloud requires an extended injection region that is most likely provided by the associated coronal shock, which is indicated by the associated TII radio burst, shown in Fig. 14. However, diverging magnetic field lines in the low corona could also provide this extent. A further analysis of the event, showing (1) connection to the shock in agreement with the particle injection time; (2) harder electron spectra of the measured in situ energetic particles as compared to those reconstructed from flare HXR spectra; (3) hard electron in situ spectra similar to events related to coronal pressure waves; (4) particle composition typical of large gradual events; and (5) the presence of TIII bursts starting at 80 MHz and being co-temporal with the TII, indicates that \nthe main accelerator of the particles might be the CME-driven shock.', '8. Conclusions': "This work illustrates how important is the preconditioning of the heliosphere and the interplanetary magnetic field in the transport and spread of SEPs. Our main conclusions can be summarized as follows: \n- · Solar source: The solar source associated with the widespread SEP event on 2022 January 20 is likely the shock driven by the CME eruption observed near the west side from Earth's perspective.\n- · Particle injection: The energetic particles are injected over a wide angular region into and outside of a previous MC ejected on 2022 January 16 present in the heliosphere at the time of the particle onset on January 20. The sunward propagation particles measured by Solar Orbiter are produced by the injection of particles in the longer (western) leg of the MC, which is still anchored to the Sun. \nAcknowledgements. The UAH team acknowledges the financial support by the Spanish Ministerio de Ciencia, Innovación y Universidades FEDER / MCIU / AEI Projects ESP2017-88436-R and PID2019-104863RBI00 / AEI / 10.13039 / 501100011033 and by the European Union's Horizon 2020 research and innovation program under grant agreement No. 101004159 (SERPENTINE). ND is grateful for support by the Research Council of Finland (SHOCKSEE, grant No. 346902). ND and CP acknowledge funding by the European Union's Horizon Europe research and innovation program under grant agreement No. 101134999 (SOLER). LAB acknowledges the support from the NASA program NNH17ZDA001N-LWS (Awards Nr. 80NSSC19K0069 and 80NSSC19K1235). EP acknowledges support from NASA's LWS (grant no. 80NSSC19K0067) and LWS-SC (grant no. 80NSSC22K0893) programmes. AK acknowledges financial support from NASA NNN06AA01C (SO-SIS Phase-E) contract. ICJ acknowledges the support of Academy of Finland (SHOCKSEE, grant 346902). JG acknowledges the National Natural Science Foundation of China (42188101, 42074222 and 42130204). COL acknowledges support from the NASA LWS program (grant no. 80NSSC21K1325) and the MAVEN project funded through the NASA Mars Exploration Program. The authors acknowledge the di ff erent SOHO, STEREO instrument teams, and the STEREO and ACE science centers for providing the data used in this paper. Solar Orbiter is a space mission of international collaboration between ESA and NASA, operated by ESA. This research has used PyThea v0.7.3, an open-source and free Python package to reconstruct the 3D structure of CMEs and shock waves (GCS and ellipsoid model). ENLIL simulation results have been provided by the CCMC at NASA Goddard Space Flight Center (GSFC) through their public Runs on Request system ( http://ccmc.gsfc.nasa.gov ; run ID Laura\\_Rodriguez-Garcia\\_121523\\_SH\\_1). The WSA model was developed by N. Arge, currently at GSFC, and the ENLIL Model was developed by D. Odstrcil, currently at George Mason University.", 'ORCID iDs': 'Laura Rodríguez-García \nhttps: // orcid.org / 0000-0003-2361-5510 \nRaúl Gómez-Herrero https: // orcid.org / 0000-0002-5705-9236 \nNina Dresing https: // orcid.org / 0000-0003-3903-4649 \nLaura A. Balmaceda https: // orcid.org / 0000-0003-1162-5498 Erika Palmerio https: // orcid.org / 0000-0001-6590-3479 \nAthanasios Kouloumvakos \nhttps: // orcid.org / 0000-0001-6589-4509 \nFrancisco Espinosa Lara \nhttps: // orcid.org / 0000-0001-9039-8822 \nChristian Palmroos https: // orcid.org / 0000-0002-7778-5454 \nImmanuel C. Jebaraj https: // orcid.org / 0000-0002-0606-7172 \nAlexander Warmuth \nhttps: \n// \norcid.org \n/ \n0000-0003-1439-3610 \nGeorgios Nicolaou https: // orcid.org / 0000-0003-3623-4928 \nIgnacio Cernuda https: // orcid.org / 0000-0001-8432-5379 \nTeresa Nieves-Chinchilla https: // orcid.org / 0000-0003-0565-4890 Annamaria Fedeli https: // orcid.org / 0000-0001-9449-4782 Christina O. Lee https: // orcid.org / 0000-0002-1604-3326 Christina M. S. Cohen https: // orcid.org / 0000-0002-0978-8127 0000-0003-2169-9618 \nJingnan Guo https: // orcid.org / 0000-0002-8707-076X Timo Laitinen https: // orcid.org / 0000-0002-7719-7783 Glenn M. Mason https: // orcid.org / George C. Ho https: // orcid.org / 0000-0003-1093-2066 Olga Malandraki https: // orcid.org / 0000-0002-4751-6835 Rami Vainio https: // orcid.org / 0000-0002-3298-2067 Javier Rodríguez-Pacheco \nhttps: // orcid.org / 0000-0002-4240-1115', 'References': "Acuña, M. H., Curtis, D., Scheifele, J. L., et al. 2008, Space Sci. Rev., 136, 203 Al-Haddad, N., Nieves-Chinchilla, T., Savani, N. P., et al. 2013, Sol. Phys., 284, 129 \n- Altschuler, M. D. & Newkirk, G. 1969, Sol. Phys., 9, 131\n- Badman, S. T., Carley, E., Cañizares, L. A., et al. 2022, ApJ, 938, 95\n- Band, D., Matteson, J., Ford, L., et al. 1993, ApJ, 413, 281\n- Benkho ff , J., Murakami, G., Baumjohann, W., et al. 2021, Space Sci. Rev., 217, 90\n- Benz, A. 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ENLIL-modeled CMEs have an artificially higher thermal pressure to compensate for the lack of a strong magnetic field (Odstrcil et al. 2004, and references therein). To improve the characterization of the heliosphere, multipoint coronagraph observations are used to infer CME parameters, using the GCS model described in Sect. 4.3.1. The inner boundary condition is given by the Wang-Sheeley-Arge (WSA) V5.2 model, using inputs from the standard quick-reduce zero-point corrected magnetograms from GONG (GONGZ), available on the National Solar Observatory website 8 . The reliability of the CME arrival predictions depends strongly on the initial CME input parameters, such as speed, direction, and width (e.g. Mays et al. 2015; Kay et al. 2020; Palmerio et al. 2022), but also on the errors that can arise in the ambient model parameters and on the accuracy of the solar wind background derived from magnetograms and coronal field modelling assumptions (e.g. Lee et al. 2013; Jin et al. 2022; Ledvina et al. 2023). Based on Wold et al. (2018), the mean absolute arrival-time prediction error in ENLIL is expected to lie around 10.4 ± 0.9 hours, with a tendency to an early prediction of -4 . 0 hours. \nThe preconditioning of the heliosphere and the interaction of the IP structures that might be present at the onset time can actively influence the magnetic connectivity of the di ff erent spacecraft. Therefore, the ENLIL simulation time ranges from January 15 to January 25 (i.e. from five days before to five days after the SEP event). This interval encompasses the possible previous CME that may influence the particle propagation at the onset time, and the ICME evolution through the IP medium up to 2.1 au. For this purpose, the GCS 3D reconstruction process presented in Sect. 4.3.1 was also used for the ten relevant prior CMEs erupting in the time range of January 15 to January 20. We used the CME LE parameters (position and speed) rather than the bulk (bright core, if present) as they often capture the overall and great impact of the high-pressure structures better. The CME and model set-up parameters, and the results of the simulations are available on the Community Coordinated Modeling Center (CCMC) website. 9', 'Appendix B: Additional SEP pitch-angle distributions': 'Figure B.1 shows the electron PAD as measured by Solar Orbiter (top) and Wind (bottom). The top panel shows the reconstruction of the electron pitch-angle distribution as observed by Solar Orbiter in the energy range of 30 to 50 keV, using data from EPT and STEP. EPT provides wider pitch-angle coverage while STEP, with its segmented detector, o ff ers finer resolution. The top-right panel shows three slices through the pitch-angle distribution at di ff erent times during the event, indicated by labels 1, 2 and 3 on the top-left panel. During the prompt phase of the event (label 1), the pitch-angle distribution exhibits a discontinuity around 60 · . As the event progresses (labels 2 and 3), the distribution becomes more isotropic. This might be related to the particle population streaming from the anti-Sun direction into the backward pitch-angle hemisphere uniformly, not filling the whole pitch-angle range evenly with particles. \nThe lower panel of Fig. B.1 shows respectively from top to bottom the electron intensity in the eight sectors of Wind / 3DP, corresponding pitch angles of the bin centers, combined pitch-angle distribution with electron intensities marked by color-coding, magnetic field magnitude and RTN-components, magnetic field latitudinal and azimuthal angles, and first-order anisotropy. On the right we show the 2-dimensional pitch-angle distributions at the times marked by vertical lines in the plot on the left. These show that the particle beam forms a plateau over a µ -range from -1 to -0.5, but not until µ = 0. Therefore, there is a high di ff erence between the two hemispheres in pitch angle space. The presence of the ICME that was ejected on January 16 could be a reason of this discontinuity in the pitch-angle hemispheres, as it might block part of the backward streaming SEP distribution.', 'Appendix C: VDA analysis: definition and methods used': 'Under the assumptions of being injected simultaneously and propagate scatter-free and without adiabatic cooling, the onset times of the energetic particles follow a velocity dispersion pattern tonset ( v ) = tin j + L / ( c ∗ β ( v )), where tin j and tonset are the SEP injection time at the Sun and observation time at the spacecraft, respectively, L the e ff ective path length, and β = v / c , where v represents the particle velocity (e.g. Vainio et al. 2013). Thus, when the onset times, determined at a number of energies, and plotted as a function of the reciprocal of the particle velocities at respective energies, the slope of a curve fitted to the data indicates the e ff ective path length L and the intercept with the y-axis gives the release time tin j .', 'Appendix C.1: Poisson-CUSUM-bootstrap hybrid method': 'The Poisson-CUSUM-bootstrap hybrid method finds distributions of particle onset times by taking random samples from the preevent background and mappping the CUSUM parameters (mean and standard deviation of the pre-event background) of the samples to the onset times. The modified hybrid method is explained in detailed in Palmroos et al. 2024 (under review in A&A). The method also applies this bootstrapping on the data while varying the integration time, in order to find the most probable onset time regardless of time resolution used, accompanied by the respective 95% confidence intervals. \nThe background window for Wind data from which the parameters for the hybrid method were calculated was set to 01:30-05:40 UT on 2022 January 20. This window starts after the previous ICME has left near-Earth spacecraft and the elevated electron levels due to ion contamination decreased, as shown in Fig. 4 left (blue shaded area and horizontal line in panel 1). For ERNE protons \nwe used a background window from 16:00 UT on 2022 January 19 to 05:00 UT on January 20. We note that ERNE protons were not a ff ected by the previous ICME that arrived to Earth, as discussed above. We used the proton channels between 13 and 50 MeV, where velocity dispersion was observed in the onset times and the peak-to-background intensity ratios were 860-3890.', 'Appendix C.2: Sigma-threshold-bootstrap method': "We used the following procedure to estimate the onsets: (1) We integrated the intensities of pairs of consecutive energy channels to enhance the statistics; (2) For each pair of channels and using the time series of the particle intensities, we defined a sliding window of 9 minutes width, in which we averaged the intensity (mean) and calculated uncertainty using the error propagation (sigma); (3) We defined a threshold value calculated as the mean value plus 4 sigma above the background level; (4) If the intensities of the five following time stamps after the window were above the threshold value, we considered the first one as an onset candidate; (5) The sampling window advances one time step and the onset condition is tested again. This is repeated until the end of the time series; (6) To avoid choosing a candidate within the background level, we added the restriction in which for each onset candidate, the following consecutive two time stamps should be also onset candidates. Thus, we created a series of new onset candidates that fulfills the aforementioned restriction; (7) Choosing the first one as the final onset. \nWe define the lower and higher uncertainties for the x-axis by looking at the time series and taking into account the di ff erent scenarios depending on the background level, statistics and rising phase. In case of channels with previous background almost nonexistent (it usually happens at higher energies), we sometimes find by eye a few counts before the onset which are very likely onset candidates but they are not found by the method. We consider the earliest of these counts as the lower uncertainty, while the upper uncertainty is the time resolution of the time series. Other cases are high background level and / or slow rising phase. In one of these two cases or combination of both, we consider the lower uncertainty as the earliest time when we see by eye that the SEP event start to increase but still not detected by the method, while the upper uncertainty is considered as the point where the increase is very clear to have started due to its steepness. \nTo determine more reliable mean values and uncertainties for the path length and injection time, we used bootstrapping. In this process, for each pair of channels, we modify randomly the value of the onset as one of these three: the value calculated by the method described above, and the lower and upper limits of the value based on the uncertainties. Moreover, we deleted a random number of onsets between zero and four. Then we did the fit with ODR and repeated this process 10.000 times to obtain the Gaussian distributions for the path length and injection time. We considered the mean of these Gaussians as the final values for path length and injection time. For the uncertainty we multiplied the standard deviation by the Student's t for a confidence level of 95%.", 'Appendix D: SIS spectra and spectrograms': 'We fitted the spectra in Fig. 11 with the 2-slope Band functional form which has been used in surveys of large SEP events (Band et al. 1993; Desai et al. 2016; Mewaldt et al. 2012). The fit coe ffi cients are listed in Table D.1, following the notation used by Desai et al. (2016). In the table, column 2 is a normalization constant C, columns 3 and 4 are γα and γβ , the low and high energy power law indices, and column 5 is the spectral break energy in MeV / nucleon. The values from the spectral fittings are similar to the survey results of Desai et al. (2016), for example their mean values for O measured in 36 events was γα = 1 . 21 ± 0 . 10, and γβ = 3 . 74 ± 0 . 17. The Oxygen γα in Table D.1 is higher than the mean shown by Desai et al. (2016), but lies with the distribution of results from their survey (Desai et al. 2016, see their Figure 4(b)). The Oxygen γβ in the Table D.1 is close to the mean by Desai et al. (2006) survey. The spectral break energies in Table D.1 decreases with increasing particle mass, as observed in previous studies (Cohen et al. 2021; Desai et al. 2006; Mewaldt et al. 2005). The fits to H, 4He, O, and Fe are shown as dotted lines in Fig. 11. Li et al. (2009) modeled the energy dependence of spectral breaks, finding that a dependence on the break energy can be ordered by (Q / A) α where Q is the ion charge state and A is the atomic number, and α depends on the shock geometry. The partially ionized state of elements O and above leads to a decrease in the break energy. \nFigure D.1 shows spectrograms for H and 4 He for the sunward- and anti-sunward pointing SIS telescopes. At energies above a few MeV / nucleon this event showed highly unusual intensities wherein the anti-sunward telescope intensities exceeded those of the sunward looking telescope. The implications of this are discussed in Sect. 7. At energies below ∼ 1 MeV / nucleon the intensity variations were typical for large SEP events with an initial large (factor of 10 or more) sunward / anti-sunward anisotropy that decayed after the initial rise phase of the event. \nTable D.1. Band spectral fit parameters', 'Appendix E: January 16 ICME reconstruction': "Table E.1. EC model fit parameters in RTN coordinates \nNotes. Column 1: Spacecraft. Column 2: MFR axis longitude ( ϕ = [0...360] · ). Column 3: inclination of the flux rope with respect to the equatorial plane ( θ = [-90... + 90] · ). Column 4: MFR rotation about its central axis ( ξ = [0...180] · ). Column 5: MFR distortion (ratio between major and minor ellipse axis, δ = [0...1]). Column 6: MFR size. Column 7: distance from the spacecraft trajectory to the MFR axis (negative value means that the spacecraft is crossing the upper part of the structure). Column 8: goodness of the fitting ( χ 2 = [0...1]). Column 9: MFR handedness. Column 10: average solar wind speed used for the fitting. \nSeveral models for reconstructing MCs from in-situ observations have been established, such as the concept of a flux rope in a force-free configuration (Burlaga 1988; Lepping et al. 1990) or models that relax the force-free conditions (e.g. Owens 2006). Nieves-Chinchilla et al. (2018) developed the Elliptical-Cylindrical analytical MFR model for MCs (hereafter the EC model) as an approach to consider the distorted cross-section of the magnetic field topology as a possible e ff ect of the MFR interaction with the solar wind. However, all the models describe a limited subset of the properties of an MC as they are based on one-dimensional measurements along a line cutting through the structure, and it is not uncommon for di ff erent reconstruction techniques to display discrepant results (e.g., Al-Haddad et al. 2013; Lynch et al. 2022). \nThe analytical MFR model or EC model was applied to reconstruct the MC present within the ICME at the locations of Wind and Solar Orbiter. The MC reconstructions are local, based on the magnetic field measured in situ at each location. The EC model assumes an MFR magnetic topology, that is, an axially symmetric magnetic field cylinder with twisted magnetic field lines of elliptical cross-section. Therefore, the EC model allows us to consider cross-section distortion as a consequence of the interaction of the flux rope with the solar wind. The MC time intervals chosen for the EC model analysis correspond to the blue shadings in Fig. 4 left (Wind) and Fig. 5 left (Solar Orbiter). Column (10) in Table E.1 shows the average solar wind speed used for the fitting. The trajectory of the spacecraft through the MC is inferred by using the minimization of the χ 2 function to obtain a set of parameters that best fit the measured data (Nieves-Chinchilla 2018). Table E.1 lists the obtained χ 2 function and the EC model fit parameters in RTN coordinates. The MFR orientation in space is given by three angles: the central magnetic field longitude, ϕ (equal to 0 · in the spacecraft-Sun direction), the tilt angle, θ (where positive values represent north of the equatorial plane), and the MFR rotation about its central axis, ξ . The geometry of the flux rope is given by the ratio between the major and minor ellipse axis, δ , and the size by the cross-section major radius, R. Y 0 is the impact parameter, which represents the closest approach to the MFR axis, where a positive value means that the spacecraft is crossing the lower part of the structure. Finally, the chirality or handedness of the flux rope is shown in Col. 9. In Fig. E.1, the magnetic field data from Solar Orbiter (left) and Wind (right) are shown, along with the EC model fitting (smooth pink lines). The changes in the magnetic field components are not well captured, especially at the rear part of the MC. \nAccording to Table E.1, Wind observes the MFR axis approximately between the perpendicular and the radial direction, based on the magnetic field longitude value ( ϕ = 214 · ), close to 270 · , while Solar Orbiter, with a longitude angle closer to 180 · ( ϕ = 157 · ) might observe the flux rope closer to a flank. We note that the central magnetic field are pointing to opposite directions. The tilt angle ( θ ) shows a di ff erence between the observatories, with a northwards tilt in Wind ( θ = 17 · ) and a southwards tilt in Solar Orbiter ( θ = -10 · ). The disagreement in the MFR rotation about its central axis, ξ , for Wind and Solar Orbiter means that the orientation of the ellipse's major axis is dissimilar in space. In the context of the other two angles, the respective ξ value of 7 · and 70 · for Wind and Solar Orbiter, means that the distorted structure is parallel and perpendicular to the spacecraft trajectory. \nThe radius of the MFR cross-section, R is higher at Wind location than at Solar Orbiter, which is expected due to the expansion of the structure. The closest distance to the MFR axis, Y0, is positive (negative) for both Wind (Solar Orbiter), so that Wind (Solar Orbiter) spacecraft would be crossing the lower (upper) part of the structure. Wind is crossing further to the MFR axis than Solar Orbiter. The chirality for Wind (Solar Orbiter) is positive (negative) corresponding to RH (LH) flux ropes. The fitting results based on χ 2 (Col. 8 in Table E.1) give satisfactory results. However, visual inspection of Fig. E.1, which shows the comparison between the fitting in pink and the magnetic field observations by Wind (left) and Solar Orbiter (right), and the final interpretation of the position of the clouds lead to nonphysical results. This is probably related to the boundaries selection for the fitting, the flank arrival of the cloud to both locations of Wind and Solar Orbiter, and the potential deformation of the shape of the ICME in the heliosphere during propagation in the heliosphere. \nFig. B.1. Pitch-angle space. Top : Reconstruction of the electron pitch-angle distribution as observed by Solar Orbiter in the energy range of 30 to 50 keV, using data from EPT and STEP. The top-right panel shows three slices through the pitch-angle distribution at di ff erent times during the event, indicated by labels 1, 2 and 3 on the top-left panel. EPT and STEP measurements are represented by filled squares and empty circles, respectively. Bottom : Left from top to bottom: electron intensity in the eight sectors of Wind / 3DP, corresponding pitch angles of the bin centers, combined pitch-angle distribution with electron intensities marked by color-coding, magnetic field magnitude and RTN-components, magnetic field latitudinal and azimuthal angles, and first-order anisotropy. Right: 2-dimensional pitch-angle distributions at the times marked by vertical lines in the plot on the left. \n<!-- image --> \nDate \nTime in year 2022 \nsolo, GSE System \nWind, GSE System", 'A & A proofs: manuscript no. main': 'Fig. D.1. Intensity spectrograms for H and He from the sunward and anti-sunward pointing telescopes, with the energy / nucleon scale multiplied by energy to increase the clarity of the higher energies. We note the anti-sunward telescope saw higher intensities than the sunward pointing telescope during the early portion of the event, as observed by the electrons discussed in Sect. 3.3. \n<!-- image --> \n6 = 0.788 \ny0 (%) \n6 = 0.704 \n78.2 [R = 0.096AU] \n<!-- image --> \nFig. E.1. Comparison of the EC model fitting results (pink) with Wind ( left ) and Solar Orbiter ( right ) magnetic field observations spanning the MC. From the top , the panels display the magnetic field strength and the three magnetic field BRTN components, respectively. \n<!-- image -->'}
2024ApJ...975..191W	A common envelope CE is proposed as the origin of the early postoutburst spectra of many novae. A simple model is proposed to explain the properties of the CE based on the emission line strengths and an assumed density distribution. Rapid changes in the spectrum during postoutburst decline are suggested as possible evidence for a CE. Timeresolved spectra from the ARAS group show sudden spectral shifts that are correlated with detected ray emission suggestive of its possible origin on the white dwarf that produces a change in condition within the CE. Episodic mass loss formation of transient heavy element absorption systems and dissipation of the CE may possibly be triggered by ray emission.	2024-11-01T00:00:00Z	['10.3847/1538-4357/ad7bae', 'arXiv:2409.09565', '2024ApJ...975..191W', '2024arXiv240909565W', '10.48550/arXiv.2409.09565']	['Novae', '1127', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena']	Common Envelopes Gamma Rays and Sudden Spectral Changes of Novae	2,024	194	0.51	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.09565.pdf	{'Robert Williams': 'Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218 Department of Astronomy & Astrophysics, Univ. of California/Santa Cruz, 1156 High Street, Santa Cruz, CA 95064', 'Russell Ryan': 'Space Telescope Science institute, 3700 San Martin Dr., Baltimore, MD 21218', 'Richard Rudy': 'Kookoosint Scientific, 1530 Calle Portada, Camarillo, CA 93010 \nAbstract : A common envelope (CE) is proposed as the origin of the early postoutburst spectra of many novae. A simple model is proposed to explain the properties of the CE based on the emission line strengths and an assumed density distribution. Rapid changes in the spectrum during postoutburst decline are suggested as possible evidence for a CE. Time-resolved spectra from the ARAS group show sudden spectral shifts that are correlated with detected g -ray emission, suggestive of its possible origin on the WD that produces a change in condition within the CE. Episodic mass loss, formation of thea transient heavy element absorption systems, and dissipation of the CE may be triggered by g -ray emission.', '1. Introduction': 'Early studies of the spectra of galactic novae were undertaken by Beals (1931), McLaughlin (1942), and Payne-Gaposchkin (1957), each of whom described the spectra in terms of distinct phases through which the spectra pass after outburst. Subsequently, atomic physics processes were incorporated into radiative transfer calculations to predict the spectra from an expanding gas, based on input parameters that could be obtained from photometry and time-resolved spectral observations (Sobolev 1960, 1969; Castor 1972). Because close mass transfer binaries are spatially unresolved with current instruments, one of the parameters most difficult to specify for outburst ejecta is their geometry. It is feasible to treat complex geometry when using Monte Carlo techniques to predict the resulting spectrum from assumed input parameters, however the problem is made more tractable by assuming a regular structure that can be solved analytically. \nStudies of individual novae have been successful in representing the ejecta by evolving winds (cf. Hauschildt et al. 1992, 1995) and discrete shells of gas ejected by the outburst, whose changing properties from expansion drive the evolution of its spectrum (Shore 2012; Shore et al. 2016; Hachisu & Kato 2022). Analysis of time-series high resolution spectra of novae during postoutburst decline, Aydi et al. (2020a, 2020b) has demonstrated that discrete episodes of ejecta that collide due to their different velocities explain key features of observed spectra. The above studies have all been successful in characterizing the spectra of selected novae. \nAn additional geometrical configuration that merits consideration in modeling the formation of novae spectra is that of a common envelope. Independent hydrodynamical \nstudies of the outburst have demonstrated that a circumbinary common envelope (CE) is very likely to form shortly after a nova outburst from the interaction of the WD ejecta with the secondary star (MacDonald, 1986; Livio et al. 1990; Kley et al. 1995; MacLeod et al. 2017; Ropke & De Marco 2023). Such an envelope would be a prominent contributor to the spectrum in the epoch of early decline. One of the key characteristics of an envelope is that, like a stellar wind, absorption and emission are linked through radiative transfer processes determined by conditions within the envelope. \nRecently, Williams et al. (2022) analyzed the photometry and spectra obtained during decline of the unusual nova V5856 Sgr/2016, which is unique in having remained close to its peak outburst luminosity for more than six years following outburst. Different aspects of its spectra were described in general terms for a common envelope that might explain the unusual behavior of the nova after outburst. However, whereas the models of winds and expanding ejecta that Hauschildt, Shore, Aydi and associates have calculated have been used in the interpretation of novae spectra, little has been done to calculate the expected spectrum of a CE after a nova outburst. We propose here a simplified structure for a CE that could serve as a starting point for treating emission and absorption processes, and that might explain some of the observed spectral characteristics after outburst. We focus on the visible region of the spectrum because that is where the majority of time-resolved data exist.', '2. Common Envelope Geometry and Emission': "Shortly after outburst the visible spectra of most novae are characterized by a continuum with superposed emission lines, some of which have blueward absorption components, i.e., classic P Cygni profiles. The absorption features can be understood as arising from absorption and re-distribution of continuum radiation by an optically thick transition in an expanding gas (Sobolev 1960; Mihalas 1978). If the emission component exceeds the absorption component, creation of photons for that transition, rather than scattering of continuum radiation, is indicated. For a centrally energized, coherent mass of gas the emission lines are formed above the continuum region, where the photosphere radius r ph is defined by where t cont » 1 along the line of sight into the object. This configuration is very similar to that of a normal stellar atmosphere, augmented by the presence of an optically thin 'chromosphere' overlying the continuum and absorption-line formation region. \nAbsorption lines are produced by an outwardly decreasing temperature gradient with a stellar density distribution determined by hydrostatic equilibrium (Mihalas 1978). Emission lines are produced by an extended, optically thin outer region that can be in the form of a stellar wind that originates in the outer layers of the photosphere, or a detached outer shell as in the case of Be stars. Such is the case for common envelopes, where in eruptive variables the outer layers experience mass loss and are not in hydrostatic equilibrium (Ivanova, Justham, & Ricker 2020). The emission line intensities depend upon the temperature and extent of the overlying optically thin gas, whereas the \ncontinuum brightness depends on the surface area and temperature of the photospheric region. \nWe propose a simple CE model that represents conditions in the postoutburst emitting regions of novae by an initially dense gas that expands. Conceptually, this geometry is similar to the winds treated by Hauschildt et al. (1992, 1994), however its structure corresponds more closely to the CE models that have been computed by Livio et al. (1990), Kley et al. (1995), and MacLeod et al. (2017), in that their density profiles are very different from those dictated by a wind that follows the equation of continuity for a positive velocity gradient. It should be noted that although the MacLeod et al. models represent a luminous red nova mass transfer binary, the primary star is not a degenerate WD. \nFor our assumed CE geometry, the luminosity of the nova is determined by the Planck function continuum intensity, where L cont µ r 2 ph T 4 ph , and r ph is the effective photosphere radius. The photosphere effective temperature T ph is set by energetics originating from the inner regions of the gas that produce an outwardly decreasing temperature gradient, thereby accounting for the formation of absorption lines. Simplifying our common envelope geometry in order to define the regions where absorption and emission lines predominate, we consider the effective photosphere region to be overlayed by an optically thin region, i.e., a chromosphere. This allows a basic relation to be established that connects emission line strengths to that of the continuum. Emission from the outer optically thin 'chromosphere' gas surrounding the photosphere consists predominantly of emission lines from ions i whose luminosities L em are \nL em µ h n i ò n e n i a i eff dV, (1) \nwhere V is the volume of the optically thin gas extending beyond the effective photosphere, n e and n i are the electron and ion densities, and a i eff is the sum of the recombination and collisional excitation coefficient of the emission line (Osterbrock & Ferland 2006). \nFor the outer optically thin region, rather than postulate its density distribution ad hoc we take the emission-line region to be an extension of the photosphere region by representing its density by a power-law distribution, \nn(r) = n ph ' (r ph /r) x , (2) \nwhere the effective photosphere density n ph and radius r ph are defined by the point where the continuum optical depth into the envelope at 5000 Å, t vis =1. Since r ph is defined by an integral over distance that involves temperature, ionization, and density, n ph cannot be specified independently of r ph . Self-consistent values of n ph and r ph need to be found iteratively, and may be done straightforwardly via the procedure described in Appendix A. The condition that we invoke for common envelopes, viz., that the density of the optically thin outer region of gas is related to the inner optically thick gas producing the continuum, i.e., not isolated from it, is for us a defining feature for a nova CE. It defines the general structure of the CE, and previous models of CE's associated \nwith mass transfer binaries have invoked this condition (Ivanova, Justham, & Ricker 2020; Ropke & De Marco 2023). \nThe effective photosphere region is defined where t vis =1 into the ejecta, with \n¥ t vis = ò r ph n(r) S x i a i dr , (3) \nwhere x i is the fractional abundance of ion i relative to the total density, and a i is the absorption cross section at 5000 Å . The spectra of novae in early decline normally exhibit low ionization species, e.g., Na o , O o , and Fe + , even when higher ionization and excitation transitions of H o and He o also appear. In the latter case, P Cygni profiles of the higher excitation species can show different radial velocity absorption features than the lower ionization species. Detailed calculations of ionization and temperature are necessary to determine the sources of optical depth into the CE. For the relatively dense region of the gas where the visible continuum and absorption lines are formed, we assume initially that local thermodynamic equilibrium (LTE) conditions apply. For densities taken from the calculations of novae CE models by Livio et al. (1990) and Kley et al. (1995), the continuous absorption coefficients expected to predominate in the visible are most likely H -bound-free, H o Balmer continuum, and H + free-free absorption, as has been found for the accretion disks and ejecta of cataclysmic variables with similar physical conditions (Williams 1980, Fig. 4; Metzger et al. 2021). \nThe H -cross section in the visible is a(H -) ~ 3 ' 10 -17 cm 2 (Wishart 1979), and the cross sections for both H o Balmer and H + free-free absorption are of order 6 ' 10 -18 cm 2 (Mihalas 1978). Much less certain are the relative abundances of the particles involved because the absorption coefficients depend upon the presence of free electrons, and these likely come from low ionization level metals. Analyses of novae continua in early decline normally indicate photospheric temperatures around 8,000 K (Bath & Harkness 1989), although this temperature is not necessarily applicable to the optically thin emission region. The excitation of H o to the n=2 level will produce H(n=2)/H £ 10 -3 unless the temperature exceeds 10 4 K. The consistent strengths of O I ll 7773 and 8446 in early decline suggests relatively low ionization of H and He, so the free electrons accessible to form H -are likely due primarily to Fe + , the most abundant of the heavy elements with low ionization potentials. Depending on the Fe/H abundance, an approximate value of H -/H £ 10 -3 may be typical. Thus, allowing for uncertainties in the temperature and ionization, whose values do require construction of detailed models, we adopt the combined fractional abundance of absorbers to be ~10 -3 of the total density, and the combined absorption cross sections in the visible to be <a i >=10 -17 cm 2 . \nUsing the density distribution from eqn. (2) and the approximate parameter values above, the t vis =1 requirement constrains the density-radius relationship for the effective photosphere to be, \nn \nph r ph » 10 20 ( x1). (4) \nFor a typical nova the Livio et al. and Kley et al. models show the densities of common envelopes to be highest by more than an order of magnitude within the radius of the secondary star photosphere. As the secondary sweeps around its orbit it therefore creates an effective photosphere for the CE at its orbital radius, i.e., with r ph » 3 ' 10 10 cm for a low mass main sequence star with an orbital period of some hours. Treating the CE as a single coherent entity rather than a collection of discrete clumps, this leads to a photosphere density of n ph » 10 10 cm -3 for low values of x <5. This density is too low to justify the assumption of LTE, and is slightly lower than that necessary to explain the frequently observed O I l 7773 emission line, whose excitation via H I Lyb resonance fluorescence of O I quintet levels requires a density of at least 10 10.5 cm -3 (Kastner & Bhatia 1995), suggesting that higher values of x are appropriate. In their model atmosphere calculations with a wind, Hauschildt et al. (1994) came to this same conclusion, i.e., a steep density gradient with x ≈ 15 was necessary for the outer optically thin gas in order to produce an acceptable fit to the nova V1974 Cygni/1992 early spectra. If a common envelope forms shortly after outburst, steep density gradients are required for the outer region of the CE in order to allow sufficiently high densities for continuum and absorption line formation with emission lines also prominent. \nIf the density of gas in the general region of the effective photosphere can be represented adequately by the eqn (2) power-law distribution, a straightforward relationship exists between the equivalent width of an emission line and the radius and density of the photosphere. Taking the Balmer H b line as an example, the equivalent width of the line, i.e., the intensity of the line compared with the continuum intensity, is defined as EW H b = L H b /(L cont Dn 1 ), where Dn 1 is the frequency width of a 1 Å interval at H b . The continuum luminosity is approximately L cont = 4 p 2 r ph 2 B n (T ph ), where B n is the Planck function, although it should be noted that at densities below ~10 13-15 cm -3 , the assumption of LTE is likely to break down and the continuum intensity will deviate from the Planck function. \nUsing eqn. (1) for the luminosity of H b in terms of its effective recombination coefficient a H b eff (Osterbrock & Ferland 2006) and the eqn. (2) power-law density distribution, the equivalent width can be expressed in terms of the temperature, radiation field, abundances, and ionization level of the gas, as \nEW H b = h n b x H 2 a b eff /[(2 x -3) p B n (T ph ) Dn 1 ] ' n ph 2 r ph , (5) \nwhere x H is the fraction of the density that is ionized hydrogen, H + . The n ph and r ph are self-consistent values of the photosphere density and radius that satisfy the requirement that t vis =1. For a power-law density distribution the integrand for optical depth in eqn. (3) is most dependent on the region having highest density, which is close to the photosphere. This results in a power-law density distribution strongly favoring emission lines being strongest near the photosphere continuum----a feature that may explain how \nearly decline spectra can change rapidly. The determination of emission line equivalent widths serves as an important test of the validity of a common envelope model.", '3. Time-Resolved Spectral Features': "Models based on different geometries have been proposed for novae, including discrete non-interacting clumps, collimated jets, and colliding shells. It is clear that separate components of ejecta do collide with each other in decline, producing shocks that power postoutburst activity (Steinberg & Metzger 2020). Spectra obtained when novae are in the Fe II spectroscopic phase frequently show lines having prominent, multiple absorption components with different radial velocities (Izzo et al. 2015; Aydi et al. 2020a). Multiple components having different velocities may be the result of episodic mass loss events from a common envelope as it dissipates into distinct ejecta components. \nWe have examined a collection of time-resolved novae spectra to look for characteristics that might be signatures of ejecta geometry. Over the past several decades two extensive databases of postoutburst novae spectra have been posted online. The first is F. Walter's SMARTS Atlas of Southern Novae 1 (Walter et al. 2012). The second is the world-wide volunteer Astronomical Ring for Amateur Spectroscopy 2 (ARAS) group (Teyssier 2019). Using different telescopes and instruments, they have amassed a useful collection of spectra that are publicly available. Walter and colleagues have used different telescopes and spectrographs on Cerros Tololo and Pachon in Chile. ARAS spectra have been obtained by many observers using different instruments and spectral resolutions and in varying photometric conditions. A unique value of the ARAS database is that with its numerous observers situated globally, there can be more than ten separate spectra of a nova obtained in a single night. \nOf the many characteristics displayed by novae spectra after outburst several may be supportive of their formation in a CE. The first is the presence in early spectra of Balmer and Fe II lines that have prominent P Cygni profiles, with both emission and absorption components having a range of strengths and widths. A substantial literature exists on the formation of P Cygni line profiles for different geometries and conditions (Rottenberg 1952; Lucy 1971; Klein & Castor 1978; Castor & Lamers 1979; & Hillier 1991). The calculations show such profiles to be a consequence of an optically thick expanding medium, with the profile shapes determined primarily by the assumed density and velocity laws with distance. In comparing the high-resolution profiles observed by Aydi et al. (2020a), displayed in their Figs. 1 and 2, with the P Cygni profiles computed in the above cited studies, all have shapes that can be accounted for by an expanding common envelope. Multiple absorption components are created by density peaks with different velocities, which may be indicative of multiple ejection or mass loss episodes. \nIn particular, for a nova CE after outburst that is not in hydrostatic equilibrium and likely to be losing mass, one does expect strong absorption to occur at velocities in the range of the CE escape velocity, which for a radius of 1 R ¤ with a 1 M ¤ primary WD should be in the range 4 -6 Í 10 2 km/s (Kley et al. 1995). Of the dozen novae displayed by Aydi et al., half of them do have absorption in this velocity range at the epochs displayed. Absorption in the other half occurs at higher velocities, which may mean that postoutburst activity on the WD has filled the CE with high-energy radiation and ejecta, causing sudden high-velocity mass loss from the CE. \nA second notable feature of postoutburst spectra that could be a favorable indicator of novae CE's is that fundamental changes in the basic spectra of novae occasionally occur quite rapidly, often in only a day or two. These changes take place generally in the period of early decline, primarily when the brightness of the nova is still within a few magnitudes of its peak visible luminosity. In these instances, the spectrum typically converts from predominantly emission lines to absorption lines that strengthen in a matter of a few days, and with the exception of H a , the emission lines fade into the continuum. Often the spectrum then reverts back to a similar emission-line configuration after an interval of a week or more. Rapid spectrum changes, which for convenience we will call shifts , contrast with the normal evolution of novae spectra after forbidden lines first appear, when change proceeds gradually via a regular evolution in the relative intensities of emission lines. \nDuring rapid shifts some novae change visible brightness, but there is no systematic trend evident in either the direction or extent of brightness variations during shifts for the majority of objects. In Figures 1-5 we show examples of novae whose spectra experienced relatively rapid shifts at a time the brightness was near peak visible luminosity. The spectra displayed are reproductions of FITS files posted on the public ARAS website, where information is also provided for each observation on the day it was obtained. The range in spectral resolution used for some of the displayed spectra may cause detection of some of the weakest, narrowest spectral features to be compromised in some spectra. The light curves of the five novae are displayed in Fig. 6, with data taken from the AAVSO website 3 . Vertical marks denote the dates when the spectra shown in the figures were taken. \nThere are some common features that novae exhibit related to shifts. Initially, the prominent features in the spectra are Balmer and Fe II multiplet 42 emission lines that often have P Cygni profiles with emission components dominant. That is, shifts normally occur when novae are in the 'Fe II' spectral phase (Williams 2012; Aydi et al. 2024). Within 1-3 days absorption lines appear and become stronger as the emission components diminish in strength, often leaving H a as the only prominent emission feature. \nExceptions do occur, e.g., the Balmer and Fe II lines for V1405 Cas retained their P Cygni profiles with strong absorption components before, during, and after the entire shift. Also, Nova V613 Sct deviates from the procedure described above in that the sudden transformation of its spectrum from predominant absorption lines to emission lines that occurred within 24 hours on 1 July was not preceded by an emission spectrum. This may be due to the fact that the nova was discovered after peak brightness. The 30 already declining light curve suggests that the nova was likely emitting an Fe II emission \nJune spectrum shown in Fig. 3 was taken just 30 hours after the nova's discovery, and the spectrum before its discovery. \nDuring spectral shifts the absorption lines tend to be the same transitions found in transient heavy element absorption ( thea ) systems (Williams et al. 2008), although the radial velocities of thea systems often differ from that of the strongest absorption component of lines having P Cygni profiles. Spectral shifts may be the origin of thea systems in most circumstances. The strong, dominant absorption components of Balmer and Fe II P Cygni profiles represent absorption by the expanding, dispersing common envelope. Thea systems consisting of transitions of Sr, Ba, Y, Ti and other Fe-peak elements may represent ejection events from the CE driven by episodic bursts of g -rays, strengthening their direct association with the secondary star. \nThe main phase for each shift is the interval where absorption lines are the dominant feature of the spectrum, which typically last of order 1-2 weeks. The spectrum then changes over an interval of a few days, back to a predominant emission spectrum that is similar to that which prevailed before the shift occurred. The most interesting aspect of spectrum shifts remains the suddenness with which the visible spectra of novae change their basic structure.", '4. Gamma Ray Emission and Spectrum Changes': "The cause of sudden spectrum shifts is clearly due to major changes in the emitting \nregion conditions. One possible source of energy that could initiate shifts is the emission of high energy g -rays that have been detected in postoutburst novae. Since its launch in 2008, the Large Area Telescope (LAT) on Fermi Gamma-ray Telescope has conducted a sky survey to identify sources that emit gamma rays. Of the roughly 165 Galactic novae that have been confirmed spectroscopically since launch, approximately 20 novae have been identified as sources of g -ray emission in the list maintained by K. Mukai 4 . Characteristics of the g -ray novae have been discussed by Franckowiak et al. (2018) and Chomiuk, Metzger, & Shen (2021), with the striking fact that most novae detections have been made just above the LAT detection limit. Detections have often required integration on the object field of view over several days in order to achieve sufficient signal-to-noise to be designated as statistically valid sources. The LAT detection limit \ndoes discriminate against the detection of g -rays from the more distant novae. \nObserved fluxes over the 20 MeV-300 GeV energy range of LAT for all detections represent a small fraction, <1%, of the radiant energy of the novae, which generally amounts to roughly the Eddington luminosity of a 1 M ¤ object. The g -ray luminosities of the detected novae show a wide range of values, such that a majority of novae could generate g -rays following outburst at luminosities up to 1% the luminosity of the nova, yet still remain undetected with current facilities (Franckowiak et al. 2018). \nA key correlation has been found between the g -ray intensity and optical brightness of novae (Li et al. 2017; Aydi et al. 2020b). For novae for which g -ray flux has been detected, the positive correlation with visible brightness has been suggested to be due to both g -rays and visible radiation being produced by interactions within the ejecta, so that 'the majority of the optical light comes from reprocessed emission from shocks rather than the white dwarf' (Li et al. 2017). The inevitable collision between ejecta components that have different velocities makes this hypothesis highly likely (Metzger & Pejcha 2017; Aydi et al. 2020a). \nIn addition to the above calculations that support the feasibility of ejecta shocks producing observed gamma rays, a correlation between spectrum changes and gamma ray intensity could be indicative of a more centralized source for the gamma rays. We suggest that consideration be given to the fact that postoutburst g -rays might originate from activity on or near the white dwarf. There has been no expectation that novae white dwarfs produce g -rays, but the same could be said of the sun, where they have been observed at energies exceeding 100 Gev. Linden et al. (2022) recently completed a survey of Fermi LAT solar observations over a complete 11 yr solar cycle, and found an anti-correlation between solar activity and g -ray emission that was interpreted in terms of solar magnetic fields producing the observed fluxes. Further analysis of the data by Arsioli & Orlando (2024) revealed an asymmetry in solar disk emission that further linked observed g -ray emission to solar magnetic fields, rather than the re-processing of galactic high energy cosmic rays. Banik et al. (2023) have proposed that acoustic-shock disturbances move upward in the solar atmosphere, producing the g -rays. Such a process could take place above the surface of the nova WD. \nThe magnetic fields of novae WDs are expected to be many orders of magnitude greater than that of the sun. In polars like AM Her, and novae as V1500 Cygni/1975 and DQ Her/1934, observations have demonstrated field strengths in excess of 10 7 gauss (Schmidt, Stockman, & Grandi 1983; Schmidt & Stockman 1991). Thus, centrally localized g -ray emission associated with activity on novae WDs does have credibility. It could ionize and heat the gas in the outer layers of the common envelope. Such a situation would also produce a positive correlation between g -ray and visible brightness. When incident upon the ejecta, whether in the form of a CE or discrete shells or globules, g -rays could definitely generate rapid spectroscopic changes because the small cross section of matter to g -rays allows them to reach every region throughout a CE and produce residual \nionization that can dictate both the position of and conditions in the effective photosphere and overlying emission-line region. \nAs a test of this hypothesis, we have checked for possible g -ray detections the novae shown in Figs. 1-5 that experienced rapid spectrum shifts: \n- 1. V5855 Sgr/ 2016c : After discovery on 20 October 2016, g -rays were detected between 28 October - 1 November 2016 (Li & Chomiuk 2016). It is clear from Fig. 1 that a major spectrum shift occurred early in decline, between 22-26 October 2016, shortly before the g -rays were detected. By 23 October the spectrum was entirely in absorption, with the exception of an H a emission component. The shift concluded with a very rapid reversal in 24 hours between 25-26 October, in which its absorption spectrum returned to being dominated by emission lines. The g -ray detection occurred within 2 days of the spectral shift. Therefore, the two events are possibly related, although not quite coincident with each other.\n- 2. V5856 Sgr/ 2016d : The spectral evolution of this nova was discussed in detail by Williams et al. (2022). As shown in Fig. 2, ARAS spectra show a spectrum shift in the interval 29 October - 18 November 2016. The initial spectra display emission with only a few absorption features present. Between 30 October - 9 November H a is the only emission feature. By 18 November a prominent emission spectrum has emerged. Li et al. (2016) detected g -rays with Fermi LAT from 8-17 November, during the time the spectrum was transitioning back to the emission spectrum. So, the spectral shift and g -ray emission for this nova did overlap in time.\n- 3. V613 Sct/ 2018 : Dominated by absorption lines on its 29 June 2018 discovery, the spectrum of this nova would be unusual if it were at peak brightness. But, because no pre-discovery observations of its immediate area of the sky were reported within weeks of its discovery, it was likely first observed after maximum luminosity, at a time when emission lines were probably prominent. The ARAS spectra in Fig. 3 show the spectrum to consist predominantly of absorption lines until early 1 July. However, later that same day within 20 hours, the absorption largely disappeared, abruptly replaced by strengthening emission lines. The spectrum retained this appearance with minimal absorption until observations ended in late July. No g -rays were detected from this object, so there is no direct evidence that they played a role in the rapid change of the visible spectrum. However, because this nova was visibly the faintest of the group, g -rays may well have been present at any time with normal luminosity, but remained undetected by the Fermi LAT.\n- 4. V1674 Her/ 2021 : Discovered on 12 June 2021, this relatively bright nova was immediately observed by Fermi LAT and by ARAS spectroscopists. Li (2021a, 2021b) reported a relatively strong g -ray flux over 12-13 June, however no further detections were reported, indicating that the g -rays weakened quickly. The ARAS spectra shown in Fig. 4 show the nova to be in the He/N phase at discovery with rather broad absorption features indicating high velocities for the emitting gas, with Fe II emission \nalso present. The rapid drop in the visible light curve shown in Fig. 6 indicates the nova might not have been discovered at peak brightness. During 13 June the spectrum began a rapid change from having strong P Cygni profiles to a dominant, broad emission spectrum. This happened at the same time as the brief, strong g -ray emission, so there is good reason to believe the g -rays and major spectral change were causally related. \n- 5. V1405 Cas/ 2021 : The correlation between high-energy g -ray and visible luminosity noted by Li et al. (2017) is quite pronounced for this nova, discovered in outburst on 18 March 2021. It was not detected as a g -ray source until 21-24 May 2021 (Buson et al. 2021), at peak visible brightness more than 60 days following outburst, as shown in Fig. 6. The nova experienced a change in spectrum between approximately 8-16 May that although not as pronounced as the shifts observed in other novae, does show absorption lines becoming prominent for more than one week before fading significantly. The detection of g -rays, the relatively rapid rise to maximum visible brightness beginning around 3 May, and the appearance and then disappearance of absorption lines, especially in the region below 4700 Å, together in May after such an abnormally long delay time following outburst, does suggest a possible correlation between the three events. \nThese data represent evidence that g -ray emission may be correlated not only with visible brightness of novae, but also with the rapid changes that are observed in their spectra.", '5. The Summing Up': "A common envelope is an expected formation for novae soon after outburst when outburst radiation and ejecta impact the secondary star. Because imaging is not yet capable of resolving the detailed structure of mass transfer binaries, spectral information is the best means of demonstrating the presence of a CE in the early decline period. We propose a simple process to model the spectra from a CE for an assumed power-law density distribution. The observed presence of strong emission lines together with absorption lines requires a step density distribution for an optically thick CE. Models with different density and velocity laws can be constructed by an iterative process, outlined in Appendix A, that allows both emission and absorption line strengths to be determined, that can be compared with observations. \nTime resolved spectroscopic data show that the spectra of novae in their early decline period often show sudden changes in their basic nature. Many of these shifts begin with predominant emission lines that change in a matter of a few days to an absorption spectrum, and then revert back to an emission-line spectrum after a period of days-toweeks. This behavior may be accounted for straightforwardly by a common envelope, especially since time-resolved spectra observe spectral shifts to occur at or near times they may be activated by g -ray emission. \nThe suggestion that shocks from collisions between ejecta components produce high energy gamma radiation and conditions that explain key aspects of novae spectra is surely valid. That does not preclude other sources of high energy from also being important, such as activity on the primary WD. The relatively quiescent sun, with magnetic fields orders of magnitude less than those of WDs in novae, produces a flux of very high energy g -rays from solar flares. It would hardly be surprising if postoutburst activity on WD's were able to provide for the g -ray luminosities observed for novae which, like that of the sun, are clearly variable. \nThe authors are indebted to Dr. Fred Walter and the world-wide group of ARAS observers for their diligent acquisition of spectra that they reduce and post online for interested researchers. We thank Dr. K. Mukai for maintaining his very useful list of galactic novae that allows samples of novae with similar characteristics to be identified straightforwardly. We also acknowledge the important work of the American Association of Variable Star Observers, who have provided the photometry for the light curves displayed in Fig. 6.", 'Appendix A': 'A simplified model can be constructed for a spherical body of gas that represents the continuum and emission line characteristics of a nova common envelope. An iterative procedure is invoked where the gas is presumed to be optically thick in the continuum, and in local thermodynamic equilibrium (LTE). It begins by arbitrarily specifying an initial radius for the CE photosphere, r ph , an initial photosphere density, n ph , and an assumed temperature distribution, T(r). The density distribution out from the photosphere is prescribed in terms of n ph by eqn. (2), with the power-law variable x initially selected. \nThe optical depth t vis at 5000 Å is computed with the above parameters along the line of sight from infinity into the center of the gas until the point where t vis =1 is reached. This requires assumed element abundances that enable the continuum absorption coefficient to be determined from ionization and excitation calculations using well known LTE relationships. The values of density n 1 and CE effective photosphere radius r 1 at this point will not agree with the initially specified values of n ph and r ph . \nThe goal of the procedure is to iterate the optical depth calculation by adjusting the initial parameter values so subsequent models converge to self-consistent values of the photosphere density and radius, and with parameter values that produce the emission line equivalent widths that are observed for novae . Determination of the temperature distribution from heating and cooling processes is important. The calculation of optical depth is sufficiently straightforward that many iterations should be able to be performed on a computer very rapidly, allowing convergence to be achieved. \nThe key is focusing on the parameter(s) that are most sensitive to determining the emission line strengths relative to the continuum. For a slowly varying, flat density distribution the optical depth builds up from the low outer densities to the point where t vis =1 before densities n ³ 10 15 cm -3 are reached that justify the assumption of LTE. This problem should be avoided by setting a lower outer limit to the assumed photosphere density, n ph . This causes the power-law density parameter x to be a key parameter to focus on when building a suite of useful common envelope models. Also, there is no reason that the eqn (2) power-law density must be adhered to. The power-law distribution can be modified by an alternative distribution.', 'References': "Metzger, B.D., Zenati, Y., Chomiuk, L. et al. 2021, ApJ, 923, 100 \nMihalas, D. 1978, Stellar Atmospheres (W.H. Freeman: San Francisco), p. 310 \nOsterbrock, D.E. & Ferland, G.J. 2006, Astrophysics of Gaseous Nebulae and Active \nGalactic Nuclei , 2 nd ed. (Sausalito: University Sci. Books) \nPayne-Gaposchkin, C. 1957, The Galactic Novae (Amsterdam: North-Holland Publ. Co.) \nRopke, F.K. & De Marco, O. 2023, LRCA, 9, 2 \nRottenberg, J.A. 1952, MNRAS, 112, 125 \nSchmidt, G.D. & Stockman, H.S. 1991, ApJ, 371, 749 \nSchmidt, G.D., Stockman, H.S., & Grandi, S.A. 1983, ApJ, 271, 735 \nShore, S.N. 2012, BASI, 40, 185 \nShore, S. N., Mason, E., Schwarz, G.J. et al. 2016, A&A, 590, 123 \nSobolev, V.V. 1960, Moving Envelopes of Stars (Cambridge: Harvard Univ. Press) \nSobolev, V.V. 1969, 'A theoretical study of galactic novae', VisAstr, 11, 181 \nSteinberg, E. & Metzger, B.D. 2020, MNRAS, 491, 4232 \nTeyssier, F. 2019, CoSka, 49,217 \nWalter, F.M., Battisti, A., & Towers, S.E. et al. 2012, PASP, 124, 1057 \nWilliams, R. 2012, AJ, 144, 98 \nWilliams, R.E. 1980, ApJ, 235, 939 \nWilliams, R., Mason, E., Della Valle, M., & Ederoclite, A. 2008, ApJ, 685, 451 \nWilliams, R., Walter, F.M., Rudy, R.J., Munari, U., Luckas, P. et al. 2022, ApJ, 941, 138 \nWishart, A. W. 1979, MNRAS, 187, 59", 'Figure 1': 'Figure 1 - ARAS spectra of V5855 Sgr shortly after outburst. Except for H a , the P Cygni profiles present in the initial 20 October spectrum have disappeared by 23 October. The spectrum suddenly emerging on 26 October shows emission lines prominent, with the large majority of absorption lines present on 25 October having disappeared within 24 hours. \n<!-- image --> \nFigure 2 - After an initial Fe II-phase emission spectrum, between 28-30 October the V5856 Sgr spectrum changed from emission to a dominant absorption spectrum. The absorption lines strengthened and persisted until 12 November, when emission components re-appeared in the form of P Cygni profiles. By 18 November the spectrum had reverted to its initial Fe II emission structure. \n<!-- image -->', 'Figure 3': "Figure 3 - ARAS spectra for V613 Sct show this nova's spectrum to have transformed significantly within a 20-hour period on 1 July from very prominent, narrow absorption lines to an emission spectrum. During the 1 July transformation interval, the visible brightness may have decreased of order 0.5 mag, as shown in Fig. 6. \n<!-- image -->", 'Figure 4': 'Figure 4 - V1674 Her displayed a He/N-phase spectrum during its rapid decline in brightness. Strong, broad absorption lines initially present in these ARAS spectra disappeared rapidly on 13 June as the spectrum became dominated by strong emission features. \n<!-- image -->', 'Figure 5': 'Figure 5 - Nova V1405 Cas displayed classic Fe II-phase spectra in these ARAS spectra with narrow Fe II P Cygni profiles having prominent absorption components until their disappearance on 16 May. Non-Balmer emission components weakened significantly between 3-8 May, as narrow absorption features appeared and strengthened, especially below 4700 Å, until their disappearance. \n<!-- image --> \nFigure 6 - Light curves for the five novae whose spectra are presented in Figs. 1-5, taken from AAVSO observations. Visible measurements are displayed in green, with r-band brightness denoted by red squares. Vertical lines mark the dates on which the ARAS spectra were obtained that are shown in Figs. 1-5. Purple horizontal brackets denote the time interval when g -ray emission was detected by the Fermi LAT. \n<!-- image -->'}
2024A&A...690A.193J	Context. 4015 WilsonHarrington hereafter WH was discovered as a comet in 1949 but has a dynamical property consistent with that of a nearEarth asteroid. Although there is a report that the 1949 activity is associated with an ion tail the cause of the activity has not yet been identified. Aims. This work aims to reveal the mysterious cometlike activity of the nearEarth asteroid. Methods. We conducted new polarimetric observations of WH from May 2022 to January 2023 reanalyses of the photographic plate images taken at the time of its discovery in 1949 and dust tail simulation modelings where the dust terminal velocity and ejection epoch are taken into account. Results. We found that this object shows polarization characteristics similar to those of lowalbedo asteroids. We derived the geometric albedo ranging from pSUBVSUB 0.076 0.010 to pSUBVSUB 0.094 0.018 from our polarimetry the values vary depending on the data used for fitting and the slopealbedo relationship coefficients. In addition the 1949 image showed an increase in brightness around the nucleus. Furthermore we found that the color of the tail is consistent with sunlight suggesting that the 1949 activity is associated with dust ejection. From the dust tail analysis 9 10SUP5SUP kg of material was ejected episodically at a low velocity equivalent to or even slower than the escape velocity. Conclusions. We conclude that WH is most likely an active asteroid of main belt origin and that the activity in 1949 was likely triggered by mass shedding due to fast rotation.	2024-10-01T00:00:00Z	['2024A&A...690A.193J', 'arXiv:2409.06448', '10.48550/arXiv.2409.06448', '10.1051/0004-6361/202451225', '2024arXiv240906448J']	['interplanetary medium', 'minor planets', 'asteroids: individual: 4015 Wilson–Harrington', 'comets: individual: 107P/Wilson–Harrington', 'Astrophysics - Earth and Planetary Astrophysics']	New evidence supporting past dust ejections from active asteroid 4015 WilsonHarrington	2,024	194	0.51	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.06448.pdf	{'New evidence supporting past dust ejections from active asteroid (4015) Wilson-Harrington': 'Sunho Jin, 1 , 2 Masateru Ishiguro, 1 , 2 Jooyeon Geem, 1 , 2 Hiroyuki Naito, 3 Jun Takahashi, 4 Hiroshi Akitaya, 5 , 6 , 7 Daisuke Kuroda, 8 Seitaro Urakawa, 8 Seiko Takagi, 9 Tatsuharu Oono, 10 Tomohiko Sekiguchi, 11 Davide Perna, 12 Simone Ieva, 12 Yoonsoo P. Bach, 13 Ryo Imazawa, 7 , 14 Koji S. Kawabata, 7 , 14 Makoto Watanabe, 15 and Hangbin Jo 1 , 2 \n- 1 Department of Physics and Astronomy, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea\n- 2 SNU Astronomy Research Center, Department of Physics and Astronomy, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea\n- 3 Nayoro Observatory, 157-1 Nisshin, Nayoro, Hokkaido, 096-0066, Japan \n4 \nCenter for Astronomy, University of Hyogo, 407-2 Nishigaichi, Sayo, Hyogo 679-5313, Japan \n- 5 Astronomical Research Center, Chiba Institute of Technology, 2-17-1 Tsudanuma, Narashino, Chiba 275-0016, Japan\n- 6 Planetary Exploration Research Center, Chiba Institute of Technology, 2-17-1 Tsudanuma, Narashino, Chiba 275-0016, Japan\n- 7 Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima, 739-8526, Japan\n- 8 Bisei Spaceguard Center, Japan Spaceguard Association, 1716-3 Okura, Bisei-cho, Ibara, Okayama 714-1411, Japan\n- 9 Department of Earth and Planetary Sciences, Faculty of Science, Hokkaido University, Kita-ku, Sapporo, Hokkaido 060-0810, Japan\n- 10 Department of Cosmosciences, Graduate School of Science, Hokkaido University, Kita-ku, Sapporo, Hokkaido 060-0810, Japan\n- 11 Asahikawa Campus, Hokkaido University of Education, Hokumon, Asahikawa, Hokkaido 070-8621, Japan\n- 12 INAF - Osservatorio Astronomico di Roma, Via Frascati 33, I-00078, Monte Porzio Catone, Italy\n- 13 Korea Astronomy and Space Science Institute (KASI), 776 Daedeok-daero, Yuseong-gu, Daejeon, 34055, Republic of Korea\n- 14 Physics Program, Graduate School of Advanced Science and Engineering, Hiroshima University, 1-3-1 Kagamiyama, HigashiHiroshima, Hiroshima 739-8526, Japan\n- 15 Department of Physics, Okayama University of Science, 1-1 Ridai-cho, Kita-ku, Okayama, Okayama 700-0005, Japan e-mail: [email protected], [email protected] \nReceived ; accepted', 'ABSTRACT': 'Context. (4015) Wilson-Harrington (hereafter, WH) was discovered as a comet in 1949 but has a dynamical property consistent with that of a near-Earth asteroid. Although there is a report that the 1949 activity is associated with an ion tail, the cause of the activity has not yet been identified. \nAims. This work aims to reveal the mysterious comet-like activity of the near-Earth asteroid. \nMethods. We conducted new polarimetric observations of WH from May 2022 to January 2023, reanalyses of the photographic plate images taken at the time of its discovery in 1949, and dust tail simulation modelings, where the dust terminal velocity and ejection epoch are taken into account. \nResults. We found that this object shows polarization characteristics similar to those of low-albedo asteroids. We derived the geometric albedo ranging from pV = 0 . 076 ± 0 . 010 to pV = 0 . 094 ± 0 . 018 from our polarimetry (the values vary depending on the data used for fitting and the slope-albedo relationship coe ffi cients). In addition, the 1949 image showed an increase in brightness around the nucleus. Furthermore, we found that the color of the tail is consistent with sunlight, suggesting that the 1949 activity is associated with dust ejection. From the dust tail analysis, ∼ 9 × 10 5 kg of material was ejected episodically at a low velocity equivalent to or even slower than the escape velocity. \nConclusions. We conclude that WH is most likely an active asteroid of main belt origin and that the activity in 1949 was likely triggered by mass shedding due to fast rotation. \nKey words. Comets: general - Minor planets, asteroids: general - Minor planets, asteroids: individual: (4015) Wilson-Harrington', '1. Introduction': "Traditionally, small Solar System bodies (SSSBs) in the inner region have been thought to be distinctively classified as either comets or asteroids. In that case, comets are icy bodies that formed beyond the snow line and display comae and tails near perihelia due to the volatile sublimation. Meanwhile, asteroids are rocky objects that originate from the inner part of the Solar System and appear as point sources in the observed images. However, discoveries have expanded this traditional classifica- \ntion to a broader continuum, with comets and asteroids as the end members (Jewitt & Hsieh 2022). Active asteroids, whose orbits and spectra are similar to those of asteroids but display comet-like activity, are a kind of object between comets and asteroids (Jewitt 2012). \n107P / (4015) Wilson-Harrington (hereafter, WH) is an intriguing object that lies between comets and active asteroids. In November 1949, when it was discovered, WH displayed a tail in photographic plate images (Cunningham 1950), but never again \n(Cunningham 1950; Bowell et al. 1992; Ishiguro et al. 2011a). Accordingly, it was classified as a dormant comet whose activity was extinct (e.g., Chamberlin et al. 1996; Coradini et al. 1997; Lupishko & Lupishko 2001). Fernández et al. (1997) conducted Finson-Probstein modeling on the tail of WH and found that the dust size would be tens to hundreds of micrometers, which contradicts the blue color ( B -R ∼ -1) of the tail reported by Bowell et al. (1992). Thus, they suggested that the observed tail is dustless and due to CO + and H2O + ion fluorescence, which are commonly found in comets. To date, it remains the only explanation for the activity of WH. \nAs Fernández et al. (1997) pointed out, however, the existence of an ion tail without the presence of dust is questionable if this object was almost a dormant comet. This is because the surface of the object would be covered with a dust mantle, and the drag force of sublimating gas from beneath the mantle would naturally carry out dust particles from the object. In addition, the spectrum indicated an asteroidal rather than cometary (i.e., C-complex asteroids or carbonaceous chondrites, Tholen 1984; Kareta & Reddy 2023). Moreover, Bottke et al. (2002) suggested from their dynamical simulation that its origin is more likely from the outer main belt (65 %) rather than the Jupiter family comet (4 %). This implies that the object could be an active asteroid that originated from the main asteroid belt. If this is indeed an active asteroid, several dust ejection mechanisms, such as impacts or rotational breakups, found in several active asteroids, could also be possible triggers for the activity (Jewitt 2012). \nIn this study, we aim to determine whether WH is a dormant comet or an active asteroid and to constrain its activity mechanism. First, we conducted the first polarimetric study on the object and compared its surface scattering properties with those of comets and asteroids. Polarimetry is a unique tool for discriminating between asteroid-like and comet-like objects, as demonstrated by Geem et al. (2022a). Moreover, polarization degrees obtained at high phase angles (i.e., Sun-object-observer's angle α > 40 · ) of the active asteroid (3200) Phaethon are significantly higher than those of a comet nucleus, further suggesting the usability of polarimetry at large phase angles (Ito et al. 2018; Kuroda et al. 2015). Thus, it is safe to say that polarimetry provides critical information for distinguishing between comets and asteroids. Next, we reanalyzed the photographic plates of WH in 1949 to inspect the color of its tail. Besides, We conducted a dust ejection simulation to determine physical quantities related to its activity, following the method in Ishiguro et al. (2007). Considering all this information (i.e., nuclear polarimetry, dust color, and dust ejection mechanism), we provide a comprehensive discussion about a possible mechanism for the 1949's activation.", '2.1. Polarimetry': "Table 1 provides a summary of our polarimetric observations of WH. We conducted polarimetric observations from May 25, 2022 to January 25, 2023 using three telescopes in Japan: the 1.6-m Pirka telescope at the Nayoro Observatory (NO) of the Faculty of Science, Hokkaido University (Minor Planet Center observatory code Q33), the 2-m Nayuta telescope at the NishiHarima Astronomical Observatory (NHAO), operated by the University of Hyogo 1 , and the 1.5-m Kanata telescope at the Higashi-Hiroshima Observatory (HHO) 2 . The observations cov- \nered a wide phase angle range, from 2.7 · to 80.4 · . At NO, we utilized the polarimetry mode of the Multi-Spectral Imager (MSI), which has a pixel scale of 0.39 '' pixel -1 (Watanabe et al. 2012). Meanwhile, at the NHAO, we employed the polarimetry mode of the Wide Field Grism Spectrograph 2 (WFGS2), which has a pixel scale of 0.198 '' pixel -1 (Uehara et al. 2004; Kawakami et al. 2021), and at the HHO, we used the Hiroshima Optical and Near-InfraRed camera (HONIR), which has a pixel scale of 0.29 '' pixel -1 (Akitaya et al. 2014). We used a standard RC-band filter and set an exposure time range of 60 to 180 seconds to obtain polarimetric accuracy of less than 1%p in non-sidereal tracking mode. The observations were conducted for 16 nights at NO, 2 nights at NHAO, and one night at HHO. \nWe conducted our data analysis using the same methods as those described in detail in Ishiguro et al. (2022) for MSI data and in Geem et al. (2022b) for WFGS2 and HONIR data. The analysis process can be outlined as follows: \n- 1. preprocessing of the data,\n- 2. masking of the field stars near the target and bad pixels,\n- 3. aperture photometry to derive the source flux of ordinary and extraordinary components,\n- 4. derivation of the Stokes parameters ( Q / I , and U / I ),\n- 5. correction of the polarization e ffi ciency, instrumental polarization, and position angle o ff set, and\n- 6. derivation of the polarization degree ( P ), the position angle ( θ P), the polarization degree with respect to the scattering plane normal( P r), and the position angle with respect to the scattering plane normal( θ r).", '2.2. Photometry': "We obtained two digitized photographic plate images of WH in 1949 from the Mikulski Archive for Space Telescopes (MAST) Digitized Sky Survey (DSS) 3 . One was taken with a photographic plate with an emulsion 103aO, sensitive to blue wavelengths (hereafter, blue plate), and the other was with an emulsion 103aE, sensitive to red wavelengths (red plate). These plates were taken on November 19, 1949, during the first Palomar Observatory Sky Survey (POSS). The coordinates of the image centers are taken from Table 1 of Fernández et al. (1997), and the size of each image is 30 ' × 30 ' . The pixel sizes of those images are 15.0 µ m (1.0 '' ) and 25.284 µ m (1.67 '' ) for blue and red plates, respectively. \nFiles downloaded from the MAST contain the photographic emulsion density multiplied by a constant factor (6 553.4) to adjust the 16-bit representation (Laidler et al. 1994). We linearized these emulsion densities to intensities following the technique in Cutri et al. (1993). Detailed procedures are given in Appendix A. Then, we subtracted the background using Source-Extractor (Bertin & Arnouts 1996). We set the mesh size for the background estimation to be 512 pixels, large enough not to be significantly a ff ected by the di ff use tail of WH. Moreover, we estimated the error of these intensities using the method described in Appendix B. The error estimated in Appendix B does not include the inherent root mean square (RMS) error of di ff use source photometry from the POSS plate. Cutri et al. (1993) suggested the amount of the RMS error to be 0.1 to 0.3. Therefore, we added the RMS error of 0.2 to our original results. \nBecause of the sidereal tracking, the nucleus appeared elongated in the images. We determined regions where the WH nucleus and tail were located from the images (see Fig. 1). We visually determined the rectangular regions and hexagonal regions \nTable 1. Summary of polarimetric observation \nNotes. ( a ) Exposure time in seconds ( b ) Number of valid exposures ( c ) Median heliocentric distance in au ( d ) Median geocentric distance in au ( e ) Median solar phase angle in degrees ( f ) Position angle of the scattering plane in the equatorial coordinate system (J2000) in degrees. \n<!-- image --> \nFig. 1. Digitized photographic plate images of WH in 1949 used for the analysis. (a) and (c) are the blue and red plate images taken with exposure times of 720 seconds and 2700 seconds, respectively. In these images, the nucleus and tails are stretched out due to the sidereal tracking mode of the telescope mount. The tail in the red plate image is not as clear as the one in the blue plate because of the lower sensitivity of the red plate. (b) and (d) are zoomed images within the white squares of images (a) and (c). These images are rotated to align the WH's velocity vectors in a horizontal direction. The areas surrounded by red rectangles indicate the region for the nucleus photometry, while those surrounded by the lower left parts of the red rectangles and yellow lines indicate the region for the tail photometry. \n<!-- image --> \nthat enclose the nucleus and the tails in each band for the photometry. The leftmost and rightmost sides of the tail area have relative position angles of 157 · and 144 · with respect to the horizontal axis of each image (i.e., the WH's velocity vector projected on the sky plane). We obtained these angles from Fig. 9 of Fernández et al. (1997). The distance between the lower side \nof the nucleus area and the tail area corresponds to 42 '' on both plates, although the horizontal lengths are di ff erent due to the di ff erent exposure times. As a result, the areas of the tail regions are 3 834.8 arcsec 2 and 8 292.2 arcsec 2 for blue and red plates, respectively. \nWe calculated the intensities and uncertainties of the nucleus and tail within these regions for blue and red plates. For the tail, which is an extended source, we derived the surface brightness by dividing the intensity by the photometric areas. Once the intensities and surface brightness were obtained, we estimated the mass of the dust tail ( M tail). It can be given as \nM tail = 4 π N 0 ρ 3 (4 + q ) h a 4 + q max -a 4 + q min i , (1) \nwhere ρ is the dust mass density. We assumed ρ = 1300 kg m -3 based on the measurement of Ryugu samples due to the similarity of the spectral type (i.e., C-complex). We supposed a simple power law size distribution of the dust particles in the radius range from a min to a max. q denotes the power index of the di ff erential size distribution, and N 0 is the reference number of dust particles with the radius a 0 = 1 m. Assuming the scattering property (i.e., the albedo and the scattering phase functions) of dust particles are the same as that of the nucleus, we derived N 0 as below: \nN 0 = R 2 WH (3 + q ) a 3 + q max -a 3 + q min I tail I WH ! , (2) \nwhere I WH and I tail are intensities in the blue plate within the enclosed areas after sky subtractions. We cited the radius of WH nucleus ( R WH) to be 2 200 m (i.e., the best-fit value obtained in Bach et al. 2017).", '2.3. Dust ejection simulation': "Assuming that the tail is associated with scattered light by ejected dust, we analyzed the tail properties. We conducted a simple simulation of the dust ejection to estimate the ejection epoch and particle size. The model descriptions are given in, for example, Ishiguro et al. (2007). \nFirst, we produced the synchrone-syndyne network, where the zero ejection speed was considered. The synchrone curves are the lines indicating the distribution of dust particles ejected at given epochs. On the other hand, the syndyne curves represent the location of dust particles with the same β value, a ratio between solar radiation pressure ( F r) and gravity ( F g). It is given as \nβ ≡ F r F g = 3 L ⊙ Q pr 16 π GM ⊙ c ρ a = KQ pr ρ a , (3) \nwhere L ⊙ , G , M ⊙ , and c are the solar luminosity, gravitational constant, solar mass, and light speed, respectively. Substituting these constant values, K is equal to 5 . 7 × 10 -4 kg m -2 . Q pr is the radiation pressure coe ffi cient averaged over the solar spectrum. We considered large absorbing particles and assumed Q pr = 1, following Ishiguro et al. (2007). The synchrone-syndyne network was drawn on the XY -plane of the digitized blue photographic plate image taken on November 19, 1949. Furthermore, we conducted a three-dimensional analysis of the dust tail, allowing consideration of the terminal ejection velocity of the particles, described in Ishiguro et al. (2007). This model assumes a terminal ejection velocity ( v ej) of particles as \nv ej = V 0 β u 1 GLYPH<18> r h 1au GLYPH<19> -u 2 , (4) \nArticle number, page 4 of 12 \nFig. 2. Polarization phase curve (PPC) of WH. Each symbol denotes observations from di ff erent observatories and instruments. A green dotted line shows a fitted line using a trigonometric function (TF, Eq. 5), and a blue dashed line shows a fitted line using a linear-exponential function (LE, Eq. 6). \n<!-- image --> \nwhere V 0 is the reference ejection velocity (m s -1 ) of the particles with β = 1 at the heliocentric distance of r h = 1 au. We set u 1 and u 2 as 0.5. With the given v ej and β , we solved Kepler's equation to determine the spatial distribution of dust particles. Moreover, we assumed the di ff erential size distribution of dust particles in the size range between a min and a max and the power index q .", '3. Results': 'In this section, we report our polarimetric results in Sect. 3.1, photometric results from the 1949 observation images in Sect. 3.2, and the tail analysis in Sect. 3.3, as described below.', '3.1. Polarimetric results': "Throughout our observations, we did not detect any signatures of comet-like activity (i.e., a tail and a coma) in our polarimetric images in 2022-2023. Therefore, we characterize the polarimetric properties of the nuclear surface. \nWesummarized nightly-averaged linear polarization degrees in Fig. 2 and Table 2. Similar to other SSSBs, it indicates the negative branch at α ≲ 20 · and the positive branch at α ≳ 20 · (Ishiguro et al. 2022). Although we cannot determine the maximum polarization degree because of insu ffi cient phase angle coverage, we find Pr ( α ) > 30% at α ∼ 80 · . \nWe compared the polarization phase curve ( Pr ( α ), PPC) of WH with other asteroids (Fig. 3). At a glance, the PPC's slope near the inversion angle is predominantly steeper than that of S-complex asteroids and consistent with those of C-complex asteroids. A careful look reveals that the WH's PPC is more moderate than that of Ryugu and steeper than that of Phaethon around the low phase angles. This fact suggests that the WH's \nTable 2. Summary of nightly-averaged linear polarization degrees \nNotes. ( a ) Median solar phase angle in degrees, ( b ) Nightly averaged linear polarization degree in percent, ( c ) Error of P in percent, ( d ) Direction of the semi-major axis of polarization ellipse with respect to the celestial north in the equatorial coordinate system (J2000) in degrees, ( e ) Error of θ P in degrees, ( f ) Polarization degree referring to the scattering plane normal in percent, ( g ) Direction of the semi-major axis of polarization ellipse with respect to the scattering plane normal in degrees \ngeometric albedo value would be between these two asteroids. We also compared WH's PPC with those of a comet nucleus (209P / LINEAR) and cometary dust (Fig. 4). Although there is only one comet whose nuclear PPC is available, we find that WH's PPC indicates a higher polarization degree than 209P. Considering most SSSBs have polarization maxima around α > 90 · while WH was observed at α << 90 · but indicated the large polarization degree ( Pr (80 · ) > 30 %), this object likely shows a polarization maximum larger than that of comet nuclei. \nTo determine the polarimetric parameters quantitatively, we employed PyMC3 (Salvatier et al. 2016) to fit the PPC using the empirical trigonometric function (TF) shown in Ishiguro et al. (2022), modified from Lumme & Muinonen (1993) as shown below: \nPr ( α ) = h sin α sin α 0 ! c 1 cos α 2 cos α 0 2 ! c 2 sin ( α -α 0) , (5) \nwhere α 0 is an inversion angle where Pr ( α 0) = 0 % is satisfied, and h is a polarimetric slope at α = α 0. c 1 and c 2 are constant values obtained from fitting. Meanwhile, We also fitted PPC using a modified linear-exponential function (LE) from Bach et al. (2024) modified from the original forms in Muinonen et al. (2002); Kaasalainen et al. (2003), \nPr ( α ) = h (1 -e -α 0 / k ) α -(1 -e -α 0 / k ) α 0 1 -(1 + α 0 / k ) e -α 0 / k , (6) \nsuch that \nα min = -k ln ( k α 0 GLYPH<16> 1 -e -α 0 / k GLYPH<17> ) , (7) \nwhere α 0 and h are the same parameter to Eq. 5, k is a scaling constant, and α min is a phase angle where minimum polarization \ndegree ( P min) locates. The best-fit parameters and their 1-sigma uncertainties derived from the fitting using various phase angle ranges are summarized in Table C.1. From the fitting using all observation data, we also derive the minimum polarization degree of P min = -0 . 96 + 0 . 32 -0 . 34 % at the phase angle of α min = 10 . 7 · + 2 . 2 · -2 . 8 · from TF and P min = 1 . 46 ± 0 . 23 % and α min = 8 . 9 · + 0 . 4 · -0 . 5 · from LE function. \nThe polarimetric slope, h , is known to have a strong correlation to the geometric albedo in the V band ( pV ) (Widorn 1967). The relation is given as \nlog 10 ( pV ) = C 1 log 10 ( h ) + C 2 , (8) \nwhere C 1 and C 2 are constant values: C 1 = -1 . 111 ± 0 . 031 and C 2 = -1 . 781 ± 0 . 025 in Cellino et al. (2015), and C 1 = -1 . 016 ± 0 . 010 and C 2 = -1 . 719 ± 0 . 012 in Lupishko (2018). We assumed this relation and constant parameters are applicable to R C-band data and derive the geometric albedo as shown in Table 3. We converted the R C-band geometric albedo ( pR C ) into the standard V -band geometric albedo ( pV ) using the color index: \npV = pR C × 10 0 . 4[( V -R C) ⊙-( V -R C)WH] , (9) \nwhere ( V -R C) ⊙ = 0 . 356 ± 0 . 003 (Ramírez et al. 2012) and ( V -R C)WH = 0 . 378 ± 0 . 025 (converted from a r ' -i ' color using relations from Jester et al. 2005) are V -R C colors of the Sun and WH, respectively (Urakawa et al. 2011). The estimated albedo values using h values from the TF fitting are 0 . 094 ± 0 . 018 with coe ffi cients in Cellino et al. (2015) and 0 . 093 ± 0 . 015 with those in Lupishko (2018), while those from the LE relation fitting are 0 . 076 ± 0 . 010 and 0 . 077 ± 0 . 008 with coe ffi cients from Cellino et al. (2015) and Lupishko (2018), respectively. Therefore, it is safe to say that the possible range of pV of WH is 0.066-0.112. These results are also summarized in Table 3, together with results from data with di ff erent α ranges used for the fitting. \nNotes. ( a ) Numbers of data used for fitting ( b ) Reference for coe ffi cients C 1 and C 2 in Eq. 8 ( c ) Cellino et al. (2015) ( d ) Lupishko (2018) \nFig. 3. Comparison in polarization phase curves (PPCs) of WH (red circles) with fitted curve using Eq. 5 and other asteroids: two B-type asteroids (3200) Phaethon (blue squares, Ito et al. 2018; Shinnaka et al. 2018; Devogèle et al. 2018), (155140) 2005 UD (cyan diamonds, Ishiguro et al. 2022), a C-type asteroid (162173) Ryugu (cyan upper triangles, Kuroda et al. 2015), two S-type asteroids (1566) Icarus (green crosses, Ishiguro et al. 2017), and (4179) Toutatis (lime pluses, Lupishko et al. 1995; Mukai et al. 1997), and a L-type asteroid (85989) 1999 JD6 (orange lower triangles, Kuroda et al. 2021). \n<!-- image -->", '3.2. Photometric results': "From the photometry, we find that the nuclear magnitudes in 1949 are 14 . 86 ± 0 . 20 in the B band and 14 . 05 ± 0 . 20 in the R C band. We compared our R C-band magnitude of the nucleus to the phase function given in Ishiguro et al. (2011a) using the heliocentric distance r h = 1 . 148 au, the geocentric distance ∆ = 0 . 025 au, and the phase angle α = 40 . 1 · , where WH was active in 1949. From the phase function, the expected R C-band reduced magnitude at the time of 1949 observation is 14 . 4 ± 0 . 1. We find that this R C-band reduced magnitude is ∼ 0 . 4 magnitude brighter than the \nFig. 4. Comparison in polarization phase curves (PPCs) of WH (red circles) with fitted curve using Eq. 5, a nucleus of Jupiter family comet 209P / Linear (black stars, Kuroda et al. 2015), dust particles in comae of Jupiter Family Comets (grey circles, Kiselev et al. 2005), 1P / Halley (cyan squares, Kikuchi et al. 1987; Chernova et al. 1993), and polarization degrees from dust continuum of 2P / Encke (green upper triangles, Jockers et al. 2005; Kwon et al. 2018). \n<!-- image --> \nexpected nuclear brightness based on the phase function of the WH's nucleus (Ishiguro et al. 2011a). Considering the magnitude error (0.2), we conclude that WH's nucleus was brightened when it showed the comet-like tail. \nIn addition, we find B -R C of the nucleus in 1949 was 0 . 80 ± 0 . 29, consistent with previous reports by Lowry & Weissman (2003) (0 . 81 ± 0 . 06) and Urakawa et al. (2011) (0 . 80 ± 0 . 04, with the magnitude converted from SDSS g', r' and i' to JohnsonCousins system using Jester et al. 2005) when it was inactive. We also derive the total signal within the tail region in the B band and the R C band to be (1 . 04 ± 0 . 09) × 10 -15 W m -2 and (1 . 78 ± 0 . 09) × 10 -15 W m -2 , respectively. The corresponding surface brightnesses of the tail are 25 . 96 ± 0 . 22 in the blue plate \nFig. 5. Synchrone (green dotted lines) and syndyne (violet solid lines) network drawn on the blue plate image on November 19, 1949. This image has a standard orientation, that is, north is up and east to the left. The numbers on the synchrone curves correspond to days of dust ejection before the observed day, while the numbers on the syndyne curves indicate the corresponding β values. \n<!-- image --> \nand 25 . 44 ± 0 . 21 magnitudes arcseconds -2 in the red plate. We confirm that the B -R C = 0 . 52 ± 0 . 30 color of the tail is consistent with the nucleus within a 1-sigma confidence interval. Although the tail color is bluer than the Sun, the similarity in color between the tail and the nucleus may suggest that the tail consists of dust grains with a similar optical color to the nucleus, and was detectable via the scattered sunlight.", '3.3. Results of dust tail analysis': "As shown in Sect. 3.2, the color of the tail is similar to the Sun and the nucleus. We also find that the nuclear magnitude in the 1949 images was brighter than the expected magnitude of the nucleus. This evidence suggests that WH ejected dust particles around the time of the 1949 observation, and the associated dust tail was imaged. \nBased on the idea above, we investigated the ejection epoch and size of the dust particles from the tail morphology (see Sect. 2.3 for model description). Fig. 5 compares the observed images with a synchrone-syndyne network. Although the synchronesyndyne analysis is a simple model that assumes a zero initial velocity, this analysis derives a rough estimate of the particle size and the ejection time. The position angle of the observed tail is 105 . 5 · ± 6 . 5 · . The best-match synchrone curve corresponds to 25-32 days before the observed date (November 19, 1949). In other words, if the ejection velocity was small, the dust particles were likely ejected within 6 days on October 18-25, 1949. This period of activity is likely shorter than 7 days in the realistic case (i.e., non-zero ejection velocity). Moreover, the intersection of the synchrone and syndyne curves indicates β is ranging from ∼ 3 × 10 -4 to ∼ 1 × 10 -2 , which correspond to a = 44 µ m-1.5 mm when we assume the Ryugu-like dust mass density ( ρ d = 1 300 kg m -3 , Miyazaki et al. 2023). This maximum radius ( a = 1.5 mm) \nFig. 6. Simulated images of WH with di ff erent V 0. We assumed that dust ejection happened for one day impulsively around 29.5 days before the observation. The field-of-view of each image is identical to the one of Fig. 1 (b). The comet-like ejection velocity, V ( a ) = V 0 × √ β , was used for this model. We assumed hemispherical dust ejection with fixed β range (3 × 10 -4 ≤ β ≤ 1 × 10 -2 ) and q for the di ff erential size distribution. \n<!-- image --> \nis a lower limit. It would be possible that larger dust particles might be present near the nucleus and enhance the nuclear magnitude. \nFigure 6 shows the dust simulation results, where we take into account the dust ejection velocity. Since our synchrone analysis suggests that the ejection duration was short, for only 7 days or less, we considered an impulsive dust ejection. In this model, we assumed that dust ejection happened for only ∆ t = 1 day on October 29.5, 1949 (the middle day from our synchrone analysis) from the entire Sun-lit hemisphere. We also assumed a power index q = -3 . 5 for the di ff erential size distribution of the dust particles. These parameters ( ∆ t and q ) cannot be determined from the observed images not only because of the faintness of the tail but also because of the lack of time-series images. In Fig. 6, four di ff erent terminal velocities ( V 0 = 10, 20, 30, and 40 m s -1 ) were tested. By visual inspection, the model image with V 0 < 20 ms -1 matches the observation. The visible region of the tail has a width of ≈ 8 '' , which matches the model image with V 0 ∼ 10 m s -1 . With Eq. 4 and V 0 ∼ 10 m s -1 , our model suggests a terminal ejection velocity of 1 mm-sized dust particles ( β = 4 . 4 × 10 -4 ) would be 0.2 m s -1 . This velocity is even slower than the escape velocity from WH (a bulk density of 1 190 kg m 3 and a diameter of ∼ 4 000 m are assumed (Watanabe et al. 2019; Bach et al. 2017)). \nFinally, we derived the dust mass. We obtained the flux ratio of the dust tail with respect to the nucleus as I tail I WH = 0 . 139 from the blue plate. Substituting maximum (1.5 mm) and minimum (44 µ m) dust sizes and assuming q = -3 . 5, we find the dust total mass in the tail was 9 . 27 × 10 5 kg using Eqs. (1)-(2). The estimated dust mass is only 1 . 8 × 10 -6 % of WH nucleus mass (when we assume that the bulk density of WH is 1 190 kg m 3 , similar to another C-complex asteroid (162173) Ryugu (Watan- \net al. 2019). We note that this estimated dust mass would be a lower limit since large particles residing near the nucleus were not included in the calculation.", '4. Discussion': "We have investigated the WH's origin and its mysterious activity in 1949 from multiple perspectives based on photometry, spectroscopy, and dust dynamical properties. As we mentioned in Sect. 1, the di ff erence between comets and asteroids has become unclear nowadays because of recent findings of objects having the hybrid natures of comets and asteroids (Jewitt &Hsieh 2022). However, we adopt a conventional classification based on the origins in the following discussion. We thus refer to a comet as an object originating from the trans-Neptunian region (Levison & Duncan 1997), while an asteroid is an object in or from the main belt. In the following subsections, we discuss each feature and consider the possible mechanism for the 1949 activity.", '4.1. The optical nature': "First, we discuss the optical nature. We derived the colors of the nucleus B -R C = 0 . 80 ± 0 . 29 and the tail B -R C = 0 . 52 ± 0 . 30. Although these errors are not small enough to discuss the origins, it is safe to say that the color of the tail is consistent with the nucleus. From this consistency, we conclude that the tail is dustscattered light. The 0.4 magnitude enhancement of the nucleus can be explained by large dust particles distributed around the Hill radius. \nFrom previous research, it is known that the nucleus color is slightly bluer than that of the Sun (Chamberlin et al. 1996). In general, comets exhibit red spectral characteristics compared to the solar spectrum (Jewitt 2002). However, the blue color of the nucleus is not necessarily a condition to reject the possibility that WHisacometary origin. In fact, some comets (e.g., 95P / Chiron) indicated blue properties (Luu 1993; Boehnhardt et al. 1999). More comprehensive spectroscopic research in the optical and near-infrared wavelengths suggested that the WH spectrum is in agreement with carbonaceous chondrite meteorites (Kareta & Reddy 2023), which originate from C-type asteroids. Accordingly, from the reflectance color and spectrum, WH likely originated from the main belt. \nWealso investigated polarization properties. No polarimetric information was reported before our work. Based on our data, we estimated the range of geometric albedo ( p V) to be 0.066 - 0.112. The derived geometric albedo is marginally consistent with or slightly higher than the previous estimates: p V = 0 . 055 ± 0 . 012 (Ishiguro et al. 2011a), 0 . 040 -0 . 055 (Bach et al. 2017), and 0 . 059 ± 0 . 011 (Licandro et al. 2009). This slight discrepancy may be due to a slight underestimation of the inherent errors in the models used in each study (including our research). Geem et al. (2022a) proposed a technique for discriminating between comet-like and asteroid-like objects by combining a polarimetric slope ( h ) and color index ( V -R C). In Fig. 7, we applied the method in Geem et al. (2022a) to our target. Indeed, WH is consistent with C- and B-type asteroids. However, the albedo of comets and some dark asteroids (C-complex and D-types) are largely overlapped with each other (Lamy et al. 2004; Usui et al. 2011; Mainzer et al. 2014), implying the geometric albedo tightly related to the polarimetric slope ( h in Geem et al. 2022a) cannot be a 'conclusive' determinant. Also, as noted above, some exceptional cometary samples have unusually blue color indices. In other words, although the method of Geem et al. \nFig. 7. Polarimetric slope h and V -R C plot of WH in comparison with di ff erent types of asteroids and comet nuclei. The comparison data are taken from Geem et al. (2022a). Points with S, C, B, and D denote S-, C-, B-, and D-type asteroids. \n<!-- image --> \n(2022a) is a useful tool that can make rough discrimination of comets and asteroids, it is not necessarily possible to conclude the origin with only h and V -R C. Further careful consideration is needed, as given in Sect. 4.2. \nFurthermore, we show that the WH's large polarization degrees at large phase angles ( α > 40 · ). These values are higher than those of the nucleus of a nearly dormant comet 209P / LINEAR. It is, however, true that the number of polarization observation samples of cometary nuclei at large phase angles is insu ffi cient because the nuclei should be enclosed by thick dust comae at the large phase angles (i.e., the small heliocentric distances). In addition, Kwon et al. (2018) speculated that large dust might have accumulated on the old comet nuclei, and the e ff ect is responsible for large polarization degrees at high phase angles. Indeed, WH has been in a stable orbit for the past 4 000 years or more, maintaining a solar distance of ∼ 1 au (Kareta & Reddy 2023). Therefore, it may be possible that the surface materials are thermally metamorphosed or sorted to extract only large dust particles. \nSummarizing the discussion regarding the optical nature, WH has a fairly high probability of being of main-belt origin. This fact is also in accordance with dynamical studies (Bottke et al. 2002; Granvik et al. 2018), although dynamical research has an intrinsic di ffi culty due to the chaotic orbit of near-Earth objects.", '4.2. The cause of the dust ejection in 1949': "We subsequently discuss the essential question of why WH exhibited comet-like activity in 1949 based on the abovementioned fact that WH is likely the main belt origin. Fernández et al. (1997) pointed out that the color of the tail was remarkably blue. Our analysis shows that the color of the tail is consistent with the color of the nucleus within the error bar. Actually, our intensity \nestimate of the tail (1 . 20 × 10 -15 W m -2 and 1 . 86 × 10 -15 W m -2 ) is consistent with Fernández et al. (1997) within a factor of 2. The di ff erence would be caused by background detection since the signal of the tail is very weak compared to the background. For reference, we recorded our method for photometric analysis in Appendix A. Since the details of the tail color analysis are not provided in Bowell et al. (1992) (where the authors describe the very blue color), it is di ffi cult to compare our results to their report, and we thus proceed with the discussion based on our findings. \nBecause of the color similarity between the tail and the Sun, we assume that the tail was responsible for the scattered sunlight by the ejected dust particles. Under this assumption, we investigated the particle size, total mass, and epoch / duration of dust ejection. From synchrone-syndyne analysis, we found that the particle size is significantly larger than the wavelength, having the size parameter X = 2 π a λ ≳ 100. This large dust size is also consistent with the neutral tail color, which is similar to sunlight. \nAfter the publication of Fernández et al. (1997), a new concept of active asteroids emerged. Asteroids eject their mass through lofting via electrostatic repulsion or thermal radiation pressure, thermal fatigue, impact, ice sublimation, and rotational disruption (Jewitt 2012; Bach & Ishiguro 2021; Molaro et al. 2020). Various recent observations widely support these ideas. Therefore, it is natural to assume that WH is also an active asteroid of main-belt origin, and it indicated an activity in 1949 for a reason. \nFirst, we consider the possibility of dust lofting by either electrostatic repulsion or thermal radiation pressure. Jewitt (2012) estimated dust size that can be ejected from a 1 km asteroid around 1 au as 1.5-5 µ m, suggesting that only ≲ 1 µ m-sized dust can be lifted on the WH's surface by electrostatic repulsion. In addition, Bach & Ishiguro (2021) examined dust lofting via thermal radiation pressure from the heated surface with various sizes at various solar distances. For the WH's size asteroid at its perihelion, no dust particles can be lifted up according to their calculation. Therefore, the hypothesis of dust lofting via these mechanisms can be ruled out since the particle size from our estimation is larger by at least an order of magnitude. \nSecond, thermal fatigue can break surface boulders and trigger exfoliation to launch some fragments. This mechanism was suggested as one of the most probable causes of the (101955) Bennu's activity (Molaro et al. 2020). However, WH was active only once in 1949, although there were > 15 observational opportunities (i.e., the perihelion passages) since 1949. Furthermore, it is di ffi cult to explain WH activity in terms of the mass of the ejecta by thermal fatigue. We estimated that 9 . 27 × 10 5 kg of dust was ejected from the WH within about one day. This is equivalent to a dust ejection rate of 10.7 kg s -1 . Since the mass ejection rate observed at Bennu is 10 -7 kg s -1 (Hergenrother et al. 2020), WH ejected nearly 8 orders of magnitude more dust than Bennu. The short duration of the ejection is also quite di ff erent from the nature of dust ejections observed at Bennu. Therefore, thermal fatigue is a mechanism less likely to cause the WH's 1949 activity. \nThird, an impact may trigger the activity of an asteroid (see, e.g., Ishiguro et al. 2011b). We estimated the diameter of a hypothetical impactor, which can produce ejecta whose mass is equivalent to tail mass, using a model in Holsapple (2022). We find that a ∼ 1m-sized object should collide with WH to generate the ejected mass. From interplanetary dust particle impact flux by Grün et al. (1985) and surface area estimated from a diameter by Bach et al. (2017), such impact would occur with a timescale of 10 8 years, longer than a typical lifetime of near-Earth asteroids. \nAlthough the impact probability is not zero (once per 10 8 years), the possibility of impact-triggered activity is considerably low. \nFourth, we consider volatile sublimation as observed in main-belt comets (MBCs, Hsieh & Jewitt 2006). The activities of MBC samples due to ice sublimation usually continued for several months (Hsieh et al. 2011a,b; Jewitt et al. 2014). The dust ejection velocity from MBCs is typically close to the escape velocity (Jewitt et al. 2014), which is similar to the WH's 1949 event. Although we derived the duration of the dust ejection of < 7 days from our non-zero ejection velocity model, this estimate should be significantly overestimated because the zero ejection velocity is unrealistic. Adopting the non-zero ejection velocity model, the duration of the activity would be ∼ 1 days. Considering such an impulsive dust ejection, the possibility of volatile sublimation for the 1949 activity is less likely. \nThe remaining possibility is that the WH tail was formed by rotational instability (see e.g., Jewitt et al. 2013). The low ejection velocity close to the escape velocity and the presence of large particles are consistent with the mechanism. We derived the total dust mass of 9 . 27 × 10 5 kg, which is only ∼ 10 -6 %of the nuclear mass ( ∼ 5 × 10 13 kg). Such rotation-induced mass shedding has been observed in various asteroids (Jewitt 2009). In the case of mass shedding, because the centrifugal force accelerates the surface material, large particles can also be ejected with nearly an escape velocity. Therefore, rotation-induced mass shedding is the most likely scenario for dust ejection based on our tail analysis. \nOur inference can be supported by the light curve analyses (Harris & Young 1983; Osip et al. 1995; Urakawa et al. 2011). The rotation period varies widely from 3.556 to 7.15 hours in the literature. Harris & Young (1983) reported a rotation period of 3.556 hours, while Osip et al. (1995) reported 6 . 1 ± 0 . 2. Urakawa et al. (2011) conducted a comprehensive analysis of the shape model, assuming the derived rotational period of 7.15 hours. However, Urakawa et al. (2011) also suggested the possibility of the solution for 3.58 hours. If the shorter solution is correct, WH still maintains a rapid rotation period. Such fast-rotating asteroids may create top shape, as observed for, for example, (66391) Moshup, (101955) Bennu, (162173) Ryugu (Ostro et al. 2006; Lauretta et al. 2019; Watanabe et al. 2019), and the shape reduces the rotational variability of the magnitude because of the axis-symmetrical shape. \nIn summary, WH is very likely an active asteroid of main belt origin. Although the possibility of volatile sublimation cannot be ruled out from our analysis, the possibility of rotational mass shedding is most likely a scenario for the 1949 activity based on our data and analyses, as well as the light curve properties derived by other research groups. \nAcknowledgements. We deeply appreciate an anonymous reviewer and Dr. Emmanuel Lellouch for the valuable comments and suggestions. This research at SNU was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (No. 2023R1A2C1006180). The Pirka, Nayuta, and Kanata telescopes are operated by the Graduate School of Science, Hokkaido University; the Center for Astronomy, University of Hyogo; and the Hiroshima Astrophysical Science Center, Hiroshima University, respectively. These telescopes are partially supported by the Optical and Infrared Synergetic Telescopes for Education and Research (OISTER) program funded by the MEXT of Japan. SI acknowledges financial support from INAF - Call for fundamental research 2022, Minigrant RSN3.", 'References': 'Akitaya, H., Moritani, Y., Ui, T., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9147, Ground-based and \nAirborne Instrumentation for Astronomy V, ed. S. K. Ramsay, I. S. McLean, &H. Takami, 91474O \n- Bach, Y. P. & Ishiguro, M. 2021, A&A, 654, A113\n- Bach, Y. P., Ishiguro, M., Takahashi, J., et al. 2024, A&A, 684, A81\n- Bach, Y. 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The opacitance can be calculated from the density of the emulsion δ as below: \nω = 10 δ -1 . (A.2) \nAccordingly, the opacitance of the region with a zero emulsion density becomes 0. Then, the magnitude of a point source from aperture photometry ( m ) is given by \nm = -2 . 5log 10 Z I source d Ω (A.3) \n= -2 . 5log 10 Z GLYPH<16> I aper -I sky GLYPH<17> d Ω (A.4) \n= -2 . 5log 10 A -2 . 5log 10 Z GLYPH<16> ω n aper -ω n sky GLYPH<17> , (A.5) \nwhere I source is the intensity from the source, and I aper, I sky, ω n aper and ω n sky are intensities and opacitances within the aperture and the sky background, respectively. \nWefirst set n in Eq. A.1 to 1 as an initial guess and conducted aperture photometry on the image using an aperture with the size of ∼ 2 × full-width half maximum of the point spread function in the image. Then, we compared the magnitudes of stars from plates to those of equivalent stars at the Pan-STARRS DR1 catalog, whose location is consistent within 3 '' di ff erence. (Chambers et al. 2016). We converted Pan-STARRS g , r , and i -band magnitudes into Johnson B - and Cousins R C-band using the relation from Kostov & Bonev (2017) to derive catalog magnitudes ( m cat). Cutri et al. (1993) showed the consistency between signals measured from Blue and Red plate images and those from photometry on CCD images with Johnson B - and Cousins R Cband, and their di ff erence is up to 0.3 magnitudes. For the blue plate image, we used 79 stars whose magnitude ranges from 17 to 19 magnitudes. Meanwhile, we used 120 stars with a magnitude range of 16 - 18 magnitudes for the red plate. Then, we plotted the catalog magnitude ( m cat) derived above on the x-axis and the plate magnitude ( m plate) on the y-axis, defined as \nm plate = -2 . 5log 10 Z GLYPH<16> ω n aper -ω n sky GLYPH<17> . (A.6) \nWe calculated the slopes a ( n ) and their intercepts b ( n ) of a linear regression line using the least square approximation. We determined the n value for each plate that makes the slope to be unity using the chi-square minimization technique. We calculated the reduced chi-square value using the formula below, \nχ 2 red = N X i = 1 m cat , i -m plate , i -b σ i ! 2 (A.7) \nwhere m cat , i and m plate , i are catalog and plate magnitudes of the i -th star within the image. 1-sigma confidence intervals of n is estimated by χ 2 ( n ) < χ 2 ( n ) + q 1 . \nred red, min N -2 \nWe derived the A values using the formula in Cutri et al. (1993), \nA = 10 b -2 . 5 I 0 ∆ λ e ff , (A.8) \nwhere I 0 is zero magnitude intensity of a filter, and ∆ λ e ff is its effective width of the filter. We used zero magnitude intensities and e ff ective widths of Johnson B and Cousins R C from the Spanish Virtual Observatory (SVO) Filter Profile Service 4 . 1-sigma confidence intervals of the A values are calculated from those of n . Finally, the linearization formulae derived for blue and red plate images are shown below: \nI Blue = (9 . 972 ± 3 . 682) × 10 -16 × ω 0 . 594 ± 0 . 066 , (A.9) \nI Red = (7 . 564 ± 1 . 183) × 10 -15 × ω 0 . 951 ± 0 . 029 . (A.10)", 'Appendix B: Error estimation of photographic plate images': 'In this study, we considered three sources of uncertainty regarding digitized photographic plate images. The first uncertainty source is from the linearization coe ffi cients ( σ A , σ n ) we derived in Sect. A. The second one is the inhomogeneous thickness of the emulsion, which causes background fluctuation in the image. The level of background fluctuation is estimated by background RMS value ( σ bg, RMS) calculated by Source-Extractor (Bertin & Arnouts 1996). The last one is the Poisson noise of incident photons, σ Poisson = √ N , where N is the number of incident photons to the plate. We estimated N from intensities using the following equation \nN = It exp h ν e ff , (B.1) \nwhere t exp, h , and ν e ff are the exposure time, Planck constant, and the e ff ective frequency of each band, respectively. From Eq. A.1, the Poisson noise of ω ( σω, Poisson) is given as \nσω, Poisson = ω n √ N . (B.2) \nWith σ A , σ n , and σω, Poisson, we can derive the RMS values of ω n and I using the following equations \nσω n = ω n r GLYPH<18> n ω σω, Poisson GLYPH<19> 2 + (ln ω σ n ) 2 , (B.3) \nσ I = I r GLYPH<18> σ A A GLYPH<19> 2 + GLYPH<18> σω n ω n GLYPH<19> 2 . (B.4) \nFinally, an uncertainty of a background-subtracted signal ( I sig = I -I bg), σ I sig , is given as \nσ I sig = q σ 2 I + σ 2 I bg + σ 2 I bg, RMS . (B.5)', 'Appendix C: Result of fitting for PPCs': 'The results from the fitting of PPCs are summarized in Table C.1. \nTable C.1. Summary of best-fit parameters and their 1-sigma uncertainties. \nNotes. ( a ) Numbers of data used for fitting'}
2024MNRAS.535.2651K	We present a study of the detached eclipsing binary TV Mon using spectra from the LAMOST Large Sky Area MultiObject Fiber Spectroscopic Telescope mediumresolution survey ASASSN All Sky Automated Survey for SuperNovae and CoRoT Convection Rotation and planetary Transits photometry. We apply multipleepochs spectral fitting to derive radial velocities and spectral parameters. The analysis of eclipses in CoRoT data shows the relative sizes of the stellar components and almost edgeon circular orbit. Combining the spectral and photometrical solutions we estimate masses and radii of the components inlineformulatexmath idTM0002 notationLaTeXMrm AB2.063pm 0.033rm stat.pm 0.095rm syst. 0.218pm 0.004rm stat.pm 0.018rm syst. mathrm Modottexmathinlineformula inlineformulatexmath idTM0003 notationLaTeXRrm AB2.394pm 0.014 2.860pm 0.016 mathrm Rodottexmathinlineformula. Spectral energy distribution analysis and Gaia parallax allow us to get an estimation of temperatures inlineformulatexmath idTM0004 notationLaTeXTrm effrm AB7624194174 5184130123 mathrm Ktexmathinlineformula and distance inlineformulatexmath idTM0005 notationLaTeXd907pm 11 mathrm pctexmathinlineformula. We identify three inlineformulatexmath idTM0006 notationLaTeXdeltatexmathinlineformula Scutitype pulsation frequencies in the primary component while we also suspect TV Mon having a spot activity in the secondary component. This system experienced intensive mass transfer and mass ratio reversal in the past but currently shows no signs of mass transfer in the spectra. The lowmass component will lose its outer envelope and shrink to the helium white dwarf the mass and orbital period of which are in good agreement with evolutionary model predictions.	2024-12-01T00:00:00Z	['2024arXiv240909902K', '10.1093/mnras/stae2494', '10.48550/arXiv.2409.09902', 'arXiv:2409.09902', '2024MNRAS.535.2651K', '2024MNRAS.tmp.2436K']	['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics']	TV Mon postmass transfer Algoltype binary with Scuti pulsations in primary component	2,024	194	0.52	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.09902.pdf	{'TV Mon - post mass transfer Algol type binary with 𝛿 Scuti pulsations in primary component.': 'Mikhail Kovalev, 1 , 2 , 3 ★ Zhenwei Li, 1 , 2 Jianping Xiong, 1 , 2 Azizbek Matekov, 1 , 4 , 5 Zhang Bo, 5 Xuefei Chen 1 , 2 , 6 and Zhanwen Han 1 , 2 , 6 \n- 1 Yunnan Observatories, China Academy of Sciences, Kunming 650216, China\n- 2 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650011, China\n- 3 International Centre of Supernovae, Yunnan Key Laboratory, Kunming 650216, China\n- 4 University of Chinese Academy of Sciences, Yuquan Road 19, Beĳing 100049 Sĳingshang Block, China\n- 5 Ulugh Beg Astronomical Institute, Uzbekistan Academy of Sciences, 33 Astronomicheskaya str., Tashkent, 100052, Uzbekistan\n- 6 Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beĳing 100101, China\n- 7 Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beĳing 100012, China \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': 'Wepresent a study of the detached eclipsing binary TV Mon using spectra from the LAMOST medium-resolution survey and ASAS-SN, CoRoT photometry. We apply multiple-epochs spectral fitting to derive RV and spectral parameters. The analysis of eclipses in CoRoT data show the relative sizes of the stellar components and almost edge-on circular orbit. Combining the spectral and photometrical solutions we estimate masses and radii of the components: 𝑀 𝐴,𝐵 = 2 . 063 ± 0 . 033 ( stat . ) ± 0 . 095 ( syst . ) , 0 . 218 ± 0 . 004 ( stat . ) ± 0 . 018 ( syst . ) 𝑀 ⊙ , 𝑅 𝐴,𝐵 = 2 . 394 ± 0 . 014 , 2 . 860 ± 0 . 016 𝑅 ⊙ . SED analysis and Gaia parallax allow us to get estimation of temperatures 𝑇 eff 𝐴,𝐵 = 7624 + 194 -174 , 5184 + 130 -123 Kand distance 𝑑 = 907 ± 11 pc. We identify three 𝛿 Scuti type pulsation frequencies in the primary component, while we also suspect TV Mon having a spot activity in the secondary component. This system experienced intensive mass transfer and mass ratio reversal in the past, but currently shows no signs of mass transfer in the spectra. The low mass component will lose its outer envelope and shrink to the helium white dwarf, the mass and orbital period of which are in good agreement with evolutionary models predictions. \nKey words: stars : fundamental parameters - binaries : eclipsing - binaries : spectroscopic stars individual: TV Mon', '1 INTRODUCTION': "tometry and multiple-epoch spectra fitting, developed in Kovalev et al. (2022a, 2023). \nDetached eclipsing binaries (dEBs) are very important astronomical objects, which allow us to study stellar structure and evolution (Torres et al. 2010; Han et al. 2020). Many close eclipsing binaries have significantly different masses of the components, which indicates there was mass-transfer in the past (Chen et al. 2020). Quite often, high-frequency pulsations can be found in these systems using precise space-based photometry (Chen et al. 2021; Wang et al. 2023). Recently Xiong et al. (2023) derived masses and radii for 56 dEBs by using the LAMOST (Large Sky Area Multi-Object fiber Spectroscopic Telescope, also known as Guoshoujing telescope) medium-resolution survey (MRS) spectra and ground-based timedomain photometry. We select one of these stars, TV Mon, due to it's large mass ratio and analyse it using precise space-based pho- \nTV Mon was discovered on photographic plates by Sergei Iwanovitsch Beljawsky and classified as Algol-type eclipsing binary (Guthnick et al. 1924). The catalogue by Svechnikov & Kuznetsova (1990) reported it as a system with semi-detached configuration. With a location close to the Rosette nebula, many studies consider it is being a member of the associated open cluster NGC 2237 (Sahade & Berón Dàvila 1963; Popova & Kraicheva 1984; Zakirov & Shaidullin 1985). However the most recent analysis rejects membership with probability of association ∼ 0 . 018 (Mužić et al. 2022). \nThe paper is organised as follows: in Sections 2 and 3 we describe the observations and methods. Section 4 presents our results. In Section 7 we discuss the results in context of binary system evolution. In Section 8 we summarise the paper and draw conclusions.", '2.1 Spectra': 'LAMOST is a 4-meter quasi-meridian reflective Schmidt telescope with 4000 fibers installed on its 5 · field of view focal plane. These configurations allow it to observe spectra for at most 4000 celestial objects simultaneously (Cui et al. 2012; Zhao et al. 2012). All available spectra were downloaded from www.lamost.org/dr9/ under the designation J062822.72+051257.2. We use the spectra taken at a resolving power of R = 𝜆 / Δ 𝜆 ∼ 7 500. Each spectrum is divided into two arms: blue from 4950 Å to 5350 Å and red from 6300 Å to 6800 Å. We convert the heliocentric wavelength scale in the observed spectra from vacuum to air using P/y.pcA/s.pc/t.pc/r.pc/o.pc/n.pc/o.pc/m.pc/y.pc (Czesla et al. 2019) and apply zero point correction using values from Zhang et al. (2021), when it is possible. Observations are carried out from 2019-02-16 till 2020-03-05, covering 12 nights with time base of 373 days. The period is short ( 𝑃 = 4 . 179742 d Kreiner (2004)) therefore we analysed spectra taken during short 20 minutes exposures, unlike Kovalev et al. (2022a,b), where spectra stacked for the whole night were used. In total we have 35 spectra, where the average signal-to-noise ratio (S/N) of a spectrum ranges from 9 to 60 pix -1 for the blue arm and from 10 to 82 pix -1 for the red arm of the spectrum, with the majority of the spectra having S/N around 40 pix -1 .', '2.2 Photometry': 'We extracted publicly available 𝑔 and 𝑉 band light curves (LCs) from the ASAS-SN portal 1 . They contain 598 and 4396 datapoints respectively and covers a timebase of 3596 days. \nThe CoRoT (Convection, Rotation and planetary Transit) satellite mission(Barge et al. 2008) provides very high-quality LCs, due to lack of atmospheric noise and high sensitivity. We download it 2 and use systematic-corrected \'WHITEFLUXSYS" data, with \'STATUS" field equals to zero (3443 datapoints in total, time coverage of ∼ 32 days). We normalise this dataset by dividing it by median values in the out-of-eclipse regions of the two time intervals: BJD<2454753.5 days (exposure time 512 𝑠 ) and BJD>2454753.5 days (exposure time 32 𝑠 ). The phase-folded LC shows two eclipses with short, flat minima ( ∼ 40 min), while many small gaps and high-frequency pulsations are clearly visible, see Figure 1. These gaps are artefacts of removal of the \'jumps", caused by the cosmic radiation, which mostly occurred when the satellite crossed the South Atlantic Anomaly (SAA) (Auvergne et al. 2009; Sokolovsky et al. 2010). \nTimeseries from the WISE satellite(Cutri et al. 2013) are available for this system in 𝑊 1and 𝑊 2filters, see Appendix B. In infrared light the secondary eclipse is much deeper than in visual, which indicates a larger contribution of the secondary component to the total light of the system in this regime. However these LCs contain only 78 datapoints each and limb darkening coefficients are unavailable for these filters, so we don\'t use these data for the LC analysis. \nUnfortunately the Transiting Exoplanet Survey Satellite (TESS Ricker et al. 2015) mission did not observe this system yet, because it is located close to the ecliptic plane, exactly in the gap between two sectors of observations. However the TESS input catalogue \n(Stassun et al. 2019) contains it as TIC 234726486, so one can expect a new observations in the future.', '3.1 Spectral fitting': 'Our spectroscopic analysis includes two consecutive stages: \n- (i) analysis of individual observations by binary and single-star spectral models, where we normalise the spectra and make a rough estimation of the spectral parameters, see brief description in Section 3.1.1. \n(ii) simultaneous fitting of multiple-epochs with a binary spectral model, using constraints from binary dynamics and values from the previous stage as an input, see Section 3.1.2.', '3.1.1 Individual spectra.': 'The single-star spectral model is the same as in Kovalev et al. (2024) and described in Appendix A. The normalised binary model spectrum is generated as a sum of the two Doppler-shifted, normalised single-star model spectra 𝑓 𝜆,𝑖 scaled according to the difference in luminosity, which is a function of the 𝑇 eff and stellar size. We use following equation: \n𝑓 𝜆, binary = 𝑓 𝜆, 2 + 𝑘 𝜆 𝑓 𝜆, 1 1 + 𝑘 𝜆 , 𝑘 𝜆 = 𝐵 𝜆 ( 𝑇 eff , 1 ) 𝐵 𝜆 ( 𝑇 eff , 2 ) 𝑘 𝑅 , (1) \nwhere 𝑘 𝜆 is the luminosity ratio per wavelength unit, 𝐵 𝜆 is the blackbody radiation (Plank function), 𝑇 eff is the effective temperature, 𝑘 𝑅 - light ratio coefficient. Throughout the paper we always assume the primary star to be brighter. \nThe binary model spectrum is later multiplied by the normalisation function, which is a linear combination of the first four Chebyshev polynomials (similar to Kovalev et al. 2019), defined separately for blue and red arms of the spectrum. The resulting spectrum is compared with the observed one using the scipy.optimise.curve\\_fit function, which provides optimal spectral parameters, radial velocities (RV) of each component plus the light ratio and two sets of four coefficients of Chebyshev polynomials. We keep metallicity equal for both components. In total we have 18 free parameters for a binary fit.', '3.1.2 Multiple-epochs fitting.': "We explore the results from the fitting of the individual epochs and find that a result's quality clearly depends on separation of RVs. Clear double-lined spectra show that spectral lines are significantly broadened ( 𝑉 sin 𝑖 ∼ 30 kms -1 ), thus the Rossiter-McLaughlin (RM) effect (Rossiter 1924; McLaughlin 1924) can be observed during eclipses. \nIf two components in our binary system are gravitationaly bound, their radial velocities should agree with the following equation: \nRV A = 𝛾 ( 1 + 𝑞 ) -𝑞 RV B , (2) \nwhere 𝑞 is the mass ratio and 𝛾 is the systemic velocity. Using this equation we can directly measure the systemic velocity and mass ratio. We should note that this equation is valid assuming the of absence of the RM effect, which can be justified by excluding spectral observations during the eclipses, and no difference in \nFigure 1. Normalised CoRoT light curve, with inline plot showing gaps and high-frequency pulsations (top panel) and phased eclipses, showing flat regions (bottom panel). \n<!-- image --> \ngravitational redshifts for two components Δ 𝛾 𝑖 . Given high mass ratio, we estimate this difference Δ 𝛾 𝐴 -Δ 𝛾 𝐵 ∼ 0 . 5 kms -1 using system parameters from Xiong et al. (2023) and Equation 1 from El-Badry (2022), which is comparable to the usual precision of our RV measurements and the typical zero point offsets of LAMOSTMRS (Zhang et al. 2021). Thus this effect can be neglected in our analysis. \nToreduce the number of free parameters in the multiple-epochs fitting we can select only one component RV for fitting and compute values for another one using Equation 2. \nWe fit previously normalised individual epoch's spectra, using their binary spectral parameters values for initialisation. We repeat this five times using only manually selected best individual epoch's fit initialisation. We select the spectroscopic solution with minimal 𝜒 2 as a final result. Surface gravity log ( g ) is poorly constrained by the spectra alone, thus we assign a value for log ( g ) : initially we use to values from Xiong et al. (2023), and then to new estimation from the LC fit, see Section 3.2. We repeat this iteratively until both spectra and LC were reasonably fitted. \nFinally we get new RV measurements for selected component, 𝑞, 𝛾 and spectral parameters for both components. We can repeat these calculations taking another component's RVs as a fitting parameters, but derived spectral parameters will be swapped and \nmass/light ratio will be inverted. Agreement between two solutions can validate assumptions of Equation 2. Then we can use these RV measurements of both components in further analysis.", '3.2 Light curve fitting': 'Visual inspection of CoRoT LC indicates that there is a short total eclipse during the main minimum and a transit during the secondary minimum, separated by half of the period. Thus it is reasonable to assume an edge-on circular orbit with \'central" eclipse (inclination 𝑖 = 90 · ) and compute approximations for relative radii 𝑅 𝑖 / 𝑎 using pure geometry of spherical stars, see Formulae 12,13 from the great book by Tsesevich (1980): \n𝑅 𝐴 / 𝑎 + 𝑅 𝐵 / 𝑎 = sin Θ 1 , (3) \n𝑅 𝐵 / 𝑎 -𝑅 𝐴 / 𝑎 = sin Θ 2 , (4) \n2 Θ 1 = 𝐷, 2 Θ 2 = 𝐷 tot , (5) \nwhere 𝐷 and 𝐷 tot are durations of eclipse and it\'s total (flat) part, both divided by period, respectively, and 𝑎 is a radius of a circular orbit. This gives us 𝑅 𝐴,𝐵 / 𝑎 ∼ 0 . 163 , 0 . 190. These values can be used to initialize detailed LC modelling. \nWe analyse the available LC datasets in two steps using different codes. Originally CoRoT LC and RV data were fitted by \nJKTEBOP and Transit Light Curve Modeler code (TLCM). These two codes assume a spherical shape of the stars and works extremely fast, although i is unable to fit several LC simultaneously. We use their solutions to check the consistency of our data and to derive period, mass ratio and orbital parameters of the system. Details about TLCM modelling can be found in Appendix D. Subsequently the /e.pc/l.pc/l.pc/c.pc code was used to simultaneously fit ASAS-SN and CoRoT LCs with fixed period, mass ratio and orbit from the previous step.', '3.2.1 JKTEBOP': 'We used the JKTEBOP code (version 40) 3 by Southworth (2013) to simultaneously fit the LC and RV timeseries. We used quadratic limb darkening coefficients provided by the JKTLD code 4 and linearly interpolated them for the spectral parameters. The eccentricity was set to zero and the systemic velocity was fitted for both binary components. In total we fit for 16 parameters: 𝐽 the central surface brightness ratio, ( 𝑅 𝐴 + 𝑅 𝐵 )/ 𝑎 the ratio of the sum of stellar radii to the semimajor axis, 𝑅 𝐵 / 𝑅 𝐴 the ratio of the radii, 𝑖 the inclination, 𝑆 0 nuisance parameter for the out-of-eclipse magnitude, reflected light 𝐴,𝐵 , 𝑃 the period, 𝑡 0 , semiamplitudes and systemic velocities 𝐾 𝐴,𝐵 , 𝛾 𝐴,𝐵 , and 𝑆 0 . We use integration ring size 1 · . \nAt first, we run the JKTEBOP code in the mode \'Task 4\' to discard outliers larger than three sigma (only 26 datapoints were removed from LC dataset, while all 23 RV were kept for both components) and allow it adjusting observational errors for LC and RV data, through several iterations until the reduced 𝜒 2 reaches unity. Then we run JKTEBOP in the Monte Carlo mode (Task 8 \'MC") to estimate uncertainties using 10000 simulations.', '3.2.2 ELLC': 'The /e.pc/l.pc/l.pc/c.pc code (Maxted 2016), uses triaxial ellipsoids for each component\'s shape, which is better approximation than the sphere, especially when \'roche" shape is chosen. In order to save computational time we use only LC data from CoRoT and ASAS-SN without RV and fix 𝑎 sin 𝑖, 𝑞, 𝑃 to the values from the previous solution. Limb and gravity darkening coefficients were computed using interpolator built-in to /e.pc/l.pc/l.pc/c.pc. Both LCs were fitted simultaneously with fractional radii 𝑅 𝐴,𝐵 / 𝑎 , inclination 𝑖 , 𝑡 0 , third light 𝐿 3 𝑖 , surface brightness ratios for all datasets 𝐽 𝑔 , 𝐽 𝑉 , 𝐽 𝐶 5 and \'heat" parameters ℎ 𝐴,𝐵 which take into account reflection effects for both components. Synchronicity parameters 𝐹 1 , 2 were set to 1, after several trial runs with different values, which couldn\'t reproduce flat minima. After a series of sets and trials we found out that the code is unable to fit flat parts of minima when whole CoRoT LC was used, thus we restrict our analysis only to eclipses from the time interval BJD=2454755.5:2454765.5 d (281 datapoints), which have less noise and shows clear flat minima with no signs of long-period variability, see Figure 1. The out-of-eclipse regions were fitted using ASAS-SN LCs, which are less affected by stellar pulsations, due to lower photometrical precision. Additionally, we found that the ℎ 𝐴 parameter was always zero, thus we fix it to this value. This is possibly due to insufficient precision of ASAS-SN LC to get an estimation of the very weak reflection effect. To get uncertainties \nTable 1. Spectral parameters from multiple-epoch fitting \nwe sample the solution using /e.pc/m.pc/c.pc/e.pc/e.pc (Foreman-Mackey et al. 2013) with 50 walkers and 2000 iterations.', '4 RESULTS': "In the Figure 2 we show the best fit by multiple-epochs binary model for two epochs with large RV separation. We zoom into the wavelength range around the magnesium triplet, H 𝛼 and in a 70 Å interval in the red arm, where many double lines are clearly visible. We can see that the primary star ( 𝑇 eff = 7232 𝐾, log ( g ) = 3 . 96 cgs, [Fe/H] = 0 . 2 dex, 𝑉 sin 𝑖 = 32 kms -1 ) contributes around 75% in the spectrum, while the secondary star ( 𝑇 eff = 5803 𝐾, log ( g ) = 2 . 90 cgs, [Fe/H] = 0 . 2 dex, 𝑉 sin 𝑖 = 41 kms -1 ) contributes to the remaining 25%. The derived mass ratio is 𝑞 ∼ 0 . 11 and systemic radial velocity 𝛾 = 31 . 70 kms -1 . Additionally we make another multiple-epoch fit with the secondary component's RV, finding the same spectral parameters with inverted mass ratio 1 / 𝑞 = 9 . 47. We present all spectroscopic results in the Table 1. The errors in the spectral parameters provided by scipy.optimise.curve\\_fit are nominal and largely underestimated. Based on simulations with synthetic spectra in Kovalev et al. (2022a, 2023), typical errors of multiple-epoch spectral analysis are Δ 𝑇 eff 𝐴,𝐵 ∼ 240 , 360 𝐾, Δ log ( g ) 𝐴,𝐵 = 0 . 15 , 0 . 20 cgs, Δ [Fe/H] = 0 . 2 dex, Δ 𝑉 sin 𝑖 𝐴,𝐵 = 10 , 30 kms -1 . In principle, one can use spectroscopic 𝑉 sin 𝑖 measurements to constrain the synchronicity parameters for LC modeling; however, our 𝑉 sin 𝑖 can be an overestimate, since we assume rotation as only one source of line broadening. In reality one also needs to take into account the macroturbulence velocity (Gray 2005). \nIn Figure 3 we show the best fits of the CoRoT LC (top) and RV(bottom) by JKTEBOP, while the zoomed version which covers only eclipses can be found in Figure 4. The fit residuals (O-C) are typically small: ≤ 0 . 02 mag in the total eclipse, ≤ 0 . 01 mag for LCs outside of eclipse regions ( mostly due to high-frequency oscillations), and ≤ 4 kms -1 for RVs. Only seven measurements for RV 𝐵 are off by ≥ 1 kms -1 and correspond to relatively low S/N spectra. The derived systemic velocities and mass ratio are similar to the values derived from a multiple-epoch spectral fit. Note 𝛾 𝐴 > 𝛾 𝐵 , as one would expect after taking into account the gravitational red shift, although this difference is well below uncertainties. We present results from JKTEBOP in the Table 2. \nThe oblateness of the components (0 . 0008 , 0 . 0869) is small, but for the secondary it is larger than the limit of 0 . 04 (Popper & Etzel 1981), which can explain the relatively large residuals in total eclipse; therefore the JKTEBOP solution should be supplemented by another code, for example /e.pc/l.pc/l.pc/c.pc. \nThere is another set of RV measurements from our spectra in Zhang et al. (2021), which we can use to check for possible systematic errors of our analysis. These RV were derived independently \nFigure 2. Example of the multiple-epoch fitting for spectra taken at 𝜙 = 0 . 274, MJD=58909.566 d (top) and 𝜙 = 0 . 749, MJD=58911.552 d (bottom). We zoom into the wavelength range around the magnesium triplet, H 𝛼 and in a 70 Å interval in the red arm. The observed spectrum is shown as a gray line, the best fit is shown as red line. The primary component is shown as the orange line, the secondary as a blue line. The fit residuals are shown as a green line. \n<!-- image --> \nfor the blue and red arms of the spectrum. We only used values derived from the blue arm of the spectra, because results for the red arm often are bad for the secondary component. We also excluded RV values from spectra taken during eclipses, to be consistent with multiple-epoch measurements. The results of JKTEBOP for these RVs are presented in the right column of the Table 2. 𝐾 𝐴,𝐵 are slightly smaller than for the multiple-epochs solution, which propagates into the derived masses and sizes of components. The root mean square of the residuals is smaller for RVs from simultaneous fit of multiple-epochs, thus we prefer them for further analysis. \nThe results derived by /e.pc/l.pc/l.pc/c.pc are collected in Table 3. They are the median values and standard deviations for all fitted parameters. The related solution is depicted in Figure 5. This model provides an excellent fit for eclipses in CoRoT LC, typically resulting in residuals of ≤ 0 . 01 mag. The ASAS-SN LCs are also well fitted, with typical residuals of ≤ 0 . 05 and ≤ 0 . 04 mag for 𝑔 and 𝑉 filters respectively. If we compute fit residuals for the CoRoT datapoints outside the time interval BJD=2454755.5:2454756.5 d, they are \nsignificantly larger - up to 0.03 mag during the total eclipse, while there is almost no difference for the secondary minimum. This can indicate a slight long period variability of the secondary component (see Section 6), which is not included in our LC model. The resulting values of radii differ from the previous solution primary being slightly smaller 𝑅 𝐴 = 2 . 39 𝑅 ⊙ and the secondary is slightly larger 𝑅 𝐵 = 2 . 86 𝑅 ⊙ . The secondary radius is less than the critical value: 𝑅 𝐵 / 𝑎 = 0 . 2020 < 𝑅 lim 𝐵 / 𝑎 = 0 . 2869. The inclination is very similar to that in the JKTEBOP solution, thus the masses are almost the same as in the previous solution, which provided us 𝑎 sin 𝑖 and 𝑞 . The corner plot for the fitted parameters can be found in Figure E1.", '4.1 Verification with W-D': 'We use the Wilson-Devinney method W-D (Wilson & Devinney 1971; Wilson 1979) with PYWD2015 (Güzel & Özdarcan 2020) user interface to verify the /e.pc/l.pc/l.pc/c.pc solution. Same datasets were fitted by differential correction program with dimensionless potentials Ω , \nTable 2. JKTEBOP solutions using two RV datasets. Error estimates are from MC simulations. \ninclination, 𝑇 eff 𝐵 , the albedo of the secondary 𝐴 𝐵 as free parameters. The bandpass luminosity 𝐿 1 of the primary component and third light was also fitted for two LC, while the parameter 𝐿 2 was computed by W-D, based on the temperatures of the components. All other parameters were fixed, including the synchronicity parameters 𝐹 1 , 2 = 1. We show the solution derived after 21 iterations in Figure 6 and Table 3. Residuals for the eclipses are ≤ 5 mmag for CoRoT dataset, ASAS-SN 𝑔, 𝑉 mostly have residuals ≤ 0 . 05 mag, with larger residuals at the primary eclipse. Overall the residuals are smaller than in /e.pc/l.pc/l.pc/c.pc possibly due to usage of the proper Roche geometry. The derived parameters are slightly different from /e.pc/l.pc/l.pc/c.pc, with the secondary component being a bit smaller and the primary is slightly larger. Also, one can get a good fit, only with non-zero third light contribution. \nSince W-D provides no errors for many parameters and the difference between /e.pc/l.pc/l.pc/c.pc and W-D solutions is relatively small, we choose /e.pc/l.pc/l.pc/c.pc solution as a final result. Additionally we run W-D using all available data from the CoRoT dataset, together with both LCs from ASAS-SN finding similar results, see Appendix F.', '5 SED FITTING': 'The spectral energy distribution (SED) offers an independent approach to estimating system parameters like 𝑇 eff . We employ available photometry and aim to utilize out-of-eclipse data whenever feasible. We utilize the /s.pc/p.pc/e.pc/e.pc/d.pc/y.pc/f.pc/i.pc/t.pc package 6 for SED fitting, using Kurucz (1979) models in conjunction with a Markov chain Monte Carlo method to identify the optimal fit and estimate the errors associated with the fitting parameters. We impose constraints on the solution by employing the parallax 𝜛 = 1 . 0654 ± 0 . 0201 mas from Gaia DR3(Gaia Collaboration et al. 2023), along with masses and radii derived from the LC solution. Errors in the constraints are included as priors in the Bayesian fitting process and are thus propagated to the final results. A more detailed description of the fitting process is given in Vos et al. (2017). In Figure 7, we present the resulting fit, while Table 4 lists the parameters. The estimated distance of 𝑑 = 907 ± 11 pc is less than the single-star model value of 𝑑 = 982 1051 943 pc from Gaia DR3. The 𝑇 eff values deviate somewhat from the spectroscopic measurements, with the secondary component being cooler and the primary component slightly warmer in \nFigure 3. Phase-folded LC (top) and orbit (bottom) fits with JKTEBOP. The magnitudes are not calibrated. \n<!-- image --> \nFigure 4. Same as Figure 3 for phases around eclipses. \n<!-- image --> \nthe SED fit. Most photometric measurements did not account for eclipse-related observations, apart from 𝑊 1 and 𝑊 2, which may explain the discrepancy, although alternative explanations could exist.', '6 PULSATIONS ANALYSIS': 'To explore pulsations in TV Mon we use a different CoRoT photometry dataset \'WHITEFLUXFIL", available in the same datafile, which was not corrected for systematics, but has significantly more datapoints. These data have an exposure time of 32 𝑠 and contain 68103 datapoints with \'STATUS" field equal to zero, which ensures that artifacts like \'jumps" due to crossing of SAA by satellite were excluded. We normalise and flatten this LC by fitting a parabola to out-of-eclipse regions. Next, we use a similar approach to the \nTable 3. LC solutions by /e.pc/l.pc/l.pc/c.pc and W-D. Error estimates are from median and standard deviations of /e.pc/m.pc/c.pc/e.pc/e.pc sampling for /e.pc/l.pc/l.pc/c.pc, while for W-D they are standard errors from the differential correction program. \nChen et al. (2021) analysis of OO Dra. As our LC model unable to capture all possible variations of the LC for all time intervals, we computed an averaged curve for the phase-folded LC and then subtract it from the original LC. We use 301 phase bins, to ensure that each bin contains enough datapoints (typically 100-300). We show the mean LC and eclipses from the start and end of the original LC in Figure 8. We see no pulsations in the primary eclipse, although it\'s flat part depth changes with time, which can indicate long period variability of the secondary component, due to fast changing \nFigure 5. The LC solution from the /e.pc/l.pc/l.pc/c.pc code. We fit whole ASAS-SN datasets, but only a part of CoRoT data from eclipses in the time interval BJD=2454755.5:2454765.5 was used in fitting (top panels). Full LCs are shown in bottom panel, while the fit residuals are shown below each panel. For the CoRoT data we also show the residuals computed for each datapoint, which were not used during fitting (gray/black open circles). \n<!-- image --> \nspots or some instrumental effect in the CoRoT observations. The secondary eclipse shows clear pulsations, with the the flat part of the minima having similar depth. Thus, high-frequency pulsations come only from the primary component and LC data with phases from -0.06 till 0.06 were excluded from further analysis, leaving us with dataset of 60150 points. \nWe explore pulsations using the P/e.pc/r.pc/i.pc/o.pc/d.pc04 software (Lenz & Breger 2005). We follow the steps described in Chen et al. (2021) and Wang et al. (2023) to iteratively find frequencies in the range from 0 to 50 cycles per day (no strong peaks were found with larger frequencies), until the residual Fourier spectrum has no peaks with S/N > 4. We identified 29 frequencies which are listed in Table 5 and shown in the top panel of Figure 9, while the best fit pulsations model is shown in the bottom panel. None of the frequencies match to 𝑓 sat = 1 / 𝑃 sat = 86400 𝑠 / 6182 𝑠 = 13 . 9715 day -1 and it\'s multiples, where 𝑃 sat is the CoRoT satellite orbital period(Auvergne et al. 2009). Thus the CoRoT team successfully removed artifacts which \nFigure 6. The LC solution by W-D using the same data as the /e.pc/l.pc/l.pc/c.pc code. Top panels show fit of the data, bottom panels show fit residuals. \n<!-- image --> \nmight have arisen due to SAA crossing 7 . With a frequency resolution of 0 . 055 day -18 wealso searched for possible orbital harmonics ( 𝑓 𝑖 = 𝑁 𝑓 orb , 𝑓 orb = 1 / 𝑃 = 0 . 239249 ± 0 . 000001 day -1 ) and combination frequencies. Unfortunately, many peaks (i.e. 𝑓 1 , 𝑓 3 , 𝑓 5 , 𝑓 9 ) possibly originate from the imperfect removal of binarity-induced light variations, although some of them can be the result of instrumental effects or long period variability of the secondary \nFigure 7. SED fitting with /s.pc/p.pc/e.pc/e.pc/d.pc/y.pc/f.pc/i.pc/t.pc. The top panels shows the observations and spectral energy distribution of the system (red line), primary (blue line) and secondary (green line). The fit residuals are shown on the bottom panel. \n<!-- image --> \nFigure 8. Mean LC and data for eclipses taken in the start and end parts of LC. \n<!-- image --> \nTable 4. SED fitting results and observed photometry from Gaia DR3 Gaia Collaboration et al. (2023), APASS Henden et al. (2015), SKYMAPPER Wolf et al. (2018), 2MASS Skrutskie et al. (2006) and WISE Cutri et al. (2013). WISE values were computed as the mean and standard deviation of out-of-eclipse parts of the time-series. \ncomponent. For example, the period value 𝑃 17 ∼ 128 d, corresponding to 𝑓 17 , is similar to the long period 𝑃 long ∼ 30 𝑃 orb , detected in many Algol-type systems (Mennickent 2017). Only 𝑓 2 = 23 . 57 , 𝑓 4 = 27 . 53 , 𝑓 6 = 20 . 87 day -1 can be considered as independent frequencies, possibly 𝛿 Scuti pressure modes. Although 𝑓 5 is close to 115 𝑓 orb , it has a very strong peak in the Fourier spectrum and small error, which is enough to conclude that it is just an accidental agreement. Star A is likely to be a source of these pulsations, so we computed pulsation constants 𝑄 = 𝑃 pul √︁ 𝜌 / 𝜌 ⊙ , where 𝑃 pul is the pulsation period, 𝜌 is the mean density of the star. We calculate 𝑄 = 0 . 014 , 0 . 016 , 0 . 018 day for 𝑓 4 , 𝑓 2 , 𝑓 6 respectively. We leave detailed physical modelling of these pulsations (see an example in Chen et al. (2021)) to future studies.', '7.1 System parameters': 'This is an Algol-like eclipsing binary system, which experienced mass transfer and mass-ratio reversal in the past. The massive primary component is hot, while the much lighter secondary star is a cool red giant. Based on the absence of emission lines in H 𝛼 region and secondary radius smaller than 𝑅 lim 𝐵 we can conclude that mass transfer is not active anymore. \nWe checked the \'O-C gateway" (Paschke & Brat 2006) to ensure stability of the period. It contains primary minima times from 1914 till 2019, based on visual, photographic and charge-coupled device (CCD) observations. When taking into account only the most accurate CCD based measurements, the period is constant, but a larger period ( 𝑃 ∼ 4 . 17977 𝑑 ) is needed to fit earliest photographic measurements. If we trust these measurements, the period \nhas slightly decreased and then become constant, although accuracy of the old photographic measurements can be very small and is hard to be evaluated now. Also TV Mon have total eclipse with duration ∼ 40 min, which can cause additional bias if not taken into account. \nAnalysis of pulsations revealed three distinct frequencies of 𝛿 Scuti type associated with the primary component. Several low frequency pulsations were also identified, which can be interpreted as imperfect removal of binary flux variations or/and slow variability caused by spots. We checked VizieR for other CoRoT targets observed simultaneously with TV Mon and found five in 2 \' cone. Two of them belong to spectral type K3II and show irregular variability, while two others have the same spectral type as TV Mon (A5IV) and have stable LCs. Thus we think that slow variability is real and is not caused by some instrumental effect. The secondary component is a cool red giant, which can have magnetic activity, required to produce spots. As these spots seems to change very fast (see lower panel in Figure 9), we don\'t model them in our LC solutions. \nPrevious work by Xiong et al. (2023) used ASAS-SN photometry together with LAMOST-MRS RV from Zhang et al. (2021), while we utilise better photometry and updated RV, based on the same spectra. They estimate 𝑀 𝐴,𝐵 = 2 . 024 ± 0 . 040 , 0 . 21 ± 0 . 04 , 𝑀 ⊙ , 𝑅 𝐴,𝐵 = 2 . 357 ± 0 . 187 , 2 . 869 ± 0 . 151 , 𝑅 ⊙ .As we show in Section 4, a systematic difference in RV will lead to slightly different stellar masses. This difference ( ∼ 5 per cent) can be seen as a realistic estimation of the errors in masses, because our MC errors can be too optimistic for masses derived from a limited number of medium resolution spectra with moderate S/N. \nIn the future, the red giant will become a low mass white dwarf (WD). Thus TV Mon will eventually become an EL CVn type eclipsing binary system, like WASP 0346-21, which has similar mass ratio, but shorter period (Lee et al. 2024).', '7.2 Binary evolution simulation': 'The low-mass component of TV Mon has a mass of 0 . 218 ± 0 . 004 𝑀 ⊙ , which is likely one type of envelope-stripped star, i.e. pre-ELM WD. The formation of (pre-)ELM WD binaries have been well studied in recent years (e.g., Althaus et al. 2013; Istrate et al. 2016; Chen et al. 2017; Sun & Arras 2018; Li et al. 2019). Following the work of Li et al. (2019), we try to construct the evolutionary history of TV Mon based on the observed parameters. The binary evolution simulations are performed via the detailed stellar evolution code \'Modules for Experiments in Stellar Astrophysics" (MESA, version 12115; Paxton et al. 2011, 2013, 2015). We adopt the Ledoux criterion and semi-convection mixing for the convection treatment, where the mixing length parameter is set to be 1.5, and semi-convection is modeled with an efficiency parameter of 𝛼 sc = 0 . 01. We adopted the element abundances of Population I stars, i.e., metallicity 𝑍 = 0 . 02 and hydrogen mass fraction 𝑋 = 0 . 70. The mass transfer processes are simulated based on the Ritter scheme (Ritter 1988). \nWe first try to evolve both components simultaneously. Here we assumed that the binary would merge once the accretor fills its Roche lobe. The typical progenitors of pre-ELM WDs are generally in the range of 1 . 0 -2 . 0 𝑀 ⊙ (Li et al. 2019). According to the total mass of TV Mon, i.e., ∼ 2 . 3 𝑀 ⊙ , the progenitor of 𝑀 𝐵 should be larger than 1 . 15 𝑀 ⊙ (since the initial donor mass is larger than the initial accretor mass). For a given initial donor mass and the chosen accretion efficiency, one can approximately determine the initial accretor mass (initial mass ratio) according to the observational constraints on TV Mon. Then we test a series of binary models with varying initial binary parameters, i.e. donor mass, mass ratio, \nBJD-2400000.5 , days \nFigure 9. Fourier spectrum, with inline plot showing the region with possible 𝛿 -Scuti pulsations (top) and best fit pulsation model (bottom). Vertical red lines indicate the position of all identified frequencies. The orange line shows the Fourier spectrum after the removal of 29 frequencies. \n<!-- image --> \nTable 5. Pulsation frequencies, sorted based on their Fourier amplitudes. Errors are from P/e.pc/r.pc/i.pc/o.pc/d.pc04 based on Montgomery & O\'Donoghue (1999). \norbital period, accretion efficiency. Unfortunately, none of these models can reproduce the observation parameters of TV Mon. The binaries would either merge due to the Roche lobe overflow (RLO) from the accretor or produce pre-ELM WDs moremassive than 𝑀 B . Therefore, in the subsequent simulations, we take the accretor as a mass point. \nThe accretion efficiency is a free parameter and cannot be limited in our simulations. As suggested by our previous work on Algol-type binary (Kovalev et al. 2022a), the accretion efficiency of 0 . 3 can support the observed parameters of TYC 2990-127-1 well. Therefore, our binary simulations adopt a fixed value of 0 . 3 for the accretion efficiency. Then the progenitor of 𝑀 B should be more massive than 1 . 64 𝑀 ⊙ . We test four groups of binary models with initial donor masses ranging from 1 . 7 to 2 . 1 𝑀 ⊙ , the initial accretor masses and orbital separations adjust correspondingly to match the observations. The binary evolution results are shown in Figure 10. In the panel (a), we put the observed and simulated samples in the He WDmass-orbital period plane, where the theoretically fitted curve taken from Lin et al. (2011) is also shown for comparison. It is clear the He WD binaries produced from stable RLO processes show a strong correlation between the orbital period and He WD mass. The observational parameters of TV Mon support such a relation pretty well. In panel (b), we present the possible evolutionary history of TV Mon. The thick black line is for the binary evolution model and the thin grey lines are for single evolutionary tracks with masses from 1 . 7 to 2 . 1 𝑀 ⊙ (from right to the left), respectively. In our best model, the initial binary contains a 1 . 8 𝑀 ⊙ donor star and a 1 . 61 𝑀 ⊙ \naccretor, and the initial orbital period is 0 . 626 d. The donor star fills its Roche lobe at the early main sequence stage, as shown in the red open circle. The donor then ascends the red giant branch in the late mass transfer phase. The pre-ELM WD is born after the termination of mass transfer processes, as shown in the red open square. The pre-ELM WD would not enter into the cooling stage immediately because the residual hydrogen layer is still burning, sustaining a relatively high luminosity (Istrate et al. 2014; Chen et al. 2017; Li et al. 2019). The mass of the pre-ELM WD in panel (b) is 0 . 218 𝑀 ⊙ , which coincides with the observations. The accretor mass after the mass transfer is 2 . 05 𝑀 ⊙ . The detailed structure of the accretor is not considered in our simulations. According to the single stellar tracks (thin grey lines), we see that the inferred mass of 𝑀 A is in the range of ∼ 1 . 7 -1 . 9 𝑀 ⊙ , which is slighly lower than the observationally derived mass of 𝑀 A (2 . 063 ± 0 . 033 𝑀 ⊙ ). The discrepancy may originate from the fact that the accretor\'s rejuvenation process may alter the stellar structure comparing with the same mass single star (e.g. Zhao et al. 2024; Lau et al. 2024). In panel (c), we present the masses of the two components as a function of evolutionary age. The onset and termination of the mass transfer processes are shown in red open circles and squares, respectively. The age of current state of TV Mon is approximately shown in grey dotted line. We see that the time elapsed since the end of mass transfer is about 250 Myr. Above all, our simulation could reproduce most of the important observed parameters of TV Mon. Our results suggest that TV Mon contains an envelope-stripped star born via binary interaction. \n0.28 \nFigure 10. The evolutionary models of TV Mon. Panel (a): The WD massorbital period plane in the simulations. The black solid line is for the theoretically fitted curve of Lin et al. (2011). Panel (b): The evolutionary track in 𝑇 eff -log 𝑔 plane for the best model. The black solid line is for the donor in the binary evolution and the grey lines are for the single evolution tracks with masses from 1 . 7 to 2 . 1 𝑀 ⊙ , from right to left. The observed parameters for the two components of TV Mon are shown in colored stars. The onset and detachment of Roche lobe overflow (RLO) are shown in red open circle and square, respectively. Panel (c): The masses of the two components as a function of the evolutionary age. The grey dotted line is the approximate age in the current state of TV Mon according to the results in panel (a). \n<!-- image -->', '8 CONCLUSIONS': 'In this paper we use available observations to characterise dEB system TV Mon. Here we summarise our results: \n- (i) weapplied multiple-epochs spectral fitting to LAMOST-MRS data to derive RV and spectral parameters;\n- (ii) we checked the CoRoT LC and found high frequency pulsations of the primary component and signs of long period variability of the secondary component. The analysis of eclipses reveal the relative sizes of the stellar components and edge-on circular orbit; \n(iii) the combined LC and RV solution provided us estimations for masses and radii of the components: 𝑀 𝐴,𝐵 = 2 . 063 ± 0 . 033 ( stat . ) ± 0 . 095 ( syst . ) , 0 . 218 ± 0 . 004 ( stat . ) ± 0 . 018 ( syst . ) 𝑀 ⊙ , 𝑅 𝐴,𝐵 = 2 . 394 ± 0 . 014 , 2 . 860 ± 0 . 016 𝑅 ⊙ ; \n(iv) SED analysis and Gaia parallax allowed us to estimate the temperatures 𝑇 eff 𝐴,𝐵 = 7624 + 194 -174 , 5184 + 130 -123 K and distance 𝑑 = 907 ± 11 pc. \n- (v) we measured three 𝛿 Scuti type frequencies in the primary component, while we also suspect TV Mon having a long period variability with period 𝑃 long ∼ 128 days, although it should be proven by the future long term observations; \n(vi) this system experienced intensive mass transfer and mass ratio reversal in the past. Now it shows no signs of mass transfer in the spectra. The low mass component will shrink to a He WD, which mass and orbital period are in good agreement with evolutionary models predictions. Eventually it will be EL CVn type eclipsing binary system. \nTV Mon is a very interesting laboratory for studying postmasstransfer binary systems. It will benefit from future observations like high-resolution spectroscopy and space-based photometry. New spectra will be able to confirm our mass estimations with good accuracy. A relatively long (40 min) total eclipse phase allows for a extraction of a detailed spectra of the red giant component, without other component contribution. Future analysis should take into account gravitational red shift and possibly convectional blue shift effects, when new spectra become available. Longer photometrical time series will allow more detailed analysis of 𝛿 Scuti pulsations in the primary and confirmation of possible long period variability of the secondary.', 'ACKNOWLEDGEMENTS': "We are grateful to the anonymous referee for a constructive report. His suggestions have significantly improved this article. We thank Hans Bähr (MPIA) for his careful proof-reading of the manuscript. We thank Hans-Ludwig (LSW) for useful discussions. MK dedicates this article to Dr. Xiang Liu, no mater if she cares about it or not, work on this paper helped him to pass breakup. \nGuoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences. This work is supported by the National Key R&D Program of China (grant Nos. 2021YFA1600403, 2021YFA1600400), the Natural Science Foundation of China (grant Nos. 12125303, 12288102, 12090040/3, 12473034, 12103086, 12273105, 11703081, 11422324, 12073070), Yunnan Fundamental Research Projects (Nos. 202401BC070007 and 202201BC070003), the Yunnan Revitalization Talent Support Program-Science & Technology Champion Project (No. 202305AB350003), the Key Research Program of Frontier Sciences of CAS (No. ZDBSLY-7005), Yunnan Fundamental Research Projects (grant Nos. 202301AT070314, 202101AU070276, 202101AV070001), and the International Centre of Supernovae, Yunnan Key Laboratory (No. 202302AN360001). We also acknowledge the science research grant from the China Manned Space Project with Nos. CMS-CSST2021-A10 and CMS-CSST-2021-A08. The authors gratefully acknowledge the 'PHOENIX Supercomputing Platform' jointly operated by the Binary Population Synthesis Group and the Stellar Astrophysics Group at Yunnan Observatories, Chinese Academy of Sciences. This research has made use of NASA's Astrophysics Data System, the SIMBAD data base, and the VizieR catalogue access tool, operated at CDS, Strasbourg, France. It also made \nuse of TOPCAT, an interactive graphical viewer and editor for tabular data (Taylor 2005). Gaia ( https://www.cosmos.esa. int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/ gaia/dpac/consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.", 'DATA AVAILABILITY': 'The data underlying this article will be shared on reasonable request to the corresponding author.', 'APPENDIX A: SPECTRAL MODELS': 'The synthetic spectra are generated using NLTE MPIA onlineinterface https://nlte.mpia.de (see Chapter 4 in Kovalev 2019) on wavelength intervals 4870:5430 Å for the blue arm and 6200:6900 Å for the red arm with spectral resolution 𝑅 = 7500. We \nFigure B1. WISE light curves. \n<!-- image --> \nuse NLTE (non-local thermodynamic equilibrium) spectral synthesis for H, Mg I, Si I, Ca I, Ti I, Fe I and Fe II lines (see Chapter 4 in Kovalev 2019, for references). \nThe grid of models (6200 in total) is computed for points randomly selected in a range of 𝑇 eff between 4600 and 8800 K, log ( g ) between 1.0 and 4.8 (cgs units), 𝑉 sin 𝑖 from 1 to 300 km s -1 and [Fe/H] 9 between -0.9 and + 0.9 dex. The model is computed only if linear interpolation of the MAFAGS-OS(Grupp 2004a,b) stellar atmosphere is possible for a given point in parameter space. Microturbulence is fixed to 𝑉 mic = 2 kms -1 for all models.', 'APPENDIX B: WISE LIGHT CURVES': 'We show WISE light curves in Figure B1. Eclipses (parts between black vertical lines) were excluded, when we recompute photometry for SED fitting.', 'APPENDIX C: RV MEASUREMENTS': 'We provide RV measurements from the multiple-epoch fit in Table C1.', 'APPENDIX D: TLCM MODELLING': 'The presence of a total eclipse and transit in the CoRoT LC allows us to use TLCM, which was developed and actively used to fit exoplanet transits in CoRoT data (Csizmadia 2020). It allows for the use of RV data from two components in \'SB2" mode and estimate parameters uncertainties using Monte Carlo simulations. Reflection, ellipsoidal and beaming effects are included in modelling for both components. Wefixperiod 𝑃 = 4 . 179741 day and eccentricity 𝑒 = 0, while fitting for all other parameters, including limb darkening coefficients for both stars 𝑢 + , -and parabolic trend with time. The final solution is determined as a median of 25000 MC simulations with 10 walkers. \nResults derived by TLCM are collected in Table D1 and shown in Figure D1. They are consistent with the JKTEBOP solution, \nTable C1. Radial velocity measurements. Errors were adjusted by JKTEBOP to reach unity in reduced 𝜒 2 . We subtract 2400000.5 days from time values. \nalthough slightly different. The total eclipse is well fitted, while for the secondary the minimum residuals are symmetric relative to phase 0.5. This code was designed to fit exoplanet transits, so it can be expected to perform worse for stars. Anyway residuals are comparable to the amplitude of high-frequency pulsations. The out-of-eclipse part of the LC is dominated by ellipsoidal variability (Ell. max 𝐴,𝐵 ∼ 0 . 25 , 6 . 0 per cent of total light) and reflection (Refl. max 𝐴,𝐵 ∼ 0 . 3 , 1 . 7 per cent of total light), while the light beaming effect is negligible (Beam. max 𝐴,𝐵 ∼ 0 . 016 , 0 . 06 per cent of total light).', 'APPENDIX E: EMCEE SAMPLE OF ELLC SOLUTION': 'In Figure E1 we show corner plot (Foreman-Mackey 2016) with /e.pc/m.pc/c.pc/e.pc/e.pc sampling results for /e.pc/l.pc/l.pc/c.pc solution.', 'APPENDIX F: FULL DATASETS WITH W-D': 'We show the solution derived after 21 iterations using three full datasets in Figures F1, F2, F3 and Table F1. Big residuals in the total eclipse are from the part of the CoRoT LC, which was not used during previous analysis by the /e.pc/l.pc/l.pc/c.pc and W-D codes. In comparison with Figure 5 the out-eclipse part is fitted well by W-D. Both ASASSN LCs are also fitted well, although for 𝑉 band secondary eclipse is slightly shallower in comparison with the best fit model. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \n<!-- image --> \nFigure D1. Phase-folded LC (top) and RV (bottom) fits with TLCM. The periodic structure is obvious in the LC residuals around phase=0.5. \nTable D1. TLCMsolution. Error estimates are from 25000 MC simulations. \nFigure E1. Corner plot for the sampling of the /e.pc/l.pc/l.pc/c.pc solution. Titles show 16, 50 and 84 percentiles. \n<!-- image --> \nTable F1. W-D solution for full three datasets. \nFigure F1. The LC solution by W-D using full datasets (CoRoT). Top panels show fit of the data, bottom panels show fit residuals. \n<!-- image --> \nFigure F3. The LC solution by W-D using full datasets (ASAS-SN 𝑔 ). Top panels show fit of the data, bottom panels show fit residuals. \n<!-- image --> \nFigure F2. The LC solution by W-D using full datasets (ASAS-SN 𝑉 ). Top panels show fit of the data, bottom panels show fit residuals. \n<!-- image -->'}
2024arXiv240907211T	Computational chemistry plays a relevant role in many astrochemical research fields either by complementing experimental measurements or by deriving parameters difficult to be reproduced by laboratories. While the role of computational spectroscopy in assisting new observations in space is described the core of the chapter is the investigation of the collisional radiative transfer and the bimolecular reactive processes occurring in the gasphase conditions of the interstellar medium using as a guide the contributions presented by the authors at the Second Italian National Congress on Protoplanetary Astrochemistry held in Trieste in September 2023. In particular the need for accurate datasets of collisional coefficients to model molecular abundances will be discussed. Then the role of quantum chemistry in the investigation of interstellarrelevant potential energy surfaces will be described focusing on accurate thermodynamic quantities for the estimate of rate coefficients.	2024-09-01T00:00:00Z	['2024arXiv240907211T', '10.48550/arXiv.2409.07211', 'arXiv:2409.07211']	['Astrophysics - Astrophysics of Galaxies']	Ab initio Calculations for Astrochemistry	2,024	194	0.43	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.07211.pdf	{'No Header': 'Mem. S.A.It. Vol. 00, 1 © SAIt 2023 \n<!-- image -->', 'Ab initio Calculations for Astrochemistry': "F. Tonolo and S. Alessandrini \nDipartimento di Chimica 'Giacomo Ciamician', Alma Mater Studiorum - Universit'a di Bologna, via Francesco Selmi 2, Bologna, 40126, Italy e-mail: [email protected], [email protected] \nReceived: XX-XX-XXXX; Accepted: XX-XX-XXXX \nAbstract. Computational chemistry plays a relevant role in many astrochemical research fields, either by complementing experimental measurements or by deriving parameters difficult to be reproduced by laboratories. While the role of computational spectroscopy in assisting new observations in space is described, the core of the chapter is the investigation of the collisional radiative transfer and the bimolecular reactive processes occurring in the gas-phase conditions of the interstellar medium, using as a guide the contributions presented by the authors at the 'Second Italian National Congress on Proto(-planetary) Astrochemistry', held in Trieste in September 2023. In particular, the need for accurate datasets of collisional coe ffi cients to model molecular abundances will be discussed. Then, the role of quantum chemistry in the investigation of interstellar-relevant potential energy surfaces will be described, focusing on accurate thermodynamic quantities for the estimate of rate coe ffi cients. \nKey words. quantum-chemistry, collision dynamics, gas-phase reactivity, molecular abundances, kinetics, radiative transfer modeling", '1. Introduction': "Quantum-chemical approaches are a powerful tool for astrochemists, with ab initio methodologies widely used to obtain experimentally inaccessible information and insights into processes occurring in the gas phase of the interstellar medium (ISM). Here, the temperature can reach values of a few tens of K and densities are usually between 10 2 and 10 6 molecules · cm -3 , thus posing constraints on both reactive and dynamic processes (Tielens 2013; Yamamoto 2017). However, the nature of quantum-chemical calculations in describing isolated molecules well matches with the extremely low-density environment of the gas phase, where molecules are mostly observed. \nObservations of molecules in the ISM are based on radioastronomical techniques and these have flourished in the recent years thanks to facilities like the ALMA (Wootten & Thompson 2009), Yebes (Tercero et al. 2021), IRAM (Baars et al. 1987) and GBT (White et al. 2022) telescopes and the associated large surveys(McGuire et al. 2020; Cernicharo et al. 2022). While the present chapter does not consider ices and reactions occurring on their surfaces, as the main focus is on gas-phase processes, it should be mentioned that the recent launch of the JWST started also to provide observations of icy dust mantles by means of the vibrational signals in the infrared region. However, the species identified are still very \nFig. 1. The role of ab initio calculations in astrochemistry: from the interplay between theory and experiment in rotational spectroscopy to the derivation of molecular abundances and gas-phase reaction profiles for kinetic calculations. \n<!-- image --> \nsimple, like CH3 + (Bern'e et al. 2023; McClure et al. 2023). \nThe radioastronomical detection of a molecule relies on the observation of its rotational transitions, which should be known with great accuracy by means of a thorough experimental work. Only a few species, like C5N -(Cernicharo et al. 2008), have been observed based only on their accurate theoretical predictions. This is due to two reasons. First, the simulation of rotational spectroscopic parameters is strongly a ff ected by the level of theory employed (Puzzarini & Stanton 2023; Puzzarini et al. 2010; Alessandrini et al. 2018). For example, rotational constants, the leading term of rotational spectroscopy, obtained with an error of about 0.1% with respect to the experimental data, translates into computed geometries with an accuracy of 0.0005-0.001 Å (Puzzarini & Stanton 2023). Secondly, the rotational transitions simulated with the rotational constants are then a ff ected by several interactions, among which the vibration-rotation one, always present. These interactions shift the computational predictions by an unknown quantity, prohibiting the direct comparison with radioastronomical data. Computational methods able to reproduce the experimental results within the accuracy mentioned above are \ncomputationally a ff ordable only for very small molecules (up to 5 / 6 atoms), thus they cannot be employed for complex-organic molecules, i . e . , the main target of modern astrochemistry. In this regard, a large e ff ort has been made by theoretical chemists to propose new tools which lower the computational cost and still retain a good accuracy. For example, composite schemes for energies and proprieties (Puzzarini et al. 2010; Heckert et al. 2005; Tajti et al. 2004), or explicitly correlated F12 methods Gruneis et al. 2017). E ff orts have been made also in the implementation of new techniques, like the Cholesky decomposition (Aquilante et al. 2011; Nottoli et al. 2021). In addition, the possibility of adopting 'experimentally accurate' data to correct computational quantities has been used to improve the spectroscopic predictions of larger species, like PAHs or long carbon chains (Melli et al. 2021; Ye et al. 2022; Puzzarini et al. 2023). Nevertheless computational spectroscopic quantities cannot be employed to analyze radioastronomical data most often, but they have a prominent role in rotational spectroscopy. Indeed, they are used to guide experimental measurements of unstable species (Melosso et al. 2022; Puzzarini et al. 2023), not known in the literature, and to constrain \nparameters that the experiment is not able to determine (Cazzoli et al. 2014). \nThe role of accurate computational methods does not end with rotational spectroscopy and assumes a prominent role in the derivation of accurate collisional coe ffi cients and kinetic estimates of gas-phase neutral-neutral reactions. As illustrated in Figure 1, the previous quantities are fundamental to derive molecular abundances and build reaction networks, respectively. The outcomes of these investigations are not stand-alone results, but their interplay is essential to support the physical and chemical modeling of the ISM. For an extended description of the most relevant processes occurring in the ISM, the reader is referred to key reviews and books such as Dalgarno & Black (1976); Watson (1976); Tennyson (2003); Wakelam et al. (2010); Bates (2012) and Yamamoto (2017). In addition, to ease the interplay of these investigations for astrochemical modeling purposes, the available information is now centralized in several public databases. To cite a few: the BASECOL (Dubernet et al. 2024), LAMDA (Schoier et al. 2005), and EMAA ( https://emaa.osug. fr// ) databases furnish the collision dynamics outcomes. For kinetics rate constants, the KIDA (Wakelam et al. 2012) and UMIST (Millar et al. 2024) databases are the most widely known, while for spectroscopy di ff erent sets have been developed depending on the frequency range. In particular, the CDMS database collects rotational lines (Endres et al. 2016), the LIDA one IR bands of molecules in ice mixtures (Rocha et al. 2022), EDIBLES (Cox et al. 2017) di ff use interstellar bands, and ExoMol (Tennyson et al. 2016) is specifically contaning line list useful for the modeling of exoplanets and hot atmospheres. \nFor more details on computational spectroscopic parameters and their accuracy the reader is referred to Puzzarini et al. (2010) while in the following collision dynamics and reactive potential energy surfaces (PES) will be addressed. Section 2 will focus on the accurate determination of collisional coe ffi cients that are used to model the abundances of molecules observed in the ISM, where local thermodynamic equilibrium (LTE) conditions are rarely \nfulfilled. The last part of the chapter (section 3) will explore the role of ab initio methods for the derivation of accurate kinetic rate constants of gas-phase for neutral-neutral reactions. \nIn the following sections, the examples provided are the ones presented during the 'Second Italian National Congress on Proto(planetary) Astrochemistry', held in Trieste in September 2023.", '2. The role of collision dynamics calculations to model molecular abundances in space': "The derivation of molecular abundances in space needs to account for the sparse physical conditions that can be found in astrophysical environments. Indeed, the radiative transfer equations used to model astrophysical observations and derive molecular column densities exhibit large sensitivity to the processes that a ff ect the population distributions among molecular levels. Such processes strongly depend on the physical conditions of the targeted environment. For example, in the ISM the density is so low ( ∼ 10 2 -10 6 molecule · cm -3 ) that molecular energy level populations are not in LTE. Under such conditions, the derivation of molecular abundances from spectral lines requires the knowledge of the collisional rate coe ffi cients of the molecule under consideration for the most abundant perturbing species, i . e . , H2, H and He (Roue ff & Lique 2013; Lique & Faure 2019). The variation in accuracy of the collisional rate coe ffi cients can cause differences up to a factor of 10 in the line intensities, which reflects to a significant change in the prediction of the molecular abundances ( e . g . , Sarrasin et al. 2010; Lanza et al. 2014). Therefore, the progress in radiative transfer calculations needs to go hand in hand with the improvement in the accuracy of the collisional predictions. This is also reflected in the major e ff orts that are currently devoted to reproduce these coe ffi cients from an experimental point of view (Yang et al. 2010, 2011; Chefdeville et al. 2012; Brouard et al. 2014; Bergeat et al. 2015, 2020). Nowadays, given the paucity of experimental setups able to probe collision dynamics, the knowledge of \nthe collisional rate coe ffi cients strongly relies on theoretical calculations (Roue ff & Lique 2013; Lique & Faure 2019). Thus, an accurate yet a ff ordable computational procedure for the characterization of the collisional properties of astrochemical molecules, and subsequent modeling of their spectroscopic transitions in terms of abundance, needs to be validated. In the following, an illustrative protocol that is particularly suited to inspect the collisional behavior of small ionic systems in both a reliable and undemanding manner is presented. The latter consists of four main steps. \nThe first one is the investigation of the collisional PES by means of ab initio calculations. This step requires particular attention because uncertainties in the PES have a significant impact on the accuracy of the collisional parameters (see Faure 2021 for an illustrative example). Hence, high computational accuracies need to be achieved. This usually exploits the excellent performances of explicitly correlated coupled cluster methods (Adler et al. 2007; Knizia et al. 2009) to describe interaction energies (Ajili et al. 2013; Tonolo et al. 2021). When ionic systems are involved, the use of fully augmented basis sets is particularly promising in describing the energetics of the long-range regions of the potential, where dispersive interactions are more relevant (Kendall et al. 1992). \nThe second step consists of expressing the potential as an expansion over angular functions. This often requires to resort to some approximations to reduce the computational cost of scattering calculations. This applies, in particular, to systems involving a collisional partner with a rotational structure, such as H2. In these cases, the e ff ects due to the coupling between the di ff erent rotational states of the collider (indexed using the J quantum number) on the inelastic cross sections need to be preliminarily assessed. This brings important hints on the feasibility of neglecting the J > 0 rotational states of the collider (the so-called 'spherical approximation'). The reduced computational cost of this approximation, if properly validated, permits to extend this procedure to larger molecular systems and to a wide range of astrochemical conditions. The spherical ap- \nproximation has been proved to be particularly suited to small ionic systems interacting with H2 (Spielfiedel et al. 2015; Balanc¸a et al. 2020; Cabrera-Gonz'alez et al. 2020). For instance, for the HC 17 O + and PO + targets (Tonolo et al. 2022, 2024) the impact of the J > 0 rotational levels of H2 resulted to be considerably weak (the inclusion of the J = 2 state of H2 led to average deviations of ∼ 7% in the values of the cross sections, while the cross sections of para -H2 and ortho -H2 agreed within ∼ 12% on average). \nThe third step is the solution of the closecoupling scattering equations in a range of energies that allows to derive the corresponding collisional parameters in the conditions of interest. For example, the pressure broadening and pressure shift parameters can be computed and can be subsequently used to infer the quality of the potential (see, as an example, Tonolo et al. 2021). From the scattering calculations, the inelastic state-to-state rate coe ffi cients are also obtained. \nThe last step consists in using the collisional rate coe ffi cients to model the rotational transitions observed in the interstellar environments by means of radiative transfer calculations. For example, the computed collisional dataset for the PO + / H2 system helped to test the reliability of the LTE approximation and to refine the column density value of PO + obtained from the observations of the G + 0.693-0.027 molecular cloud (Rivilla et al. 2022). Additionally, radiative transfer calculations indicated maser behavior for the first rotational transitions of PO + at various kinetic temperatures and densities typically found in interstellar sources. This emphasizes the importance of accurate collisional coe ffi cients for the precise modeling of molecular abundances in the ISM.", '3. Simple organics from gas-phase reactions: a computational view': "The observation of new small organic molecules claims for innovative formation routes either on grains or in the gas phase of the ISM (McGuire 2022; Puzzarini 2022). The latter class of reactions is challenging due to \nthe prohibitive conditions of the ISM and the three main points that guide neutral-neutral gas-phase reactions are: (i) only bi-molecular reactions can occur, and due to the lack of a third-body stabilization only bi-molecular products can be formed(Yamamoto 2017); (ii) no external energetic input is provided for the process, so the products must be exothermic and have to involve submerged barrier with respect to the reactants; (iii) to make a collision reactive, the process has to involve a radical species, like CN, NH or CH. Still, also other gas-phase processes occur in the ISM, like photodissociation or ion-molecule reactions (Yamamoto 2017; Tielens 2013). \nBased on these three guidelines, it is possible to select reactions based on thermodynamic estimates and move to the kinetic study only for a small selection of processes. Indeed, the critical step in the gas-phase chemistry of the ISM is the estimation of rate coe ffi cients that should be at least in the order of 10 -11 / 10 -12 cm 3 molecule -1 s -1 (Yamamoto 2017; Tielens 2013). The latter can be obtained from experimental set-ups or using first principle computational methodologies. In both cases, limitations are involved: experimental measurements can su ff er from the impossibility of reproducing interstellar-medium-like conditions. At the same time, theoretical kinetic estimates are strongly a ff ected by the energetic quantities employed for their calculation and the approximation introduced in the procedure. A discussion on the accurate derivation of rate constants is out of the scope of the present chapter, but the need for accurate reaction barrier heights has to be mentioned and it is where ab initio computations play the main role. Indeed, meaningful kinetic rate constants can be obtained only if post-Hartree Fock methods, like coupled-cluster (CC) methodologies (Shavitt & Bartlett 2009) or multi-reference (MR) formulations, like MR configuration interaction (MRCI) (Buenker & Peyerimho ff 1974) are used. In the case of CC techniques, one has to refer to the CCSD(T) method that includes singles and doubles excitations and a perturbative treatment of triples excitations (Stanton 1997)(Zheng et al. 2009; Klippenstein 2017). For example, it has been pointed out that the \nuse of hybrid functional led to the wrong theoretical outcome in the case of the reaction between CH3CN + CN (Sleiman et al. 2018; Puzzarini et al. 2020) and in this particular case the CCSD(T) method in conjunction with standard basis set was not able to reproduce more accurate methodologies (Lupi et al. 2020). \nHowever, the accuracy can be pushed to its nowadays limit by employing explicitly correlated F12 methodologies (Adler et al. 2007; Knizia et al. 2009) in combination also with composite schemes, where several terms are computed at the best compromise between accuracy and computational cost and then combined to reach a better estimate of a property, like the energy (Tajti et al. 2004; Barone et al. 2021; Ventura et al. 2021; Alessandrini et al. 2019; Lupi et al. 2021; Barone et al. 2023). If accurate methods are employed, the reference geometry for the calculations can be obtained from density functional methods (Zheng et al. 2009), either using hybrid or doublehybrid functionals (Becke 1988; Kohn & Sham 1965; Santra et al. 2019). This is a good compromise between accuracy and cost, considering DFT geometries provide structures that are as accurate as those obtainable from more expensive methods, like the CCSD(T) one, but can be employed on extremely vast PES, like those mapped for the CH radical (Nikolayev et al. 2021; He et al. 2022). \nThe use of computational means has also highlighted the possibility of studying several reaction mechanisms between the same stable species and di ff erent radicals. This is useful to understand if common reaction paths occur in the ISM, an hypothesis was first done for methanimine that is considered the precursor of complex imines in the ISM CH2NH (Puzzarini & Barone 2021; Barone & Puzzarini 2022; Puzzarini 2022). The reaction between methanimine and CN was analyzed by Vazart et al. (2015) and leads to cyanomethanimine, (HNC--CHCN as well as CH2NCN species), all observed in the ISM (Zaleski et al. 2013; San Andr'es et al. 2024). The same reaction with the CCH radical forms propargylimine (HNC = CHCCH), another molecule detected in recent years (Bizzocchi et al. 2020), while the OH radical seems to lead \nto the formation of formamide (Vazart et al. 2016; Rubin et al. 1971). Do the CH2NH + CN, CH2NH + CCH and, CH2NH + OH reactions have something in common? The PES involves a radical in the doublet electronic state and points to identical exothermic reaction paths with slightly di ff erent energy barriers (even if still submerged). To confirm the presence of a general reaction mechanism, the processes occurring between CH2NH and a large set of radicals having an unpaired electron (HCO,HCS,SH,NO,NS,SiN,C3N,andCP) were analyzed (Alessandrini et al. 2021; Ye et al. 2024). Both H-abstraction and radical addition were considered and the products were hypothesized using the general reaction mechanism. Thus, the first step was the derivation of accurate energetic quantities to asses whether exothermic products are formed. This step was carried out using double-hybrid functional for geometries and vibrational frequencies (harmonic) then, the junChS (Alessandrini et al. 2019) composite scheme for the energies. Only the C3N and the CP radicals can lead to an exothermic path after the addition of the radical, while open channels for H-abstraction seem to occur for these species, but also for SH and HCS. For all the exothermic paths, the full potential energy surface has been explored with the double-hybrid functional and the energy was refined with the junChS composite scheme. A common reaction mechanism was observed for radical addition and H-abstraction, confirming the validity of the general mechanism. \nStill, kinetics estimates are those able to point out if a reaction in the ISM is feasible. In the case of the methanimine-plus-radical reactions, the kinetic estimate where obtained using variational transition state theory with the Master Equation System Solver program (MESS, Georgievskii et al. 2013) and using phase space theory for the derivation of the the kinetic term for the barrierless approach. According to our results, the processes leading to the addition of the radical on the C atom of CH2NH are fast even in the ISM conditions, thus further pointing out HNCHCP and HNCHC3N species might be present in the ISM and only laboratory characterization \nmight be missing for their observation. Even if the work here discussed employs statistical approached to obtain reaction rates, it should be mentioned that other routes are available to derive the rate coe ffi cients. Among them, quantum dynamics is the reference strategy if one aims at experimental accuracy based on theoretical means. However, this method can only be used for systems with a reduced dimension, like the destruction of CH + (Bovino et al. 2015) or the reaction H + H2 (Ghosh et al. 2021).", '4. Conclusions': "In this chapter, an overview of the role of ab initio strategies as a support of gas-phase astrochemical investigations is presented, from the prediction of the spectroscopic parameters to assist the experimental characterization of interstellar molecules, to reactivity and collision dynamics calculations. Specifically, the authors brought into focus the latter two topics and the results presented during the 'Second Italian National Congress on Proto(-planetary) Astrochemistry'. First, a description of the various steps required to compute the collisional coe ffi cients and derive non-LTE abundances of interstellar molecules has been outlined. Secondly, we have discussed how the investigation of general mechanisms might be useful to describe interstellar chemistry and suggest new interstellar molecules to observe. In doing so, the need for accurate ab initio methodologies to derive reliable kinetics rate coe ffi cients was mentioned. \nAcknowledgements. Both authors acknowledge the PRIN grant No. 202082CE3T 'ARES-A Road from Earth to the Stars' for financial support and the ROT&Comp Lab for the inspiring discussions over the past years.", 'References': "Adler, T. B., Knizia, G., & Werner, H.-J. 2007, J. Chem. Phys., 127 \nAjili, Y., Hammami, K., Jaidane, N. E., et al. 2013, Phys. Chem. Chem. Phys., 15, 10062 Alessandrini, S., Barone, V., & Puzzarini, C. 2019, J. Chem. 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2024arXiv240907502C	We construct a solution operator for the linearized constant scalar curvature equation at hyperbolic space in space dimension larger than or equal to two. The solution operator has good support propagation properties and gains two derivatives relative to standard norms. It can be used for CorvinoSchoentype hyperbolic gluing partly extending the recently introduced MaoOhTao gluing method to the hyperbolic setting.	2024-09-01T00:00:00Z	['arXiv:2409.07502', '10.48550/arXiv.2409.07502', '2024arXiv240907502C']	['General Relativity and Quantum Cosmology', 'Mathematics - Differential Geometry']	A Bogovskitype operator for CorvinoSchoen hyperbolic gluing	2,024	195	0.06	['EPRINT_HTML', 'EPRINT_PDF']	1	https://arxiv.org/pdf/2409.07502.pdf	{'A Bogovski i-type operator for Corvino-Schoen hyperbolic gluing': 'Piotr T. Chru´sciel, Albachiara Cogo, Andrea N¨utzi', 'Abstract': 'We construct a solution operator for the linearized constant scalar curvature equation at hyperbolic space in space dimension larger than or equal to two. The solution operator has good support propagation properties and gains two derivatives relative to standard norms. It can be used for Corvino-Schoen-type hyperbolic gluing, partly extending the recently introduced Mao-Oh-Tao gluing method to the hyperbolic setting.', '1 Introduction': "Mao, Oh and Tao [11] have shown how to simplify the gluing constructions of Corvino and Schoen [9,10] for three-dimensional asymptotically flat general relativistic vacuum initial data sets. The simplification comes from using Bogovski ˇ i-type solution operators for the linearized equations, which preserve the support of the sources and have good differentiability properties. Similar methods [12] led to simplifications and improvements of the gluing constructions of Carlotto and Schoen [4] in dimensions larger than or equal to three, in particular they provide better asymptotics of the glued solutions. \nThe aim of this note is to point out the existence of a Bogovski ˇ i-type solution operator for the linearization of the constant scalar curvature equation at hyperbolic space, with the same good support and differentiability properties. It thus can be used in nonlinear problems, e.g. in Corvino-type gluing near hyperbolic space, and in particular to provide simpler proofs of some of the small-data gluing results of [7]. See Section 4 for details. \nThe constant scalar curvature equation for an n -dimensional asymptotically hyperbolic Riemannian manifold ( M,g ) is \nR( g ) = -n ( n -1) , (1.1) \nwhere R( g ) is the scalar curvature of the metric g . Recall that (1.1) is equivalent to the time-symmetric general relativistic constraint equations with cosmological constant 2Λ = -n ( n -1). \nHere we consider the case where the manifold is the upper half-space \nM = /C0 n = { x = ( x 1 , . . . , x n ) \| x 1 > 0 } . \nThen a trivial solution of (1.1) is given by the hyperbolic metric \nb = 1 ( x 1 ) 2 ( dx 1 ⊗ dx 1 + · · · + dx n ⊗ dx n ) . \nWe consider the linearization of (1.1) at b . This is given by the underdetermined elliptic linear operator P : C ∞ ( /C0 n , S 2 ( T ∗ /C0 n )) → C ∞ ( /C0 n , /CA ), \nP( h ) = D i D j h ij -D i D i h j j -R ij h ij , (1.2) \nwhere S 2 ( T ∗ /C0 n ) denotes the bundle of two-covariant symmetric tensors, D i and R ij are the covariant derivative, respectively the Ricci curvature, of b , and where we use the index conventions detailed in Remark 2.1. \nOur main proposition provides a solution operator for the linear equation \nP( h ) = f (1.3) \nwith good support and differentiability properties. We discuss necessary integrability conditions of this equation when the source f has compact support in /C0 n , and one seeks a solution h with compact support. Recall that the space of static KIDs for the hyperbolic metric is given by the kernel of the formal adjoint of P. Explicitly, this is spanned over /CA by the n +1 functions \n1 + \| x \| 2 2 x 1 , 1 -\| x \| 2 2 x 1 , x 2 x 1 , . . . , x n x 1 , (1.4) \nwhich we will denote by κ 1 , . . . , κ n +1 . For all h and all a = 1 . . . n + 1 the product κ a P( h ) is a divergence, in fact the identity \nκ a P( h ) = D i V ( a ) i ( h ) (1.5) \nholds where by definition \nV ( a ) i ( h ) = κ a ( D j h ij ) -( D j κ a ) h ij -κ a ( D i h j j ) + ( D i κ a ) h j j . \nTherefore for all h with compact support, \n∫ /C0 n κ a P( h ) dµ b = 0 (1.6) \nwith dµ b = 1 ( x 1 ) n dx 1 · · · dx n . These are n +1 necessary integrability conditions for (1.3), in the compact support case. We now state the main proposition, which shows in particular that these integrability conditions are sufficient for existence of a solution with compact support. \nFor an open subset Ω ⊆ /C0 n we denote by C ∞ c (Ω , /CA ) the space of smooth functions with compact support in Ω, analogously for sections of S 2 ( T ∗ Ω). \nProposition 1.1. Let Ω ⊆ /C0 n be a bounded connected open set with smooth boundary such that ¯ Ω ⊆ /C0 n . Then there exists an /CA -linear operator \nG : C ∞ c (Ω , /CA ) → C ∞ c (Ω , S 2 ( T ∗ Ω)) \nwith the following properties: \n(a1) The map Π : C ∞ (Ω , /CA ) → C ∞ (Ω , /CA ) defined by \nc c PG = /BD -Π \nhas rank n + 1 , is a projection ( Π 2 = Π ), and is explicitly given as follows: Let κ 1 , . . . , κ n +1 be the static KIDs (1.4) . There exist functions ϕ 1 , . . . , ϕ n +1 ∈ C ∞ c (Ω , /CA ) such that for all f ∈ C ∞ c (Ω , /CA ) , \nΠ( f ) = n +1 ∑ a =1 ϕ a ∫ Ω κ a f dµ b . (1.7) \n(a2) For all s ∈ /CA and 1 < p < ∞ there exists a constant C > 0 such that for all i = 1 . . . n and f ∈ C ∞ c (Ω , /CA ) , \n‖ G( f ) ‖ W s,p ≤ C ‖ f ‖ W s -2 ,p , ‖ [G , ∂ x i ]( f ) ‖ W s,p ≤ C ‖ f ‖ W s -2 ,p , \nwhere ‖ · ‖ W s,p are the standard L p -based Sobolev norms. Analogous estimates hold for standard Holder norms with exponent 0 < α < 1 . \nThe proof appears at the end of Section 3.2. \nAcknowledgement. Many useful discussions with Bobby Beig and Erwann Delay are acknowledged. We are thankful to the Erwinn Schrodinger Institute in Vienna for hospitality during an initial stage of this work. AC was supported by the DAAD 'Forschungsstipendien fur Doktorandinnen und Doktoranden' during her research stay in Vienna. AN is supported by the Swiss National Science Foundation, project number P500PT-214470. PTC's research was supported in part by the NSF under Grant No. DMS-1928930 while the author was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley during the Fall 2024 semester.", '2 Linearized scalar curvature equation': 'In this section we derive a simple formula for the linear operator P in (1.2). We will use the conventions detailed in the following remark. \nRemark 2.1. For a symmetric two-tensor h ∈ C ∞ ( /C0 n , S 2 ( T ∗ /C0 n )) we denote by h ij = h ( ∂ x i , ∂ x j ) its components relative to the coordinates x . Indices are lowered with b ij = b ( ∂ x i , ∂ x j ) and raised with its inverse b ij . All repeated indices are summed over 1 . . . n , regardless of their position. We denote by δ ij = δ ij = δ i j the Kronecker delta. ✷ \nLet h ∈ C ∞ ( /C0 n , S 2 ( T ∗ /C0 n )). Using the fact that R ij = (1 -n ) b ij we have \nP( h ) = D i D j ( h ij -h /lscript /lscript b ij ) -(1 -n ) h /lscript /lscript \n. \nThus if we define ˜ h = h -h /lscript /lscript b then we obtain \nP( h ) = D i D j ˜ h ij -˜ h i i . \nThe non-vanishing Christoffel symbols of the hyperbolic metric are \nΓ 1 11 = -1 x 1 , Γ 1 pq = 1 x 1 δ pq , Γ p 1 q = -1 x 1 δ p q , \nwhere p, q = 2 . . . n . Then by direct calculation, \nD i D j ˜ h ij = ( x 1 ) n +1 ∂ i ∂ j ( 1 ( x 1 ) n +1 ˜ h ij ) +( x 1 ) n ∂ 1 ( 1 ( x 1 ) n +1 δ ij ˜ h ij ) , \nwhere ∂ i = ∂ x i . Thus, using ˜ h i i = 1 ( x 1 ) 2 δ ij ˜ h ij , \nP( h ) = ( x 1 ) n +1 ∂ i ∂ j ( 1 ( x 1 ) n +1 ˜ h ij ) +( x 1 ) n +1 ∂ 1 ( 1 ( x 1 ) n +2 ( δ ij ˜ h ij ) ) . \nWe have therefore proved the following lemma: \nLemma 2.2. Let h ∈ C ∞ ( /C0 n , S 2 ( T ∗ /C0 n )) . Define \nA = 1 ( x 1 ) n +1 ( h -h i i b ) . (2.1) \nThen h = ( x 1 ) n +1 ( A + 1 1 -n A i i b ) and \nP( h ) = ( x 1 ) n +1 ( ∂ i ∂ j A ij + ∂ 1 ( 1 x 1 A ii ) ) .', '3 Construction of the solution operator': 'In this section we construct the operator G of Proposition 1.1. The main tool is a right inverse for the divergence operator on Euclidean space originally introduced by Bogovski ˇ i [2, 3]. We recall it in the next lemma. \nLemma 3.1. Let Ω ⊆ /C0 n be as in Proposition 1.1. There exists an /CA -linear map B : C ∞ c (Ω , /CA ) → C ∞ c (Ω , /CA n ) with the following properties: \n- (b1) There exists a function ϕ ∈ C ∞ c (Ω , /CA ) such that ∫ Ω ϕdx = 1 where dx is the Lebesgue measure, and such that for all f ∈ C ∞ c (Ω , /CA ) , \n∂ i B i ( f ) = f -ϕ ∫ Ω f dx . (3.1) \n- (b2) For all s ∈ /CA and 1 < p < ∞ there exists a constant C > 0 such that for all i = 1 . . . n and f ∈ C ∞ c (Ω , /CA ) , \n‖ B( f ) ‖ W s,p ≤ C ‖ f ‖ W s -1 ,p , ‖ [B , ∂ i ]( f ) ‖ W s,p ≤ C ‖ f ‖ W s -1 ,p . \nAnalogous estimates hold for Holder norms with exponent 0 < α < 1 . \n- (b3) For all i = 1 . . . n and f ∈ C ∞ c (Ω , /CA ) , \n∫ Ω B i ( f ) dx = -∫ Ω x i f dx + m i ∫ Ω f dx , (3.2) \nwhere by definition m i = ∫ Ω x i ϕdx and m ij = ∫ Ω x i x j ϕdx , and where we use the summation convention in Remark 2.1. \n∫ Ω x j B j ( f ) dx = -1 2 ∫ Ω \| x \| 2 f dx + 1 2 m jj ∫ Ω f dx , (3.3) \nProof. Such an operator exists by [14, Section 4.1] (note that Ω can be covered by finitely many open sets that are star-shaped with respect to a ball), or alternatively by [13, Theorem 1]. More precisely, the references \nimply that there exists an operator ˜ B : C ∞ c ( /CA n , /CA ) → C ∞ c ( /CA n , /CA n ) that satisfies (b1), ˜ B( C ∞ c (Ω , /CA )) ⊆ C ∞ c (Ω , /CA n ), and such that if χ ∈ C ∞ c ( /CA n , /CA ) then ˜ B χ is a pseudodifferential operator of order -1 on /CA n . Define B = ˜ B \| C ∞ c (Ω , /CA ) . Then B satisfies (b1), and (b2) using standard mapping properties for pseudodifferential operators, see [15, Chapter 13.6, Proposition 6.5] for Sobolev norms, and [15, Chapter 13.8, Proposition 8.5] for Holder norms. \nWe show (b3). For (3.2) multiply (3.1) with x i and integrate, which yields \n∫ Ω x i ∂ j B j ( f ) dx = ∫ Ω x i f dx -m i ∫ Ω f dx . \nIntegrating by parts yields (3.2). For (3.3) multiply (3.1) with \| x \| 2 and proceed analogously. ✷ \nGiven such a right inverse B for the divergence operator, we obtain the solution operator G for P( h ) = f in two steps. We first use B to obtain a right inverse for the double divergence on symmetric trace-free two-tensors (Section 3.1). Using this operator one can then solve separately for the trace of h and the trace-free part of h (Section 3.2). In particular, given B the construction of G requires no further analysis.', '3.1 Trace-free symmetric double divergence': 'Let C ∞ c (Ω , S 2 /CA n ) be the space of symmetric n × n matrix valued functions with compact support in Ω. In this section we use the right inverse B of the divergence from Lemma 3.1 to obtain a solution operator for the equation \n∂ i ∂ j A ij = f , (3.4) \nwhere the source f has compact support in Ω and where we seek A ∈ C ∞ c (Ω , S 2 /CA n ) with vanishing matrix trace, tr( A ) = A ii = 0. Define \n· Q ( f ) = ∫ Ω ( 1 , x 1 , x 2 , . . . , x n , \| x \| 2 ) f dx ∈ /CA n +2 . \nIntegrating by parts one obtains that for all A ∈ C ∞ c (Ω , S 2 /CA n ) with A ii = 0, \n· Q ( ∂ i ∂ j A ij ) = 0 . (3.5) \nThus · Q ( f ) = 0 are n +2 necessary integrability conditions for (3.4). \nAs a first step we need the next lemma, this is similar to [11, Lemma 2.2]. \nLemma 3.2. Let Ω , B , ϕ , m i , m ij be as in Lemma 3.1. Define the operator S : C ∞ c (Ω , /CA ) → C ∞ c (Ω , S 2 /CA n ) by \nS ij ( f ) = 1 2 B i (B j ( f )) + 1 2 B j (B i ( f )) . \nThen: \n(c1) For all f ∈ C ∞ c (Ω , /CA ) , \n∂ i ∂ j S ij ( f ) = f -( ( ϕ + m j ∂ j ϕ ) ∫ Ω f dx -∂ j ϕ ∫ Ω x j f dx ) . \n- (c2) The estimates in point (a2) of Proposition 1.1 hold for S .\n- (c3) For all f ∈ C ∞ c (Ω , /CA ) , \n∫ Ω S ii ( f ) dx = ( m i m i -1 2 m ii ) ∫ Ω f dx -m i ∫ Ω x i f dx + 1 2 ∫ Ω \| x \| 2 f dx . \nProof. (c1): Using point (b1) of Lemma 3.1 we have \n∂ i ∂ j S ij ( f ) = ∂ j ∂ i B i (B j ( f )) = ∂ j B j ( f ) -∂ j ϕ ∫ Ω B j ( f ) dx = f -ϕ ∫ Ω f dx -∂ j ϕ ∫ Ω B j ( f ) dx. \nTogether with (b3) one obtains (c1). \n(c2): This follows directly from (b2), for the commutator estimate also use [B k B /lscript , ∂ i ] = B k [B /lscript , ∂ i ] + [B k , ∂ i ] B /lscript for all k, /lscript, i = 1 . . . n . \n(c3): This follows directly from (b3). ✷ \nSuppose that f ∈ C ∞ c (Ω , /CA ) satisfies the integrability conditions · Q ( f ) = 0 from (3.5). Then by (c1), respectively (c3), \n∂ i ∂ j S ij ( f ) = f , ∫ Ω S ii ( f ) dx = 0 . \nThe second identity and point (b1) of Lemma 3.1 imply S ii ( f ) = ∂ j B j (S ii ( f )). This motivates making the following ansatz for a trace-free solution of (3.4): \nS ij ( f ) + ( X /lscript k ) ij ∂ /lscript B k ( T ) with T = S ii ( f ) , (3.6) \nwhere ( X /lscript k ) ij ∈ /CA is symmetric in ij , ( X /lscript k ) ij = ( X /lscript k ) ji . We require that: \n- · The total symmetrization of ( X /lscript k ) ij in /lscript, i, j vanishes, for all k . Then ∂ i ∂ j ( ( X /lscript k ) ij ∂ /lscript B k ( T ) ) = 0, hence (3.6) still solves (3.4).\n- · ( X /lscript k ) ii = -δ /lscript k for all k, /lscript . Then ( X /lscript k ) ii ∂ /lscript B k ( T ) = -∂ /lscript B /lscript ( T ) = -T , hence (3.6) is trace-free. \nThe following choice for ( X /lscript k ) ij satisfies these properties: \n( X /lscript k ) ij = 1 n -1 ( 1 2 ( δ /lscripti δ j k + δ /lscriptj δ i k ) -δ ij δ /lscript k ) . (3.7) \nLemma 3.3. Let Ω , ϕ , B , S be as in Lemma 3.1 and Lemma 3.2, and let ( X /lscript k ) ij be as in (3.7) . Define · S : C ∞ c (Ω , /CA ) → C ∞ c (Ω , S 2 /CA n ) by \n· S ij ( f ) = S ij ( f ) + ( X /lscript k ) ij ∂ /lscript B k ( S pp ( f ) ) . (3.8) \nThen: \n- (d1) For all f ∈ C ∞ c (Ω , /CA ) one has ∂ i ∂ j · S ij ( f ) = ∂ i ∂ j S ij ( f ) .\n- (d2) The estimates of point (a2) of Proposition 1.1 hold for · S .\n- (d3) For all f ∈ C ∞ c (Ω , /CA ) one has · S ii ( f ) = ϕ ∫ Ω S ii ( f ) dx . \nIn particular, if · Q ( f ) = 0 then A = · S( f ) is trace-free and solves (3.4). \nProof. Denote the second term on the right of (3.8) by X ij ( f ). \n- (d1): By ∂ i ∂ j X ij ( f ) = 0, using our choice of ( X /lscript k ) ij .\n- (d2): This follows from points (b2) of Lemma 3.1 and (c2) of Lemma 3.2.\n- (d3): We have \nX ii ( f ) = -∂ /lscript B /lscript (S ii ( f )) = -S ii ( f ) + ϕ ∫ Ω S ii ( f ) dx, \nwhere for the last equality we use (b1). This implies (d3). ✷', '3.2 Linearized constant scalar curvature equation': "Define the linear operator ˜ P : C ∞ (Ω , S 2 /CA n ) → C ∞ (Ω , /CA ), \n˜ P( A ) = ∂ i ∂ j A ij + ∂ 1 ( 1 x 1 A ii ) . \nBy Lemma 2.2 this agrees with the linearized scalar curvature operator P up to the change of variables (2.1) and the overall factor ( x 1 ) n +1 . Here we use the operator · S from Lemma 3.3 to obtain a solution operator for \n˜ P( A ) = f , (3.9) \nwhere f has compact support in Ω and we seek A ∈ C ∞ c (Ω , S 2 /CA n ). Define \nQ KID ( f ) = ∫ Ω ( 1 , x 2 , . . . , x n , \| x \| 2 ) f dx ∈ /CA n +1 . \nBy (1.6), for all A with compact support we have \nQ KID ( ˜ P( A )) = 0 (3.10) \nand thus Q KID ( f ) = 0 are n +1 necessary integrability conditions for (3.9). We indicate how one can solve (3.9) when Q KID ( f ) = 0. Decompose \nA = · A + 1 n τ /BD \nwhere · A is trace-free symmetric and τ is a function, and where /BD is the n × n identity matrix. Then (3.9) reads \n∂ i ∂ j · A ij = f -˜ P ( 1 n τ /BD ) , (3.11) \nwith ˜ P( 1 n τ /BD ) = 1 n ∂ i ∂ i τ + ∂ 1 ( 1 x 1 τ ). Note that for all τ with compact support, \n∫ Ω x 1 ˜ P ( 1 n τ /BD ) dx = -∫ Ω 1 x 1 τ dx . (3.12) \nThus by choosing τ appropriately one can achieve that f ' = f -˜ P( 1 n τ /BD ) satisfies both Q KID ( f ' ) = 0 and ∫ Ω x 1 f ' dx = 0, which means that f ' satisfies the n + 2 integrability conditions · Q ( f ' ) = 0 for the trace-free symmetric double divergence in (3.5). Thus · A = · S( f ' ) is trace-free and solves (3.11). \nThis is implemented in the following proposition. \nLemma 3.4. Let Ω , ϕ , · S be as in Lemma 3.3. Set ψ = -x 1 ϕ . Define ˜ G : C ∞ c (Ω , /CA ) → C ∞ c (Ω , S 2 /CA n ) by \n˜ G( f ) = · S ( f -˜ P ( 1 n ψ ( ∫ Ω x 1 f dx ) /BD ) ) + 1 n ψ ( ∫ Ω x 1 f dx ) /BD . \nThen: \n(e1) There exist ϕ 1 , . . . , ϕ n +1 ∈ C ∞ c (Ω , /CA ) such that for all f ∈ C ∞ c (Ω , /CA ) , \n˜ P( ˜ G( f )) = f -n +1 ∑ a =1 ϕ a Q a KID ( f ) . \n(e2) The estimates of point (a2) of Proposition 1.1 hold for ˜ G . \nProof. (e1): Abbreviate τ = ψ ∫ Ω x 1 f dx and f ' = f -˜ P( 1 n τ /BD ). Then \n∫ Ω x 1 f ' dx = ∫ Ω x 1 f dx -∫ Ω x 1 ˜ P ( 1 n τ /BD ) dx (3.14) \nQ KID ( f ' ) = Q KID ( f ) -Q KID ( ˜ P ( 1 n τ /BD )) = Q KID ( f ) , (3.13) \n= ( 1 + ∫ Ω 1 x 1 ψdx ) ∫ Ω x 1 f dx = 0 . \nFor (3.13) we use (3.10), and for (3.14) we use (3.12) and ∫ Ω 1 x 1 ψdx = -∫ Ω ϕdx = -1. By definition of ˜ P and ˜ G we have \nwhere for the second equality we use (d1) and (d3). We now rewrite the first term using (c1), and the second term using (c3), then we simplify using (3.13) and (3.14). This yields (e1) with \n˜ P( ˜ G( f )) = ∂ i ∂ j · S ij ( f ' ) + ∂ 1 ( 1 x 1 · S ii ( f ' ) ) + ˜ P ( 1 n τ /BD ) = ∂ i ∂ j S ij ( f ' ) + ∂ 1 ( 1 x 1 ϕ ) ∫ Ω S ii ( f ' ) dx + ˜ P ( 1 n τ /BD ) \nϕ a = ϕ + m i ∂ i ϕ -( m i m i -1 2 m ii ) ∂ 1 ( 1 x 1 ϕ ) if a = 1 , -∂ a ϕ + m a ∂ 1 ( 1 x 1 ϕ ) if a = 2 . . . n , -1 2 ∂ 1 ( 1 x 1 ϕ ) if a = n +1 . \n(e2): By linearity of · S we can write \n˜ G( f ) = · S( f ) + ( 1 n ψ /BD -· S ( ˜ P ( 1 n ψ /BD )) ) ∫ Ω x 1 f dx . \nThe first term satisfies the estimates by (d2), for the second they are clear. ✷ \nWe are now ready to pass to the \nProof (of Proposition 1.1). Define G by \nG ij ( f ) = ( x 1 ) n +1 ( ˜ G ij + 1 1 -n b ij b k/lscript ˜ G k/lscript ) ( 1 ( x 1 ) n +1 f ) . \nWe check (a1): We have (1.7) by Lemma 2.2 and (e1); the map Π has rank n + 1 since by (1.7) the rank is ≤ n + 1 and by (1.6) the rank is ≥ n + 1; we have Π 2 = Π since applying Π to (1.7) and using (1.6) yields 0 = Π -Π 2 . We check (a2): This holds by (e2), note that multiplying and dividing by x 1 does not affect the estimates since ¯ Ω ⊆ /C0 n . ✷", '4 Application to Corvino-Schoen-type gluing': "Proposition 1.1 can be used to prove the following nonlinear result. We give an informal statement, since the details are routine. \nProposition 4.1 (Informal). Let n ≥ 2 and s > n/ 2 . Let Ω ⊆ /C0 n be an open annulus-type region with ¯ Ω ⊆ /C0 n . Let g in and g out be two smooth (or H s = W s, 2 ) metrics on ¯ Ω that have constant scalar curvature -n ( n -1) and that are sufficiently close in H s to the hyperbolic metric b . Then there exists a smooth (or H s ) metric g on ¯ Ω such that: \n- · R( g ) = -n ( n -1) mod ran(Π) , where Π is defined in Proposition 1.1.\n- · g = g in in a neighborhood of the interior boundary of ¯ Ω .\n- · g = g out in a neighborhood of the exterior boundary of ¯ Ω . \nProof. This is similar to [11, Proof of Theorem 1.3, Step 1 and 2], hence we only give a sketch. Write R( b + h ) = -n ( n -1) + P( h ) + N( h ) where N( h ) is the nonlinearity, given by terms of quadratic and higher order. Fix an interpolation h between g in -b and g out -b , then make the ansatz g = b + h + h ' where the correction h ' has compact support in Ω. It is constructed as the solution of the fixed point \nh ' = -G ( P( h ) + N( h + h ' ) ) . (4.1) \nThen g satisfies all properties stated in the proposition. \n✷ \nIn Proposition 4.1, the constant scalar curvature equation is only solved modulo the range of Π, which has dimension n +1. The full equation R( g ) = -n ( n -1) holds if and only if the argument of G in (4.1) is in the kernel of Π, equivalently, if and only if for all a = 1 , . . . , n +1, \n∫ Ω κ a ( P( h ) + N( h + h ' ) ) dµ b = 0 \nwhere κ a are the static KIDs defined in (1.4). By (1.5) and the divergence theorem, this is equivalent to \n∫ ∂ out Ω V ( a ) i ( g out -b ) dS i b -∫ ∂ in Ω V ( a ) i ( g in -b ) dS i b = -∫ Ω κ a N( h + h ' ) dµ b (4.2) \nwhere the integrals on the left are flux integrals over the exterior boundary ∂ out Ω and interior boundary ∂ in Ω respectively, both in the outward direction (in the low regularity regime one can use averaged flux integrals). \nOf particular interest is the case when g in is given and one looks for g out in the family of boosted Schwarzschild-AdS metrics, which depends on n + 1 mass and boost parameters θ = ( θ 1 , . . . , θ n +1 ), see e.g. [7]. The idea is to apply Proposition 4.1 with θ as parameters, and to then adjust the parameters such that the n +1 conditions (4.2) hold. \nSince the right hand side of (4.2) is small, it is necessary that the flux integrals of g in are approximately equal to the flux integrals of some member of the family of boosted Schwarzschild-AdS metrics. If this is the case, and under appropriate smallness conditions, one can then construct the exact \nparameters θ for which (4.2) holds by reformulating (4.2) as a fixed point equation for θ . Arguments of this kind are worked out in detail in [7] in the hyperbolic setting, and in [5,6,9,11] in the Euclidean setting. A particularly simple argument (which also shows that the smallness conditions needed can be realised in some cases) can be given for parity-invariant metrics, see Example 4.2 below. \nIn this manner, Proposition 1.1 can be used to glue a constant-scalarcurvature metric g in to a boosted Schwarzschild-AdS metric, in such a way that the glued metric has constant scalar curvature. This gives a simpler proof of a similar perturbative result in [7], under the much weaker differentiability conditions of Proposition 4.1, and extends the analysis of [7] to include dimension n = 2. \nExample 4.2 (Gluing parity-invariant metrics). Let Z : /C0 n → /C0 n be the parity isometry of the hyperbolic metric about the point (1 , 0 , . . . , 0). In Proposition 4.1, if the annulus Ω and the metrics g in and g out are Z -invariant, then the glued metric g will be Z -invariant, by using instead of G the Z -invariant 1 2 (G+ Z ∗ G Z ∗ ). Then n of the n +1 equations in (4.2) hold trivially, and (4.2) becomes a scalar equation. Then: \nIf g in ,/epsilon1 is a one-parameter family of Z -invariant constant-scalarcurvature metrics depending smoothly on a parameter /epsilon1 ∈ ( -1 , 1) , and with g in , 0 = b the hyperbolic metric, then for all /epsilon1 sufficiently close to zero, the metric g in ,/epsilon1 can be glued, with constant scalar curvature, to a Z -invariant Schwarzschild AdS-metric g out . \n/negationslash \nTo see this, let g out ,m be the unboosted and Z -invariant Schwarzschild AdSmetric with mass parameter m , in particular g out , 0 = b . By Proposition 4.1 one obtains a Z -invariant glued metric g m,/epsilon1 that depends parametrically on m and /epsilon1 . The scalar equation (4.2) is of the form u ( m,/epsilon1 ) = 0, where u is a smooth function near the origin that satisfies u (0 , 0) = 0. Furthermore ∂u ∂m (0 , 0) = 0 since only the flux integral over ∂ out Ω contributes to this dervative. Hence by the implicit function theorem, u ( m,/epsilon1 ) = 0 can be locally solved for m = m ( /epsilon1 ), implying the claim. ✷ \nRemark 4.3. 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2024PSJ.....5..221C	46PWirtanen is a Jupiterfamily comet probably originating from the solar systems Kuiper Belt that now resides on a 5.4 yr elliptical orbit. During its 2018 apparition comet 46P passed unusually close to the Earth within 0.08 au presenting an outstanding opportunity for closeup observations of its inner coma. Here we present observations of HCN HSUP13SUPCN and HCSUP15SUPN emission from 46P using the Atacama Compact Array. The data were analyzed using the SUBLIME nonLTE radiative transfer code to derive SUP12SUPCSUP13SUPC and SUP14SUPNSUP15SUPN ratios. The HCNHSUP13SUPCN ratio is found to be consistent with a lack of significant SUP13SUPC fractionation whereas the HCNHCSUP15SUPN ratio of 68 27 using our most conservative 1 uncertainties indicates a strong enhancement in SUP15SUPN compared with the solar and terrestrial values. The observed SUP14SUPNSUP15SUPN ratio is also significantly lower than the values of 140 found in previous comets implying a strong SUP15SUPN enrichment in 46Ps HCN. This indicates that the nitrogen in Jupiterfamily comets could reach larger isotopic enrichments than previously thought with implications for the diversity of SUP14SUPNSUP15SUPN ratios imprinted into icy bodies at the birth of the solar system.	2024-10-01T00:00:00Z	['10.48550/arXiv.2409.05711', '10.3847/PSJ/ad7829', 'arXiv:2409.05711', '2024arXiv240905711C', '2024PSJ.....5..221C']	['Comets', 'Neutral coma gases', 'Comae', 'Small Solar System bodies', 'Astrochemistry', 'Cosmochemistry', 'Isotopic abundances', 'Radio interferometry', 'Submillimeter astronomy', '280', '2158', '271', '1469', '75', '331', '867', '1346', '1647', 'Astrophysics - Earth and Planetary Astrophysics']	Evidence for Surprising Heavy Nitrogen Isotopic Enrichment in Comet 46PWirtanens Hydrogen Cyanide	2,024	195	0.52	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.05711.pdf	{"Evidence for Surprising Heavy Nitrogen Isotopic Enrichment in Comet 46P/Wirtanen's Hydrogen Cyanide": "M. A. Cordiner, 1, 2 K. Darnell, 1, 2 D. Bockel'ee-Morvan, 3 N. X. Roth, 1, 2 N. Biver, 3 S. N. Milam, 1 S. B. Charnley, 1 J. Boissier, 4 B. P. Bonev, 5 C. Qi, 6 J. Crovisier, 3 and A. J. Remijan 7 \n1 Astrochemistry Laboratory, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA. \n2 Department of Physics, Catholic University of America, Washington, DC 20064, USA. \n3 LESIA, Observatoire de Paris, Universit'e PSL, CNRS, Sorbonne Universit'e, Universit'e de Paris, 5 place Jules Janssen, F-92195 Meudon, France. \n4 Institut de Radioastronomie Millimetrique, 300 rue de la Piscine, F-38406, Saint Martin d'Heres, France. \n5 Department of Physics, American University, Washington D.C., USA. \n6 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS 42, Cambridge, MA 02138, USA. \n7 National Radio Astronomy Observatory, Charlottesville, VA 22903, USA.", 'ABSTRACT': "46P/Wirtanen is a Jupiter-family comet, probably originating from the Solar System's Kuiper belt, that now resides on a 5.4 year elliptical orbit. During its 2018 apparition, comet 46P passed unusually close to the Earth (within 0.08 au), presenting an outstanding opportunity for close-up observations of its inner coma. Here we present observations of HCN, H 13 CN and HC 15 N emission from 46P using the Atacama Compact Array (ACA). The data were analyzed using the SUBLIME non-LTE radiative transfer code to derive 12 C/ 13 C and 14 N/ 15 N ratios. The HCN/H 13 CN ratio is found to be consistent with a lack of significant 13 C fractionation, whereas the HCN/HC 15 N ratio of 68 ± 27 (using our most conservative 1 σ uncertainties), indicates a strong enhancement in 15 N compared with the solar and terrestrial values. The observed 14 N/ 15 N ratio is also significantly lower than the values of ∼ 140 found in previous comets, implying a strong 15 N enrichment in 46P's HCN. This indicates that the nitrogen in Jupiter-family comets could reach larger isotopic enrichments than previously thought, with implications for the diversity of 14 N/ 15 N ratios imprinted into icy bodies at the birth of the Solar System. \nKeywords: Comets, individual: 46P/Wirtanen - Radio interferometry - Molecular lines - Astrochemistry", '1. INTRODUCTION': 'Comets consist of a mixture of ice, dust and pebbles, which are thought to have accreted in the vicinity of the giant planets around 4.5 Gyr ago, and have remained relatively unaltered ever since. Measurements of their compositions therefore provide a unique tool for investigating chemical and physical processes that occurred in the protosolar accretion disk during (and prior to) the epoch of planet formation. Due to the difficulty of protoplanetary disk midplane observations using even the most powerful ground and space-based telescopes, important details regarding the chemistry of star and planet formation remain unknown ( Oberg et al. 2023). Cometary observations are uniquely useful in their ability to provide fundamental, quantitative constraints on astrochemical models for star and planet forming regions, in particular, regarding the chemistry that occurred during the earliest history of our Solar System. \nIsotopic ratios such as D/H and 15 N/ 14 N within cometary molecules are especially sensitive to the physical conditions prevalent during the formation and accretion of cometary matter. Isotopic fractionation is the process by which different isotopes of a given atom can become concentrated in a (gas or solid phase) molecular reservoir, leading to isotopic', 'Cordiner et al.': "Table 1. Observed Spectral Line Details \nabundance ratios that differ from the elemental ratios of the bulk reservoir. As described in the review by Nomura et al. (2023), isotopic fractionation occurs in interstellar, protostellar, protoplanetary disk, and planetary environments through a broad range of gas- and solid-phase processes. \nIn dense interstellar clouds, strong depletion of 15 N in N 2 H + gas is commonly found relative to the local interstellar medium (ISM) (Bizzocchi et al. 2013; Redaelli et al. 2018). On the other hand, Hily-Blant et al. (2018) found evidence for 15 N enrichment in HC 3 N towards the L1544 prestellar core. ALMA observations of protoplanetary disks have recently revealed significant 15 N enrichment in gas-phase HCN (Guzm'an et al. 2017; Hily-Blant et al. 2019), and this can be explained as a result of isotope-selective photodissociation of N 2 (Nomura et al. 2023). As shown by Visser et al. (2018), self shielding of the dominant N 2 isotopologue leads to a region of the disk enriched in gas-phase 15 N, which becomes incorporated into other gas-phase molecules, resulting in enhanced 15 N/ 14 N ratios. When the density is high enough and the temperature is low enough, such isotopically enriched gas-phase molecules freeze out onto dust grain surfaces to form ice mantles, which are later incorporated into comets and other icy bodies. \nConsistent with this picture, the observed protoplanetary disk HC 14 N/HC 15 Nratios of ∼ 100-200 are similar to those found in comets (Nomura et al. 2023), which corroborates our basic understanding of the genetic relationship between protoplanetary disk and cometary compositions. Unlike protoplanetary disks, however, comets show a surprising degree of uniformity in their 14 N/ 15 N ratios (among different molecules, and across different comets), with a weighted average value of 144 ± 3 from HCN, CN and NH 2 in 31 comets (Hily-Blant et al. 2017). A similar nitrogen isotopic fingerprint was also found in the molecular nitrogen gas emitted by comet 67P ( 14 N/ 15 N ∼ 130 ± 30; Altwegg et al. 2019). \nContinued studies of cometary isotopic ratios are therefore of interest, to explore the distribution of 14 N/ 15 N values for comparison with observations of protoplanetary disks and models for the formation of our own Solar System, with the aim of constraining the physics and chemistry of these crucial planet-forming environments. In this article, we present new results on the HC 14 N/HC 15 N abundance ratio in Jupiter family comet 46P/Wirtanen, which was observed by Cordiner et al. (2023) using the Atacama Large Millimeter/submillimeter Array (ALMA) during its exceptional 2018 apparition. The unusually close Earth-comet distance allowed the detection of weak spectral lines not typically detectable in Jupiter family comets from the ground, resulting in the first map of HC 15 N in a Jupiter family comet, and new insights into the possible diversity of 14 N/ 15 N ratios among the comet population.", '2. OBSERVATIONS': "Observations of comet 46P/Wirtanen were conducted using ALMA during 2018 December 2-7, when the comet was around 0.1 au from Earth and 1.06-1.07 au from the Sun (the comet's perihelion date was 2018-12-12). This study focuses on the Atacama Compact Array (ACA) data, which incorporated 12 × 7 m antennas covering baselines in the range 9-50 m. The shorter baselines of the ACA compared with the main (12 m) ALMA array make it more sensitive to extended coma emission; the resulting synthesized beam size (angular resolution) was θ B = 4 . 5 '' × 2 . 8 '' at 354 GHz. Observations were conducted of the HCN ( J = 4 -3), H 13 CN ( J = 3 -2) and HC 15 N ( J = 3 -2) transitions, using the Band 6 and 7 receivers. Multiple lines of the CH 3 OH J K = 5 K -4 K band were observed in the range 241-242 GHz in order to derive the coma kinetic temperature. Additional observational parameters are given in Table 1, including the spectral resolution (∆ ν ), Geocentric distance (∆) and spectrally integrated line intensity ( ∫ S ν dv , with ± 1 σ statistical errors derived from the actual noise level inside a ± 90 kms -1 region adjacent to each spectral line). For CH 3 OH, the integrated line intensity was summed over the 14 detected transitions of the J K = 5 K -4 K band. \nTable 2. Production rates and abundances for H 2 O and the observed 46P HCN isotopologues \nData flagging, calibration and continuum subtraction were performed as described by Cordiner et al. (2023). Imaging was performed using the CASA tclean (Hogbom) algorithm with natural weighting and a pixel size of 0 . 5 '' . Deconvolution was carried out within a 30 '' -diameter circular mask centered on the comet, with a flux threshold of twice the RMS noise level ( σ ).", '3. RESULTS': "Spectrally integrated flux maps for the three HCN isotopologues are shown in Fig. 1, integrated over velocity ranges ± 1 . 2 kms -1 with respect to the line rest velocities. Angular distances on the sky have been converted to spatial coordinates at the distance of the comet, with the origin at the HCN peak (for HCN), and at the CH 3 OH peak for HC 15 N and H 13 CN (observed simultaneously with CH 3 OH). For the weaker (HC 15 N and H 13 CN) lines, the spectral integration ranges were determined based on the velocity width of the stronger (HCN) line. \nSpectra were extracted from the (0,0) position in each map (shown in Fig. 2). Based on the spectrally integrated line fluxes, HCN is detected at a high significance (76 σ ), while HC 15 N and H 13 CN are detected at 4 . 6 σ and 3 . 2 σ , respectively. The observed CH 3 OH spectrum is shown in Fig. 3. The spectral line profiles are well resolved for HCN and CH 3 OH, showing a characteristic double-peaked sub-structure due to Doppler motion of the quasi-isotropically expanding coma along the line of sight, whereas no sub-structure is expected for the lower-resolution H 13 CN and HC 15 N observations. \nThe raw statistical significance of our HC 15 N and H 13 CN detections is less than optimal, but the evidence for both molecules is strengthened by the properties of the spectral line profiles, which match (within the noise) the rest velocity and FWHM of the (high-significance) HCN line. In the case of HC 15 N, the four spectral channels that make up the line peak are all at a significance of greater than 3 σ (where σ = 4 . 7 mJy). Furthermore, the HC 15 N emission peak coincides with the known location of the comet's nucleus, as derived from the peak position of the spectrally integrated CH 3 OH data. As a general rule of thumb in radio interferometry, a detection can be considered real if appears above the 3 σ level for a source with a known location (as is the case here), or above 5 σ if the location is unknown; both HC 15 N and H 13 CN therefore fulfill the detection criteria. \nSpectral modeling was performed using the spherically symmetric (1D) version of SUBLIME: a time-dependent, non-LTE radiative transfer code for cometary comae (Cordiner et al. 2022), and the 1D version of the model was found to provide a sufficiently good fit to these ACA data. In the molecular excitation calculation, collision rates between CH 3 OH and H 2 O were assumed to be the same as CH 3 OH with H 2 (Rabli & Flower 2010). HCN-H 2 O collision rates are from Dubernet & Quintas-S'anchez (2019), and were assumed to apply equally for all three HCN isotopologues. Rovibrational pumping due to the solar radiation field was calculated using molecular data for HCN and CH 3 OH from the HITRAN and Planetary Spectrum Generator databases (see Cordiner et al. 2023; Villanueva et al. 2018, for details). Hyperfine structure was included in the model spectra for all three HCN isotopologues, assuming equilibrium line-strength ratios among the hyperfine components in a given J state. \nH 2 Oproduction rates for comet 46P ( Q (H 2 O)) were obtained for our observation times from cubic spline interpolation of the SOHO Lyα -derived measurements by Combi et al. (2020), and are given in Table 2. These Q (H 2 O) values are consistent with the average water production rate of 8 × 10 27 s -1 measured by Lis et al. (2019) using the SOFIA telescope a few days closer to perihelion (between December 14-20), and with the average value of 7 × 10 27 s -1 obtained using IRTF between December 6-21 (Khan et al. 2023). The fact that the values from three different observatories (spanning ultraviolet to far-infrared wavelengths) are consistent despite the large differences in their beam sizes, implies that the SOHO water production rates should be applicable at the spatial resolution of our ACA data. \nPrior to spectral modeling, the measured ACA spectra were corrected for interferometric flux loss factors of 0.79 for HCN and 0.73 for HC 15 N and H 13 CN. These were derived based on initial best-fit SUBLIME model image cubes, which were spectrally integrated then processed using CASA simobserve according to the particular sky position and observation time of each line, then subject to the same cleaning and deconvolution as the observations. The purpose of \n<!-- image --> \n46P HC \n<!-- image --> \n15 \nN \nUT 2018-12-07 23:57 \nFigure 1. ALMA ACA maps of spectrally integrated HCN, H 13 CN and HC 15 N emission from comet 46P/Wirtanen. Contour intervals are in units of 5 σ for HCN and 1 σ for H 13 CN and HC 15 N. Beam size (angular resolution) is shown lower left; skyprojected comet-sun vectors are shown lower right. The HCN map is centered on the emission peak, whereas the H 13 CN and HC 15 N maps are centered on the stronger (simultaneously observed) CH 3 OH emission peak (not shown). \n<!-- image --> \n46P/Wirtanen HCN \nJ \n=4 3 \nFigure 2. ALMA ACA spectra of HCN, H 13 CN and HC 15 N in comet 46P/Wirtanen, extracted from the central position(s) shown in Figure 1. Best fitting SUBLIME radiative transfer models (including hyperfine structure) are overlaid in orange. The 3 σ feature in the HC 15 N spectrum around -5 kms -1 is interpreted as an unusually strong ( > 3 σ ) noise spike. \n<!-- image --> \nFigure 3. ALMA ACA spectrum of CH 3 OHobserved in comet 46P/Wirtanen (extracted from the CH 3 OHspectrally integrated brightness peak). Best fitting SUBLIME radiative transfer model is overlaid in orange, demonstrating a gas kinetic temperature of T kin = 93 ± 3 K. \n<!-- image --> \nthis loss factor is to account for the fact that the the ACA images are missing a portion of the flux from the extended coma because the interferometer is only sensitive to structures less than λ/D in spatial extent, where λ is the observed wavelength and D is the minimum antenna separation (9 m in this case). \nA beam-averaged gas kinetic temperature of T kin = 93 ± 3 K was derived using a least-squares fit to the CH 3 OH spectrum (Fig. 3). By fitting the HCN J = 4 -3 line profile, a coma outflow velocity of 0 . 53 ± 0 . 01 kms -1 , and Doppler shift of -0 . 097 ± 0 . 006 kms -1 was derived; these values were used in subsequent fits to the spectrally less well resolved (and lower signal-to-noise) HCN isotopologue lines. Best fitting production rates and abundances (relative to H 2 O) for the three HCN isotopologues are given in Table 2. Formal (1 σ ) uncertainties were derived from the diagonal elements of the covariance matrix of the least-squares fit. To account for the correlation between adjacent spectral channels introduced by the ACA correlator, the RMS noise measurement on each spectrum was multiplied by a factor of 1.29, following Nixon et al. (2020). \nThe maximum optical depth of our radiative transfer model at the HCN central peak is 0.2, but since the coma opacity falls rapidly with nucleocentric distance, the mean optical depth inside the ACA beam is only 0.04. Hence, the HCN J = 4 -3 line is largely optically thin, as demonstrated by the ratio of the main HCN line peak with respect to the hyperfine satellites, which are too weak to be clearly detected, consistent with the optically thin limit (see Figure 2). Considering the cometary HCN data are typically well reproduced by a spherically-symmetric coma model (see also Cordiner et al. 2014, 2019; Roth et al. 2021; Cordiner et al. 2023), it is therefore unlikely that the presence of spurious, high-opacity HCN clumps or jets would significantly impact our results.", '4. DISCUSSION': "46P/Wirtanen is only the second Jupiter-family comet to-date in which the minor ( 15 N and 13 C) isotopologues of HCN have been detected. The first was 17P/Holmes, which had HCN/HC 15 N = 139 ± 26 (observed using the IRAM 30-m telescope during the comet's major outburst in October 2007; Bockel'ee-Morvan et al. 2008). Accounting for purely statistical errors, the HCN/HC 15 N production rate ratio in comet 46P is 67 ± 16, and the HCN/H 13 CN ratio is 90 ± 28. Within the uncertainties, the 12 C/ 13 C ratio is consistent with previous cometary observations (Bockel'eeMorvan et al. 2015; Cordiner et al. 2019), whereas the 14 N/ 15 N ratio is surprisingly enriched in the minor ( 15 N) isotope - the (error-weighted) average of prior HCN/HC 15 N measurements in four previous comets is 146 ± 11 (Bockel'eeMorvan et al. 2008; Biver et al. 2016). A tentative (3 σ ) detection of HC 15 N was also obtained in comet 46P by Biver et al. (2021) using the IRAM 30-m telescope between 2018-12-12 and 2018-12-18, leading to an HCN/HC 15 N ratio of 77 ± 26. The combination of these IRAM and ALMA results add credibility to the conclusion that comet 46P's HCN was surprisingly enhanced in 15 N. \nBecause HCN and its minor isotopologues were observed in comet 46P on different dates (almost 6 days apart), variations in the HCN/H 2 O production rate ratio, as well as uncertainties in Q (H 2 O) should be incorporated into our isotopic ratio uncertainties. Based on six infrared spectroscopic measurements of Q (HCN) and Q (H 2 O) in comet 46P between December 6-21 (Bonev et al. 2021; Khan et al. 2023), the HCN abundance relative to H 2 O remained apparently constant, with a standard deviation on the Q (HCN)/ Q (H 2 O) ratio of only 0.00013 (corresponding to 6% of the HCN/H 2 O value), and the error-weighted mean was Q (HCN)/ Q (H 2 O) = 0 . 0020 ± 0 . 0001. It is therefore reasonable to assume that variations in the comet's HCN/H 2 O ratio contribute a negligible source of uncertainty to our result. Errors on Q (H 2 O) may be more significant, however, considering the scatter in Combi et al. (2020)'s measurements as a function of time, which amount to an RMS of 7 . 6 × 10 26 s -1 ( ∼ 10%) with respect to the best-fitting linear trend in Q (H 2 O) between 2018-11-28 and 2018-12-11. Adding this fractional uncertainty in quadrature with the statistical uncertainty on Q (HC 15 N) /Q (H 2 O), combined with a further 10% uncertainty on Q (HCN) and Q (HC 15 N) to account for possible inaccuracies in the absolute ACA flux scale, gives a value of HCN/HC 15 N = 67 ± 20. \nThe possibility of more extreme temporal variability in Q (H 2 O) (or greater variability in HCN/H 2 O) as a function of time cannot be ruled out however. In that case, we combine the error on the observed H 13 CN abundance ((1 . 03 ± 0 . 32) × 10 -5 ) with the expected HCN/H 13 CN ratio ( ≈ 90; based on measurements of 12 C/ 13 C ratios in a diverse range of comets and other solar system bodies; Nomura et al. 2023) to obtain a more conservative HCN abundance (and associated uncertainty) of (9 . 27 ± 2 . 88) × 10 -3 at the time of our HC 15 N observation. This leads to a more conservative HCN/HC 15 N ratio (and uncertainty) of 68 ± 27. For comparison with meteoritic measurements, whereby the isotopic ratio is typically expressed as a fractional enhancement of the minor isotope, δ 15 N, with respect to the terrestrial standard ratio ( 14 N/ 15 N) Earth (Nier 1950), we calculate δ 15 N = ( 14 N/ 15 N)/( 14 N/ 15 N) Earth -1 = (3015 ± 1200) h (per mil). \nIn the context of prior 14 N/ 15 N ratios measured throughout the Solar System and beyond, our value in comet 46P is statistically unusual. Cometary 14 N/ 15 N ratios are known to be systematically lower than those found in the terrestrial and giant planets, as well as the Sun, but the distribution of cometary values measured to-date appears surprisingly uniform. The weighted average 14 N/ 15 N ratio for HCN, CN and NH 2 in a sample of 31 comets is 144 ± 3 (Hily-Blant et al. 2017), which is significantly enriched in 15 N compared to the bulk terrestrial and Solar values of 273 and 459, respectively (Nier 1950; Marty et al. 2011). Indeed, within the error bars, all prior cometary 14 N/ 15 N measurements (within various molecules) may be consistent with a value ≈ 140, including the in-situ mass spectrometry measurements of the 67P coma made by the Rosetta spacecraft (see Fig. 4). Our new 14 N/ 15 N value in comet 46P is 2 . 8 σ less than the weighted average of 144, and therefore represents an unexpected outlier with respect to the overall comet population, as well as to other (bulk) Solar System bodies. \nMeteoritic organics typically have 14 N/ 15 N ratios in the range 200-270 (somewhat enriched relative to the Earth; see Fig. 4). Carbonaceous chondrite meteorites also contain small, micron-sized isotopic 'hot spots', which exhibit strong 15 N enrichment (with 14 N/ 15 N values as low as 65 ± 14; Busemann et al. 2006), which is similar to our 46P value. Considering the anomalous nature of our value compared with the numerous previous cometary 14 N/ 15 N measurements, it is interesting to speculate that comets (or comet 46P in particular) could also contain isotopically heterogeneous material. Our HC 15 N observation occurred over a time period of 1 h (between UT 2018-12-07 23:58 and 2018-12-08 00:59) with a beam size ≈ 3 . 5 '' (probing radial distances ∼ 114 km from the nucleus). For an outflow velocity of 0.53 km s -1 and volatile mass loss rate of 320 kg s -1 (based on Q (H 2 O); Table 2, and a typical CO 2 /H 2 O ratio of 17%; Ootsubo et al. 2012), the mass of volatiles within the ACA beam was ∼ 69 , 000 kg. This is indeed small compared with the total mass of the comet ( ∼ 4 × 10 14 kg, adopting a mean radius of 560 m and density of 0.6 g cm -3 ), so it is plausible that our ACA measurement is representative of a spatially isolated 15 N enhancement confined to a relatively small part of the comet's nucleus. This idea is supported by the fact that Moulane et al. (2023) measured a nominal CN/C 15 N ratio of 150 ± 30 using VLT ultraviolet spectroscopy only one day later (on 2018-12-09 at UT 00:41), which could be more indicative of the comet's bulk 14 N/ 15 N value. However, it should be noted that CN in cometary comae often has an additional source that cannot be explained by HCN photolysis alone (Fray et al. 2005; Cottin & Fray 2008), so its nitrogen isotopic ratio need not necessarily be the same as HCN. As shown by Fig. 4, there is no prior evidence for differing CN/C 15 N and HCN/HC 15 N ratios in comets, but this possibility should be investigated further, and could help shed light on the extent of the chemical link between CN and HCN in comets. \nGiven the 9.1 h rotation period of the 46P nucleus (Farnham et al. 2021), the time difference of ≈ 24 h between our ACA HCN observations and the VLT CN observations of Moulane et al. (2023) (corresponding to 2.6 nucleus \nFigure 4. Nitrogen isotope measurements in primitive Solar System materials, expressed as 14 N/ 15 N (lower x axis, with smaller values indicating greater 15 N enrichment towards the right); the fractional isotopic enrichment δ 15 N is shown on the upper x axis. This Figure was adapted from (Bockel'ee-Morvan et al. 2015). The meteoritic bulk value is from CR chondrite insoluble organic matter (IOM) measurements (Nomura et al. 2023), while the 15 N hotspots are regions that present strong isotopic enrichments relative to the surrounding meteoritic material. Grey ellipses represent the range of values from multiple laboratory sample analysis measurements. The CN, HCN and NH 2 measurements in cometary gases are the averages from Manfroid et al. (2009), Shinnaka et al. (2016), Bockel'ee-Morvan et al. (2008) and Biver et al. (2016). 67P values are from Altwegg et al. (2019), and the 46P CN value is from Moulane et al. (2023). \n<!-- image --> \nrotations) also allows for the possibility that different regions of the nucleus have different nitrile 14 N/ 15 N ratios, in the case of outgassing dominated by a solar-facing jet (as was deduced for this comet by Cordiner et al. 2023). \n15 N enrichment in cometary nitriles and meteoritic organics probably occurred as a consequence of isotope-selective photodissociation of N 2 in the protosolar nebula (Visser et al. 2018; Lee et al. 2021). Although such fractionation appears to have operated to produce relatively uniform 14 N/ 15 N ratios (in the range 140 ∼ 170) within the bulk of the icy materials found in the comets and moons of the outer Solar System (Furi & Marty 2015), the full range of \n14 N/ 15 N values present in the planet forming regions of our protosolar disk remains unknown. The surprisingly low HCN/HC 15 N ratio in comet 46P therefore presents a new challenge for our understanding of the physical and chemical processes that occurred during Solar System formation. \nAlthough the 12 C/ 13 C values measured in molecules in interstellar clouds and star-forming regions exhibit some genuine diversity, the 12 C/ 13 C ratios measured throughout the Solar System are relatively uniform (Nomura et al. 2023). In contrast to HC 15 N, our HCN/H 13 CN ratio in 46P is consistent with the values of 12 C/ 13 C ∼ 90 found in meteorites, terrestrial and giant planets, and their icy satellites, as well as other comets. Consequently, we confirm the findings of Cordiner et al. (2019) suggesting a lack of strong carbon isotopic fractionation in cometary HCN during the formation of the Solar System.", '5. CONCLUSION': "We observed rotational emission lines from HCN and its two minor isotopologues HC 15 N and H 13 CN in the coma of comet 46P during its exceptionally close December 2018 apparition, using the Atacama Compact Array. The HC 15 N ( J = 3 -2) line was surprisingly strong compared with the H 13 CN ( J = 3 -2) line, allowing HC 15 N to be mapped for the first time in a comet. The spectral line data were subject to non-LTE radiative transfer modeling, from which we derived a HCN/HC 15 N production rate ratio of 67 ± 17 (or a more conservative value of 68 ± 27, based on the observed H 13 CN and HC 15 N abundances, with an assumed HCN/H 13 CN ratio of 90). These values are significantly lower than previously measured in any N-bearing molecules in comets or in any other large Solar System bodies. Within the Solar System, comet 46P's HCN was therefore surprisingly enriched in 15 N, reaching a value similar to those found in the most 15 N-enriched 'hotspots' in spatially isolated meteoritic samples. This result implies that cometary 14 N/ 15 N ratios could be more diverse, and potentially 15 N rich, than previously thought. More studies of 14 N/ 15 N in comets are warranted to better understand the diversity of isotopic fractionation processes taking place in protoplanetary disks. \nThis work was supported by the National Science Foundation under grant Nos. AST-2009253, AST-2009398, and by NASA's Planetary Science Division Internal Scientist Funding Program through the Fundamental Laboratory Research work package (FLaRe). This work makes use of ALMA data set ADS/JAO.ALMA#2018.1.01114.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSTC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.", 'REFERENCES': "Altwegg, K., Balsiger, H., & Fuselier, S. 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2024NIMPA106969819B	The Probe Of Extreme MultiMessenger Astrophysics POEMMA is a proposed dualsatellite mission to observe UltraHighEnergy Cosmic Rays UHECRs increasing the statistics at the highest energies and VeryHighEnergy Neutrinos VHENs following multimessenger alerts of astrophysical transient events throughout the universe such as gammaray bursts and gravitational wave events. POEMMABalloon with Radio PBR is a scaleddown version of the POEMMA design adapted to be flown as a payload on one of NASAs suborbital Super Pressure Balloons SPBs circling over the Southern Ocean for up to 100 days after a launch from Wanaka New Zealand. This overview will provide a summary of the mission with its science goals the instruments and the current status of PBR.	2024-12-01T00:00:00Z	['arXiv:2409.06753', '2024NIMPA106969819B', '10.48550/arXiv.2409.06753', '2024arXiv240906753B', '10.1016/j.nima.2024.169819']	['Ultra-high-energy cosmic rays', 'High-energy neutrinos', 'Orbital experiment', 'Multi-messenger astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena']	POEMMABalloon with Radio A balloonborn multimessenger multidetector observatory	2,024	195	0.44	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	1	https://arxiv.org/pdf/2409.06753.pdf	{'POEMMA-Balloon with Radio: a balloon-born multi-messenger multi-detector observatory': "Matteo Battisti a, ∗ , Johannes Eser b , Angela Olinto c , Giuseppe Osteria d , for the JEM-EUSO Collaboration \na Universit'e Paris Cit'e, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France, b University of Chicago, Illinois, U.S.A. c Columbia University, New York, U.S.A d INFN Napoli, Italy", 'Abstract': "The Probe Of Extreme Multi-Messenger Astrophysics (POEMMA) is a proposed dual-satellite mission to observe Ultra-HighEnergy Cosmic Rays (UHECRs), increasing the statistics at the highest energies, and Very-High-Energy Neutrinos (VHENs), following multi-messenger alerts of astrophysical transient events throughout the universe such as gamma-ray bursts and gravitational wave events. POEMMA-Balloon with Radio (PBR) is a scaled-down version of the POEMMA design, adapted to be flown as a payload on one of NASA's sub-orbital Super Pressure Balloons (SPBs) circling over the Southern Ocean for up to 100 days after a launch from Wanaka, New Zealand. This overview will provide a summary of the mission with its science goals, the instruments, and the current status of PBR. \nKeywords: Ultra-High-Energy Cosmic Rays, High-Energy Neutrinos, orbital experiment, multi-messenger astrophysics.", '1. The Science': 'The POEMMA-Balloon with Radio (PBR) is a scientific mission optimized for a flight on a NASA Super-Pressure Balloon (SPB), flying at a nominal altitude of 33 km. \nPBR will explore multi-messenger science from a unique perspective (from sub-orbital altitudes), featuring a unique set of detectors and detection techniques, with a secondary goal to advance the technological readiness level (TRL) of a future multi-messenger space mission such as POEMMA [1]. \nThe main science objectives of PBR are: (1) to observe UHECRs via the fluorescence technique from sub-orbital space; (2) to observe High-Altitude Horizontal Air showers (HAHAs) with energies above the cosmic ray knee (E > 0.5 PeV) using the optical and radio detection for the first time; and (3) to follow astrophysical event alerts in search of VHENs. The PBR design and main objectives are illustrated in Fig. 1.', '2. The detector': 'PBR is built upon the experience of the previous balloon iterations, EUSO-SPB1 [2] and EUSO-SPB2 [3]. The PBR instrument consists of a 1.1 m aperture Schmidt telescope similar to the POEMMA design with two cameras in its focal surface: a Fluorescence Camera (FC) and a Cherenkov Camera (CC). In addition, PBR has a Radio Instrument (RI) optimized for the detection of EASs (covering the 50-500 Mhz range). The entire structure can freely rotate by 360 · in azimuth and can be \nFigure 1: PBR design. The main types of researched events are highlighted, along with the main detector used to observe them. \n<!-- image --> \ntilted from nadir up to 13 · above the horizon in zenith. The FC, devoted to the search of UHECR-induced EASs, is made of a matrix of 4 Photo Detection Modules (PDMs) arranged in a 2 × 2 configuration. A PDM consists of a 6 × 6 array of 64channel MAPMTs Photo Multi-Anode Photo-Multiplier Tubes (MAPMTs) [Hamamatsu R11265 1 , 64 pixels each] for a total of 9216 pixels imaged every 1.05 µ s. The CC, devoted to the observation of cosmic-ray-induced HAHAs and search for neutrino-induced upward-going EAS, is made of a matrix of 32 Silicon Photo-Multiplier (SiPM) arrays [Hamamatsu S13361- \nFigure 3: The potential all-flavor sensitivity range for PBR for a 100-day flight. \n<!-- image --> \nFigure 2: Left: HAHA trajectories with di ff erent inclination angles relative to PBR Nadir. Right: angular distribution of accepted HAHA events for di ff erent primary cosmic ray energies. \n<!-- image --> \n3050NE-08 2 , 64 channels each] for a total of 2048 pixels. The CC covers a spectral range of 320-900 nm with an integration time of 10 ns. The RI is mounted at the bottom of the telescope and moves solidly with the two cameras, resulting in a wide FoV that covers those of the FC and CC. It is based on the Low-Frequency (LF) instrument for the Payload for Ultrahigh Energy Observations (PUEO) [4]. In addition, a number of ancillary detectors will complete the PBR payload, including a set of infrared cameras for cloud monitoring and a charged particle, X-ray and gamma-ray detector that will work in conjunction with the CC. It will point in the same direction of the CC (overlapping FoV), with self-triggering capabilities as well as with the possibility to receive external triggers from the CC. It will be therefore possible to measure the shower emissions in the optical, radio, X and gamma bands at the same time.', '3. High-Altitude Horizontal Air showers (HAHAs)': "HAHAs refer to EASs induced by cosmic rays that skim the Earth's atmosphere and traverse the telescope FoV, never intersecting the ground. PBR will be a unique laboratory to study the development of cosmic rays in such a peculiar condition. The majority of HAHA shower development occurs, in fact, above altitudes of 20 km, where the atmosphere is rarified, allowing for propagation over hundreds of kilometers (Fig. 2, left). HAHAs will be observed by PBR in tilted mode, looking above the limb, mainly by the CC, with additional information provided by the RI, X and gamma-ray detectors. PBR will observe HAHAs ranging from Earth's limb (84.2°) to horizontal, at a rate of ∼ 1 event per minute. The Earth's atmosphere acts as an energy filter, therefore the angular acceptance is energy dependent (Fig. 2, right). PBR might also be able to provide chemical identification on a statistical basis around the cosmic ray knee energy.", '4. Neutrino search from Targets of Opportunity': 'With the Earth as a neutrino converter, high energy tau neutrinos can produce τ -leptons that emerge from the Earth and \ninitiate showers from their decay. When pointing below the limb, the CC can detect the Cherenkov light produced by the up-going showers [5] starting from ∼ 0.5 PeV (above 300 PeV the shower could be detected by the RI as well). Simulations show that the sensitivity of PBR to transient VHEN sources (supernovae, binary neutron star (BNS) mergers, tidal disruption events, blazar flares, and gamma-ray bursts) is comparable to current ground-based neutrino telescopes and can constrain some transient source scenarios (Fig. 3).', '5. Status and timeline': "PBR has been selected by NASA as part of the Astrophysics Research Analysis (APRA) Program. The design of the instrument is well underway with the main elements already defined. The collaboration is currently going through the component's procurement, testing and prototyping phases. The full integration of the instrument is planned for early 2026, while the final field test will take place for a few months in the spring at the Telescope Array [6] site in Utah. The flight is planned for the spring of 2027 from NASA's Wanaka launch site in New Zealand, with an expected duration of up to 100 days.", 'Acknowledgements': 'The authors would like to acknowledge the support by NASA award 80NSSC22K1488, by the French space agency CNES and the Italian Space agency ASI. We also acknowledge the invaluable contributions of the administrative and technical sta ff s at our home institutions. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy O ffi ce of Science User Facility operated under Contract No. DE-AC02-05CH11231.', 'References': "- [1] A. V. Olinto, et al. (POEMMA), JCAP 06 (2021) 007. doi: 10.1088/ 1475-7516/2021/06/007 . arXiv:2012.07945 .\n- [2] G. Abdellaou, et al., Astroaarticle Physics 154 (2024) 102891. doi: 10. 1016/j.astropartphys.2023.102891 .\n- [3] J. Eser, A. V. Olinto, L. Wiencke (JEM-EUSO), PoS ICRC2023 (2023) 397. doi: 10.22323/1.444.0397 . arXiv:2308.15693 .\n- [4] K. A. Hughes, et al. ('PUEO'), PoS ICRC2023 (2023) 1027. doi: 10. 22323/1.44.1027 . \n- [5] T. Heibges, et al. (JEM-EUSO), PoS ICRC2023 (2023) 1134. doi: 10. 22323/1.444.1134 . arXiv:2310.12310 .\n- [6] Y. Tameda, Telescope Array Experiment, Nuclear Physics B - Proceedings Supplements 196 (2009) 74-79. doi: https://doi.org/10.1016/ j.nuclphysbps.2009.09.011 ."}
2024PhRvD.110f3552T	The Starobinsky model of cosmological inflation in four spacetime dimensions is reviewed with the emphasis on the impact of quantum gravity corrections. As a specific example of the quantum corrections the GrisaruZanon quartic curvature terms in the gravitational effective action of closed superstrings are chosen. Those quartic curvature terms are compared to the BelRobinson tensor squared in a flat Friedman universe and the upper bound on the effective string coupling constant is found by demanding unitarity causality and the absence of ghosts. It is found that the quantum corrections to the observables tilts of the cosmic microwave background radiation in the Starobinsky inflation may be of the same order of magnitude as the nexttonexttonext classical contributions in the Starobinsky model with respect to the inverse powers of the efolding number at the horizon crossing.	2024-09-01T00:00:00Z	['arXiv:2407.21349', '2024arXiv240721349T', '2024PhRvD.110f3552T', '10.48550/arXiv.2407.21349', '10.1103/PhysRevD.110.063552']	['Cosmology', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']	Starobinsky inflation beyond the leading order	2,024	195	0.24	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	1	https://arxiv.org/pdf/2407.21349.pdf	{'Starobinsky inflation beyond the leading order': 'Shunsuke Toyama a, & and Sergei V. Ketov a,b,c, # , 1 \n- a Department of Physics, Tokyo Metropolitan University,\n- 1-1 Minami-ohsawa, Hachioji-shi, Tokyo 192-0397, Japan\n- b Research School of High-Energy Physics, Tomsk Polytechnic University, \nTomsk 634028, Russian Federation \n- c Kavli Institute for the Physics and Mathematics of the Universe (WPI), \nThe University of Tokyo Institutes for Advanced Study, Kashiwa 277-8583, Japan \n& [email protected], # [email protected]', 'Abstract': 'The Starobinsky model of cosmological inflation in four spacetime dimensions is reviewed with the emphasis on impact of quantum gravity corrections. As a specific example of the quantum corrections, the Grisaru-Zanon quartic curvature terms in the gravitational effective action of closed superstrings are chosen. Those quartic curvature terms are compared to the Bel-Robinson tensor squared in a flat Friedman universe, and the upper bound on the effective string coupling constant is found by demanding unitarity (causality) and the absence of ghosts. It is found that the quantum corrections to the observables (tilts) of the cosmic microwave background radiation in the Starobinsky inflation may be of the same order of magnitude as the next-to-next-to-next classical contributions in the Starobinsky model with respect to the inverse powers of the e-folding number at the horizon crossing.', '1 Introduction': "Ultra-violet (UV) completion of phenomenologically viable field-theoretical models of cosmological inflation is important because inflation is sensitive to high energy physics in the early Universe and, hence, quantum corrections to classical description of inflation may be important. The existence of the UV completion to a particular inflation model allows one to treat it as the effective field theory (EFT) originating from quantum gravity. The UV completion is even more relevant to large-field inflation models, while the Starobinsky inflation model [1] is one of them. Unfortunately, little is known about quantum gravity, so UV-completion is often discussed in the framework of Swampland conjectures [2], see also Refs. [3, 4, 5] for the applications of the Swampland conjectures to the Starobinsky inflation. \nString theory is a good candidate for the theory of quantum gravity, so it is natural to seek an UV completion of the Starobinsky model in string theory, see e.g., Ref. [6] for the earlier attempts. This task turned out to be difficult because the Starobinsky model is defined in four space-time dimensions, whereas string theory needs to be compactified from higher dimensions to four dimensions, while the gravitational low-energy EFT in string theory is subject to large ambiguities related to field redefinitions of spacetime metric, see e.g., Ref. [7] for more. \nThe Starobinsky model of inflation is based on the R 2 gravity and can be destabilized by the higher-order curvature terms in the gravitational EFT if those terms have large coefficients. Inflation provides the mechanism for generation of cosmological perturbations, while the Starobinsky model is in excellent agreement with current measurements of the cosmic microwave background (CMB) radiation, so the higher order curvature terms must be subleading during inflation. Nevertheless, it makes sense to investigate robustness of the Starobsinky inflation against specific quantum gravity corrections derived from superstring theory. In the case of closed (type II) superstrings, the leading ( α ' ) 3 -correction beyond the Einstein-Hilbert term is given by the terms quartic in the space-time curvature, which were first derived by Grisaru and Zanon in 1985 from the vanishing four-loop renormalization group beta-function of the supersymmetric non-linear sigma model in two dimensions, describing propagation of a test superstring in the gravitational background [8]. The same quartic curvature terms arise from M-theory [9, 10] after dimensional reduction down to four dimensions. The dependence of the gravitational EFT in string theory upon the Ricci scalar curvature (and the Ricci tensor also) is known to be ambiguous, see Ref. [7] for a review, because the perturbative string theory is consistently defined only on Ricci-flat backgrounds. \nIn this paper, we combined the Starobinsky model with the Grisaru-Zanon (GZ) quantum gravity (superstring) correction, called the Starobinsky-Grisaru-Zanon (SGZ) gravity, that is the generalization of the Einstein-Grisaru-Zanon gravity introduced in Ref. [11]. It allowed us not only derive the restrictions on the effective superstring coupling constant in front of the quartic curvature terms but also compare the contributions of those quantum corrections to the subleading terms (beyond the leading order with respect to the e-folding number) in the inflationary (CMB) observables such as cosmological tilts and their running, as well as derive the leading quantum gravity (superstring) corrections to the Starobinsky solution. \nThe paper is organized as follows. In Sec. 2 the Starobinsky model of inflation is reviewed both in the original (Jordan) frame and in the Einstein (quintessence) frame. We do not follow historical developments but introduce the Starobinsky model from the modern perspective. In Sec. 3 we define the SGZ gravity in the perturbative setup with respect to the GZ term, and compare it to the similar but different Starobinsky-Bel-Robinson (SBR) gravity [12, 13, 14] also having the quartic curvature terms in its action. In Sec. 4 we \nderive the leading quantum corrections to the inflationary dynamics due to the GZ term in the Jordan frame, in the first order with respect to its effective (string) coupling constant. In Sec. 5 we find the upper limits on the effective string coupling constant by demanding the absence of ghosts, unitarity and causality. The leading quantum corrections to the CMB observables are derived and compared against the classical subleading contributions in Sec. 6. We conclude in Sec. 7.", '2 Review of Starobinsky inflation': "The Starobinsky model of inflation is the generally covariant and nonperturbative extension of the Einstein-Hilbert (EH) gravity theory by the term quadratic in the Ricci scalar curvature R . All the curvature-dependent terms beyond the EH one are irrelevant in the Solar system, while they may also be negligible during reheating after inflation in the weakgravity regime. However, it is not the case during inflation in the high curvature regime where the R 2 term is the leading contribution (see below). \nThe Starobinsky model is the particular case of modified F ( R ) gravity, and it is geometrical because only gravitational interactions are used. A modified gravity action has the higher derivatives and generically suffers from Ostrogradsky instabilities and ghosts. However, in the most general modified gravity action, whose Lagrangian is quadratic in the spacetime curvature, the only ghost-free term is just given by R 2 with a positive coefficient, which leads to the Starobinsky model with the action \nS Star . = α ∫ d 4 x √ -gR 2 + M 2 Pl 2 ∫ d 4 x √ -gR , α ≡ M 2 Pl 12 M 2 , (1) \nhaving the only parameter α or M , where M Pl = 1 / √ 8 πG N ≈ 2 . 4 × 10 18 GeV, the spacetime signature is ( -, + , + , + , ) and the natural units are used, ℏ = c = 1. The first term in this action is scale invariant with the dimensionless parameter α . \nThe origin of the R 2 term was originally proposed due to contributions of quantized matter fields in the EH gravity [1]. However, because the EH term is subleading during inflation, we adopt the opposite interpretation, namely, with the EH term being originated from the scale-invariant gravity. For instance, when starting from the scale-invariant action for gravity and a scalar field ϕ as [15, 16, 17] \nS [ g µν , ϕ ] = ∫ d 4 x √ -g [ αR 2 + ξϕ 2 R -1 2 ( ∂ϕ ) 2 -λϕ 4 ] , (2) \none finds that it can undergo a phase transition (called dimensional transmutation) due to quantum corrections, known as the Coleman-Weinberg mechanism of spontaneous symmetry breaking [18]. It leads to the massive scalar field ϕ that may be identified with dilaton or Higgs field having a non-vanishing vacuum expectation value (VEV) in the effective action, as can be demonstrated in the one-loop perturbation theory [16, 17]. As a result, both the Planck mass and the EH term are generated with \n1 2 M 2 Pl = ξ ⟨ ϕ ⟩ 2 , (3) \nthough this cannot be considered as the UV-completion of the Starobinsky gravity. \nThe metric of a flat Friedman universe is given by \nds 2 = -dt 2 + a 2 ( dx 2 1 + dx 2 2 + dx 2 3 ) . (4) \nThen the action (2) leads to equations of motion in the form \n2 H H -( ˙ H ) 2 + H 2 ( 6 ˙ H + M 2 ) = 0 , H = ˙ a/a , (5) \nknown as the Starobinsky equation in the literature, where the dots stand for the time derivatives and H ( t ) is Hubble function. \nWhen searching for a solution to the Starobinsky equation in the form of left Painlev'e series, H ( t ) = ∑ k = p k = -∞ c k ( t 0 -t ) k , one finds the Hubble function (see e.g., Ref. [14]) \nH ( t ) M = M 6 ( t 0 -t ) + 1 6 M ( t 0 -t ) -4 9 M 3 ( t 0 -t ) 3 + 146 45 M 5 ( t 0 -t ) 5 -11752 315 M 7 ( t 0 -t ) 7 + O ( M -9 ( t 0 -t ) -9 ) (6) \nvalid for M ( t 0 -t ) > 1. This special solution is an attractor, while R = 12 H 2 +6 ˙ H . \nIn the slow-roll (SR) approximation defined by ∣ ∣ ∣ H ∣ ∣ ∣ ≪ ∣ ∣ ∣ H ˙ H ∣ ∣ ∣ and ∣ ∣ ∣ ˙ H ∣ ∣ ∣ ≪ H 2 , one gets the leading term in the Starobinsky solution as \nH ( t ) ≈ ( M 2 6 ) ( t 0 -t ) (7) \nthat is entirely due to the R 2 -term in the action. The attractor solution spontaneously breaks the scale invariance of the R 2 -gravity and, therefore, implies the existence of the Nambu-Goldstone boson (called scalaron) that is the physical excitation of the higherderivative gravity. It can be made manifest by rewriting the Starobinsky action into the quintessence form after the field redefinition (or Legendre-Weyl transform) [19] \nφ = √ 3 2 M Pl ln F ' ( χ ) and g µν → 2 M 2 Pl F ' ( χ ) g µν , χ = R . (8) \nIt yields \nS [ g µν , φ ] = M 2 Pl 2 ∫ d 4 x √ -gR -∫ d 4 x √ -g [ 1 2 g µν ∂ µ φ∂ ν φ + V ( φ ) ] , (9) \nin terms of the canonical inflaton φ with the scalar potential \nV ( φ ) = 3 4 M 2 Pl M 2 [ 1 -exp ( -√ 2 3 φ/M Pl )] 2 . (10) \nThis potential has the infinite plateau (for the large ϕ -field values of the order M Pl and beyond) that implies the approximate shift symmetry of the inflaton field as the consequence of the scale invariance of the R 2 gravity or due to the approximate scale invariance of the action (2) in the large-curvature regime. The potential (10) also has the positive 'cosmological constant' given by the first term in the square brackets, induced by the R 2 term in the action (1), which can be physically interpreted as the energy driving inflation. The scale of inflation is determined by the parameter M that is identified with the inflaton mass. The universality class of inflationary models is determined by the critical parameter √ 2 / 3 in the exponential term [20]. \nThe equivalent actions (1) and (9) are usually referred to the Jordan frame and the Einstein frame, respectively. The approximate shift symmetry of the potential (10) is the consequence of the approximate scale invariance of the R 2 gravity, which requires the presence of the R 2 term in any viable model of inflation based on modified F ( R )-gravity. It becomes even more transparent by using the inverse transformation from the Einstein frame to the Jordan frame, having the parametric form [21] \nR = ( √ 6 M Pl dV dφ + 4 V M 2 Pl ) e √ 2 3 φ/M Pl , F = ( √ 6 M Pl dV dφ + 2 V M 2 Pl ) e 2 √ 2 3 φ/M Pl . (11) \nAs is clear from these equations, in the SR approximation (chaotic inflation) the first term in the brackets is much less than the second term, which immediately implies F ( R ) ∼ R 2 . \nThe gravitational EFT during inflation does not have to be limited to the terms given in Eq. (1) but should also include the higher-order curvature terms. Those terms eliminate the infinite plateau in the inflaton potential (10). The fact that the Starobinsky model of inflation is in excellent agreement with the current CMB measurements (see below) implies that those terms do not destabilize the Starobinsky inflation, which put restrictions on their contributions. \nIt is convenient to use the e-foldings number N instead of time t , which are related by \nN = ∫ t 0 t H ( ˜ t ) d ˜ t , (12) \nand the co-moving wavenumber k = 2 π/λ related to N ( t ) by the equation d ln k = -dN . The SR (running) parameters in the Einstein frame are defined by \nε sr ( φ ) = M 2 Pl 2 ( V ' V ) 2 and η sr ( φ ) = M 2 Pl ( V '' V ) (13) \nin terms of the quintessence scalar potential V , where the primes denote the derivatives with respect to φ . In the Jordan frame, one uses the Hubble flow functions, \nϵ H = -˙ H H 2 , η H = ϵ H -˙ ϵ H 2 ϵ H H . (14) \nThe amplitude of scalar perturbations at the horizon crossing with the pivot scale k ∗ = 0 . 05 Mpc -1 is known from CMB measurements (called WMAP normalization) as \nA s = V 3 ∗ 12 π 2 M 6 Pl ( V ∗ ' ) 2 = 3 M 2 8 π 2 M 2 Pl sinh 4 ( φ ∗ √ 6 M Pl ) ≈ 2 · 10 -9 , (15) \nwhere subscript (*) refers to the CMB pivot scale, in the case of Starobinsky inflation. This allows us to fix the only parameter M (or α ) and the scale of inflation, H inf . , in the Starobinsky model as \nM M Pl ≈ O (10 -5 ) , α ≈ O (10 9 ) , H ≈ O (10 14 ) GeV , R M 2 Pl ≈ 12 H 2 M 2 Pl ≈ 10 -7 . (16) \nIt is worth mentioning here that the higher-order curvature terms in the gravitational EFT beyond the Starobinsky model are given by power series with respect to H 2 /M 2 Pl ∼ 10 -8 , so they must be sub-leading during inflation unless they have very large coefficients. It is also worth noticing that the large value of α required by CMB does not speak in favor of generating the R 2 -term by quantum matter contributions because a single quantized matter field contributes in the 1-loop approximation about 10 -3 to the α -parameter, so one needs about 10 12 quantized matter fields in order to achieve the desired result. \nThe primordial spectrum P ζ ( k ) of 3-dimensional scalar (density) perturbations ζ ( x ) in a flat Friedman universe is defined by the 2-point correlation function \n〈 δζ ( x ) ζ δζ ( y ) ζ 〉 = ∫ d 3 k k 3 e ik · ( x -y ) P ζ ( k ) P 0 , (17) \nwhere k = 2 π/λ is the co-moving number. Similarly, one defines the primordial spectrum P t ( k ) of tensor perturbations, see e.g., Ref. [20] for more details The scale k is simply related to the e-folds number N via N = -∫ k d ˜ k/ ˜ k . The power spectra coincide with the corresponding amplitudes A s ( k ) and A t ( k ), respectively. \nGiven the power spectra P ζ ( k ) and P t ( k ), one defines the scalar tilt n s ( k ), its running parameter α s ( k ), the tensor tilt n t ( k ) and its running parameter α t ( k ) (all dimensionless) as \nn s = 1 + d ln P ζ ( k ) d ln k , α s = d 2 ln P ζ ( k ) ( d ln k ) 2 , n t = d ln P t ( k ) d ln k , α t = d 2 ln P t ( k ) ( d ln k ) 2 , (18) \nas well as the tensor-to-scalar ratio \nr ( k ) = P t P ζ = 8 \| n t \| . (19) \nThe Starobinsky model gives simple predictions for the cosmological tilts of the scalar and tensor power spectra in the leading order with respect to the e-folds N ∗ evaluated when perturbations left the horizon (at the horizon crossing) as [22, 23] \nn s ≈ 1 -2 N ∗ , α s ≈ -2 N 2 ∗ , α t ≈ -3 N 3 ∗ , r ≈ 12 N 2 ∗ . (20) \nTherefore, tensor perturbations are suppressed with respect to scalar perturbations by the extra factor of N -1 ∗ , whose value can be estimated by comparing those predictions with CMB measurements [24, 25] , \nn s ≈ 0 . 9649 ± 0 . 0042 (68%CL) and r < 0 . 032 (95%CL) , (21) \nthat fit the Starobinsky model predictions for \nN ∗ = 56 ± 8 . (22) \nThis prediction for the duration of inflation agrees with our calculations in the Jordan frame, based on the solution (6). The corresponding times for the end and the beginning of inflation are M ( t 0 -t end ) ≈ 2 . 5 and M ( t 0 -t start ) ≈ 27 . 7, respectively. \nIn particular, excluding N ∗ from Eqs. (20) yields the sharp prediction of the Starobinsky model for the tensor-to-scalar ratio as \nr ≈ 3(1 -n s ) 2 . (23) \nThe Starobinsky inflation does not exclude the higher-order curvature terms in the action (1), though it implies that those terms should be subleading during inflation, being suppressed by the powers of H 2 /M 2 Pl ∼ 10 -8 . The Starobinsky model is sensitive to quantum (UV) corrections because of its high inflation scale and the inflaton field values near the Planck scale during inflation. Hence, it is important to determine its UV-cutoff Λ UV of the Starobinsky model by studying scaling of scattering amplitudes with respect to energy, E/ Λ UV . A careful calculation yields [26] \nΛ UV = M Pl . (24) \nTherefore, the predictions of the Starobinsky model for inflation and CMB make sense and the model itself can be considered as a trustable effective field theory after decoupling of heavy modes expected at the Planck scale [5].", '3 Starobinsky-Grisaru-Zanon (SGZ) gravity': 'The SGZ action is defined by \nS SGZ [ g ] = M 2 Pl 2 ∫ d 4 x √ -g ( R + 1 6 M 2 R 2 -72 γ M 6 Z ) , (25) \nwhere we have added the GZ (quantum) superstring correction [8] \n72 Z = ( R µρσν R λρστ + 1 2 R µνρσ R λτρσ ) R αβλ µ R τ αβν (26) \nto the Starobinsky action (1) with the new dimensionless coupling constant γ > 0 2 . \nThe value of γ cannot be calculated from string theory because the action (25) is in four space-time dimensions, so γ depends upon compactification from ten to four dimensions and the unknown vacuum expectation value of the string dilaton. \nThe SGZ gravity (25) is different from the SBR gravity defined by the action [12, 14] \nS SBR = M 2 Pl 2 ∫ d 4 x √ -g ( R + 1 6 M 2 R 2 -β 8 M 6 T 2 ) , (27) \nwhere β is another dimensionless coupling constant, the T 2 stands for the Bel-Robinson (BR) tensor squared, T 2 ≡ T ρνλµ T ρνλµ , with \nT ρνλµ ≡ R ρσηλ R ν ση µ + ∗ R ρσηλ ∗ R ν ση µ = R ρσηλ R ν ση µ + R ρσηµ R ν ση λ -1 2 g ρν R σηξλ R σηξ µ , (28) \nand the star denoting the Hodge dual tensor in four dimensions. \nThe BR tensor was introduced by Bel and Robinson [27, 28] by analogy with the energymomentum tensor of Maxwell theory of electromagnetism, \nT Maxwell µν = F µρ F ν ρ + ∗ F µρ ∗ F ν ρ , F µν = ∂ µ A ν -∂ ν A µ . (29) \nThere is an identity [29, 30] \nT 2 = -1 4 ( ∗ R µνλρ ∗ R µνλρ ) 2 + 1 4 ( ∗ R µνλρ R µνλρ ) 2 = 1 4 ( P 2 4 -E 2 4 ) = 1 4 ( P 4 + E 4 )( P 4 -E 4 ) (30) \nrelating the BR tensor squared to the Euler and Pontryagin topological densities in four dimensions, E 4 and P 4 , respectively. There is another well-known identity \nE 4 = G GB ≡ R µνλρ R µνλρ -4 R µν R µν + R 2 (31) \nthat relates the Euler density to the Gauss-Bonnet invariant G GB in four dimensions. \nWe consider the first two terms in the SGZ and SBR actions nonperturbatively, but the last (quartic) curvature (BR or GZ) terms only perturbatively, in the first order with respect to the coupling constants. Therefore, no Ostrogradski ghosts arise. The difference between the GZ and BR terms in four dimensions, in the context of superstrings/M-theory, was first noticed in Ref. [31]. \nIn a flat Friedman universe (4) we find \nZ = H 8 +2 H 6 ˙ H + 11 6 H 4 ˙ H 2 + 2 3 H 2 ˙ H 3 + 1 12 ˙ H 4 , (32) \nwhereas \n1 144 T 2 = H 8 +2 H 6 ˙ H + H 4 ˙ H 2 , (33) \nwhich makes the difference manifest, though the first two leading terms (relevant to the SR approximation) are the same.', '4 GZ quantum corrections to Starobinsky inflation': 'The SGZ gravity equation of motion in a flat Friedman universe is given by \nm 6 H 2 +6 m 4 H 2 ˙ H +2 m 4 H H -m 4 ˙ H 2 -12 γH 8 +132 γH 6 ˙ H +44 γH 5 H +138 γH 4 ˙ H 2 +48 γH 3 ˙ H H +28 γH 2 ˙ H 3 +12 γH ˙ H 2 H -3 γ ˙ H 4 = 0 (34) \nthat extends the Starobinsky equation (5) by the γ -dependent terms. A solution to this equation in the first order with respect to the (small) γ -parameter, similarly to Eq. (6) reads \nH ( t ) = H 0 ( t ) + γH 1 ( t ) , (35) \nwhere \nH 1 ( t ) M = -1 163296 M 7 ( t 0 -t ) 7 -2 2835 M 5 ( t 0 -t ) 5 -391 90720 M 3 ( t 0 -t ) 3 -9061 306180 M ( t 0 -t ) -127 7776 M ( t 0 -t ) -1931203 5358150 M 3 ( t 0 -t ) 3 + O ( M -5 ( t 0 -t ) -5 ) , (36) \nand the leading contribution during inflation comes from the 2nd term above. \nAccordingly, the scalar curvature is given by \nR M 2 = M 2 ( t 0 -t ) 2 3 -1 3 -4 9 M 2 ( t 0 -t ) 2 + 16 5 M 4 ( t 0 -t ) 4 -6908 189 M 6 ( t 0 -t ) 6 + γ [ -M 8 ( t 0 -t ) 8 40824 -151 M 6 ( t 0 -t ) 6 58320 + 143 M 4 ( t 0 -t ) 4 122472 -4163 M 2 ( t 0 -t ) 2 81648 -68713 7144200 -5109281 5143824 M 2 ( t 0 -t ) 2 + 17584432631 964467000 M 4 ( t 0 -t ) 4 -75802186291 300056400 M 6 ( t 0 -t ) 6 + O ( M -8 ( t 0 -t ) -8 ) ] , (37) \nwhere the leading γ -dependent contribution during inflation is due to the 2nd term in the square brackets also. \nThe Hubble flow functions (14) are given by \nϵ H = 6 M 2 ( t 0 -t ) 2 -6 M 4 ( t 0 -t ) 4 + 48 M 6 ( t 0 -t ) 6 + γ [ -5 M 4 ( t 0 -t ) 4 4536 -869 M 2 ( t 0 -t ) 2 11340 -4591 22680 + 38051 34020 M 2 ( t 0 -t ) 2 -1385 648 M 4 ( t 0 -t ) 4 + 51649511 396900 M 6 ( t 0 -t ) 6 ] + O ( M -8 ( t 0 -t ) -8 ) (38) \nand \nη H = 6 M 4 ( t 0 -t ) 4 -96 M 6 ( t 0 -t ) 6 + γ [ -5 M 4 ( t 0 -t ) 4 1512 -869 M 2 ( t 0 -t ) 2 5670 -4591 22680 + 1385 648 M 4 ( t 0 -t ) 4 -51649511 198450 M 6 ( t 0 -t ) 6 ] + O ( M -8 ( t 0 -t ) -8 ) , (39) \nwhere the leading γ -dependent contributions during inflation are due to the 2nd terms in the square brackets too.', '5 The upper bounds on γ': "Some modified gravity models of inflation can be described by the effective function F ( H 2 ) entering equations of motion in a flat Friedman universe [32]. Our equations of motion (34) are not of that type because they include the higher time derivatives of the Hubble function, but they fall into that type in the SR approximation, see Ref. [14] for applications to the SBR gravity theory. In the case of SGZ gravity, we find \nR ( R 12 -H 2 ) -H ˙ R = 3 M 2 ( H 2 -12 γH 8 M 6 + 22 γH 6 M 4 ) ≡ 3 M 2 F ( H 2 ) , (40) \nso we have \nF ( H 2 ) = H 2 -22 γ M 4 ( H 2 ) 3 -12 γ m 6 ( H 2 ) 4 . (41) \nThe derivatives of F ( H 2 ) with respect to H 2 are as follows: \nF ' ( H 2 ) ≡ dF d ( H 2 ) = 1 -66 γ ( H M ) 4 -48 γ ( H M ) 6 (42) \nand \nF '' ( H 2 ) = -132 γ M 4 H 2 -144 γ M 6 H 4 . (43) \nThe effective Newton constant in the higher-derivative gravity theories studied in Ref. [32] must obey the condition \nG eff . = 1 8 πM 2 Pl [ F ' ( H 2 ) + 4( H 2 /M 2 )] > 0 , (44) \nthat implies \nF ' ( H 2 ) + 4 H 2 M 2 > 0 or 1 -66 γh 4 -48 γh 6 +4 h 2 > 0 , (45) \nwhere h = H/M . The maximal value of h in the Starobinsky inflation is h max . ≈ 4 . 6, which implies \nγ < 1 . 74 × 10 -4 . (46) \nAnother condition proposed in Ref. [32] from demanding the absence of negative energy fluxes (or unitarity and causality constraints) reads \n-4 ≤ 210 H 2 F '' ( H 2 ) F ' ( H 2 ) + 4( H 2 /M 2 ) ≤ 4 , (47) \nthat in our case is given by \n-1 ≤ 210( -33 γh 4 -38 γh 6 ) 1 -66 γh 4 -48 γh 6 +4 h 2 ≤ 1 . (48) \nIt implies \nγ ≤ 1 + 4 h 2 12 h 4 (634 h 2 +583) , (49) \nand, therefore, the upper bound \nγ ≤ 1 . 12 × 10 -6 . (50) \nBoth bounds (46) and (50) are slightly stronger than those found in Ref. [14]", '6 Quantum corrections versus classical corrections to CMB observables beyond the leading order': 'It is instructive to compare contributions of the GZ quantum correction to the CMB observables against the classical contributions beyond the leading order given by Eq. (20) within the possible range of N ∗ in Eq. (22). \nThe subleading terms for the predicted CMB observables in the Starobinsky inflation are given by [33, 34] \nn s = 1 -2 N ∗ + 2 . 4 N 2 ∗ -ln(2 N ∗ ) 6 N 2 ∗ + O ( ln 2 N ∗ N 3 ∗ ) , α s = -2 N 2 ∗ + O ( ln 2 N ∗ N 3 ∗ ) , r = 12 N 2 ∗ + 2 ln(2 N ∗ ) N 3 ∗ -56 . 76 N 3 ∗ + O ( N -4 ∗ ) , α t = -3 3 + O ( N -4 ∗ ) , (51) \nN ∗ \nwhere ln 2 N ∗ /N 3 ∗ < 4 . 1 · 10 -5 and N -4 ∗ < 2 · 10 -7 . The subleading contribution to the scalar tilt n s in the 3rd term above can increase the n s -value by 1 . 0 · 10 -3 , while the subleading contribution to the tensor-to-scalar-ratio r , given by the 3rd term above, can decrease the r -value by 5 · 10 -4 . All the subleading contributions are within the observational errors given in Eq. (21). \nOn the other hand, when using Eqs. (38) and (39) in the first order with respect to the string parameter γ and taking their first-order contributions to the scalar tilt n s and the tensor-to-scalar-ratio r at the horizon crossing, \nn s ≈ 1 -4 ϵ H +2 η H , r = 16 ϵ H , (52) \nwith the maximal value of γ from Eq. (50), we get the quantum contributions to the n s and r up to +2 . 5 · 10 -4 and -4 . 9 · 10 -5 , respectively. \nTherefore, the quantum contributions to the CMB tilts are smaller than the subleading terms proportional to N -2 ∗ by one order of magnitude but may be of the same order of magnitude as the classical N -3 ∗ contributions. The same conclusion also applies to the running parameters α s and α t .', '7 Conclusion': 'The upcoming CMB measurements by the CORE Collaboration [35], S4 Collaboration [36], LiteBIRD Collaboration [37], NASA PICO Collaboration [38], the Simons Observatory survey [39] and EUCLID Collaboration [40] are expected to probe the tensor-to-scalar ratio r in the range of 10 -3 and improve the precision value of n s . \nShould those measurements confirm Eq. (23), it would be a triumph of the Starobinsky model of inflation. Should the predicted relation (23) be ruled out, the question would arise about the origin of disagreement. If the disagreement will be significant, it would rule out the Starobinsky inflation. If, however, the disagreement will be small (say, within one order of magnitude), one may expect that due to the sub-leading corrections to the leading order predictions in Eq. (20). Then another question about the origin of those small corrections would arise. The latter may be due to the sub-leading terms in the classical Starobinsky model or due to quantum gravity corrections to the gravitational EFT, or they have a very different origin, say, due to reheating or new physics, see e.g., Ref. [41]. \nThe main new result of this paper is about possible superstring (as quantum gravity) corrections that may be of the same size as the next-to-next-to-next classical corrections (in the N -3 ∗ -terms). Of course, the value of the effective string coupling γ may be much lower than the estimate found in Eq. (50), which may reduce the size of quantum corrections even further.', 'Acknowledgements': 'One of the authors (SVK) is grateful to Ignatios Antoniadis, Eugenio Bianchi, Norma Borstnik, Gia Dvali, Maxim Khlopov, Elias Kiritsis and Holger Nielsen for discussions. SVK was supported by Tokyo Metropolitan University, the Japanese Society for Promotion of Science under the grant No. 22K03624, the World Premier International Research Center Initiative (MEXT, Japan), and the Tomsk Polytechnic University development program Priority-2030-NIP/EB-004-375-2024.', 'References': "- [1] A. A. Starobinsky, 'A New Type of Isotropic Cosmological Models Without Singularity,' Phys. Lett. B 91 (1980) 99-102.\n- [2] E. Palti, 'The Swampland: Introduction and Review,' Fortsch. Phys. 67 no. 6, (2019) 1900037, arXiv:1903.06239 [hep-th] .\n- [3] M. Brinkmann, M. Cicoli, and P. Zito, 'Starobinsky inflation from string theory?,' JHEP 09 (2023) 038, arXiv:2305.05703 [hep-th] .\n- [4] D. Lust, J. Masias, B. Muntz, and M. Scalisi, 'Starobinsky Inflation in the Swampland,' arXiv:2312.13210 [hep-th] .\n- [5] S. V. Ketov, 'Starobinsky inflation and Swampland conjectures,' arXiv:2406.06923 [hep-th] .\n- [6] R. Blumenhagen, A. Font, M. Fuchs, D. Herschmann, and E. Plauschinn, 'Towards Axionic Starobinsky-like Inflation in String Theory,' Phys. Lett. B 746 (2015) 217-222, arXiv:1503.01607 [hep-th] .\n- [7] S. V. Ketov, Quantum nonlinear sigma models: From quantum field theory to supersymmetry, conformal field theory, black holes and strings . 2000.\n- [8] M. T. Grisaru and D. Zanon, ' σ Model Superstring Corrections to the Einstein-hilbert Action,' Phys. Lett. B 177 (1986) 347-351.\n- [9] M. B. Green, M. Gutperle, and P. Vanhove, 'One loop in eleven-dimensions,' Phys. Lett. B 409 (1997) 177-184, arXiv:hep-th/9706175 .\n- [10] R. Blumenhagen, N. Cribiori, A. Gligovic, and A. Paraskevopoulou, 'Emergence of R 4 -terms in M-theory,' arXiv:2404.01371 [hep-th] .\n- [11] R. Campos Delgado and S. V. Ketov, 'Einstein-Grisaru-Zanon gravity,' Phys. Lett. B 855 (2024) 138811, arXiv:2405.03925 [hep-th] .\n- [12] S. V. Ketov, 'Starobinsky-Bel-Robinson Gravity,' Universe 8 no. 7, (2022) 351, arXiv:2205.13172 [gr-qc] ."}
2024arXiv240910116M	Pulsating stars have been studied using nonlinear hydrodynamic codes since the pioneering work of Robert Christy in the 1960s. Modern codes include improvements such as allowing for convection but there is a penalty in terms of computation speed and for some stars convection is not significant. In this work a new version of the Christy program has been developed which can run hundreds of star models to convergence in a day or two of computer time. This allows overall patterns of behaviour to be studied and suitable models for individual case stars to be identified. The star SZ Lyn was chosen as a test case for the model. Light curve and radial velocity data were obtained for this star using amateur equipment. A run of 625 parameter sets mass luminosity effective temperature and hydrogen fraction identified the best fit parameters for SZ Lyn. Model results show a good fit to the observed data in terms of amplitude period and shape of the light and velocity curves. In this paper we report on the model developed by PM and radial velocity observations by RL.	2024-09-01T00:00:00Z	['arXiv:2409.10116', '2024arXiv240910116M', '10.48550/arXiv.2409.10116']	['Astrophysics - Solar and Stellar Astrophysics']	Modelling Pulsating Stars	2,024	196	0.47	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.10116.pdf	{'Modelling Pulsating Stars': 'Philip Masding ([email protected]) Robin Leadbeater \nBritish Astronomical Association', 'Subject Keywords': 'Hydrodynamic codes; radial velocity measurement; spectroscopy; pulsating stars; SZ Lyn', 'Abstract': "Pulsating stars have been studied using non-linear hydrodynamic codes since the pioneering work of Robert Christy in the 1960's. Modern codes include improvements such as allowing for convection but there is a penalty in terms of computation speed and for some stars convection is not significant. In this work a new version of the Christy program has been developed which can run hundreds of star models to convergence in a day or two of computer time. This allows overall patterns of behaviour to be studied and suitable models for individual case stars to be identified. \nThe star SZ Lyn was chosen as a test case for the model. Light curve and radial velocity data were obtained for this star using amateur equipment. \nA run of 625 parameter sets (mass, luminosity, effective temperature and hydrogen fraction) identified the best fit parameters for SZ Lyn. Model results show a good fit to the observed data in terms of amplitude, period and shape of the light and velocity curves. \nIn this paper we report on the model developed by PM and radial velocity observations by RL.", 'Introduction': "Pulsating variable stars are very important in astronomy particularly because the period luminosity relationship is vital for determining distances. The first Cepheid was discovered by John Goodricke in 1784 but little was known about the fundamental mechanism of variation until the early 20 th century. Eclipses by binary companion stars were suggested but did not explain the shape of the light curves. This led to the proposal by Shapley that the variability was due to intrinsic pulsation of the star (Shapley, 1914). Beginning in 1918 the scientist and mathematician Arthur Eddington developed the heat engine theory to explain the pulsation (Eddington, 1918). \nWith their importance in distance determination, as exemplified by Hubble using Cepheids to establish that the Andromeda galaxy was a separate star system to the Milky Way, there was an obvious incentive to reproduce the behaviour of real stars using Eddington's theory. Unfortunately, the equations cannot be solved analytically and linear solutions can only explain the amplitude and period of the brightness curves rather than their shape. However, with the advent of powerful computers in the 1960's and the numerical mathematical techniques developed during the Manhattan project in the second world war it became possible to calculate a solution based on Eddington's theory. The pioneer in this area was Robert Christy. His work showed that it was possible to predict many of the complex features seen in variable star brightness and radial velocity curves (Christy, 1964). \nThe motivation of this work is to develop a modern version of Christy's code and exploit the speed of modern computers to run hundreds of trial models to match observations. Amateur astronomers have a long history of observing variable star light curves but in recent years it has become possible to observe the radial velocity as well using a spectrograph on a small telescope. The combination of a \nmodel which can run quickly on a personal computer and observational data to go with it means that amateurs can contribute to the understanding of pulsating stars.", 'The mechanism of variability': "Figure 1 Illustration of Eddington's heat engine theory \n<!-- image --> \nB \nA simple illustration of Eddington's theory is shown in Figure 1. In 1A a zone of a star is not in hydrostatic equilibrium such that the gravity force exceeds the supporting pressure so the zone falls. As it falls it is compressed becomes denser and more opaque trapping radiation in the zone below. Eventually at 1C the trapped radiation increases the pressure below the zone so much that it is forced upwards where upon it becomes relatively transparent allowing the radiation to escape and the cycle will repeat. \nA problem with the initial version of the theory is that as the zone falls and is compressed it becomes denser and hotter. Although increasing density is associated with higher opacity, increasing temperature causes lower opacity and the temperature effect will predominate. An explanation was provided by Zhevakin (Zhevakin, 1953) who showed that in some stars temperatures at critical depths are just right to cause ionisation of hydrogen and helium. This ionisation absorbs energy without a temperature rise leaving the increasing density to raise opacity. This is the so-called kappa mechanism which explains why only some stars pulsate.", 'The computer model': "By the 1960's computers had become powerful enough to calculate a numerical solution to the pulsating star theory. Also, new methods from the atom bomb project could now be used to solve the equations for gases under extreme conditions. One of the people who had worked on the bomb project, Robert Christy, had by this time moved to Caltech and he developed a model using the IBM 7090 computer (Figure 2) at the university. \nFigure 2 An IBM 7090 computer in the 1960's (credit NASA) \n<!-- image --> \nIn this project a new version of Christy's code was developed in C++ to take advantage of the speed of a modern PC which is typically 10,000 times faster than the IBM7090. \nThis type of model is 1 dimensional which means that the star can pulsate in the radial direction only. In order to allow this the star is divided up into a number of mass zones or shells from the surface down to a depth where any pulsation is negligible which is typically 15% of the total radius. The inner boundary condition is then zero velocity and a constant radiation flux from the nuclear reactions in the core. A simplified schematic of the new code is shown in Figure 3. \nThere are 3 main parts to the solution beginning with the static solver. Solving the static model requires finding a set of temperatures and zone radiuses which achieve hydrostatic equilibrium or a balance of gravity and pressure forces. The other one-off calculation is to create a 2D look up table of properties of the gas mixture at various temperature and densities. This calculation requires a solution of the Saha equations to determine the state of ionisation of the hydrogen, helium and metal mixture. \nIn the dynamic stage the model steps forward through time with a variable step length dependent on the Courant condition or time for sound to cross the narrowest zone. Each cycle begins by solving the energy balance for the temperature using an iterative technique. The velocity and radius are then updated by explicit numerical integration. \nOnce only calculation of properties for 2D interpolation tables: \nOpacity (from OP Project) Mean particle weight Total particle number density Ionisation energy \nStatic model solver \neffective temperature and composition \nDivide mass into appropriate zones Solve for radius and temperature at hydrostatic equilibrium \nFigure 3 The model structure \nInterpolated properties \nDensity and temperature \nMass division, initial temperature and radius \nDynamic model runs repeatedly for multiple time steps \nSolve coupled energy balance equations (First Law of Thermodynamics) for all zones to obtain temperatures Energy balance based on previous values of velocity and radius \nCalculate acceleration due to gravity and pressure forces: \nIntegrate acceleration to update velocity and radius. Cycle back to energy balance", 'Observations for a test star SZ Lyn': 'In order to test the model, the authors obtained light curve and radial velocity data for the high amplitude 𝛿 Scuti star SZ Lyn. This star is conveniently placed for northern hemisphere observers in the spring and has a short 2.892 hour period which means all observations can be obtained in one night. At magnitude 9.08 to 9.72 it also bright enough to allow the radial velocity to be measured. \nData for the light curve was obtained unfiltered using an 80mm refractor and Atik 313+ camera. These observations were reduced using standard techniques. The radial velocity measurements are more involved and are the subject of the next section.', 'Measurement of radial velocity': 'The star-centric radial pulsation velocity data for SZ Lyn used in this paper were derived from radial velocity measurements on spectra recorded using a LHIRES III spectrograph (resolving power R =7000) mounted on a 0.28m aperture telescope. The wavelength covered was 443-458 nm and the exposure time for each measurement was 600s. To minimise the effects of any drift in the spectrograph, the wavelength calibration was based on the average of calibration lamp spectra taken before and after each star spectrum. \nFor SZ Lyn, 34 spectra were recorded during the night of 7-8 th March 2023 covering 2 complete pulsation cycles. The signal/noise ratio in each spectrum was typically 35 per resolution interval. The spectra were individually reduced using ISIS software isis-software (astrosurf.com). Typical spectra recorded around maximum and minimum radial velocity are shown in Figure 4. All spectra are available to view and download from the British Astronomical Association Spectroscopy Database. \nhttps://britastro.org/specdb/ \nFigure 4 Spectra of SZ Lyn \n<!-- image --> \nThe relative radial velocities were measured by the cross-correlation method, using the tool in ISIS. The first spectrum in the series was used as the reference template. The velocities and observation times were then heliocentric corrected. \nThe data were folded into a single cycle using a period 0.12053 day for SZ Lyn as published in the International Variable Star Index. The International Variable Star Index (VSX) (aavso.org) \nand the phase referenced relative to the time of maximum brightness. The velocity offset (due to the motion of the star and the arbitrary choice of reference spectrum) was removed. The resulting measured radial velocities are plotted in Figure 5 \nFigure 5 Radial velocity for SZ Lyn \n<!-- image --> \nA projection factor was applied to account for the fact that the measured velocity is an average over the stellar disc of the component of the star-centric radial pulsation velocity in our direction. (This component reduces towards the limb and hence the average measured velocity is smaller than the pulsation velocity.) This factor also includes the effect of limb darkening among other factors and its precise value remains of some debate and may to some degree be star specific (Borgniet, 2019). Here we have adopted a value of 1.35 with an uncertainty of ~5%. \nFinally, the sign of the velocity was reversed to change the origin of the reference coordinates from heliocentric to star centric. \nThe stochastic uncertainty of the individual measured pulsation velocities is 2 km/s for SZ Lyn, estimated from the residuals to a quadratic fit over the falling part of the pulsation velocity curve. There is an additional systematic uncertainty arising from the uncertainty in the value of the projection factor.', 'Note:': 'SZ Lyn is a binary system with a period of 3.1 years and so the orbital motion will have an effect on the observation times and measured radial velocities and any light from the secondary would be included in the brightness measurements. (Moffett, et al., 1988). The timescale covering these observations was sufficiently short compared with the orbital period such that effect on the timings and radial velocity are negligible and were not corrected for. There is no evidence of the secondary in the visible spectrum which implies the secondary is at least 2 magnitudes fainter. (Bardin & Imbert, 1984). A 2-magnitude fainter secondary would reduce the observed magnitude range by 0.09. Here we have assumed no significant contribution from the secondary in the brightness measurements.', 'Model results': 'The model requires a number of key parameters to define a star. For SZ Lyn suitable literature values are 7200𝐾 < 𝑇 𝑒 < 7800𝐾 , 0.91𝑀 ⊙ < 𝑀 < 1.74𝑀 ⊙ and , 2.3𝑅 ⊙ < 𝑅 < 2.87𝑅 ⊙ (Adadduriya, et al., 2020 ). In order to find the best fit to the observational data the model was run with 5 values of each parameter spanning the quoted range of uncertainty. In addition, the composition was varied with 5 values of hydrogen fraction in the range 0.6 < 𝑋 < 0.73 at fixed Z=0.015. As a result, the total number of cases computed was 625. The model used a 74-zone envelope with 20km/s initial velocity imposed at the surface. \nBest fit physical parameters from all these cases are 𝑇 𝑒 = 7500𝐾, 𝑅 = 2.81𝑅 ⊙ , M= 1.57𝑀 ⊙ and 𝑋 = 0.61 . With these values the model result show excellent agreement with observations made by the authors in March 2023. \nFigure 6 Model and Observation for SZ Lyn \n<!-- image -->', 'Inside the star': 'Observations are restricted to what happens at the surface of the star, but with the model we can predict what is happening throughout the envelope. In Figure 7 we see the velocity of the zones, with yellow colours indicating positive velocity as the star expands and blue indicating negative velocity as it contracts. Notice that significant velocity only extends down to about 75% of the total radius. \nFigure 7 Radial velocity inside the star \n<!-- image --> \nFigure 8 shows the variation of HeII as a fraction of total helium. As such the bright yellow band represents 100% HeII and this band is broadest at maximum radius. The transition from HeI to HeII always takes place over a short distance near the surface whereas the transition from HeII to HeII is at a deeper level and takes place over a larger distance. \nFigure 8 Variation of HeII fraction inside the star \n<!-- image --> \nThe variation of opacity within the star has interesting patterns as shown by Figure 9. On this scale the cycling of hydrogen ionisation is poorly resolved since it takes place so close to the surface. In deeper zones the variation in opacity due to helium and the z-bump leads to complex cycling. \nFigure 9 Opacity \n<!-- image -->', 'Conclusions': 'This work has demonstrated that a new implementation of the Christy hydrodynamic code can run extremely fast on a modern computer. This speed opens up the possibility to test hundreds of sets of the basic input parameters of mass, luminosity, effective temperature and composition and obtain a best fit model for observed light and radial velocity curves for a given star. \nAmateur astronomers have a long history of observing light curves but in recent years spectrographs such as the LHIRES III have become available which are capable of measuring the radial velocity of stars down to at least magnitude 10 using telescopes of around 0.3m aperture. This means that amateur data can be used to fully test the model. \nWe obtained observational results for SZ Lyn as a test case for the model. This star is relatively bright and has a short period making it ideal to observe. In this case we screened several hundred parameter sets and found a set which very accurately fits the observed data and is within error bounds for each parameter as quoted in the literature. \nIn future work it is hoped to observe and fit models to more stars. The original Christy model does not include convection which does limit its applicability so a future plan is to include a simple representation of convection in the new code.', 'Acknowledgements': 'PM would like to thank Joyce Guzik for her encouragement in submitting this work to the AAVSO conference and also thank Edward Masding for his help with C++ coding and building a fast computer. \nThis research has made use of the International Variable Star Index (VSX) database, operated at AAVSO, Cambridge, Massachusetts, USA.', 'References': 'Adadduriya, J. et al., 2020 . Asteroseismology of SZ Lyn using multiband high time resolution photometry from ground and space. Mon Not R Astron So, 502(1). \nBorgniet, A., 2019. Consistent radial velocities of classical Cepheids from the cross-correlation technique. Astronomy & Astrophysics, p. 631. \nChristy, R. F., 1964. The Calculation of Stellar Pulsation. Rev Mod Phys 36, p. 555. \nEddington, A. S., 1918. On the pulsation of gaseous stars and the problem of the Cepheid variables. Monthly notes of the Royal Astronomical Society , Volume 79, pp. 2 - 22. \nMoffett, T. J. et al., 1988. Orbital and photometric properties of SZ Lyncis. The Astronomical Journal, Volume 95. \nShapley, H., 1914. On the Nature and Cause of Cepheid Variation. Astrophysical Journal , Volume 40, pp. 448-465. \nZhevakin, S. A., 1953. Theory of Cepheids. Astron J Sov. Union.'}
2024arXiv240709584R	This is a review of some recent developments on quantum gravity aspects of black hole physics. In particular we focus on a scenario leading to the prediction of the existence of a Planckmass quasistable object that could form a component of dark matter.	2024-07-01T00:00:00Z	['2024arXiv240709584R', 'arXiv:2407.09584', '10.48550/arXiv.2407.09584']	['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	Planck stars White Holes Remnants and Planckmass quasiparticles. The quantum gravity phase in black holes evolution and its manifestations	2,024	196	0.36	['EPRINT_HTML', 'EPRINT_PDF']	7	https://arxiv.org/pdf/2407.09584.pdf	{"Planck stars, White Holes, Remnants and Planck-mass quasi-particles The quantum gravity phase in black holes' evolution and its manifestations": "Carlo Rovelli abcd , Francesca Vidotto def \n- a Aix-Marseille University, Universit'e de Toulon, CPT-CNRS, 13288 Marseille, France. b Perimeter Institute, N2L2Y5 Waterloo, Ontario, Canada c Santa Fe Institute, 87501 Santa Fe, New Mexico, USA\n- d Philosophy Department and Rotman Institute, Western University, N6A5B7 London, Ontario, Canada e Physics and Astronomy Department, Western University, N6A5B7 London, Ontario, Canada and f Instituto de Estructura de la Materia, IEM-CSIC, Serrano 121, 28006 Madrid, Spain \nThis is a review of some recent developments on quantum gravity aspects of black hole physics. In particular, we focus on a scenario leading to the prediction of the existence of a Planck-mass quasi-stable object, that could form a component of dark matter.", 'I. Introduction': '1 \n- A. A sketch of the scenario \n2 \n- B. The domain of validity of classical gravity \n3 \n- II. Non dissipative aspects of the transition \n4 \n- A. Planck stars \n4 \n- B. Black-to-white transition \n5 \n- C. The exterior metric \n6 \n- D. The Boundary Region \n7 \n- E. The LQG transition amplitude \n8 \n- F. White holes \n10', 'III. Dissipative aspects of the transition': '11 \n- A. Black hole lifetime \n11 \n- B. How big is the interior of a black hole? \n11 \n- C. Remnants and their lifetime \n12 \n- D. Instability \n13 \n- E. Planckian Remnants \n14 \n- F. Remnants and quantum spread of the extrinsic curvature \n14 \n- G. There is no information paradox \n15', 'IV. Elements of Phenomenology': '17 \n- A. Dark Matter \n17 \n- B. Direct detection \n17 \n- C. Cosmological implications\n- 18\n- D. Erebons \n18 \n- E. Modeling remnants emission \n18', 'V. Acknowledgments': '20 \nReferences \n20', 'I. INTRODUCTION': "Quantum gravity is a theory with a mass scale: m P = √ c ℏ /G , a fraction of milligram. This is a small scale in astrophysics and a large scale in high-energy physics. It is reasonable to study the possibility that the spectrum \nof the theory could include a stable or semi-stable nonperturbative object at this scale: a Planck-mass quasiparticle. Recent developments in classical general relativity (GR) and in LQG (LQG) add plausibility to this possibility and suggest the existence of a semi-stable object at the mass \nm = √ √ 3 γc ℏ 4 G ∼ 14 √ γ µg (1) \nwhere γ is the Barbero-Immizi parameter, a free parameter in LQG, akin to the θ QCD angle in quantum chromodynamics, and presumed of order unity. These developments emerged from studying the dynamics of black holes. \nWe expect black holes to evolve into spacetime regions dominated by strong quantum gravity effects. A number of recent lines of research have studied these regions, and found an interesting scenario for the full evolution of a black hole [1]. Different ingredients contribute to this scenario. These include a new solution of the Einstein equations [2] showing that a trapping horizon can evolve into an anti-trapping one, a better understanding of the interior of white holes and black holes [3], and numerous applications of a variety of LQG techniques -canonical, covariant, and numerical- to describe the genuinely nonperturbative regions [4-15]. \nThree aspects of this scenario are particularly appealing. (a) It offers a natural solution to the black hole information 'paradox'. (b) It is in principle, and perhaps even in practice, directly testable [16]. And (c) it provides a natural candidate for dark matter that does not require any new physics (such as new fields, particles, or modifications of the field equations): just general relativity and its quantum properties [17]. This scenario is compatible with and possibly corroborating the general idea that primordial black holes could play a key role for explaining dark matter, see [18, 19]. \nThis scenario combines distinct quantum phenomena happening in different spacetime regions. It also includes dissipative as well as non-dissipative aspects. The analysis of its various aspects employs different approximations and truncations for treating different phenomena. \nBecause of this complexity, it can only be addressed ''a la Fermi', estimating the relevance and the import of the various physical effects, rather than within a single mathematical-physics idealization. This complexity motivates the present review paper, which brings together several ingredients for this scenario, scattered in the literature. The only truly original part of this review is Section III F, which present an argument supporting the idea that a Planck-mass quasi-particle produced as a black hole remnant could be understood as a superposition of a black and a white horizon. \nWe start with a quick sketch of the scenario (Section I A) and an analysis of the regions where classical GR is unreliable (Section I B). Then we break the presentation into two parts: a first part (Section II) where we discuss the non-dissipative (time-reversal symmetric) aspects of the global dynamics of a black hole, and a second part (Section III) where we take the dissipative (time oriented) effects into account. \nIn the first part, we discuss the bounce of a collapsing matter distribution (Section II A), namely the 'Planck Star' phenomenon, and the external metric of the global black hole dynamics (Section II B). A side section discusses the counter-intuitive classical geometry of white holes, because this play a role in global picture (Section II F). This part ends presenting the current state of the calculations of the black-to-white transition probability [20]. \nIn the second part, based on these calculations, we discuss the relevant temporal regimes, the black hole information paradox (Section III G), and the structure of and stability of the remnants (Section III E). The last part (Section IV A) focuses on the possibility of observations supporting the scenario described. Unless indicated, we use Planck units c = G = ℏ = k = 1.", 'A. A sketch of the scenario': "Perhaps the most interesting discovery of the last decades is the abundance and variety of the astrophysical objects we now call black holes. Current direct and indirect observations of these objects are all well accounted for by classical GR. But classical GR is insufficient to account for the phenomena that happen in their highcurvature regions, where genuine quantum gravity effects are most likely non-negligible. Therefore GR does not provide a reliable global picture of the dynamics of these objects. \nIn a black hole produced by a collapsing matter distribution there are three distinct spatiotemporal regions where we expect quantum gravity effects to dominate: (i) the region when the matter distribution reaches Planck density; (ii) the region, outside the matter distribution but inside the horizon, when Planck curvature is reached; (iii) the region outside the horizon, where the Planck curvature is reached because of the shrinking of the horizon due to the Hawking radiation. These events are space- \nlike with respect to one another. Each region needs to be treated separately and understood on its own terms. But all of them can be described using LQG as a guide, utilizing tools and results from various branches of LQG, in particular the spinfoam amplitudes of covariant LQG [21, 22], Loop Quantum Cosmology [23, 24] and numerical calculations [25, 26]. \nThe intuition underpinning the scenario that emerges from combining these results [8] is that quantumgravitational pressure can stop gravitational collapse and cause a bounce [6], permitting the entire content of the black hole to eventually dissipate [5]. This is the same phenomenon as in loop quantum cosmology, that indicates that the dominant quantum effect at high density is a quantum pressure sufficient to counterbalance weight and reverse collapse. \nThis quantum gravitational pressure stops gravitational collapse when the energy density becomes of Planckian order, yielding a new phase in the life of gravitationally collapsed objects. At the bounce itself, the matter distribution has maximal density and is called a 'Planck star' [6]. After this, the collapsing object bounces out. In the meanwhile, the geometry outside the star, including the horizon, also bounces tunnelling [27] from a black to a white hole geometry. This process can be short in proper time, as well as, due to the huge gravitational time dilation, very long for an external observer. \nThe process may seem incompatible with Birkhoff's theorem (in the spherically symmetric case), and this may be the reason for which the possibility for it to happen was long been missed. As detailed later on, there is no incompatibility, because the Birkhoff's theorem is local but not global, hence quantum tunnelling in a compact region can circumvent the theorem [2] and permit a black hole to evolve into an anti-trapped region, that is, a white hole. This scenario has been explored in [8, 20, 2831]. \nIf the transition happens at the end of the evaporation, when quantum gravity becomes relevant also outside the horizon, and where quantum gravitational transition amplitudes indicate its probability to approach unity, most of the energy of the black hole has already been radiated away into Hawking radiation, which is a dissipative phenomenon that breaks time-reversal invariance. Therefore, even if the transition itself can be described as an elastic non-dissipative phenomenon, the overall life of a black hole is very far from being time-reversal symmetric, and we may expect the white hole emerging from the transition to be a remnant with mass at the Planck scale. \nThe internal geometry of these objects is a remarkably precise realization, fully consistent with classical GR, of the 'cornucopia' intuition considered in the 1980's and later abandoned on the basis of arguments that (as we shall discuss below) are no longer cogent [32, 33]. \nThere is a well-known instability affecting eternal and macroscopic white holes: they can easily fall into a black hole as in the process described by the classical Kruskal \ngeometry. On the other hand, there are arguments indicating that a Plank-mass white hole can be stabilized by quantum gravity [17], by shifting it into a quantum superposition of black and white geometries. As we show below, the key of this stabilization is the LQG area gap, because of which there is likely no black (or white) hole with a mass smaller than the Planck mass. \nHence the scenario predicts the existence of quasistable remnants of a known mass, determined by the LQG area gap. A large number of these, of primordial or possibly even pre-big-bang origin, would behave precisely as dark matter. The possibility of direct detection of these objects has been explored in [16] and we review it below. \nOnce quantum gravity phenomena are taken into account, the notion itself of horizon is necessarily different than the one relevant for the classical theory. Event horizons lose their relevance, because they are likely not to form at all. The Hawking radiation drags the dynamical evolution of the horizon into regions where the Einstein equations do not hold anymore. Accordingly, the discussion on the so called black-hole information paradox needs to be reconsidered. \nEven more dramatically, the traditional definition of a black hole as an object characterized by an event horizon becomes misleading: in the light of quantum gravity, the horizon of the realistic astrophysical black holes is likely not to be an event horizon, because while the Kerr geometry and its perturbations used in modelling astrophysical objects may be appropriate, its future time evolution (which determine whether the horizon is an event horizon) is not the one determined by the Einstein equations. Of course the local properties of the horizon (described as apparent [34], trapping [35], isolated [36], dynamical [37], and similar, see this last reference for a review) remain the same. \nWhile the scenario illustrated takes fully into account quantum gravitational phenomena in the high curvature region, it assumes that classical GR and quantum field theory on curved spacetimes are good approximations outside these regions. Contrary to what sometimes claimed, GR and quantum field theory taken together, yield no inconsistencies in the entire region where quantum gravity can be neglected. As argued in detail below, the common claim that inconsistencies appear already at the Page time depends on strong versions of the holographic hypothesis [38], or from confusion between ADM mass and Bondi mass, between thermodynamic and von Neumann entropies, or between event and dynamical horizons. On the contrary, the scenario considered here is only based on conventional GR and conventional quantum ideas. \nYet, this scenario implies a number or remarkable new phenomena: (i) Due to the huge time dilation in a black hole, the process can last micro-seconds in local proper time, but billions of light-years observed from the outside; (ii) the internal volume of the hole can remain huge even as the size of the horizon shrinks; (iii) the star's \nFigure 1. The Carter-Penrose diagram of a Schwarzshild black hole until the onset of quantum gravity. The light grey region is the collapsing star. The dark grey region is where quantum gravity becomes relevant. \n<!-- image --> \nbounce volume is much larger than Planckian, because the onset of quantum-gravity effects is governed by density, not size; (iv) the interior of an evaporating hole can keep memory of the initial state, without information loss; (v) information can follow a different path from energy, and be slowly released by the remnants, while most of the energy was previously lost into the Hawking radiation. All this shows that unpalatable phenomena like 'firewalls' [39] are not to be expected on the basis of GR and quantum theory alone, unless one adopts arbitrary holographic assumptions that are at best stringmotivated only.", 'B. The domain of validity of classical gravity': "What happens inside a black hole is dynamical: it cannot be understood in terms of any static or stationary model. The Carter-Penrose diagram of a (classical) Schwarzschild black hole created from a gravitational collapse is depicted on Figure 1. The dark grey region is where the classical theory becomes unreliable, due to quantum gravitational effects. We expect this to happen when the curvature becomes Planckian, for instance when the Kretschmann scalar \nK 2 = R αβγδ R αβγδ = 48 M 2 r 6 (2) \nbecomes of order 1 in natural units. Here M is the black hole mass and r is the Schwarzschild radius. This happens before the r = 0 singularity inside the black hole. Notice that if M is macroscopic, it happens at a radius of order r ∼ M 1 3 much larger than the Planck radius ( r ∼ 1). \nImportantly the surface where this happens is spacelike. \nJust outside the horizon, where r ∼ 2 M , K ∼ M r 3 ∼ 1 M 2 . The Hawking evaporation steadily decreases the mass M of an isolated black hole, bringing K up to \nPlanckian values, hence the quantum region extends outside the horizon . As we will see later on, results in [13, 20] indicate that the transition probability P from black holes to white holes is proportional to \nP ∼ e -M 2 . (3) \nThus becoming dominant at the end of the evaporation, where M ∼ 1. \nThe possibility that quantum effects could appear earlier, at a (retarded) time of order M 2 after the collapse, has been considered in [2, 40] and in [10, 31, 41] by considering the interplay with the collapsing matter. The phenomenology this could give rise to has been explored extensively in [42-47]. Here we will not consider this hypothesis. \nThe region of the onset of quantum gravitational phenomena can thus be organized into three sub-regions [48] (see Figure 1 ): \n- · Region a : The region which is neither directly causally connected to the horizon nor to the collapsing matter distribution (it is amid them).\n- · Region b : The region in the vicinity of the horizon ( the boundary ).\n- · Region c : The collapsing matter distribution region ( the center ). \nThe onset of the phenomena in these three regions are causally disconnected, as they are separated by spacelike distance. For a black hole of (initial) mass M , this distance is of the order [48] \nL ≃ M 10 3 , (4) \nwhich is colossal for a macroscopic black hole. Notice that the locus of the onset of quantum gravity is not 'at a point', but rather 'at a time'. It is not 'somewhere' but rather, 'at some time', which, in relativistic terms, means in a set of space-like related regions. \nThe fact that the various onsets of quantum gravity are spatially related implies by locality that what happens in a region of this locus is causally disconnected, hence independent from what happens in other regions. To imagine that quantum gravity could violate this macroscopic causal disconnection is in gross contradiction with what we know about reality, not supported or suggest by anything we know, as far as we can see. To understand the physics of the end of a black hole, we have to understand the quantum evolution of each of these three regions independently.", 'A. Planck stars': "Let us start by studying what happens in the interior of a collapsing star, after it enters its horizon, when \nFigure 2. The scale factor a ( T ) in eq. (9) that gives the standard LQC bounce. \n<!-- image --> \nit reaches Planckian density. The simplest modelling of a collapsing matter distribution is provided by the Oppenheimer-Snyder model [49]: a spherically symmetric pressure-less homogeneous matter distribution freefalling under its own weight. Assuming the matter distribution to start at rest in the distant past, the metric inside a such a matter distribution distribution can be written in co-moving and proper time coordinates ( T, R ) as \nd s 2 = -d T 2 + a 2 ( T )(d R 2 + R 2 dΩ 2 ) , (5) \nwhere the T = constant slices define the homogeneity foliation, dΩ 2 is the metric of the unit 2-sphere, R ∈ [0 , R boundary ] and a ( T ) is known as the scale factor. The radial co-moving coordinate of the boundary of the matter distribution distribution can be chosen to be R boundary = 1 without loss of generality. The uniform density of the matter distribution distribution is then ρ = m/ 4 3 πa 3 , where m is the total mass. Inserting this metric in the Einstein field equations gives the Friedmann equation for a ( t ): \n˙ a 2 a 2 = 8 π 3 ρ, (6) \nwhere the over-dot means differentiation with respect to T , which coincides with the proper time. Eq. (6) can be solved: \na ( T ) = ( 9 m ( T -T 0 ) 2 2 R 3 star ) 1 / 3 . (7) \nWithout loss of generality we can take the time at which the matter distribution distribution collapses to zero physical radius to be T = 0. \nHow does quantum gravity affects this dynamics? A major result in loop quantum cosmology [50-53] is that the Friedmann equation for the scale factor, Eq. (6), is modified by quantum gravity effect into \n˙ a 2 a 2 = 8 π 3 ρ ( 1 -ρ ρ c ) , (8) \nwhere ρ c = √ 3 c 2 / (32 π 2 γ 3 ℏ G 2 ) ∼ c 2 / ℏ G 2 , γ being the Barbero-Immirzi parameter, is the critical density [54], a \nconstant with the dimension of a density and Planckian value. This equation can be integrated to give \na ( T ) = ( 9 mT 2 + Am 2 R 3 star ) 1 / 3 , (9) \nwhere A = 3 / (2 πρ c ) is a parameter of Planckian value. In units where G = c = 1, the constant is A ∼ ℏ ∼ m 2 Pl . Notice that a ( T ) is positive for the whole range T ∈ [ -∞ , ∞ ]: it decreases for T < 0, reaches a minimum a 0 = 3 √ Am/ 2 for T = 0 and then increases for T > 0. See Fig. 2. This is the characteristic bounce of loop quantum cosmology. This result shows that the line element in eq. (5) is well defined everywhere, without singular collapse: the star reaches a maximal density, where it is called a 'Planck star', and then bounces and expands. \nOf course the time-reversal symmetric bounce is an approximation, because dissipative effects break this symmetry. The Hawking radiation generates an ingoing flux of negative energy that is likely to compensate the energy of the star, in the interior of the hole. We will address these phenomena later on, when discussing the dissipative aspects of the dynamics of the black hole. \nTo get a feeling for the physics of the bounce, we can rewrite (8) in the form \n˙ a 2 = 2 m a -Am 2 a 4 . (10) \nThe coordinate T is the proper time along the comoving worldlines, hence it is also the proper time on the boundary of the star. This means that Eq. (10) gives also the evolution of the physical radius r b ( T ) = a ( T ) of the matter distributionin its own proper time, hence \n˙ r 2 b = 2 m r b -Am 2 r 4 b . (11) \nThis shows that a mass element on the boundary of the matter distributionfalling in its own proper time feels a potential that is the Newtonian one, precisely as in classical GR, but corrected by a repulsive quantum term proportional to the inverse of the 4th power of the radius. This is the short scale repulsive force due to the quantum pressure, that grows stronger at smaller radius. \nIs this dynamics compatible with the dynamics of the surrounding geometry?", 'B. Black-to-white transition': "In the classical Oppenheimer-Snyder model, the geometry of the collapsing matteris compatible with a surrounding Schwarzschild geometry. That is, matching conditions between the two geometries are satisfied on the boundary of the matter distribution. To confirm this, consider the surface bounding the collapsing matter as a free falling spherically symmetric shell in a Schwarzschild metric \nd s 2 = -F ( r ) dt 2 + dr 2 /F ( r ) + r 2 dΩ 2 , (12) \nwhere F ( r ) = 1 -2 m/r . The shell follows a radial timelike geodetic and therefore satisfies \n-1 = -F ( r ) ˙ t 2 + ˙ r 2 /F ( r ) (13) \nThe conservation law associated to the time-like Killing symmetry gives F ( r ) ˙ t = E where E is a constant that can be taken equal to unit if the shell starts at rest at infinite radius. Hence -F ( r ) = -E 2 + ˙ r 2 , that is \n˙ r 2 = 2 m r . (14) \nA comparison with (11) shows that this is exactly the change of the Schwarzschild radius in proper time of the boundary of the star, in the classical ( A =0) case. \nBut this suggest immediately the form of a metric which is compatible with the bouncing star, in the quantum case: it is again (12) but with \nF ( r ) = 1 -2 m r + Am r 4 . (15) \nThis is an interesting metric. (For earlier attempts to write the black hole metric in the quantum region see for instance [53, 55-65]) It was suggested already in [6] and has been derived by various quantum gravity research groups, using different methods [9, 31, 64-68], as a credible candidate for the effective metric in the high curvature of a spherically symmetric black hole. For instance, it is uniquely determined by requiring it to satisfy matching conditions with the collapsing matter distribution and keep the Killing symmetry of the Schwarzschild geometry (which in the interior of the horizon is spacelike, not time-like as in the exterior.) \nThis metric has inner and outer Killing horizons. See [11] for full details. For m ≫ m Pl , that is m 2 ≫ A , the outer one is at \nr + = 2 m + O ( A/m ) ∼ r Schwarzschild (16) \nin the classical region. If m is large, this is a negligible modification of the usual Schwarzschild horizon. While the inner one is at \nr -= 3 √ Am/ 2 + O ( A 2 / 3 /m 1 / 3 ) ∼ 3 √ m/m Pl l Pl (17) \ndeep inside the quantum region, where the spacetime curvature has Planckian size. These are all also apparent horizons: they separate trapped, non-trapped and antitrapped regions [11]. \nThe Carter-Penrose diagram of the maximal extension of this metric is depicted in Figure 3, which shows the relative location of these horizons. \nInterestingly this geometry has a global structure (including the inner horizons) similar to the Kerr, the Reissner-Nordstrom, and the general Kerr-Newman black holes. (The full bounce in the Reissner-Nordstrom case has been described in [69], while the Kerr case has not been constructed yet.) \nFigure 3. Conformal diagram of the maximal extension of the spacetime representing the matter distribution and the exterior region defined by eqs. (12) and (15). \n<!-- image --> \n<!-- image --> \ndown \nE \nThe maximal extension of this geometry has (i) two asymptotic regions and (ii) a time-like singularity in a high curvature region. These two features make it an unlikely candidate for describing the actual physics of realistic black holes. But a surprising result on spherically symmetric solutions of the Einstein's field equations, solves both these difficulties. This is illustrated in the next section.", 'C. The exterior metric': "The Birkhoff's theorem is sometimes presented as stating that the only spherically symmetric spacetime compatible with the vacuum Einstein's field equations is the Kruskal geometry. At a sufficient distance from a black hole, we expect quantum effects to be negligible and therefore the theorem to hold. Hence we might expect that whatever happens where quantum effects are negligible must form a subset of a Kruskal geometry. In the Kruskal metric, there is an anti-trapped region, but it is before, not after the trapped region. Hence there seems no way to join the metric described above to a single external asymptotic region. But the above is wrong. The reason is that Birkhoff's theorem is local, not global: it states that a spherically symmetric solution of the vacuum Einstein field equations is locally , and not necessarily globally, isometric to the Kruskal metric. This seems a red herring, but has momentous consequences for understanding the dynamics of quantum black holes. \nIn fact, as surprisingly realized in [2], there is an exact solution of the Einstein's field equations, with a single asymptotic region, that can surround a quantum transition from a black to a white hole. The Carter-Penrose diagram of this solution is depicted in the left hand side of Figure 4, while the right hand side shows how this can be locally isometrically mapped onto the Kruskal space- \nE \nE \ntime. The point is that the map is not injective, hence the (pink) classical part of the spacetime is locally isomorphic but not a subset of the Kruskal geometry. \nE \ndown \nFigure 4. The spacetime discovered in [2], describing the collapse of a null shell into a black hole and its bounce out as a white hole (left); and the way it can be locally isometrically mapped onto the Kruskal spacetime (right). The pink region is an exact solution of the classical Einstein equations. \n<!-- image --> \nThis solution describes a null shell collapsing into a black hole and bouncing out from a white hole. The key point is that the black hole (the trapped region) is in the past of the white hole (the anti-trapped region) while in the Kruskal spacetime it is the opposite. This shows that an exact solution of the Einstein equations is compatible with a quantum process happening in a compact high curvature region tunnelling a black into a white hole. \nThis same idea solves the two difficulties mentioned at the end of last section, providing a spacetime with a single asymptotic region and no singularities for the bouncing star. This can be constructed by cutting and pasting the maximal extension of the metric defined by eqs. (12) and (15), as follows. \nFirst, pick a point α in the interior region, on the surface invariant under time reversal and a point β L outside the horizons, in the first asymptotic region, as in the left panel of Figure 5 This choice depends on three parameters: the advanced times v ( α ) and v ( β L ) and the Schwarzschild radius of β L . Next, consider the blue line of the figure. This has a null portion joining α and β L with their last common past event and a spacelike portion joining β L to spacelike infinity along a constant Schwarzschild time. Next, draw the line symmetric to this under time reversal, as in the right panel of Figure 5. \nNext, delete the entire spacetime region enclosed within the blue line, and paste its two spacelike portions as in Figure 6. This is clearly possible since they are both constant Schwarzschild time surfaces. \nThe resulting spacetime has a single asymptotic region, is everywhere locally isomorphic to the maximal extension of the metric (12-15), and has a hole, depicted as the B diamond of Figure 6. In [11] it is shown that the B region admits a regular Lorentzian metric compatible \nE \nE \nup \ndown \nPhysics of the horizon (region B) \nPhysics of the horizon (region B) \nHan, Rovelli, FS (2023) \nFigure 5. The construction of the single asymptotic region spacetime, I: choosing the cut and glue surface \n<!-- image --> \nPhysics of the horizon (region B) \nHan, Rovelli, FS (2023) \nFigure 6. The construction of the single asymptotic region spacetime, II: deleting the singular region and gluing the two equal time exterior surfaces. \n<!-- image --> \nwith rest of the spacetime geometry. The B region and its physics will be discussed later on. For the moment, let us focus on the rest of the spacetime. \nThe popular idea that at the end of the evaporation a black hole disappears is not supported by any theory and contradicts unitarity. The above construction offers a far more plausible alternative. See Figure 7. \nThe relation of the resulting geometry with the various quantum gravity phenomena mentioned above is depicted in Figure 8. (See also Figure 8 in [70].) \nWe have constructed the above metric introducing three parameters ( v ( α ) , v ( β L )) and the Schwarzschild radius of β L . The resulting geometry (apart from the choice of the metric inside B ) depends then on these parameters, plus the mass of the star. Of the three parameters, two simply locate the B region: they determine the minimal and maximal radius of the diamond defining it. The last one is far more interesting. It determines the global geometry of the spacetime, in the sense in which the radius of a cylinder determines the global geometry of a locally flat cylinder. Its geometrical and physical inter- \nHan, Rovelli, FS (2023) \npretation can be seen as follows: it determines the time T at which an observer at large distance from the hole sees the duration of the process. \nThis is illustrated in Figure 9, where the red line represents an observer at a fixed radial distance R . As shown in [11], this duration is \nT = 2 R +4 m ln r -2 m -4 m ln δ. (18) \n15 15 The first term on the right-hand side can be interpreted as the back and forth travel time for the light from the observer to the hole. The second term is the standard relativistic correction. The last term is independent of the radius and represents a property of the transition itself. The quantity δ can be measured by distant observers by measuring T at different values of R . A little geometry shows that it depends on the choice of β L as follows: r = 2 m + δ is the radius at which the constant Schwarzschild-time surface passing by β L intersects the surface of the falling star. This is clearly very close to the collapse (the point where the surface enters entirely in its horizon), if the lifetime of the hole is long. Thus \nτ = -4 m ln δ (19) \ncan be taken as a definition if the duration of the quantum process, as observed from infinity. In our construction, it is determined by how the geometry has been constructed by cutting and pasting. Operationally, it is the observed duration of the process from very large distance. Physically, it is determined by the quantum theory of the geometry of the B region, to which we now turn. \n15", 'D. The Boundary Region': "In the previous sections, we have constructed a spacetime geometry that could describe the evolution of a black hole past the region where quantum gravitational effects dominate. How well supported by known physics is this picture? \nFigure 7. Left: a popular idea of what happens at the end of the evaporation. This is not supported by any theory and contradicts unitarity. On the right, the plausible scenario. \n<!-- image --> \nFigure 8. The Carter-Penrose diagram of the black-to-white transition. The dark grey region is the quantum gravity region. The black hole (trapped region) is below the quantum gravity region while the white hole is above. The trapping horizons are the dashed lines. \n<!-- image --> \nThe bounce of the matter distribution is supported by the LQG modelling of the quantum dynamics of symmetric spacetimes. The physical plausibility of the effective metric outside the matter distribution but inside the outer horizons is also supported by various LQG modelling of the quantum dynamics of symmetric spacetimes. The structure of the B region, on the other hand, has only been guessed as a plausible solution to the requirement of a global dynamics compatible with the external geometry dictated by the classical Einstein equations. Does it follow from quantum gravity? Is it allowed by it? \nTo address this question we need to step back and think more precisely what we are doing. First, let us remember once more that for the moment we are disregarding the dissipative effects that break time reversal invariance. These will be studied in the next section. Then, consider the fact that we are describing a quantum process. There are different ways of describing quantum processes. In some regimes, these can be described as corrections to the classical equations of motion. This is not viable for 118 3. The black-to-white hole spacetime \nFigure 3.9: In red is the worldline of an observer moving at a constant distance R /greatermuch 2 m in the qualitative Carter-Penrose diagram of the black-to-white hole spacetime. Figure 9. The definition of the duration of the process. The red line represents an observer at fixed radius. \n<!-- image --> \nthe quantum transition of the horizon, which is a nonperturbative process. One cannot describe quantum tunnelling as a simple ℏ correctios to classical trajectories. Tunnelling is non-analytical in ℏ . \nAnother possibility is to describe the evolution of a quantum state. Quantum states that approximate classical configurations, such as coherent states, spread. We could therefore consider the full quantum state of the geometry after the actual quantum process as a quantum state, namely a quantum superposition of geometries. This is analogous to describing nuclear radioactivity in terms of a wave function of the emitted particle widely spread in space and time. Correct, but incomplete. What we observe in nuclear radioactivity is not the wave function of the emitted particle, widely spread in space and time. Rather, it is a specific space and time location where the particle is detected, say by a Geiger counter. \nIn Copenhagen terms, we observe the result of a measurement. In Many World terms, we observe the position of the particle in one branch, after decoherence. In Relational Quantum Mechanics, we observe the metric relative to us, again after decoherence. The theory, of course, does not predict any specific outcome, but it does predict the probability of the alternatives. \nTo compute these, we need to compute transition amplitudes. This is how we can address the problem of describing the B region in LQG. \nThat is, we can compute the transition amplitude for the geometry to make a quantum transition from its value in the past boundary of B to its future boundary (as opposed to not yet making it, as in radioactivity). Notice that the geometry (both intrinsic and extrinsic) of these two boundaries is known from the construction of the above section. We have simply to compute the LQG transition amplitude between quantum states approximating these geometries. \nThe covariant formulation of LQG is precisely formulated to do this calculation, using approximations provided by truncation of the degrees of freedom. The calculation has been performed, under some drastic simplifications by Christodoulou and D'Ambrosio using analytical techniques in [20, 48, 71]. The result has been confirmed numerically in [12] and [13]. Numerical calculations are been currently developed to go beyond these approximations. The way these calculations are performed is summarized in the next section.", 'E. The LQG transition amplitude': "The covariant formulation of LQG defines transition amplitudes between quantum states of the geometry, in suitable truncations of the numbers of degrees of freedom. The number of degrees of freedom can be truncated by approximating the 3d geometry of the boundary by means of a cellular decomposition. The two-skeleton of the dual of the decomposition form a graph and the trun- \ncated variables can be taken to be SU (2) group elements U l on the links l of this graph. The quantum states of the geometry can then be taken to be functions of the U l 's in the space L 2 [ SU (2) L /SU (2) N ], where L and N are respectively the numbers of links and nodes of Γ, as in lattice Yang Mills theory. The algebra formed by the group elements U l themselves and the left-invariant operators is the observable algebra, which has a natural interpretation in terms of functions of the discretized geometry. A large literature has developed a theory of coherent states for this algebra, representing semi-classical geometrical states. \nTransition amplitudes can be equally understood as Feynman sums over geometries, or as analogous to lattice Yang-Mills calculations. Concretely, they are defined in a truncation. A truncation is here given by a two-complex bounded by the boundary graph. The amplitude is defined by the product of the vertex amplitudes \nA v ( ψ ) = P SL (2 ,C ) Y γ ψ (11) , (20) \nwhere P SL (2 ,C ) is the projector on the SL (2 , C ) invariant part and Y γ is the 'simplicity map' from SU (2) to SL (2 , C ) representations defined, in the canonical basis, by \n\| m ; j ⟩ ↦→ \| γj, j ; j, m ⟩ . (21) \nFigure 8 \n<!-- image --> \n<!-- image --> \ntime interval and a finite radial interval and it is delimited by an exterior two-sphere S + that surrounds the horizon and by an interior two-sphere S -surrounded by the horizon (sitting on the bounce radius of the transition of the internal geometry of the black hole). The two two-spheres S + and S -split the boundary Σ of B into a past component Σ p and a future component Σ f . \nConsidering instead the entire quantum region simplifies the topology. The boundary of the quantum region can be take to be as depicted in blue Figure 10. The tip, at the largest radius, is a two-sphere, chosen outside the Planckian curvature region. The lowest (past) part of the boundary and the upper (future) one are both 3d balls. Each of these can be triangulated with four tetrahedra and the interior can be triangulated with two foursimplices sharing an internal tetrahedron, as depicted in Figure 11. \ncrossing ti \nsion as \nT \nc \nestimate in \n(left \ndle links (faces) carry the boundary dat \nC (left) and its oriented (right) chosen in [6]. The four midFIG. 4. The spinfoam 2-complex C boundary graph Γ = ∂ C (right) chosen in Figure 11. The graph of the spin network and the twocomplex of the spinfoam for the calculation of the transition. \nTo apply this general theory to the B region we need to find a suitably simple cellular decomposition of this region. Steps in this direction have been taken in [73], and a numerical calculation has been achieved in [13]. An analytical calculation, on the other hand, has been completed only in a simpler setting, where the entire quantum region is discretized. dle links (faces) carry the boundary data intersection of C ± . carry the boundary data of the surfaces C ± \nω ∆ and ζ ∆ that correspond to a discretization of the sphere ∆, defined as the The six upper and six lower links (faces) ω ± and ζ ± respectively, that correspond to a particularly rough discretization of the remaining while the surfaces F ± were disregarded. It is striking that this rough discretization gives exactly the behavior for the bounce time T c and lifetime τ expected on general grounds from the analysis in Section V. This should be taken as an indication that the relevant physics happen in Thus the l 24 ( ± 2 These re correspond to a discretization of the spher intersection of C ± . The six upper and six carry the boundary data ω ± and ζ ± respe spond to a particularly rough discretizatio of the surfaces C ± while the surfaces F ± It is striking that this rough discretizatio behavior for the bounce time T c general grounds from the analysis in Secti be taken as an indication that the relevant the vicinity of the sphere ∆, see [90] for a We refer to [20, 48, 71] for the analytical calculation. The numerical calculations have been performed using different techniques. [12] has analyzed the simpler geometry, utilizing the sl2cfoam-next code for computing spinfoam transition amplitudes. A good introduction to this, and full references, is in [74]. The code can explore the low-spin (small-mass) regime of the amplitude. On the other hand, [13] utilizes the complex saddle point method for computing spinfoam amplitudes, which can explore the high-spin (large-mass) regime [75-77]. A common result of all these investigations (that contradicts some previous expectations) is that the transition amplitude gives a transition probability proportional to \nζ \n) \n≈ \nand lifeti \nthe vicinity of the sphere ∆, see [90] for a detailed argument. low. We br tails given glued along one of their five tetrahedra P ∼ e -Gm 2 c ℏ ∼ e -( m m P ) 2 . (22) \nSee [21, 22] for the full details. This amplitude has been shown to give the Einstein dynamics in suitable limits [72] and can be taken as a definition of the covariant LQG theory. FIG. 4. The spinfoam 2-complex boundary graph Γ = ∂ \nFigure 10. The boundary of the quantum region for the simplified calculation of the transition amplitude. \n<!-- image --> \nfrom \nω \nand \nk \n, \nThe complexity of the calculation on the B region alone is given by the topology of B , which is the product of a two-sphere and a disk. The disk is the product of a finite it vanishes. The vanishing of the 4-volume follows from \nthe fact that the triangulation is taken to be intrinsically \nflat: \nthe five tetrahedra making up each four simplex \nglued along one of their five tetrahedra so that they correspond to a simplicial manifold dual to the spinfoam in Figure 4, have zero 4-volume. This can be checked explicitly by calculating the edge lengths of the 4-simplices /lscript /lscript n and then calculating their 4-volume written as a Cayley-Menger determinant, verifying that is shown i present in the estima crossing ti T = 2 π/γ . lifetime sc stant. Inst respond to a simplicial manifold dual t Figure 4, have zero 4-volume. This ca plicitly by calculating the edge lengths from ω /lscript and k /lscript n , and then calculatin written as a Cayley-Menger determina it vanishes. The vanishing of the 4-vol the fact that the triangulation is taken flat: the five tetrahedra making up e Notice that this is indeed non-analytic in the Planck constant. Therefore it is a genuine non-perturbative result that cannot be obtained in conventional perturbation theory expanding around a classical solution. The exponential is characteristic of quantum tunnelling phenomena. The transition, in fact, is forbidden classically (classically the geometry evolves into a singularity) and is a characteristic quantum tunnelling effect. Another way of understanding this result is to connect to the asymptotic behaviour of the amplitudes \nIn Fig \nin the mas \nm . glue properly when embedded in a 3d W ∼ e iS Regge ∼ e i ∑ f j f Θ f ( j ) , (23) \nThey correspond to a tetrahedron split i \nwith all deficit angles on the interior ed \nThus, when promoted to a 4-simplex, t \ninverse of \n4-simplex. For an analogy in one dime \nglue properly when embedded in a 3d Euclidean space. \nThey correspond to a tetrahedron split in four tetrahedra \nC \nh \ne \nfir \ns \nt \n. Thi \nt \ny of cla \nss \nical g \no \ns \ni \nt \ni v \ne \nr \n. No \ns \na \nt \ni m \ne \nlin \ne \no \nt \nh \ne t \nwo ou \ni m \ne \na \ns \n(p \ns \ne \ne \nyond \nr \n= \no \ns \nt \nof \nt \nh \ne \nblack hol \ne \ns \nw \ne \ns \nee \ni n \nt \nh \ne \ns \nky w \ne \nr \ne \nli k \ne \nl y form \ne \nd by \nt \nh \ne \ncollap \ns \ne \nof a \nwhere S Regge is the Regge action: a sum over the faces of the face spin times the dihedral angle of the face, which is a function of all the spins, determined by the interpolating flat geometry. Here there is no interpolating geometry and the angles turn out to be imaginary, giving \nW ∼ e -c ∑ f j f ∼ e -c ∑ f A f , ∼ e -c m 2 . (24) \nThe simplest way to interpret this result is in analogy to nuclear radioactivity: as a transition probability per unit of (here Planck-) time. The immediate consequence of this result is that the transition probability is exponentially suppressed if the mass of the hole is larger than the Planck mass. \nAnticipating the discussion about dissipative effects, we can see that toward the end of the Hawking radiation, m decreases and approaches m P . At this stage the transition becomes increasingly probable, going to probability unity when the mass becomes actually Planckian. \nIn other words, the calculation indicates that the transition can happen and it is likely to happen, but only at the end of the Hawking evaporation, giving birth to a white hole with near Planckian mass.", 'F. White holes': "Before addressing the dissipative aspects of the life of a hole, let us pause to clarify a few properties of the white holes. White holes, like black holes, are exact solutions of the Einstein's equation. Precisely like black holes, they have long been expected to not play any role in our universe, by a majority of physicists. The situation has changed for black holes in the last decades. It has not yet equally changed for white holes, but it might soon. \nA white hole spacetime is simply the time reversal of a black hole spacetime. For instance, a classical black hole formed by a collapsing matter distribution and its time reversal, a white hole from which matter distribution emerges, are depicted in Figure 12. White holes 108 Black holes \n<!-- image --> \nTh \ne \nblack hol \ne \ng \ne \nom \net \nry of a collap \ns \ni ng \ns \nt \nar \ne \nnd \ns \ni n \nt \no a quan \nt \nhole. The end of a black hole is definitely a quantum phenomenon, because it involves high curvature regions, where quantum theory cannot be neglected. The same must be true for the birth of a white hole. \nAll this points to a very simple possibility: the end of a black hole is the birth of a white hole, via a quantum transition process. This is precisely the scenario we are studying here. A white hole can be originated by a dying black hole. \nThe difference between a black hole and a white hole is not very pronounced. In fact, observed from the outside (say from the exterior of a sphere of radius r = 2 m + ϵ > 2 m , where m is the mass of the hole) and for a finite amount of time, a white hole cannot be distinguished from a black hole. \nThis is clear from the usual Schwarzschild line element, which is symmetric under time reversal, and therefore describes equally well the exterior of a black hole and the exterior of a white hole. Equivalently, zone II of the maximal extension of the Schwarzschild solution is equally the outside of a black hole and the outside of a white hole (see Fig. 1, Left). Analogous considerations hold for the Kerr solution. The continuation of the external metric of a stationary Kerr or Schwarzschild spacetime inside the radius r = 2 m + ϵ contains both a trapped region (a black hole) and an anti-trapped region (a white hole). \nFigure 13. Left: in the extended Schwarzschild spacetime, which is stationary, the (light grey) region outside r = 2 m + ϵ (dotted line) is equally the outside of a black and a white hole. Center: A collapsing matter distribution (dark grey) replaces the white hole region ('WH') in the non-stationary collapse metric. Right: The time-revered process. The difference between the last two can only be detected looking at the past, or the future. \n<!-- image --> \nWhite hole r=0 r=2m he part of the Kruskal spacetime that is relevant for the geometry around a ollapsing star. The dotted line is the surface of the star. /a116 Figure 11.8 The part of the Kruskal spacetime that is relevant for the geometry around an exploding white hole. The dotted line is the surface of the exploding matter. Figure 12. The spacetimes of a black and a white hole (outside the respective stars, as subsets of the Kruskal geometry). \nR \ne \nmarkably, in \nt hi s g e om et ry t h e r e i s a s e cond e x te rior r e gion, s e para te d from s s e cond r e gion ha s a s e para te a s ymp t o t ic infini t y. t um ph e nom e na, approaching r = 0 w e l e av e t h e domain of valide n e ral r e la t i v i t y; t h e r e for e w e mu s t li mi t our s e l v e s t o t h e r e gion s of te t ha t t h e t wo r e gion s r = 0 ar e s pa t ial r e gion s (in Minkow s ki r = 0 lin e ). Th e g e om et ry form e d by t h e black and whi te hol e s , in addi t ion te r r e gion s , i s t h e larg e s t po ss ibl e e x te n s ion of Schwarz s child s pac e -udo-) Ri e mannian g e om et ry. Q uan t um t h e ory could e x te nd i t fur t h e r, 0. whi te hol e g e om et ry e m e rg e s from a quan t um r e gion. I t i s po ss ibl whi te hol e s could e m e rg e from t h e s am e quan t um r e e nd. I s hall t ouch on t hi s po ss ibili t y in t h e la s t Chap te r, on quan t The reason black holes have been traditionally taken more seriously as candidates for real objects than white holes is that we have a well understood classical scenario for how a black hole can form, but not so for a white hole. Since the two are the time reversal of one another, this implies that we have a well understood classical scenario for how a white hole can end, but not so for a black \nblack holes formed by a collapsed star \ne t \no \num r e gion. Th e s u s p e c t t ha t gion in which black hol e s um gravi t y. What distinguishes then the objects we call 'black holes' from 'white holes'? The objects in the sky we call 'black holes' are described by a stationary metric only approximately and for a limited time. We expect that (at least) in the past their metric was definitely non-stationary, and they were produced by gravitational collapse. The energy contained inside r = 2 m + ϵ was less than m in the past and the continuation of the metric inside this radius contains the trapped region, but not the anti-trapped region, which is instead replaced by the region describing the collapsing star. Therefore seen from the outside a 'black hole' (as opposite to a 'white hole') is only characterized by the fact that in the past it does not have an anti-trapped region (see Fig. 1, Center). Vice versa, a white hole is an object that from the exterior and for a finite time is indistinguishable from a black hole, but in the future ceases to be stationary, the amount of energy inside r = 2 m + ϵ decreases, and there \nDu \ne t \no quan \nt \nar, wh \ne \nn \nt \nh \ne \nh \ne \na \nt \nof nucl \ne \nar fu \ns \nion i \ns \nno long \ne \nr \ns \nuffici \ne \nn \nt t \no produc \ne t \nh \ne \npr \ne \nss \nur \ne \nha \nt \ncoun \nte \nrbalanc \ne \ns \ngravi \nt \ny. \nOnly \nt \nh \ne e \nx \nte \nrior of \nt \nh \ne \ns \nt \nar i \ns \nd \ne \ns \ncrib \ne \nd by a par \nt \nof \nt \nh \ne e \nx \nte \nnd \ne \nd Schwarz \ns \nchild \nis no trapped region in the future (see Fig.1, Right). All this regards classical gravity only.", 'A. Black hole lifetime': "The most relevant dissipative phenomenon in the life of a black hole is the Hawking radiation. This is a markedly irreversible process that dissipates the energy of the collapsed matter distribution into the heat of the emitted thermal radiation. \nThe lifetime τ BH of a black hole is known by Hawking radiation theory. It can be estimated as follows. The Hawking radiation is thermal, namely it has a Planck spectrum. The peak of the spectrum is on a wavelength λ that is determined by the only length parameter in the problem: the Schwarzschild radius 2 Gm/c 2 . Therefore the radiation is mostly composed by quanta of frequency \nν = c/λ ∼ c 3 Gm (25) \nThese have an energy E = hν and therefore (being thermal) a temperature T given by \nkT = E ∼ c 3 ℏ Gm (26) \nThis is (up to a numerical factor) Hawking's temperature. The emission of the horizon can be modelled as the emission of a sphere at this temperature with the area of the horizon. This gives an emitted power (going back to natural units) \nP = dm dt ∼ AT ∼ m 2 m -4 = m -2 . (27) \nThis is a differential equation for m ( t ) that can be immediately integrated giving m 3 ∼ t . Therefore we expect that the finite lifetime of a black hole due to the Hawking evaporation is of the order \nτ BH ∼ m 3 o (28) \nwhere m o is the initial mass of the hole. 1 \nThe evaporation shrinks the horizon, bringing it close to Planckian size, where the black-to-white transition has high probability to happen. The white hole generated by the process then has a horizon of Planckian size. (This lifetime can be shorter if quantum gravity fluctuations trigger an earlier tunnelling. This could be as early as τ BH ∼ m 2 o as suggested in [2, 40]. Here we only focus on \nthe possibility that the transition happens at the end of the evaporation.) \nHowever, the fact that the black hole has evaporated does not mean that it is 'small' in any sense of the word. In fact, its interior is vast. To understand this point, a detour on the size of the interior of black holes is important, as this is a frequently misunderstood and misleading issue.", 'B. How big is the interior of a black hole?': "In Minkowski spacetime, we say that a space-like 2sphere S of radius r encloses a space-like ball of volume 4 3 πr 3 . As a 3d surface in Minkowski space, this ball is characterized by two properties: it is linear, and it is the surface that maximizes the volume among those bounded by S . Any deformation, in fact, is time-like and decreases the volume. \nIn a curved Lorentzian manifold, linearity is meaningless, but we can still talk about the volume enclosed in a spacelike (topological) 2-sphere S by defining it as the volume of the 3d surface that maximizes the volume, among all 3d surfaces bounded by S . This definition, introduced in [3], allows us to talk of the interior volume of a black hole in a rigorous manner, and provides a simple intuition of the internal geometry of a black hole by defining a natural foliation of this geometry. \nConsider therefore the spherical symmetric spacetime of a collapsing matter distribution and choose a retarded time v . The intersection of the retarded time with the horizon defines a 2-sphere S v . Let V ( v ) be the volume of the maximal-volume spacelike ball bounded by S v . This is the volume of the interior of the black hole at the retarded time v on its horizon. The dependence of this volume on v has been computed both for an eternal horizon and for an evaporating one [3, 28, 79-82]: it grows linearly in v . \nSpecifically [3, 82] that, at advanced time v after the collapse, a black hole with mass m (disregarding evaporation) has interior volume \nV ∼ 3 √ 3 π m 2 v (29) \nfor v ≫ m . \nRecall that the interior of the horizon is never stationary: it is dynamical. (The common expression 'stationary black hole' means a black hole whose external geometry is stationary.) In the natural foliation considered above, the internal volume of the black hole increases steadily in time. More specifically, the interior is like a tube whose radial dimensions shrink in time while its longitudinal dimension increases, in such a way that the total volume increases. \nThis implies that an old evaporated black hole has a small horizon but a huge internal volume . Ignoring this fact has nourished a wrong intuition about the geometry of evaporated black holes. Contrary to what is sometimes \nbelieved, these are genuinely different from young black holes with the same mass. \nIn other words, we must not be fooled by the nohair theorem. This theorem states that two black holes that have the same mass, charge and angular momentum, classically settle on the same external geometry, but certainly does not say that they have the same interior! Older ones are bigger. In fact they can have vastly different interiors. Accordingly, in the quantum theory, the full quantum state of a black hole will certainly not be determined by external quantities like mass, charges and angular momentum only: there will be quantum numbers v i distinguishing holes with the same exterior but (possibly vastly) different interiors. 74 CARLO ROVELLI This was Hal's first idea: the inside of the black hole is \nAt the moment of the quantum transition therefore an old (previously) macroscopic black hole is an extremely long geometrical entity. It is radially shrinking, but growing in length. The transition is a bounce that reverses the process: the interior of a white hole is expanding radially, and shortening. See Figure 14. able to cross the zone forbidden by Einstein's equationsthe gray zone in the figures above- and jump, by tunnel effect, 'to the other side.' The quantum properties of space and time allow the inside of the black hole to 'leap' beyond the singularity, when classical equations would have time stop. \nFigure 14. A sketch of evolution of the internal geometry of the hole at the bounce (up to some quantum gravity phenomena, see later). \n<!-- image --> \nWhite\\_9780593545447\\_all\\_2p\\_r1.indd 74 \n7/10/23 \n11:16 AM", 'C. Remnants and their lifetime': "The argument of the previous section suggests that at the end of the evaporation a black hole undergoes a quantum transition to a white hole with a Planckian-size horizon and a vast interior. \nThe possibility of remnants with a similar structure was considered in the 1990's [83]. What was not realized \nat that time is that classical GR does in fact predict the existence of objects with these properties: white holes; and quantum theory could account for their formation at the end of the evaporation and (as we shall see below) for their stability. \nBefore studying in more detail the quantum structure of these remnants, let us ask what could be their lifetime. \nA number of arguments suggest that this should be long. First, the Einstein equations state that it takes a long (advanced) time to generate a black hole with a large interior. Time reversing, they also state that it takes a long (retarded) time to dissipate a white hole with a large interior. \nThe second argument concerns the information trapped inside the hole. The Hawking radiation is genuinely thermal: it is describe by a quantum state which is not pure: it is a density matrix. This is because the Hawking radiation that escapes to infinity forms together with negative radiation that falls inside the hole. The two are entangled, therefore the part escaping to infinity has von Newman entropy. The total entropy of the Hawking radiation is of the order of the Bekenstein-Hawking entropy \nS BG = A 4 . (30) \nwhere A ∼ m 2 o is the initial area of the black hole, before the beginning of the evaporation. Some scientists expect that the total entropy of the Hawking radiation starts decreasing at Page time because late Hawking quanta are correlated with early ones. We are not convinced by this expectation for reasons that we explain in detail later on. Hence an amount of information of the order S BH is trapped inside the hole during the quantum transition. This must be emitted later in the form or radiation by the remnant. The remnant has total energy of the order of the Planck mass, namely of order unity in natural units. It must therefore be able to emit a large amount of information by emitting very small energy. \nTo emit a lot of information within small energy, we need a large number of very low-energy quanta. These must have very low frequency, and hence require a long time. As shown in [84-86] (and with an argument detailed in Section IV E) this leads to a lower bound in emission time: \nτ WH ∼ m 4 o . (31) \nHere m o is the mass of the hole before the evaporation (which determines the amount of information in the hole). \nNotice that the state of the black hole at some given (advanced) time, which determines its full future evolution, is not specified only by its current mass m , but also by the internal geometry, which in turn is determined by the initial mass of the black hole. \nWe can account for this by writing the quantum state of the black hole at some given time in the form \| m o , m ⟩ B , \nwhere the first quantum number is the initial mass, which determines the size of the interior, while the second is the current mass which determines the area A = 16 πm 2 of the horizon on the given time slice and decreases in the evaporation. We do not need to keep track of other possible quantum numbers here. \nAt formation, the hole is in the state \| m o , m o ⟩ B , \nthen m decreases by Hawking evaporation until the state \| m o , m Pℓ ⟩ B . This states tunnels to a white hole state with the same quantum numbers, which we denote \| m o , m Pℓ ⟩ W . The tunnelling process itself from black to white is short and takes a time of the order of the current mass [20, 87]. Here is therefore a first account (which we correct below) of the full life cycle of a gravitationaly collapsed object \n-----→ collapse \| m o , m o ⟩ B τ WH ∼ m 3 o ------→ black hole \| m o , m Pℓ ⟩ B τ T ∼ m Pℓ ------→ tunnelling \| m o , m Pℓ ⟩ W τ WH ∼ m 4 o ------→ white hole \| m Pℓ , m Pℓ ⟩ W ---------→ full dissipation . (32) \nThese are time scales for a distant observer. If m o is large, the process is very long in the time of a distant observer. But it is extremely short (of order m o , which is the time light takes to cross the radius of the star) if measured on the bouncing matter distribution itself. The huge difference is due to the extreme gravitational time dilation [2]. Time slows down near high density mass. An observer (if capable of resisting the tidal forces) landing on a Planck matter distribution will find herself nearly immediately in the distant future, at the time where the black hole ends its evaporation. The proper lifetime of a Planck matter distribution is short: from its own perspective, the matter distribution is essentially a bounce. A black hole is a shortcut to the distant future. \nThis is a compelling scenario, but it still needs to be corrected.", 'D. Instability': "Aclassic macroscopic white hole is an unstable solution of the Einstein equations (see Chapter 15 in [88] and references therein). This means that there are solutions with initial data that are arbitrarily close to the white hole initial data, but have a qualitatively different future. \nThe qualitative different future is the formation of a black hole in the future of the white hole. That is, a white hole is unstable toward becoming a black hole. The transformation of a white hole into a black hole is similar to the process described by the vacuum Kruskal spacetime. \nTo see why there is this instability it is easier to address the time reversed scenario first, because we have a better intuition about it: to show that solutions with final data arbitrarily close to those of a black hole, must have a qualitatively different past from the black hole. Or in other words, that the data on future null infinity on a black hole spacetime cannot admit certain arbitrary small variations. \nTo see this, consider a black hole of mass m , formed by a collapsed star. In the unperturbed solution, no energy reaches future null infinity (here we are considering classical physics: no Hawking radiation). Now, consider a perturbation of this spacetime in which there is a small spherically symmetric pulse of null radiation with total \nenergy ϵ arriving at null infinity. Here ϵ can be arbitrarily small. In order to arrive at null infinity, this radiation must be emitted by the surface of the matter distribution before this enters the horizon. Say it was emitted when the radius of the matter distribution was 2 m + δ . But this must also be outside the Schwarzschild radius of the matter distribution plus the energy to be emitted, otherwise it could not reach null infinity either. Hence necessarily 2 m + δ > 2 m + ϵ that is, we must have δ > ϵ . Now let u δ be the retarded time of the emission point. Necessarily, the energy pulse of energy ϵ must reach null infinity before u δ . Therefore, no perturbation with energy ϵ can reach null infinity after u δ . That is, we cannot perturb the final data arbitrarily. However small is ϵ there are locations where it cannot get to. \nCan a perturbation with energy ϵ reach null infinity after u δ , with some different past? Yes it does! A white hole of mass 2 m + ϵ can emit the pulse and then become a black hole of mass m ! In this case, the pulse is never at a radius where it can trigger an increase of the size of the blackhole. \nTime reversing this scenario shows immediately why a classical eternal white hole of mass m is unstable: any arbitrary small pulse originating from past null infinity sufficiently early in time will bring enough energy to be inside the Schwarzschild radius before reaching the matter distribution emerging from the white hole, thus triggering the formation of a black hole. \nIn other words, the spacetime depicted in the Center panel of Figure 13 does not change much under a small arbitrary modification of its initial conditions on past null infinity; but it is drastically modified if we modify its final conditions on future null infinity. This is intuitively simple to grasp: if we sit on future null infinity and look back towards the hole, we see a black disk. This is the final condition. A slightly perturbed final condition includes the possibility of seeing radiation arriving from this disk. This is impossible in the spacetime of the Center panel of Figure 13, because of the huge red shift of the radiation moving next to the horizon, but it is possible in the Left spacetime, because the radiation may have crossed over from the other asymptotic region. \nThe same is true for a white hole, reversing the time direction. In the spacetime depicted in the Right panel, with some radiation, there is necessarily a dark spot in the incoming radiation from past null infinity. If we per- \nturb this configuration, and add some incoming radiation to this dark spot, the evolution generically gives the spacetime of the Left panel. \nPhysically, what happens is that this radiation moves along the horizon, is blue shifted, can meet radiation coming out of the white hole and this is more mass than m at a radius r ∼ 2 m : it is mass inside its Schwarzschild radius. At this point the region is trapped, and a black hole forms. Consequently the evolution of the perturbed initial conditions yields the spacetime of the Left, not the one of the Right: the white hole is unstable and decays into a black hole. This is the standard 'instability of white holes' in classical GR.", 'E. Planckian Remnants': "How does the instability discussed in the last paragraph affect the remnants formed at the end of a black hole evaporation? \nAs observed in [8], the wavelength of the perturbation needed to trigger the instability must be smaller than the size of the hole. To make a Planck size black hole unstable, therefore, we would need trans-Planckian radiation, and this is likely not be allowed by quantum gravity. \nBut let's nevertheless assume that a Planck-scale white hole is unstable. Its decay mode, as explained above, is into a Planck-scale black hole. Using the notation introduced above, this is the process \n\| m o , m ⟩ W ------→ instability \| m o , m ⟩ B . (33) \nThe quantum gravity process discussed earlier in the paper is \n\| m o , m ⟩ B ------→ tunnelling \| m o , m ⟩ W . (34) \nIn quantum mechanics, two such processes imply that a system free to lose energy into radiation will settle on its lowest available energy eigen-state, which will be a superposition of the two states, of the form \n\| m o , m ⟩ = α \| m o , m ⟩ W + β \| m o , m ⟩ B . (35) \nThat is, the remnant will settle into a superposition of black and white hole. 2 \nSince the black and white holes are indistinguishable from the exterior, this superposition has no effect on the exterior. It is only a quantum spread of the geometry on and just around the horizon. In the next section we describe it more precisely. \nMinimal energy, for the horizon, means minimal area. However, vanishing area means flat space, namely total dissipation. The transition to this state is likely to be strongly suppressed by the arguments in the last section. Intuitively, this is because it takes time to re-absorb the huge interior, and its entropic content. Hence, the system will settle and remain on the lowest nonvanishing eigenstate of the area for a long time. \nLQG predicts the area to be quantized [91, 92]. Its eigenvalues are \nA min = 8 πγ ℏ G c 3 √ j ( j +1) . (36) \nThe lowest non vanishing eigenvalue, or 'area gap', is given by j = 1 2 . Given the relation between area and mass of the horizon, this gives the mass (1). \nThat is, LQG predicts that black holes end up as longliving particles with a mass of a few micro-grams. \nOnce a remnant has attained this minimal size, it is semi-stable. It can still radiate away its energy, and dissipate into flat space. But this last transition \n\| m o , m P ⟩ → \| 0 ⟩ (37) \nis strongly suppressed and therefore takes a long time, at least of order m 4 o . \nThe existence of these objects contradicts the intuition of many particle-physics or string string theorists. This is because in flat space physics, a small volume with a small energy can contain only a finite number of different states. This is what prevents the blackbody UV catastrophe. Intuitively, to have many states we need many particles, and to have many particles with small total energy we need to have them with long wavelength, but this is not available in the volume contained within a small sphere, because this has a small volume. This is the basis of the intuition of many particle-physics and string theorists. \nIn general-relativistic physics, however, these constraints do not hold, because on a curved geometry an arbitrarily large volume can be enclosed into an arbitrarily small sphere. Hence an arbitrarily small horizon can enclose any number of different states, limited only by the maximal size m o of the parent black hole and by the finiteness of the time during which the black hole interior has had the opportunity to grow.", 'F. Remnants and quantum spread of the extrinsic curvature': 'Let us look more closely at the physics of the remnants. Consider the geometry of a black hole in coordinates where (unlike Schwarzschild) t = constant surfaces properly cut the horizon at different moments. In Lemaitre time, for instance, \nds 2 = -dt 2 +( dr + √ 2 m/rdt ) 2 + r 2 d Ω 2 . (38) \nThe horizon r = 2 m has area A = 16 πm 2 . The variable conjugate to the area is the extrinsic curvature, which has a single (radial-radial) component k = a/m 2 , where a here is a numerical constant. In quantum theory, A and k are conjugate variables and cannot both be sharp. We expect a Heisenberg relation of the form \n∆ A ∆ k > ℏ G. (39) \nClassical black holes are described by semiclassical quantum states that are peaked on large values of A and small values of k , with ∆ A and ∆ k satisfying the Heisenberg relation but all relative spreads being small. Now consider an isolated black hole that radiates away all its energy, but is stuck on the minimal non-vanishing area because of the suppression of the transition to Minkowski due to its large interior. Then the system settles on an eigenstate of A . This is the common situation of all quantum systems, left alone and free to equilibrate by radiating away energy: they settle on their minimal reachable energy eigenstate. An eigenstate of the area will have k maximally spread. Therefore we expect Planck scale remnants to be in states where k is maximally spread. Now consider a white hole. Its metric is easily obtained time reversing the black hole metric, that is \nds 2 = -dt 2 +( dr -√ 2 m/rdt ) 2 + r 2 d Ω 2 . (40) \nThe area of the horizon is the same as the black hole, but k has opposite sign. An eigenstate of A , having k maximally spread does not distinguish between a black hole and a white hole. Hence remnants must be superpositions between the two.', 'G. There is no information paradox': "We close this part of the paper by discussing the vexata questio of the black hole information paradox, in light of the above results. This Section follows closely [38] and [93]. \nThe black hole information problem, as formulated by Don Page in 1993 [94], regards the physics of the spacetime region before the quantum gravity onset. Figure 15 is the Carter-Penrose diagram of a spherically symmetric spacetime geometry around a collapsing star, on which there is an evolving quantum field ϕ . The geometry takes into account the back-reaction of the Hawking radiation of the field. The collapsed matter distribution generates a trapped region: the black hole. The boundary of this region is the (trapping) horizon. In the limit in which we disregard the back-reaction of the Hawking radiation, the horizon is null, but taking back-reaction into account, it is time-like. The notion of event horizon is not defined, because no assumption is made about the distant future, which depends on quantum gravity and is not relevant. Considerations on this region alone are sufficient for Page's argument for the information problem. \nHere is Page's key observation. Consider a sphere S on the horizon at retarded time u (see Figure 15). The \nregion of future null infinity preceding the time u receives the Hawking radiation emitted until the horizon has reached S . Consider the case where the black hole is 'old' at S , namely the area A of S is much smaller than the the initial horizon area A 0 at S 0 . The Hawking radiation arriving at future infinity around u is in a mixed state. If the initial state of the field was pure and evolution is unitary, this radiation must be correlated with something else: with what? \nThere are two reasonable possibilities: \n(a) it is correlated with degrees of freedom inside the horizon; \n(b) it is correlated with degrees of freedom outside the horizon. In particular, late Hawking quanta may be correlated with early Hawking quanta. \nA part of the theoretical community has got convinced that the correct answer must be (b) because (a) is ruled out by the following argument by Don Page, based on a specific assumption. Let N be the number of states of a black hole with which external degrees of freedom can be entangled at some given time and let \nN BH = e A/ 4 . (41) \nwhere A is the area of the horizon at that time. Consider the following \nAssumption : N ∼ N BH . (42) \nThen, necessarily (a) is ruled out, because for sufficiently late times A is small and therefore N is insufficient for purifying the Hawking radiation. \nBut is the assumption justified? \nThe thermodynamic interaction between a black hole and its surroundings is well described by treating the black hole as a system with N BH = e A/ 4 (orthogonal) states, where A is the horizon area. But a system can have many more degrees of freedom than those determining its thermodynamic interactions. For instance, this is the case when some degrees of freedom are physically constrained from acting back on the exterior, as is precisely the case for a black hole. It is only for ergodic systems that the von Neumann entangled entropy (which purifies the external state) must be equal or smaller than the thermodynamic entropy, and certainly a black hole is not ergodic. So, there is definitely no compelling reason based on known physics for assuming (42). \nSome theoretical hypotheses, like AdS-CFT or some String formulations elevate (42) into a postulate. Juan Maldacena has called this assumption a 'central dogma', and, like all dogmas, it can well be false. As many dogmas, this one as well leads to bizarre consequences, like firewalls on the horizon [39, 85]. \nThe possibility that N ≫ N BH is supported by the fact that according to classical GR the interaction between a black hole and its surroundings is entirely determined by what happens in the vicinity of the horizon, not by the interior. The number N BH is likely to count only horizon states that (before the transition) can be distinguishable from the exterior. These are 'surface' states. On \nFigure 15. The portion of spacetime relevant for the information 'paradox'. \n<!-- image --> \nthe other hand, N counts also states that can be distinguished by local observables inside the horizon. These do not contribute to the thermodynamical entropy S = A/ 4, but can contribute to the von Neumann entropy, which can therefore remain high even when A shrinks. \nThese generic considerations can be sharpened. The fact that a blackhole can have more states than N BH , in fact, follows from elementary considerations of causality. \nTo show this, consider a gravitationally collapsed object and let Σ 1 be a Cauchy surface that crosses the horizon but does not hit the singularity, see Figure 16. Let Σ 2 be a later similar Cauchy surface and i = 1 , 2. Let A i be the area of the intersection of Σ i with the horizon. Assume that no positive energy falls into the horizon during \nFigure 16. The (lowest part) of the conformal diagram of a gravitational collapse. The clear grey region is the object, the dotted line is the horizon, the thick upper line is the singularity, the dark upper region is where quantum gravity effects may become relevant (this region plays no role in this paper.) The two Cauchy surfaces used in the paper are the dashed lines. \n<!-- image --> \nthe interval between the two surfaces. Let quantum fields live on this geometry, back-reacting on it [95]. Finally, let Σ in i be the (open) portions of Σ i inside the horizon. \nCare is required in specifying what is meant here by 'horizon', since there are several such notions (event horizon, trapping horizon, apparent horizon, dynamical horizon...) which in this context may give tiny differences in location. For precision, by 'horizon' we mean here the event horizon, if this exists. If it doesn't, we mean the boundary of the past of a late-time spacelike region lying outside the black hole (say outside the trapping region). With this definition, the horizon is light-like. \nBecause of the back-reaction of the Hawking radiation, the area of the horizon shrinks and therefore \nA 2 < A 1 . (43) \nNow consider the evolution of the quantum fields from Σ 1 to Σ 2 . We are in a region far away from the singularity and therefore (assuming the black hole is large) from high curvature. Therefore we expect conventional quantum field theory to hold here, without strange quantum gravity effects, at least up to high energy scales. Since the horizon is light-like, Σ in 1 is in the causal past of Σ in 2 . This implies that any local observable on Σ in 1 is fully determined by observables on Σ in 2 . That is, if A i is the local algebra of observables on Σ in i then A 1 is a subalgebra of A 2 : \nA 1 ⊂ A 2 . (44) \nTherefore any state on A 2 is also a state on A 1 and if two such states can be distinguished by observables in A 1 they certainly can be distinguished by observables in A 2 as the former are included in the latter. Therefore the states that can be distinguished by A 1 -which is to say: on Σ in 1 - can also be distinguished by A 2 -which is to say: on Σ in 2 . Therefore the distinguishable states on Σ in 1 are a subset of those in Σ in 2 . How many are they? Either there is an infinite number of them, or a finite number due to some high-energy (say Planckian) cut-off. If there is an infinite number of them, then immediately the number of states distinguishable from inside the black hole is larger that N NB , which is finite. If there is a finite number of them, then the number N 2 of distinguishable states on Σ in 2 must be equal or larger than the number N 2 of states distinguishable on Σ in 1 , because the second is a subset of the first. That is \nN 2 ≥ N 1 . (45) \nComparing equations (43) and (45) shows immediately that it is impossible that N i = e A i / 4 , as the exponential is a monotonic function. \nThis follows from only assuming the validity of quantum field theory in regions of low curvature. In other words, the 'dogma' (42) necessarily violates known physics in a region where we have no reasons to assume that it should fail. In our opinion, this is not good scientific method. \nThe conclusion is that the number of states distinguishable from the interior of the black hole must be different from the number N BH = e A/ 4 of the states contributing to the Bekenstein-Hawking entropy. Since the second is shrinking to zero with the evaporation, the first must overcome the second at some point. Therefore in the interior of a black hole there are more possible states than e A/ 4 . \nThe physical interpretation of the conclusion is simple: the thermal behavior of the black hole described by the Bekenstein-Hawking entropy S = A/ 4 is determined by the physics of the vicinity of the horizon. \nA vivid manifestation of the fact that in classical GR the effect of a black hole on its surroundings is independent of the black hole interior is in the numerical simulations of black hole merging and radiation emission by oscillating black holes: in writing the numerical code, it is routine to cut away a region inside the (trapping) horizon: it is irrelevant for whatever happens outside! This is true in classical GR, and there is no compelling reason to suppose it to fail if quantum fields are around. \nThe idea of interpreting of S BH as determined by the number of states of near-surface degrees of freedom, and not interior ones is of course not a new idea. It has a long history [96-102]. See in particular [103], [104] in support of this idea from two different research camps, loops and strings. \nImportantly, this conclusion is not in contrast with the various arguments leading to identify BekensteinHawking entropy with a counting of states. To the opposite, evidence from it comes from the membrane paradigm [105] and from Loop Quantum Gravity [103, 106-108], which both show explicitly that the relevant states are surface states, but also from the string theory counting [109, 110], because the counting is in a context where the relevant state space is identified with the scattering state space, which could be blind to interior observables. For a classic discussion on different viewpoints about these alternatives, see [111] \nIn conclusion, if there are more states available in a black hole than e A/ 4 , then the Page argument for the information loss paradox fails. States purifying the Hawking radiation are inside the hole even when the horizon shrinks. The information can later escape from the remnant emission (see below). \nNotice that in popular accounts (for a recent one, see for instance [112]), three assumptions are erroneously said to be proven incompatible: (i) unitarity of the quantum evolution, (ii) equivalence principle (absence of firewalls), and (iii) quantum field theory on curved spacetimes. This is wrong. It is only the further assumption of the dogma, that leads to problems.", 'IV. ELEMENTS OF PHENOMENOLOGY': 'Measuring quantum gravity effects is notoriously difficult [113]. Still, some intriguing possibilities of obser- \nvation are opened by the scenario described above. The first is the possibility that the remnants described above are the constituents of, or contribute to, dark matter [17]. In particular, remnants forming dark matter could be produced by primordial (or pre-big bang) black holes. Direct detection of these remnants has been studied in [16]. A number of possible astrophysical implications of this scenario or variants have been tentatively explored in [17, 45, 114-120].', 'A. Dark Matter': 'Remnants are a dark matter candidate that does not require exotic assumptions of new forces, or particles or corrections to the Einstein equations, or physics beyond the standard model. It only requires general relativity and quantum theory to hold together. \nThe possibility that remnants of evaporated black holes of primordial origin could form a component of dark matter was suggested by MacGibbon [121] thirty years ago and has been explored by many authors [122132]. Since there are no strong observational constraints on this potential contribution to dark matter [133], the weak point of the scenario has been, until now, the question of the physical nature of the remnants. The scenario discussed in these notes has changed the picture: conventional physics provides a candidate for remnants.', 'B. Direct detection': "Alocal dark matter density of the order of 0 . 01 M ⊙ /pc 3 corresponds to approximately one Planck-scale white hole per each 10 . 000 Km 3 . These objects are presumably moving fast with respect to our local frame, since we are rotating with the galaxy at hundreds of Km per second, while dark matter models suggest that it isn't. This gives a very rough estimates of a few Planck scale \nFigure 17. A particle of mass m in a superposition state with separation ϵ . The DM particle passes by with velocity v and a closest approach distance d . \n<!-- image --> \n. \nparticles per m 2 per year flying by us. Can we detect them? \nIf the sole interaction of remnants is gravitational, they are very good dark matter candidates, but for this same reason direct detection using classical sensing is challenging [133], due to the extreme weakness of the gravitational interaction. However, there may be a quantum technology that could open a window to do the detection. In fact, recent developments in the area of table-top experiments involving gravity and quantum phenomena (see for instance [134] for up-do-date references) open the theoretical possibility of direct detection of purelygravitationally-interacting dark matter particles. An idealized detector where the center of detector mass is set in a superposition of locations and a more concrete tentative protocol, which employs Josephson junctions, have been illustrated [16]. \nFigure 17 illustrates the conceptual setting. The dark matter particle flies by the device, passing closer to one of the two locations in which a quantum particle is quantum split. The different momentum transfer determines a phase difference between the two branches, that can be picked up by interferometry, or as a current change in an array of Josephson junctions. See [16] for details.", 'C. Cosmological implications': 'Remnants could be produced by primordial black holes formed in the early universe. The lower bound of the lifetime of the remnants is of the order of or lower than m 4 o . If we assume that lifetime is precisely of the order of the lower bound m 4 o (as mentioned, this may not be correct), then for these objects to still be present now we need their lifetime to be larger or equal than the Hubble time T H , that is \nm 4 o ≥ T H . (46) \nOn the other hand, we expect these to be produced by evaporated black holes, therefore the lifetime of the black hole must be shorter than the Hubble time. Therefore \nm 3 o < T H . (47) \nThis gives an estimate on the possible value of m 0 : \n10 10 gr ≤ m o < 10 15 gr. (48) \nThese are the masses of primordial black holes that could have given origin to dark matter present today in the form of remnants. Their Schwarzschild radius is in the range \n10 -18 cm ≤ R o < 10 -13 cm. (49) \nAccording to primordial black hole formation theory, black holes of a given mass could have formed when their Schwarzschild radius was of the order of the horizon. Remarkably, the horizon was presumably in this range at \nthe end of inflation, during or just after reheating. A preliminary phenomenological analysis of this particular scenario was carried out in [43]. \nIf the lifetime can be longer, constraints are less stringent. Furthermore, an intriguing possibility opens up. Remarkably, the strength of the interaction of such particles, combined with the assumption of a sufficiently hot big bang, leads to a density of these objects at decoupling whose order of magnitude is compatible with the present dark matter density [135, 136].', 'D. Erebons': 'An alternative possibility for the generation of remnants is that they were formed in a contracting phase before the current expanding one, in a big bounce scenario (for a review of classical and quantum bouncing cosmologies, see [137, 138] and references therein). \nThe possibility that black holes could live across the big bounce and represent a component of dark matter has been considered in [139]. Roger Penrose has coined the name erebons , from the Greek god of darkness Erebos, to refer to matter crossing over from one eon to the successive one [140] in his cyclic cosmology [141]. Large black holes evaporated before the bounce could have given rise to a population of Planck-size white hole remnants that has crossed the bounce and formed what we see today as dark matter. For this to happen, their density should have been sufficient to balance the huge dilution in an eventual inflationary phase. \nAn intriguing aspect of this scenario is the speculative idea that it might address the apparent low-entropy of the initial cosmological state [142]. This can be consistent and can represent a concrete realization of the perspectival interpretation of entropy suggested in [143]. If the cosmos at the big bounce was finely dotted by white holes with large interiors, then the gravitational field was not in the very improbable low entropy homogeneous or nearly homogeneous configuration. It was in a high entropy crumpled configuration. But being outside all white holes, we are in a special place, and from the special perspective of this place we see the universe under a coarse grain which defines an entropy that was low in the past.', 'E. Modeling remnants emission': "Finally, remnants must themselves emit, although very softly, in order to release the information that was trapped with the in-falling Hawking radiation and dissipate (as white holes do). \nClassically, this a very low power emission. Quantum mechanically, it is given by a weak decay probability, akin to radioactivity. This process was tentatively modelled in [86], which we summarize here. \nSay a black hole of initial mass m o evaporates via Hawking evaporation, leaving a remnant of Planckian mass which contains an amount of information sufficient to purify its Hawking radiation, namely of order \nS ∼ A 4 = 4 πm 2 o (50) \nHere A is the area of the horizon at formation . Information is emitted in the form of radiation. Since the radiation is emitted radially, it can be modelled as a uniform one-dimensional gas of photons. Assume for simplicity that the radiation emitted by the surface of the remnant during the lifetime τ , is in thermal equilibrium [85]. At the end of the remnant lifetime the radiation covers a length L = τ . The energy E available for this gas is only that of the mass of the remnant, which is of the order of the Planck mass, namely unity in natural units. \nE ∼ 1 , (51) \nwhile its total entropy, needed to purify the Hawking radiation is (50). Entropy S and energy E of a 1d photon gas of temperature T in a space of length L are [144] \nS = 2 π 3 LT, E = 1 6 LT 2 , (52) \nwhich gives \nL = 6 m 4 , T = 1 m 2 , (53) \nand a number of quanta (say photons) emitted \nN γ ∼ m 2 . (54) \nThe lifetime of a white hole would be equal to the time required for the photons to travel a distant L . We therefore have \nτ W ∼ 6 m 4 (55) \nas already mentioned, while the temperature of the white hole emission is be much lower than the initial temperature ∼ 1 /m of the Hawking temperature of the parent black hole. \nA population of black holes formed at a time t = 0, with mass m and uniformly distributed in space that evaporate around time τ B ∼ m 3 as predicted by Hawking radiation theory, and survive as white hole remnants for a time τ W as in (55) emits a steady radiation between times τ B and τ B + τ W . For τ B < t < τ B + τ W an observer would observe that the radiation density would evolve in time as \nρ ( t ) = 0 for t < m 3 , = ( t -τ B τ W -τ B ) Ω for m 3 < t < 6 m 4 , = Ω for t > 6 m 4 . (56) \nIn other words, the emission process is a steady (linear in time) transformation of dust into radiation, on a m 4 timescale. See Figure 18. \nFigure 18. Background white hole radiation as a function of time. The solid black line represents a classical linear emission while the dashed red line represents a quantum emission. \n<!-- image --> \nQuantum mechanics alters this picture a bit. A continuous energy emission as the one described above would imply a continuous decrease of the white hole horizon area, below the Planck area. But this is not permitted by LQG, because area is quantized. \nRather, a remnant with near-Planckian mass and area must make a single quantum leap into radiation, as in conventional nuclear radioactivity, where emission is realized by individual quantized quanta governed by a probability distribution [145]. \nIn other words, in first order perturbation theory the only allowed transition is the emission of the entire Planck energy of the remnant. \nIn the language of quantum field theory, this is given by a vertex between the remnant and a large number of low energy quanta. A vertex coupling a remnant to a single or a few photons, indeed, is forbidden by conservation of information (unitarity), because a few photons do not have enough degrees of freedom to match the large number of quantum numbers describing the white hole interior. Few photons cannot carry the entire information that can be stored in the remnant. Hence the transition must be of the form remnant → γ 1 ...γ n to a large number of low energy photons: \n<!-- image --> \nThe number of photons emitted by a single remnant is given in (54). This conclusion is relevant in view of an old objection to the remnant scenario, that contributed to its abandonment in the Nineties. The objection was that the large number of remnant internal states would make them too easy to produce in particle physics experiments. Here we see why that conclusion was too quick. The effective vertex responsible for a remnant production would be \n<!-- image --> \nin order to create a long living Planck size remnant. If the number of photons was small, these could be high energy, but the remnant produced would correspond to a remnant whose parent is a black hole of Planckian size, but short lived. The process would not be distinguishable by the standard collapse predicted by conventional quantum gravity. To produce an actual long living remnant, on the other hand, we need m to be large, and hence we would need to focus a large number of low energy photons . \nIn the cosmological standard model, primordial black holes may have formed at reheating. To get a sense of the characteristic of the diffuse radiation that remnants may emit, one can estimate its parameter in the simplest case [86]. This model is entirely determined by a single parameter of order of unity, that can be taken to be \nx = log 10 ( m/m Pl ) ∈ [15 , 20] . 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2024arXiv240910397S	The hot phase of the circumgalactic medium CGM allows us to probe the inflow and outflow of gas within a galaxy which is responsible for dictating the evolution of the galaxy. Studying the hot CGM sheds light on a better understanding of gas physics which is crucial to inform and constrain simulation models. With the recent advances in observational measurements probing the hot CGM in Xrays and tSZ we have a new avenue for widening our knowledge of gas physics and feedback by exploiting the information from currentfuture observations. In this paper we use the TNG300 hydrodynamical simulations to build a fully selfconsistent forward model for the hot CGM. We construct a lightcone and generate mock Xray observations. We quantify the projection effects namely the locally correlated largescale structure in Xrays and the effect due to satellite galaxies misclassified as centrals which affects the measured hot CGM galactocentric profiles in stacking experiments. We present an analytical model that describes the intrinsic Xray surface brightness profile across the stellar and halo mass bins. The increasing stellar mass bins result in decreasing values of beta the exponent quantifying the slope of the intrinsic galactocentric profiles. We carry forward the current stateoftheart by also showing the impact of the locally correlated environment on the measured Xray surface brightness profiles. We also present for the first time the effect of misclassified centrals in stacking experiments for three stellar mass bins 1010.511 Modot 101111.2 Modot and 1011.211.5 Modot. We find that the contaminating effect of the misclassified centrals on the stacked profiles increases when the stellar mass decreases.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.10397', '2024arXiv240910397S', 'arXiv:2409.10397']	['Astrophysics - Astrophysics of Galaxies', 'Astrophysics - High Energy Astrophysical Phenomena']	Quantifying Observational Projection Effects with a Simulationbased hot CGM model	2,024	196	0.52	['EPRINT_HTML', 'EPRINT_PDF']	2	https://arxiv.org/pdf/2409.10397.pdf	{'Quantifying Observational Projection Effects with a Simulation-based hot CGM model': 'Soumya Shreeram 1 ⋆ , Johan Comparat 1 , Andrea Merloni 1 , Yi Zhang 1 , Gabriele Ponti 1 , 2 , Kirpal Nandra 1 , John ZuHone 3 , Ilaria Marini 4 , Stephan Vladutescu-Zopp 5 , Paola Popesso 3 , Ruediger Pakmor 6 , Riccardo Seppi 7 , Celine Peroux 3 , and Daniele Sorini 8 \n- 1 Max Planck Institute for Extraterrestrial Physics (MPE), Gießenbachstraße 1, 85748 Garching, Munich, Germany\n- 2 INAF-Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate (LC), Italy\n- 3 Smithsonian Astrophysical Observatory, Observatory Building E, 60 Garden St, Cambridge, MA 02138, United States\n- 4 European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching, Munich, Germany\n- 5 Universitäts-Sternwarte München, Scheinerstraße 1, 81679 München, Germany\n- 6 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Straße 1, 85748 Garching, Munich, Germany\n- 7 University of Geneva, 1205 Geneva, Switzerland\n- 8 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham, DH1 3LE, United Kingdom \nReceived ZZZZ, ZZZZ; accepted XXXX, XXXX; published YYYY, YYYY', 'ABSTRACT': 'Aims. The hot phase of the circumgalactic medium (CGM) allows us to probe the inflow and outflow of gas within a galaxy, which is responsible for dictating the evolution of the galaxy. Studying the hot CGM sheds light on a better understanding of gas physics, which is crucial to inform and constrain simulation models. With the recent advances in observational measurements probing the hot CGM in X-rays and tSZ, we have a new avenue for widening our knowledge of gas physics and feedback by exploiting the information from current / future observations. \nMethods. In this paper, we use the TNG300 hydrodynamical simulations to build a fully self-consistent forward model for the hot CGM. We construct a lightcone and generate mock X-ray observations. We quantify the projection e ff ects, namely the locally correlated large-scale structure in X-rays and the e ff ect due to satellite galaxies misclassified as centrals, which a ff ect the measured hot CGM galactocentric profiles in stacking experiments. \nResults. We present an analytical model that describes the intrinsic X-ray surface brightness profile across the stellar and halo mass bins. The increasing stellar mass bins result in decreasing values of β , the exponent quantifying the slope of the intrinsic galactocentric profiles. We carry forward the current state-of-the-art by also showing the impact of the locally correlated environment on the measured X-ray surface brightness profiles. We also present, for the first time, the e ff ect of misclassified centrals in stacking experiments for three stellar mass bins: 10 10 . 5 -11 M ⊙ , 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ . We find that the contaminating e ff ect of the misclassified centrals on the stacked profiles increases when the stellar mass decreases. When stacking galaxies of Milky-Way-like stellar mass, this e ff ect is non-negligible already at a low level of contamination: in particular, misclassified centrals contributing 30%, 10%, or 1% of a sample dominate the measured surface brightness profile at radii ≥ 0 . 06 × R 200 m , ≥ 0 . 2 × R 200 m , and ≥ 0 . 3 × R 200 m , respectively. \nKey words. Hot circumgalactic medium - galaxies: evolution - methods: numerical', '1. Introduction': "The Circumgalactic medium (CGM) plays a crucial role at interfacing the gas between the interstellar medium (ISM) within galaxies and the external intergalactic medium (IGM) outside them. The CGM is commonly defined as the gas within the virial radii of the galaxy and outside their galactic disks. It is the gravitationally bound gas that encompasses the fossil imprints of the physical mechanisms, such as outflows, inflows, and feedback processes, dictating the evolution of the galaxy (see Tumlinson et al. 2017 for a review). These mechanisms are sensitive to the galaxy's environment and its halo properties, as shown by the complex Stellar-to-Halo-Mass-Relation (SHMR); see the review from Wechsler & Tinker (2018). According to most models, the SHMR indicates that the low-halo-mass end is sensitive \nto stellar and supernovae (SN) driven feedback (Dekel & Woo 2003; Benson et al. 2003), causing heating of the gas, hot bubbles (McKee & Ostriker 1977), galactic wind outflows (Dekel & Silk 1986), and turbulence (Ostriker & Shetty 2011; Strickland & Heckman 2009). In contrast, Active Galactic Nuclei (AGN) should be the main drivers of feedback in the high-halo-mass end (Silk & Rees 1998; Fabian 2012; Eckert et al. 2021). Of particular interest is the peak of the SHMR relation at the pivotal halo mass M h ∼ 10 12 M ⊙ , similar to our Milky-Way (MW) mass, corresponding to the mass scale where star formation e ffi ciency reaches its maximum. Thus, studying the volume-filling hot gas component of the CGM, especially in the MW halo mass regime, encapsulating a range of physical processes, is crucial for testing galaxy formation models (Faucher-Giguère & Oh 2023). \nThe most general scenario for the presence of such hot CGM assumes that infalling gas within halos ≳ 10 12 M ⊙ is shock heated \nup to the virial temperature, T vir ≳ 10 6 K, resulting in X-ray emission (White & Rees 1978). This hot gas is probed with the thermal Sunyaev-Zeldovich (Sunyaev & Zeldovich 1972, tSZ) e ff ect at mm-wavelengths (Lim et al. 2021; Das et al. 2023; Oren et al. 2024) and in X-rays via absorption and emission studies. X-ray absorption studies use sightlines with bright background sources to probe the hot CGM via absorption lines like OVI or OVII K α (Galeazzi et al. 2007; Bhattacharyya et al. 2023; Mathur et al. 2023). The major challenge in this technique is constructing statistical samples and mapping the largescale extent of the CGM, which will improve with future X-ray telescopes with microcalorimeters (Wijers et al. 2020; Bogdán et al. 2023). On the emission side, studies use narrow-band and broad-band observations. The narrow-band emission comprising of OVII and OVIII metal lines dominates the lower halo mass regime 10 12 -10 13 M ⊙ (Bertone et al. 2010; van de Voort 2013; Wijers 2022) and is well-studied within our MW (Zheng et al. 2024; Locatelli et al. 2024; Ponti et al. 2023; Koutroumpa et al. 2007). However, for extragalactic studies, future high-spectral resolution instruments are required to distinguish the extragalactic emission from that of the MW foreground (Nelson et al. 2023; Truong et al. 2023; Schellenberger et al. 2024; ZuHone et al. 2024). \nMultiple extragalactic single-object broad-band X-ray emission studies have been published (Bogdán et al. 2013a,b; Anderson et al. 2016; Bogdán et al. 2017; Li et al. 2017; Das et al. 2019). However, due to the X-ray emissivity scaling with the square of the density, detections are limited to the densest parts of the CGM, within 40 -150 kpc from the galaxy centre. To map the extended CGM out to the halo virial radius, R vir, Anderson et al. (2015) stacked SDSS galaxies with the full-sky X-ray data from the ROSAT survey. With the advent of SRG / eROSITA, we have unprecedented statistics for studying the hot CGM in Xrays with stacking analysis, as first shown by Oppenheimer et al. (2020), who generated mock observations with IllustrisTNG and EAGLE hydrodynamical simulations. Comparat et al. (2022) and Chadayammuri et al. (2022), and more recently, Zhang et al. (2024, hereafter Z + 24) conduct such experiments with the eROSITA data and detect the hot CGM out to the virial radius of Milky-Way and M31-like galaxies. X-ray stacking analysis provides the most sensitive state-of-the-art observations of the hot CGM in the current observational landscape. Still, they are subject to various observational e ff ects that a ff ect the interpretation of the hot CGM. Consequently, we need a complete theoretical framework for describing the intrinsic hot CGM emission to disentangle the intrinsic signal from other observational e ff ects. \nA plethora of theoretical models (e.g., Faerman et al. 2017; Voit et al. 2019; Pal Choudhury et al. 2019; Stern et al. 2024; Singh et al. 2021; Pandya et al. 2023; Faerman & Werk 2023) have been used to describe and predict the intrinsic properties of the hot CGM; Singh et al. (2024) discuss the comparison of some idealised cases. Similarly, Oppenheimer et al. (2020); Vladutescu-Zopp et al. (2024) made explicit predictions of the intrinsic hot CGM profiles obtained by assigning mock X-ray emission to the hot halos found in cosmological hydrodynamical simulations. However, before testing theoretical and numerical predictions of the hot CGM against state-of-the-art observations, we must quantify all the e ff ects that influence the hot CGMobservations in stacking experiments. Among the primary sources of contamination in X-ray stacking experiments, inhibiting us from retrieving the physical properties of the detected CGM emission, are (1) the unresolved AGN and X-ray binaries (XRB) population of galaxies (Bi ffi et al. 2018; VladutescuZopp et al. 2023), and (2) the projection e ff ects within the hot gas \nemission of the cosmic web. These projection e ff ects include the contribution from the Large-Scale Structure (LSS) halo environment (locally correlated X-ray emission), the e ff ect of satellite galaxies being misclassified as central due to limitations in the (photometric) redshift accuracy for the galaxies in large surveys (see e.g., Weng et al. 2024), and the Line-of sight (LoS) projection of uncorrelated X-ray emission of fore- and background structures. Z + 24 tried to model these e ff ects empirically for the first time; however, we need forward models based on cosmological hydrodynamical simulations for a complete description and full understanding of these contaminating e ff ects. \nIn this paper, we analyse numerical simulations to empirically quantify the e ff ects of the locally correlated X-ray emission and the misclassified centrals, also called the satellite-boost e ff ect, relevant for measuring the hot CGM in stacking experiments. To do this, we use the TNG300 hydrodynamical simulations (Pillepich et al. 2018a; Marinacci et al. 2018; Naiman et al. 2018; Nelson et al. 2015; Springel et al. 2018) to construct a lightcone and generate mock X-ray observations. We focus here on the projection e ff ects of hot gas a ff ecting the X-ray surface brightness profiles. The structure of the paper is as follows. Sec. 2 details the construction of the TNG300 lightcone (LCTNG300) used for modelling the hot CGM in this study. We explain all the projection e ff ects in detail in Sec. 2.1. In Sec. 3, we describe the process for generating mock X-ray observations that mimic observational data and provide the machinery for computing the surface brightness profiles. Sec. 4 presents the main results of this work done with the TNG300 lightcone. Lastly, Sec. 5 discusses the main findings and Sec. 6 provides an overview with prospects.", '2. Simulated lightcone with IllustrisTNG: LC-TNG300': "Since we are interested in studying the impact of the locally correlated X-ray emission within the LSS environment of the halo, we require a simulation that contains all the complexities introduced by feedback and cooling, which are imprinted in X-ray measurements. We also require a large box size to encompass the e ff ects of the cosmological LSS itself. Therefore, we use cosmological hydrodynamical simulations that self-consistently predict the LSS and its impact on gas dynamics within a halo. We use the IllustrisTNG cosmological hydrodynamical simulation with the box of side length 302 . 6 Mpc (TNG300; Nelson et al. 2019) 1 . IllustrisTNG accounts for many physical processes, among which the most notable ones are star formation regulated by a subgrid ISM model (Springel & Hernquist 2003), metal enrichment (Naiman et al. 2018), radiative gas cooling, galactic wind outflows (Pillepich et al. 2018b), magnetic fields and diffuse radio emissions (Marinacci et al. 2018), supermassive black hole growth with Bondi accretion and mergers, thermal and kinetic modes for black hole feedback (Weinberger et al. 2018). IllustrisTNG also reproduces correlation functions and power spectra of particles and halos (Springel et al. 2018). TNG300 contains 2500 3 dark matter particles, with a baryonic mass resolution of 1 . 1 × 10 7 M ⊙ , a comoving value of the adaptive gas gravitational softening length for gas cells of 370 comoving parsec, gravitational softening of the collisionless component of 1 . 48 kpc, and dark matter mass resolution of 5 . 9 × 10 7 M ⊙ . The TNG simulations adopt the Planck Collaboration XIII 2016 cosmological parameters, with the matter density parameter Ω m = Ω dm + Ω b = 0 . 3089, baryonic density parameter Ω b = 0 . 0486, \nTable 1. For every stellar mass bin, we present the mean stellar mass within twice the stellar half mass radius, the total number of central galaxies, the total number of satellite galaxies, mean halo mass, the minimum and maximum values of halo masses for the given stellar mass bin, the mean R 200 m , and the mean R 500 c .Table 2. For every halo mass bin used to generate the X-ray surface brightness profiles, we present the mean halo mass, the total number of distinct halos, the mean stellar mass, the mean R 200m and the mean R 500c, for every halo mass bin. \nFig. 1. Illustration of the lightcone built using TNG300 in the x-y plane. The figure shows the subhalos within LC-TNG300 at 0 . 03 ≤ z ≤ 0 . 3 remapped using boxremap (Carlson & White 2010). The observer is set at the (0 , 0 , 0) location. The lightcone goes out to 1231 cMpc along the x-axis, subtending an area of 47 . 28 deg 2 on the sky in the y-z plane. The subhalos are colour-coded with their stellar masses. \n<!-- image --> \nMpc \nHubble constant H0 = 100 h km / s / Mpc with h = 0 . 6774, and ΩΛ = 0 . 6911. \nThe Friends-of-Friends (FoF) algorithm is applied to the dark matter particles with linking length b = 0 . 2 to obtain the halos. The subhalos are retrieved with subfind (Springel et al. 2001; Dolag et al. 2009) that detects gravitationally bound substructures, which are equivalent to galaxies in observations. Additionally, subfind also classified the subhalos into centrals and satellites, where centrals are the most massive substructure within a distinct halo. \nTo model the hot gas emission up to z ∼ 0 . 3, as done in observations (e.g. Comparat et al. 2022; Chadayammuri et al. 2022; Zhang et al. 2024), we need four boxes lined up behind each other to arrive at the comoving distance, dC(z = 0 . 3), of 1231 Mpc. For a full-sky lightcone going up to z = 0 . 3, we must replicate and stack 512 boxes; however, in this technique, the large-scale projection e ff ects cannot be estimated properly; see Merson et al. (2013). Therefore, we box remap 2 all snapshots into a configuration where the longest length of one of \nthe sides is ∼ 4 × the original length (Carlson & White 2010). This technique ensures that the new elongated box has a oneto-one remapping, remains volume-preserving, and keeps local structures intact. For a box whose original dimensions are normalized to (1 , 1 , 1), the unique solution for the transformed box sides is (4 . 1231 , 0 . 7276 , 0 . 3333), respectively. We remap the coordinates of particles (gas, dark matter, and stars) and the halo and their subhalo catalogues. Therefore, we obtain 22 remapped particle cuboids and 22 remapped galaxy catalog cuboids at redshift z < 0 . 3. We define the observer's position as being at a corner of the smallest face. The opening angles for the observer are ( θ LC obs , ϕ LC obs) = (10 . 16 , 4 . 64) degrees. The area subtended on the sky for a given observer is 47 . 28 deg 2 . To study the impact of the projection e ff ects for MW-like galaxies (M ⋆ = 10 10 . 5 -11 M ⊙ ), we have 5 , 109 centrals and 7 , 826 satellites in the light cone, which is ample to statistically model the projection e ff ects within Milky-Way-sized halos. We illustrate the lightcone constructed with the box remap technique in Fig. 1. We also publically release LightGen , the code used for generating the lightcone in this work. \n- \n- \n2 \n· \n00 \n1 \n· \n00 \n0 \n· \n00 \n01 \n· \n00 \n02 \n· \n00 \n- \n2 \n0 \n02 \n- \n<!-- image --> \nh \n23 h 40 23 50 0 h 00 0 10 0 h 20 Fig. 2. The projected rest-frame X-ray events from the TNG300 ligthcone in the 0 . 5 -2 . 0 keV band for a telescope with energy-independent collecting area 1000 cm 2 and exposure time of 1000 ks. The events are generated using the hot gas cells within the TNG300 lightcone at redshifts 0 . 03 -0 . 3 using pyXsim (ZuHone & Hallman 2016). The centre for the projection onto the sky is chosen as R.A., Dec. ≡ (0 ., 0 . ) degrees. The contours represent the two X-ray surface brightness levels of 5 × 10 -14 kev s -1 cm -2 arcmin -2 and 2 × 10 -16 kev s -1 cm -2 arcmin -2 . \n- \nh \n<!-- image --> \nh \n23 40 23 50 0 00 0 10 0 h 20 Fig. 3. The projected halos ( centrals ) are overplotted with their corresponding scaled R 500 c at z = 0 . 3; the R 500 c of the halos are represented by the size of the circles. We also show the projected rest-frame X-ray events from the TNG300 ligthcone in the 0 . 5 -2 . 0 keV band for a telescope with energy-independent collecting area 1000 cm 2 and exposure time of 1000 ks at the redshift slice of 0 . 284 ≤ z ≤ 0 . 3. The halos with M 200 m ∈ 10 14 -14 . 5 M ⊙ are shown in red, M 200 m ∈ 10 13 . 5 -14 M ⊙ in grey, M 200 m ∈ 10 13 -13 . 5 M ⊙ in cyan, and M 200 m ∈ 10 12 . 5 -13 M ⊙ in blue. \nh \nThe distinct halos, CEN sim halo , and subhalos within the LCTNG300 are cataloged with their physical properties and binned in stellar mass and halo mass bins, as shown in Tab. 1 and Tab. 2. This paper defines the stellar mass used from TNG300 as the mass within twice the stellar half-mass radius. We present quantities relative to both critical and mean densities because R 500 c 3 is the radius most commonly used by X-ray astronomers (see e.g., Lyskova et al. (2023) and references therein), and R 200 m 4 repre- \nh \nh \nsents the halo's viral radius and is theoretically more relevant as it presents quantities within the virialized halo. The galaxy catalog is divided into central, CEN sim , and satellites, SAT sim for the concerned stellar mass bins. To construct the SAT sim catalogue, we refer to the halo / subhalo classification, where we match all the central galaxies with the distinct halos, thereby leaving behind all the secondary subhalos or satellites.", '2.1. Contributions to observed X-ray surface brightness profiles': 'The main objective of this study is to model the various sources of contamination that come into play when measuring the hot gas component of the CGM. Here, we list all the components contributing to the observed X-ray surface brightness profile. \n- 1. Intrinsic emission from the halo. This corresponds to the emission within the radius, R 200m, of the central halo in 3D. We associate this X-ray emission with being intrinsic to the galaxy CGM. It must be noted that the definition of a halo boundary at which the gas is bound is non-trivial; we refer to Diemer et al. (2017) and references therein for more details.\n- 2. Locally correlated environment. This component encapsulates the surrounding emission of the galaxy with the LSS in which it resides. Also known as the 2-halo term (Cooray & Sheth 2002; Kravtsov et al. 2004), this corresponds to the contribution arising due to the local background changing with the size of the halo. This e ff ect is explored in tSZE studies, where Vikram et al. (2017), Lim et al. (2021) show the impact of the two-halo component, where the one-halo contribution at mass scales M 200 ≤ 10 13 -13 . 5 h -1 M ⊙ is swamped by the two-halo term due to nearby massive systems dominating the measured signal. Given our focus on MW-size halos, modelling the contribution from the locally-correlated background is crucial for disentangling intrinsic hot CGM emission from the local background.\n- 3. Intrinsic emission from satellites. The X-ray emission from the satellite galaxy contributes to the total CGM emission. As the host galaxy is more massive, with a deeper potential well, the contribution of this e ff ect is negligible in stacking experiments (see Rohr et al. (2024) for further insights on the detectability of the satellite emission).\n- 4. Contamination from misclassified centrals. In observations that use photometric surveys, classifications between centrals and satellites are inhibited due to limitations in the redshift accuracy. This e ff ect is large and unavoidable for photometric surveys ( ∼ 30% for MW-like galaxies; Sec 3.5 in Z + 24) and significantly mitigated for spectroscopic surveys ( ∼ 1% for MW-like galaxies; Sec 3.7 in Z + 24), but not completely removed due to survey incompleteness or residual uncertainty in the central / satellite classification for systems with a low number of galaxies (see also Weng et al. 2024 who quantify the e ff ect for absorbers in cold gas with TNG50). This implies that the measured X-ray surface brightness profiles in stacked samples of galaxies classified as central contain the intrinsic emission around truly central galaxies but are contaminated by the emission measured around misclassified centrals. In conclusion, including the emission around satellites in the stacking analysis alters the recovered profiles and, therefore, must be modelled.\n- 5. Contamination from other X-ray sources. Other X-ray emitting sources like XRB and AGN contaminate the measured hot CGM. XRB emission is distributed on the scale of the stellar body of a galaxy; however, for an instrument with a Point Spread Function (PSF) like eROSITA, it is unresolved and appears as a point source. Z + 24 take special precautions to account for this by masking the eROSITA detected point sources within the X-ray data. They also model additional contributions from the unresolved XRB and AGN. For this study, we leave the modelling of the AGN and XRB resolved / unresolved emissions with the LC-TNG300 for future work.\n- 6. Line of Sight (LoS) projection. Objects along the line of sight, which do not reside near the galaxy, also contaminate the detected signal. Nevertheless, this uncorrelated contamination is well-modelled as the large-scale background and foreground in observations and, therefore, is not discussed further in this work. \nThis work focuses on modelling the locally correlated environment and the e ff ect due to misclassified-centrals (the satelliteboost e ff ect).', '3. Method': 'We detail the process to create mock X-ray observations in Sec. 3.1 and introduce the formalism used to fit the X-ray surface brightness profiles in Sec. 3.2. We detail the data products generated to study the projection e ff ects in Sec. 3.3.', '3.1. Mock X-ray observation': 'The photons are simulated in the 0 . 5 -2 . 0 keV intrinsic band with pyXsim (ZuHone & Hallman 2016), which is based on phox (Bi ffi et al. 2013; Bi ffi et al. 2018), by assuming an input emission model where the hot X-ray emitting gas is in collisional ionization equilibrium. The spectral model computations of hot plasma use the Astrophysical Plasma Emission Code, apec 5 code (Smith et al. 2001) with atomic data from atomdb v3.0.9 (Foster et al. 2012). This model requires the plasma temperature of the gas cells (in keV), the metal abundances, the redshift z and the normalization, \nN = 10 -14 4 π [ D A(1 + z )] 2 Z n e n HdV , (1) \nwhere D A is the angular diameter distance to the source (cm), dV is the volume element (cm 3 ), n e and n H are the electron and hydrogen densities (cm -3 ), respectively. The temperature is calculated from the internal energy u and the electron abundance x e( = n e / n H) of the gas cells 6 . The temperature T for every gas cell is defined as \nT = ( γ -1) u k B µ (2) \nwhere k B is Boltzmann\'s constant in CGS units and γ = 5 / 3 is the adiabatic index. The mean molecular weight µ is given as \nµ = 4 1 + 3 X H + 4 X H x e m p (3) \nwhere X H = 0 . 76 is the hydrogen mass fraction and m p is the proton mass in grams. The metal abundances within TNG are provided for the snapshots at redshift intervals of every 0 . 1. As a result, 19 of the 22 snapshots within the lightcone constructed in this work lack metallicity information. Given the lack of metallicity information and the inaccuracies introduced by extrapolation of metallicity values between the 0.1 redshift intervals due to the evolution of gas particles, we assume a constant metallicity of 0 . 3 Z ⊙ ; this is consistent with measurements for our MW(Miller & Bregman 2015; Bregman et al. 2018; Kaaret et al. 2020; Ponti et al. 2023). This work uses the solar abundance values from Anders & Grevesse (1989). \nFig. 4. Mean X-ray surface brightness profiles in the stellar mass bins: M ⋆ = 10 10 . 5 -11 M ⊙ , corresponding to MW-like galaxies ( left ), M ⋆ = 10 11 -11 . 2 M ⊙ ( centre ), and M ⋆ = 10 11 . 2 -11 . 5 M ⊙ ( right ). The vertical dashed line is the mean R 500 c at 242 . 75 kpc ( left ), 369 . 01 kpc ( centre ), and 484 . 52 kpc ( right ) of the respective galaxy stellar mass bin with the shaded area corresponding to the minimum and maximum values. The black line is the analytical model, shown in Eq. 7, fit to the LC-TNG300 mean X-ray surface brightness profiles. The best-fitting parameters for the model are given in Tab. 3. We find decreasing values for the slope, β , with increasing stellar mass bins. \n<!-- image --> \nFig. 5. Mean X-ray surface brightness profiles in the halo mass bins: M 200m = 10 12 . 5 -13 M ⊙ , corresponding to MW-like galaxies ( left ), M 200m = 10 13 -13 . 5 M ⊙ ( centre ), and M 200m = 10 13 . 5 -14 M ⊙ ( right ). The vertical dashed line is the mean R 500 c at 246 . 35 kpc ( left ), 353 . 35 kpc ( centre ), and 519 . 34 kpc ( right ) of the respective halo mass bin with the shaded area corresponding to the minimum and maximum values. The black line is the analytical model, shown in Eq. 7, that fits the LC-TNG300 mean X-ray surface brightness profiles. The best-fitting parameters for the model are given in Tab. 4. We find decreasing values for the slope, β , with increasing halo mass bins. \n<!-- image --> \nThe TNG star formation model is based on the subgrid twophase model proposed by Springel & Hernquist (2003), with some modifications (see Pillepich et al. 2018a, and references therein). The gas cells that emit e ffi ciently in the soft X-ray band (i.e. [0.3-5] keV) are due to the SN-driven kinetic decoupled winds, which ultimately deposit energy into non-star-forming gas cells. Additionally, the hot component of this two-phase model exhibits typically high temperature ( > 10 5 K). As further explained in Truong et al. (2020) (see Appendix B-C), because this multiphase structure within the TNG model is not resolved, and it is instead modelled by a simplistic two-phase structure with unrealistic assumptions, we cannot make a sensible estimate of the X-ray emission from the unresolved phases of the ISM. Therefore, by excluding the parameter space of the warmneutral ISM (Le Brun et al. 2014; Rahmati et al. 2016; Wijers et al. 2019), namely: (1) excluding star-forming gas cells, (2) ignoring gas cells below 10 5 K, and (3) ignoring gas cells with densities above 10 -25 g / cm 3 , we ensure the gas particles used in this work are physically emitting X-rays. \nWithin pyXSim , the number of photons generated depends on the specified collecting area of the assumed X-ray telescope, its exposure time, and redshift. We generate su ffi cient photons by \nassuming a telescope with an energy-independent collecting area of 1000 cm 2 and an exposure time of 1000 ks. The photon-list is generated in the observed frame of the X-ray emitting gas cells and is corrected to rest frame energies. The LoS direction determines the event\'s position in the sky. We define the LoS along the x-axis within the lightcone. By applying a θ LC obs / 2 = 5 . 08 deg rotation along the z-axis and -ϕ LC obs / 2 = -2 . 32 deg rotation along the y-axis, we centre the y-z plane at (0 , 0) degrees. The photons generated by the gas particles are projected onto the sky; the resulting image is shown in Fig. 2. \nWe also project the halo and subhalo positions on the sky, as shown in Fig. 3 for a redshift slice of z = 0 . 3. The X-ray events are then stacked around the projected halo / subhalo positions. The surface brightness profiles 7 are calculated S ( r ) as a function of the projected radius on the sky within the 0 . 5 -2 keV energy band. The number of photons with rest frame energy E [erg] in each radially outward bin r , N ( E , r ), is weighted by the area, A [kpc 2 ], of the 2D shell between r and r + dr, the fraction of photons collected from the source, f A [cm 2 ], and the exposure \nTable 3. For every stellar mass bins, we present the best-fitting parameters of the model (see Eq. 7): S 0,:the central surface brightness; r c, the core radius; β , the exponent quantifying the slope of the profile; rs , the scale radius at which the slope changes to ϵ . \n. \n. \n. \n. \nTable 4. For every halo mass bins, we present the best-fitting parameters of the model (see Eq. 7): S 0, the central surface brightness; r c, the core radius; β , the exponent quantifying the slope of the profile; rs , the scale radius at which the slope changes to ϵ . \ntime, t exp [seconds], \nS X( r ) = P 2 keV Ei = 0 . 5 keV N ( Ei , r ) Ei f A t exp A ( r ) " erg sec kpc 2 # . (4) \nHere, f A is the fraction of the source photons collected by the synthesized telescope that has a collecting area A : \nf A = A 4 π d L( z ) 2 (5) \nwhere d L( z ) is the luminosity distance to the source.', '3.2. X-ray surface brightness profiles and their analytic modelling': 'The hot CGM surface brightness profile can be analytically described by the β model (Cavaliere & Fusco-Femiano 1976); \n¯ S X , 0 . 5 -2 . 0 keV[ r ] = S 0 1 + r r c ! 2 -3 β + 1 2 , (6) \nwhere S 0 is the central surface brightness, r c is the core radius at which the profile slope becomes steeper, and β is the exponent quantifying the slope of the profile. In cases where the outskirt steepens more than the slope defined for the inner radii by the β -model, following Vikhlinin et al. (2006), we introduce a new slope-parameter as follows: \n¯ S X , 0 . 5 -2 . 0 keV[ r ] = S 0 1 + r r c ! 2 -3 β + 1 2 × " 1 + r r s ! γ # -ϵ/γ , (7) \nwhere rs is the scale radius at which the slope changes to ϵ and γ defines the width of the transition region. We fix γ = 3 and restrict the priors on rs > rc and ϵ < 5, as suggested by Vikhlinin et al. (2006), in the fitting procedure.', '3.3. Prerequisite data products for quantifying projection effects': 'To obtain contributions from the large-scale structure, the locally correlated X-ray emission, we generate cubes and profiles with the CEN sim catalog for the two cases as follows: \n- 1. R 200m cubes and profiles: every central galaxy within the halo catalog is assigned X-ray events within R 200m of the parent halo. We define these profiles as the intrinsic hot gas emission profiles.\n- 2. R ± 3Mpc, R ± 9Mpc and R ± 27Mpc cubes and profiles: all the X-ray events, intrinsic and locally extrinsic, irrespective of whether they belong to the galaxy but within 3 Mpc, 9 Mpc, and 27 Mpc of the source centre, are selected in 3D. We construct the profiles and cubes to quantify the impact of contamination from the local vicinity on the intrinsic source emission. \nTo quantify contributions from the emission associated with misclassified centrals, we stack the emission around the galaxies in the SAT sim catalogue in the same stellar mass bin as the central galaxies. For galaxies in the three stellar mass bins: 10 10 . 5 -11 M ⊙ , 10 11 -11 . 25 M ⊙ , and 10 11 . 25 -11 . 5 M ⊙ , we construct a total galaxy sample, N T = N CEN + N SAT with varying fractions of satellites; namely f sat = 0 . 01 , 0 . 1 , and 0 . 3.', '4. Results': 'Here, we present the main results of this work. In Sec. 4.1, we discuss the outcome of fitting the analytical model to the stacked LC-TNG300 intrinsic profiles. In Sec. 4.2, we carry forward the current state-of-the-art to understand the impact of the locally correlated environment on the intrinsic X-ray surface brightness profile. Lastly, in Sec. 4.3, we show the prominence of the e ff ect due to misclassified centrals.', '4.1. Fitting analytic models to the intrinsic X-ray surface brightness profiles': 'Fig. 4 and Fig. 5 show the mean X-ray surface brightness profiles in three stellar mass and halo mass bins, respectively. We fit the surface brightness profiles with the analytic model introduced in Eq. 7 and provide the best-fit parameters in Tab. 3 and Tab. 4 for the stellar and halo mass bins, respectively. A simple β model, described by Eq. 6, describes the lowest stellar and halo mass bins. However, the β model does not describe the more massive stellar and halo mass bins. Therefore, we implement Eq. 7, which very well describes the LC-TNG300 profiles in the stellar and halo mass bins across all masses. \nThe model has two break radii, the core radius rc and the scale radius rs , where rc ≪ rs . The scale radius rs a ff ects the profile at radii beyond R 500c, the mean R 500c of the stacked mass bin. The model also introduces two slope parameters, β , which influences the X-ray surface brightness profile at radii r < rs and ϵ , the slope that a ff ects the profile at r > rs . As R 500c is the radius used most commonly in observations, and β quantifies the shape \nFig. 6. Mean X-ray surface brightness profiles in the stellar mass bin M ⋆ = GLYPH<2> 10 10 . 5 , 10 11 GLYPH<3> M ⊙ , corresponding to MW-like galaxies. The locally correlated large-scale structure contributions are shown by comparing the profiles obtained with photons selected within ± R 200 m ( purple ) and those obtained within ± 3 Mpc ( solid black ), ± 9 Mpc ( solid grey ), ± 27 Mpc ( dashed grey ) away from the halo centre. The crosses are from previous work by Oppenheimer et al. (2020), where they generate mock X-ray observations using TNG-100. The vertical dashed line at 242 kpc and the dotted line at 525 kpc is the mean R 500 c and R 200 m , respectively, of the 5 , 109 galaxies used in the mass bin with the shaded area corresponding to the minimum and maximum R 500 c values. The shaded uncertainties on the profiles represent the variance obtained by bootstrapping. \n<!-- image --> \nof the profile at radii ≤ R 500c, we discuss here the variations in β across the stellar and halo mass bins. \nThe increasing stellar mass bins 10 10 . 5 -11 M ⊙ , 10 11 . 2 -11 . 5 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ result in decreasing values of β from 0 . 4, 0 . 28, and 0 . 25, respectively. The same trend holds from the halo mass bins, where increasing halo mass bins: 10 12 . 5 -13 M ⊙ , 10 13 -13 . 5 M ⊙ , and 10 13 . 5 -14 M ⊙ have decreasing values of β from 0 . 57, 0 . 30, to 0 . 24, respectively. One of the reasons for the decrease is attributed to the mass-dependence of the feedback prescriptions in the TNG300 model, i.e., stellar mode dominating at lower masses, and the kinetic and thermal modes of energy injection by AGN dominating galaxies with M ⋆ ≥ 10 10 . 5 M ⊙ (Weinberger et al. 2016). As discussed in Weinberger et al. (2018), the two AGN-related feedback channels depend on the black-hole (BH) accretion rates. For the high accretion rates, the thermal mode causes the gas cells close to the galactic centre to heat, eventually releasing the energy radiatively in X-rays. However, for lower accretion rates, the kinetic mode kicks in, depositing energy through winds and jets in random directions away from the BH. This causes gas heating and, hence, X-ray emission via cooling at a larger distance away from the galaxy centre, overall causing a flattening of the radial X-ray surface brightness profile as shown by the slope β in Fig. 4 and 5. This reasoning is further consolidated with studies detailing the e ff ects of the kinetic mode of BH feedback in the TNG model (Terrazas et al. 2020), its impact on the gas properties, i.e., temperature, entropy, density, and CGM frac- \n020; Davies et al. 2020), and its correlations with X-ray emission (Truong et al. 2020; Oppenheimer et al. 2020; Truong et al. 2021; Ayromlou et al. 2023). Sorini et al. (2024) show a similar trend of decreasing slope for the gas density profiles with the SIMBA (Davé et al. 2019) suite of simulations. With simulation comparison projects like camels simulations (Villaescusa-Navarro et al. 2021), it is possible to study the correlation between X-ray emission around galaxies with di ff erent feedback implementations, the exploration of which we leave for a future study. \nIn the following two sections, we focus on how the projection e ff ects result in deviations from the intrinsic profile. We discuss the e ff ects on the stellar mass bins as it is an observationally available mass proxy.', '4.2. Locally correlated environment': "For the MW-mass bin, M ⋆ = 10 10 . 5 -11 M ⊙ , with mean R 200m = 525 kpc, we generate cubes within ± 3 Mpc, ± 9 Mpc and ± 27 Mpc from the galaxy center. This corresponds to photons selected within ∼ 5 . 7 × R 200m, ∼ 17 × R 200m, and ∼ 51 . 4 × R 200m, respectively. Fig. 6 presents the impact of the locally correlated LSS for MW-like halos. We show the intrinsic profile that is obtained by selecting events within ± R 200 m of the galaxy centre in purple. We compare our result with previous work from Oppenheimer et al. (2020), shown with the crosses in Fig. 6. They predict the profile between 10 -242 kpc for a stellar mass sample \nFig. 7. E ff ect of centrals ( purple ) and misclassified-centrals ( green ), in the stellar mass bin M ⋆ = GLYPH<2> 10 10 . 5 , 10 11 GLYPH<3> M ⊙ , on the total X-ray surface brightness profiles ( black ). The total sample is constructed such that there are 1% satellites and 99% centrals ( left panel ), 10% satellites and 90% centrals ( middle panel ), and 30% satellites and 70% centrals ( right panel ). The vertical dashed line at 242 kpc is the mean R 500 c of the 5109 galaxies used in the mass bin with the shaded area corresponding to the minimum and maximum values. The shaded green region is the uncertainty on the misclassified central profile obtained by bootstrapping. \n<!-- image --> \nFig. 8. E ff ect of centrals ( purple ) and misclassified-centrals ( green ), in the stellar mass bin M ⋆ = GLYPH<2> 10 11 , 10 11 . 25 GLYPH<3> M ⊙ , on the total X-ray surface brightness profiles ( black ). The total sample is constructed with 1% ( left panel ), 10% ( middle panel ), and 30% ( right panel ) satellites. The vertical dashed line at 369 kpc is the mean R 500 c of the 680 galaxies used in the mass bin with the shaded area corresponding to the minimum and maximum values. \n<!-- image --> \nFig. 9. E ff ect of centrals ( purple ) and misclassified-centrals ( green ), in the stellar mass bin M ⋆ = GLYPH<2> 10 11 . 25 , 10 11 . 5 GLYPH<3> M ⊙ , on the total X-ray surface brightness profiles ( black ). The total sample is constructed with 1% ( left panel ), 10% ( middle panel ), and 30% ( right panel ) satellites. The vertical dashed line at 484 . 5 kpc is the mean R 500 c of the 305 galaxies used in the mass bin with the shaded area corresponding to the minimum and maximum values. \n<!-- image --> \nin the mass range 10 8 . 2 -11 . 39 M ⊙ containing ∼ 400 galaxies. Their stellar mass sample is divided into low sSFR and high sSFR. We take the mean of the low sSFR and high sSFR profiles generated in their work and compare them with ours. The halo mass range corresponding to their stellar mass bin is 10 ∼ 12 . 3 -13 M ⊙ , which entails the mean halo mass M 200m = 10 12 . 7 M ⊙ , same as that of the MW mass bin used in our work. Oppenheimer et al. (2020) use the TNG-100 simulation and synthesize mock X-ray observations for individual halos as opposed to this work that uses TNG300 and synthesizes mock X-ray observations within the lightcone. Despite these di ff erences, this work's predicted X-ray intrinsic emission profiles are in good agreement with Oppenheimer et al. (2020). \nWe find that increasing the volume by 4 π 3 (5 . 7 3 -1 3 ) R 3 200 m = 771 . 5 × R 3 200 m , i.e., by including events in ± 3 Mpc, boosts the Xray surface brightness profile at R 200 m by a factor of 5.2. We show this with the thick black line in Fig. 6. We find deviations from the true intrinsic profile due to events selected within ± 3 Mpc at ∼ 150 kpc, which corresponds to ≈ 0 . 6 × R 500 c and ≈ 0 . 3 × R 200 m for the MW-stellar mass bin. \nWhen considering the events in ± 9 Mpc, i.e., increasing the volume used to compute the X-ray surface brightness profiles by 4 π 3 (17 3 -1 3 ) R 3 200 m = (2 . 0 × 10 4 ) R 3 200 m , we find that the X-ray surface brightness profile at R 200 m is boosted by a factor of 16.8 This is shown by the grey curve in Fig. 6. We find deviations from the true intrinsic profile due to events selected within ± 9 Mpc, at ∼ 100 kpc, which corresponds to ≈ 0 . 4 × R 500 c and ≈ 0 . 2 × R 200 m for the MW-stellar mass bin. \nFor events selected in ± 27 Mpc or by including events in a volume of (8 . 2 × 10 4 ) R 3 200 m , the X-ray surface brightness profile at R 200 m is boosted by 47 . 3. This is shown by the light grey dashed curve in Fig. 6. The profile, in this case, remains unchanged only at radii ≤ 40 kpc, which corresponds to ≈ 0 . 2 × R 500 c and ≈ 0 . 08 × R 200 m for the MW-stellar mass bin. \nIn conclusion, we show for the first time the e ff ect of the local environment on the true intrinsic profile of a mean MWlike stacked X-ray surface brightness profile. Namely, we find deviations from the true profile at ≈ 0 . 3 × R 200 m , ≈ 0 . 2 × R 200 m , and ≈ 0 . 08 × R 200 m by including events out to ∼ 5 . 7 × R 200m, ∼ 17 × R 200m, and ∼ 51 . 4 × R 200m, respectively. The black, grey, and dashed-grey curves in Fig. 6 show that increasing the integration volume swamps the features of the intrinsic profile at radii closer to the galaxy centre. \nIn observations, one is sensitive to the complete line of sight towards the observer. Therefore, this e ff ect can be corrected by subtracting a background emission level determined empirically from the observed surface brightness at a large distance from the halos of interest (assuming spherical symmetry in the largescale emission). Nevertheless, given the setup used in this work, we can, for the first time, probe this e ff ect locally around the halo and show its stark impact in stacking experiments.", '4.3. The Effect of Misclassified Centrals': 'In simulations, we completely and accurately classify central and satellite galaxies within a given stellar mass bin. Using this to our advantage, here, we make a precise prediction of the average emission arising from stacking around satellite galaxies by considering the following fraction of satellite (or misclassified centrals) contaminating the total galaxy sample: 0 . 01, 0 . 1 and 0 . 3. We select these contamination fractions of satellites by bootstrapping over the entire satellite galaxy sample. Tab. 1 details the number of centrals and satellites in the three stellar mass \nbins presented here. We show our findings in Fig. 7, 8, and 9, corresponding to stellar mass bins 10 10 . 5 -11 M ⊙ , 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ , respectively. \nFig. 7 quantifies the e ff ect of misclassified centrals on the total surface brightness profile, shown with the solid black line, in the MW-stellar mass bin of 10 10 . 5 -11 M ⊙ . We show the integrated profile obtained by stacking only the satellite galaxies misclassified as centrals in green and the intrinsic profile due to central galaxies in purple. The point at which the profile due to misclassified centrals ( green ) intersects the intrinsic central galaxy ( purple ) profile represents the point at which the e ff ect due to misclassified centrals contributes ∼ 50% to the total (centrals and satellites) emission. For a sample with 1% ( left panel ) and 10% ( central panel ) contaminating satellites, the e ff ect due to misclassified centrals dominates over the intrinsic central galaxy emission at radii ≥ 1 . 2 × R 500 c ( ∼ 300 kpc) and ≥ 0 . 2 × R 500 c ( ∼ 84 kpc), respectively. In a sample with 30% satellites ( right panel ), the e ff ect due to misclassified centrals dominates over the intrinsic central galaxy emission at radii ≥ 0 . 06 × R 500 c ( ∼ 34 kpc). \nAnalogously, Fig. 8 quantifies the e ff ect of misclassified centrals on the total surface brightness profile, shown with the solid black line, for galaxies in the stellar mass bin 10 11 -11 . 2 M ⊙ . For a sample with 1% ( left panel ) contaminating satellites, we find that the e ff ect of misclassified centrals has negligible impact within radii ≤ R 500 c . For 10% satellite contamination ( central panel ), the e ff ect due to misclassified centrals dominates over the intrinsic central galaxy emission at radii ≥ 1 . 6 × R 500 c ( ∼ 600 kpc) and for 30% satellites contamination ( right panel ), the e ff ect due to misclassified centrals dominates over the intrinsic central galaxy emission at radii ≥ 0 . 8 × R 500 c ( ∼ 300 kpc). \nFinally, Fig. 9 quantifies this e ff ect for the largest mass bin we are considering, 10 11 . 2 -11 . 5 M ⊙ . For a sample with 1% ( left panel ), 10% ( centre panel ) and 30% satellite contamination ( right panel ), the e ff ect due to misclassified centrals remains negligible at all radii ≤ R 500 c . More precisely, the 1% satellite contamination has a negligible impact on the total surface brightness profile. The 10% and 30% satellite contamination dominate over the intrinsic central galaxy profile at ≥ 2 × R 500 c ( ∼ 1000 kpc) and ≥ 1 . 4 × R 500 c ( ∼ 700 kpc), respectively. \nAs expected, we find from Fig. 7, 8, and 9 that the e ff ect due to misclassified centrals becomes increasingly important as we probe lower stellar mass bins in stacking experiments. As we go to higher mass bins, the decreasing impact of the e ff ect of misclassified centrals is attributed to the satellites - of the same stellar mass bin - residing in less massive parent halos (see further explanation in Sec. 5). \nFig. 10 presents the radial fraction of the X-ray emission from the intrinsic CGM - for four di ff erent levels of misclassified centrals contamination - in three stellar mass bins: 10 10 . 5 -11 M ⊙ , 10 11 -11 . 25 M ⊙ , and 10 11 . 25 -11 . 5 M ⊙ , respectively. We complement the conclusions from Fig. 7, 8, and 9 by showing that for MWlike galaxies, the e ff ect of misclassified centrals at the lowest contamination fraction of 0 . 01 results in the intrinsic central galaxy emission contributing only ∼ 62% of the total emission at R 500 c . This further deteriorates with the increasing fraction of misclassified centrals in the galaxy sample, where fractions of 0 . 1, 0 . 3, or 0 . 5 result in the intrinsic emission contributing ∼ 14%, ∼ 5% and ∼ 3% of the total emission at R 500 c , respectively. This e ff ect is less pronounced for the intermediate and most massive stellar mass bins of 10 11 -11 . 25 M ⊙ and 10 11 . 25 -11 . 5 M ⊙ compared to the MW-like stellar mass bin. In the case of the 10 11 -11 . 25 M ⊙ mass bin, for a fraction of misclassified centrals of 0 . 01, 0 . 1, 0 . 3, or 0 . 5, the central intrinsic emission \nFig. 10. The fraction of X-ray emission from the intrinsic hot CGM for di ff erent levels of satellite contamination (misclassified centrals) in the total galaxy sample. The fraction of X-ray emission from the intrinsic hot CGM for misclassified central contamination fractions of 0 . 01 (dotted line), 0 . 1 (dash-dotted line), 0 . 3 (dashed line), and 0 . 5 (solid line) are shown for three di ff erent mass bins: MW-like galaxies with stellar-masses in 10 10 . 5 -11 M ⊙ (left panel), 10 11 -11 . 2 M ⊙ (middle panel), and 10 11 . 2 -11 . 5 M ⊙ (right panel). The shaded regions around each curve correspond to the uncertainty of the mean profile obtained by bootstrapping. The vertical black solid line at 242 kpc in the left panel, 369 kpc in the middle panel, and 484 . 5 kpc in the right panel corresponds to the mean R 500 c of the respective mass bins with the shaded area signifying the minimum and maximum values. We find that the contaminating e ff ect of the misclassified centrals on the stacked profiles increases when the stellar mass decreases. \n<!-- image --> \ncontributes ∼ 96%, ∼ 71%, ∼ 45% and ∼ 33% of the total emission at R 500 c , respectively. Analogously, for the stellar mass bin of 10 11 . 25 -11 . 5 M ⊙ , for a fraction of misclassified centrals of 0 . 01, 0 . 1, 0 . 3, or 0 . 5, the central intrinsic emission contributes ∼ 99%, ∼ 87%, ∼ 68% and ∼ 57% of the total emission at R 500 c , respectively. Therefore, we present a clear trend of the increasing importance of the e ff ect of misclassified centrals, not only due to the increasing fraction of satellite contamination in the galaxy sample but also due to the decreasing stellar mass bins.', '5. Discussion': 'This work uses a TNG300-based forward model for the hot CGM emission and presents, for the first time, the e ff ect of locally correlated large-scale structure around a halo and the e ff ect of misclassified centrals in stacked hot CGM galactocentric profiles. Our findings are vital in light of our explorations for studying X-ray emission around lower mass galaxies with stacking experiments. We divide the discussion of our results by focusing on the projection e ff ects due to the locally correlated environment, first, in Sec. 5.1 and second, the e ff ect of misclassified centrals in Sec. 5.2.', '5.1. Locally correlated environment': 'The locally correlated environment for MW-like stellar mass galaxies boosts the mean galactocentric X-ray emission at radii ≤ R 200 m . More precisely, at R 200 m the X-ray emission is boosted 5 . 2 × , 16 . 8 × and 47 . 3 × by including emission ± 3 Mpc, ± 9 Mpc, ± 27 Mpc away from the halo centre, respectively. \nIncreasing the volume over which the intrinsic galactocentric profiles are measured leads to increased X-ray emission at smaller radii. This increased X-ray emission, in projection, is attributed to the local environment in which the galaxy resides. The upturn in the black line in Fig. 6, corresponding to including events within ± 3 Mpc ( ∼ 5 . 7 R 200 m ) of the galaxy centre, signifies the presence of other X-ray emitting halos ≳ 1 Mpc away from MW-like galaxies, i.e., ≈ 2 × R 200 m or ≈ 4 × R 500 c . This "upturn" \nfeature is washed out as we start including events in even larger volumes, as shown by grey and dashed-grey lines (corresponding to events selected within ± 9 Mpc and ± 27 Mpc, respectively) in Fig. 6. This supports our finding that an observer starts probing the averaged hot gas emission from all galaxies in projection along the line of sight, drowning out the features due to the local environment when integrating events over larger volumes around a galaxy centre. \nFurther improvements to study this e ff ect would involve including events to even larger radii around the galaxy. Given the limited area of the lightcone, 47 . 28 deg 2 , we are constrained in studying this e ff ect out to ± 27 Mpc. To quantify the effects of the locally correlated large-scale environment to even larger distances than those explored here, we need larger-volume lightcones, which is possible with larger cosmological hydrodynamical simulations like Magneticum (Dolag 2015), MillenniumTNG (Pakmor et al. 2023; Hernández-Aguayo et al. 2023), and FLAMINGO (Schaye et al. 2023). Additionally, future studies could study how di ff erent simulation feedback prescriptions can impact the trend observed in the X-ray surface brightness profiles by including events in larger volumes. It is particularly interesting to understand how the physics implemented in hydrodynamical simulations impacts the baryon spread around galaxies, i.e., the radius at which all the emissions converge to the mean background level in X-rays. We leave the study of these aspects to future work, which is made possible with the advent of projects like the camels simulations (Villaescusa-Navarro et al. 2021); see, e.g., Gebhardt et al. (2024) and Sorini et al. (2022).', '5.2. The Effect of Misclassified Centrals': 'We quantify the impact of the misclassified centrals (satelliteboost) residing in the same stellar mass bin as the central galaxies in Fig. 7, 8, and 9 for the stellar mass bins 10 10 . 5 -11 , 10 11 -11 . 2 , and 10 11 . 2 -11 . 5 , respectively. For satellite galaxies with stellar masses of MW-like galaxies, the emission around satellites dominates over central emission because the satellites of the stellar mass range 10 10 . 5 -11 M ⊙ reside in distinct halos with masses \nM 200m = 10 11 . 9 -15 . 1 , corresponding to a mean halo mass M 200 m = 10 13 . 7 M ⊙ . We find that the satellite subhalos probe the emission of their parent halos, contributing to the stack with o ff -centred surface brightness profiles of their more massive parent halo. This contribution causes shallower slopes for the total emission (central and satellites) in the MW-stellar mass bin. In cases where 30%, 10%, or 1% of the satellites contribute to the total emission, we have shown that they dominate the measured total profile at radii ≥ 0 . 06 × R 200 m , ≥ 0 . 2 × R 200 m , and ≥ 0 . 3 × R 200 m . We further investigate the e ff ect of misclassified centrals in the more massive stellar mass bins of 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ in Fig. 8 and Fig. 9, respectively. We find the e ff ect of misclassified centrals, with contamination fractions of 0 . 01, 0 . 1, and 0 . 3, less a ff ects the 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ mass bins as opposed to the MW-stellar mass bin. We rea ffi rm this with the trend shown in Fig. 10, where the fraction of intrinsic emission at the mean R 500 c is lowest for the MW stellar mass bin for all contamination levels, as opposed to the more massive stellar mass bins. \nTo understand why the MW-stellar mass bin is most a ff ected by the e ff ect of misclassified centrals, we first look at the mean halo masses of the satellites in the higher stellar mass bins of 10 11 -11 . 2 M ⊙ and 10 11 . 2 -11 . 5 M ⊙ . The satellites reside in mean distinct halos of ∼ 10 13 . 8 M ⊙ and ∼ 10 13 . 9 M ⊙ , respectively. This is a direct implication arising from the di ff erence in the stellarto-halo-mass relation for central and satellite galaxies (Shuntov et al. 2022). Given that the mean halo mass of the central galaxies in the LC-TNG300 stellar mass bins of 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ is 10 13 . 3 M ⊙ and 10 13 . 6 M ⊙ , respectively, we require large fraction of contamination to see the e ff ect of misclassified centrals compared to that for the MW-stellar mass bin. More precisely, we find that the e ff ect of misclassified centrals dominates over the central galaxy emission at R 500 c with satellite contamination fractions, f cont , sat, of 0 . 02 , 0 . 28, and 0 . 72 for the stellar mass bins 10 10 . 5 -11 M ⊙ , 10 11 -11 . 25 M ⊙ , and 10 11 . 25 -11 . 5 M ⊙ , respectively. Hence, the contamination fractions probed here ( f cont , sat = 0 . 01 , 0 . 1 , and 0 . 3), which are closer to the realistic f cont , sat in observations, do not significantly a ff ect the stellar mass bins 10 11 -11 . 25 , and 10 11 . 25 -11 . 5 , respectively. \nGiven these findings, we conclude that the contribution of satellites in MW-like galaxy samples containing satellites must not be neglected in stacking analysis. Observationally, to disentangle the contamination of the misclassified centrals, we must fit the profiles contributing to the intrinsic central galaxy jointly with the component capturing the e ff ect of misclassified centrals for a given fraction of satellite contamination. The analytical model encapsulating the central profile and the e ff ect of misclassified centrals will be addressed in future work.', '6. Summary': 'The main conclusions from this work are summarised as follows. \n- 1. We present an analytical model (Eq. 7) that well-describes the intrinsic S X -profile in LC-TNG300 across the stellar mass bins 10 10 . 5 -11 M ⊙ , 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ and halo mass bins of 10 12 . 5 -13 M ⊙ , 10 13 -13 . 5 M ⊙ , and 10 13 . 5 -14 M ⊙ . We provide the best-fitting parameters for the analytical model in Tab. 3 and Tab. 4 for the stellar and halo mass bins explored in this work.\n- 2. We carry forward the current state-of-the-art modelling analysis presented in Oppenheimer et al. (2020) by also showing the impact of the locally correlated environment on the measured X-ray surface brightness profiles. \n- 3. We present, for the first time, the e ff ect of misclassified centrals in stacking experiments for three stellar mass bins: 10 10 . 5 -11 M ⊙ , 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ . We find that the contaminating e ff ect of the misclassified centrals on the stacked profiles increases when the stellar mass decreases.\n- 4. For the MW-like galaxies, we conclude that the contribution of satellites (or misclassified centrals) can not be neglected in stacking analysis (see Fig. 10). In cases where 30%, 10%, or 1% of the satellites contribute to the total emission of MWlike galaxies, we have shown that they dominate the measured total S X profile at radii ≥ 0 . 06 × R 200 m , ≥ 0 . 2 × R 200 m , and ≥ 0 . 3 × R 200 m , respectively. \nModelling observed CGM profiles and comparing them with simulations is crucial to constrain the di ff erent galaxy formation models and to understand how the feedback and physics prescriptions a ff ect the hot CGM profile. Current state-of-the-art cosmological hydrodynamical simulations are calibrated on the stellar mass function and successfully reproduce realistic galaxy populations. Despite this, Davies et al. (2020) show that EAGLE and IllustrisTNG predict di ff erent total gas mass fractions, a ff ecting the observed CGM properties at MW-masses. Similarly, Khrykin et al. (2024) shows how the hot gas is sensitive to the di ff erent feedback variants within SIMBA. The model and methodology presented here provide the machinery to compare the measured hot CGM profiles among current simulations in future works. \nFuture X-ray missions, on the observation side, like Athena (Nandra et al. 2013), HUBS (Cui et al. 2020), LEM (Kraft et al. 2022) will push our current detection limits and provide us with (i) the spatial resolution to reach higher redshifts, (ii) better quantify point-like source contamination within the hot CGM, (iii) the spectral resolution to allow disentangling components via spectral fitting and (iv) the grasp to reach even fainter surface brightness levels. This work, focusing on modelling the projection e ff ects, is a step towards exploiting the information provided by the next-generation telescopes to better understand the intrinsic hot CGM emission. The prospects of this framework would be to extend it to interpret hot X-ray CGM measurements in stacking experiments by accounting for all projection e ff ects, i.e., not only the impact due to the local environment of the halo and the e ff ect due to misclassified centrals but also the emission from other X-ray sources such as AGN and XRB. \nAcknowledgements. SS would like to thank Fulvio Ferlito, Matteo Guardiani, Christian Garrel, Emre Bahar, and Jeremy Sanders for all the useful scientific discussions. \nforzamento delle Eccellenze (FARE) per la ricerca in Italia (R20L5S39T9). \nComputations were performed on the HPC system Raven at the Max Planck Computing and Data Facility. We acknowledge the project support by the Max Planck Computing and Data Facility.', 'Data Availability': 'The lightcone generation code, LightGen , will be publicly available on GitLab 8 upon acceptance of this paper. 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2024arXiv240904875C	We analyze lensing of photons and neutrinos in a gravitational field proposing a method to include radiative effects in classical lens equations. The study uses Schwarzschild and a ReissnerNordstrom metrics expanded at second post Newtonian order in the Newtonian potential employing a semiclassical approach to compare oneloop corrections from the Standard Model with Einsteins deflection formula via an impact parameter representation. We also explore the energy dependence of deflection due to quantum corrections and integrate these with classical lens equations.	2024-09-01T00:00:00Z	['2024arXiv240904875C', '10.48550/arXiv.2409.04875', 'arXiv:2409.04875']	['General Relativity and Quantum Cosmology', 'Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Phenomenology']	Semiclassical Lensing and Radiative Lens Equations	2,024	196	0.22	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.04875.pdf	{'Semiclassical Lensing and Radiative Lens Equations 1': '(1) Claudio Corianò, (1) , (2) Mario Cretì and (1) Leonardo Torcellini \n(1) \nDipartimento di Matematica e Fisica, Università del Salento and INFN Sezione di Lecce,Via Arnesano 73100 Lecce, Italy National Center for HPC, Big Data and Quantum Computing (2) Center for Biomolecular Nanotechnologies, \nIstituto Italiano di Tecnologia, Via Barsanti 14, 73010 Arnesano, Lecce, Italy', 'Abstract': "We analyze lensing of photons and neutrinos in a gravitational field, proposing a method to include radiative effects in classical lens equations. The study uses Schwarzschild and a Reissner-Nordstrom metrics expanded at second post Newtonian order in the Newtonian potential, employing a semiclassical approach to compare one-loop corrections from the Standard Model with Einstein's deflection formula via an impact parameter representation. We also explore the energy dependence of deflection due to quantum corrections and integrate these with classical lens equations.", '1 Introduction': "Classical photon deflections can be compared at classical and quantum levels by matching the classical gravitational cross section, expressed via the photon's impact parameter, with the perturbative cross section that includes radiative corrections [1, 2, 3]. This comparison, leads to a differential equation for the beam's impact parameter, linking the classical and quantum descriptions, as proposed in [4, 5]. Specifically, the energy dependence introduced at one-loop order allows for a new formula that connects the impact parameter b h ≡ b/ (2 GM ) , measured in units of the Schwarzschild radius ( 2 GM , the horizon scale) to the beam's energy E and the angle of deflection α . This energy dependence, absent in Einstein's classical formula, influences all lensing observables, including magnifications, cosmic shears, microlensing light curves, and Shapiro time delays. This indicates that radiative corrections lead to a violation of the classical equivalence principle in General Relativity, a principle that is inherently classical and conflicts with quantum mechanics due to the Heisenberg uncertainty principle. \nIn this framework, gravity is treated as an external background, and the quantum transition amplitude involves the TV V vertex (where T is the Standard Model's energy-momentum tensor and V the electromagnetic current) for photons [1], or the Tff vertex [3, 2] (with f representing a neutrino) for fermions. Numerical comparisons between the classical and semiclassical deflection formulas reveal that the energy dependence of the bending angle, while small, becomes more significant at higher energies due to the logarithmic growth of electroweak corrections [1, 2]. \nThe aim of this approach is to propose a method for integrating quantum effects into conventional lens equations and to illustrate the applicability of this method through a comprehensive numerical study. Photon scattering and gravitational lensing have been pivotal areas of research in astrophysics and cosmology, especially for testing Einstein's General Relativity (GR) and investigating the distribution of matter, including dark matter, in the universe [6]. In extreme scenarios, such as when photons travel near a black hole, the deflections can become highly significant, sometimes even causing photons to orbit the black hole before escaping. This leads to intricate lensing effects and the formation of multiple images. These phenomena have been rigorously studied across various spacetime geometries, including those of Schwarzschild, Reissner-Nordström, and rotating black holes at classical level. \nWhile classical photon deflection by gravity is well understood, the impact of full electroweak corrections had not been thoroughly explored before [1, 2, 3]. As just mentioned, previous studies have focused on QED corrections in weak lensing, but the electroweak corrections, though small, become significant at high energies or near massive gravitational sources. This is particularly relevant for very high energy gamma rays.", '2 The method': 'The approach of [4, 5] leads to a differential equation that links the cross section defined in terms of the impact parameter b , to the deflection angle α , aligning well with Einstein\'s predictions for weak lensing. The fundamental relation \nb sin θ db dθ = dσ d Ω , (2.1) \nwhere dσ/d Ω is computed perturbatively at quantum level and θ is the scattering angle of the quantum cross section, can be viewed as a differential equation satisfied by the classical impact parameter. The solution of (2.1) takes the general form \nb 2 h ( α ) = b 2 h ( ¯ θ ) + 2 ∫ ¯ θ α dθ \' sin θ \' d ˜ σ d Ω \' , (2.2) \nwith b 2 h ( ¯ θ ) denoting the constant of integration. The semiclassical scattering angle α is obtained from (2.2) as a boundary value of the integral in θ of the quantum cross section. As discussed in [1], the integration constant derived from (2.2) has to be set to zero for ¯ θ = π , in order for the solution of (2.1) to match the classical GR α ∼ 2 /b h at very large b h . \nWhen radiative corrections are included, the deflection angle becomes energy-dependent, creating a "gravitational rainbow." This phenomenon, absent in both classical and quantum calculations at the Born level, highlights a key difference between the two approaches. Though its impact is typically small, it becomes significant near the event horizon of a massive and very compact object. \nThe formalism employs a retarded graviton propagator, ignoring the back-reaction of the scattered beam on the source, similar to a typical scattering problem with a static external potential. Due to the presence of a horizon, we identify a lower bound on the impact parameter where classical General Relativity (GR) and quantum predictions align. This bound is around 20 b h , close to the horizon of the source, where the two descriptions agree. For smaller impact parameters ( 4 < b h < 20 ), discrepancies arise between the approaches (see the analysis in [1, 2]). As the beam nears the photon sphere, the logarithmic singularity in the deflection angle becomes significant, reflecting the limitations of the weak field approximation used in the metric.', '2.1 The neutrino case': "In the neutrino case the one-loop cross section in the electroweak theory is computed from the Tff vertex, with one graviton and two external fermion lines, giving \ndσ d Ω = G 2 M 2 cos 2 θ/ 2 sin 4 θ/ 2 { 1 + 4 G F 16 π 2 √ 2 [ f 1 W ( E,θ ) + f 1 Z ( E,θ ) -1 4 Σ L Z -1 4 Σ L W ]} , (2.3) \nwhose explicit expression has been given in [2]. G F is the Fermi constant, while f W f Z , Σ W , Σ Z are computed perturbatively and denote contributions generated by the exchange of W ± and Z gauge bosons in the loops. For massless neutrinos, loop corrections do not induce flavor transition vertices, such as those computed in [3]. In the case of a point-like gravitational source and of neutrino deflection, at Born level one obtains from (2.2) and (2.3) the differential equation \ndb 2 dθ = -2 ( GM sin 2 θ 2 ) 2 cos 2 θ 2 sin θ (2.4) \nthat in the small α (i.e. large b ) limit takes the asymptotic form \nb = GM ( 4 α + α 3 (1 + ln 8 -3 ln α ) ) + O ( α 2 ) , (2.5) \nFigure 1: Lens geometry, showing the primary I p and secondary I s images generated by the two geodesics of the isotropic emission from the source plane. D LS denotes the distances between the lens and the source planes, and D OS the distance between the observer and the source, measured along the optical axis of the lens. \n<!-- image --> \nwhich allows us to identify the deflection angle as α ∼ 4 GM/b for large b (or b h ), in agreement with Einstein's prediction for the angular deflection. If we consider the complete epxression of the related cross section (2.3), the expansion of the quantum corrections are organised in terms of the angle of deflection α and of coefficients derived from the quantum corrections. In the neutrino case they provide the relation \nb h ( E,α ) = 2 α + c ( E ) α + d ( E ) α ln( α ) + f ( E ) α 3 + g ( E ) α 3 ln α + h ( E ) α 3 ln 2 α + O ( α 5 ) (2.6) \nthat we can invert in order to get the angle of deflection in terms of the energy of the incoming neutrino and the impact parameter α ( E,b h ) \nα ( E,b h ) = 2 b h -1 b 3 h [ ( 2 + 4 C 1 ( E ) ) log b h + A ( E ) ] + O (1 /b 5 h ) . (2.7) \nwhose explicit expression can be found in [2].", '2.2 The photon case': 'In the photon case, the relevant interaction is the one-loop TV V , with one stress energy tensor coupled to the external gravitational fluctuations h µν , and two photon currents. V µναβ ( p 1 , p 2 ) is the Born level interaction of \nthe graviton vertex with the two photons and Γ µναβ (1) indicate the renormalized quantum corrections at one-loop computed in the complete electroweak theory. The matrix element for the scattering of a photon in an external gravitational field is given by \ni S (1) if = 2 π δ ( q 0 ) ( κM 2 \| ⃗q \| 2 ) S µν ( V µναβ ( p 1 , p 2 ) + Γ µναβ (1) ( p 1 , p 2 ) ) A i α ( p 1 ) A f β ( p 2 ) \n(2.8) \nwhere A i,f are normalized plane waves of the incoming state and final state photon, while ¯ S µν ≡ η µν -2 δ 0 µ δ 0 ν . At Born level the photon cross section is then given by \ndσ d Ω ∣ ∣ ∣ ∣ (0) γ = ( GM ) 2 cot 4 ( θ/ 2) . (2.9) \nAt this level, from Eq.(2.1) we obtain the differential equation \ndb 2 0 dθ = -2 ( GM ) 2 cot 4 ( θ 2 ) sin θ , (2.10) \nwith b 0 denoting the value of b computed at this order. The equation is separable and determines b 0 as a function of α , modulo an integration constant. If we set this constant to zero we obtain the solution \nb 2 0 ( α ) = 4 G 2 M 2 ( csc 2 ( α 2 ) +4 log sin ( α 2 ) -sin 2 α 2 ) . (2.11) \nIn the small α limit (i.e. for a large impact parameter) the solution above becomes \nb 0 ∼ GM ( 4 α + α 6 ( 1 + 12 log α 2 ) ) , (2.12) \nwhich allows us to identify the deflection angle α as α ∼ 4 GM/b 0 , in agreement with the classical GR result. The remaining corrections arise from the Born-level result but are not dependent on energy. A discussion of the energy dependence of the semiclassical deflection, solving for the impact parameter as a function of α can be found in [2]. Here we report the simpler expression obtained analyzing the post newtonial corrections to the Born level result, which become energy dependent \nα \| (0) γ, 1PN = 2 b h -1 b 2 h π 2 E ( GM ) -1 b 3 h ( ln b h ( 4 -1 16 π 2 E 2 ( GM ) 2 ) -1 64 π 2 E 2 ( GM ) 2 -4 3 π E ( GM ) -1 3 ) + O ( b 4 h ) . (2.13) \nThese corrected expressions can be inserted into the lens equation (see Fig. 1). The equation takes the scalar form \nβ = θ I -α ( E ) D LS D OS , (2.14) \nwhich can be extended to the case of stronger lensing by the inclusion of the contributions of the 1 /b n corrections contained in α ( E ) . D LS and D OS denote the distance between the lens and the source, and the source and the observer, respectively.', '3 Conclusions': 'In conclusion, we note that this approach can be extended to other lensing observables, such as lensing magnification and others, where the deflection angle plays a direct or indirect role, including the analysis of Shapiro time delays.', 'Acknowledgements': 'This work is partially funded by the European Union, Next Generation EU, PNRR project "National Centre for HPC, Big Data and Quantum Computing", project code CN00000013; by INFN, inziativa specifica QG-sky and by the grant PRIN 2022BP52A MUR "The Holographic Universe for all Lambdas" Lecce-Naples.', 'References': "- [1] C. Corianò, L. Delle Rose, M. M. Maglio, and M. Serino. Electroweak corrections to photon scattering, polarization and lensing in a gravitational background and the near horizon limit. JHEP , 01:091, 2015. doi:10.1007/JHEP01(2015)091. arXiv:1411.2804 [hep-ph].\n- [2] C. Corianò, A. Costantini, M. Dell'Atti, and L. Delle Rose. Neutrino and photon lensing by black holes: Radiative lens equations and post-newtonian contributions. JHEP , 07:160, 2015. doi:10.1007/JHEP07(2015)160. arXiv:1504.01322 [hep-ph].\n- [3] C. Corianò, L. Delle Rose, E. Gabrielli, and L. Trentadue, 'Fermion Scattering in a Gravitational Background: Electroweak Corrections and Flavour Transitions,' JHEP 03 , 136 (2014) [arXiv:1312.7657 [hepph]].\n- [4] R. Delbourgo and P. Phocas-Cosmetatos, Phys.Lett. B41 , 533 (1972).\n- [5] F. A. Berends and R. Gastmans, Nucl. Phys. B88 , 99 (1975).\n- [6] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses , Astronomy and Astrophysics Library, Springer-Verlag, Berlin, Heidelberg (1992)."}
2024arXiv240904528A	We provide a modesum prescription to directly compute the renormalized stressenergy tensor RSET for scalar fields in the Boulware vacuum. The method generalizes the recently developed extended coordinate method which was previously only applicable to HartleHawking states. We exhibit the accuracy and efficiency of the method by calculating the RSET in subextremal and extremal ReissnerNordstrm spacetimes. We find numerical evidence for the regularity of the RSET at the extremal horizon regardless of the field mass and its coupling. We employ our numerical results of the RSET to source the semiclassical Einstein equations demonstrating that if the RSET is considered as a static perturbation it will either deextremalize the black hole or convert it into a horizonless object.	2024-09-01T00:00:00Z	['arXiv:2409.04528', '10.48550/arXiv.2409.04528', '2024arXiv240904528A']	['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory', 'Quantum Physics']	The renormalized stressenergy tensor for scalar fields in the Boulware state with applications to extremal black holes	2,024	197	0.27	['EPRINT_HTML', 'EPRINT_PDF']	3	https://arxiv.org/pdf/2409.04528.pdf	{'The renormalized stress-energy tensor for scalar fields in the Boulware state with applications to extremal black holes': 'Julio Arrechea ∗ \nIFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34014 Trieste, Italy SISSA, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy', 'Cormac Breen †': 'School of Mathematics and Statistics, Technological University Dublin, Grangegorman, Dublin 7, Ireland', 'Adrian Ottewill ‡': 'School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland \nLorenzo Pisani § and Peter Taylor ¶ \nCenter for Astrophysics and Relativity, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland (Dated: September 10, 2024) \nWe provide a mode-sum prescription to directly compute the renormalized stress-energy tensor (RSET) for scalar fields in the Boulware vacuum. The method generalizes the recently developed extended coordinate method which was previously only applicable to Hartle-Hawking states. We exhibit the accuracy and efficiency of the method by calculating the RSET in sub-extremal and extremal Reissner-Nordstrom spacetimes. We find numerical evidence for the regularity of the RSET at the extremal horizon regardless of the field mass and its coupling. We employ our numerical results of the RSET to source the semi-classical Einstein equations, demonstrating that if the RSET is considered as a static perturbation, it will either de-extremalize the black hole, or convert it into a horizonless object.', 'I. INTRODUCTION': "In the semiclassical approximation to quantum gravity, one replaces the stress-energy tensor in the gravitational field equations with the corresponding expectation value for the quantized matter fields, including ostensibly quantized graviton fields considered as linear perturbations about a background geometry. The gravitational field equations in this framework are \nG µν +Λ g µν + α 1 H (1) µν + α 2 H (2) µν = 8 π ⟨ ˆ T µν ⟩ ren , (1) \nwhere g µν is the metric of spacetime, Λ is the cosmological constant, G µν is the Einstein tensor and ⟨ ˆ T µν ⟩ ren is the renormalized expectation value of the stress-energy tensor (RSET) of the quantum fields in some quantum state. The tensors H (1) µν , H (2) µν are geometrical and include terms that are quadratic in the curvature; their inclusion arises through the point-splitting regularization process that yields ⟨ ˆ T µν ⟩ ren . This regularization process allows for the calculation of the finite physical RSET with an infinite renormalization of the constants Λ, α 1 and α 2 . It is difficult to conceive how to practically solve Eq. (1) in \na general self-consistent way since the right-hand side depends on the unknown solution g µν , and yet to compute the stress-energy tensor explicitly requires that we know the geometry of spacetime a priori . It is only by considering highly symmetric spacetimes [1] or by subjecting the RSET to additional approximations [2-4] that the backreaction problem becomes tractable. In black hole spacetimes, one usually proceeds by a perturbative expansion of the metric about a background classical solution. This leads to the reduced order semiclassical equations which describe, to first order in ℏ , the quantum backreaction induced by a quantum field on the classical geometry [5-7]. In these equations, the source term is the renormalized expectation value for a quantum field in a given quantum state on a fixed classical background spacetime. \nIt turns out that in most situations of interest, even the calculation of the renormalized expectation value of the stress-energy tensor (RSET) for a particular quantum state on a fixed background is a technically challenging endeavour. For black hole spacetimes, there are three main approaches to this calculation, the CandelasHoward approach [8] and its extensions (see for example [9-11]), the so-called 'pragmatic mode-sum prescription' [12-14] and the method developed by authors of this paper called the 'extended coordinate method' [15-18]. While all these methods have different approaches and advantages, one drawback of both the original CandelasHoward approach and the extended coordinate approach is that they were developed specifically for the field in the Hartle-Hawking state [19]. Working with this state \noffers a computational advantage since one can employ Euclidean techniques that enforce a discretization of the frequency spectrum. Anderson, Hiscock and Samuel [10] generalized the Candelas-Howard method to include a direct method for computing the RSET for a scalar field in the Boulware state [20]. Although the RSET in the Unruh, Boulware and Hartle-Hawking state were computed in Ref. [18], only the Hartle-Hawking state was directly renormalized, the RSET for the other two states were computed by leveraging the fact that the difference between two quantum states is regular. In this paper, we generalize the extended coordinate method to directly compute the RSET for a scalar field in the Boulware state. \nA natural question then is: if the RSET for a field in the Boulware state can be computed by first renormalizing in the Hartle-Hawking state and employing a state subtraction scheme, then why do we need a direct method for renormalizing in the Boulware state? There are several justifications for a direct renormalization prescription for Boulware. First, when employing the indirect method one needs to compute both the Euclidean modes (to do the renormalization in the Hartle-Hawking state) and the Lorentzian modes (to do the state subtraction scheme). Employing a direct method for Boulware requires only one set of modes and hence is computationally less expensive. Second, there are black hole spacetimes that do not admit a Hartle-Hawking state, for example, the extreme Reissner-Nordstrom spacetime, and hence one requires a direct method for computing the RSET in such situations. Finally, moving away from black hole spacetimes, if we consider semiclassical effects in the spacetime of a star, then the natural vacuum state is the Boulware state (there is, in fact, no other reference state). Hence this paper sets the foundation for renormalization in those contexts. \nThe approach we take is similar to that taken in the extended coordinate method for the Hartle-Hawking state [15-17]. We take an arbitrarily high-order expansion of the Hadamard parametrix which encodes the shortdistance behaviour of the two-point function, and we rewrite the terms in this expansion as mode-sums of the same form as the Boulware propagator. While the spirit is the same, the mathematical development is significantly trickier because the frequency spectrum is continuous, rather than the discrete frequency spectrum that one obtains in the Euclidean approach to defining the Hartle-Hawking state. As well as presenting a computational challenge, the continuous frequency spectrum also means that the Fourier transform of the terms in the Hadamard parametrix, which is needed to express these as mode-sums, yields functions of the frequency that are not ordinary functions but rather generalized functions. When we treat these mode-sums rigorously as generalized functions, it becomes completely transparent how an infra-red cutoff appears in these ω integrals and indeed it is completely explicit that the RSET is independent of the choice of this cutoff. A similar cutoff was introduced \nin Ref. [10] but it was introduced by necessity to eliminate an infra-red divergence in the WKB approximation. While the approaches are ultimately equivalent, it is clear here that the source of the cutoff is a consequence of treating rigorously the generalized functions that arise in the Fourier transform of the Hadamard parametrix. \nAs a consistency check, we contrasted our numerical results with previously known results for the sub-extremal and extremal Reissner-Nordstrom spacetime [18, 21] and found excellent agreement. We pay particular attention to the regularity of the RSET at the horizon of the extreme Reissner-Nordstrom black hole. Numerical evidence suggesting its regularity was found in [21] for massless fields (see [22-24] for discussions about backreaction effects). A clear advantage of the extended coordinate method over some previous approaches is that it does not invoke the Wentzel-Kramers-Brillouin (WKB) approximation in any step of the calculation, allowing to generate very accurate results for the RSET near the horizon. Our results reinforce the existing numerical evidence of the regularity of the RSET in extremal black holes, and extends those results to massive fields as well. \nThis paper is organized as follows. In Sec. II, we show how we express the expansion of the Hadamard parametrix as a mode-sum of the same type as the Boulware propagator for a scalar field in a static, spherically symmetric spacetime. In Sec. III, we give the explicit expressions for the renormalized expectation values for the vacuum polarization and the stress-energy tensor for a scalar field in the Boulware state. In Sec. IV, we apply the new method to compute the RSET for a scalar field on the Reissner-Nordstrom spacetime, including the extremal case where a direct calculation is necessary. In Sec. V, we employ our accurate RSET results to solve the reduced-order semiclassical equations to compute the backreaction on the extremal Reissner-Nordstrom black hole spacetime. Finally, in Sec. VI, we conclude with some discussion and future prospects.", 'II. RENORMALIZATION PRESCRIPTION IN THE BOULWARE STATE': "In this section, we will briefly outline the extended coordinate approach to calculating the RSET for a quantum scalar field in the Boulware quantum state [20], propagating on a static, spherically symmetric black hole spacetime. The Boulware state is vacuum according to a stationary observer far from the black hole. It is useful to treat the Boulware state as a thermal state at zero temperature and to employ Euclidean methods. We consider Euclidean line elements of the form: \nds 2 = f ( r ) dτ 2 + dr 2 /f ( r ) + r 2 ( dθ 2 +sin 2 θdϕ 2 ) . (2) \nIf this line element described a black hole spacetime with event horizon r = r + and surface gravity κ = 1 2 f ' ( r + ), then for the Hartle-Hawking state we would impose the periodicity condition τ = τ +2 π/κ + on the Euclidean \ntime. Imposing this periodicity discretizes the frequency spectrum of the field modes which is a major computational advantage. However, in the Boulware case, the field temperature is zero and so no such identification is made in the Euclidean time. As a result, the Euclidean geometry for a non-extremal black hole contains a conical singularity at the horizon r = r + and indeed the quantum expectation values will diverge there as a result. Notwithstanding that we do not have a discrete frequency, the Euclidean approach is still useful here since the Green function does not require an ' i ϵ ' prescription to avoid singularities on the lightcone; there are no divergences on the lightcone since there is no lightcone in a Euclidean space. In particular, the wave equation now satisfies an elliptic equation \n( □ -µ 2 -ξ R ) ϕ = 0 , (3) \nwhere □ is the d'Alembertian operator with respect to the Euclidean metric, µ is the field mass with dimensions of inverse length, R is the Ricci curvature scalar of the background spacetime and ξ is the coupling strength between the field and the background geometry. The corresponding Euclidean Green function has the following mode-sum representation (with r = r ' for simplicity) on this black hole spacetime, \nG ( x, x ' ) = 1 8 π 2 ∞ ∑ l =0 (2 l +1) P l (cos γ ) ∫ ∞ -∞ dω e iω ∆ τ g ωl ( r ) , (4) \nwhere ∆ x ≡ x ' -x ∼ O ( ϵ ) is the coordinate separation, γ is the geodesic distance on the 2-sphere and P l ( z ) is the Legendre polynomial of the first kind. We have denoted by g ωl ( r ) := p ωl ( r ) q ωl ( r ) /N ωl the one-dimensional radial Green function evaluated at the same radius. The radial modes p ωl ( r ), q ωl ( r ) are solutions of the homogeneous equation: \n[ d dr ( r 2 f ( r ) d dr ) -r 2 ( ω 2 f ( r ) +( µ 2 + ξ R ) ) -l ( l +1) ] χ ωl ( r ) = 0 , (5) \nwith p ωl ( r ) the solution regular at the horizon and q ωl ( r ) regular at the outer boundary (usually spatial infinity or the cosmological horizon). The normalization constant is given by \nN ωl = -r 2 f ( r ) W{ p ωl ( r ) , q ωl ( r ) } , (6) \nwhere W{ p, q } denotes the Wronskian of the two solutions. \nIn the coincidence limit ∆ x → 0 (i.e. γ → 0 and ∆ τ → 0), the mode sum (4) diverges. To renormalize this mode sum, we will adapt the so-called extended coordinate implementation of Hadamard renormalization developed in Refs. [15-17]. For a quantum state satisfying the Hadamard condition, the local singularity structure \nof the two-point function when x and x ' are sufficiently close is encapsulated in the Hadamard parametrix \nK ( x, x ' ) = 1 8 π 2 ( ∆ 1 / 2 ( x, x ' ) σ ( x, x ' ) + V ( x, x ' ) log ( 2 σ ( x, x ' ) ℓ 2 ) ) (7) \nwhere σ ( x, x ' ) is Synge's world function which measures half the square of the geodesic distance between x and x ' , ∆ 1 / 2 ( x, x ' ) and V ( x, x ' ) are symmetric biscalars that are regular in the coincidence limit x ' → x . Moreover, these biscalars depend only on the local geometry and the field parameters, not on any global considerations such as the quantum state. The lengthscale ℓ here is an arbitrary lengthscale associated with the renormalization ambiguity. It arises since V ( x, x ' ) is a solution to the homogeneous wave equation and so one is free to add arbitrary factors of V ( x, x ' ) to K ( x, x ' ) and it will remain a parametrix for the wave operator. Since σ ( x, x ' ) → 0 as x ' → x , it is clear that K ( x, x ' ) is singular at coincidence. \nThe fact that the mode-sum (4) does not converge at coincidence is because the two-point function possesses singularities precisely of the type in Eq. (7) in the shortdistance limit. The goal is to subtract K ( x, x ' ) from G ( x, x ' ) in such a way that the coincidence limit can be meaningfully taken. The only way to proceed in this approach is to express K ( x, x ' ) as a mode-sum of the same type as the mode-sum representation of Eq. (4) and then to subtract mode-by-mode. This renormalized mode-sum will converge in the coincidence limit. \nExpressing the Hadamard parametrix as a mode-sum is non-trivial. The approach of Refs. [15, 16] is to expand the parametrix in a judiciously chosen set of extended coordinates. In that context, it was assumed the field was in the Hartle-Hawking state and the extended coordinates were adapted to that state by ensuring the time dependence had the appropriate thermal periodicity. This must be amended here for the Boulware state. We define \ns 2 = f ( r ) ∆ τ 2 +2 r 2 (1 -cos γ ) , (8) \nand then, rather than expand the parametrix in coordinate separation ∆ x , we expand in s and ∆ τ (we take radial points together ∆ r = 0). The result of expanding the parametrix up to O ( ϵ 2 m log ϵ ) in these new extended \ncoordinates is \nK ( x, x ' ) = 1 8 π 2 ( m ∑ a =0 a ∑ b =0 D (r) ab ( r ) ∆ τ 2 a +2 b s 2 b +2 + m ∑ a =1 a ∑ b =1 D (p) ab ( r )∆ τ 2 a -2 b s 2 b -2 + m -1 ∑ a =1 a -1 ∑ b =0 T (r) ab ( r ) ∆ τ 2 a +2 b +2 s 2 b +2 + m -1 ∑ a =0 a ∑ b =0 T (l) ab ( r ) s 2 a -2 b ∆ τ 2 b log ( s 2 ℓ 2 ) + m -1 ∑ a =1 a ∑ b =0 T (p) ab ( r ) s 2 a -2 b ∆ τ 2 b ) + O ( ϵ 2 m log ϵ ) . (9) \nThe coefficients D (r) ab ( r ) and D (p) ab ( r ) come from the expansion of the direct part of the parametrix U/σ and correspond to the coefficients of terms in the expansion that are rational in ∆ τ 2 and s 2 and those that are polynomial in ∆ τ 2 and s 2 , respectively. Similarly, T (r) ab ( r ) and T (p) ab ( r ) are the coefficients of terms that are rational and polynomial coming from the tail V log(2 σ/ℓ 2 ) of the parametrix, while T (l) ab ( r ) is the coefficient of terms in the tail that contain a logarithm. Expressions for the first few of these coefficients are given in Appendix C. \nFormally speaking, this expansion need only be computed up to m = 1 in order to remove the divergences from the Green function. Nevertheless, by expanding up to higher order terms, including terms that formally vanish in the coincidence limit, and expressing these terms as a mode-sum permits us to have precise control over the rate of convergence of the renormalized mode sum. On the other hand, if we only subtract the singular terms, the mode sum would indeed converge, however, it would converge slowly. Furthermore, the distinction between the different types of terms that occur in the expansion is an important one. In particular, the terms that are polynomial in the variables ∆ τ 2 and s 2 do not require a mode-sum representation. For example, a multipole expansion of terms that are polynomial in s 2 would involve finite multipole moments and hence cannot affect the large l behaviour. Hence the terms above that are polynomial in the expansion parameters are left as-is. The rational and logarithmic terms will be expressed as mode sums. This is the focus of the next section.", 'A. Regularization Parameters for the Direct Part': 'Starting with the direct part of the Hadamard parametrix, which involves terms of the form ∆ τ 2 a +2 b /s 2+2 b for a ≥ b ≥ 0. We take as an ansatz for \nthe mode-sum representation of this term the following \n∆ τ 2 a +2 b s 2+2 b = ∞ ∑ l =0 (2 l +1) P l (cos γ ) ∫ ∞ -∞ dωe iω ∆ τ Ψ ωl ( a, b \| r ) , (10) \nwhere the generalized functions Ψ ωl ( a, b \| r ) we call the regularization parameters for the direct part. The goal is to invert the above ansatz and hence obtain an explicit closed-form representation for the regularization parameters. A double integral expression for the regularization parameters follows immediately from the completeness of the Legendre polynomials and the Fourier transform of the delta distribution, yielding \nΨ ωl ( a, b \| r ) = 1 4 π ∫ 1 -1 dx ∫ ∞ -∞ dt e -iωt t 2 a +2 b P l ( x ) ( f t 2 +2 r 2 (1 -x )) 1+ b , (11) \nwhere we have relabeled the integration variables by x = cos γ and t = ∆ τ . If we now define \nρ 2 = f 2 r 2 , (12) \nthen we can rewrite our integral expression above as \nΨ ωl ( a, b \| r ) = 1 4 π 1 (2 r 2 ) b +1 ∫ 1 -1 dxP l ( x ) ( -1) b b !2 b × ( 1 ρ ∂ ∂ρ ) b ∫ ∞ -∞ dt e -iωt t 2 a ( ρ 2 t 2 +1 -x ) . (13) \nNow it is this t -integral that distinguishes the regularization parameters in the Hartle-Hawking and the Boulware states. In the former, the t -integral above would be over a finite domain on account of the time periodicity of a thermal state. But in this case, the integral is over the entire real line which means that for a ≥ 1, the integral does not converge in the sense of ordinary functions and must be considered in the sense of generalized functions. The most straightforward way to proceed is to start with the a = 0 integral which is an ordinary function and is given by \n∫ ∞ -∞ dt e -iωt ( ρ 2 t 2 +1 -x ) = π e -\| ω \| √ 1 -x/ρ ρ √ 1 -x . (14) \nFor a ≥ 1, we can take (2 a )-derivatives with respect to ω of the result above. Since the exponent on the right-hand side involves \| ω \| and not ω , we pick up delta distributions and their derivatives. The result is \n∫ ∞ -∞ dt e -iωt t 2 a ( ρ 2 t 2 +1 -x ) = π ( -1) a e -\| ω \| √ 1 -x/ρ (1 -x ) a -1 / 2 ρ 2 a +1 -2 π ( -1) a a ∑ p =1 (1 -x ) a -p ρ 2 a -2 p +2 δ (2 p -2) ( ω ) , (15) \nwhich is clearly not an ordinary function. Substituting this back into (13) gives \nΨ ωl ( a, b \| r ) = 1 (2 r 2 ) b +1 ( -1) a + b b !2 b +2 ( 1 ρ ∂ ∂ρ ) b { 1 ρ 2 a +1 A ωl ( a \| ρ ) -2 a ∑ p =1 δ (2 p -2) ( ω ) ρ 2 a -2 p +2 B l ( a, p ) } , (16) \nwhere \nA ωl ( a \| ρ ) = ∫ 1 -1 P l ( x ) e -\| ω \| √ 1 -x/ρ (1 -x ) a -1 / 2 dx, (17) B l ( a, p ) = ∫ 1 -1 P l ( x )(1 -x ) a -p dx. (18) \nThe second integral here is a standard one. The result is that integral vanishes whenever l ≥ a -p + 1 and \notherwise gives \nB l ( a, p ) = ( -1) l 2 a -p +1 Γ( a -p +1) 2 Γ( a -p + l +2)Γ( a -p +1 -l ) . (19) \nThe A ωl ( a \| ρ ) integral can also be easily performed and there are several different equivalent representations. The tidiest form involves derivatives of the modified Bessel functions \nA ωl ( a \| ρ ) = 2 a +3 / 2 ∂ 2 a ∂z 2 a I l +1 / 2 ( z/ 2) K l +1 / 2 ( z/ 2) , (20) \nwhere \nz = \| ω \| √ 2 ρ = \| ω \| r √ f . (21) \nHowever, this is not the best representation for numerically computing this object. For numerical purposes, except when z >> 1, we find the following representation much more efficient \nA ωl ( a \| ρ ) = 2 a √ 2 π ( -1) l { Γ( a + 1 2 ) 2 2 ˆ F 3 ( { a + 1 2 , a + 1 2 } { a + l + 3 2 , a -l + 1 2 , 1 2 } ω 2 2 ρ 2 ) -Γ( a +1) 2 \| ω \| √ 2 ρ 2 ˆ F 3 ( { a +1 , a +1 } { 3 2 , a + l +2 , a -l +1 } ω 2 2 ρ 2 ) } , (22) \nwhere 2 ˆ F 3 is the regularized general hypergeometric function [25]. Now using the fact that, \n( 1 ρ ∂ ∂ρ ) b ρ -α = ( -1) b 2 b Γ( α/ 2 + b ) Γ( α/ 2) ρ -α -2 b , (23) \nwe can easily express the derivatives in terms of hypergeometric functions. Putting all of the above together, and after some simplification, we obtain the following expression for the regularization parameters \nΨ ωl ( a, b \| r ) = I ωl ( a, b \| r ) -(2 r 2 ) a f a + b +1 ( -1) a + l b ! a ∑ p =1 2 a -p ( a -p )!( a -p + b )! ( a -p + l +1)!( a -p -l )! ( f 2 r 2 ) p δ (2 p -2) ( ω ) , (24) \nwhere, for later convenience, we have defined \nI ωl ( a, b \| r ) = (2 r 2 ) a -1 / 2 f a + b +1 / 2 2 a -2 √ 2 π ( -1) a + l Γ( b +1) { Γ( a + 1 2 )Γ( a + b + 1 2 ) 2 ˆ F 3 ( { a + 1 2 , a + b + 1 2 } { a + l + 3 2 , a -l + 1 2 , 1 2 } ω 2 r 2 f ) -Γ( a +1)Γ( a + b +1) \| ω \| r √ f 2 ˆ F 3 ( { a +1 , a + b +1 } { 3 2 , a + l +2 , a -l +1 } ω 2 r 2 f ) } . (25) \nSubstituting this back into (10) and performing the ω integration in the terms involving delta distributions gives \n∆ τ 2 a +2 b s 2+2 b = ∞ ∑ l =0 (2 l +1) P l (cos γ ) ∫ ∞ -∞ dω e iω ∆ τ I ωl ( a, b \| r ) + 1 f b +1 a ∑ p =1 ( a -p + b )! b !( a -p )! ( 2 r 2 f ) a -p ∆ τ 2 p -2 (cos γ -1) a -p . (26) \nIn arriving at the last line of this expression, we made use of the identity \nn ∑ l =0 ( -1) l (2 l +1) P l (cos γ ) ( n + l +1)!( n -l )! = (1 -cos γ ) n 2 n ( n !) 2 , (27) \nwhich can be proved by induction. In the coincidence \nlimit, the last line of (26) is nonzero only when p = a = 1.', 'B. Regularization Parameters for the Logarithmic Terms in the Tail': "In this subsection, we derive a mode-sum representation for the terms in the tail part of the Hadamard parametrix which in the partial coincidence limit ∆ r = 0 involves terms of the form ∆ τ 2 b s 2 a -2 b log( s 2 /ℓ 2 ). For later convenience, we will write the logarithmic terms as \n∆ τ 2 b s 2 a -2 b log( s 2 /ℓ 2 ) = ∆ τ 2 b s 2 a -2 b log( ρ 2 ∆ τ 2 +1 -x ) +∆ τ 2 b s 2 a -2 b log(2 r 2 /ℓ 2 ) . (28) \nThe second term here is polynomial in ∆ τ 2 and s 2 and requires no mode-sum representation. This term contains the arbitrary lengthscale ℓ which encodes the renormalization ambiguity. This term does not contribute to the vacuum polarization since a ≥ 1 in these terms and hence they all vanish in the coincidence limit. For the first term above, we assume the mode-sum ansatz \n∆ τ 2 b s 2 a -2 b log( ρ 2 ∆ τ 2 +1 -cos γ ) = ∞ ∑ l =0 (2 l +1) P l (cos γ ) ∫ ∞ -∞ dω e iω ∆ τ χ ωl ( a, b \| r ) (29) \nand as before, we can invert to obtain a formal double integral expression for the regularization parameters \nχ ωl ( a, b \| r ) = (2 r 2 ) a -b 4 π ∫ 1 -1 dxP l ( x ) × lim α → a ∂ ∂α ∫ ∞ -∞ e -iωt t 2 b ( ρ 2 t 2 +1 -x ) α -b dt, (30) \nwhere we have expressed the log terms as the limit of a derivative with respect to an exponent α , and where t and x are defined as before. The time integral above is a complicated generalized function but we can express it in terms of the derivative of a simpler integral as \nχ ωl ( a, b \| r ) = (2 r 2 ) a -b 4 π 1 2 b ( 1 ρ ∂ ∂ρ ) b ∫ 1 -1 dxP l ( x ) × lim α → a ∂ ∂α Γ( α -b +1) Γ( α +1) ∫ ∞ -∞ e -iωt ( ρ 2 t 2 +1 -x ) α dt. (31) \nThe advantage of this form is that the time integral can now be performed explicitly yielding, \nχ ωl ( a, b \| r ) = (2 r 2 ) a -b √ 2 π 1 2 b ( 1 ρ ∂ ∂ρ ) b × ∫ 1 -1 dxP l ( x ) F ωl ( a, b \| x, r ) (32) \nwhere \nF ωl ( a, b \| x, r ) = lim α → a ∂ ∂α { 2 α ρ α -1 / 2 Γ( α -b +1) \| ω \| α +1 / 2 Γ( -α )Γ( α +1) (1 -x ) α/ 2+1 / 4 K α +1 / 2 ( \| ω \| √ 1 -x ρ )} , (33) \nand K α +1 / 2 ( z ) is the modified Bessel function of the second kind. From the series representation of the Bessel function, we obtain \nF ωl ( a, b \| x, r ) = π √ 2 lim α → a ∂ ∂α { Γ( α -b +1) Γ( α +1) sec πα (2 ρ ) 2 α Γ( -α ) \| ω \| 2 α +1 ∞ ∑ k =0 1 k !Γ( -α + k + 1 2 ) ω 2 k (1 -x ) k (2 ρ ) 2 k -Γ( α -b +1) Γ( α +1) sec πα (1 -x ) α +1 / 2 Γ( -α )2 ρ ∞ ∑ k =0 1 k !Γ( α + k + 3 2 ) ω 2 k (1 -x ) k (2 ρ ) 2 k } . (34) \nThe second term here is straightforward to deal with in the limit as α → a . In particular, it is evidently an entire function of α . In fact, if we consider the Taylor series for 1 / Γ( -α ) about a positive integer a , we have \n1 Γ( -α ) =( -1) a +1 a !( α -a ) + ( -1) a +1 a ! ψ ( a +1)( α -a ) 2 + O ( α -a ) 3 , (35) \nwhere ψ ( z ) = Γ ' ( z ) / Γ( z ) is the polygamma function, and so it is clear that the only term in the last line of (34) \nthat survives the limit α → a is when the α -derivative acts on 1 / Γ( -α ). \nThe first term in (34) is more subtle owing to the presence of the \| ω \| -2 α -1 term. Clearly this is not an ordinary function since it is singular at ω = 0. It is a generalized function whose definition involves a regularization at ω = 0 (not to be confused with the renormalization prescription required to regularize the coincidence limit of the point-split two-point function). One way to treat this is to simply introduce an infra-red cutoff in the ω - \nral in (28), but this is not the rigorous approach. We will see that, if we treat \| ω \| -2 α -1 rigorously as a generalized function, a cutoff will emerge naturally and indeed it will be clear that the overall result will be independent of this choice. The salient point is that, as a function of the exponent α , the definition (which involves its regularization as a distribution) of \| ω \| -2 α -1 has a simple pole at integer values of α . However, this pole is canceled by the zero in 1 / Γ( -α ) at integer values (see Eq. (35)) so that the combination 1 / (Γ( -α ) \| ω \| 2 α +1 ) is actually an entire function. We require an expansion of this combination about the integer α = a up to order ( α -a ) 2 if we are to compute the limit appearing in (34). We already have a series for 1 / Γ( -α ). Next, we note the Laurent series for \| ω \| -2 α -1 which can be found in Ref. [26], \n\| ω \| -2 α -1 = -δ (2 a ) ( ω ) (2 a )!( α -a ) + \| ω \| -2 a -1 + δ (2 a ) ( ω ) (2 a )! 2 log λ + O ( α -a ) , (36) \nwhere the λ appearing here is an arbitrary inverse lengthscale required because ω has dimensions of inverse length. This inverse lengthscale will be shown later to be related to an infra-red cutoff in certain integrals over the frequency. The corresponding result for Eq. (36) in Ref. [26] is actually for a dimensionless generalized function and \nso doesn't contain the additional log λ term, but applying the result from [26] to ˆ ω = ω/λ gives Eq. (36). It is essential to understand that \| ω \| -2 a -1 appearing on the right-hand side is to be understood in terms of its regularization as a generalized function. For example, for φ ( ω ) a test function of compact support, we have \n( \| ω \| -2 a -1 , φ ( ω )) = ∫ ∞ 0 dω ω -2 a -1 [ φ ( ω ) + φ ( -ω ) -2 a -1 ∑ m =0 φ (2 m ) (0) (2 m )! ω 2 m -2 φ (2 a ) (0) (2 a )! ω 2 a θ ( λ -ω ) ] , (37) \nwhere θ ( z ) is the step function. The last term in the regularization requires this step function since otherwise the integral would not converge at infinity. \nNow combining (35) and (36) gives us the desired expansion \n1 Γ( -α ) \| ω \| 2 α +1 = ( -1) a a ! (2 a )! δ (2 a ) ( ω ) +( α -a )( -1) a a ! [ ( ψ ( a +1) -2 log λ ) (2 a )! δ (2 a ) ( ω ) -1 \| ω \| 2 a +1 ] + O ( α -a ) 2 . (38) \nEmploying these limits in Eq. (34) gives, after some tedious simplifications, \nF ωl ( a, b \| x, r ) = π √ 2 ( a -b )! { (( ψ ( a -b +1) + ψ ( 1 2 -a + k ) + 2 ln(2 ρ/λ )) a ∑ k =0 δ (2 a -2 k ) ( ω )(1 -x ) k (2 ρ ) 2 a -2 k k !(2 a -2 k )!Γ( k -a + 1 2 ) -∞ ∑ k =0 \| ω \| 2 k -2 a -1 (1 -x ) k (2 ρ ) 2 a -2 k k !Γ( k -a + 1 2 ) + ∞ ∑ k =0 ω 2 k (1 -x ) k + a +1 / 2 k !Γ( k + a + 3 2 )(2 ρ ) 2 k +1 } , (39) \nwhere we also made use of the distributional identity \nω 2 k δ (2 a ) ( ω ) = { (2 a )! (2 a -2 k )! δ (2 a -2 k ) ( ω ) k ≤ a 0 k > a (40) \nto truncate the infinite sums involving delta distributions. Substituting this expression back into (32) and performing the integration yields \nχ ωl ( a, b \| r ) = √ π (2 r 2 ) a -b ( a -b )! ( -1) l 2 b ( 1 ρ ∂ ∂ρ ) b { ∞ ∑ k =0 Γ( k + a + 3 2 )2 k + a +3 / 2 ω 2 k k ! Γ( k + a + l + 5 2 ) Γ( k + a -l + 3 2 )(2 ρ ) 2 k +1 -∞ ∑ k =0 k !2 k (2 ρ ) 2 a -2 k \| ω \| 2 k -2 a -1 ( k + l +1)!( k -l )!Γ( k -a + 1 2 ) +[ ψ ( a -b +1) + ψ ( k -a + 1 2 ) + 2 log(2 ρ/λ )] a ∑ k =0 k !2 k (2 ρ ) 2 a -2 k δ (2 a -2 k ) ( ω ) (2 a -2 k )!( k + l +1)!( k -l )!Γ( k -a + 1 2 ) } . (41) \nThe second term is a generalized function of ω whenever k ≤ a and an ordinary function otherwise. In the former case, it is understood in the sense of (37). It would be better if we could could express the definition in (37) in a way that is independent of the test function. It can be shown that the regularization in Eq. (37) is equivalent to defining \nthe generalized function as \n2 ∑ k =0 ( a -k )! k ! ( a -k -b )!Γ( k -a + 1 2 )( k + l +1)!( k -l )!(2 ρ 2 ) k ∑ p =0 p = a -k δ (2 p ) ( ω ) λ 2 p -2 a +2 k (2 p )!(2 p -2 a +2 k ) , \n̸ \n1 \| ω \| 2 j +1 = θ ( ω 2 -λ 2 ) \| ω \| 2 j +1 +2 ∞ ∑ k = j λ 2 k -2 j δ (2 k ) ( ω ) (2 k )!(2 k -2 j ) , j > 0 ∈ Z (42) \n̸ \nwhere λ > 0 is the arbitrary cut-off. This can be straightforwardly proven by splitting the integral in (37) into intervals (0 , λ ) and ( λ, ∞ ), and then employing a Taylor series about ω = 0 for the test function on the interval (0 , λ ). Employing definition (42) in the second term of (41) for k ≤ a gives \n̸ \nχ ωl ( a, b \| r ) = √ π (2 r 2 ) a -b ( a -b )! ( -1) l 2 b ( 1 ρ ∂ ∂ρ ) b { ∞ ∑ k =0 Γ( k + a + 3 2 )2 k + a +3 / 2 ω 2 k k ! Γ( k + a + l + 5 2 ) Γ( k + a -l + 3 2 )(2 ρ ) 2 k +1 -θ ( ω 2 -λ 2 ) a ∑ k =0 k !2 k (2 ρ ) 2 a -2 k \| ω \| 2 k -2 a -1 ( k + l +1)!( k -l )!Γ( k -a + 1 2 ) -∞ ∑ k =0 ( k + a +1)!2 k + a +1 \| ω \| 2 k +1 ( k + a + l +2)!( k + a +1 -l )!Γ( k + 3 2 )(2 ρ ) 2 k +2 -2 a ∑ k =0 k !2 k (2 ρ ) 2 a -2 k ( k + l +1)!( k -l )!Γ( k -a + 1 2 ) ∞ ∑ p = a -k λ 2 p +2 k -2 a δ (2 p ) ( ω ) (2 p )!(2 p +2 k -2 a ) +[ ψ ( a -b +1) + ψ ( k -a + 1 2 ) + 2 log(2 ρ/λ )] a ∑ k =0 k !2 k (2 ρ ) 2 a -2 k δ (2 a -2 k ) ( ω ) (2 a -2 k )!( k + l +1)!( k -l )!Γ( k -a + 1 2 ) } . \nFinally computing the derivatives and simplifying gives \nχ \nωl \n( \na, b \n\| \nr \nwhere \nJ ωl ( a, b \| r ) = √ π (2 r 2 ρ 2 ) a -b ( a -b )!2 2 a ( -1) l { -a -b ∑ k =0 θ ( ω 2 -λ 2 ) ( a -k )! k ! \| ω \| 2 k -2 a -1 ( a -k -b )!Γ( k -a + 1 2 )( k + l +1)!( k -l )!(2 ρ 2 ) k -( -1) b (2 ρ 2 ) a +1 \| ω \| ( a +1)! b ! 2 ˆ F 3 ( { a +2 , b +1 } { 3 2 , a -l +2 , a + l +3 } ω 2 2 ρ 2 ) + ( -1) b 2 a +1 / 2 ρ 2 a +1 Γ( a + 3 2 )Γ( b + 1 2 ) 2 ˆ F 3 ( { a + 3 2 , b + 1 2 } { 1 2 , a -l + 3 2 , a + l + 5 2 } ω 2 2 ρ 2 ) } . (45) \nPutting this together gives \n∞ ∞ \n∆ τ 2 b s 2 a -2 b log( ρ 2 ∆ τ +1 -cos γ ) = ∫ -∞ dω e iω ∆ τ ∑ l =0 (2 l +1) P l (cos γ ) χ ωl ( r ) = ∫ ∞ -∞ dω e iω ∆ τ ∞ ∑ l =0 (2 l +1) P l (cos γ ) J ωl ( a, b \| r ) + √ π (2 r 2 ρ 2 ) a -b ( a -b )!2 2 a { a -b ∑ k =0 ( -1) a -k ( a -k )!∆ τ 2 a -2 k (1 -cos γ ) k 2 k k !( a -k -b )!(2 a -2 k )!Γ( k -a + 1 2 )(2 ρ 2 ) k [ [ ψ ( a -b +1) -ψ ( a -b -k +1) + ψ ( a -k +1) + ψ ( k -a + 1 2 ) + 2 log(2 ρ/λ ) ] -b ∑ k =1 ( -1) b ( b -k )!( k -1)!∆ τ 2 b -2 k (1 -cos γ ) k + a -b 2 k + a -b ( k + a -b )!(2 b -2 k )!Γ( k -b + 1 2 )(2 ρ 2 ) k + a -b -2 a -b ∑ k =0 ( a -k )!(1 -cos γ ) k 2 k k !( a -k -b )!Γ( k -a + 1 2 )(2 ρ 2 ) k m -k ∑ p =0 p = a -k ( -1) p ∆ τ 2 p λ 2 p -2 a +2 k (2 p )!(2 p -2 a +2 k ) } . (46) \n) = J ωl ( a, b \| r ) + √ π (2 r 2 ρ 2 ) a -b ( a -b )! 2 2 a ( -1) l × { a ∑ k =0 k !( a -k )! δ (2 a -2 k ) ( ω ) [ ψ ( a -b +1) -ψ ( a -b -k +1) + ψ ( a -k +1) + ψ ( k -a + 1 2 ) + 2 log( 2 ρ λ ) ] ( a -k -b )!(2 a -2 k )!Γ( k -a + 1 2 )( k + l +1)!( k -l )!(2 ρ 2 ) k -a ∞ } (44) \n̸ \n(43) \nEven though the sum over p in equation (46) is technically up to ∞ , we actually only need terms up to order ϵ 2 m in the Hadamard parametrix. In that expression, the order of the expansion is given by (1 -cos γ ) k ∆ τ 2 p and is ϵ 2 k +2 p . Hence, we have truncated the sum at p = m -k . In arriving at the expression above, we have also made \nuse of the identity (27).", 'C. Mode-sum expression for the parametrix': "We now have mode-sum representations of all the singular terms in (9) yielding the following mode-sum representation of the Hadamard parametrix \n(47) \nK ( x, x ' ) = 1 8 π 2 ∫ ∞ -∞ dω e iω ∆ τ ∞ ∑ l =0 (2 l +1) P l (cos γ ) k ( m ) ωl ( r ) + 1 8 π 2 { m ∑ a =1 a ∑ b =1 D (p) ab ( r )∆ τ 2 a -2 b s 2 b -2 + m -1 ∑ a =1 a ∑ b =0 T (p) ab ( r )∆ τ 2 b s 2 a -2 b + m -1 ∑ a =0 a ∑ b =0 T (l) ab ( r )∆ τ 2 b s 2 a -2 b log(4 f/ ( ℓ 2 λ 2 )) + m ∑ a =1 a ∑ b =0 D (r) ab ( r ) (2 r 2 ) -b -1 b ! ( ρ 2 ) a + b +1 a ∑ p =1 ( -1) a + p ( a -p + b )! ρ 2 p ( a -p )! ∆ τ 2 p -2 (1 -cos γ ) a -p + m -1 ∑ a =1 a -1 ∑ b =0 T (r) ab ( r ) (2 r 2 ) -b -1 b ! ( ρ 2 ) a + b +2 a +1 ∑ p =1 ( -1) a +1+ p ( a +1 -p + b )! ρ 2 p ( a +1 -p )! ∆ τ 2 p -2 (1 -cos γ ) a +1 -p + m -1 ∑ a =0 a ∑ b =0 T (l) ab ( r ) √ π (2 r 2 ρ 2 ) a -b ( a -b )!2 2 a [ a -b ∑ k =0 ( -1) a -k ( a -k )!∆ τ 2 a -2 k (1 -cos γ ) k 2 k k !( a -k -b )!(2 a -2 k )!Γ( k -a + 1 2 )(2 ρ 2 ) k [ ψ ( a -b +1) -ψ ( a -b -k +1) + ψ ( a -k +1) + ψ ( k -a + 1 2 ) ] -b ∑ k =1 ( -1) b ( b -k )!( k -1)!∆ τ 2 b -2 k (1 -cos γ ) k + a -b 2 k + a -b ( k + a -b )!(2 b -2 k )!Γ( k -b + 1 2 )(2 ρ 2 ) k + a -b -2 a -b ∑ k =0 ( a -k )!(1 -cos γ ) k 2 k k !( a -k -b )!Γ( k -a + 1 2 )(2 ρ 2 ) k m -k ∑ p =0 p = a -k ( -1) p ∆ τ 2 p λ 2 p -2 a +2 k (2 p )!(2 p -2 a +2 k ) ]} + O ( ϵ 2 m log ϵ ) , \nwhere we have defined for compactness \nk ( m ) ωl ( r ) = m ∑ a =0 a ∑ b =0 D (r) ab ( r ) I ωl ( a, b \| r ) + m -1 ∑ a =1 a -1 ∑ b =0 T (r) ab ( r ) I ωl ( a +1 , b \| r ) + m -1 ∑ a =0 a ∑ b =0 T (l) ab ( r ) J ωl ( a, b \| r ) . (48) \nWhile the expression (47) looks very complicated, we note that many of the terms vanish in the coincidence limit.", 'III. VACUUM POLARIZATION AND COMPONENTS OF THE RSET': "When computing vacuum polarization, the coincidence limit kills almost all of the finite terms in the Hadamard parametrix above, we obtain \n⟨ ˆ ϕ 2 ⟩ ren = 1 8 π 2 { ∫ ∞ -∞ ∞ ∑ l =0 (2 l +1) ( g ωl ( r ) -k ( m ) ωl ( r ) ) dω -√ π m -1 ∑ a =1 a ∑ b =0 T (l) ab ( r ) ( a -1)!2 2 a f a -b Γ( -a + 1 2 ) λ 2 a } -1 8 π 2 [ D (p) 11 ( r ) + D (r) 10 ( r ) f + D (r) 11 ( r ) f 2 + T (l) 00 ( r )(log( f/ ( ℓ 2 λ 2 )) -2 γ E ) ] . (49) \n̸ \nThe mode-sum terms above can be made to converge as fast as we like by choosing m appropriately large. It can be shown that the renormalized vacuum polarization is completely independent on the choice of the cutoff (we report the calculations in appendix A for clarity). This property can also be proven for the regularization terms for the stress energy tensor. \nThe components of the RSET may be written in the form [17]: \n⟨ ˆ T µ ν ⟩ ren = -w µ ν -( ξ -1 2 ) w ; µ ; ν +( ξ -1 4 ) □ w δ µ ν + ξR µ ν w -1 8 π 2 v 1 δ µ ν . (50) \nwhere: \nw ( r ) = ⟨ ˆ ϕ 2 ⟩ ren ( r ) , w µν ( x ) ≡ lim x → x ' [ W ( x, x ' ) ; µν ] , W ( x, x ' ) = G ( x, x ' ) -K ( x, x ' ) , (51) \nand \nv 1 = 1 720 R µνρσ R µνρσ -1 720 R µν R µν -1 24 ( ξ -1 5 ) □ R + 1 8 ( ξ -1 6 ) 2 R 2 + 1 4 µ 2 ( ξ -1 6 ) R + 1 8 µ 4 , (52) \nwhich must be included in (50) to ensure the RSET is conserved [27]. \nDue to the symmetry of the background spacetime w is a function of r only, therefore once it has been calculated numerically to high accuracy on a suitably dense grid, the derivatives of w appearing in Eq. (50) are now easily and accurately obtained by differentiating an interpolation function for w . Considering the components of w a b , we have, by virtue of the wave equation satisified by W ( x, x ' ), that [28]: \nw r r = -w τ τ -w θ θ -w ϕ ϕ -ξR w -µ 2 w -3 4 π 2 v 1 . (53) \nFor the remaining non-zero components, w τ τ and w θ θ = w ϕ ϕ , it is advantageous to express these in terms of mixed derivatives at x and x ' of W ( x, x ' ) and derivatives of w ( r ) = ⟨ ˆ ϕ 2 ⟩ ren , using Synge's Rule [17]: \nlim x → x ' [ W ( x, x ' ) ; µ ' ν ] = 1 2 w ; µν ( x ) -w µν ( x ) . (54) \nThe required mixed time derivatives and mixed angular derivatives may be obtained by taking appropriate derivatives of Eq (9). Therefore, the RSET is determined by the calculation of three mode sums Eq.(49) along with \nthe following two mode sums: \n̸ \n[ g ττ ' W ,ττ ' ] = 1 8 π 2 f { ∫ ∞ -∞ ∞ ∑ l =0 (2 l +1) ω 2 ( g ωl -k ( m ) ωl ) dω -√ π m -1 ∑ a =0 a =1 a ∑ b =0 T (l) ab ( r ) a !2 2 a f a -b Γ( 1 2 -a )( a -1) λ 2 a -2 } + 1 4 π 2 [ T (p) 10 + D (p) 22 + 1 f ( T (p) 11 + D (p) 21 ) + D (r) 20 f 2 + D (r) 21 f 3 + D (r) 22 f 4 + T (r) 10 f 2 +( f T (l) 10 + T (l) 11 )(log( f/ ( ℓ 2 λ 2 )) -2 γ E +3) ] , (55) \n[ g ϕϕ ' W ,ϕϕ ' ] = 1 16 π 2 r 2 { ∫ ∞ -∞ ∞ ∑ l =0 (2 l +1) l ( l +1) ( g ωl -k ( m ) ωl ) dω + √ π m -1 ∑ a =2 a ∑ b =0 T (l) ab r 2 ( a -b )( a -2)!2 2 a f a -b -1 Γ( 3 2 -a ) λ 2 a -2 } + 1 4 π 2 [ D (p) 22 + T (p) 10 -D (r) 20 f 2 -2 D (r) 21 f 3 -3 D (r) 22 f 4 -T (r) 10 f 2 + T (l) 11 f + r 2 T (l) 10 (log( f/ ( ℓ 2 λ 2 )) -2 γ E +1) ] . (56) \nInserting the expressions in Eqs. (49), (55), (56) into Eq. (50) results in a natural splitting of the RSET components into a numeric and an analytic part: \n⟨ ˆ T µ ν ⟩ ren = ⟨ ˆ T µ ν ⟩ numeric + ⟨ ˆ T µ ν ⟩ analytic , (57) \nwhere ⟨ ˆ T µ ν ⟩ numeric contains the terms in Eqs. (49), (55), (56) that are in curly brackets, while ⟨ ˆ T µ ν ⟩ analytic is comprised of those terms in square brackets. We give explicitly the expression for ⟨ ˆ T µ ν ⟩ analytic in Appendix D. With these definitions, both ⟨ ˆ T µ ν ⟩ numeric and ⟨ ˆ T µ ν ⟩ analytic are independently conserved. In fact ⟨ ˆ T µ ν ⟩ analytic is equal, up to the addition of a conserved tensor, to the equivalent analytic component obtained by the AHS method [10], which in turn reproduces Page's approximation for a massless field. The fact that our direct method reproduces the AHS results, whose derivation depended on the use of the WKB approximation to the radial Green function, is surprising, particularly when contrasted with the results of the extended coordinates approach for a thermal field [17]. In that case the equivalent analytic component does not reproduce the AHS equivalent and neither the analytic nor numerical components are independently conserved, rather only their combination is conserved.", 'IV. APPLICATION TO THE REISSNER-NORDSTR OM SPACE-TIME': 'In this section, we apply the renormalization scheme presented developed herein to the case of scalar fields in the Boulware state in the Reissner-Nordstrom black hole space-time: \nds 2 = -( r -r + ) ( r -r -) r 2 dt 2 + r 2 ( r -r + ) ( r -r -) dr 2 + r 2 ( dθ 2 +sin θ 2 dϕ 2 ) , (58) \nwhere r ± = M ± √ M 2 -Q 2 \nWe will first verify this new method by reproducing results for the non extremal case, which were obtained via the state subtraction method in [18]. We then apply the method to the case of an extremal black hole, where we investigate the regularity of the Boulware state on the event horizon for both a massless and massive scalar field. While the regularity of the RSET has previously been examined by Anderson et al. [21], we are not aware of any previous exact results for a massive field. \nBefore presenting results for both the non-extremal and extremal cases, we will discuss the numerical implementation of the method for both cases.', '1. Non-Extremal Modes': "As seen from Eqs. (49-56), the mode sums required to compute the RSET can be expressed in terms of derivatives of ⟨ ˆ ϕ 2 ⟩ ren . In particular, the only mode sum we need to calculate is that for the one-dimensional Green function g ωl (4), defined as the normalized product of solutions to the radial equation (5). \nThus, to obtain the RSET with sufficient accuracy (as, for example, to check its regularity at the horizon) we need to generate highly-precise results for the vacuum polarization and, in turn, for the Euclidean modes. \nFor the non-extremal case the computation of the p ωl ( r ) modes is simplified by recasting the radial equation, (5), into a confluent Heun form. This is achieved by expressing the dependent variable χ ωl ( r ) as: \nχ ωl ( r ) = e -( ω -¯ ω ) r e ωr ∗ Y ( r ) \nand introducing a new independent variable: \nz = r + -r r + -r -. \nThe radial equation then takes the form of the confluent Heun equation: \nz ( z -1) Y '' ( z ) + ( δ 1 ( z -1) + δ 2 z + z ( z -1) δ 3 ) Y ' ( z ) +( q 2 z -q 1 ) Y ( z ) = 0 (59) \nwhere \n(60) \nq 1 = l ( l +1) + r 2 + ( ω -¯ ω ) 2 -( r + + r -)( ω -¯ ω ) -2 r + ¯ ω q 2 = ( r 2 + -r 2 -)( ω -¯ ω ) 2 -2( r + -r -)¯ ω δ 1 = 1 + 2 ωr 2 + r + -r -δ 2 = 1 -2 ωr 2 -r + -r -δ 3 = -2¯ ω ( r + -r -) z = r + -r r + -r - \nand with \n¯ ω = √ ω 2 + µ 2 . (61) \nIf we let H ( q 1 , q 2 , δ 1 , δ 2 , δ 3 ; z ) denote the solution to Eq. (59) that is analytic in the vicinity of z = 0 and normalized to unity there, then we may express the p ωl solution as: \np ωl ( r ) = e -( ω -¯ ω ) r e ωr ∗ H ( q 1 , q 2 , δ 1 , δ 2 , δ 3 ; z ) . (62) \nComputing the q ωl ( r ) modes is computationally harder. While these modes can still be written in confluent Heun form, the Heun functions with the appropriate boundary conditions for q ωl ( r ) are not built into Mathematica or Maple. There are several options one can consider for computing q ωl ( r ) but we found it most efficient to simply numerically integrate the radial equation inwards from a large r value using an asymptotic expansion for the initial conditions (see for example [10] for details of the asymptotic expansion). The initial conditions were optimized so that the asymptotic expansion solved the wave equation to our working precision with the least number of terms in the asymptotic expansion and for the smallest reasonable r value at which this precision could be achieved. Using this approach, the mode solutions and their derivatives that were generated were accurate to at least 30 significant digits. We tested this accuracy by checking the constancy of the Wronskian over the radial grid for the solution pairs { p ωl ( r ) , q ωl ( r ) } .", '2. Extremal Modes': "The extremal black hole metric in its Euclideanized form is \nds 2 = ( 1 -M r ) 2 dτ 2 + ( 1 -M r ) -2 dr 2 + r 2 ( dθ 2 +sin 2 θdϕ 2 ) , (63) \nfor which the mode equation (5) amounts to \nΦ '' ωl + 2Φ ' ωl r -M -[ r 4 ω 2 ( r -M ) 4 + µ 2 r 2 + l ( l +1) ( r -M ) 2 ] Φ ωl = 0 . (64) \nFor convenience, we follow [29] and adopt the dimensionless coordinate η = r/M -1, for which the wave equation is \nη 4 Φ '' ωl +2 η 3 Φ ' ωl -[ ω 2 M 2 ( η +1) 4 + l ( l +1) η 2 + µ 2 M 2 η 2 ( η +1) 2 ] Φ ωl = 0 . (65) \nThe radial solutions regular at the horizon and at radial infinity obey, respectively, the asymptotic expansions \np ωl = η 2 ωM e -ωM/η ∞ ∑ k =0 a k η k , η → 0 (66) \nand \nq ωl = η -2 ωM e ωM/η e -√ ω 2 + µ 2 Mη ∞ ∑ k =0 b k η -k -β 0 , η →∞ . (67) \nReplacing these relations in (65) we find the following recursion relations for the a k , b k coefficients, \na 1 = [ ˜ µ 2 + l ( l +1) + 2˜ ω (˜ ω -1) ] a 0 2˜ ω , a 2 = [ ˜ µ 2 +2˜ ω 2 ] a 0 2˜ ω + [ ˜ µ 2 + l ( l +1) + 2˜ ω (˜ ω -3) -2 ] a 1 4˜ ω , a k = ( ˜ µ 2 + ˜ ω 2 ) a k -3 2˜ ωu + ( ˜ µ 2 +2˜ ω 2 ) a k -2 ˜ ωu + [ k (1 -k ) + ˜ µ 2 + l ( l +1) -4˜ ωk +2˜ ω (1 + ˜ ω ) ] a k -1 2˜ ωk , (68) \nfor the p ωl mode, where ˜ ω = ωM , ˜ µ = µM , with a 0 = 1. For the q ωl mode we have, instead, \nβ 0 =1 -2˜ ω + √ ˜ ω 2 + ˜ µ 2 + ˜ ω 2 √ ˜ ω 2 + ˜ µ 2 , b 1 = [ ˜ µ 2 + l ( l +1) + β 0 (1 -β 0 -4˜ ω ) +2˜ ω ( 1 + ˜ ω -√ ˜ ω 2 + ˜ µ 2 )] b 0 × [ 2˜ µ 2 -2 β 0 √ ˜ ω 2 + ˜ µ 2 +4˜ ω ( ˜ ω -√ ˜ ω 2 + ˜ µ 2 )] -1 , b k = ( β 0 + v -2) ˜ ωb k -2 ( β 0 + k -1) √ ˜ ω 2 + ˜ µ 2 -2˜ ω ( ˜ ω -√ ˜ ω 2 + ˜ µ 2 ) -˜ µ 2 [ -k 2 + k (3 -2 β 0 -4˜ ω ) + l ( l +1) + ˜ µ 2 -β 2 0 + β 0 (3 -4˜ ω ) + 2˜ ω ( 3 + ˜ ω -√ ˜ ω 2 + ˜ µ 2 ) -2 ] b k -1 × [ 2 ( β 0 + k -1) √ ˜ ω 2 + ˜ µ 2 -4˜ ω ( ˜ ω -√ ˜ ω 2 + ˜ µ 2 ) -2˜ µ 2 ] -1 , (69) \nwith b 0 = 1.", '3. Accuracy Issues': 'The numerical implementation of the direct method is more challenging than the extended coordinate method developed for thermal states, which uses a discretized frequency spectrum. \nMost of the issues are related to the sampling of the continuous frequency spectrum. To numerically perform the integral in ω , one needs to interpolate the integrand over a predefined ω -grid. This grid must be fine enough so that the interpolation function works well, ensuring the result is independent of the grid choice. This becomes challenging when the integrand is very large and rapidly decaying. In this context, we found it necessary to push the cutoff λ to values no smaller than 1 / 10 and to increase the density of the ω -grid immediately after the cutoff. This is done to ensure the interpolation function can properly approximate the rapidly decaying behavior of the cutoff dependent terms in the integrand, which are much larger at smaller ω . \nAnother issue is related to the cancellation of the cutoffdependent terms. We have shown analytically that the renormalized vacuum polarization does not depend on the choice of the cutoff, as the cutoff-dependent terms ultimately cancel each other. However, achieving this outcome numerically is more challenging and can significantly impact the accuracy of the results. For the case of a thermal state, one can increase the accuracy of the results, by increasing the expansion order of the singular field ( m in Eq. (9)). Moreover, this is extremely efficient numerically as only a very modest number of modes are required. However for the direct approach, as the magnitude of the cutoff-dependent terms increases with increasing order, so does the potential error introduced by the numerical cancellation of these terms. To counteract this, as the expansion order increases, one has to also increase the density in the grid in ω immediately after the cutoff. In effect if one increases the expansion order, one has to also significantly increase the number of numerical modes required, making the process increasingly inefficient. Hence, there is a balance to be struck between the accuracy of the results and the efficiency of the method. For the results contained in this paper, we have found that a third order expansion ( m = 3) and summing/integrating up to l = 40/ ω = 5 gives sufficiently accurate results, with the conservation equation satisfied to approximately 9-10 decimal places. \nThere is an important exception to this choice and that is in the vicinity of the event horizon. All the cutoffdependent terms vanish on the event horizon and therefore as we approach the horizon the error introduced by these terms also decreases. This is particularly true in the extremal case, where the metric function f ( r ) has a double root. Therefore, for points close to the horizon (up at a distance of approximately M/ 10) we employ a sixth order expansion to obtain our results and find that the conservation equation is satisfied to approximately 13-15 decimal places for a massive field and 10-11 deci- \nmal places for the massive case. In addition, particularly near the horizon, the integral is dominated by the low ω behavior. Hence, it is crucial to select the smallest point in the ω grid accordingly and extrapolate the integrand to ω = 0 to avoid losing important contributions. \nBesides the problems related to the continuous ω spectrum, we have found different representations for I ωl ( a, b \| r ) and J ωl ( a, b \| r ). The most amenable to numerical implementation are the ones reported in the derivations above, which involve differences of two hypergeometric functions. However, these representations fail whenever z = \| ω \| r √ f ≫ 1, due to catastrophic cancellation. For the region of the domain where z ≫ 1, the representations in terms of derivatives of modified Bessel functions and generalized incomplete gamma functions are more efficient and not subject to numerical errors. We include them in appendix B for convenience.', 'B. Non-Extremal Results': 'In Fig. (1) we compare the results of the direct method outlined in this paper to the corresponding results obtained via a state-subtraction approach in [18], for the calculation of the RSET for a massless, minimally coupled scalar field in the Boulware state. The background spacetime on which this calculation is performed is the non-extremal Reissner-Nordstrom spacetime with Q = 2 M/ 10. As is evident from the plot in Fig. (1), there is good agreement between the two approaches, verifying the new method presented in this paper. However, the state-subtraction method appears to be more accurate, as measured by the conservation equation, in the vicinity of the event horizon. This suggests that for space-times where there exists a Hartle-Hawking state to use as a reference state, taking the state-subtraction approach outlined in [18] is the appropriate choice. However, for other space-times, such as the extremal Reissner-Nordstrom spacetime considered in the next section, where we do not have a Hartle-Hawking state to exploit, the direct approach of this paper is required for the RSET calculation for scalar field in the Boulware state. \nReturning to the splitting into independently conserved tensors in (57), we observe that the leading-order divergence of the RSET at the horizon is determined entirely by ⟨ ˆ T µ ν ⟩ analytic ∝ f -2 , with ⟨ ˆ T µ ν ⟩ numeric ∝ f -1 amounting to a sub-leading correction. This indicates that ⟨ ˆ T µ ν ⟩ analytic indeed captures exactly the singular behaviour of the Boulware state at event horizons, and further justifies its use in backreaction analyses as an analytic approximation to the exact RSET [30].', 'C. Extremal Results': "In this section we present the results of the extended coordinates approach to the calculation of the RSET for a \nFIG. 1: RSET components in the Boulware state for a massless, minimally coupled scalar field calculated via the direct method (line) and the state subtraction method (dots). The background spacetime is the non-extremal Reissner-Nordstrom spacetime with Q = 2 M/ 10. \n<!-- image --> \nscalar field in the Boulware quantum state on an extremal Reissner-Nordstrom spacetime. \nIn Figs. (2) and (3) we present the components of the RSET in the exterior region of the black hole for a massless and massive field ( µM = 1 / 10) respectively. For the massless case, on the horizon ⟨ ˆ T a b ⟩ numeric vanishes and the RSET is given precisely by ⟨ ˆ T a b ⟩ analytic with \n⟨ ˆ T µ ν ⟩\| r = M = ⟨ ˆ T µ ν ⟩ analytic \| r = M = 1 2880 π 2 M 4 δ µ ν , (70) \nwhich is equal to the corresponding RSET in the Bertotti-Robinson spacetime [21]. The fact that ⟨ ˆ T µ ν ⟩ analytic is exact on the horizon, suggests that it might serve as a good approximation for the full RSET, at least in the massless case. However this is not the case, with ⟨ ˆ T µ ν ⟩ analytic failing to capture the main features of the full results, except in the immediate vicinity of the horizon. Finally, as r increases we see that all RSET components tend to 0, as would be expected for a zero-temperature state. \nFor the massive case, the situation is quite different. Here both ⟨ ˆ T µ ν ⟩ analytic and ⟨ ˆ T µ ν ⟩ numeric diverge logarithmically on the horizon, however these divergences cancel to render each of the components of ⟨ ˆ T µ ν ⟩ finite there. Specifically, the analytic terms become, in the near-horizon limit, \n⟨ ˆ T µ ν ⟩ analytic \| r = M = µ 2 8 π 2 [ ( ξ -1 6 ) M 2 diag (1 , 1 , -1 , -1) + µ 2 4 δ µ ν ] log ( r M -1 ) . (71) \nIn contrast to the massless case, only the ⟨ ˆ T r r ⟩ and ⟨ ˆ T t t ⟩ components agree on the horizon with ⟨ ˆ T θ θ ⟩ taking on a distinct value. For massive fields, the r →∞ behaviour of the RSET components depends on the renormalisation lengthscale ℓ . Hence, there is a natural choice of the \nFIG. 2: Plot of the components ⟨ ˆ T µ ν ⟩ ren for a conformally coupled massless field in the Boulware state, on an extremal Reissner-Nordstrom spacetime. \n<!-- image --> \nlengthscale for massive fields for which the RSET tends to 0 as r →∞ . This choice is given by: \nℓ = 1 2 µ exp { 3 / 4 + 5 γ E } , \nand is the value chosen for the results presented in Figs. (3) and (5). For the massless case, there is no such natural choice, so we simply set ℓ = M . \nIn Figs. (4) and (5) we focus on the near horizon region and investigate the regularity of the Boulware state in an extremal Reissner-Nordstrom spacetime for a massless and massive field ( µM = 1 / 10) respectively. The quantity ⟨ ˆ T µ ν ⟩ analytic is not regular on the event horizon resulting in an infinite energy density there as perceived by a freely falling observer: \n⟨ ˆ E⟩ analytic = ⟨ ˆ T r r ⟩ analytic -⟨ ˆ T t t ⟩ analytic f = -µ 2 [ 3 M 2 µ 2 +12 ( ξ -1 6 )] 96 π 2 M 2 ( r M -1 ) -2 + 15 M 4 µ 4 -2 240 π 2 M 4 ( r M -1 ) -1 -1 60 π 2 M 4 log ( r M -1 ) + O ( r M -1 ) 0 . (72) \nThis quantity possesses a ( r -M ) -1 divergence for a massless field and a ( r -M ) -2 for a massive field. In Figs. (4) and (5), we show that this divergence appears to be precisely canceled by a corresponding divergence from ⟨ ˆ T µ ν ⟩ numeric , leaving a total ⟨ ˆ E⟩ that is finite on the event horizon. Of course, to prove this definitively would require a closed form expression for the quantity ⟨ ˆ E⟩ on the horizon (or equivalently a near horizon expression for ⟨ ˆ T µ ν ⟩ to O ( r -M ) 2 ). However, we feel the numerical results presented here provide compelling evidence for the regularity of the Boulware state. \nWe have therefore used the novel method presented in the paper to confirm the results of [21] for a massless field \nFIG. 3: Plot of the components ⟨ ˆ T µ ν ⟩ ren for a minimally coupled field with mass µM = 1 / 10 in the Boulware state, on an extremal Reissner-Nordstrom spacetime. \n<!-- image --> \nFIG. 4: Plot of the cancellation of the divergence in the numerical and analytical contributions to the quantity ⟨ ˆ E⟩ for a minimally coupled massless field, leaving a total result that appears to be finite on the event horizon of the extremal Reissner-Nordstrom spacetime \n<!-- image --> \n- \nand provide, to the best of the authors' knowledge, the first exact results for the RSET of a massive scalar field in the extremal Reissner-Nordstrom spacetime.", 'V. BACKREACTION IN EXTREMAL BLACK HOLES': "Computing the RSET in the background of an extremal black hole naturally begs the question of whether extremal horizons are stable under quantum corrections. Extremal black holes, despite being in a static zerotemperature vacuum state and thus not suffering from any evaporation process, generate a RSET which is (generically) non-vanishing at the horizon and will backreact on the background metric. In this section we assume the RSET is perfectly regular and take this effect into consideration by expanding the Einstein equations \nFIG. 5: Plot of the cancellation of the divergence in the numerical and analytical contributions to the quantity ⟨ ˆ E⟩ for a minimally coupled field with mass µ = M/ 10, leaving a total result that appears to be finite on the event horizon of the extremal Reissner-Nordstrom spacetime \n<!-- image --> \n- \nperturbatively in ℏ , \nG µ (0) ν + ℏ G µ (1) ν + O ( ℏ ) 2 = 8 π ( T µ (em) ν + ℏ ⟨ ˆ T µ ν ⟩ ) (73) \nwhere G µ (0) ν is the Einstein tensor evaluated in the background spacetime and T µ (em) ν is the stress-energy tensor of the electromagnetic field. Since we are considering a real quantum scalar field, the electromagnetic stressenergy tensor is not affected by semiclassical contributions [21]. Following [5] we express the metric components as \ng tt = -e 2 ψ ( r ) (1 -2 m ( r ) /r ) , g rr = (1 -2 m ( r ) /r ) -1 , (74) \nwhere m ( r ) denotes the Misner-Sharp mass, and expand the metric functions as \nψ ( r ) ≃ log [1 + ϵ R ( r )] , m ≃ M ( 1 -M 2 r ) [1 + ϵ M ( r )] , (75) \nwith ϵ = ℏ /M 2 . To linear order in ϵ , we write the semiclassical Einstein equations in the form \n2 ( z -1) 2 R ' ( z ) M 2 z 3 = 8 πM 2 ( ⟨ ˆ T r r ⟩ - ⟨ ˆ T t t ⟩ ) , -M ( z ) M 2 z 4 -(2 z -1) M ' ( z ) M 2 z 3 = 8 πM 2 ⟨ ˆ T t t ⟩ , (76) \nwhere we have used the dimensionless coordinate z = r/M . The first equation is clearly the difference between the radial and temporal equations, and the angular equations are such that the Bianchi identities are satisfied. Once the RSET is specified (here we replace it by the exact, numerical values for massless and massive fields shown in the previous section), we can integrate Eqs. (76) for M and R . The full solutions for the perturbations are numeric and we can only calculate them \nwithin the region for which we have numerical RSET values at our disposal. \nFor massless fields in the near-horizon limit z → z e , we can make use of the fact that the energy density and pressures of the exact RSET are given, to leading order in z -z e , by (70) to obtain the following analytic solutions for the perturbations \nM = -C 0 1080 π + O ( z -z e ) , R = K 0 1080 π + O ( z -z e ) , (77) \nwhere { C 0 , K 0 } are arbitrary integration constants. For massive fields, however, we cannot follow a similar argument since we ignore the exact values the RSET must reach at the extremal horizon, and extrapolating our numerical results introduces spurious errors. Written in ingoing Eddington-Finkelstein coordinates, the metric components become \ng vv = g tt = -ϵC 0 1080 π + O ( z -z e ) , g vr = e ψ = 1 + ϵK 0 1080 π + O ( z -z e ) . (78) \nIn light of these expressions, we observe that the metric receives a correction so that g tt is no longer vanishing at z = z e . For negative C 0 we have g tt > 0, hence the surface z = z e appears to lie inside a trapped region. In this case perturbative corrections have shifted the position of the horizon from its classical value z e = 1 to a new position z H > z e which must be found numerically. The surface gravity associated to this quantum-corrected horizon is equal to \nκ H = e ψ 2 M ∂ ∂z ( 1 -2 Mm z ) + e ψ M ( 1 -2 Mm z ) ∂ψ ∂z ∣ ∣ ∣ ∣ z H . (79) \nExtremal horizons are characterized by a vanishing surface gravity, i.e., a quadratic root in g vv . In the Reissner-Nordstrom metric, this comes from the fact that Q = M . The RSET makes the Misner-Sharp mass acquire a radial dependence, but does not affect the charge, allowing for the possibility of breaking the delicate balance of scales required by extremal horizons. In practice what we observe is that the backreacted spacetime, locally around z = z e , becomes like the sub-extremal or super-extremal Reissner-Nordstrom spacetimes depending which sign we choose for the mass perturbations. \nIn particular, together with shifting the position of the horizon outwards, taking ϵC 0 < 0 makes it acquire a positive surface gravity. The relative position and surface gravity of the quantum-corrected horizon are shown in Fig. 6. For massless fields, we see this non-extremal horizon smoothly tends to a horizon of the extremal kind in the C 0 → 0 limit. In the massive case, since we lack RSET values arbitrarily close to z e , we have not been able to generate reliable results in the C 0 → 0 limit. Nonetheless, for large and negative C 0 values, the curves for the massless and massive cases are indistinguishable. \nFIG. 6: Position of the perturbed horizon ∆ z = z H -z e and its surface gravity κ H for a semiclassically corrected extremal black hole. Corrections from massive and massless fields shift the Misner-Sharp mass by a constant value, which either breaks the extremal horizon into a pair of inner and outer horizons (if C 0 < 0), or eliminates any horizon entirely (if C 0 > 0). Our results do not change qualitatively as long as ϵK 0 ≪ 1, so we set K 0 = 0 here. \n<!-- image --> \nSince we do not have exact results for z < z e , we cannot obtain the quantum-corrected metric in the region z ≪ z H . However, it is clear (since the corrections are perturbative) that we are observing a splitting of the extremal horizon into a pair of outer and inner horizons. By solving (73) to the next order in ℏ , we would find: i) that the vacuum state in which this RSET is evaluated is no longer a zero-temperature state, and ii) that this state is singular at the newly formed inner horizon [31]. For positive C 0 the function g vv has zero roots, hence the extremal horizon disappears entirely and the perturbed metric describes a horizonless (and singular) object, whose associated vacuum state has zero temperature. Finally, between both regimes there is a separatrix solution with C 0 = 0 that preserves both the position and the surface gravity of the extremal horizon. This solution could potentially exist for all orders in ℏ only if we fine-tune the perturbations so that their contribution to the Misner-Sharp mass vanishes at every order in ϵ . Of course, the treatment adopted in this section only deals with static solutions from the start, hence we remain agnostic in regard to the stability of extremal horizons under time-dependent perturbations.", 'VI. DISCUSSION AND CONCLUSIONS': "In this paper, we have provided a generalisation of the extended coordinate prescription to directly compute the renormalized stress-energy tensor (RSET) of scalar fields in the Boulware vacuum. While the extended coordinates approach was first developed for the calculation of expectation values for a scalar field in the Hartle-Hawking state, it can be used to obtain results for other quantum states using a state subtraction approach [18]. However, \nfor spacetimes where one does not have a Hartle-Hawking state to leverage as a reference state, such as an extremal black hole or the spacetime of a star then a direct approach to compute the RSET is required. \nIn the extended coordinate approach we take an expansion of the Hadamard parametrix in extended coordinates, which encodes the short-distance behaviour of the two-point function, and we rewrite the terms in this expansion as mode-sums of the same form as those in the unrenormalized Green function, which may then be renormalized in a mode-by-mode manner. The main technical challenge in applying this approach to a scalar field at zero temperature arises from having a continuous frequency spectrum and hence the mode sums involve integrals over ω (as opposed to the Hartle-Hawking case where we sums over discrete values of ω ). As result, when we express terms in the Hadamard parametrix as mode-sums, we find that the integrands are not ordinary functions of ω and we must treat these mode-sums rigorously as generalized functions. When we do this, we see that an infra-red cutoff naturally appears in the integrals over ω in such a way so that the final results are independent of this cut-off. \nHaving developed this approach, we first applied it to the non-extremal Reissner-Nordstrom spactime. In this case we reproduced the results obtained via the state subtraction approach in [18], thereby verifying the new direct approach. We then applied the direct approach the the calculation of the RSET in an extremal ReissnerNordstrom spacetime, for both a massless and massive scalar field. The massless case had previously been considered in [21], however, to the best of the authors' knowledge, the results in this paper are first results obtained for a massive field. \nOnce we have calculated the RSET components we used these results to investigate the regularity of the Boulware state by calculating the energy density on the horizon as perceived by a freely falling observer. Here we find strong numerical evidence that the Boulware state is regular for both the massless and massive case. However, we stress that these results are indicative only and to say anything conclusive requires exact results on the horizon. We hope to report on this in the near future. \nDespite lacking a definite proof of the RSET regularity at extremal horizons, we can nonetheless assume that it is indeed regular to give a hint on the backreaction effects it would entail. By solving the static semiclassical equations perturbatively in ℏ [5] we have seen that, unless perturbations are fine-tuned to every order in the expansion, extremal horizons are unstable. If these perturbations contribute positively to the Misner-Sharp mass, the extremal horizon splits into a pair of outer and inner horizons, while in the opposite scenario the horizon disappears completely, leaving no marginally trapped surfaces in the spacetime. \nThe generalization of the extended coordinate method to the Boulware state opens the window towards obtaining the RSET of scalar fields in previously unexplored sit- \nuations, particularly, in stellar spacetimes. Recent works that considered perfect fluid stars show indications that the RSET becomes a dominant contribution in stellar interiors as these stars approach their maximum compactness limits [32-34]. Since previous works were based on different RSET approximations, a calculation of the exact RSET will allow to address their validity. These explorations have great implications for the theoretical plausibility of ultracompact stars sourced by semiclassical effects.", 'ACKNOWLEDGMENTS': 'The authors thank the Institute for Fundamental Physics of the Universe for hosting their visit under the Team Research program Dark Compact Objects and The Quantum Vacuum . JA acknowledges funding from the Italian Ministry of Education and Scientific Research (MIUR) under the grant PRIN MIUR 2017-MB8AEZ. The authors would like to thank Marc Casals for helpful conversations.', 'Appendix A: Cut-Off Independence': 'There is a subtle cancellation happening between the terms dependent on λ in J ωl ( a, b \| r ) and the finite terms in (47) which also depend on λ . We can make this cancellation explicit by analyzing the terms in the renormalized vacuum polarization that contain the cutoff, which we refer to as w ( λ ) : \nw ( λ ) = T (l) 00 ( r ) 8 π 2 [ ∫ ∞ -∞ dω θ ( ω 2 -λ 2 ) \| ω \| -1 +log( λ 2 ) ] + √ π 8 π 2 m -1 ∑ a =1 a ∑ b =0 T (l) ab ( r ) f a -b 2 2 a Γ( -a + 1 2 ) [ a ! ∫ ∞ -∞ dω θ ( ω 2 -λ 2 ) \| ω \| -2 a -1 -( a -1)! λ 2 a ] . (A1) \nWe have separated the T ( l ) 00 ( r ) and a > 0 terms in J ωl ( a, b \| r ) and used the identity (27) to truncate the l sum as well as introduced k = 0, only condition for these terms being nonzero at coincidence. We can show that the renormalized vacuum polarization is independent on the choice of the cutoff by verifying that \n∂ ∂λ ( w ( λ ) ) = T (l) 00 ( r ) 8 π 2 ( -2 λ + 2 λ ) + √ π 8 π 2 m -1 ∑ a =1 a ∑ b =0 T (l) ab ( r ) f a -b 2 2 a Γ( -a + 1 2 ) ( -2 a ! λ -2 a -1 +2 a ! λ -2 a -1 ) = 0 , (A2) \nwhere we made use of \n∂ ∂λ ( θ ( ω 2 -λ 2 )) = -( δ ( ω + λ ) + δ ( ω -λ ) ) . (A3) \nSimilarly, it can be demonstrated that the RSET is also completely independent of the choice of the cutoff.', 'Appendix B: Other representations for direct and tail regularization parameters': "In this appendix, we report different but equivalent representations for the functions I ωl ( a, b \| r ) and J ωl ( a, b \| r ). Above, we have presented the derivation for these expressions in terms of hypergeometric functions. The alternative representations below involve the modified Bessel functions I ν ( z ) and K ν ( z ) and the generalized incomplete gamma \nfunction Γ( a, z 0 , z 1 ) = ∫ z 1 z 0 t a -1 e -t dt . \nI ωl ( a, b \| r ) = r 2 b +1 f -b -1 / 2 (2 b b !) (2 r 2 ) b +1 b ∑ p =0 ( -1) a 2 p ( p ) 2 b -2 p Γ( b -p +1) ∂ 2 a + p ∂ω 2 a + p ( ω p I l + 1 2 ( ωr √ f ) K l + 1 2 ( ωr √ f )) = 1 2 ( -1) a 2 2 b b !( f ) b +1 / 2 r b ∑ p =0 2 p ( p ) 2 b -2 p Γ( b -p +1) 2 a + p ∑ k =0 ( 2 a + p k ) p ! ω -2 a + k ( -2 a + k )! k ∑ d =0 ( k d )( r √ f ) k ( -1) d 2 k × 2 k -2 d ∑ c =0 ( k -d c ) I l +1 / 2 -k + d +2 c ( ωr √ f ) 2 d ∑ n =0 ( d n ) K l +1 / 2 -d +2 n ( ωr √ f ) = lim x →-1 2 l ( -1) a + b ( 2 r 2 ) a -1 2 2 b ! f a + b + 1 2 { b ∑ k =1 l ∑ j =0 ( -1) j +1 j ! ( l j )( 1 2 ( l + j -1) l ) 2( -k + j +1)! ∂ b -k ∂x b -k ( (1 -x ) a + b -1 2 e -ω √ 2(1 -x ) r 2 f ) + l ∑ k = b k -b ∑ j =0 k ! ( l k )( 1 2 ( k + l -1) l ) ( -1) -2 b + k -j ( k -b )! ( k -b j ) ( ω √ 2 r 2 f ) -2 a -2 k +2 j -1 Γ ( 2 a +2 k -2 j +1 , 0 , 2 rω √ f ) } , (B1) \n(B2) \nJ ωl ( a, b \| r ) = -√ π (2 r 2 ρ 2 ) a -b ( a -b )!2 2 a ( -1) l a -b ∑ k =0 θ ( ω 2 -λ 2 ) ( a -k )! k ! \| ω \| 2 k -2 a -1 ( a -k -b )!Γ( k -a + 1 2 )( k + l +1)!( k -l )!(2 ρ 2 ) k +(2 r 2 ) a -b ( -1) b Γ(1 + a -b ) r f 1 / 2 a -b +1 ∑ k =0 ( -1) k 2 a -b +2 ( a -b +1 k ) (2 l +2( a -b ) -4 k +3) ( l -k + 1 2 ) a -b +2 × ∂ 2 b ∂ω 2 b I l + 3 2 + a -b -2 k ( ωr √ f ) K l + 3 2 + a -b -2 k ( ωr √ f ) = -√ π (2 r 2 ρ 2 ) a -b ( a -b )!2 2 a ( -1) l a -b ∑ k =0 θ ( ω 2 -λ 2 ) ( a -k )! k ! \| ω \| 2 k -2 a -1 ( a -k -b )!Γ( k -a + 1 2 )( k + l +1)!( k -l )!(2 ρ 2 ) k +(2 r 2 ) i -j ( -1) j ( r √ f ) 2 j +1 Γ(1 + i -j ) i -j +1 ∑ k =0 ( -1) k 2 i -j +2 ( i -j +1 k ) (2 l +2( i -j ) -4 k +3) ( l -k + 1 2 ) i -j +2 × 2 j ∑ p =0 ( 2 j p ) ( -1) p 2 2 j 4 j -2 p ∑ q =0 ( 2 j -p q ) I l +3 / 2+ i -3 j -2 k + p +2 q ( ωr √ f ) 2 p ∑ n =0 ( p n ) K l +3 / 2+ i -j -2 k -p +2 n ( ωr √ f ) = -lim ω ' → ω f a -b 4 π a -b ∑ k =0 2 l +2 ( -1) a -k ( a -b k ) ∂ 2 a -2 k ∂ω ' 2 a -2 k l ∑ n =0 n ∑ j =0 π ω ' ( -1) n -j ( l n )( 1 2 ( l + n -1) l )( n j ) × ( 2 r 2 f ) k Γ ( 2( k -j + n +1) , 0 , 2 ω ' r √ f ) ( √ 2 ω ' r √ f ) 2( k -j + n +1) -2 k -j + n θ ( λ 2 -ω 2 ) k -j + n +1 .", 'Appendix C: Hadamard Coefficients': "Below we list the coefficients D (p) ij , D (r) ij , T (l) ij , T (r) ij and T (p) ij to 2nd order. \nD (r) 00 = 2 , D (r) 10 = -f ( r 2 f '' -2 rf ' +2 f -2 ) 12 r 2 , \nD (r) 11 = f ( r 2 ( f ' 2 ) -4 f ( rf ' +1) + 4 f 2 ) 2 \n24 r , D (r) 20 = 1 2880 r 4 ( f [ -5 r 2 ( -f ' 2 ) ( r 2 f '' -2 rf ' -2 ) -8 f 2 ( 3 r 3 f (3) -7 r 2 f '' +19 rf ' +10 ) + f ( 9 r 4 f '' 2 -20 r 2 f '' +86 r 2 f ' 2 +4 rf ' ( 3 r 3 f (3) -14 r 2 f '' +20 ) +4 ) +76 f 3 ] ) , D (r) 21 = -1 2880 r 4 ( f [ r 4 ( f ' 2 + f ' 4 ) + r 2 f ( -30 rf ' 3 + f ' 2 ( 11 r 2 f '' -30 )) +4 f 3 ( 11 r 2 f '' -52 rf ' -40 ) -2 f 2 ( 10 r 2 f '' -67 r 2 f ' 2 + f ' ( 22 r 3 f '' -80 r ) -28 ) +104 f 4 ] ) , D (r) 22 = f 2 ( r 2 f ' 2 -4 f ( rf ' +1) + 4 f 2 ) 2 1152 r 4 , D (p) 11 = -f ' 6 r , D (p) 21 = f ( -9 rf ' 2 +6 rf ( rf (3) -2 f '' ) +2 f ' ( 7 r 2 f '' + f +5 )) 720 r 3 , D (p) 22 = 7 r 2 f ' 2 -10 rf ' + rf (9 rf '' +4 f ' ) -3 f 2 +3 720 r 4 , T (l) 00 = 6 µ 2 r 2 -(6 ξ -1) ( r 2 f '' +4 rf ' +2 f -2 ) 12 r 2 , T (l) 10 = 1 480 r 4 ( r 2 f '' ( (10 ξ (3 ξ -1) + 1) r 2 f '' -10(6 ξ -1) ( 2 ξ + µ 2 r 2 )) +4(5 ξ (24 ξ -5) + 1) r 2 f ' 2 +2 rf ' ( (5 ξ -1) r 3 f (3) +2 ( 60 ξ 2 -5 ξ -1 ) r 2 f '' +40(1 -6 ξ ) ξ +10 µ 2 (1 -12 ξ ) r 2 ) +2 rf ( (80 ξ (3 ξ -1) + 6) f ' + r ( (20 ξ (3 ξ +2) -8) f '' + r ( (5 ξ -1) rf (4) ( r ) + (40 ξ -7) f (3) ))) -40 fξ ( 6 ξ +3 µ 2 r 2 -1 ) +4(10 ξ (3 ξ -1) + 1) f 2 +2 ( 15 ( 2 ξ + µ 2 r 2 ) 2 -2 ) ) T (l) 11 = f 480 r 4 ( -2 ( r 2 f '' +2 ) ( (1 -5 ξ ) r 2 f '' +10 ξ +5 µ 2 r 2 -2 ) +2(1 -10 ξ ) r 2 f ' 2 + rf ( (16 -80 ξ ) f ' + r ( (12 -80 ξ ) f '' + r ( rf (4) ( r ) + (6 -20 ξ ) f (3) ) +20 µ 2 )) +8(5 ξ -1) f 2 + rf ' ( (10 ξ -1) r 3 f (3) +4(20 ξ -3) r 2 f '' +20(6 ξ -1) ) ) T (r) 10 = f ( r 2 f ' 2 -4 f ( rf ' +1) + 4 f 2 ) ( (1 -6 ξ ) (2 -r ( rf '' +4 f ' ) -2 f +2) -6 µ 2 r 2 ) 576 r 4 , T (p) 10 = ( f -1) ( (1 -6 ξ ) (2 -r ( rf '' +4 f ' ) -2 f ) -6 µ 2 r 2 ) 144 r 4 , T (p) 11 = f ( rf ' -2 f +2) ( (1 -6 ξ ) (2 -r ( rf '' +4 f ' ) -2 f ) -6 µ 2 r 2 ) 144 r 4 . \n,", 'Appendix D: Analytical Part of the RSET': "In this appendix, we give explicit expressions for ⟨ ˆ T µ ν ⟩ analytical that arises in the split in Eq. 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2024ApJ...975..168W	Coronal mass ejections CMEs from pseudostreamers represent a significant fraction of largescale eruptions from the Sun. In some cases these CMEs take a narrow jetlike form reminiscent of coronal jets in others they have a much broader fanshaped morphology like CMEs from helmet streamers. We present results from a magnetohydrodynamic simulation of a broad pseudostreamer CME. The early evolution of the eruption is initiated through a combination of breakout interchange reconnection at the overlying null point and ideal instability of the flux rope that forms within the pseudostreamer. This stage is characterized by a rolling motion and deflection of the flux rope toward the breakout current layer. The stretching out of the strapping field forms a flare current sheet below the flux rope reconnection onset there forms lowlying flare arcade loops and the tworibbon flare footprint. Once the CME flux rope breaches the rising breakout current layer interchange reconnection with the external open field disconnects one leg from the Sun. This induces a whiplike rotation of the flux rope generating the unstructured fan shape characteristic of pseudostreamer CMEs. Interchange reconnection behind the CME releases torsional Alfvn waves and bursty dense outflows into the solar wind. Our results demonstrate that pseudostreamer CMEs follow the same overall magnetic evolution as coronal jets although they present different morphologies of their ejecta. We conclude that pseudostreamer CMEs should be considered a class of eruptions that are distinct from helmetstreamer CMEs in agreement with previous observational studies.	2024-11-01T00:00:00Z	['10.3847/1538-4357/ad7941', '2024arXiv240908126W', '10.48550/arXiv.2409.08126', 'arXiv:2409.08126', '2024ApJ...975..168W']	['Solar coronal mass ejections', 'Active solar corona', 'Solar magnetic reconnection', '310', '1988', '1504', 'Astrophysics - Solar and Stellar Astrophysics']	A Model for Flux Rope Formation and Disconnection in Pseudostreamer Coronal Mass Ejections	2,024	197	0.53	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.08126.pdf	{'A Model for Flux Rope Formation and Disconnection in Pseudostreamer Coronal Mass Ejections': 'P. F. Wyper, 1 B. J. Lynch, 2 C. R. DeVore, 3 P. Kumar, 3 S. K. Antiochos, 4 and L. K. S. Daldorff 5, 3 \n1 Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK \n- 2 Department of Earth, Planetary, and Space Sciences, University of California-Los Angeles, Los Angeles, CA 90056, USA\n- 3 Heliophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771\n- 4 Department of Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109, USA 5 The Catholic University of America, 620 Michigan Ave., N.E. Washington, DC 20064 USA', 'ABSTRACT': "Coronal mass ejections (CMEs) from pseudostreamers represent a significant fraction of large-scale eruptions from the Sun. In some cases, these CMEs take a narrow jet-like form reminiscent of coronal jets; in others, they have a much broader fan-shaped morphology like CMEs from helmet streamers. We present results from a magnetohydrodynamic simulation of a broad pseudostreamer CME. The early evolution of the eruption is initiated through a combination of breakout interchange reconnection at the overlying null point and ideal instability of the flux rope that forms within the pseudostreamer. This stage is characterised by a rolling motion and deflection of the flux rope toward the breakout current layer. The stretching out of the strapping field forms a flare current sheet below the flux rope; reconnection onset there forms low-lying flare arcade loops and the two-ribbon flare footprint. Once the CME flux rope breaches the rising breakout current layer, interchange reconnection with the external open field disconnects one leg from the Sun. This induces a whip-like rotation of the flux rope, generating the unstructured fan shape characteristic of pseudostreamer CMEs. Interchange reconnection behind the CME releases torsional Alfv'en waves and bursty dense outflows into the solar wind. Our results demonstrate that pseudostreamer CMEs follow the same overall magnetic evolution as coronal jets, although they present different morphologies of their ejecta. We conclude that pseudostreamer CMEs should be considered a class of eruptions that are distinct from helmetstreamer CMEs, in agreement with previous observational studies. \nKeywords: Sun: corona; Sun: magnetic fields", '1. INTRODUCTION': "Coronal mass ejections (CMEs) exhibit a variety of morphologies in coronograph images. In recent years it has become increasingly recognised that these different morphologies are closely tied to the large-scale structures that define the open-closed magnetic boundary overlying the source regions of the events. The structures are of two types: helmet streamers and pseudostreamers. Helmet streamers lie between coronal holes of opposite magnetic polarity and taper into the base of the heliospheric current sheet. Pseudostreamers by contrast lie between or within coronal holes of a single polarity, are associated with at least one coronal null \[email protected] \npoint, and have no large-scale current sheet (Titov et al. 2012; Kumar et al. 2021). \nHelmet-streamer CMEs are well studied as most active regions lie beneath helmet streamers. Large activeregion or quiet-Sun filament eruptions generally create classic 3-part CMEs with bubble-like shapes (e.g., Riley et al. 2008; Webb & Howard 2012). Streamer-blowout CMEs, which form from the large-scale expansion and pinch-off of magnetic loops from the streamer and sometimes are associated with filament ejections from the streamer base (Vourlidas & Webb 2018), are slower but share a similar bubble-like morphology. Many 'stealth' CMEs (Lynch et al. 2016; Bhowmik et al. 2022) also fall into this category, as they are the extreme end of the continual pinch-off and blob-formation process occurring at the tips of helmet streamers (e.g., Sheeley et al. 1999; Higginson et al. 2017). Helmet-streamer \nCMEs therefore all share a generally bubble-like morphology, although they vary considerably in the amount of magnetic flux and plasma ejected. \nCMEs originating from pseudostreamers, in contrast, are more varied in morphology. Wang (2015) and Wang & Hess (2018) classified them as either fan-shaped or jet-like. Fan-shaped CMEs have an unstructured core with typical widths up to 30 · , while jet-like CMEs are more collimated with narrower widths nearer 10 · . Examples of each type are shown in Figure 1(a)-(d). Both types generally travel at a steady ejection speed once underway. Some fan-shaped CMEs exhibit a V-shape suggestive of concave-up field lines beneath the flux rope. Wang & Hess (2023) compared a variety of pseudostreamer CMEs, concluding that all are laterally confined by the adjacent open field and are likely different manifestations of large-scale coronal jets. In contrast, Kumar et al. (2021) analyzed three pseudostreamer CMEs that did not fit the Wang & Hess pattern. Their more energetic events were much wider than 40 · , were more bubble-like in morphology, and clearly had a shock front ahead of them, Fig. 1(f). These characteristics are much more typical of helmet-streamer CMEs. In addition, Kumar et al. observed pre-eruption jets (Fig. 1(c) & (e)) and dimmings associated with slow reconnection/opening near the null points about 1-3 hr prior to the filament/flux-rope eruptions. \nAll pseudostreamer eruptions involve certain fundamental constituent parts. Filament channels (with or without filament material) form slowly over the course of days to weeks under the arcades of the pseudostreamer before becoming unstable and erupting. These filament channels appear as cavities when viewed on the limb (e.g., Guennou et al. 2016; Karna et al. 2019) and can contain either a flux rope or sheared arcade. In the latter case, a flux rope will form once flare reconnection commences beneath the eruption. The novel constituent part of a pseudostreamer (compared to a helmet streamer) is its overlying coronal magnetic null point(s) and background unipolar open flux. Clearly, the subsequent interaction of the rising flux rope with the magnetic topology could be the key determinant of the different pseudostreamer CME morphologies. \nJet-like CMEs are the most straightforward to explain. In this case there is relatively little expansion of the flux rope, which launches a jet-like CME when it reconnects at the pseudostreamer null point. Exactly the same evolution occurs in coronal jets associated with mini-filament eruptions, which share a similar null-point structure but on a much smaller scale (e.g., Sterling et al. 2015; Kumar et al. 2018, 2019a,b). This correspondence suggests that all these events are part of a continuum of \nFigure 1. LASCO C2 running difference images (Brueckner et al. 1995) showing the different morphologies of CMEs originating from pseudostreamers. (a) & (b): two narrow jet-like events. (c) & (d) a fan-shaped CME and its precursor jet. (e) & (f) a broad bubble-like CME and its precursor jet. Adapted from Kumar et al. (2021). \n<!-- image --> \njet-like eruptions (Wyper et al. 2017, 2021; Kumar et al. 2021). \nIn Wyper et al. (2017), we presented a model for coronal jets with mini-filaments. The model generalises the breakout mechanism for CMEs (Antiochos et al. 1999; Lynch et al. 2008) to the null-point topology of coronal jets in the open field of coronal holes. Sustained breakout reconnection is key to removing all of the overlying magnetic flux, allowing the erupting flux rope to reach the breakout current layer and reconnect with the external field. Furthermore, the quasi-uniform strength of the open field strongly suppresses the expansion of the flux rope during its rise. Generalisations of this model with varied inclinations of the open field (Wyper et al. 2018a, 2019) and manner of energisation (Wyper et al. 2018b) have revealed these features of the evolution to be quite general. In certain cases, coupling of the breakout feedback mechanism to an ideal instability of the flux rope was found to initiate the eruption (Wyper et al. 2019). \nThe internal magnetic structure and evolution with height of the broader, fan-like pseudostreamer CMEs is less well understood. In Wyper et al. (2021), we showed that if the pseudostreamer is topologically connected to a helmet streamer, then a cou- \npled pseudostreamer/helmet-streamer blowout eruption can occur and produce a broad CME. However, that eruption was jet-like in the low corona, whereas most broad pseudostreamer CMEs appear to have a CME-like lifting-off of the flux rope at low heights. The lift-off is often accompanied by a rolling motion of the flux rope, e.g. Panasenco et al. (2011); Kumar et al. (2021). Ultimately, the erupting flux rope is expected to reach the breakout current sheet where it will reconnect with the open field as in a jet; but when and where in the evolution this occurs is not well understood. Does the fan-like ejecta represent a flux rope still connected at both ends to the solar surface? Or has the flux rope reconnected after reaching a certain height? If so, where does the reconnection occur? \nIn this paper we present a magnetohydrodynamic (MHD) simulation model of a fan-shaped pseudostreamer CME designed to address these questions. The setup is a generalisation of our model for coronal jets and reproduces many of the observed features of fan-shaped pseudostreamer CMEs. In Wyper et al. (2022), we focused on the interchange reconnection dynamics of the early breakout process at the top of a model pseudostreamer. Here we have extended the run time of that simulation in order to investigate the subsequent eruption. Most importantly, we demonstrate that the magnetic evolution is exactly the same as that of mini-filament coronal jets. The model demonstrates that despite their differences in ejecta morphologies and scale, coronal jets and pseudostreamer CMEs belong to a continuum of eruptions unified by the pseudostreamer topology. \nIn § 2 we describe the simulation setup. § 3 gives an overview of the eruption and § 4 describes the energy release and reconnection process in more detail. In § 5 we discuss our results in the context of recent observations. Our conclusions are given in § 6.", '2. SIMULATION SETUP': "The ideal, compressible MHD equations are solved by the Adaptively Refined Magnetohydrodynamics Solver (ARMS; DeVore & Antiochos 2008) in the following form: \n∂ρ ∂t + ∇ · ρ v = 0 , (1) \n∂ρ v ∂t + ∇ · ρ vv = -∇ p + 1 4 π ( ∇ × B ) × B + ρ g , (2) \n∂ B ∂t = ∇ × ( v × B ) , (3) \nwhere ρ is the mass density, v the plasma velocity, and B is the magnetic field. Gravity takes the form g = -GM ⊙ /r 2 e r . We assume that the plasma is an \nideal gas with p = 2( ρ/m p ) k B T , where k B is Boltzmann's constant and m p is proton mass. The temperature is assumed to be constant and uniform throughout the volume with T = 1MK. \nThe domain is a spherical wedge with radius r ∈ [1 R ⊙ , 20 R ⊙ ] and latitude/longitude θ, ϕ ∈ [ -50 . 4 · , 50 . 4 · ]. The magnetic field is initialised as a monopolar radial magnetic field of strength b 0 at r = R ⊙ together with 16 sub-surface radially aligned dipoles, such that \nB = b 0 R 2 ⊙ r 2 e r + ∑ i M i ( d \| r -r i \| ) 3 [3( m i · e r ) -1] e r , (4) \nwhere b 0 = -2 . 5 G, R ⊙ = 7 × 10 10 cm, m i is the unit vector in the direction of r -r i , and d = 8 × 10 9 cm. The values used for M i and r i are given in Table 1. The field is shown in Figure 2(a) and takes the form of a bipolar surface flux distribution supporting a largescale coronal null-point topology (null height ≈ 0 . 25 R ⊙ above the surface). The spacing between the dipoles and relative strengths of the dipoles and monopolar field closely matches our cartesian jet simulation model with vertical background field (Wyper et al. 2018a), but on a much larger scale and in spherical geometry. The key difference between the two setups is the radial expansion of the monopolar field, which we will demonstrate plays a key role in the eruption evolution. \nTable 1. Dipole parameters: r i = ( r i , θ i , ϕ i ); M i (G); r i (cm); and θ i , ϕ i (degrees). \nFigure 2(b) shows the computational grid. A volume of fixed maximum refinement is centered around the closed field region. Outside of this volume, the grid refines adaptively as needed with the refinement criterion depending upon the local electric current density (Karpen et al. 2012). The base grid level was set to 16 × 8 × 8 grid blocks (each block contains 8 × 8 × 8 \nFigure 4 shows the onset and early development of the eruption, which is triggered at t ≈ 8 hr when the flux rope begins to rise rapidly. This coincides with a transition from slow to rapid reconnection at the HFT and the formation of a vertical exhaust outflow, Figure \n<!-- image --> \n\|v\| (km/s) \nFigure 2. (a) The initial magnetic field. (b) Side view of the initial simulation grid blocks (each block contains 8 × 8 × 8 grid cells). (c) The surface driving profile. The PIL is shown in grey in each panel. \ngrid cells), with up to 4 additional levels of refinement in this simulation (2 fewer than in Wyper et al. 2022, where the aim was to track the evolution of small-scale plasmoids). The atmosphere is initialised with a 1D isothermal Parker (1958) wind solution and relaxed to a quasisteady state over 4 × 10 4 s (see Fig. 1(b) of Wyper et al. (2022) for the resulting wind profile). All times in the rest of this paper are quoted from that point on, i.e., t = 0 corresponds to the end of the relaxation and the start of the surface driving. \nThe driving profile is the same as that used in our Cartesian jet setup (Wyper et al. 2018a) and is given by \nv ⊥ = v 0 g ( B r ) e r × ∇ B r , (5) g ( B r ) = k b b r -b l B r tanh ( k b B r -b l b r -b l ) , b l ≤ B r ≤ b r , 0 , otherwise (6) \nwhere B r is the normal field component on the lower boundary, b r = 30 and b l = 1 . 6 define the contours of B r within which the flow is restricted, and the constants k b and v 0 are set to 5 and 3 . 079 × 10 13 , respectively. By design the flow follows the contours of B r , so it does not change the B r surface distribution. The spatial profile is shown in Figure 2(c). The driving is ramped up over 1000 s, held constant for 24000 s, and then ramped down just before the onset of the CME (halting at t = 2 . 5 × 10 4 s or 6 hr 57 min). The driving speed peaks at v ⊥ ≈ 30 km s -1 in the center of the surface bipolar flux distribution. This speed is chosen for numerical convenience. Although relatively fast compared to solar surface flows, the flow is still substantially sub-Alfv'enic and sub-sonic, hence the pre-eruption closed magnetic field evolves quasistatically.", '3.1. Pre-eruption Reconnection': "The evolution during the driving phase is similar to that in the jet model, wherein the surface driving forms a quasi-circular filament channel overlying the PIL, Figure 3(a)-(b). The closed field expands asymmetrically and stresses the null point, forming a breakout current layer and inducing interchange reconnection there. The reconnection is resolved well enough that plasmoids form in the breakout current layer, and the reconnection eventually enters a bursty regime. This launches a plasma jet modulated by plasmoid ejections (Fig. 3(c)) and torsional Alfv'enic waves into the solar wind (Wyper et al. 2022). Aside from the jet itself, another observable signature of the onset of breakout reconnection is a dimming in synthetic white-light base-difference images shown in Figure 3(d). This is consistent with our findings of EUV dimmings in previous observational (Kumar et al. 2021) and modeling (Wyper et al. 2021) work. \nThe breakout reconnection at this point is selfsustaining due to feedback between the outward expansion of the filament channel and the removal of overlying strapping field by the breakout reconnection. Well before this time, a twisted flux rope formed within the filament channel. This also occurred in the jet simulations: it arises from gradients in the surface driving profile creating a thin current layer inside the channel. Tethercutting reconnection in this layer forms the twisted flux rope as part of a hyperbolic flux tube (HFT), but it does not trigger the eruption (see § 4 for further discussion).", '3.2. Eruption Onset and Early Evolution': 'Figure 3. (a) and (b): Two views of the flux rope (yellow field lines) formed within the sheared filament channel at the end of the driving period ( t = 6hr 57 min). Magenta field lines show short flare loop field lines. (c) \| v \| showing the pre-eruption plasma jet (analyzed by Wyper et al. (2022)). (d) Synthetic white-light base-difference image (7 hr 30 min - 6 hr 32 min) showing the pre-eruption dimming. \n<!-- image --> \nFigure 4. Top panels: radial velocity, v r . Bottom panels: the logarithm of normalised current density R ⊙ \| J \| /c . (a,e): t = 7hr 38 min. (b,f): t = 8hr 37 min. (c,g): t = 9hr 10 min. (d,h): t = 9hr 35 min. BCS = breakout current sheet; FCS = flare current sheet. An animation of panels (e) to (h) is available showing the formation and eruption of the flux rope and the subsequent interchange reconnection. Key features are highlighted in the static figure. The duration is 8 s and runs from t = 0 to 16 hr 23 min. \n<!-- image --> \n4(b,f). We denote this time (8 hr 37 min) as the onset of fast flare reconnection. Unlike the jet model, in which rapid reconnection is triggered only when the flux rope reaches the breakout current layer, in this case substantial strapping field remains above the flux rope when the fast reconnection turns on. The strapping field is carried out along with the flux rope as it continues to rise and accelerate, in the manner of a typical breakout CME (e.g., Antiochos et al. 1999; Lynch et al. 2008). As is typical in pseudostreamer CMEs (e.g., Panasenco et al. 2011; Lynch & Edmondson 2013; Sahade et al. 2022), the \nerupting flux rope deflects toward the null-point breakout current sheet where the magnetic field strength is lowest, Figure 4(c,g). This deflection plus the exhaust jet from the reconnecting flare current sheet combine to create a rolling motion of the rising flux rope, Figure 4(c,d). See also the animation of this figure.', '3.3. Flux Rope Disconnection': 'As the flux rope rises, the breakout reconnection continues to erode the strapping field while the flare reconnection continues to strengthen the flux rope. The null \nFigure 5. (a) Isosurface of normalised current density magnitude ( R ⊙ J/c = 1 . 5 G) shaded by the radial component ( R ⊙ J r /c ). (b)-(d) Field line evolution showing the flux rope disconnection. An animation of panels (b) to (d) is available showing the dynamic evolution of the field lines. The duration is 1 s and runs from t = 7hr 55 min to 10 hr 33 min. \n<!-- image --> \nFigure 6. Flux rope disconnection in the jet simulation. (a) Isosurface of normalised current density ( J/c = 1 . 2 G) shaded by the radial component ( J r /c ). (b)-(d) Field line evolution showing the flux rope disconnection. \n<!-- image --> \npoint and breakout current layer are pushed out to several solar radii before the strapping field is exhausted and the flux rope itself starts to reconnect. Despite the greatly expanded size of the open-closed separatrix sur- \nFigure 7. Second stage of CME flux rope disconnection. The top panels show field lines extending into the CME traced from the surface. The bottom panels show a closeup view (from the side) of the same field lines at the same times showing the re-closing of the yellow field lines and the shift in the CME flux rope footpoints to the cyan field lines. \n<!-- image --> \nface, the flux rope reconnection proceeds in exactly the same manner as in the jet simulation. The breakout and flare current layers combine into one long current sheet that wraps around the flux rope, Figure 5(a). The null point then moves within this sheet, sliding from the top of the separatrix and down the side to ultimately end up below the flux rope. At the end point, the flux rope has reconnected completely and is comprised entirely of open field lines. \nFigure 5(b) to (d) shows the field line evolution during this phase. The majority of the flux rope disconnection occurs when the null point (i.e., the main interchange reconnection site) is on the side/flank of the flux rope rather than at the apex as one might anticipate. As the flux rope expands outwards and the legs become radially oriented, one of the legs will have field lines antiparallel to the adjacent coronal hole field lines, shown in blue in Figure 5. As a result, there must be a current layer separating these field regions and, in general, flux rope disconnection will occur along that leg of the CME. As shown in Figure 5(d), this leads to the formation of a transient V-shape in the flux rope shortly after disconnection. This kink straightens out as the flux rope continues to rise (see the animation of this figure). \nFor comparison, Figure 6 shows the combined current sheets and flux rope disconnection in the Cartesian jet simulation with vertical background field (Wyper et al. 2018a). As this flux rope is less expanded, the disconnection occurs nearer its apex, but otherwise the disconnection process proceeds in exactly the same manner. \nFigure 8. Flux ropes formed following the second phase of interchange reconnection at t = 12hr 30min. Light blue/cyan shows the flux tube with the twist of the original rope. Other twisted flux tubes are formed from plasmoids ejected as part of the bursty interchange reconnection process. (a) Close-in view of the flux ropes; note that the yellow field lines showing the original rope footpoints are now closed. (b) Farther-out view showing how the flux ropes wrap into the CME structure. (c) Cut showing the large-scale rotation of the field lines. \n<!-- image -->', '3.4. CME Magnetic Structure': "In the wake of the flux rope disconnection, interchange reconnection continues to sequentially open sheared closed field lines while closing down unsheared open field lines. This process progresses around the circular PIL (see § 4.2). Ultimately, the closed-field footprint returns to approximately where it started. As part of this subsequent evolution, the erupting flux rope undergoes a second leg reconnection event. However, this one is much less dramatic, occurring much closer to the surface and taking the form of a shift in the flux rope footpoints, Figure 7. The yellow field lines of the erupted flux rope close down, while the neighboring closed cyan field lines open up. In effect, the erupting flux rope footpoints shift from one side of the pseudostreamer to the other (note the change in color of the propagating CME field lines from yellow to cyan). This connectivity of the field lines in the CME is maintained thereafter. \nThe magnetic structure of the CME as a whole, on the other hand, is more complex. As in the breakout reconnection phase, the interchange reconnection during the disconnection phase is also bursty and dominated by plasmoid ejection. This dynamically creates further twisted flux tubes within the null-point current layer that propagate into the open field in the wake of the disconnection. Figure 8 shows a selection of these flux tubes. The original erupting flux rope is shown in cyan, while plasmoid flux tubes are shown in the order in which they were launched sequentially: orange, green, blue, pink, and then red. The twist within the flux tubes propagates as torsional Alfv'en waves behind the main CME, while their field lines map into the main \nbody of the CME and thread through the sheath that surrounds the original flux rope, Figure 8(b) and (c). \nFollowing the two-stage disconnection and reconnection of the original coronal flux rope, the erupting structure reaches its final magnetic configuration. The main body of the CME contains an embedded twisted flux tube that is the remnant of the original erupting flux rope. This is surrounded by a sheath of highly distorted but untwisted field. Following behind in the trailing wake of the CME are bursty outflows and twisted flux tubes launched by the interchange reconnection process. All of these structures are comprised of open field lines.", '3.5. CME Plasma Evolution': 'The evolution of some additional properties of the CME are shown in Figure 9. The embedded flux rope (top of the CME in this view) is visible as a region of depleted density, Figure 9(a). Outside of this feature, the CME appears broadly unstructured in nature. In a follow-up paper we will make a detailed exploration of the white-light properties of the CME structure from different viewing angles. \nFigures 9(c), (f), and (i) show that the main body of the CME rotates as it propagates. It is tempting to equate this rotation to that of the outflowing plasma about the spire in helical coronal jets resulting from minifilament/filament channel eruptions. However, this rotation is qualitatively different. In the case of jets it is the untwisting of the erupted flux rope reconnected onto open field lines that drives the rotation. Here, the rotation is driven by the whip-like motion of the flux rope axis and the sheath of field lines surrounding it, Figure 8(b). It is the precessing axis of the flux rope - i.e., \nFigure 9. ρ (left), v r (middle), and v ϕ (right) in a cut ( ϕ = 6 · ) through the CME at different times. \n<!-- image --> \nthe evolution of the writhe - that drives the large-scale rotation, rather than its twist. Due to the expanded horseshoe-like shape of the flux rope prior to disconnection, the axis of the twisted open flux tube created by the disconnection is highly kinked. The straightening out of the axis, along with the straightening out of the surrounding sheath field lines, creates the large-scale rotation of the CME. This evolution is actually closer in nature to the whip-like field-line motion in the helical jet model of Pariat et al. (2009) for jets without filament channels. The twist of the flux rope also propagates along its axis, as is true of the plasmoid-generated flux tubes in the wake of the CME. In this sense there are torsional waves within waves involved in the CME evolution. \nAlso notable is the MHD shock launched ahead of the CME by the initial flux rope expansion and the reconnection outflow from the breakout reconnection. The \nlatter is most visible in the v ϕ component, Figure 9(c), and is eventually overtaken by the CME at later times. Wyper et al. (2021) and Kumar et al. (2021) noted similar pre-eruption jets in their simulation and observational studies, respectively, of pseudostreamer CMEs. \nIn Figure 10 we show two measures of the speed of the ejecta. The initial rise of the flux rope is well captured by following the highest point on a field line (purple) traced through the axis of the rope. Initially the flux rope accelerates before plateauing at v ≈ 600 km s -1 once the disconnection begins; panel (c), blue curve. Beyond this point the changing connectivity of the CME makes it difficult to follow individual magnetic field structures. To estimate the overall CME speed beyond this time, a radial sample was taken (white line; panel (a)) and the shock front was identified. Height/time and speed curves for the front are shown in red in panels (b) and (c). Evidently, once the disconnection begins \nFigure 10. (a) Shading shows v r in the plane ϕ = 0 at t = 9hr 35 min. The white line shows the path along which the CME front speed is calculated. The red arrow shows the position of the front at this time. The purple field line shows the flux rope axis. The blue arrow shows the highest point of this field line. (b) Radial positions of the front (red) and highest point of the axis field line (blue). (c) Radial speeds: V front (red), V axis (blue). \n<!-- image --> \nthe acceleration of the erupting material ceases, with the CME front propagating at a nearly constant speed thereafter. Qualitatively, the height/time plot in panel (b) closely matches the height/time plots of essentially all pseudostreamer CMEs (e.g., Wang 2015). The high speed of our explosive, fast CME is near the upper end of the 250-700 km s -1 range observed.', '4.1. Energetics': 'Having summarised the main evolutionary features of the eruption, we now explore the manner of the energy release and its relation to reconnection. To do so, we first define the free magnetic and kinetic energies of the system to be \nE m = ∫∫∫ V 1 8 π B 2 dV -(∫∫∫ V 1 8 π B 2 dV ) t =0 , (7) \nE k = ∫∫∫ V 1 2 ρ v 2 dV -(∫∫∫ V 1 2 ρ v 2 dV ) t =0 . (8) \nTime t = 0 corresponds to the end of the relaxation period when both quantities have stopped varying to within a few percent. Their subsequent evolution is shown in Figure 11(a). The early stages of the simulation ( t < 6 . 5 hr) are marked by a gradual increase in free magnetic energy as the closed field is sheared. By contrast there is negligible additional kinetic energy in the system, including when the HFT forms. It is \nclear that the HFT formation in our simulation is a lowenergy process, not an explosive one. This differs from most previous breakout CME studies where the eruption commences when flare reconnection begins and the HFT first forms (e.g., Karpen et al. 2012). \nThe start of the breakout phase overlaps with the end of the driving phase. Breakout reconnection starts at t ≈ 5 hr, but becomes more rapid and self-sustaining around the time the HFT forms at t ≈ 6 hr. Once the driving ceases, this leads to a steady, slow decrease in E m and a small increase in E k . The evolution switches to a rapid increase in E k and drop in E m , characteristic of a breakout CME-like evolution, in the early phase of the eruption ( t ≈ 8 hr to t ≈ 9 hr). The near-exponential rise in E k (and drop in E m ) then slows slightly throughout the flux rope disconnection, before tapering off once the disconnection finishes. By comparison, the disconnection in the jet simulation occurs much more rapidly and initiates the rapid rise in E k (and drop in E m ). Following the disconnection of our pseudostreamer CME flux rope, E k continues to rise as more mass is ejected and the drop in E m tapers off as the interchange reconnection relaxes the closed field toward a new equilibrium. In this end state, the closed field still retains a small amount of free magnetic energy, as the opening/closing process is not 100% efficient in transferring the free energy and helicity to the open field. This is another general property of jet-like eruptions (e.g., Pariat et al. 2009; Wyper & DeVore 2016; Karpen et al. 2017; Wyper et al. 2018a).', '4.2. Surface Connectivity Evolution': "To relate the energy release to the reconnection process, it is instructive to consider how the surface connectivity changes during the eruption. Figure 12 shows the evolution of the squashing factor Q (Titov 2007) (grey scale) on the surface throughout the simulation. The squashing factor shows the surface imprint of magnetic (quasi-)topological boundaries in the volume, and has been shown to closely correlate with flare ribbons in observed flares (e.g., Janvier et al. 2013; Savcheva et al. 2016). To calculate Q we implemented the method of Tassev & Savcheva (2017) on adaptive grids and applied it to the data from ARMS. \nFigure 12(a) shows the surface connectivity near the end of the driving period. The fan plane footprint is a closed ring of Q at the boundary between open (yellow shading) and closed field regions. The footpoint of the inner spine is also highlighted. Two small hooks of Q denote the formation of the first flux rope field lines (and simultaneously the HFT) in the simulation volume. Figure 12(b) shows how this has evolved by \nFigure 11. (a) Free magnetic ( E m , blue) and kinetic ( E k , red) energies. Grey: Time profile of the driving (scaled to fit on the plot). Blue shading highlights the flux rope disconnection period. (b) Average number of turns within the flux rope ( ⟨ T w ⟩ , black) and the decay index at the flux rope axis ( n , red). (c) flux rope flux (Φ FR , black) and cumulative interchange reconnected flux (Φ int , red), normalised by the total closed-field flux (Φ tot ). (d) Rates of change of these normalised fluxes (same colour scheme). Green: Normalised closed-field reconnection rate obtained by calculating the swept-over flare-ribbon flux (see text for details). HFT = hyperbolic flux tube; FR = flux rope. \n<!-- image --> \nthe the early stages of the eruption. The flux rope footpoints have grown considerably and the hooked ends of the quasi-separatrix layer (QSL) ribbons are now clearly discernible. Furthermore, the hooks have spread farther around the polarity inversion line (PIL) and two parallel ribbons are spreading out from the right side of the PIL (see also the animation). Both patterns follow from reconnection at the HFT below the flux rope. In fact, this ribbon evolution is exactly that of the standard tworibbon flare model (e.g., Aulanier et al. 2012; Janvier et al. 2013), here embedded within the spine-fan topology of the pseudostreamer for a filament channel formed above a circular PIL. Additionally, the straight sections of the ribbon exhibit a corrugated structure, which is the imprint of plasmoid flux ropes formed within the flare current layer (Wyper et al. 2022; Dahlin et al. 2022). \nFigure 12(c) shows the beginning of the flux rope disconnection, whichs occurs when the breakout and flare current sheets combine into one long sheet, Figure 5(a). The corresponding imprint of this merger in the QSL ribbons is when the two inner ribbons reach the circular ribbon (on the negative side) and the inner spine (on the positive side). The two ribbon systems meet as there is now no intervening flux between the erupting flux rope and the open/closed boundary. The 'flare' reconnection at this point becomes interchange reconnection, and the null point moves below the flux rope. This is exactly the same surface-connectivity evolution \nseen in our jet model (Pariat et al. 2023). Furthermore, as in the jet simulation the formation of plasmoids in the null-point current layer imparts spiral structure to the circular ribbon (e.g., Pontin & Wyper 2015; Wyper et al. 2016). \nThe negative footpoint of the erupting flux rope now rapidly opens up while the ribbons on the other side of the closed-field region continue to spread apart, Figure 12(d). In the aftermath of the disconnection, the interchange reconnection continues around the PIL, with the fan plane first shifting up (Fig. 12(e)), and then left and down (Fig. 12(f)); see also the animation of this figure. This latter shift closes the field back down over the negative flux rope footpoint, moving the CME footprint as shown in Figure 7. Ultimately, this puts the closed-field region roughly back where it was before the eruption, with the surrounding open flux now having received most of the twist/helicity that was injected into the closed field by the driving. Therefore, the footpoints of the open field lines threading the CME are adjacent to the closed field, and are seen as a broad region of complex Q structure in the open field, Figure 12(f). Analogous features form in the jet simulations (Wyper et al. 2016).", '4.3. Flux Rope Identification': 'To complement the evolution of reconnection in the system inferred from the surface connectivity, we also isolated the flux rope itself. After some experimenta- \nFigure 13(a) shows the flux rope identified using this procedure in a vertical cut; the threshold τ w = 6 is contoured in black. Figure 13(b) shows that this threshold fully captures the flux rope (shown in yellow), which in this plane should be contained within a closed loop of high Q above the HFT (e.g., Savcheva et al. 2016). Noting from Figure 13 that the average twist within the \n<!-- image --> \nFigure 12. Evolution of the squashing factor ( Q ) on the surface throughout the evolution. Yellow shading shows the open field regions. An animation is available showing the motion of the QSLs and the closed-field region across the surface. The static figure highlights the main features in 6 panels. The duration is 4 s and runs from t = 4hr 52 min to 13 hr 3 min. \n<!-- image --> \ntion, we found that a reliable non-dimensional field-lineintegrated quantity for identifying the flux rope is \nτ w = T w L L PIL , (9) \nwhere T w is defined as (Berger & Prior 2006; Liu et al. 2016) \nT w = ∫ L ∇ × B · B 4 π \| B \| 2 dl, (10) \nand L and L PIL are, respectively, the field-line length and the length of the PIL along which the flux rope forms (here this is the entire circular PIL). T w is the average number of turns of neighbouring field lines around the given field line; it reduces to evaluating the forcefree parameter on the field line if the field is locally \nFigure 13. Top: Turns parameter T w at t = 8hr 12min; the black contour shows the length-weighted twist parameter τ w = 6 identifying the flux rope. Bottom: Q ; the flux rope is shaded yellow. \nforce-free. Once the eruption gets underway, the field is far from force free and includes many small-scale flux ropes within the breakout current sheet. The weighting of L/L PIL ensures that the main coronal flux rope is preferentially identified. Furthermore, we limit the calculation of turns T w to closed field lines with welldefined, finite lengths L . This procedure enables us to identify flux-rope field lines unambiguously by monitoring τ w prior to their disconnection. \ncontoured region is T w ≈ 3, this indicates that the flux rope field lines are roughly twice the length of the PIL, L/L PIL ≈ 2.', '4.4. Reconnection Rates': 'Figure 11(c) shows the increase in the toroidal magnetic flux (Φ FR ) contained within the flux rope versus time, calculated by integrating the field component through the plane over the region identified in Figure 13. The value is normalised by the total flux (Φ tot ; all closed) of the minority polarity. The plot shows that the flux rope contains as much as about 30% of the magnetic flux within the closed field. By comparison, when we apply the same analysis to the vertical jet from Wyper et al. (2018a), the flux rope accumulates a maximum of only about 10% of the closed flux. This explains, at least in part, why the erupting CME flux rope takes a comparatively long time to disconnect: three times the amount of flux must be processed. \nThe magnetic flux cumulatively reconnected by interchange reconnection (Φ int ) is shown in red in Figure 11(c) for comparison. This shows the total flux that is opened (or closed) over time, again normalised by the total closed flux. By the time the flux rope leg has completely disconnected ( t just over 10 hours), the cumulatively opened/closed flux equals the entire amount of closed-field flux; i.e., by this time all of the closed flux has likely interchange-reconnected once. The subsequent re-closing down of the opened field then grows the total Φ int well past the amount of flux in the closed field. \nThe rates of change of the two quantities are shown in Figure 11(d). The red curve is the interchange reconnection rate, representing the breakout reconnection rate before the flux rope disconnects and the fast flare-like reconnection rate after. The black curve represents the rate of closed/closed flare reconnection occurring at the HFT prior to the flux rope disconnection. The negative rate after disconnection is an artifact of identifying only closed flux-rope field lines. The breakout reconnection starts out the fastest and is well underway when the fast flare reconnection is triggered at t ≈ 8 hr. Shortly after this time, the HFT reconnection rate rapidly surpasses the interchange reconnection rate. It is notable that the interchange reconnection rate does not increase significantly after onset of the rapid HFT reconnection and flux-rope rise around t ≈ 8 hr. This indicates that the fast HFT flare reconnection is driven by the rapid rise of the flux rope, rather than by an increase in the removal of strapping flux via breakout reconnection (which occurs in the jet simulation). The breakout reconnection rate increases later in reaction to the flux rope driving \ninto the breakout current sheet from below, which initiates the disconnection of the flux rope from the surface. \nThe amount of flux contained within the flux rope also can be determined by calculating the surface flux swept out by the flare reconnection, in the manner of two-ribbon flares (e.g., Kazachenko et al. 2022). Field lines undergoing closed/closed flare reconnection were identified if their length changed more than 40% from one time to the next. Care was taken to not doublecount the flux and to exclude changes due to the interchange reconnection. The green curve in Figure 11(d) shows one-half the normalised rate of flux swept out in the negative (majority) polarity closed-field footprint. The result closely matches the rate of toroidal flux accumulation within the flux rope (black curve) prior to disconnection. This close agreement suggests that there are equal increases in the poloidal and toroidal fluxes within the flux rope at this time. Qualitatively, it is consistent with the presence of a strong guide field within the erupting filament channel where the flux rope forms. Quantitatively, it shows that the fluxes swept out by the hooks and straight sections of the QSL ribbon are the same, as they are conjugate footpoints of the reconnecting field.', '4.5. Ideal Flux Rope Evolution': 'Here we focus on the ideal evolution of the flux rope by averaging the twist on each field line within the region shown in Figure 13(b). This gives an approximation to the overall twist of the flux rope (e.g., Liu et al. 2016). The black curve in Figure 11(b) shows that the flux rope is highly twisted at formation, averaging ⟨ T w ⟩ ≈ 3, or nearly three turns along its length. This is well into the unstable range of the kink instability (e.g., Torok & Kliem 2005). Notably, the number of turns begins to decrease after eruption onset at t ≈ 8 hr, suggesting that twist in the flux rope is being converted to writhe of its axis at this late stage. This result implies that the flux rope does, indeed, kink, but not before the eruption onset. \nThe red curve in Figure 11(b) shows the decay index \nn = -d (ln( B ex ) d (ln( h )) , (11) \nwhere h is the height above the lower boundary of the highest point on a field line approximating the flux rope axis and B ex is the field external to the flux rope. The decay index is difficult to calculate accurately; see Zuccarello et al. (2015) for an in-depth discussion. The index may be approximated by taking as the external field ( B ex ) the horizontal component of the potential field at the flux rope axis (Zuccarello et al. 2015), and we do so \nhere. Figure 11(b) shows that the decay index starts at quite a high value n ≈ 2 . 0 and increases steadily until reaching a value n ≈ 2 . 5 when the eruption gets underway. This value is consistent with values in simulations of CMEs triggered by the torus instability (e.g., Kliem & Torok 2006; Aulanier et al. 2012), although it is substantially higher than the value n ≈ 1 . 5 often quoted as the threshold for instability. \nThe onset of fast flux rope acceleration ( t ≈ 8 . 5 hr) occurs well after the onset of self-sustaining breakout reconnection and flux-rope formation due to tethercutting reconnection ( t ≈ 6 hr). The flux rope survives an extended interval apparently unstable to the kink mode, as measured by its average twist ⟨ T w ⟩ , and to the torus mode, as measured by its decay index n . Taken together, our analysis suggests that neither mechanism is responsible for the transition from slow to fast rise and eruption onset in the pseudostreamer. In contrast, this transition is clearly concurrent with the onset of fast flare reconnection below the coronal flux rope, as evidenced by the abrupt turning up of the FR flux curve in Figure 10(c) and the FR reconnection rate curve in Figure 10(d).', '5.1. Implications for Theory': 'To better understand the role played by the ideal flux-rope evolution in the pseudostreamer CME, we revisited findings from previous coronal-hole jet studies. For the vertical jet simulation reported in Wyper et al. (2018a), we calculated the average twist ⟨ T w ⟩ and local decay index n of the flux rope in the same manner as described above. The results are that ⟨ T w ⟩ ≈ 2 . 2 and n ≈ 2 . 0 around the time that the vertical jet was launched. Both values are high enough to imply linear instability, and are comparable to those obtained for the CME. The breakout reconnection already had become self-sustaining for the jet, just as it had for the CME. In both cases, the upward flux-rope motion was slow, not explosive, and clearly was set by the rate at which breakout reconnection removed the strapping field above. The key signature associated with the transition to strong upward acceleration of the flux ropes was the onset of fast flare reconnection in the corona below them. \nA contrast to these behaviors was found in a related study of active-region periphery jets, in which the ambient magnetic field is highly inclined from the vertical (Wyper et al. 2019). In such configurations, the null point resides in the low corona off to the side of the filament-channel flux rope, rather than in the high corona above it. We found in that case that the breakout feedback was inhibited; the flux rope rose more or \nless vertically within the pseudostreamer dome rather than pressing its strapping field horizontally against the breakout current layer. Lacking effective feedback between the flux-rope rise and the breakout reconnection, eventually the flux rope suffered a classic kink instability. Its internal twist converted to writhe of its axis, distorting the flux rope into an inverseγ shape as it kinked, and its apex tilted toward the null point and accelerated to and, later, through the breakout current layer there (see Figure 9 in Wyper et al. 2019). The internal twist peaked at T w ≈ 3 and averaged ⟨ T w ⟩ ≈ 1 . 5 (see Figure 10 and text in Wyper et al. 2019), values that are similar to those for the vertical jet and the CME. The rapid rise of the newly kinked flux rope in this third case was accompanied, rather than preceded, by the onset of fast flare reconnection below it. \nThe absence of the expected signatures of kinkinstability onset, which are so clear in the active-regionperiphery (ARP) jet, from the vertical jet and the CME suggest that the instability plays no critical role in either case. The evidence is less clear-cut for the lack of a critical role for torus instability, but as noted, the critical index n measured at our flux ropes is well above the typically cited threshold n = 1 . 5 (Kliem & Torok 2006). The same is true for the twist parameter T w and its profile-dependent threshold T w ≈ 1 . 5 for the kink instability. What do these facts imply? \nOne possibility is that the analytically derived thresholds are too small, and the actual thresholds are significantly higher. This is plausible, given the special symmetry of the equilibrium configurations and the simplifying assumptions required by the analyses. Our configurations possess no special symmetries whatsoever, and the assuredly stabilizing effects of line-tying the overlying strapping fields must be taken into account but greatly complicate the analyses. The main argument against this explanation is the occurrence of classic kink signatures in the ARP jet, although it certainly is possible that only this particular example among our three cases actually reaches its true instability threshold. \nA second possibility is that either instability has, or both have, in fact, reached the true threshold for onset; but the evolving system has adjusted to attain a quasi-static state in which the mode(s) saturated. The overlying strapping fields have more freedom to expand upward and accommodate the strengthening flux rope in the vertical jet and the CME, with the null point high above, than in the ARP jet, with the null point low and off to the side. The last case may simply constrain the flux rope so much more effectively that it becomes strongly unstable and kinks violently, unlike the other two cases. \nFigure 14. Schematic summary of pseudostreamer CMEs combining our latest results with previous observational and simulation studies. Top row: low-expansion, jet-like CME events. Bottom row: high-expansion, fan-shaped CME events. \n<!-- image --> \nIndirect support for the above explanations is provided by the translationally symmetric simulation of pseudostreamer CMEs by Lynch & Edmondson (2013). Their eruptions were driven by sheared arcades: fluxrope instabilities played no role because there were no flux ropes in the system prior to eruption. Due to the special symmetry, the CME flux ropes formed only upon onset of the flare reconnection, and they were untethered to the Sun at creation. The ideal expansion of the increasingly energized sheared arcade field eventually induced breakout reconnection of the strapping fields at the null point above. The expansion turned explosive when fast flare reconnection switched on below the rising arcade, rapidly accelerating both the ideal upward motion and the breakout reconnection at the apex of the pseudostreamer, and forming the untethered CME flux rope in the process. \nA fully definitive resolution of the role of ideal instability in these simulated eruptions cannot be achieved with Eulerian MHD models, such as ARMS, alone. All such computational models have irreducible amounts of numerical diffusion in them to stabilize their solutions and make them monotone. The consequence is that it is impossible to eliminate all nonideal evolution from their calculations, including magnetic reconnection. A purely ideal model is needed to simulate these configurations, and others, to firmly determine whether ideal instability is essential or inconsequential to the initiation of coronal jets and CMEs. The Lagrangian Field-Line Universal relaXer (FLUX; DeForest & Kankelborg 2007; Lowder \net al. 2024) would be a possible tool to apply to such studies (e.g., Rachmeler et al. 2010).', '5.2. Implications for Observations': "The present simulation explains many observational features of fan-shaped pseudostreamer CMEs. Prior to eruption, the model predicts that both a faint jet and a base-difference dimming should be produced along the open spine as breakout reconnection launches closedfield plasma into the heliosphere, as has been described previously (Kumar et al. 2021; Wyper et al. 2021). The model also reproduces the bursty outflows and dense plasmoid signatures often observed in the breakout and post-eruption flare current sheets (Kumar et al. 2019b, 2021, 2023). The model further predicts a rolling motion of the erupting flux rope in the low corona. The flux rope is deflected toward the lower field strength at the null point, as has been noted in previous studies (Panasenco et al. 2011; Lynch & Edmondson 2013; Sahade et al. 2022), but also rotates due to the flare reconnection jet becoming oriented along the side of the flux rope, Figure 4(c). Such rolling motions are common in the early stages of fan-shaped pseudostreamer CMEs (Wang & Hess 2018; Kumar et al. 2021). \nThe simulation produces V-shaped features that may explain those noted by Wang & Hess (2018). First, Vshaped retracting field lines are formed on the underside of the flux rope in the early stages of the eruption; e.g., Figure 4(g). Second, much larger V-shaped field lines are formed when the flux rope disconnects; e.g., Figure 5(d). The disconnection process also produces \nretracting high-lying cusp structures, which Wang & Hess (2018) concluded were evidence of interchange reconnection. The simulation reveals that the fan-shaped CME is an open-field magnetic structure, with an embedded twisted flux tube that is the remnant of the original coronal flux rope. Embedded bubble-like structures are sometimes observed in these events (cf. Fig. 2, third row, in Wang 2015). Furthermore, Wang & Hess (2018) noted that the CME exhibited a 'twisting' motion concurrent with the formation of the high-lying cusp structures. Our simulation shows that this twisting is likely the whip-like motion of the disconnected flux-rope and sheath field lines. This differs from the spire rotation along a fixed spine seen in jet-like CMEs and coronal jets associated with filament channel eruptions. Finally, the simulation predicts that a shock is created and propagates out ahead of the CME body. Such shocks have been observed ahead of blowout jets and pseudostreamer CMEs in white-light images (e.g., Vourlidas et al. 2003; Miao et al. 2018; Kumar et al. 2021). In a follow-up study we will explore in greater detail the synthetic white-light signatures of this simulation. \nOur simulation also offers insight into how high-energy particles could be released into the heliosphere in these events. The opening of the flux rope here is similar to the scenario established by Masson et al. (2013, 2019) for the release of high-energy particles. In their case, the null-point topology was fully closed beneath a helmet streamer initially, then dynamically opened during the eruption. In our case, the null-point topology is surrounded by open field from the outset. In both cases, however, high-energy particles accelerated by closed/closed flare reconnection in the early stages of the eruption will be trapped within the flux rope. Some will mirror back and forth between the two footpoints. When the disconnection occurs, these particles will be promptly released out along the newly opened field. Some particles may be accelerated further by the intense interchange reconnection associated with the disconnection. Such particle bursts should be detectable in situ by missions such as Solar Orbiter or Parker Solar Probe. They may also be associated type-III radio bursts (e.g., Kumar et al. 2017; Chen et al. 2018) in the same way that many coronal jets are. \nThe trailing tail of the CME is dominated by torsional Alfv'enic waves and denser field-aligned flows associated with the interchange reconnection that enables the flux disconnection. The Alfv'enic waves might steepen to form switchbacks (Squire et al. 2020; Wyper et al. 2021) that could be detected as a switchback patch by Parker Solar Probe. Furthermore, the enhanced den- \nty in the field-aligned flows should be observable with high-cadence white-light coronagraphs on missions such as Solar Orbiter. Production of these waves and flows will continue for several hours after the CME has concluded. Similar post-eruption interchange reconnection, but on the much smaller scales of coronal jets, leads to the formation of transient plumes (Raouafi & Stenborg 2014). This correspondence also highlights the similarities between jets and pseudostreamer CMEs.", '6. SUMMARY': "We have presented an analysis of a simulated broad, fan-shaped pseudostreamer CME. Based on our findings, in Figure 14 we summarise the key features of these eruptions and how they compare with narrow, jetlike pseudostreamer CMEs. Both types are ultimately constrained by the adjacent open field and have a discernible element of rotation. In jet-like CMEs, the rotation manifests in the propagation along the spire of twist from the flux rope in the form of helical outflows, whereas in fan-shaped CMEs it manifests as a whip-like motion of the embedded twisted flux tube (the remnant of the original coronal flux rope) and its sheath field lines. Both CME types are comprised of open field lines following the disconnection of one end of the flux rope due to interchange reconnection. In addition, both are preceded and followed by bursty interchange reconnection that launches torsional Alfv'enic waves and episodic field-aligned dense outflows. \nThe key difference between the two types of pseudostreamer CMEs is the greater expansion of the erupting flux rope in broad, fan-shaped versus narrow, jet-like eruptions. At larger scales where the spherically expanding geometry is important, the field strength falls off with height allowing for greater transverse and vertical expansion of the developing flux rope. This enables the eruption to more easily push aside the ambient background field so that more of the flux rope survives intact its ascent into the high corona, forming a fan-shaped CME. At smaller scales and in a straighter, more uniform ambient field, the flux rope is more highly constrained by and interacts more strongly with the background field. More of the flux rope is consumed by reconnection as it breaches the null point, injecting its twist onto the surrounding open field and forming a collimated, narrow CME. \nDespite their differing CME morphologies, the magnetic connectivity evolution is the same in both event types: it consists of the eruption and disconnection of a flux rope from beneath the pseudostreamer topology. Therefore, our model shows that fan-shaped pseudostreamer CMEs are simply the extreme end of a con- \ntinuum of eruptive events which includes jet-like pseudostreamer CMEs and minifilament coronal jets. Moreover, these eruptive events should be considered as a separate class of CMEs from the bubble-like CMEs that originate beneath helmet streamers. In those events, the connection of the flux rope at both ends to the solar surface is maintained to much greater distances from the Sun owing to the magnetic polarity reversal across the heliospheric current sheet at the top of the helmet streamer. \nThis work highlights the need for coordinated simulation and observational studies of pseudostreamer CMEs. In a follow-up paper (Lynch et al. 2024) we will present the expected remote and in-situ observational signatures of this simulation as a guide for interpreting the latest observations from Solar Orbiter and Parker Solar Probe. We also plan to conduct future simulations to identify the conditions that govern the transition from narrow, jet-like to broad, fan-shaped CMEs in these pseudostreamer events. \nPFW was supported by an STFC (UK) consortium grant ST/W00108X/1 and a Leverhulme Trust Research Project grant. PFW & BJL were supported by NSF grant NSF AGS 2147399. BJL acknowledges support from the NASA XRP and LWS programs. CRD was supported by a NASA H-ISFM grant to Goddard Space Flight Center and the NASA XRP program. SKA was supported by a LWS grant to U Michigan. The computations were sponsored by allocations on Discover at NASA's Center for Climate Simulation and on the DiRAC Data Analytic system at the University of Cambridge, operated by the University of Cambridge High Performance Computing Service on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk) and funded by BIS National E-infrastructure capital grant (ST/K001590/1), STFC capital grants ST/H008861/1 and ST/H00887X/1, and STFC DiRAC Operations grant ST/K00333X/1. DiRAC is part of the National E-Infrastructure. The supporting ARMS dataset is published in Durham University's Collection open data repository. DOI: 10.15128/r1fb494847x \nWe thank Judy Karpen, Angelos Vourlidas, Mariana Cec'ere, and Etienne Pariat for useful discussions.", 'REFERENCES': "Antiochos, S. K., DeVore, C. R., & Klimchuk, J. A. 1999, \nApJ, 510, 485, doi: 10.1086/306563 \nAulanier, G., Janvier, M., & Schmieder, B. 2012, A&A, 543, A110, doi: 10.1051/0004-6361/201219311 \nBerger, M. 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2024arXiv240911991T	We consider theories which break the invariance under diffeomorphisms Diff down to transverse diffeomorphisms TDiff in the matter sector consisting of multiple scalar fields. In particular we regard shiftsymmetric models with two free TDiff scalar fields in a flat RobertsonWalker spacetime and use the perfect fluid approach to study their dynamics. As a consequence of the symmetry breaking an effective interaction between the fields is induced from the conservation of the total energymomentum tensor without the necessity to introduce any explicit interacting term in the Lagrangian. We study the different singlefield domination regimes and analyze the energy exchange between the fields. Thereupon we introduce an application of these models for the description of interactions in the dark sector and compare the theoretical predictions of our model to observational data from Type Ia supernovae.	2024-09-01T00:00:00Z	['arXiv:2409.11991', '2024arXiv240911991T', '10.48550/arXiv.2409.11991']	['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	Multifield TDiff theories for cosmology	2,024	197	0.32	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.11991.pdf	{'Multi-field TDi ff theories for cosmology': "Diego Tessainer a, ∗ , Antonio L. Maroto a , Prado Mart'ın-Moruno a \na Departamento de F'ısica Te'orica and Instituto de F'ısica de Part'ıculas y del Cosmos (IPARCOS-UCM), Universidad Complutense de Madrid, 28040, Madrid, Spain.", 'Abstract': 'We consider theories which break the invariance under di ff eomorphisms (Di ff ) down to transverse di ff eomorphisms (TDi ff ) in the matter sector, consisting of multiple scalar fields. In particular, we regard shift-symmetric models with two free TDi ff scalar fields in a flat Robertson-Walker spacetime and use the perfect fluid approach to study their dynamics. As a consequence of the symmetry breaking, an e ff ective interaction between the fields is induced from the conservation of the total energy-momentum tensor, without the necessity to introduce any explicit interacting term in the Lagrangian. We study the di ff erent single-field domination regimes and analyze the energy exchange between the fields. Thereupon, we introduce an application of these models for the description of interactions in the dark sector, and compare the theoretical predictions of our model to observational data from Type Ia supernovae. \nKeywords: cosmology, scalar fields, shift-symmetry, transverse di ff eomorphisms, dark energy, interacting dark sector.', '1. Introduction': 'It is widely known, as observational data indicate, that our Universe currently exhibits an accelerated expansion [1]. Many models explain this as the consequence of a dark energy component dominating the cosmic expansion, taken to be a cosmological constant in the standard model. However, there are other alternatives, such as quintessence , involving a canonical scalar field with a dynamical equation of state determined by its potential [2], and k-essence [3], which also display dynamical dark energy and can avoid fine-tuning problems, but include non-canonical kinetic terms in the action. Additionally, observational data also indicate that most of the matter composition of our Universe is dark matter [4]. Furthermore, it is nowadays recognized that there exists a tension in the Hubble parameter H 0 measurements [5], which could be alleviated by models involving dark sector interactions [6] or phantom models [7], in which the energy density of the dark energy component increases with the expansion. In addition, it has been proven that in order to ease both the H 0 and S 8 cosmological tensions simultaneously by taking into account new physics that is relevant only at late cosmic times, a dark energy component crossing the phantom limit is necessary [8]. On the other hand, as the nature of the dark sector is unknown, possible modifications of gravity at cosmological scales are often considered [9]. Regarding this possibility, multiple modified gravity theories extending upon General Relativity (GR) have been explored [10]. \nEven if GR provides a very powerful tool for studying gravity and cosmology, theories breaking invariance under di ff eomorphisms (Di ff ) have been recently gaining popularity, with one \nof the most prominent ones being Unimodular Gravity (UG) [11, 12, 13]. In UG, the metric determinant is taken to be a fixed non-dynamical field and the Di ff invariance is broken down to transverse di ff eomorphisms (TDi ff ) and Weyl rescalings. UG theories could provide a solution to the problem of vacuumenergy which does not gravitate in this type of theories [14]. Nevertheless, in this work we will focus on theories that are only invariant under TDi ff , which have lately started to be studied more deeply. Thus for instance, TDi ff models beyond UG have been studied in references [15, 16, 17, 18]. The cosmological evolution in TDi ff -invariant theories propagating a scalar graviton mode was analyzed in reference [17]. On the other hand, TDi ff invariant models with broken di ff eomorphisms in the matter sector have been analyzed in references [19, 20] for single scalar fields. There it is shown that even though on small scales such theories behave as standard Di ff models, on superHubble scales the behaviour can be drastically di ff erent, thus opening up a wide range of possibilities for cosmological model building. Thus, in particular, a simple TDi ff model for dark matter based on a free scalar field was proposed in references [19, 20]. A unified TDi ff model for the dark sector has been considered in reference [21]. A general classification of singlefield TDi ff models based on their speed of sound and equation of state was performed in reference [22]. TDi ff models for single abelian gauge fields can be found in reference [23] and their phenomenological implications for cosmic magnetic field evolution in reference [24]. \nIn this work we will extend the previous works and consider multi-scalar TDi ff invariant models in the matter sector in flat Robertson-Walker (RW) spacetimes. We will specifically regard shift-symmetric models, which are invariant under shift transformations of the field, i.e., ϕ → ϕ + C , where C is a constant. Thus, we will only consider the exact kinetic domination regime for each field. The motivation behind this approach \nlies in the fact that in this way we can avoid fine-tuning problems depending on the specific choice for the potential term in the action. On the other hand, not considering any mass or potential-like terms in the action also results in the EinsteinHilbert action only receiving higher-order radiative corrections, which also motivates our choice to only break the Di ff symmetry in the matter sector. \nUnlike the single-field case, the energy-momentum tensor (EMT) conservation will entail an e ff ective interaction between the fields as a consequence of the symmetry breaking even without introducing any interaction terms in the Lagrangian. This fact opens up a wide range of phenomenological implications for multi-field models. Particularly, we will apply this e ff ect to describe an interacting dark sector, comparing its predictions with observational data. \nThe work is organized as follows. In section 2 we briefly review the TDi ff formalism, focusing on shift-symmetric theories and lay the groundwork for our particular models. Section 3 is devoted to explain the theoretical framework for multi-scalar TDi ff models. In section 4 we perform a numerical analysis for our model, applying it to the dark sector. Results will be compared both with observations and w CDM, and physical predictions for our TDi ff model will be obtained. Finally, in section 5 we will discuss the conclusions.', '2. Single-field shift-symmetric TDi ff theories': 'In this section we will briefly recap the main results obtained for shift-symmetric TDi ff theories involving one scalar field.', '2.1. Transverse di ff eomorphisms and matter action': "Let us first consider a general infinitesimal coordinate transformation x µ 7→ x ' µ = x µ + ξ µ ( x ) given by the vector field ξ . As it is well known, the variation of metric tensor g µν ( x ) will be given by its Lie derivative, i.e., \nδ g µν = L ξ ( g µν ) = -∇ νξµ - ∇ µξν, (1) \nand thus it follows that the metric determinant ( g : = \| det( g µν ) \| ) will transform according to \nδ g = gg µν δ g µν = -2 g ∇ µξ µ . (2) \nLet us now write down our action. This is \nS = S EH[ g µν ] + S mat[ g µν, ϕ ] , (3) \nwhere S mat denotes the matter part of the action involving a single scalar field ϕ . Since we will only break the Di ff symmetry in the matter action, the geometrical part will just be the usual Einstein-Hilbert action \nS EH[ g µν ] = -1 16 π G Z d 4 x √ g R . (4) \nOn the other hand, the matter part will read \nS mat[ g µν, ϕ ] = Z d 4 x f ( g ) L ( g µν ( x ) , ϕ ( x ) , ∂µϕ ( x )) , (5) \nwhere L denotes the corresponding scalar under Di ff Lagrangian density and f ( g ) an arbitrary function of the metric determinant. Recalling (1) and (2) we can compute δξ S , which, after integration by parts and assuming that the fields vanish at infinity, reads [19] \nδξ S = Z d 4 x ∂µξ µ [ f ( g ) -2 gf ' ( g )] L . (6) \nThus, we see that the action is invariant under any infinitesimal coordinate transformation (Di ff invariant) only when f ( g ) -2 gf ' ( g ) = 0, i.e. f ( g ) ∝ √ g . However, the action is also invariant for any form of f ( g ) if the transformations satisfy ∂µξ µ = 0. This corresponds to a smaller subgroup of symmetry, the transverse di ff eomorphisms (TDi ff ).", '2.2. Single scalar-field models in the kinetic regime': 'Let us first consider the matter part of the action with a simple kinetic term [19, 20]: \nS mat = Z d 4 x 1 2 f ( g ) ∂µϕ∂ µ ϕ, (7) \nwhere f ( g ) is a positive coupling function of the metric determinant. The corresponding equation of motion reads \n∂µ GLYPH<0> f ( g ) ∂ µ ϕ GLYPH<1> = 0 , (8) \nand the EMT will be defined as usual: \nT µν : = -2 √ g δ S mat δ g µν , (9) \nwhich in this case reads \nT µν = f ( g ) √ g GLYPH<16> ∂µϕ∂νϕ -F ( g ) g µν ✷ ϕ GLYPH<17> , (10) \nwhere we have defined F ( g ) : = d ln f ( g ) / d ln g . Since we are not modifying the Einstein-Hilbert action, the Bianchi identities are preserved and thus the local conservation of the EMT will still hold [19, 20] under solutions of Einstein equations. \nIn relation to the background geometry, we will consider a spatially flat FLRW metric. Since we have less gauge freedom than in the Di ff case, we will not generally be able to perform a coordinate change that fixes the lapse function to one and we will have more physical degrees of freedom than in the Di ff case. Thus, our spacetime can be described by the following line element [25]: \nds 2 = b 2 ( τ ) d τ 2 -a 2 ( τ ) dx 2 , (11) \nwhere a ( τ ) and b ( τ ) are the independent components that will act as the scale factor and lapse function, respectively; and τ denotes the time-coordinate. Both must be computed from Einstein equations 1 . \nLet us now apply the perfect fluid approach. It is worth recalling that, when ∂µϕ is a time-like vector, the EMT (10) takes the form [20] \nT µν = ( ρ + p ) u µ u ν -p g µν, (12) \nwhere ρ = T 0 0 denotes the energy density, p = -T i j δ j i / 3 the pressure, and u µ is the four-velocity of the fluid, a time-like unit vector. Recalling (10) and using (11) we get \nρ = f ( g ) b 2 √ g GLYPH<2> 1 -F ( g ) GLYPH<3> ( ϕ \' ) 2 , (13) \np = f ( g ) b 2 √ g F ( g ) ( ϕ \' ) 2 , (14) \nwhere we have considered a homogeneous field ϕ = ϕ ( τ ). It is straightforward to see from equations (13) and (14) that \nw ϕ : = p ρ = F ( g ) 1 -F ( g ) ; (15) \nwhich will generally depend on τ and, thus, the equation of state parameter w ϕ will generally evolve throughout time. One particular case of interest takes place when the coupling function is a power-law, i.e., f ( g ) = kg α , where k and α are constants. In this case we obtain for w ϕ the following result: \nw ϕ = α 1 -α = const . (16) \nNotice how this requires α < 1 in order for the weak energy condition to be satisfied [20]. In addition, the zeroth component of the EMT conservation equation ∇ ν T µν = 0 yields the usual result [19]: \nρ \' + 3 a \' a ( ρ + p ) = 0 . (17) \nOn the other hand, the equation for the G 00 component of the Einstein tensor yields [25] \na \' a ! 2 = 8 π G 3 ρ b 2 , (18) \nwhich is the usual Friedmann equation in time τ . Notice that it recovers its original form under the coordinate transformation d t = b ( τ ) d τ , where t is the cosmological time. We will denote \' = d / d τ and · = d / d t . \nFinally, let us write the equation of motion of ϕ ( τ ) in this space-time (8): \nϕ \'\' ( τ ) + ϕ \' ( τ ) L \' ( τ ) L ( τ ) = 0 , (19) \nwhere L ( τ ) ≡ f ( g ( τ )) / b 2 ( τ ). This equation of motion implies that \nϕ \' ( τ ) = C ϕ L ( τ ) , (20) \nwith C ϕ a constant parameter. Substituting (20) in equations (13) and (14); factoring out ρ + p in equation (17) and recalling g = b 2 a 6 , the conservation law (17) reads \nd d τ ln GLYPH<16> a 6 GLYPH<17> = g \' ( τ ) d d g " ln (1 -2 F ( g )) g f ( g ) !# , (21) \nwhich provides the geometrical constraint that allows the conservation law (17) to be satisfied. This is [19]: \ng f ( g ) (1 -2 F ( g )) = Cga 6 , (22) \nwhere Cg is a constant. This geometrical constrain on the metric determinant g allows us to obtain the relation between b and a for any given coupling. For instance, if f ( g ) ∝ g α , equation (22) implies that \nb ∝ a 3 α/ (1 -α ) . (23) \nNotice that only when we take α = 1 / 2 (Di ff limit), we recover the standard sti ff -fluid behaviour ρ ( a ) ∝ a -6 of a kinetically dominated scalar field [26, 27]. In conclusion, TDi ff symmetry allows for a much wider phenomenology for simple kinetically driven scalar fields.', '3. Shift-symmetric multi-field TDi ff models': "In this section we will extend the previous results to the case of two free shift-symmetric TDi ff homogeneous scalar fields in the matter action with di ff erent coupling functions. Since both fields will be kinetically driven, our action will read \nS mat = Z d 4 x 1 2 2 X i = 1 GLYPH<16> fi ( g ) ∂µϕ i ∂ µ ϕ i GLYPH<17> . (24) \nNotice that we did not consider an interaction potential between both fields. As we will see, the energy exchange and the rich phenomenology will arise from geometrical constrains coming from the conservation of the total EMT, since the individual EMTs of each field will not be conserved as a consequence of the symmetry breaking. In fact, since our fields are free, the total EMT will simply be the sum of the individual EMTs of each field: \nT µν = T (1) µν + T (2) µν = ( ρ 1 + p 1) u µ u ν -p 1 g µν + ( ρ 2 + p 2) u µ u ν -p 2 g µν (25) \nFor homogeneous fields in a Robertson-Walker background both fields share a common velocity u µ and \nρ i = fi ( g ) b 2 √ g GLYPH<2> 1 -Fi ( g ) GLYPH<3> ( ϕ ' i ) 2 , i = 1 , 2 (26) \npi = fi ( g ) b 2 √ g Fi ( g ) ( ϕ ' i ) 2 , i = 1 , 2 (27) \nwhere very much as in the single-field case, we have defined Fi ( g ) : = d ln fi ( g ) / d ln g , so that the corresponding equations of state read \nwi : = pi ρ i = Fi ( g ) 1 -Fi ( g ) , i = 1 , 2 (28) \nThe conservation of the total energy-momentum tensor implies \n∇ µ T µν = ∇ µ T (1) µν + ∇ µ T (2) µν = 0 , (29) \nwhich in the Robertson-Walker background reads \nρ ' 1 + 3 a ' a ( ρ 1 + p 1) + ρ ' 2 + 3 a ' a ( ρ 2 + p 2) = 0 . (30) \nNotice that the previous expression does not imply the energy conservation for individual fields, but in general we will have \nρ ' 1 + 3 a ' a ( ρ 1 + p 1) = Q , (31) \nρ ' 2 + 3 a ' a ( ρ 2 + p 2) = -Q , (32) \nwhere Q is commonly referred to as the interacting kernel in the literature [28]. \nOn the other hand, the fields equations of motion read \nϕ '' i ( τ ) + ϕ ' i ( τ ) L ' i ( τ ) Li ( τ ) = 0 , i = 1 , 2 (33) \nwith Li ( τ ) ≡ fi ( g ( τ )) / b 2 ( τ ), so that very much as in the singlefield case, we can write \nϕ ' i ( τ ) = C ϕ i Li ( τ ) , i = 1 , 2 (34) \nwith C ϕ i constants. \nSubstituting these expressions into the respective pressures and energy densities (26) and (27), recalling the conservation equation (30) and proceeding analogously to the single-field case, calculations yield the following geometrical constrain: \nC 2 ϕ 1 g \| 2 F 1 -1 \| f 1 + C 2 ϕ 2 g \| 2 F 2 -1 \| f 2 = Cga 6 , (35) \nIn the case in which the coupling functions are simple power laws \nfi ( g ) = λ ig α i , i = 1 , 2; (36) \nwith λ i , α i constants, the conservation equation (35) implies \nC 1 g 1 -α 1 \| 2 α 1 -1 \| + C 2 g 1 -α 2 \| 2 α 2 -1 \| = Cga 6 ; (37) \nwhere C 1 = C 2 ϕ 1 /λ 1 and C 2 = C 2 ϕ 2 /λ 2. This is a very illuminating result, since as we observe from equation (37) it does not require the individual EMT conservation of each field and thus it will involve a geometrical-like interaction between the two components caused by the symmetry breaking. Unlike the single-field case, an explicit solution of this equation cannot be obtained even for simple power-law functions. \nLastly, here we include the expression for the energy density in the power-law coupling case, which will be of valuable use throughout the rest of the work: \nρ i ( a , b ) = Ci (1 -α i ) b 1 -2 α i a 6 α i + 3 , i = 1 , 2 (38) \nwhich is straightforwardly obtained from equation (26) using (34). Notice how the e ff ective interactions will be reflected on the particular form of b ( a ) obtained through the EMT conservation law (37).", '3.1. Approximate results: single-field domination': 'Let us first consider the case in which one of the fields, for example ϕ 1, dominates over the other, ϕ 2. We can thus neglect the contribution of ϕ 2 in (37), so \nC 1 g 1 -α 1 \| 2 α 1 -1 \| ≃ Cga 6 , (39) \nwhich can be solved as \nb ∝ a 3 w 1 (40) \nw 1 = α 1 1 -α 1 , (41) \nw 2 = α 2 1 -α 2 , (42) \nas we can see from (28). This is the same geometrical constrain one would obtain if ϕ 1 was the only field. Notice that this is just an approximation that provides the leading order of b ( a ), but it gives us valuable information concerning the evolution of the energy densities in the di ff erent domination regimes. Recalling (38) and using (40) yields \nρ 1( a ) ∝ a -3(1 + w 1) and ρ 2( a ) ∝ a -3(1 + w e ff ) , (43) \nand thus ϕ 1 decays as expected from its equation of state, but the subdominant field ϕ 2 will exhibit a decay as if it were a perfect fluid with constant equation of state parameter w e ff , w 2, where \nw e ff = 2 w 2 -w 1 + w 1 w 2 1 + w 2 . (44) \nIt is worth noting that the individual equation of state parameters wi will then depict the asymptotic decay behavior of each component when it is dominant. Fig.1 summarizes the wide range of phenomenological possibilities for the subdominant component. \nFigure 1: E ff ective equation of state parameter w e ff of the subdominant field ϕ 2 under ϕ 1 domination in terms of the individual equation of state parameters w 1 and w 2. \n<!-- image --> \nwhere \nThis result happens to be physically illuminating with regards to cosmological contexts. As we can see above, the induced interactions between perfect TDi ff fluids with di ff erent equation of state parameters allow for a wide range of possible evolutions for the subdominant component. In particular, all of the possible dark energy behaviors are plausible for the subdominant field, including phantom dark energy [29] ( w e ff < -1, where its energy density increases over time) and quintessence. We emphasize that these behaviours can be obtained without the addition of non-canonical kinetic terms [3], they are a result of breaking the Di ff symmetry down to TDi ff . Interestingly, although wi < -1 is not allowed for each individual field, in accordance to the weak energy condition [20], the dominance regimes allow for subdominant phantom behavior without violating the energy conditions. As a result, this provides a vast range of possibilities to describe an interacting dark matter-dark energy sector ( w 1 = 0, w 2 < -1 / 3) with an evolving dark energy decay given by a function w e ff ( a ) stemming from the broken Di ff invariance, exhibiting phantom decay at early times during the matter epoch. This will allow for phantom-crossing, as it will later be discussed.', '3.2. Energy exchange': "We will now analyze the exchange of energy between the fields induced by the e ff ective interaction, and its evolution through the several field domination regimes by studying the interacting kernel Q . Let us consider two kinetically-driven scalar fields ϕ 1 and ϕ 2, with constant equation of state parameters w 1 and w 2, respectively. Let us also assume that ϕ 1 dominates over ϕ 2. Using (38) on equation (32) and recalling (44) we obtain the following expression \nQ = 3 C 2(1 -α 2) a ' a a -3(1 + w e ff ) ( w e ff -w 2) . (45) \nwhich can be rewritten as \nQ = 3 ρ 2 H ( w 2 -w 1)(1 -w 2) 1 + w 2 . (46) \nwhere H = a ' / a denotes the Hubble parameter in time coordinate τ . \nLet us now study the sign of Q during the single-field domination regimes. Firstly, we observe from (46) that when ϕ 1 dominates, Q has two zeros, those being at w e ff = w 2, i.e., w 2 = w 1 and w 2 = 1. On the other hand, the analysis in the ϕ 2 domination regime is fully akin to the previous one, but we have to perform the change w 1 7→ w 2 and change the sign of Q (remember we defined Q with respect to the conservation law for ϕ 1). We show in Fig. 2 the sign of the interaction kernel in both cases ( ϕ 1 and ϕ 2 domination). \nIn light of this analysis, we distinguish three scenarios. In the first case, in which both equation of state parameters are smaller than one ( w 1, w 2 < 1), the sign of Q does not change between both domination regimes and thus the direction of the energy exchange will not be altered over time. More clearly, if we assume w 1 > w 2 we see from Fig. 2 that when ϕ 1 dominates Q < 0 and ϕ 1 loses energy in favor of ϕ 2, with the same happening as well when ϕ 2 is dominant. The same reasoning can \nbe applied to the case in which w 1 < w 2 (although in this case Q > 0), allowing us to conclude that in this case it is the field with the greater equation of state parameter who always loses energy. \nSecondly, we also have the case in which both fields are beyond sti ff fluids ( w 1, w 2 > 1). We can immediately check (see Fig. 2), similarly to how we proceeded in the previous case, that the direction of the energy exchange will not change during the interaction and it will always be the field with the larger equation of state parameter which gains energy from the other component. \nLastly, there is the case in which one of the fields is beyond a sti ff fluid and the other is not ( w 1 > 1, w 2 < 1 and vice versa). As opposed to the previous scenarios, we see from Fig. 2 that the direction of the energy flux changes between both domination regimes. For instance, if w 1 > 1 and w 2 < 1, Q will be smaller than zero under ϕ 1 domination and thus ϕ 2 will be gaining energy from ϕ 1. However, when ϕ 2 is dominant, since w 1 > 1 we can see that Q > 0 and thus it is ϕ 1 which gains energy from ϕ 2 now (the analysis is analogous if w 1 < 1, w 2 > 1). \nRegarding the potential applications for the description of the dark sector, notice that in (44) w e ff = -1 / 3 when w 2 = ( -1 + 3 w 1) / (7 + 3 w 1) ≡ A < w 1. This separates the region of w 2 values in which the subdominant field, taken to be ϕ 2 for this example, starts decaying as dark energy. Similarly, w e ff = -1 occurs at w 2 = ( -1 + w 1) / (3 + w 1) ≡ B and it corresponds to the phantom behavior boundary for ϕ 2. Hence, if w 1 < 1 we can see that if w 2 ∈ ( -1 , B ) the subdominant component will exhibit phantom dark energy behavior and the dominant field will lose energy in favor of this; and if w 2 ∈ ( B , A ) it will also gain energy from the dominant component ϕ 1, but not enough to display phantom nature. \nMore specifically, if we consider a dark sector model consisting of dark matter (DM) with w 1 = 0 and dark energy (DE), with w 2 < -1 / 3, we can observe from (44) that DE will always be phantom during the matter domination epoch due to the energy flux from DM ( Q < 0). The energy exchange will occur in the same direction when DE dominates, although it will now not be enough to keep the phantom behavior, and the DE decay will gradually transition to resemble its asymptotic value for the equation of state parameter w 2. On the other hand, DM will slowly start to exhibit a di ff erent decay than the typical a -3 as DE becomes more dominant. \nLastly, before we go on with our analysis let us briefly comment about the existence of tracking solutions in this model. Recalling (43) we see that the condition that must be satisfied in order for both fields to exhibit the same decay would be \n-3(1 + w 1) = 3 w 1 -9 w 2 -3 w 1 w 2 -3 1 + w 2 , (47) \nwhich cannot be accomplished unless we are in the trivial case in which both components are indeed the same, i.e., w 1 = w 2, and there would be no interaction. Thus, there will not be tracking solutions in this particular TDi ff model. \n<!-- image --> \nFigure 2: Sign of the interaction kernel Q when the ϕ 1 fluid dominates (left) and for ϕ 2 domination (right) \n<!-- image -->", '3.3. Analytical model': 'Solving the general constrain (37) is not a simple task, and it usually requires numerical treatment. However, there is a particular dark sector model for which equation (37) can be analytically solved, consisting of DM with w 1 = 0 ( α 1 = 0) and DE with w 2 = -1 / 2 ( α 2 = -1). Despite not being the best fitting model, as we will later see, being analytical provides us with a wide insight to further understand the physics behind shiftsymmetric multi-field TDi ff models. The constrain (37) then reads \nC 1 g + 3 C 2 g 2 = Cga 6 , (48) \nwhich is quadratic in g and can be easily solved as \ng = -C 1 6 C 2 + q C 2 1 + 12 C 2 Cga 6 6 C 2 , (49) \nwhere we have taken into account that Ci = C 2 ϕ i /λ i should be positive to avoid ghosts instabilities, so that only the positiveroot solution of equation (48) is physically sensible. This solution allows us to explicitly obtain the relation b ( a ): \nb ( a ) = r C 1 6 C 2 a -6 s 1 + 12 C 2 Cg C 2 1 a 6 -1 1 / 2 , (50) \nvalid for all values of a . As we will later see, the remote past a ≪ 1 will correspond to the matter era, and in the distant future a ≫ 1 DE will be dominant, as expected. \nFor a ≪ 1, expanding (50) in powers of a yields \nb ( a ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> a ≪ 1 ≃ p CgC 1 1 -3 2 C 2 Cg C 2 1 a 6 , (51) \nfrom which we can obtain the respective energy densities: \nρ 1( a ) ≃ p C 1 Cg a -3 -3 C 2 Cg 2 C 2 1 a 3 , (52) \nρ 2( a ) ≃ 2 Cg C 1 ! 3 / 2 C 2 a 3 . (53) \nNotice how the DM ( ρ 1) decay is governed by the a -3 term, which corresponds to the expected behavior according to w 1 = 0. Consequently, DM is dominant at early times. Besides, DE ( ρ 2) evolves with a 3 , exhibiting the phantom nature we previously discussed (in particular, w e ff = -2) as a result of it gaining energy from DM. This can be illustrated writing the conservation equations for each component in terms of the energy density of the other, which read: \nρ \' 1 + 3 a \' a ρ 1 ≃ -9 2 H ρ 2 , (54) \nρ \' 2 + 3 a \' a ( ρ 2 + p 2) ≃ + 9 C 2 g C 2 C 1 H 1 ρ 1 ; (55) \nwhere the phantom nature is exposed in (55) as a result of ρ 1 appearing in the denominator. \nOn the other hand, for a ≫ 1, expanding (50) yields \nb ( a ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> a ≫ 1 ≃ r C 1 6 C 2 √ Aa -3 / 2 -1 √ 2 A a -9 / 2 ! , (56) \nwith A ≡ q 12 C 2 Cg / C 2 1 . The energy densities thus read \nρ 1( a ) ≃ C 3 / 2 1 √ 6 C 2 √ Aa -9 / 2 -1 2 √ A a -15 / 2 ! , (57) \nρ 2( a ) ≃ C 3 / 2 1 6 3 / 2 C 1 / 2 2 ( -3 a -9 / 2 √ A + 2 a -3 / 2 A 3 / 2 ) . (58) \nThe leading order for large values of a in ρ 2 indicates that now our DE will decay as expected from its equation of state ( w 2 = -1 / 2), and DM decays faster than a -3 . This implies that at later times it is DE who becomes dominant, and from the reasoning of the previous subsection, we can see that DM \nis still giving energy to DE, but not enough to keep the phantom behavior as DM starts becoming subdominant. One could write analogous expressions to (54) and (55), but they are not as physically enlightening due to the lack of phantom nature under dark energy domination. \nWe will now write the exact expressions for both energy densities in order to discuss the whole evolution. From the previous analysis we know that both energy densities become equal at a certain time: a = a eq, ρ i = ρ eq. We can thus write ρ 1 and ρ 2 substituting (50) in (38) and equating them. We obtain \nC 1 = 5 2 ρ 2 eq a 6 eq C -1 g , C 2 = 125 16 ρ 4 eq a 6 eq 1 C 3 g ; (59) \nwhich allow us to write down the energy densities in terms of these parameters, which are easier to physically interpret than the integration constants Ci . Thus, we have: \nρ 1( z ) = 1 √ 3 ρ eq 1 + z 1 + z eq ! 6 Θ 1 / 2 ( z ) , (60) \nρ 2( z ) = 1 √ 27 ρ eq 1 + z 1 + z eq ! 6 Θ 3 / 2 ( z ); (61) \nwhere we defined Θ ( z ) ≡ q 1 + 15[(1 + z eq) / (1 + z )] 6 -1, and where z = 1 / a -1 denotes the redshift. \nWe will now obtain some physical results and compare this model to Λ CDM before analyzing the general case. Firstly, recalling (38) and the conservation law (17), we can parameterize the decay of each component with a function w e ff , i( z ) which satisfies the individual conservation law \nρ \' i + 3 a \' a [1 + w e ff , i( z )] ρ i = 0; (62) \nwhich yields the following result when recalling (38): \nw e ff , i( z ) = -1 3 " -1 + z b ( z ) (1 -2 α i ) d b d z -6 α i -3 # -1 , (63) \nwhere α i denote the exponents of the coupling functions of each component. This recovers the previously studied constant results when considering the respective field domination regimes. The functions w e ff , i( z ) can easily be simplified for the analytical case since we know b ( z ) explicitly. \nUsing the exact expression for b ( z ) (50) yields the result in Fig.3, for two di ff erent values of the free parameter z eq. Notice how in reality we only have z eq as our free parameter, since the cosmic sum rule ( ρ 1 + ρ 2) GLYPH<12> GLYPH<12> GLYPH<12> t = t 0 = (1 -Ω B) ρ c must be satisfied and, thus, it enforces an extra relation between the parameters that allows to remove the dependence on ρ eq. As a reminder, Ω B depicts the baryonic matter component of the universe, ρ c is the critical density and we are ignoring the radiation component at late times. It is worth noting that in this work we assumed that only the dark sector breaks the Di ff invariance, hence we will treat baryons as ordinary Di ff matter. As we can see from Fig.3, the DE decay behavior starts being phantom-like at early times, as expected, and then evolves in time until it reaches the \nFigure 3: w e ff ( z ) for DM and DE for various z eq. Dark energy transitions from being phantom during the matter era to decaying as usual dark energy with equation of state parameter w 2, with there being phantom crossing at recent times. Dark matter starts to decay faster than expected from w 1 = 0 as dark energy starts dominating. \n<!-- image --> \nasymptotic value reflected in the equation of state w 2 > -1 in the future, with there being phantom-crossing near the present. The parameter z eq only changes slightly the behavior in the intermediate regimes, without altering the main physical behavior. On the other hand, DM will exhibit its usual a -3 decay at early times but it will decay faster when DE starts to dominate. In light of this we can see that TDi ff models can provide a very rich phenomenology involving di ff erent time evolutions for the dark sector. This could lead to new models for interactions in the dark sector without the introduction of non-canonical terms or ghost instabilities in the action. \nLastly, we will analyze this model from the perspective of the density parameters to further understand shift-symmetric TDi ff dark sector models. We will denote the density parameters for DM and DE, respectively, as Ω DMT and Ω DET. We will also use the standard notation for the Λ CDM parameters: Ω M = Ω DM + Ω B for matter and ΩΛ for the cosmological constant. Recalling the Friedmann equation (18) and using cosmological time d t = b ( τ )d τ yields \nH 2 = 8 π G 3 ( ρ B + ρ 1 + ρ 2) . (64) \nMultiplying and dividing this expression by the Hubble parameter at t = t 0 (today) H 2 0 , and recalling that ρ c = 8 π G / (3 H 2 0 ) it is straightforward to obtain \nΩ DMT( z ) = 1 √ 3 ρ eq ρ c 1 + z 1 + z eq ! 6 Θ 1 / 2 ( z ) E 2 ( z ) , (65) \nΩ DET( z ) = 1 3 √ 3 ρ eq ρ c 1 + z 1 + z eq ! 6 Θ 3 / 2 ( z ) E 2 ( z ) , (66) \nwhere we defined \nE 2 ( z ) ≡ Ω B(1 + z ) 3 + ρ eq √ 3 ρ c 1 + z 1 + z eq ! 6 Θ 1 / 2 ( z ) " 1 + Θ ( z ) 3 # . (67) \nThis allows us to obtain the time evolution for each density parameter. This will obviously depend on the specific value of z eq, but the general behaviour will be similar. For the sake of simplicity, we included the case z eq = 1 . 1 in the Fig.4 to show qualitative results. This results in a higher DE abundance and a \nFigure 4: Evolution of the density parameters: Λ CDM (continuous lines) vs analytical TDi ff case (dashed lines), for z eq = 1 . 10. In shift-symmetric TDi ff models dark energy would be more dominant as a consequence of it being phantom at early times, gaining energy from dark matter. \n<!-- image --> \nlower DM one at t = t 0 than those from Λ CDM, which can be interpreted as a consequence of the phantom era during the DM domination regime. As we previously discussed, DM transfers part of its energy to DE, which translates into its phantom behavior and thus contributes to obtaining higher values of Ω DET. It is worth mentioning, however, that we shall not directly compare these parameters to those from Λ CDM, as Ω DMT and Ω DET may not be regarded as true DM and DE density parameters, since, as opposed to Λ CDM, this model presents an interacting dark sector and thus there may be contributions from both components to each parameter.', '4. A TDi ff model for dark sector interactions': 'We will now consider a more general case with w 1 = 0, which could play the role of DM; and arbitrary w 2 < -1 / 3, which could play the role of DE. We will then contrast the predictions of this simple model to observations to get a glimpse on the viability of shift-symmetric TDi ff models for describing the dark sector. Recalling the geometrical constrain (37) arisen from the conservation of the EMT, using α 1 = 0 and dividing by C 2 yields \nλ (1 -α 2) g + g 1 -α 2 \| 2 α 2 -1 \| = a 6 ( λ (1 -α 2) + \| 2 α 2 -1 \| ) . (68) \nwhere we have used \nρ 1( t 0) ρ 2( t 0) = C 1 C 2 (1 -α 2) -1 = Ω DMT Ω DET ≡ λ. (69) \nand normalized 1 a ( t 0) = 1, and also g ( t 0) = 1, which leads to b ( t 0) = 1. Therefore, we only have two free parameters, those \nbeing the exponent of the power-law coupling function of the DE component, α 2, and λ . However, we will use another physical parameter instead of λ in order to obtain a more direct analysis and an easier comparison to observations. In fact, recalling Friedman equation (64) in cosmological time, using (38) and noting that ( ρ 1 + ρ 2) GLYPH<12> GLYPH<12> GLYPH<12> t 0 = (1 -Ω B) ρ c as a consequence of the cosmic sum rule yields the following expression for H 2 ( z ): \nH 2 ( z ) = H 2 0 Ω B(1 + z ) 3 + (1 -Ω B) 1 + 1 λ ! -1 b ( z )(1 + z ) 3 + (1 -Ω B) 1 λ 1 + 1 λ ! -1 b ( z ) 1 -2 α 2 (1 + z ) 6 α 2 + 3 , (70) \nwhere we neglected radiation, as the purpose of this model is to study the DM and DE domination epochs. Otherwise, we should have included the corresponding Ω R(1 + z ) 4 contribution from radiation, assuming it is a Di ff component. Notice that at early times, when the ϕ 1 fluid dominates over ϕ 2, we can neglect the last term in (70). In addition, b ( a ) ∝ a 3 w 1 takes a constant value at early times b ( z ) ≃ bearly since w 1 = 0. This allows us to define the following e ff ective density parameter for total matter at high redshift \nΩ e ff M ≡ Ω B + (1 -Ω B) 1 + 1 λ ! -1 b early , (71) \nWe will use this parameter Ω e ff M instead of λ , since both are trivially related through (71). Acknowledge that b early can be directly computed from the conservation equation (68) taking into consideration that DM dominates at this time and radiation does not contribute to the geometrical constrain, since we are treating it as a Di ff component and thus its EMT is automatically conserved. Thus, using (69) we obtain \nb early = s λ (1 -α 2) + \| 2 α 2 -1 \| λ (1 -α 2) . (72) \nIf we express H -1 0 as 2997 . 9 h -1 Mpc, with h being the reduced Hubble constant, this will allow us to fit our parameters ( w 2 , Ω e ff M ) to observations and obtain physical predictions for this model. Notice that we are using w 2 instead of α 2 as the model parameter since the are trivially related by (42). We will consider the baryon density parameter obtained from the abundance of light elements Ω B h 2 = 0 . 02240 ± 0 . 00069 [30], as it is independent of the particular choice for the cosmological model, and we will use the central value of the Planck measurement for H 0 = 67 . 66 km s -1 Mpc -1 [31]. In particular, we developed a code in Python that solves the conservation law (68) for any given pair of these two parameters. Hence, we can obtain b ( z ) and H ( z ) through (70). We will then regard the Union2-database observational data coming from type Ia Supernovae [32, 33] consisting of 557 data for 0 . 015 < z < 1 . 030 \nand compare them to the theoretical distance moduli µ ( z ) predictions of our model. We will study the agreement between theory and observations using the χ 2 s statistical estimator [34]: \nχ 2 s = X i ( µ obs ( zi ) -µ th ( zi )) 2 E 2 i , (73) \nwhere the theoretical distance modulus is given by \nµ th ( z ) = 5 log 10 d L( z ) 1 Mpc ! + M = ˆ µ ( z ) + M , (74) \nwith d L( z ) the luminosity distance computed from \nd L( z ) = (1 + z ) Z z 0 d z H ( z ) , (75) \nfor flat spatial sections and M being the absolute magnitude, which we marginalized the following way: \nM = X i 1 σ µ obs ( zi ) -ˆ µ ( zi ) E 2 i . (76) \nwhere Ei denotes the error in the µ i measurement at redshift zi and σ = P i E -2 i . Numerical integration will allow us to perform the analysis in the subsequents sections.', '4.1. One-parameter fit': 'We will first start by studying which w 2 value for the dark energy equation of state parameter fits best to the observational data, when fixing Ω e ff M to the central value measured by Planck CMB observations ( Ω M = 0 . 315 ± 0 . 007) [31]. This will allow us to get a glimpse of the viability of this model before tackling the two parameter case, in which we will regard Ω e ff M as another free parameter (notice from (71) that fixing Ω e ff M leaves us with w 2 as our only free parameter). We have considered a set of values of w 2 lying in the interval ( -0 . 99 , -0 . 33) and computed the distance moduli using (74) for the respective redshift values present in Union2 data. This allows us to study the agreement between our models and observations by using the χ 2 s estimator defined above. For the sake of comparison, we will also include the results obtained from the respective one-parameter w CDM fit, for which the distance moduli are computed from (74), the Hubble parameter being the only di ff erence: \nH ( z ) w CDM = H 0 p Ω M(1 + z ) 3 + ΩΛ (1 + z ) 3(1 + w ) , (77) \nfor the DM and DE domination eras, where w denotes the constant DE equation of state. This allows us to compare our oneparameter TDi ff fit to the analogous w CDM fit when considering w as the only free parameter by fixing Ω M to the CMB measurement (we used values of w in (-2,0)). This yields the result in Tab.1. \nTable 1: One-parameter fit: TDi ff vs w CDM results. Both models fit very well to observational data. In the TDi ff case, dark energy would be phantom in the past and then it would slowly transition to decay as quintessence with equation of state parameter w 2 in the future, there being phantom crossing. The 1σ regions for each parameter are also included. \n. \nWe can thus infer that both models provide excellent concordance with observations, with the minimal χ 2 di ff ering just a 0.10% between them. As we know, in Λ CDM, dark energy is a cosmological constant with a non-dynamical equation of state. However, in our ideal TDi ff model, it would be a TDi ff component with p = -0 . 736 ρ but a dynamical decay behavior given by an e ff ective parameter w e ff ( z ), through (63). Using (63) for the best TDi ff fit allows us to obtain the DE decay behavior for all redshifts. Thus, the DE component would have been phantom in the past under DM domination and, afterwards, it would slowly transition until reaching the asymptotic decay reflected by w 2 (quintessence dark energy). Numerically we obtained that w e ff ≃ -0 . 8 today.', '4.2. Two-parameter fit': 'We will lastly analyze our TDi ff model without constraining Ω e ff M , which will allow us to conclude if the actual best TDi ff fit is compatible with type Ia supernovae observations and the CMB measurement for Ω M. On the grounds of this, we will consider a set of values for w 2 in the interval ( -0 . 993 , -0 . 60) and values of Ω e ff M in the interval (0 . 18 , 0 . 56). For each possible pair of values, we calculated the distance moduli using (74) and then computed the respective χ 2 s estimator for each case using (73). In the following analysis we will also include the direct w CDM analogue of this two-parameter fit, in which both Ω M and w are fitted (using values for w in ( -2 , 0) and Ω M in (0 . 01 , 0 . 60)), in order to compare both models. Numerical analysis thus yields the results in Tab.2. \nTable 2: Two-parameter fit: TDi ff vs w CDM. Both models are in good agreement with observations. The 1σ intervals for each parameter have also been included. \nThese results indicate that both models fit extremely well the observational data, with the TDi ff model even achieving this in a subtly better way. Therefore, we will focus on the TDi ff case from now on and present the contour plot for both parameters up to the 4σ region in Fig.5. We can infer that, not only does our TDi ff model provide excellent agreement with observational data, but it is also compatible with the Ω M CMB measurement in the 1σ region, indicating that TDi ff models provide good compatibility when it comes to type Ia supernovae observations. \nThe respective 1σ intervals for each of the parameters were obtained by marginalizing the joint likelihood \nL ( w 2 , Ω e ff M ) = N e -χ 2 s / 2 , (78) \nwhich can be done for each variable by performing the integration with respect to the other. This yields: \nL m ( w 2) = N 1 Z e -χ 2 s / 2 d Ω e ff M , L m ( Ω e ff M ) = N 2 Z e -χ 2 s / 2 d w 2; (79) \nFigure 5: Two-parameter fit: contour plot for χ 2 s up to the 4σ region using Union2 data. The 1σ region is compatible with the CMB measurement for Ω M. The red dot indicates the best fit. Notice that the 68% contour region di ff ers a bit from the marginalized 1σ intervals, which is a consequence of the non-gaussianity of the distributions. \n<!-- image --> \nwhere N 1 and N 2 are normalization constants (the way of proceeding for the w CDMcase is fully analogous). The maximization of these marginalized likelihood distributions L m allowed us to obtain the 1σ regions for both of the parameters, which has been done making use of the GetDist package for Python. 1 \nNotice how, although the model is compatible with a cosmological constant behavior in the 2σ region (see Fig.5), the best fit area lies in the range of w 2 in the interval ( -0 . 873 , -0 . 711). This indicates that, in light of observational data, the TDi ff model would favor a non-cosmological constant behavior in the asymptotic future for the dark energy component, with it being phantom in the matter era and there being phantom crossing, as we studied from the single-field dominance regimes. The 1σ region for the e ff ective matter parameter Ω e ff M is compatible with the Planck measurement from CMB data. \nFig.6 summarizes the results of the best fitting model and its comparison to w CDM, displaying excellent agreement with observations, and also to w CDM, although there start being minor di ff erences between both models at higher redshift values. \nLastly, we include the evolution of the e ff ective equation of state parameter for the DE and DM components for the best fitting TDi ff model in Fig.7. We see that today DE evolves with an e ff ective equation of state w e ff , 2( t 0) ≃ -0 . 9. As a result, TDi ff models favor the presence of a dynamical DE, starting from phantom at early times and slowly transitioning to usual quintessence DE, with an asymptotic quintessence decay dictated by w 2), roughly being a cosmological constant at current times. Similarly, DM will exhibit a faster decay than that expected from w 1 = 0 at recent times as a consequence of the symmetry breaking, without the usual a -3 decay being altered during the matter era. \nThe results obtained throughout this section indicate that this TDi ff model should be further explored in the future. Particularly, the Hubble tension problem should be taken into con- \nFigure 6: Best fit: comparison to w CDMandobservations. Both models exhibit great accordance with observational data from type Ia supernovae and do not di ff er much from each other. Minor di ff erences start appearing between the models at higher redshift values. \n<!-- image --> \nFigure 7: w e ff ( z ) for the two-parameter best fit. DE behaves as a phantom component under DM domination and its dynamical decay transitions to depict its quintessence w 2 behavior in the future. There is phantom crossing taking place near the current time, when DE approximately behaves as a cosmological constant. \n<!-- image --> \nsideration, as other models involving phantom DE have been proven to be favored by observations [7]. \nIt is worth remarking that this time-evolving DE behavior involving phantom-quintessence transitions was obtained without enforcing any type of interaction potential in the Lagrangian, and without the addition of non-canonical or ghost terms in the matter action.', '5. Conclusions and future work': 'In this work, we have considered shift-symmetric theories with two kinetically-driven scalar fields breaking the Di ff symmetry down to TDi ff and studied their cosmological consequences. We have analyzed the geometrical condition imposed by the conservation of the total EMT, which still holds as a consequence of the Bianchi identities. When working in a flat FLRWbackground, this conservation allows us to obtain a geometrical constraint that leads to a particular shape for the lapse \nfunction, which cannot be freely chosen now due to the symmetry breaking. \nThis geometrical constraint enforces an exchange of energy between both fields, as their individual EMTs are not conserved. In light of this fact, we have proposed a dark sector model involving two TDi ff scalar fields coupled to gravity through power-law functions of the metric determinant, with one field describing a DM fluid and the other DE. We have regarded the di ff erent field domination regimes and showed that, although the equation of state parameters of both fluids are constant, both components will exhibit a di ff erent dynamical decay as that corresponding to Di ff models with the same constant parameters. Particularly, when imposing that DM decays as a -3 at early times, we show that the DE component will present phantom behavior during the matter era for it to slowly transition into quintessential behavior in the future, even if its equation of state parameter takes a constant value larger than minus one. It should be emphasized that this interaction of the dark sector is obtained without including interacting potentials. Moreover, ghost or non-canonical kinetic terms have not been considered to obtain phantom behaviour. In this framework one naturally obtains an interacting dark sector with a dark energy component that crosses the phantom regime. In addition, the shift symmetry of the fields allows us to describe a dynamical interacting dark sector avoiding fine-tuning problems depending on the specific choice for the potential. \nWe have also studied the evolution of the energy exchange between the fluids and shown how in these models it is always DE which gains energy from DM. On the other hand, we have also considered a particular analytical model to understand the physics involved in this TDi ff dark sector framework. For that model, we have analysed the form of the interaction kernel, and investigated the decay of the fluids, parameterized through w e ff , i ( z ). \nBeyond the simplest dark sector model, we have studied this interacting dark matter-dark energy framework in deeper detail using numerical techniques. We have considered its parameters: w 2 (the equation of state parameter for the DE field, linked to the exponent of the coupling function) and Ω e ff M (an e ff ective density parameter at high redshift values, which can be easily compared to the CMB measurement for Ω M). Moreover, we have used the Union2 data for Ia supernovae and fitted our two parameters to these observations to get a first glance regarding the viability of these theories. Our results shows exceptional compatibility with those data and a goodness of the fit similar to that of w CDM. In addition, the best fitting region for our parameters at 1σ is compatible with the CMB measurement of the matter density parameter. It is worth mentioning that these results have been obtained considering two parameters ( w 2 and Ω e ff M ), but further and more complete results should be obtained in future work when extending to the full observational analysis using more data sets and more parameters (namely the Hubble constant and the baryonic abundance, as well as the supernovae absolute magnitude). That is, after having introduced this shiftsymmetric multi-field TDi ff model for the first time in this work and studied it from a more theoretical point of view, these first positive results definitely indicate that the model deserves fur- \nalyses. \nFinally, it is worthy to emphasize that the present work has started a new line of research based on studying multi-field TDi ff theories. Future projects include to investigate the stability under cosmological perturbations of these theories, in order to study structure formation from the TDi ff perturbation formalism perspective; the covariantized approach, studied in detail in reference [22], could be of special relevance for this purpose. Moreover, one could also analyse the models resulting from considering more general coupling functions or non-homogeneous fields, as well as going beyond the shiftsymmetric case and / or breaking the Di ff symmetry also in the Einstein-Hilbert action. From an observational point of view, future work will also be done regarding a deeper likelihood analysis using additional data sets, such as (Pantheon + SH0Es [32, 33], CMB, BAO and H ( z )) and an extended parameter space, together with the possible impact on the Hubble tension problem.', 'Acknowledgements': "The authors would like to thank Dar'ıo Jaramillo Garrido and Alfredo Delgado Miravet for useful comments and discussions regarding TDi ff theories. DTB also acknowledges financial help from the Ayudas de M'aster IPARCOSUCM / 2023. This work has been supported by the MICIN (Spain) Project No. PID2022-138263NB-I00 funded by MICIU / AEI / 10.13039 / 501100011033 and by ERDF / EU.", 'References': "- [1] A. G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009-1038. arXiv:astro-ph/9805201 , doi:10.1086/300499 .\n- [2] S. Tsujikawa, Quintessence: A Review, Class. Quant. 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2024arXiv240908833G	We investigate in this work the evolution of the collective fast neutrino flavor conversion FFC in a three dimensional 3D cubic box with periodic boundary condition for three different neutrino angular distributions that are axially asymmetric. We find that the system evolves toward a quasistationary state where the angular distribution of the spatially averaged neutrino electronminusmuon lepton number ELN does not contain any crossings. In the quasistationary state near flavor equilibration is achieved in one angular domain enclosed by the initial ELN angular crossing contour similar to the conclusion derived based on simplified one dimensional 1D system with axially symmetric neutrino angular distributions. We have also performed additional simulations in coordinates where the initial first ELN angular moment has only one nonvanishing spatial component by using the original axially asymmetric ELN angular distributions as well as the corresponding axisymmetric ELN distributions and find interesting similarity between these two sets. Finally we propose three different analytical prescriptions generalized from earlier 1D models to 3D models and evaluate their performances in predicting the postFFC moments. Our findings suggest that further development of effective classical transport model in multidimensions to capture the effect of FFC is promising.	2024-09-01T00:00:00Z	['2024arXiv240908833G', '10.48550/arXiv.2409.08833', 'arXiv:2409.08833']	['Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Phenomenology']	Evolution and the quasistationary state of collective fast neutrino flavor conversion in three dimensions without axisymmetry	2,024	197	0.37	['EPRINT_HTML', 'EPRINT_PDF']	2	https://arxiv.org/pdf/2409.08833.pdf	{'Evolution and the quasistationary state of collective fast neutrino flavor conversion in three-dimensions without axisymmetry': 'Manu George , 1, ∗ Zewei Xiong , 2, † Meng-Ru Wu , 1, 3, 4, ‡ and Chun-Yu Lin 5, § \n1 Institute of Physics, Academia Sinica, Taipei 115201, Taiwan 2 GSI Helmholtzzentrum fur Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany 3 Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 106319, Taiwan 4 Physics Division, National Center for Theoretical Sciences, Taipei 106319, Taiwan 5 National Center for High-performance Computing, Hsinchu 30076, Taiwan \n(Dated: December 11, 2024) \nWe investigate in this work the evolution of the collective fast neutrino flavor conversion (FFC) in a three-dimensional (3D) cubic box with periodic boundary condition for three different neutrino angular distributions that are axially asymmetric. We find that the system evolves toward a quasistationary state where the angular distribution of the spatially averaged neutrino electronminus-muon lepton number (ELN) does not contain any crossings. In the quasistationary state, near flavor equilibration is achieved in one angular domain enclosed by the initial ELN angular crossing contour, similar to the conclusion derived based on simplified one-dimensional (1D) system with axially symmetric neutrino angular distributions. We have also performed additional simulations in coordinates where the initial first ELN angular moment has only one nonvanishing spatial component by using the original axially asymmetric ELN angular distributions as well as the corresponding axisymmetric ELN distributions and find interesting similarity between these two sets. Finally, we propose three different analytical prescriptions generalized from earlier 1D models to 3D models and evaluate their performances in predicting the post-FFC moments. Our findings suggest that further development of effective classical transport models in multidimensions to capture the effect of FFC is promising.', 'I. INTRODUCTION': 'Neutrinos are known to have significant impacts on the dynamics and composition of astrophysical systems such as the core-collapse supernovae and the neutron star mergers. In the central regions of these systems where the neutrino densities are large, the flavor evolution of neutrinos is dominated by the nonlinear self-coupling due to the coherent neutrino-neutrino forward scattering [1, 2]. The nonlinear nature leads to various collective phenomena of neutrino flavor oscillations (see e.g. [3-5] for recent reviews) and can potentially affect our understanding of these astrophysical events [6-16]. Among these, the likely occurrence of fast flavor conversion (FFC) [17, 18] due to the presence of the angular crossing in the neutrino electron-minus-muon lepton number (ELN) in the supernova core or in the merger remnant has been identified [8-10, 19-32]. This has triggered a tremendous number of studies over the last decade on this topic; see, e.g., [33, 34] for earlier reviews. \nSince FFC typically develops within physical scales of subnanoseconds and subcentimeters, much smaller than the hydrodynamical or interaction time and length scales in supernovae or neutron star mergers, one strategy to study FFC is to numerically solve the neutrino quantum kinetic equation ( ν QKE) [2, 35, 36] in a small lo- \nal volume, within which the system is assumed to be nearly homogeneous and collisions of neutrinos may be neglected. Taking the periodic boundary condition, recent works that assume translation symmetry in two spatial dimensions and axisymmetry in the neutrino angular distributions in local one-dimensional (1D) boxes suggest that FFC drives the system to a quasistationary state where near flavor equilibration is achieved on one side of the ELN crossing when coarse-grained over the volume of the box [13, 37-44]. On the other side of the ELN crossing, the corresponding coarse-grained properties can be described by simple formulas subject to the conservation of the total ELN [45, 46]. Based on these results, Reference [47] recently showed that it is possible to effectively include FFC in the classical transport model of neutrinos by applying the quasistationary state prescription obtained from local simulations to locations where ELN crossings are found in global transport simulations under spherically symmetric and static supernova background profiles. Different methods addressing the global ν QKE simulations including FFC have also been attempted [12, 16, 43, 48-56] or proposed [57-59]. Efforts on identifying fast flavor instabilities [26, 31, 60-63] and predicting the post-FFC angular moments [64, 65] based on limited available information of the neutrino angular moments as well as simulation methods that directly evolve the neutrino angular moments for FFC [66, 67] have been investigated. \nAs simulations of FFC in multiple spatial dimensions are computationally more demanding, they have been carried out only in a handful of works [53, 56, 68]. Ref. [68] performed local simulations in two- and three- \ndimensional (2- and 3D) boxes with periodic boundary conditions for several cases. For those with zero total ELN initially, coarse-grained flavor equilibration is reached, independent of the dimensionality of the system. For the case with an initial nonzero total ELN that possesses axisymmetry in the ELN angular distribution, a similar coarse-grained quasistationary state is also obtained in simulations with different dimensions. Although these results seem to hint that the outcome of FFC in a multidimensional periodic box may be similar to that obtained in the corresponding 1D case with axisymmetry, it remains important to investigate whether such a conclusion holds for more general initial conditions 1 . It is worth noting that global 2D FFC simulations have been carried out to explore the impact of the initial condition [56] and the evolution of FFC in neutron star merger remnants [53]. \nIn this work, we perform 3D simulations of FFC in a local box with the periodic boundary condition for three cases where the system has nonzero total ELN and does not contain any axisymmetry. We find that for all these cases, the ELN angular crossings are erased in the quasistationary state on a coarse-grained level. Near flavor equilibration is also reached in one angular domain enclosed by the initial ELN crossing contour, in agreement with what were found in earlier 1D and multidimensional studies. We also perform additional simulations for the same physical systems in rotated coordinates where only one spatial component of the ELN fluxes is nonvanishing, and the corresponding auxiliary simulations for cases that possess axisymmetry. We will show that in the rotated coordinates, the numerical results based on the auxiliary axisymmetric cases can be used to accurately describe the evolution of neutrino angular moments in the axially asymmetric cases. These simulation outcomes allow one to generalize previously proposed analytical prescriptions [45, 46] to predict the post-FFC angular moment values. \nThis paper is organized as follows. In Sec. II, we describe the setup of our model, including the equation of motion, the definition of the coordinate systems, and the numerical scheme used for the simulations. In Sec. III, we present our simulation results in both coordinate systems and discuss the implications. In Sec. IV, we show how to generalize various analytical prescriptions developed based on 1D box models to 3D cases to approximately predict the post-FFC angular moments. Conclusions and outlook are given in Sec. V. Throughout this paper, natural units with ¯ h = c = 1 are adopted.', 'A. Neutrino flavor transport equations': "We consider a simplified two-flavor neutrino system in a localized 3D box in which oscillations can convert the initial ν e and ¯ ν e to the heavy lepton flavors ν x and ¯ ν x . As we focus on studying FFC in our simulation domain, we assume that the neutrino distribution functions inside the box are homogeneous (before applying perturbation seeds; see below), and neglect the collisions for neutrinos. The neutrino vacuum mixing and the neutrinomatter forward scattering potentials can also be omitted for simplicity. Under these assumptions, the space-time evolution of the normalized neutrino and antineutrino densities, ϱ v ( x ) and ¯ ϱ v ( x ), is governed by the following equations \nv η ∂ η ϱ v ( x ) = -i [ H v ( x ) , ϱ v ( x )] , (1a) \nv η ∂ η ¯ ϱ v ( x ) = -i [ H ∗ v ( x ) , ¯ ϱ v ( x )] , (1b) \nwhere x η = ( t, x ), v η = (1 , v ) with v η v η = 0. In Eq. (1a), \nH v ( x ) = µ ∫ d Γ ' (1 -v · v ' )[ g ν ( v ' ) ϱ v ' ( x ) -g ¯ ν ( v ' )¯ ϱ ∗ v ' ( x )] , (2) \nwhere µ = √ 2 G F n ν with n ν the neutrino number density and d Γ = ( dv z dϕ ) / (2 π ) with v z the z component of v and ϕ the corresponding azimuthal angle on the x -y plane. The neutrino angular distribution function g ν ( v ) is normalized by ∫ d Γ g ν ( v ) = 1, while the antineutrino one g ¯ ν ( v ) satisfies ∫ d Γ g ¯ ν ( v ) = n ¯ ν /n ν = α , which represents the number density ratio between antineutrinos and neutrinos. The ELN angular distribution function G v ( x ) is defined by \nG v ( x ) = g ν ( v )( ϱ ee -ϱ xx ) -g ¯ ν ( v )(¯ ϱ ee -¯ ϱ xx ) , (3) \nwhere ϱ ee and ϱ xx are the diagonal entries of ϱ v ( x ) in the flavor basis, and ¯ ϱ ee and ¯ ϱ xx are the corresponding ones of ¯ ϱ v ( x ), whose dependence on v and x are not displayed explicitly.", 'B. Initial ELN distributions': "For given initial angular distributions g ν ( v ) and g ¯ ν ( v ) for ν e and ¯ ν e , the corresponding flux vectors normalized by the neutrino number density can be computed by \nF 0 ν e (¯ ν e ) = ∫ d Γ v g ν (¯ ν ) ( v ) . (4) \nWithout loss of generality, we take a nonzero angle θ r = cos -1 [ F 0 ν e · F 0 ¯ ν e / ( \| F 0 ν e \|\| F 0 ¯ ν e \| )] between F 0 ν e and F 0 ¯ ν e . We assume that g ν (¯ ν ) ( v ) are axisymmetric with respect to the direction of their respective flux vectors and are given by \ng ν (¯ ν ) ( v ) ∝ exp[ -( v F ν (¯ ν ) -1) 2 / (2 σ 2 ν (¯ ν ) )] , (5) \nFIG. 1. Initial ELN angular distribution function G 0 v z ,ϕ in the original (left panel) and G 0 v ' z ,ϕ ' the rotated (middle panel) coordinate systems for θ r = 30 · . The relation of the two coordinate systems is shown in the right panel. \n<!-- image --> \nwhere v F ν (¯ ν ) = v · F 0 ν (¯ ν ) / \| F 0 ν (¯ ν ) \| are the velocity projections in the directions of the flux vectors and σ ν (¯ ν ) are width parameters that determine the degree of anisotropy of g ν (¯ ν ) . \nThroughout the rest of the paper, we adopt α = 0 . 9, σ ν = 0 . 6, σ ¯ ν = 0 . 5, and take θ r = 30 · , 45 · , and 60 · to explore three different cases without axisymmetry along any directions. As will be further discussed below, for each case, we perform simulations in two different coordinate systems for the same G v that is axially asymmetric. The first coordinate system is chosen such that F 0 ν e ,x = F 0 ν e ,y = F 0 ¯ ν e ,y = 0, i.e., F 0 ν e is along the z -axis while F 0 ¯ ν e lies on the x -z plane, similar to what taken in Ref. [68], and it is denoted as 'original' coordinates. For the second coordinate system (labeled with superscript ' ), it is rotated from the first one such that F 0 ν e ,x ' -F 0 ¯ ν e ,x ' = 0, i.e., the x ' component of the first ELN moment is initially zero and is denoted as 'rotated' coordinates for the rest of the paper. The rotation angle between the two coordinates is given by \nθ rot = tan -1 ( sin θ r \| F 0 νe \| \| F 0 ¯ νe \| -cos θ r ) . (6) \nFigure 1 shows the relation between the two coordinates (right panel) as well as the initial ELN distributions G 0 v z ,ϕ in the original (left panel) and G 0 v ' z ,ϕ ' in the rotated coordinates (middle panel) for θ r = 30 · as an example, with the ELN crossing contours indicated by the black solid curves. Clearly, the initial ELN angular distribution is less axisymmetric in the original coordinates than in the rotated coordinates. We note that the ELN angular distributions constructed here have reflection symmetry with respect to ϕ → -ϕ (or ϕ ' → -ϕ ' in the rotated coordinates), resulting in vanishing y ( y ' ) components of all flux vectors. \nBesides the axially asymmetric ELN angular distributions described above, we also perform auxiliary simulations in the rotated coordinates by taking the axisymmet- \nric angular distributions averaged over ϕ ' from g ν (¯ ν ) ( v ' ) \ng a ν (¯ ν ) ( v ' z ) = ∫ dϕ ' 2 π g ν (¯ ν ) ( v ' ) . (7) \nThe corresponding ELN angular distributions G a v ' z ( x ) can be evaluated in the same way as Eq. (3).", 'C. Simulation setup': 'We use the extended version of cose ν , which adopts a grid-based method to solve Eq. (1), to perform numerical simulations in a 3D cubic box with periodic boundary conditions in all three spatial dimensions. The box has a volume of L 3 with L = 100 µ -1 and is discretized by N 3 rectangular cell-centered grids with N = 100. For the 2D phase (angular) space, we discretize -1 ≤ v z ≤ 1 and 0 ≤ ϕ ≤ 2 π into N v z × N ϕ cell-centered bins with N v z = 32 and N ϕ = 8 (same for both the original and rotated coordinates). The spatial derivatives are evaluated using the finite-volume plus seventh-order accurate weighted essentially nonoscillatory scheme while the time integration is computed using the fourth-order RungeKutta method (see [70] for details). We have taken a fixed time step size ∆ t = C CFL × ( L/N ) = 0 . 4 µ -1 with C CFL = 0 . 4 the Courant-Friedrichs-Lewy number. The phase-space integration in Eq. (2) is taken using the simple Riemann sum. \nTo trigger the fast flavor instabilities, we assign spherical Gaussian perturbations centered at the origin of the coordinates to ϱ v and ¯ ϱ v at t = 0 with \nϱ 0 ee = ¯ ϱ 0 ee = 1 2 [ 1 + √ 1 -ϵ 2 ( x ) ] , (8a) \nϱ 0 xx = ¯ ϱ 0 xx = 1 2 [ 1 -√ 1 -ϵ 2 ( x ) ] , (8b) \nϱ 0 ex = ¯ ϱ 0 ex = ϵ ( x ) / 2 , (8c) \nwhere ϵ ( x ) = 10 -3 exp[ -( x 2 + y 2 + z 2 ) / 2 σ 2 r ], and σ r = √ 5 µ -1 . All simulations are performed up to t = 344 µ -1 \nFIG. 2. Spatial distribution of of ⟨ ϱ ee ( x ) ⟩ Γ taken at different simulation time snapshots of t = 17 . 2, 86 . 0, 172 . 0, and 344 . 0 µ -1 shown in panels (a-d) for θ r = 30 · , respectively. Flavor conversions are triggered by the initial perturbations at the center of the box [panel (a)]. A coherent wave-like feature develops when flavor conversions reach the nonlinear regime [panel (b)]. When flavor waves interact as they cross the periodic boundaries, smaller scale structures appear [panel (c)] and the system eventually settles into the final quasistationary state [panel (d)]. \n<!-- image --> \nwhen the systems have reached the coarse-grained quasistationary states.', 'A. Results in the original coordinates': 'We first discuss the evolution of the system in the original coordinates. Fig. 2 shows the volume rendering of the phase-space averaged ρ ee , defined as ⟨ ρ ee ( x ) ⟩ Γ = [ ∫ d Γ g ν ( v ) ρ ee ( v , x )] at different time snapshots of t = 17 . 2, 86 . 0, 172 . 0, and 344 . 0 µ -1 for the case with θ r = 30 · . Initially, the flavor instability drives the growth and spatial drift of the 3D Gaussian perturbation in the linearized regime [panel (a)]. When the system enters the nonlinear regime, flavor waves form and propagate, demonstrated by the coherent pattern shown in panel (b). At later times, when flavor waves interact, smaller-scale structures appear, as shown in panel (c). Eventually, when the system settles to the quasistationary state, fully developed flavor depolarization results in spatial fluctuation dominated by the length scale of ∼ 5 µ -1 in the entire simulation box [panel (d)]. The overall behavior is qualitatively very similar to what were reported in the earlier studies (e.g., [39, 68]). \nWe show in Figure 3 the time evolution of the ν e and ¯ ν e survival probabilities averaged over both the 3D box volume V and the phase space volume Γ \n⟨ P ν e ( t ) ⟩ Γ ,V = [∫ d Γ d 3 xg ν ( v ) ϱ ee ( v , x , t ) ]/ L 3 , (9a) ⟨ P ¯ ν e ( t ) ⟩ Γ ,V = [∫ d Γ d 3 xg ¯ ν ( v )¯ ϱ ee ( v , x , t ) ]/ ( αL 3 ) , (9b) \nand in Figure 4 the initial ELN angular distributions G 0 v z ,ϕ (left panels) as well as the final ELN distributions \nFIG. 3. Survival probabilities averaged over the box and the phase-space volume, ⟨ P ν e ⟩ Γ ,V (upper panel) and ⟨ P ¯ ν e ⟩ Γ ,V (lower panel) as functions of time for θ r = 30 · (red lines), 45 · (blue lines), and 60 · (green lines). We also show the evolution of ⟨\| ¯ ϱ ex \| ( t ) ⟩ Γ ,V in logarithmic scale for t < 150 µ -1 in the inset of the bottom panel for different θ r along with the maximally unstable growth rates (dotted lines) obtained from the linear stability analysis. \n<!-- image --> \n⟨ G v ⟩ V = ∫ d 3 xG v ( x ) /L 3 averaged over the entire box (middle panels) for all three cases with θ r = 30 · , 45 · , and 60 · . Fig. 3 shows that when taking a larger value of θ r , the increased amount of axial asymmetry leads to an earlier onset of flavor conversion as well as smaller quasistationary values of ⟨ P ν e (¯ ν e ) ( t ) ⟩ Γ , x . This is related to the absolute values of the positive and negative parts of \nFIG. 4. The initial ELN angular distribution G 0 v z ,ϕ (left panels), the spatially averaged final ELN angular distribution ⟨ G v ⟩ V at t = 344 µ -1 (middle panels), and the spatially averaged flavor survival probabilities ⟨ P sur ( v z , ϕ ) ⟩ V (right panels) for θ r = 30 · (upper panels), 45 · (middle panels), and 60 · (right panels), respectively. \n<!-- image --> \nthe initial ELN, defined by \nI + = 2 π ∫ d ΓΘ( G 0 v z ,ϕ ) G 0 v z ,ϕ , (10a) \nI -= 2 π ∣ ∣ ∣ ∣ ∫ d ΓΘ( -G 0 v z ,ϕ ) G 0 v z ,ϕ ∣ ∣ ∣ ∣ , (10b) \nwhere Θ is the Heaviside function. The values of I + ( I -) are 1.61(0.98), 2.18(1.55), and 2.76(2.13) for θ r = 30 · , 45 · , and 60 · , respectively. Note that I + -I -= 2 π (1 -α ) is the same for all cases. Clearly, the more asymmetric cases (with larger θ r ) have larger values of I + and I -, which can also be seen from the initial ELN distributions shown in the left panels of Fig. 4. When I + and I -are larger, the associated instability growth rate is also larger (see the inset of Fig. 3 showing the evolution of ⟨\| ¯ ϱ ex \| ( t ) ⟩ Γ ,V in logarithmic scale for t < 150 µ -1 ) 2 , re- \nFIG. 5. Time evolution of the x (red curves) and z (blue curves) components of ⟨ F ν e ⟩ V (upper panel) and ⟨ F ¯ ν e ⟩ V (lower panel) for θ = 30 · . \n<!-- image --> \nier onset of flavor conversions. What is also similar to the 1D box cases is that the ELN crossing is erased when averaging over the entire 3D box, as shown in the middle panels of Fig. 4. Since the total ELN in the box is conserved with the periodic boundary condition, more flavor conversions happen when I -is larger, resulting in a lower value of the quasistationary ⟨ P ν e (¯ ν e ) ⟩ Γ ,V . \nBesides the elimination of the ELN crossing, the middle panels of Fig. 4 also show that for the angular region where G 0 v z ,ϕ < 0 initially, the corresponding values of ⟨ G v ⟩ V = ∫ d 3 xG v ( x ) /L 3 become nearly zero, showing that near flavor equilibration on the coarse-grained level in most part of this angular domain is reached. For completeness, we show in the right panels of Fig. 4 the final coarse-grained angle-dependent flavor survival probabilities ⟨ P sur ( v z , ϕ ) ⟩ V = ∫ d 3 xρ ee ( x , v ) /L 3 taken at the final time snapshot for all three cases. Once again, these panels confirm that ⟨ P sur ( v z , ϕ ) ⟩ V ≃ 0 . 5 is obtained in the angular domain whose G 0 v z ,ϕ < 0. They also show more clearly that slight flavor overconversion with ⟨ P sur ( v z , ϕ ) ⟩ V < ∼ 0 . 5 happens in regions where G 0 v z ,ϕ are more negative around ϕ ∼ 0. All these suggest that similar conclusions derived based on the 1D box simulations hold in a more general setting where both the axisymmetry in the angular distribution and the translation symmetry in spatial dimensions are broken explicitly. We note that for all cases the initial reflection symmetry with respect to ϕ →-ϕ is preserved throughout the evolution for all cases as can be inferred from Fig. 4. \nFigure 5 shows the evolution of the spatially averaged x and z components of the ν e and ¯ ν e first angular moments \n(normalized by n ν ) \n⟨ F ν e ⟩ V = 1 L 3 ∫ d 3 xd Γ v g ν ϱ v ( x ) , (11a) \n⟨ F ¯ ν e ⟩ V = 1 L 3 ∫ d 3 xd Γ v g ¯ ν ¯ ϱ v ( x ) , (11b) \nfor θ r = 30 · . For the initially nonzero flux vector components ( ⟨ F ν e ,z ⟩ V , ⟨ F ¯ ν e ,z ⟩ V , and ⟨ F ¯ ν e ,x ⟩ V ), flavor conversions of ν e (¯ ν e ) to ν x (¯ ν x ) lead to the reduction of their values in all cases. For ⟨ F 0 ν e ,x ⟩ V , which is initially zero, since FFC converts more (less) ν e to ν x around ϕ ∼ 0 ( ϕ ∼ π ) as shown in the right panels of Fig. 4, it becomes negative over time as a result of the broken axisymmetry. Similar evolution of the flux vector components are obtained for other values of θ r .', 'B. Results in the Rotated Coordinates': "We now turn our attention to results obtained in the rotated coordinates. Since the physical evolution of the system is independent of the choice of the coordinate system, we do not repeat the flavor evolution and the properties of the quasistationary states reported in Sec. III A. Instead, we focus on comparing results obtained by taking the axially asymmetric ELN distribution G 0 v ' z ,ϕ ' and the corresponding axisymmetric ELN distribution G a, 0 v ' z = ∫ dϕ ' G 0 v ' z ,ϕ ' / (2 π ), constructed by averaging over ϕ ' from G 0 v ' z ,ϕ ' . Since this coordinate system is chosen such that the only nonzero initial first ELN angular moment ∫ d Γ ' v G 0 v ' z ,ϕ ' is along the z ' direction as introduced in Sec. II B, the initial zeroth and first ELN angular moments evaluated using axisymmetric ELN G a, 0 v ' z are identical to those evaluated using G 0 v ' ,ϕ ' . \nz Figure 6 compares the time evolution of the first ELN moments (averaged over the box volume) for the axially asymmetric ( F ELN ) and the axisymmetric ( F a ELN ) cases for θ r = 30 · , defined as \nF ELN = 1 L 3 ∫ d 3 x ' d Γ v ' G v ' z ,ϕ ' , (12a) \nF a ELN = 1 L 3 ∫ d 3 x ' d Γ v ' G a v ' z . (12b) \nIt shows that the evolution of F ELN ,z ' and F a ELN ,z ' closely follow each other. However, for the axially asymmetric cases, F ELN ,x ' becomes nonzero due to the breaking of the axisymmetry, while F a ELN ,x ' remains zero throughout the evolution. \nInterestingly, if we take the time-dependent, spatially averaged flavor survival probability computed in the axially symmetric cases ⟨ P axi ee ( t, v ' z ) ⟩ and use it together with the angular distribution functions used in the axially asymmetric cases to approximately evaluate the time evolution of the survival probability, flux-ratio vectors, or ELN moments, it results in remarkable agreements. Figure 7 shows the comparison of ⟨ F ν e ,x ' ⟩ V ' , ⟨ F ν e ,z ' ⟩ V ' , \nFIG. 6. Evolution of the first ELN angular moments obtained in the rotated coordinates with axially asymmetric initial distribution function ( F ELN ,x/z , solid curves) and with axisymmetric distribution ( F a ELN ,x/z , dotted curves with dots) for θ r = 30 · . \n<!-- image --> \nFIG. 7. Comparison of the evolution of the ν e (upper panel) and ¯ ν e (lower panel) flux vector components obtained from simulations with axially asymmetric distribution (solid curves) with that estimated using coarse-grained survival probabilities obtained in the corresponding axisymmetric simulations (dotted curves with dots) [see Eq. (13)] in the rotated coordinates with θ r = 30 · . \n<!-- image --> \n⟨ F ¯ ν e ,x ' ⟩ V ' , ⟨ F ¯ ν e ,z ' ⟩ V ' from the axially asymmetric simulations with those evaluated by \nF axi -appr ν e (¯ ν e ) = ∫ d Γ ' v ' g ν (¯ ν ) ( v ' ) ⟨ P axi ee ( t, v ' z ) ⟩ V ' , (13) \nfor the case with θ r = 30 · as an example. It clearly shows that one can use ⟨ P axi ee ( t, v ' z ) ⟩ V ' derived in the axially symmetric simulations in the rotated frame with the axially asymmetric initial angular distributions to closely predict the time evolution of the angle-integrated quantities. We have also verified that similar agreements apply to cases with different values of θ r .", 'IV. APPROXIMATED PRESCRIPTIONS FOR EVALUATING THE ANGULAR MOMENTS': "A main goal of conducting local box FFC simulations is to find simple prescriptions to characterizing the postFFC angular distributions or angular moments for practical implementation into effective classical neutrino transport models [47]. Below, we discuss how to generalize different analytical prescriptions proposed in [45, 46] for 3D cases without axisymmetry in neutrino angular distributions. \nFirst, given the fact that near flavor equilibrium is reached in one of the angular domain, it is straightforward to generalize the prescription of the 'boxlike' scheme [45, 46] to approximate the post-FFC survival probabilities for both ν e and ¯ ν e with \nP b sur ( v z , ϕ ) = { 1 2 for Γ < , 1 -I < / (2 I > ) for Γ > , (14) \nwhere I < = min( I -, I + ), I > = max( I -, I + ) and Γ < (Γ > ) denotes the corresponding angular domain that contributes to I < ( I > ). Equation (14) allows us to approximately evaluate various coarse-grained angle-integrated quantities after FFC has settled to the quasistationary state in a straightforward manner. For instance, the spatially averaged post-FFC zeroth and first angular moments in this boxlike scheme can be estimated by \nN p ν e = ∫ d Γ g ν ( v ) P b sur ( v z , ϕ ) , (15a) \nN p ¯ ν e = ∫ d Γ g ¯ ν ( v ) P b sur ( v z , ϕ ) , (15b) \nF p ν e = ∫ d Γ v g ν ( v ) P b sur ( v z , ϕ ) , (15c) \nF p ¯ ν e = ∫ d Γ v g ¯ ν ( v ) P b sur ( v z , ϕ ) . (15d) \nSecond, the excellent agreement in the evolution of moments shown in Fig. 7 in the rotated coordinates suggests that one may directly utilize the improved analytical prescription developed based on axisymmetric simulations [46] to approximately evaluate the post-FFC angular moments. Below, we show how a axisymmetric (in the rotated coordinates) prescription can be directly employed in the multidimensional condition. First, we find the angular domains corresponding to I < and I > as small and large sides, respectively. The flavor equilibration is assumed on the small side. For the large side, we will determine the distribution of survival probability as follows. We find the initial ELN flux vector F 0 ELN whose direction points to where the large side is located. We then compute the values of the projection of all ELN crossing velocity v c along ˆ F 0 ELN and find the maximal projected value as v m c , where F 0 ELN = F 0 ELN / \| F 0 ELN \| . The survival \nTABLE I. Values of the post-FFC spatially averaged ν e and ¯ ν e zeroth angular moments as well as their x and z components of the first angular moments obtained from numerical simulation (scheme 'sim'), estimated using the boxlike prescription for survival probability [scheme 'box'; see Eq. (14)], the power-1/2-s prescription [scheme 'power-1/2s'; see Eq. (16)], and the power-1/2-a prescription [scheme 'power-1/2-a'; see Eq. (19)]. \nprobability on the large side is then given by \nP p -s sur ( v ) = { 1 2 , for v · ˆ F 0 ELN ≤ v m c , 1 -1 2 h ( \| v · ˆ F 0 ELN -v m c \| a ) , for v · ˆ F 0 ELN > v m c , (16) \nwhere h ( x ) = ( x 2 +1) -1 / 2 , and the parameter a can be determined using the following equation derived from the conservation of the ELN, \nI < = ∫ Γ > d Γ G ( v ) h ( \| v · ˆ F 0 ELN -v m c \| /a ) , (17) \nwhere the integration is performed over the domain Γ > , generalized from [46] 3 . We then use Eq. (16) to replace P b sur ( v ) in Eqs. (15) to compute the corresponding values of the zeroth and the first moments. This scheme is named 'power-1/2-s'. \nThird, we can further generalize the power-1/2 prescription to include the axial asymmetric feature of the results, and use it to evaluate the post-FFC moments, dubbed as the 'power-1/2-a' scheme. Just as the original motivation for the power-1/2 prescription in 1D-box simulations, which is to provide a prescription with the survival probability smoothly transitioning across the angular crossing, we achieve the same goal for the power1/2-a scheme, by defining a shortest 'distance' to the initial ELN crossing contour as \nD ( v ) = 1 -f max ( v ) , (18) \nwhere the function f max ( v ) returns the maximal value of v · v c for all v c with a given v . The corresponding \nsurvival probability to predict the quasistationary state in this scheme is then defined by \nP p -a sur ( v ) = { 1 2 , for Γ < , 1 -1 2 h [ D ( v ) /a ] , for Γ > . (19) \nThe determination of a in the power-1/2-a scheme follows the same procedure outline above for the the power-1/2-s scheme. \nTable I compares the values of the ν e and ¯ ν e zeroth moments as well as the x and z components of their first moments derived from the simulations in the original coordinates to those evaluated using the above three different prescriptions for all three different θ r values. In all cases, the fractional differences between values obtained by numerical simulations and those with prescriptions are smaller than ∼ 10%. Unsurprisingly, the boxlike scheme generally leads to larger errors, due to its simplest form that is discontinuous. For the power-1/2-s scheme and the power-1/2-a scheme, the fractional differences are generally smaller compared to the boxlike scheme, especially for the zeroth moment as well as the z component of the flux vectors. Figure 8 shows the graphical representation of the initial ν e (black arrows) and ¯ ν e flux vectors (blue arrows), their final coarse-grained vectors from simulations (red and magenta arrows), and the values predicted using the box-like (green dots), the power1/2-s prescription (cyan dots), as well as the power-1/2-a prescription (red dots) for all three cases with different θ r to demonstrate the impact of FFC, and the difference between the numerical outcome and the approximated evaluations. Once again, it illustrates that in general the power-1/2-a scheme provides the best prediction for the post-FFC flux vectors, followed by the power-1/2-s scheme, and then the boxlike scheme. \nFinally, we note that in the power-1/2-s scheme that assumes axisymmetry in the rotated coordinates we have conservatively chosen the maximally projected velocity v m c as the critical velocity value in Eq. (16). This is to avoid having a final ELN angular distribution that still contains ELN crossings. If we have used the averaged projected velocity value as the critical velocity in Eq. (16), it will result in similar approximated numbers for the final moments, but the corresponding final ELN angular distribution will have crossings. Such a choice can lead to inconsistency when implemented into the effective classical transport model generalized from [47] in multidimensions. We also caution that this method is not guaranteed to work for very extreme cases where v c · ˆ F 0 ELN span a very wide range of values. As for the further improved power-1/2-a scheme, it does not suffer from this issue and can be universally used for any ELN distributions without axisymmetry. Also noted is that for cases with very shallow ELN angular crossings, the power-1/2-a scheme automatically converges toward the boxlike scheme.", 'V. CONCLUSION AND OUTLOOK': "In this work, we have conducted numerical simulations of collective neutrino fast flavor conversions in three spatial dimensions in a cubic box with periodic boundary conditions using the extended version of cose ν . We first examined three different physical cases with axially asymmetric neutrino ELN angular distributions in a coordinate system where the x and y components of the ν e flux vector are zero. Our results show that when the system has reached the quasistationary state, the spatially averaged ELN angular crossing is erased, and near flavor equilibration is reached in one angular domain defined by the initial ELN crossing contour, similar to what previously observed in the 1D box cases. We have further conducted the same set of axially asymmetric simulations in rotated coordinate systems where the x ' and y ' components of the ELN angular moments are zero, as well as the corresponding axisymmetric counterparts. We found that the evolution of the zeroth ELN moments and the z ' components of the first ELN moments in these two sets of simulations are nearly identical, while the evolution of the x ' components of the first ELN moments differs. Interestingly, we find that by folding the spatially averaged flavor survival probabilities computed based the axisymmetric counterpart distributions with the initially axially asymmetric angular distributions. \nBased on the simulation outcomes, we have provided a generalized boxlike scheme and two different versions (power-1/2-s and power-1/2-a) based on the power-1/2 scheme proposed in [46] to approximate the post-FFC flavor survival probabilities in 3D. The generalized power1/2-s scheme assumes axisymmetry in the rotated coordinates while the power-1/2-a scheme takes into account the axial asymmetric feature of the system. While we have found that all these schemes give rise to less than < ∼ 10% errors in the predicted post-FFC moments, the power-1/2-a scheme predicts the post-FFC moments more accurately than the other two approximated prescriptions. We note that, however, the implementation of the power-1/2-a and the power-1/2-s schemes require more computation time than the simplest boxlike scheme, for their better accuracy. \nThe numerical studies presented in this work indicate that the important feature - the erasure of the spatially averaged ELN crossing and the coarse-grained flavor equilibration in an angular domain - concluded based on 1D periodic box simulations with axisymmetric neutrino distributions remains generally valid in multidimensional cases without axisymmetry. Hence, it is expected that one can similarly implement the analytical prescriptions proposed in this paper into the effective classical transport model proposed in [47] to take into account the effect of FFC in global neutrino transport simulation in multidimensions. This aspect will be further pursued in the future. \nFIG. 8. Graphical representation of the initial ν e (black arrows) and ¯ ν e flux vectors (blue arrows), their final coarse-grained vectors from simulations (red and magenta arrows), and the values predicted using the box-like (green dots), the power-1/2-s prescription (cyan dots), as well as the power-1/2-a prescription (red dots) for θ r = 30 · (left panel), 45 · (middle panel), and 60 · (right panel) in the original coordinates, respectively. \n<!-- image -->", 'ACKNOWLEDGMENTS': "M.-R.W. and M.G. acknowledge support from the National Science and Technology Council, Taiwan under Grant No. 111-2628-M-001-003-MY4, and the Academia Sinica (Project No. AS-CDA-109-M11). 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(14)], P p -s sur [Eq. (16)], and P p -a sur [Eq. (19)], labeled by 'simulation', 'box', 'power-1/2-s', and 'power-1/2a', respectively. The right column shows the corresponding electron neutrino angular distributions, g ν P sur , for all cases. For better illustration, these distributions are interpolated onto finer velocity grids. Black dotted curves denote the initial ELN angular crossing for all panels. \nAs described in the main text, the angular distribution of the survival probability obtained from simulation shows that near flavor equilibration ( P sur ∼ 0 . 5) is achieved in one of the angular domains separated by the ELN crossing contour, with a slight flavor overconver- \non around ϕ ∼ 0 and v z ∼ 0. In the other domain, P sur increases smoothly from ∼ 0 . 5 from the ELN crossing contour to larger values of ∼ 0 . 8 at ϕ ∼ π and θ ∼ 0 . 5. For P sur given by different prescriptions, both the power1/2-s and power-1/2-a schemes qualitatively capture the smooth transition obtained in simulation, although both schemes predict fewer flavor conversions around ϕ ∼ 0 and v z < ∼ 0. As for the box-like scheme, P sur takes two distinct values, which transit abruptly at the crossing contour by construction. \nFor the ν e angular distribution, g ν P sur , since g ν is more forward peaked in v z , the values of g ν P sur are suppressed in v z < ∼ 0 in all cases as shown in the right column of Fig. 9. However, the above abrupt transition in the boxlike scheme remains clearly visible in g ν P sur , which results in larger deviation of the predicted angular moments discussed in the main text. For the two power1/2 prescriptions, the overall shapes of g ν ϱ ee agree better with the simulation results than that from the box prescription, leading to smaller deviations in angular moments discussed in the main text. \nFIG. 9. Comparison for the angular distributions of the survival probability P sur (left panels) and g ν ϱ ee (right panels) in the coarse-grained asymptotic states from the simulations with those using the box-like, power-1/2-s, and power-1/2-a prescriptions for θ r = 30 · . Black dotted curves denote the initial ELN angular crossing. \n<!-- image -->"}
2024MNRAS.533.4384I	Broad absorption line BAL outflows are commonly detected in active galactic nuclei AGNs but their driving mechanism remains poorly constrained. Here we investigate whether radiation pressure on dust can adequately explain the BAL phenomenon observed in quasars. In the framework of our AGN radiative dusty feedback scenario we show that dustdriven outflows can reach BAL windlike velocities inlineformulatexmath idTM0001 notationLaTeXv sim 104texmathinlineformula km sinlineformulatexmath idTM0002 notationLaTeX1texmathinlineformula on galactic scales inlineformulatexmath idTM0003 notationLaTeXr lesssim 1texmathinlineformula kpc. This is consistent with recent observations indicating that BAL acceleration typically occurs on scales of inlineformulatexmath idTM0004 notationLaTeXsim 10texmathinlineformula pc and that the majority of BAL outflows are located at galactocentric radii greater than inlineformulatexmath idTM0005 notationLaTeXsim 100texmathinlineformula pc. We derive the outflow radial velocity profile and compute the associated outflow momentum rate and kinetic power which are found to be in agreement with the outflow energetics measured in BAL quasars. Therefore radiation pressure on dust may account for the observed BAL outflow dynamics and energetics. Furthermore we consider BAL cloudsclumps leading to a clumpy BAL flow characterized by a wide range of outflowing velocities and we analyse how the resulting covering factors affect the shape of the absorption line profiles. We conclude that dustdriven BAL outflows may provide a significant contribution to AGN feedback on galactic scales.	2024-10-01T00:00:00Z	['10.48550/arXiv.2409.03842', '10.1093/mnras/stae2074', '2024MNRAS.tmp.2031I', '2024MNRAS.533.4384I', '2024arXiv240903842I', 'arXiv:2409.03842']	['Astrophysics - Astrophysics of Galaxies', 'Astrophysics - High Energy Astrophysical Phenomena']	Are BAL outflows powered by radiation pressure on dust	2,024	197	0.58	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	1	https://arxiv.org/pdf/2409.03842.pdf	{'W. Ishibashi 1 , 2 , 3 ⋆ , A. C. Fabian 4 and P. C. Hewett 4': "- 1 Physik-Institut, Universit a t Z u rich, Winterthurerstrasse 190, 8057 Z u rich, Switzerland\n- 2 Istituto Ricerche Solari (IRSOL), Universit ' a della Svizzera italiana (USI), 6605 Locarno Monti, Switzerland\n- 3 Euler Institute, Universit ' a della Svizzera italiana (USI), 6900 Lugano, Switzerland\n- 2 Institute of Astronomy, Madingley Road, Cambridge CB3 0HA \nAccepted ? Received ?; in original form ?", 'ABSTRACT': 'Broad absorption line (BAL) outflows are commonly detected in active galactic nuclei (AGN), but their driving mechanism remains poorly constrained. Here we investigate whether radiation pressure on dust can adequately explain the BAL phenomenon observed in quasars. In the framework of our AGN radiative dusty feedback scenario, we show that dust-driven outflows can reach BAL wind-like velocities ( v ∼ 10 4 km/s) on galactic scales ( r ≲ 1 kpc). This is consistent with recent observations indicating that BAL acceleration typically occurs on scales of ∼ 10 pc, and that the majority of BAL outflows are located at galactocentric radii greater than ∼ 100 pc. We derive the outflow radial velocity profile and compute the associated outflow momentum rate and kinetic power, which are found to be in agreement with the outflow energetics measured in BAL quasars. Therefore radiation pressure on dust may account for the observed BAL outflow dynamics and energetics. Furthermore, we consider BAL clouds/clumps (leading to a clumpy BAL flow characterised by a wide range of outflowing velocities), and we analyse how the resulting covering factors affect the shape of the absorption line profiles. We conclude that dust-driven BAL outflows may provide a significant contribution to AGN feedback on galactic scales. \nKey words: black hole physics - galaxies: active - quasars: absorption lines', '1 INTRODUCTION': 'A direct evidence of active galactic nucleus (AGN) feedback in action can be observed in the form of ionised gas outflows, detected as blueshifted absorption lines in the rest-frame ultraviolet spectra of quasars. Broad absorption line (BAL) outflows are characterised by very high velocities ( v ∼ 10 3 -10 4 km/s) and wide absorption troughs that are blueshifted with respect to the systemic redshift (e.g. Weymann et al. 1991; Laha et al. 2021, and references therein). The absorption troughs arise from ionised material: the majority from high-ionisation species, such as C IV and Si IV (HiBAL), and a minority also from low ionisation species, such as Mg II and Al III (LoBAL). A particular subclass show additional absorption from iron Fe II (FeLoBAL). \nBAL outflows are typically observed in ∼ (10 -20)% of optically selected quasars, while the intrinsic BAL fraction can be significantly higher (Hewett & Foltz 2003; Gibson et al. 2009; Allen et al. 2011). BAL quasars are usually associated with high luminosities ( L > 10 46 -10 47 erg/s), with an enhanced BAL fraction reported in e.g. the WISSH sam- \n- ⋆ E-mail: [email protected] \nple of hyper-luminous quasars (Bruni et al. 2019). Recent observational works indicate that the BAL fraction increases with redshift, from ∼ 20% at z ∼ 2 -4 to nearly ∼ 50% at z ∼ 6 (Bischetti et al. 2022, 2023). Therefore BAL outflows seem to be common in high-luminosity AGNs, with BAL quasars comprising a significant fraction of the quasar population at high redshifts. \nIt has long been recognised that BAL quasars are redder in the rest-frame UV spectra compared to non-BAL counterparts, with LoBAL being more strongly reddened than HiBAL (Weymann et al. 1991; Reichard et al. 2003; Gibson et al. 2009; Zhang et al. 2014; Krawczyk et al. 2015; Chen et al. 2022). The redder colour observed in BAL quasars is likely associated with the presence of dust grains mixed with the gas, which cause additional dust reddening. This already suggests that the dust may be somehow involved in the BAL phenomenon. \nBAL outflows have been extensively investigated over the past three decades. However, the physical mechanism powering BAL winds is still much debated. What actually drives BAL outflows? A widely adopted model is that of accretion disc winds driven by radiation pressure on UV resonance lines (Murray et al. 1995; Proga et al. 2000). In \nthe line-driven scenario, the wind is launched from the inner accretion disc at small radii ( r ∼ 0 . 01 pc) and is rapidly accelerated to high speeds on sub-pc scales. Another possibility is radiation pressure on dust, which acts on larger scales, beyond the sublimation radius for dust grains ( r ≳ 1 pc). This suggests the existence of a maximal terminal velocity set by the smallest launching radius, i.e. the dust sublimation radius (Scoville & Norman 1995). More recently, He et al. (2022) report that the kinematics observed in BAL outflows may be accounted for by dust-driving, with the launch region located close to the dusty torus. \nA key parameter that can help distinguish between the two models is the BAL outflow distance from the centre, i.e. the galactocentric radius r . Recent observational measurements indicate that BAL outflows are located at radii of tens of parsecs to a few kiloparsecs, typically at r ∼ 100 pc -1 kpc (Arav et al. 2018, 2020; Xu et al. 2019; Miller et al. 2020). Furthermore, a robust constraint on the BAL location is crucial for the determination of the outflow energetics, such as the momentum flux and the kinetic power. By comparing the observed outflow energetics with the central luminosity output, one can assess the potential contribution of BAL outflows to AGN feedback. \nHere we consider radiation pressure on dust as the driving mechanism for BAL outflows. We examine whether the observed BAL properties can be explained in terms of radiation pressure-driven dusty outflows. We also provide a first quantification of the BAL outflow dynamics and energetics in the framework of our AGN radiative dusty feedback scenario (Ishibashi & Fabian 2015, 2018; Ishibashi et al. 2021). The paper is structured as follows. We first recall the basics of AGN radiation-driven dusty outflows (Sect. 2), which we now apply to the case of BAL outflows. We analyse the resulting BAL outflow dynamics, and we derive analytic limits for the wind radial velocity profile in the case of fixed-mass shells (Sect. 3). We compute the corresponding BAL outflow energetics in Sect. 4 and compare with available observational measurements. The case of expanding shells sweeping up matter from the surrounding environment is treated in Sect. 5. In addition, we consider the case of clumpy BAL outflows, which lead to a range of covering factors and absorption line profiles (Sect. 6). The prospect of such dust-driven BAL outflows is further discussed in the broader context of AGN-galaxy co-evolution over cosmic time (Sect. 7).', '2 OUTFLOWS DRIVEN BY RADIATION PRESSURE ON DUST': "The combination of high luminosity and dust reddening observed in BAL quasars is suggestive of radiatively driven dusty winds. We have previously considered how galactic outflows can be powered by AGN radiation pressure on dust (see e.g. Ishibashi & Fabian (2015, 2018) for a more detailed discussion of the AGN radiative dusty feedback scenario). The equation of motion of the outflowing shell is given by \nd dt [ M sh ( r ) v ] = L c (1 + τ IR -e -τ UV ) -GM ( r ) M sh ( r ) r 2 , (1) \nwhere L is the central luminosity and M sh ( r ) is the outflowing shell mass. We assume that the total mass follows an isothermal distribution M ( r ) = 2 σ 2 r G (where σ is the velocity dispersion) and we initially consider a fixed-mass shell \n( M sh ( r ) = M sh ). The infrared (IR) and ultraviolet (UV) optical depths are given by τ IR , UV = ( κ IR , UV M sh ) / (4 πr 2 ), where κ IR =5cm 2 g -1 f dg , MW and κ UV =10 3 cm 2 g -1 f dg , MW are the IR and UV opacities, with the dust-to-gas ratio ( f dg ) normalised to the Milky Way value. The following values are assumed as fiducial parameters of the model: central luminosity L = 5 × 10 46 erg/s, shell mass M sh = 10 6 M ⊙ , velocity dispersion σ =200 km/s, dust-to-gas ratio f dg = 1 × f dg , MW , and initial radius r 0 = 3 pc. \nThe temporal evolution of the outflowing shell is determined by the competition between the outward radiative force and the inward gravitational force (corresponding to the first and second terms on the right hand side of equation 1). By equating these two opposite forces, we obtain a critical luminosity that can be considered as a generalised form of the Eddington luminosity \nL ' E ( r ) = 2 cσ 2 M sh r (1 + τ IR -e -τ UV ) -1 . (2) \nThe corresponding effective Eddington ratio is given by Γ ' = L/L ' E . To launch an outflow, the central luminosity must exceed this critical luminosity ( L > L ' E ); or equivalently, the effective Eddington ratio must be greater than unity Γ ' > 1. \nThree distinct physical regimes can be identified depending on the optical depth of the ambient medium: optically thick to both IR and UV, optically thick to UV but optically thin to IR (single scattering limit), and optically thin to UV. The corresponding IR and UV transparency radii are defined by R IR , UV = √ κ IR , UV M sh 4 π . The effective Eddington ratios in the three optical regimes are given by (cf. Ishibashi et al. 2018): \nΓ IR = κ IR L 8 πcσ 2 r ∝ f dg L (3) \nΓ SS = Lr 2 cσ 2 M sh ∝ L M sh (4) \nΓ UV = κ UV L 8 πcσ 2 r ∝ f dg L (5) \nWe see that the luminosity appears in all three optical depth regimes, and that the effective Eddington ratio is inversely proportional to the shell mass in the single scattering limit. On the other hand, the effective Eddington ratios are independent of the shell mass in the IR-optically-thick and UV-optically-thin regimes, while both Γ IR and Γ UV directly scale with the dust opacity and hence dust-to-gas ratio.", '3.1 Radial velocity profile': 'Integrating the equation of motion (equation 1), we obtain the radial velocity profile of the outflowing shell. In Fig. 1, we show the outflow velocity as a function of radius, for different central luminosities and outflowing shell masses. We see that outflows driven by radiation pressure on dust can reach very high velocities, v ∼ several 1000 km/s and up to v ≳ 10 4 km/s, at radial distances of r ∼ (0 . 01 -1) kpc. Thus typical BAL wind-like velocities can be attained on galactic scales. This is to be compared to BAL outflow observations with available distance measurements. \nFigure 2. Outflow radial velocity profile v ( r ) with variations by a factor of two in luminosity, shell mass, and dust-to-gas ratio: fiducial parameters (black solid), L = 10 47 erg/s (pink dash-dot), M sh = 5 × 10 5 M ⊙ (green dashed), f dg = 2 × f dg , MW (orange dotted). The two black dots mark the location of the IR and UV transparency radii ( R IR and R UV ) for the fiducial case. The yellow shaded area represents the typical outflow velocity range observed in BAL quasars. \n<!-- image --> \nFigure 1. Outflow radial velocity profile v ( r ) with variations in central luminosity and outflowing shell mass: L = 5 × 10 46 erg/s with M sh = 10 5 M ⊙ (green dotted), M sh = 10 6 M ⊙ (red solid), M sh = 10 7 M ⊙ (blue dashed); L = 10 47 erg/s and M sh = 10 6 M ⊙ (orange dash-dot-dot); L = 10 46 erg/s and M sh = 10 6 M ⊙ (violet dash-dot). The two black dots mark the location of the IR and UV transparency radii ( R IR and R UV ) for the fiducial case. The yellow shaded area represents the typical outflow velocity range observed in BAL quasars. \n<!-- image --> \nBased on the use of troughs from ionic excited states, Arav et al. (2018) report that at least ∼ 50% of BAL outflows are located at r > 100 pc from the central source. This is corroborated by results from a blind survey indicating that ∼ 75% of S IV outflows have r > 100 pc (Xu et al. 2019). Indeed, most BAL outflows are observed at large galactocentric distances, with the majority lying between r ≳ 100 pc and r ≲ few kpc (Miller et al. 2020; Arav et al. 2020). \nFrom Fig. 1, we observe that the outflow velocity increases with radial distance, reaching terminal asymptotic speeds at large radii ( r ≳ 1 kpc). Thus most of the wind acceleration phase takes place between the launch radius at ∼ few pc (beyond the dust sublimation radius) and ≲ 100 pc. This is consistent with the empirical evidence indicating that the BAL acceleration occurs on scales of tens of parsecs (He et al. 2022). Based on BAL trough variability measurements, He et al. (2022) find that the outflow distance increases with velocity -from several pc to a hundred pc- suggesting that the BAL outflow is typically accelerated around ∼ 10 pc. Such large radial distances far exceed the typical acceleration region of line-driven accretion disc winds, but are consistent with dust-driven outflows launched from the dust sublimation radius ( R sub ∼ few pc for luminous quasars). For given initial conditions, higher outflowing velocities are obtained for higher luminosities and/or lower mass shells (Fig. 1).', '3.2 Dependence on physical parameters': 'To better understand how the outflow velocity profile depends on the underlying parameters, we plot variations by a factor of two in luminosity, shell mass, and dust-to-gas ratio \n(Fig. 2). In general, we see that an increase in luminosity and dust-to-gas ratio, as well as a decrease in shell mass, lead to higher outflowing velocities. \nBreaking down into the three optical depth regimes (introduced in Sect. 2), we note that the central luminosity has the major effect on the outflow velocity, since L appears in all three regimes (equations 3-5). Such a luminosity dependence is also consistent with the empirical trend observed in BAL quasars, whereby higher UV luminosity sources reach higher wind velocities (Gibson et al. 2009; Bruni et al. 2019). \nThe dust-to-gas ratio (appearing in both Γ IR and Γ UV ), enhances the outflow velocity at small radii and large radii (corresponding to the IR-optically-thick and UV-opticallythin regimes, respectively); while it has no effect in the single scattering limit at intermediate radii. In contrast, the shell mass mainly affects the single scattering regime (as Γ SS scales inversely with M sh ) such that lower mass shells can be accelerated to higher velocities. Hence the shell mass is an important parameter determining the outflow propagation at intermediate radii ( R IR ≲ r ≲ R UV ). Moreover, as the effective Eddington ratio increases with radius in the single scattering limit (Γ SS ∝ r ), the outflowing shell tends to become increasingly super-Eddington. The highest outflow velocities are obtained for a favourable combination of high luminosity, high dust-to-gas ratio, and low mass shell. \nIn addition, variations in the initial launch radius can have a significant impact on the velocity profile, with smaller r 0 leading to higher outflowing speeds. A lower limit on the initial radius of dust-driven outflows is set by the dust sublimation radius: R sub = √ L 4 πσ SB T 4 sub , where σ SB is the StefanBoltzmann constant, and T sub is the sublimation temperature of dust grains. Assuming L ∼ 10 47 erg/s and T sub ∼ 1500 K, the dust sublimation radius is about R sub ∼ 2 pc.', '3.3 Analytic limits': 'In the case of fixed-mass shells, analytic solutions of the outflow radial velocity profile can be derived from the equation of motion (see also Ishibashi et al. (2021)). Here we provide analytic expressions for the outflow velocity as a function of radius v ( r ) in two regions, separated by the UV transparency radius R UV : the UV-optically-thick region at smaller radii (IR+SS regimes) and the UV-optically-thin region at larger radii (UV regime). \nOn small scales ( r < R UV ), the UV optical depth is much greater than unity ( τ UV ≫ 1), and the UV exponential term can be neglected. The resulting outflow velocity in the UV-optically-thick region is given by \nv I ( r ) = √ 2 L cM sh ( r -r 0 ) + κ IR L 2 πc ( 1 r 0 -1 r ) -4 σ 2 ln r r 0 , (6) \nassuming a zero initial velocity ( v 0 = 0). \nConversely, at large radii ( r > R UV ), the UV optical depth is small ( τ UV ≪ 1), and the exponential term may be approximated as e -τ UV ∼ (1 -τ UV ). The corresponding asymptotic velocity in the UV-optically-thin region is given by \nv II ( r ) = √ v 2 I ( R UV ) + κ UV L 2 πc ( 1 R UV -1 r ) -4 σ 2 ln r R UV , (7) \nwhere the expression v I ( r ) is evaluated at r = R UV (equation 6). For fiducial parameters, the outflow velocity at small radii is about v I ( r ∼ 10pc) ∼ 3700 km/s, while the terminal velocity reaches v II ( r ∼ 1kpc) ∼ 12,000 km/s on kpc-scales. \nFigure 3 shows the comparison between the analytic formulae provided in equations 6-7 (coloured curves) and the full numerical solution of the outflow radial velocity profile (black curve). We see that the analytic limits provide an accurate description of the numerical results, in excellent agreement in the respective validity domains. Therefore our analytic solutions may be used for directly estimating the wind velocity (at any given radius) of radiation-driven dusty outflows.', '4 BAL OUTFLOW ENERGETICS': 'Knowing the spatial extent of BAL winds, alongside their outflowing velocities, we can now estimate the corresponding energetics. Assuming a thin spherical shell geometry (e.g. He et al. 2019; Arav et al. 2020), the mass outflow rate, the momentum flux, and the kinetic power are respectively given by \n˙ M = 4 πm p µ Ω N sh rv = µ Ω M sh v r (8) \n˙ p = ˙ Mv = 4 πm p µ Ω N sh rv 2 = µ Ω M sh v 2 r (9) \n˙ E k = 1 2 ˙ Mv 2 = 2 πm p µ Ω N sh rv 3 = µ Ω M sh v 3 2 r (10) \nwhere N sh = M sh / (4 πm p r 2 ) is the shell column density, µ = 1 . 4 is the mean atomic mass per proton, and Ω = 0 . 2 is the global covering factor (He et al. 2019, 2022). \nFigure 3. Outflow radial velocity profile: comparison between the full numerical solution v ( r ) (black solid) and the analytic limits, v I ( r ) starting from r = R 0 (cyan dotted) and v II ( r ) starting from r = R UV (magenta dashed). The two black dots mark the location of the IR and UV transparency radii ( R IR and R UV ). \n<!-- image --> \nThe outflow energetics can be further quantified by two related quantities normalised by the AGN radiative output: the momentum ratio and the energy ratio, defined as \nζ = ˙ p L/c = 4 πm p µ Ω rN sh v 2 L/c = µ Ω M sh v 2 ( L/c ) r (11) \nϵ k = ˙ E k L = 2 πm p µ Ω rN sh v 3 L = µ Ω M sh v 3 2 Lr (12) \nIn Fig 4, we show the radial profiles of the momentum ratio ζ (left-hand panel) and energy ratio ϵ k (righthand panel) of the radiation pressure-driven outflows. We see that the outflow energetics typically span the range ζ ∼ (0 . 1 -10) and ϵ k ∼ (10 -3 -0 . 1) on radial scales of r ∼ (0 . 01 -1) kpc. Observational results indicate that the majority of absorption outflows have momentum ratios in the range ˙ p/ ( L/c ) ∼ (0 . 1 -10) and energy ratios of ˙ E k /L ∼ (0 . 001 -5) %, with most of the BAL winds being located at galactocentric distances of r ∼ (100 -1000) pc (Miller et al. 2020). Similarly, the typical kinetic-tobolometric luminosity ratio is found to be a few percent in a large sample of BAL quasars drawn from the SDSS (He et al. 2019). So the predicted range of momentum ratios and energy ratios in dust-driven outflows can quantitatively account for the energetics observed in BAL quasars. The resulting high energetics ( ϵ k ∼ few %) suggests that BAL outflows are capable of providing an important contribution to AGN feedback on galactic scales.', '5 EXPANDING SHELLS': 'Up to now, we have considered the simple case of fixedmass shells, which allowed us to derive analytic limits for the outflow dynamics (Sect. 3). In more realistic situations, the expanding shell is likely to sweep up mass from the surrounding environment during its outward propagation. The \nFigure 5 shows the radial velocity profile of the outflowing shell for different values of the external gas density n 0 . As expected, the outflow propagation is slowed down for expanding shells sweeping up mass from the surroundings, and the terminal velocity is lower compared to the case of a fixed-mass shell. As the shell sweeps up more mass, higher gas densities naturally lead to stronger outflow deceleration. We note that the shell velocity is unaffected at small radii ( r ≲ 1 kpc), while the growing mass effect becomes dominant at larger radii ( r ≳ 1 kpc). In fact, the outflow dynamics on galactic scales will be determined by the amount of swept-up material, hence the ambient gas density distribution. \n<!-- image --> \nFigure 4. Outflow energetics radial profiles: momentum ratio (left-hand panel) and energy ratio (right-band panel) for fiducial values (with µ = 1 . 4, Ω = 0 . 2). Variations in shell mass, luminosity, and dust-to-gas ratio: M sh = 10 6 M ⊙ (red solid), M sh = 10 7 M ⊙ (blue dashed), M sh = 10 5 M ⊙ (green dotted); L = 10 47 erg/s and M sh = 3 × 10 6 M ⊙ (orange dash-dot-dot); M sh = 3 × 10 5 M ⊙ and f dg = 2 × f dg , MW (violet dash-dot). The yellow shaded area represents the typical range of momentum ratio ( ζ = ˙ p/ ( L/c )) and energy ratio ( ϵ k = ˙ E k /L ) observed in BAL quasars. \n<!-- image --> \nambient gas density distribution can be parametrised as a power law of radius (cf. Ishibashi & Fabian (2018)) \nn ( r ) = n 0 ( r r 0 ) -α , (13) \nwhere n 0 is the density of the external medium and α is the power law exponent, with α = 2 for an isothermal density distribution. The expanding shell mass is then given by \nM sh ( r ) = M sh , 0 +4 πm p ∫ n ( r ) r 2 dr ≈ M sh , 0 +4 πm p n 0 r α 0 r 3 -α 3 -α (14) \nwhere M sh , 0 is the initial mass of the shell. \nIn Fig. 6, we plot the column density N sh of expanding shells (vertical left axis) as a function of the shell velocity v . The outflow velocities increase with decreasing column densities, i.e. lower column density shells can be accelerated to higher speeds. Physically, this is simply because lower column density shells have higher Eddington ratios in the single scattering regime (Γ SS ∝ 1 /N sh ) and thus attain higher terminal velocities. The column density of the outflowing BAL gas is typically observed in the range N H ∼ (10 21 -10 22 ) cm -2 , with higher column density gas located \nFigure 5. Radial velocity profile of outflowing shells with different external gas densities: fixed-mass shell with M sh , 0 = 10 6 M ⊙ or equivalently n 0 = 0 (yellow solid); expanding shells with n 0 = 1cm -3 (cyan dash-dot) and n 0 = 10cm -3 (pink dotted). \n<!-- image --> \nat smaller radii (He et al. 2022). This may also be consistent with the empirical anticorrelation between N H and BAL distance (Arav et al. 2018) and thus the outflow velocity. Figure 6 also shows the plot expressed in terms of the extinction A V (vertical right axis) obtained by assuming a linear relation between hydrogen column density and optical extinction, of the form N H (cm -2 ) = (2 . 87 ± 0 . 12) × 10 21 A V (mag) (Guver & Ozel 2009; Foight et al. 2016). \nFigure 6. Shell column density N sh (left axis) and shell extinction A V (right axis) as a function of outflowing velocity. Variations in initial shell mass and external gas density: M sh , 0 = 5 × 10 6 M ⊙ , n 0 = 0 , 10 2 , 10 3 cm -3 (blue solid, dashed, dotted); M sh , 0 = 10 6 M ⊙ , n 0 = 0 , 10 , 50 cm -3 (red solid, dashed, dotted); M sh , 0 = 10 5 M ⊙ , n 0 = 0 , 1 , 10 cm -3 (green solid, dashed, dotted). \n<!-- image -->', '6.1 BAL clouds or clumps': "Smooth spherical shells subtending 4 π have been implicitly assumed in the previous sections. We now consider the case of BAL clouds or clumps that subtend only a small fraction of the solid angle from the central source. For instance, the outflowing shell may gradually break up and fragment into clumps, or some individual clouds may be pre-existing. For such clumpy outflows, the wind dynamics will be modified with respect to the spherical shell case, because the temporal evolution of the clump column density becomes decoupled from its radial evolution (Thompson et al. 2015). The equation of motion of the clump is given by \nd dt [ M c v c ] = L c ( πR 2 c 4 πr 2 ) [ 1 -e -τ UV ] -GM ( r ) M c r 2 , (15) \nwhere M c and R c are the clump mass and radius, A c = πR 2 c is the cross section of the cloud, τ UV = κ UV M c /A c is the UV optical depth, and the IR term is neglected. \nIn the case of clumpy outflows, a new timescale is introduced into the problem: the cloud expansion time t exp = R c /c s , where c s is the cloud internal sound speed. As the cloud is accelerated outwards by radiation pressure, it will expand laterally in the direction perpendicular to the radial acceleration (due to its finite internal temperature). Assuming that the cloud expands freely, the clump radius will increase as R c = R c, 0 + c s ( t -t 0 ), where R c, 0 is the clump initial radius at t 0 = 0 (Scoville & Norman 1995). Such lateral expansion implies an increase in the effective cross section of the cloud ( πR 2 c ), leading to an increase in the amount of incident radiation impinging on it. \nAs in the case of shells (Sect. 3), the effective Eddington luminosity for the clump is obtained by equating the outward radiative force to the inward gravitational force \nL ' E , c = 8 πm p cσ 2 rN c 1 -e -τ UV , (16) \nwhere N c = M c /m p A c is the column density of the clump. The corresponding effective Eddington ratio is Γ ' c = L/L ' E , c . The condition for the outflow launch (Γ ' c > 1) requires a critical clump column density of N c < L (1 -e -τ UV ) 8 πm p cσ 2 r . \nWe note that the radiative acceleration of the clump in the single scattering regime is inversely proportional to its column density \na c , SS = L 4 πm p cr 2 N c , (17) \nwhile in the UV-optically-thin regime, the radiative acceleration becomes independent of the column density and directly scales with the UV opacity \na c , UV = κ UV L 4 πcr 2 . (18) \nIntegrating the equation of motion (equation 15), we obtain the radial velocity profile of the outflowing clumps. In Fig. 7 (left-hand panel), we show the clump velocity as a function of radius for different values of the initial clump column density N c , 0 . We see that lower column density clumps are accelerated to higher velocities and reach larger radii compared to higher column density clumps. This is expected since the radiative acceleration scales as ∝ 1 /N c in the single scattering limit, such that lower column density material is more efficiently accelerated. In this picture, lowerN c clumps might travel ahead of higherN c clumps, leading to a range of outflowing velocities. At a given radial location (say r ∼ 1 kpc), the individual clump velocities can vary from v ≳ 2000 km/s to v ≲ 10 4 km/s, covering a wide range of outflowing speeds. The velocity spread further increases with increasing radial distance. \nWe also analyse how the cloud lateral expansion affects the outflow evolution. Figure 7 (right-hand panel) shows the radial velocity profiles varying the initial clump radius R c, 0 and internal sound speed c s . We observe that an increase in the sound speed leads to higher outflowing velocities (red curves), as higher sound speeds give rise to enhanced cross sections for radiative acceleration. On the other hand, smaller initial clump radii lead to higher outflowing velocities (green curves). The fastest outflows are obtained for a combination of low N c , high c s , and small R c, 0 . Overall, the clump velocities at a given distance from the centre can differ by several thousand km/s, depending on their initial column density, initial size, and internal sound speed. This implies that clumpy BAL outflows are characterised by a wide range of outflowing velocities.", '6.2 Covering fraction and absorption line profile': 'A BAL system may be formed when outflowing clumps propagating along the line of sight- absorb the central radiation. The clump covering factor is given by \nC f = πR 2 c 4 πr 2 = ( R c, 0 + c s t ) 2 4 r 2 = ( R c, 0 + c s v r ) 2 4 r 2 . (19) \n© 2012 RAS, MNRAS 000 , 1-10 \nFigure 8 (left-hand panel) represents the normalised flux F ( v ) as a function of the outflow velocity, for different initial clump column densities. We see that the covering fraction is larger at lower velocities, leading to deep absorption at low speeds, while C f decreases with increasing velocity. Higher column densities produce deeper absorption troughs with narrower line profiles, whereas lower column densities give rise to shallower troughs with broader line profiles. Meanwhile, larger initial clump radii and higher sound speeds -implying larger covering factors- lead to deeper absorption line profiles (Fig. 8, right-hand panel). The global shape of the absorption line profile is thus determined by the individual clump properties -column density, initial size, and sound speed. The observed BAL trough widths can span thousands of km/s, suggesting that the bulk flow is indeed better described in terms of clumps accelerated to a wide \n<!-- image --> \nFigure 7. Radial velocity profiles of clumpy outflows for fiducial parameters with R c, 0 = 3 pc, c s = 3 km/s, and r 0 = 50 pc. Clump radial velocity profiles with variations in initial column density (left-hand panel): N c, 0 = 10 21 cm -2 (green dotted), N c, 0 = 3 × 10 21 cm -2 (orange dash-dot-dot), N c, 0 = 10 22 cm -2 (red solid), N c, 0 = 3 × 10 22 cm -2 (violet dash-dot), N c, 0 = 10 23 cm -2 (blue dashed). Clump radial velocity profiles with variations in initial clump radius and internal sound speed (right-hand panel): N c, 0 = 10 21 cm -2 with c s = 10 km/s and different initial clump radius: R c, 0 = 0 . 3 pc (green dotted), R c, 0 = 1 pc (green solid), R c, 0 = 3 pc (green dashed); N c, 0 = 10 22 cm -2 with R c, 0 = 1 pc and different sound speed: c s = 30 km/s (red dotted), c s = 10 km/s (red solid), c s = 3 km/s (red dashed). \n<!-- image --> \nIf the clump radius remains constant during the outward propagation ( R c = cst), the covering fraction decreases with distance as C f ∝ 1 /r 2 (geometric dilution). However, if the clump expands laterally as it accelerates outwards, the covering fraction declines more slowly. At large radii, the velocity-dependent covering factor will drop as C f ∝ 1 /v 2 . Therefore larger radii and higher velocities lead to lower covering fractions in accelerating outflows. \nThe covering fraction, together with the optical depth, determine the shape of the absorption line profile. Following most works in BAL outflows, the normalised flux in the partial covering model is defined as (e.g. Arav et al. 2018) \nF ( v ) = 1 -C f ( v ) + C f ( v ) e -τ ( v ) , (20) \nwhere C f ( v ) is the velocity-dependent covering factor, and τ ( v ) = κ UV M c /πR 2 c is the UV optical depth. \nrange of velocities rather than single shells (Thompson et al. 2015).', '7 DISCUSSION': "While the dynamics and energetics of BAL outflows have been analysed in both individual sources and large quasar samples (Arav et al. 2018, 2020; Xu et al. 2019; Miller et al. 2020; He et al. 2019, 2022), the actual BAL acceleration mechanism is still a matter of debate. At least two physical models of BAL outflow driving have been discussed in the literature: accretion disc winds and radiation pressure on dust. Recent observations indicate large galactocentric distances for the BAL location, with the majority of absorption outflows located at radii r > 100 pc (Arav et al. 2018; Xu et al. 2019; Miller et al. 2020). Furthermore, the BAL acceleration phase likely occurs on scales of tens of pc (He et al. 2022). Such large radii seem to be incompatible with winds originating from the inner accretion disc. Instead, the observational constraints favour radiation pressure on dust -operating beyond the dust sublimation radius- as the main BAL outflow launching mechanism. \nHere we explicitly show that radiation-driven dusty outflows can be efficiently accelerated to reach BAL-like velocities ( v ∼ 10 4 km/s) on galactic scales ( r ≲ 1 kpc). Higher BAL velocities are attained for higher luminosities, lower column densities, and higher dust-to-gas ratios. We also compute the corresponding outflow energetics, and obtain momentum ratios and energy ratios in the range ζ ∼ (0 . 1 -10) and ϵ k ∼ (10 -3 -0 . 1), consistent with available measurements of the outflow energetics in BAL quasars (e.g. He et al. 2019; Miller et al. 2020). Therefore BAL out- \n<!-- image --> \nFigure 8. Normalised flux as a function of outflow velocity for fiducial parameters with R c, 0 = 3 pc, c s = 30 km/s, and r 0 = 30 pc. Variations initial clump column density (left-hand panel): N c, 0 = 3 × 10 23 cm -2 (blue dashed), N c, 0 = 10 23 cm -2 (violet dash-dot), N c, 0 = 3 × 10 22 cm -2 (red solid), N c, 0 = 10 22 cm -2 (orange dash-dot-dot), N c, 0 = 3 × 10 21 cm -2 (green dotted). Variations in initial clump radius and internal sound speed (right-hand panel): N c, 0 = 3 × 10 23 cm -2 and R c, 0 = 1 pc with different sound speeds of c s = 10 km/s (violet dashed), c s = 20 km/s (violet dash-dot), c s = 30 km/s (violet dotted); N c, 0 = 3 × 10 22 cm -2 and c s = 30 km/s with different initial clump radii of R c, 0 = 1 pc (green dashed), R c, 0 = 2 pc (green dash-dot), R c, 0 = 3 pc (green dotted). \n<!-- image --> \nered by radiation pressure on dust, may provide an important contribution to AGN feedback on galactic scales. \nObservations of BAL outflows indicate typical number densities in the range n ∼ (1 . 6 × 10 3 -2 . 5 × 10 5 ) cm -3 on radial scales of ∼ 100 pc to ∼ 4 kpc (Xu et al. 2019). For a thin spherical shell model, the number density in the outflowing gas can be roughly estimated as n ( r ) ∼ ˙ M/ (4 πm p r 2 v ). As the shell moves outwards, it sweeps up mass from the surroundings, and the expanding shell mass grows as ∝ n 0 r α 0 r 3 -α 3 -α (equation 14), depending on the ambient gas density distribution. For an isothermal full spherical shell ( α = 2) with fiducial parameters and n 0 ∼ 10 6 cm -3 , the corresponding number density is about n ∼ (10 3 -10 5 ) cm -3 at r ∼ (0 . 1 -0 . 01) kpc. Larger values of the number density -at larger radii- may be obtained by considering flatter gas density distributions (e.g. α = 1 , 0). In radiation pressure confinement scenarios, the outflowing gas may be further compressed by radiation pressure (Baskin et al. 2014). In the more realistic case of a clumpy gas distribution, the cloud number density ( n c ∼ 3 M c / 4 πm p R 3 c ) will depend on the individual clump properties ( M c , R c ), leading to a wide range of densities within the outflowing gas. \nRelatedly, different ionization states can be expected in BAL outflows, giving rise to high/low ionization lines. The ionization parameter is defined as ξ = L i / ( nr 2 ), where L i = k i L is the ionizing luminosity. Assuming an ionizing fraction of k i ∼ 0 . 01 -0 . 5, the ionization parameter of the fiducial isothermal shell is log ξ ∼ (0 . 6 -2 . 3) erg cm s -1 , while the ionization parameter tends to decrease with radius for flatter ambient density distributions ( α < 2). In reality, in a clumpy inhomogeneous flow, the ionization parameter will depend on the individual cloud properties ( M c , R c ), resulting in a broad range of ionization states. This may naturally account for the wide range of ionization levels observed in BAL outflows, with the ionization parameter typically \nspanning log ξ ∼ ( -0 . 5 -2 . 5) erg cm s -1 (Laha et al. 2021). The ionization structure of BAL outflows is further affected by the presence of dust grains. Dust can absorb a significant fraction of the ionizing photons and reduce the ionizing flux, thus facilitating the formation of low ionization species. While high ionization lines may originate in lower density clouds, low ionization lines are more likely produced in high density dusty clumps. \nThe distinct spectral features of LoBAL and HiBAL outflows are also reflected in their different dust extinction and reddening properties. LoBAL quasars are found to be significantly redder than HiBAL quasars, with A V ∼ 0 . 42 mag and A V ∼ 0 . 09 mag, respectively, based on median composite spectra from SDSS (Chen et al. 2022). The greater reddening is suggestive of a higher dust content in LoBAL quasars; in fact, a particularly high LoBAL fraction is found among dust-reddened quasars (Urrutia et al. 2009). In the framework of the AGN evolutionary sequence (e.g. Sanders et al. 1988), (Lo)BAL quasars may represent a short-lived transition phase from dust-obscured ULIRGs to unobscured luminous quasars. An example is the dusty LoBAL outflow detected in a dust-reddened quasar at z ∼ 2, which is possibly caught in the act of blowing out its surrounding dust cocoon (Yi et al. 2022). Redder LoBAL quasars may then be associated with an earlier evolutionary phase, subsequently transitioning into bluer HiBAL quasars (Chen et al. 2022). \nAlternatively, dust-driven BAL winds have been interpreted in terms of geometrical configurations, such as in the failed radiatively accelerated dusty outflow (FRADO) model (Naddaf et al. 2023, and references therein). In this picture, the BAL phenomenon is only observed when the line of sight lies within the outflowing cone, and is described as an orientation effect. The enhanced reddening of (Fe)LoBAL could \nthen be explained by particular sightlines, e.g. passing close to the edge of the dusty torus (Dunn et al. 2015). \nDirect connections between BAL outflow properties and dust emission have been reported in a number of studies. A significant correlation between the outflow strength and the near-IR continuum slope ( β NIR , an indicator of the amount of hot dust emission relative to the accretion disc emission) is observed in a large sample of BAL quasars at z ∼ 2 (Zhang et al. 2014). More recently, Temple et al. (2021) report that the strength of the ∼ 2 µ m emission (corresponding to emission from hot dust at the sublimation radius) correlates with the blueshift of the C IV emission line, such that objects with stronger hot dust emission present stronger C IV outflow signatures. Given that larger C IV emission line blueshifts are associated with faster and stronger BAL troughs (Rankine et al. 2020), one may expect causal connections between dust emission and BAL outflow features. A straightforward explanation is that the dust itself is actually providing the opacity for BAL outflow acceleration. \nWe note that radiative line-driving also plays an important role in BAL quasars. Line-locked CIV absorption systems appear common in Narrow Absorption Line BALs (Bowler et al. 2014). Lewis & Chelouche (2023) have recently examined the parameter space for line locking to occur, finding that it requires a low column density N H < 10 19 cm -2 and log ionization parameter U ∼ 0. The relative contribution of the CIV doublet to the total radiation-pressure force for a dusty medium then exceeds one percent (see also Bowler et al. (2014)). This allows Line Locking between clouds along the same line of sight to the nucleus to occur. In the present work, dust driving does the 'heavy lifting' for clouds to exceed 10,000 km/s, at which point, (see Fig. 6), N H < 10 19 cm -2 , so resonant CIV absorption may be strong enough to lock clouds with a 500 km/s velocity difference together. Line-locked absorbers can be present in both BAL and non-BAL quasar samples. \nIt is interesting to note that BAL and non-BAL quasars are known to share similarities in their emission and continuum properties (Weymann et al. 1991), suggesting that the two sub-classes are drawn from a common parent population (Rankine et al. 2020; Temple et al. 2024). In other words, BAL and non-BAL features can be observed in otherwise similar quasars. In our picture, we may envision several AGN outflow stages during the quasar lifetime. A first massive outflow event, sweeping up mass at lower speeds, may clear out the gas from the host galaxy. These slower massive winds without BAL features could be associated with powerful molecular outflows observed in e.g. ULIRGs (Fluetsch et al. 2021). Subsequently, less massive, high-velocity outflows can develop and rapidly propagate into the depleted galaxy, displaying characteristic BAL signatures. The latter may correspond to the fast BAL winds observed in BAL quasars. Overall, radiation pressure on dust provides a promising mechanism for powering BAL outflows across cosmic time. \nBAL outflows are more commonly observed at higher redshifts, with the BAL fraction increasing with redshift (e.g. Allen et al. 2011). The BAL fraction is found to increase up to ∼ 50% at z ≳ 6, with BAL outflows reaching extreme velocities ( v ≳ 0 . 1 c ) (Bischetti et al. 2022). This suggests an efficient wind acceleration mechanism operating at early times, possibly boosted by radiation pressure \non dust. The redshift evolution observed in BAL quasars cannot be accounted for by differences in central luminosity and accretion rate. The higher BAL fraction in the early cosmic epoch may be explained by wider angle outflows in the orientation scenario, or longer blowout phases in the AGN evolutionary sequence (Bischetti et al. 2023). In either case, widespread and powerful dust-driven BAL outflows might play a critical role in shaping black hole-galaxy co-evolution in the early Universe.", 'ACKNOWLEDGEMENTS': 'We thank the anonymous referee for a constructive report. ACF acknowledges early work on the topic by Naoki Arakawa.', 'DATA AVAILABILITY': 'No new data were generated or analysed in support of this research.', 'REFERENCES': "Allen J. T., Hewett P. C., Maddox N., Richards G. T., Belokurov V., 2011, MNRAS, 410, 860 Arav N., Liu G., Xu X., Stidham J., Benn C., Chamberlain C., 2018, ApJ, 857, 60 Arav N., Xu X., Miller T., Kriss G. 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2024AJ....168..161T	K22 bHIP 116454 b the first exoplanet discovery by K2 during its TwoWheeled Concept Engineering Test is a subNeptune 2.5 0.1 R SUBSUB 9.7 1.2 M SUBSUB orbiting a relatively bright K SUB S SUB 8.03 Kdwarf star on a 9.1 day period. Unfortunately due to a spurious followup transit detection and ephemeris degradation the transit ephemeris for this planet was lost. In this work we recover and refine the transit ephemeris for K22 b showing a 40 discrepancy from the discovery results. To accurately measure the transit ephemeris and update the parameters of the system we jointly fit spacebased photometric observations from NASAs K2 Transiting Exoplanet Survey Satellite and Spitzer missions with new photometric observations from the ground as well as radial velocities from HARPSN that are corrected for stellar activity using a new modeling technique. Ephemerides becoming lost or significantly degraded as is the case for most transiting planets highlights the importance of systematically updating transit ephemerides with upcoming large efforts expected to characterize hundreds of exoplanet atmospheres. K22 b sits at the highmass peak of the known radius valley for subNeptunes and is now wellsuited for transmission spectroscopy with current and future facilities. Our updated transit ephemeris will ensure no more than a 13 minute uncertainty through 2030.	2024-10-01T00:00:00Z	['10.3847/1538-3881/ad60bf', 'arXiv:2409.07019', '2024arXiv240907019T', '2024AJ....168..161T', '10.48550/arXiv.2409.07019']	['Exoplanet astronomy', 'Exoplanet systems', 'Exoplanet catalogs', 'Exoplanets', 'Ephemerides', 'Transits', '486', '484', '488', '498', '464', '1711', 'Astrophysics - Earth and Planetary Astrophysics']	The K2 and TESS Synergy. III. Search and Rescue of the Lost Ephemeris for K2s First Planet	2,024	197	0.55	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.07019.pdf	{"The K2 & TESS Synergy III: search and rescue of the lost ephemeris for K2's first planet": 'ERICA THYGESEN , 1, ∗ JOSEPH E. RODRIGUEZ , 1 ZOË L. DE BEURS , 2, † ANDREW VANDERBURG , 3 JOHN H. LIVINGSTON , 4, 5, 6 JONATHON IRWIN, 7 ALEXANDER VENNER , 8 MICHAEL CRETIGNIER , 9 KAREN A. COLLINS , 10 ALLYSON BIERYLA , 10 DAVID CHARBONNEAU , 10 IAN J. M. CROSSFIELD , 11 XAVIER DUMUSQUE , 12 JOHN KIELKOPF , 13 \nDAVID W. LATHAM , 10 MICHAEL WERNER 14 \n1 Center for Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA 2 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 3 Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 4 Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 5 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 6 Astronomical Science Program, Graduate University for Advanced Studies, SOKENDAI, 2-21-1, Osawa, Mitaka, Tokyo, 181-8588, Japan 7 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, United Kingdom 8 Centre for Astrophysics, University of Southern Queensland, West Street, Toowoomba, QLD 4350, Australia 9 Department of Physics, University of Oxford, Oxford OX13RH, United Kingdom 10 Center for Astrophysics \| Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA 11 Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA 12 Geneva Observatory, University of Geneva, Chemin des Mailettes 51, 1290 Versoix, Switzerland 13 Department of Physics and Astronomy, University of Louisville, Louisville, KY 40292, USA 14 NASA Jet Propulsion Laboratory, Pasadena, CA, USA', 'ABSTRACT': "K2-2 b/HIP 116454 b, the first exoplanet discovery by K2 during its Two-Wheeled Concept Engineering Test, is a sub-Neptune (2.5 ± 0.1 R ⊕ , 9.7 ± 1.2 M ⊕ ) orbiting a relatively bright (K S = 8.03) K-dwarf on a 9.1 day period. Unfortunately, due to a spurious follow-up transit detection and ephemeris degradation, the transit ephemeris for this planet was lost. In this work, we recover and refine the transit ephemeris for K2-2 b, showing a ∼ 40 σ discrepancy from the discovery results. To accurately measure the transit ephemeris and update the parameters of the system, we jointly fit space-based photometric observations from NASA's K2, TESS, and Spitzer missions with new photometric observations from the ground, as well as radial velocities from HARPS-N that are corrected for stellar activity using a new modeling technique. Ephemerides becoming lost or significantly degraded, as is the case for most transiting planets, highlights the importance of systematically updating transit ephemerides with upcoming large efforts expected to characterize hundreds of exoplanet atmospheres. K2-2 b sits at the high-mass peak of the known radius valley for sub-Neptunes, and is now well-suited for transmission spectroscopy with current and future facilities. Our updated transit ephemeris will ensure no more than a 13-minute uncertainty through 2030.", '1. INTRODUCTION': "In the era of cutting-edge atmospheric characterization of transiting exoplanets, precise and accurate ephemerides are crucial for efficiently scheduling these expensive observations. However, over 80% of transiting exoplanets will have uncertainties on their future transit times greater than 30 minutes by the end of the decade (see Thygesen et al. 2023), \nCorresponding author: Erica Thygesen \[email protected] \n- ∗ Quad Fellow\n- †\n- National Science Foundation Graduate Research Fellow MIT Collamore-Rogers Fellow \nrendering these systems extremely challenging to observe with JWST (Gardner et al. 2006; Beichman et al. 2020), major upcoming facilities such as the Atmospheric Remotesensing Infrared Exoplanet Large-survey (ARIEL; Tinetti et al. 2018, 2021), and 30m class telescopes like the Thirty Meter Telescope (TMT; Sanders 2013), Giant Magellan Telescope (Johns et al. 2012), and the 39 m European Southern Observatory Extremely Large Telescope (ELT; Udry et al. 2014). This problem can be solved by observing new transits of these planets with current facilities. Fortunately, NASA's Transiting Exoplanet Survey Satellite (TESS) mission (Ricker et al. 2015) is observing the entire sky, providing a valuable opportunity to refine the transit ephemeris for most known planets. \nAfter a successful 4-year nominal mission, discovering thousands of exoplanets, the Kepler mission (Borucki et al. 2010) was repurposed due to a mechanical issue. Using the solar pressure to stabilize pointing of the Kepler spacecraft, the K2 mission was able to survey the ecliptic plane, finding hundreds of exciting new systems that are well-suited for detailed characterization (Howell et al. 2012; Vanderburg et al. 2016; Zink et al. 2021; Kruse et al. 2019; Pope et al. 2016; Livingston et al. 2018a; Crossfield et al. 2016; Dattilo et al. 2019). The K2 mission ended in 2019, with many of its newly-detected planets never being reobserved since their discovery campaign(s). The K2 & TESS Synergy project is an effort to provide the community with updated and accurate transit times and system parameters for exoplanets originally discovered by the K2 mission that have been recently observed by TESS (Ricker et al. 2015). Following a successful pilot study (Ikwut-Ukwa et al. 2020), the second paper in this series revisited 26 K2 single-planet systems that TESS reobserved during its prime mission (Thygesen et al. 2023). This work improved the average ephemeris uncertainties by multiple orders of magnitude due to the addition of new TESS transits. Additionally, we identified systems where the original ephemeris has been completely lost (See K2-260; Thygesen et al. 2023), which is similar to this work on K2-2 b, K2's first exoplanet discovery. \nK2-2 b was identified during the Two-Wheeled Concept Engineering Test (campaign 0) of the K2 mission. K2-2 b is a sub-Neptune (2.5 ± 0.1 R ⊕ , 9.7 ± 1.2 M ⊕ ) on a 9.1-day orbit around a bright ( V = 10.2, J = 8.6, HIP 116454) K-dwarf (Vanderburg et al. 2015). At discovery, a single clear transit was detected in the K2 observations, along with a marginal ( ∼ 3 σ ) detection from the Microvariablity and Oscillations of Stars (MOST) Space Telescope (Walker et al. 2003). Followup observations were scheduled with Spitzer (P.I. Werner, AOR 57185280) and the Hubble Space Telescope (P.I. Bourrier, proposal I.D. 15127), however, the transit was not seen during the predicted window from the discovery ephemeris. It was then determined that the MOST transit was likely not a real transit of K2-2 b, having skewed the period enough to cause subsequent transits to be missed. \nIn this work, we combine the discovery observations from Vanderburg et al. (2015) with new observations from NASA's TESS mission, follow up ground-based photometry, and improved radial velocities to accurately measure the ephemeris of K2-2 b for the first time, proving the original detection from MOST to be a false positive. In Section 2 we describe the observations used and the relevant reduction and analysis methods, including the reanalysis of radial velocities from the High Accuracy Radial Velocity Planet Searcher-North (HARPS-N; Cosentino et al. (2012)) on the 3.58m Telescopio Nazionale Galileo at the Roque de los Muchachos Observatory. Section 3 outlines the methodology used in running the \nEXOFASTv2 global fit of all observations and archival information. We present our results and discuss the importance of ephemeris refinement in the context of future characterization of K2-2 b in Section 4.", '2. OBSERVATIONS AND ARCHIVAL DATA': 'The discovery analysis for K2-2 b included a 47 day long light curve from MOST (Walker et al. 2003), which was thought to contain a marginal ∼ 3 σ detection of the transit, but future follow up attempts to reobserve the transit with Spitzer and HST showed no transit during or near the predicted window. This ultimately led to the idea that the MOST observations were not reliably constraining the transit ephemeris. While it is not clear why this happened, it is possible that Gaussian noise or satellite systematics caused an already marginal detection to be anchored to a different time of transit. Our new observations from MEarth, ULMT, Spitzer and TESS (Figure 1) confirm this hypothesis. In the near decade since its discovery, a variety of follow up observations have been conducted to better characterize the K2-2 system and to recover the transit ephemeris. In the following sections, we describe the new and archival observations used in our analysis. The magnitudes and literature values for K22 are listed in Table 1, and the photometric data sets we used are outlined in Table 2.', '2.1. Ground-based archival imaging': "At the discovery of of K2-2 b, Vanderburg et al. (2015) used multiple archival from the National Geographic Society-Palomar Observatory Sky Survey (POSS-I, van Leeuwen 2007) and Sloan Digital Sky Survey (SDSS, Abazajian et al. 2009), and newly acquired images from Robo-AO on Palomar (Baranec et al. 2014; Law et al. 2014) and Natural Guide Star Adaptive Optics (NGSAO) system on Keck to rule out nearby close companions that might be contaminating the K2 aperture. A nearby white dwarf with a separation of around 8 '' was identified to share a similar proper motion to K2-2, suggesting that they exist in a gravitationally bound system (this is discussed more in Section 4.2). The white dwarf is within the K2 aperture, but is 6-7 magnitudes dimmer than K2-2, which would not affect the final transit depth of K2-2 b. No other nearby companions were found to a 7 σ significance in the H band to the limits of 3.0 mag at 0 '' .1 separation, 9.2 mag at 1 '' .0 and 12.7 mag at 5 '' .0.", '2.2. K2 Photometry': 'A single transit of K2-2 b was observed at 30-minute cadence during the Kepler Two-Wheel Concept Engineering Test during February 2014. Due to the loss of two of the four reaction wheels on the spacecraft, significant systematics were introduced to the light curves of the K2 mission. We corrected for these using the methods described in Vanderburg & Johnson (2014) and Vanderburg et al. (2016), which \nutilize a series of 20 apertures to extract raw light curves used to perform the corrections. Short timescale variations in each of these light curves are correlated with the roll angle of the spacecraft, with the latter being subtracted from the light curves. This process is repeated iteratively until the light curve is free of any variations associated with the roll of the spacecraft. The most precise light curve out of the 20 following the corrections is selected for final analysis. We performed further corrections by fitting the transit and correcting for the systematics and any low-frequency stellar variability, prior to the global fit.', '2.3. MEarth': "MEarth was used to initially recover the transit of K22 b and constrain the ephemeris, observing multiple partial and full transits. MEarth consists of 16 separate 0.4 m telescopes using custom 715 nm longpass filters designed to find Earth-sized planets around M dwarfs (Nutzman & Charbonneau 2008; Irwin et al. 2015). Telescopes 1-8 are a part of the MEarth-North Observatory at Fred Lawrence Whipple Observatory (FLWO) on Mount Hopkins, Arizona, while the other eight telescopes (numbered as 11-18) are part of the MEarth-South Observatory located at Cerro Tololo InterAmerican Observatory (CTIO) on Cerro Tololo, Chile. K2-2 was observed using a subset of four telescopes from each observatory (see Table 2) with 1 minute cadence on UT 2016 September 21 and 30, and UT 2016 October 09. Light curves from MEarth are automatically extracted through a pipeline (see Irwin et al. 2007; Berta et al. 2011) that calibrates the images using flat fields, dark current frames and bias exposures. We combined the light curves across multiple nights for each telescope, so within the global fit the variance can be determined independently for each instrument. We sliced the light curves such that we only included one full transit duration before and after the transit, and detrended against airmass in the global fit. While the original observations also included telescopes 4, 5 and 8, we did not use these in our analysis as the light curves did not contain full transits and would not contribute significant value to the global fit. The transit was also missed during the night of UT 2016 September 11 due to the incorrect ephemeris. \nThese observations were the first use of the defocus observing mode of MEarth for transit follow-up, and served as the prototype for a large number of observations of TESS objects of interest done in later years. Here we describe the modifications made to the system to implement this mode. Prior to implementation of defocus, MEarth observations of bright stars were limited by scintillation noise due to the short maximum exposure times possible before detector saturation, combined with high overheads (approximately 15s, most of which was consumed by CCD readout and download over USB2 connection to the host computer), resulting \nin a low duty cycle. For scintillation limited observations of events of fixed duration such as transits, the overall transitaveraged photometric noise is determined by the duty cycle (e.g. Young 1967) so the goal of implementing defocus was to improve this by substantially lengthening the exposure times possible before saturation. \nThe scheduling and telescope control software were modified to allow each observation request to specify defocus as half flux diameter (HFD), in pixels. For these first observations of K2-2, we used HFD = 6.0 pixels, where the pixel scales are 0.76 arcsec/pix for MEarth-North and 0.84 arcsec/pix for MEarth-South. The telescope focus was offset by the scheduler prior to commencing observations of each target by the appropriate number of focus encoder counts, where the scaling factor was determined from the calibration curve of HFD versus focus encoder counts used by the standard automatic focus routine (normally used for focusing the telescope at the start of the night). \nMEarth did not have autoguiders, and guiding to stabilize the target star position on the detector (vital for precise transit work) had to be done using the science exposures themselves, which were 36s for K2-2. The standard MEarth target acquisition and guiding system for normal in-focus images consisted of astrometric analysis of the images after readout to determine their center in celestial coordinates, followed by offsetting of the telescope to center the target based on its calculated position. Target acquisition was done by applying the full offset, and guiding by passing these measurements into a standard proportional-integral-derivative (PID) control loop with an overall gain less than unity to provide damping and avoid overshoot and oscillation during guiding. \nTo implement the defocus observing mode, the image analysis part of this astrometric routine was replaced with a custom source detection routine using a standard matched filter approach (e.g. Irwin 1985), where in the case of defocused images, rather than using a standard approximately Gaussian filter kernel, the filter kernel was instead a model of the defocused telescope PSF. This technique is appropriate for analysis of images with mild amounts of defocus, such as needed on MEarth. Previous work (e.g. McCormac et al. 2013) has usually concentrated on the case of severe defocus, where different analysis techniques are needed. \nThe PSF model was constructed by approximating the telescope entrance pupil as a circular annulus, and introducing defocus by setting the complex phase of this function to a multiple of the Z 0 2 Zernike mode. The resulting PSF was computed by taking the inverse Fourier transform of this function. In practice, it was also convolved by a Moffat profile (Moffat 1969) with parameters chosen based on standard in-focus MEarth observations to approximate seeing and any effects other than diffraction that contribute to the system's \nFigure 1. The discovery and follow up phase-folded transits of K2-2 b used in the EXOFASTv2 (see Section 3) analysis. The observations from K2 (black), TESS (purple), Spitzer (blue), MEarth (green), and ULMT (yellow) are shown in open colored circles with the solid colored line representing the EXOFASTv2 model for that dataset. The closed colored circles represent 30-minute bins. East transit is offset by a constant for clarity. \n<!-- image --> \nnormal in-focus PSF spot size. The relationship between the Z 0 2 Zernike coefficient and HFD was determined empirically. \nThe PSF model was also used to compute exposure times and set photometric aperture radii for the automatic extraction pipeline. We found that these theoretical estimates of exposure times based on the idealised PSF models were rather optimistic, and in practice it was necessary to use shorter exposures (or equivalently, somewhat more defocus for a given desired exposure time) to avoid the risk of saturation due to non-uniformity of the resulting defocused star image. This can be caused by atmospheric turbulence (particularly in short exposures), but also other optical aberrations affecting the defocused star image, such as coma, which causes an asymmetric distribution of brightness around the resulting ring shaped PSF, and can cause one side of the ring to become too bright. Being remotely operated robotic telescopes, it was not always possible to maintain optimal collimation of the MEarth telescope optics, and while this had minimal effect on the normal in-focus images used for the majority of the survey, it did noticeably affect the defocused PSFs. \nWith an appropriate detection threshold, this source detection procedure was found to produce quite robust results, albeit at reduced sensitivity to faint sources, and with a practical upper limit to the defocus HFD of approximately 15 pixels. Given the field of view of the MEarth telescopes of approximately 27x27 arcmin the number of detected sources was found to still be sufficient for accurate multi-star guiding using the astrometric solutions on nearly all of the targets observed over several years of observations, including hundreds of TESS objects of interest.", '2.4. ULMT': "Once the ephemeris was refined from the MEarth observations, an ingress of K2-2 b was observed using the University of Louisville Manner Telescope (ULMT; formerly MVRC) at the Mt. Lemmon summit of Steward Observatory, Arizona. The observation was made in the r ' band with 50 second exposure time on UT 2016 October 10. The setup used for the observation included a 0.6 m f/8 RC Optical Systems Ritchey-Chrétien telescope and SBIG STX-16803 CCD camera with a 4k × 4k array of 9 µ m pixels, which yielded a 26.6' × 26.6' field of view and 0.39 pixel-1 plate scale. The images were calibrated and photometric data were extracted using AstroImageJ (Collins et al. 2017), and the light curves were detrended against airmass in the global fit.", '2.5. Spitzer': 'With the ephemeris more precisely constrained from the MEarth and ULMT transits, Spitzer was used to observe a single transit of K2-2 b on UT 2017 April 1 (P.I. M. Werner, observing program 13052, AOR 62428416; Werner et al. \nTable 1. Literature values for K2-2. \nNotes. The uncertainties of the photometry have a systematic error floor applied. Proper motions taken from the Gaia EDR3 archive and are in J2016. Parallaxes from Gaia EDR3 have a correction applied according to Lindegren et al. (2021). \n2016). The observation was 10.5 hours long, and was taken with the InfraRed Array Camera (IRAC; Fazio et al. 2004) channel 2 (4.5 µm) with a 2-second exposure time. We used the technique described in Livingston et al. (2018b) to extract the light curve. In brief, we extracted an optimal light curve by selecting the photometric aperture that minimized both white and red noise, and then corrected for systematics using pixel-level decorrelation (PLD; Deming et al. 2015). \nAs Spitzer can have correlated noise due to spacecraft systematics, we scaled the per point errors so that we did not underestimate the uncertainties. We followed the procedure from Winn et al. (2008), where a scaling factor, β , is applied to the measured standard deviation to account for timecorrelated noise. We first calculated the out-of-transit standard deviation for the unbinned data, σ 1 (for this calculation we conservatively defined out-of-transit as being outside of a full transit duration centered at the transit midpoint). We then binned the out-of-transit data points to a series of 10 temporal bin widths ranging from 4.2 minutes to 8.8 minutes, increasing in equal steps of 0.46 minutes. The limits on the bin widths correspond to the 1 σ range of the ingress/egress duration based on a preliminary fit using K2 and TESS light curves. \nTable 2. Photometry used in this analysis. \nNOTES: Each telescope caught one full transit, except for ULMT which observed the ingress and partial transit. Observations with MEarth North used Telescopes 1, 2, 3 and 6, while MEarth South included Telescopes 11, 12, 16 and 18. \nWe then calculated the standard deviation for each set of binned data. In general, this should be equivalent to σ N = σ 1 / √ N × √ M / ( M -1 ) , where M is number of bins and N is data points per bin, if there is no time-correlated noise. However, the measured σ N can be larger than the expected value (by the factor β ). We calculated this factor for each bin width, then used the mean value across all widths as the final value for β . Finally, we scaled the original unbinned, out-oftransit error bars by the factor β = 1.19, which is used as the per point uncertainty in our global fit.', '2.6. TESS Photometry': 'A single transit was observed by the Transiting Exoplanet Survey Satellite (TESS) in each of Sectors 42 and 70. We used the 120 second cadence lightcurves in our global fits. We retrieved the light curve through the Python package Lightkurve (Lightkurve Collaboration et al. 2018), selecting the light curve processed through the Science Processing Operations Center (SPOC) pipeline at the NASA Ames Research Center (Jenkins et al. 2016), which corrects for various systematics and identifies transits. The light curves were created from the Pre-search Data Conditioned Simple Aperture Photometry (PDCSAP) flux, which uses the optimal TESS aperture to extract the flux and corrects the target for systematics using the PDC module (Stumpe et al. 2012, 2014; Smith et al. 2012). To correct for stellar variability and any remaining systematics based on the out-of-transit photometry, we used the spline-fitting routine keplerspline 1 (Vanderburg & Johnson 2014). We applied an initial estimate on the per-point errors for the corrected light curves as being the median absolute deviation of the out-of-transit photometry. We note that the perpoint error is optimized through a fitted jitter term in the EXOFASTv2 global fit (See Section 3).', '2.7. Archival Spectroscopy': 'We included archival spectroscopy to determine the host star properties and to refine the mass measurement of K2-2 b. In particular, to better characterize the host star in the global fit, we used metallicity measurements of K2-2 from the Tillinghast Reflector Echelle Spectrograph (TRES; F"urész 2008) on the 1.5m Tillinghast Reflector at the Fred L. Whipple Observatory (FLWO). This is in keeping with our procedure for the larger Synergy catalog, where we are using TRES metallicities where available. The stellar parameters using TRES spectra were derived using the Stellar Parameter Classification (SPC; Buchhave et al. 2012). Three measurements from TRES ([M/H] = -0.193 ± 0.086, -0.191 ± 0.08, 0.009 ± 0.08) were available through the ExoFOP website 2 . We used the mean value to place a Gaussian prior on metallicity ([Fe/H]) of -0.125 ± 0.08. \nWe used a total of 105 spectra of K2-2, including those used in Vanderburg et al. (2015) and Bonomo et al. (2023), acquired using the High Accuracy Radial velocity Planet Searcher for the Northern hemisphere (HARPS-N) on the 3.6m Telescopio Nazionale Galileo (TNG) at the Roque de los Muchachos Observatory (Cosentino et al. 2012), in order to better characterize the mass of K2-2 b (Figure 2). Each observation had either 15 or 30 minutes exposure time, with a resolving power of R = 115,000. We followed the procedure of Dumusque et al. (2021) to reduce the RVs that were used in our global fits. The observations occurred in two main blocks, separated by ∼ 2.5 years; the first run was from UT 2014 July 7 to December 6 2017, and the second from UT 2020 June 25 to 2023 November 27. The second series of RVs was significantly offset to the earlier measurements, which led us to apply post-processing systematics corrections to investigate whether the offset was instrumental or physical in nature.', '2.7.1. YARARA processing to correct remaining systematics': 'YARARA (Cretignier et al. 2021) is a post-processing methodology that aims to perform correction of the spectra by the analysis of the spectra time-series. While a more advanced version of the pipeline has been presented recently in Cretignier et al. (2023) (sometimes referred to as the YARARA V2 or YV2 datasets), the SNR of the target was too low to apply those advanced methods of correction (such as the SHELL presented in Cretignier et al. (2022)) and we remained with the YARARA V1 or YV1 version of the products. \nThe corrections available in YARARA cover as much as the telluric lines, as instrumental systematics or stellar activity. The pipeline usually starts from the S1D order-merged \nspectra produced by official DRS that have been continuum normalized by RASSINE (Cretignier et al. 2020b). The method then consists of a multi-linear decorrelation by fitting a basis of vectors that are designed to correct for some dedicated effects, either obtained by optimized extraction (see e.g. Stalport et al. (2023)) or by principal component analysis (PCA) as initially presented in Cretignier et al. (2021). For a dataset around SNR ∼ 50, the main corrections that are possible to perform consist of removing cosmic, telluric lines, and the change of the instrumental PSF (Stalport et al. 2023). Even if a clear and strong emission is detected in the core of the CaII H&K lines, no reliable and precise extraction of the signal could be achieved and the stellar activity correction that mainly relies on this proxy (which contains most of the information from active regions (Cretignier et al. 2024)) was therefore skipped. The RVs were obtained with a crosscorrelation function (CCF) on the corrected spectra using a line list optimised for the star following the line centre procedure described in Cretignier et al. (2020a). \nAfter the application of YARARA, we still detect the longtrend signal which discards any potential effects from telluric or change of the instrumental PSF at the precision level of our data.', '2.7.2. CCF Activity Linear Model (CALM) to model stellar variability': "To model stellar variability in the radial velocities, we used activity indicators derived using the CCF Activity Linear Model (CALM) (de Beurs et al. 2024). CALM is a linear regression method which exploits the shape changes that stellar variability introduces into the cross-correlation functions (CCFs) computed from stellar spectra. Since CCFs represent an average of all line shapes in a star's spectrum, CALM is especially sensitive to line shape changes that persist in most spectral lines. In this method, we do not include the entire CCF in our model since CCFs are comprised of 49element arrays and we only have 105 RVs. Including the entire CCF would lead to overfitting. We experimented with sampling various fractions of the CCFs and across random locations within the CCF. We found that using 5 CCF locations provides a balance between preventing overfitting and optimizing goodness-of-fit. These 5 CCF locations are then used to decorrelate against in the global fit performed using EXOFASTv2 . We visualize the CCFs for K2-2 and the specific 5 CCF locations in Figure 3, where we observe a clear pattern in the stellar variability and the CCF shape changes. This pattern allows us to use CALM to probe and predict stellar activity contributions to the RVs. In Figure 4, we plot the CALM model predicted stellar activity contributions to the RVs both in time and in the fourier domain. These activity indexes are able to probe both short- and long-term activity signals while preserving the planetary reflex motion. The ∼ 270 day signal that is predicted by the CCF4 parameter was also \nFigure 2. Archival HARPS-N radial velocities for K2-2 from Vanderburg et al. (2015) and Bonomo et al. (2023). The left panel shows the phased-folded RVs, and the right panel shows the long-term trend in the unphased RVs. \n<!-- image --> \nFigure 3. Residual CCFs ( ∆ CCFs) computed from HARPS-N spectra. The residual CCFs are computed by subtracting a median CCF. The CALM model-predicted stellar activity signal is indicated by the color (red = redshifted RVs, blue = blue-shifted RVs). The 5 CCF indexes used in our stellar activity model are indicated by black lines. \n<!-- image --> \nfound by Bonomo et al. (2023) and they noted that this signal is also seen in the periodograms of s-index and FWHM. This suggests that this signal corresponds to stellar variability and may be on a timescale longer than the stellar rotation period for K2-2.", '3. GLOBAL FITS': 'Following the method described in Thygesen et al. (2023), we used the differential evolution Markov Chain Monte Carlo (DE-MCMC) exoplanet fitting software EXOFASTv2 (Eastman et al. 2013, 2019) to simultaneously fit the parameters of K2-2 b and its host star. For a global fit to be accepted as converged, we required that the Gelmin-Rubin statistic be less that 1.01 and the number of independent draws, T z , greater than 1000. The global fits use MCMC sampling to find the best fit parameters for the system based on the photometric and spectroscopic data. We placed priors on several \nparameters as follows: a uniform prior from 0 to an upper bound of 0.09858 on the line-of-sight extinction ( A v ) from Schlegel et al. (1998) and Schlafly & Finkbeiner (2011); a Gaussian prior on parallax of 16.0044 ± 0.0456 from Gaia Early Data Release 3 (accounting for the small systematic offset; EDR3; Gaia Collaboration et al. 2016, 2021; Lindegren et al. 2021); and a Gaussian prior on metallicity ([Fe/H]) of -0.125 ± 0.08 based on measurements from TRES (see Section 2.7). The fit also included the spectral energy distribution (SED) photometry as reported by Gaia EDR3 (Gaia Collaboration et al. 2021), WISE (Cutri et al. 2012) and 2MASS (Cutri et al. 2003) (see Table 1). To better characerize the host star, the MESA Isochrones and Stellar Tracks (MIST) stellar evolution models (Paxton et al. 2011, 2013, 2015; Choi et al. 2016; Dotter 2016) were used within the EXOFASTv2 fits. Within EXOFASTv2 , limb darkening is constrained via priors derived from models by Claret & Bloemen (2011) and Claret (2017), with physical bounds from Kipping (2013) (see Section 3 of Eastman et al. (2019) for more details on how EXOFASTv2 constrains limb darkening). \nAlthough the TESS PDCSAP light curves generally have a correction applied for any contaminating sources, we fitted for a dilution term in case of any sources that may have been missed, based on the contamination ratio (CR) for K2-2 of 0.002101 as reported in the TESS input catalog (TICv8, Stassun et al. 2018). We used placed a 10% Gaussian prior on the dilution centered about CR/(1+CR) = 0.0021. However, the fitted dilution was consistent with zero in all the fits we ran. \nTo account for any residual correlated noise in the systematics-corrected Spitzer data within the EXOFASTv2 fit (see Section 2.5), we followed the procedure outlined in §3 of Rodriguez et al. (2020). We scaled the uncertainties by the factor β = 1.19 before using the light curve in the \nFigure 4. Timeseries and periodograms of the CALM predicted stellar variability. In the left panels, the DRS pipeline radial velocities and the stellar variability predictions from CCF index 1, 2, 3, 4, and 5 are plotted as a function of time. The location of these CCF indexes are indicated in Figure 3. On the right panel, the Lomb-Scargle periodograms of the corresponding RV timeseries are plotted. In yellow, the Keplerian period of K2-2 is indicated in the periodograms. We do not see signals at this planetary period, which provides reassurance that CALM is not absorbing or creating planetary signals. \n<!-- image --> \nglobal fit. To ensure EXOFASTv2 did not reduce the perpoint uncertainties on the Spitzer photometry within the fit, we enforced a lower bound on the variance of zero, otherwise the global fit could over-correct the scaled uncertainties to be consistent with pure white noise.', '3.1. RV model selection': 'As the RVs still exhibited an offset in the second observing block after all processing (see Section 2.7), we compared five different models that attempt to model this long-term change and evaluated their goodness-of-fit with EXOFASTv2 , while keeping all other inputs and priors the same. For each of \nthese models, we first performed a fit using CALM since these long-term trends could be caused by stellar variability. We then took the initial CALM fit to the RVs for each model and ran a global fit with EXOFASTv2 . The five models are listed in Table 3 and each include the CALM model, but differ in their modeling of the long-term trends where they include some combination of a linear ( ˙ γ ) trend with time, a quadratic ( ¨ γ ) trend with time, and/or an offset D between the two observing blocks. In particular, our models include (i) a CALMmodel with a linear and quadratic trend with time that treats the RV timeseries as one RV observing season without an offsets between the two observing blocks, (ii) a CALM \nTable 3. Models tested for long-term RV trend. \nFigure 5. Projected difference in the time of transit for K2-2 b to the year 2030 using the original ephemeris (gray) and the new ephemeris from this work (purple). Shaded regions indicating up to the 3 σ level uncertainty are shown. The inset shows the updated ephemeris, zoomed in for clarity. \n<!-- image --> \nand linear trend model that treats the RV timeseries as one RV observing season without an offset, (iii) a CALM model with an offset D between the two observing blocks, (iv) a CALM model with a linear trend and an offset D , and (v) a CALM model with a linear and quadratic trend and an offset D . For the models where we treated the two observing blocks as separate seasons, this allows for different zero-points to be determined for each season. Comparing the Bayesian Information Criterion (BIC) of the models, we found that those including an offset component (i.e. two observing seasons) are heavily disfavored as seen in Table 3. The single-season models perform comparabley and we adopt the quadratic-trend model as it has the lowest BIC.', '4. RESULTS AND DISCUSSION': "In this work, we have combined multiple new observations with existing data available for K2-2 b to produce the most accurate and precise system parameters and transit ephemeris (transit time uncertainty < 13 minutes in 2030). The period of K2-2 b has been updated to 9.1004157 + 4.1 E -06 -4.5 E -05 days and T 0 to 2458072.29291 + 0.00062 -0.00061 BJD (Figure 5). The solutions for the stellar and planetary parameters are shown in Tables 4 and 5, respectively. Table 6 contains the radial velocity parameters, including the detrending parameters we used, and Table 7 lists the parameters of the photometric models for each light \nTable 4. Median values and 68% confidence interval for K2-2 stellar parameters from the EXOFASTv2 global fit. \nNotes. See Table 3 in Eastman et al. (2019) for a detailed description of all parameters. Gaussian and uniform priors are indicated as G [ mean, σ ] and U [ lower bound,upper bound ] , respectively. The metallicity prior is adopted from the average of three TRES measurements: [M/H] = -0.193, -0.191, 0.009 (see Section 2.7 for details). 1 The metallicity of the star at birth. 2 Corresponds to static points in a star's evolutionary history. See §2 in Dotter (2016). \ncurve. We included the MOST light curve in a preliminary fit, as the transit window was observed four times in the full light curve. However, this did not add value to the fit, and the transit was not detectable even with the updated ephemeris, so we did not include the MOST data in the final global fit. The discovery period (Vanderburg et al. 2015) we determined to be 28.8 minutes ( ∼ 40 σ ) from the true period. For context, if someone attempted an observation in 2025 of a K2-2 b transit using the original ephemeris, it would be ∼ 200 hours from the correct time. We note that this would only result in an offset of ∼ 18 hours from a transit of K2-2 b since the offset would be quite close to the orbital period of the planet by then, resulting in catching the next adjacent transit. \nK2-2 b has a radius of 2.47 + 0.10 -0.09 R ⊕ and a mass of 9.7 ± 1.2 M ⊕ . This yields a bulk density of 3.53 + 0.63 -0.57 g cm -3 , which is twice that of Neptune (1.638 g cm -3 ). According to the composition models from Zeng et al. (2016), it is likely K2-2 b has a high water content (Figure 6). While it is consistent with 100% water, a more physically motivated solution would be a rocky core with an extended envelope of volatiles", 'THYGESEN ET AL.': 'Table 5. Median values and 68% confidence interval for K2-2 b planetary parameters from the EXOFASTv2 global fit. \nNotes. See Table 3 in Eastman et al. (2019) for a detailed description of all parameters. \n1 Optimal time of conjunction minimizes the covariance between TC and Period. 2 Note that due to the low significance of the eccentricity, this is consistent with e = 0 when considering the Lucy-Sweeney bias (Lucy & Sweeney 1971). 3 Assumes no albedo and perfect redistribution. 4,5 Within the fits, these are parameterized as √ e cos ω ∗ and √ e sin ω ∗ , respectively, to ensure a uniform prior on eccentricity. \nTable 6. Median values and 68% confidence interval for radial velocity parameters. \nNotes. Reference epoch = 2458561.069744 BJD. Five additive detrending parameters were included to account for stellar activity (see Section 3). \nincluding a H/He envelope. More observations are needed to place further constraints on the planetary composition. \nThe mass of K2-2 b was updated in a recent in-depth radial velocity study of Kepler and K2 systems (Bonomo et al. 2023) to refine planet masses and identify cold Jupiters in systems containing small planets. Bonomo et al. (2023) refined important planetary parameters such as the period (to 9.0949 ± 0.0026 days) and mass (to 10.1 + 1.2 -1.1 M ⊕ ), and did not find any long-term trends in the RVs that could correspond to a long-period companion. We used the same RV observations from this work (in addition to those from Vanderburg et al. 2015) but with improved precision from improved modeling of the stellar activity using the CALM technique (see Section 2.7.2) in our global fit, and when combined with the other photometric and spectroscopic data, we were able to refine these measurements and uncover a potential outer companion due to a long-term trend in the RVs.', '4.1. RV trend': 'As mentioned in Section 2.7, there is a long-term trend in the radial velocities (see Figure 2) after correcting for stellar variability. To test the possibility of a second planet or star within the system, we reran the fit described in Section 3 but allowed EXOFASTv2 to fit for a second planet within the RVs only. We note that there is no additional transit signal detected in any photometric data sets used in this analysis. However, preliminary fits did not converge nor provide any useful constraint on the period of a potential companion, even with improved constraints on K2-2 b. Figure 2 shows the long-term trend in the RVs, and our resulting best-fit model from EXOFASTv2 . It is clear that the period \nFigure 6. Mass-radius diagram for K2 sub-Neptunes ( R P = 2.0 -3.0 R ⊕ ). The large black circle is K2-2 b, while the small gray circles are other sub-Neptunes with measured masses from the NASA Exoplanet Archive. The lines represent composition tracks from Zeng et al. (2016). \n<!-- image --> \nof this secondary companion is much longer term than the extent of our RV data set from HARPS-N ( ∼ 2500 days). We instead model the long-term trend with a quadratic acceleration term. Our best-fit results find a linear slope in the RVs of 0.0024 ± 0.0004 m s -1 with a quadratic term of 1.33 E -06 ± 3.6 E -07 m s -1 day -2 to best represent the longterm RV trend. \nThe observed RV trend may correspond to an additional companion to K2-2 with an orbital separation of several AU. Vanderburg et al. (2015) acquired high-resolution imaging observations of the star and did not detect any stellar companions between 0.1 -5.0" ( ≈ 6 -310 AU). This non-detection, combined with the relatively small amplitude of the RV acceleration, suggests that this outer companion could be a planet or a brown dwarf. \nAs K2-2 was observed by Hipparcos , it is possible to place additional constraints on any outer companions using Hipparcos-Gaia astrometry (Brandt 2018, 2021). If a massive companion exists at a separation of several AU from K2-2, it would likely generate a significant astrometric acceleration between Hipparcos and Gaia . However, no significant acceleration is detected in the Hipparcos-Gaia as- \nTable 7. Median values and 68% confidence intervals for the photometric models. \nNotes. † Linear limb-darkening coefficient. ‡ Quadratic limb-darkening coefficient. ⋆ Added variance. ∗ Baseline flux. \ntrometry, with χ 2 = 2.3 for a constant proper motion (Brandt 2021). The astrometric precision for K2-2 is ∼ 0.07 mas yr -1 , equivalent to ∼ 20 m s -1 at the 62.48 ± 0.18 pc distance of the system. This means that a net Hipparcos-Gaia velocity change greater than ≳ 100 m s -1 can be excluded at 5 σ confidence. This non-detection largely excludes the existence of massive companions ( ≳ 10 M J ) orbiting K2-2 within several AU. However, a planetary-mass companion could be reconciled with the astrometric non-detection. \nContinued RV monitoring of the K2-2 system is needed to constrain the further evolution of the RV trend, providing some constraints on the fundamental parameters of the possible second planet in the system.', '4.2. Future work': 'The K2 mission was driven by the community, which led to planets orbiting much brighter host stars than the original Kepler mission, targets well suited for detailed characterization. Although characterization might be challenging with current facilities, K2-2 b is a worthwhile target for ongoing monitoring and targeted observations. Following the Kempton et al. (2018) prescription for the transmission spectroscopy metric (TSM), we find that K2-2 b has a TSM of 50.0 9.2 8.7 , which falls just below the lowest value suggested for target prioritization for JWST. However, when compared to the other ∼ 160 sub-Neptunes ( R P = 2.0 -3.0 R ⊕ ) in the K2 catalog, the TSM of K2-2 b is the fifth highest, suggesting that it is a suitable candidate for studying sub-Neptunes in closer detail. Monitoring the radial velocities of K2-2 would allow for more refined constraints on the stellar activity, and possibly uncover additional long-period and/or low-mass candidates in the system. \nThe co-moving white dwarf (WD) companion to K2-2 provides an avenue to measure a precise age for the system if the mass and age for the WD can be determined. The stellar parameters were calculated as part of a catalog of all WDs within Gaia EDR3 3 by Gentile Fusillo et al. (2021). The mass, effective temperature, and surface gravity were determined for three different atmospheric compositions: pure H, pure He, and a mix of H and He (see Table 8). Assuming the highest mass value from the models (pure-H, 0.52 ± 0.04 M ⊙ ), we find a lower limit on the cooling age of 1.13 ± 0.13 Gyr. While this current age estimate does not constrain the system age further, more precise photometry and measuring the spectrum of the WD would constrain the mass (and system age) more reliably than Gaia photometry alone. \nTable 8. Stellar parameters for the white dwarf companion of K2-2 from Gentile Fusillo et al. (2021).', '5. CONCLUSION': 'With thousands of exoplanets discovered to date, some will inevitably be \'lost" (unconstrained ephemerides) or forgotten as newer discoveries peak the interest of the community. Unfortunately, these lost planets may be excellent targets for detailed characterization with JWST (Gardner et al. 2006), but are not accessible due to large uncertainties in future transit times. K2-2 b was the first planet discovered during the TwoWheeled Concept Engineering Test of the K2 mission (Howell et al. 2014), showing very quickly that K2 would be a successful repurposing of the Kepler spacecraft. By combining observations from multiple NASA missions along with key ground-based follow up that span nearly a decade, we have recovered the lost transit ephemeris of K2-2 b. In addition to being the first K2 planet, it is also well-suited for studying the atmosphere of a hot sub-Neptune as it orbits a bright ( K ∼ 8.03) K-dwarf. This would be a valuable measurement since it sits on the high-mass peak of the sub-Neptune radius valley (Owen & Jackson 2012; Fulton et al. 2017) and could provide insight to the formation and evolution of subNeptunes. Our updated ephemeris ( P = 9.1004157 + 4.1 E -06 -4.5 E -06 days, T 0 = 2458072.29291 + 0.00062 -0.00061 BJD) confirms the false detection from the MOST satellite (Vanderburg et al. 2015) that led to a ∼ 40 σ offset to the true period. Systems like K2-2 show the importance of continued monitoring of exoplanet systems and dedicated ephemeris refinement efforts like the K2 & TESS Synergy project (Ikwut-Ukwa et al. 2020; Thygesen et al. 2023), ExoClock (Kokori et al. 2021, 2022, 2023), Exoplanet Watch (Zellem et al. 2019, 2020), and ORBYTS (Edwards et al. 2019, 2020, 2021). \nSoftware: Lightkurve (Lightkurve Collaboration et al. 2018), EXOFASTv2 (Eastman et al. 2013, 2019), AstroImageJ (Collins et al. 2017) \nFacilities: TESS, K2, Spitzer, MEarth, University of Louisville Manner Telescope (ULMT), Telescopio Nazionale Gailieo 3.58 m (HARPS-N), FLWO 1.5m (Tillinghast Reflector Echelle Spectrograph; TRES), Gaia, MAST.', 'ACKNOWLEDGMENTS': "We thank the referee for the feedback that greatly improved the manuscript. ET and JER acknowledge support for this project from NASA'S TESS Guest Investigator program (G04205, P.I. Rodriguez). We thank Jason Eastman for the lengthy discussions on the inner workings of EXOFASTv2 . ET would like to thank the Quad Fellowship for support. Z.L.D. would like to thank the generous support of the MIT Presidential Fellowship and to acknowledge that this material is based upon work supported by the National Science Foundation Graduate Research Fellowship under grant No. 1745302. Z.L.D would like to acknowlegde the MIT Collamore-Rogers Fellowship. \nThe MEarth Team gratefully acknowledges funding from the David and Lucile Packard Fellowship for Science and Engineering (awarded to D.C.). This material is based upon work supported by the National Science Foundation under grants AST-0807690, AST-1109468, AST-1004488 (Alan T. Waterman Award), and AST-1616624, and upon work supported by the National Aeronautics and Space Administration under Grant No. 80NSSC18K0476 issued through the XRP Program. This work is made possible by a grant from them John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. \nThis research has made use of SAO/NASA's Astrophysics Data System Bibliographic Services. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www. cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa. int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This work makes use of observations from the LCO network. This work is based [in part] on observations made with the Spitzer Space Telescope, which was operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This research made use of Lightkurve, a Python package for Kepler and TESS data analysis. The data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. Data from TESS Sectors 42 and 70 can be accessed at doi:10.17909/t9-nmc8f686 and K2 Campaign 0 at doi:10.17909/T9F88F. Some data in this work were accessed at ExoFOP, accessible at doi:10.26134/ExoFOP3. The Spitzer data used in this work can be found at doi:10.26131/IRSA430. \nFunding for the TESS mission is provided by NASA's Science Mission directorate. We acknowledge the use of public TESS Alert data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products."}
2024arXiv240908074B	Understanding the interaction between hydrogen cyanide HCN and silicate surfaces is crucial for elucidating the prebiotic processes occurring on interstellar grain cores as well as in cometary and meteoritic matrices. In this study we characterized the adsorption features of HCN on crystalline forsterite Mg2SiO4 surfaces one of the most abundant cosmic silicates by combining experimental infrared spectra at low temperatures 100150 K with periodic DFT simulations. Results showed the coexistence of both molecular and dissociative HCN adsorption complexes as a function of the considered forsterite crystalline face. Molecular adsorptions dominate on the most stable surfaces while dissociative adsorptions occur predominantly on surfaces of lower stability catalyzed by the enhanced Lewis acidbase behavior of surfaceexposed Mg2O2 ion pairs. On the whole set of adsorption cases harmonic frequency calculations were carried out and compared with the experimental infrared bands. To disentangle each vibrational mode contributing to the experimental broad bands we run a best nonlinear fit between the predicted set of frequencies and the experimental bands. The outcome of this procedure allowed us to i deconvolute the experimental IR spectrum by assigning computed normal modes of vibrations to the main features of each band ii reveal which crystal faces are responsible of the largest contribution to the adsorbate vibrational bands giving information about the morphology of the samples. The present straigthforward procedure is quite general and of broad interest in the fine characterization of the infrared spectra of adsorbates on complex inorganic material surfaces.	2024-09-01T00:00:00Z	['2024arXiv240908074B', 'arXiv:2409.08074', '10.48550/arXiv.2409.08074']	['Condensed Matter - Materials Science', 'Astrophysics - Astrophysics of Galaxies']	Unraveling the Interface Chemistry between HCN and Cosmic Silicates by the Interplay of Infrared Spectroscopy and Quantum Chemical Modeling	2,024	197	0.26	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.08074.pdf	{'Unraveling the interface chemistry between HCN and cosmic silicates by the interplay of infrared spectroscopy and quantum chemical modeling': 'Niccolò Bancone , † , ‡ Rosangela Santalucia , ‡ Stefano Pantaleone , ‡ Piero Ugliengo , ‡ Lorenzo Mino , ‡ Albert Rimola , ∗ , † and Marta Corno ∗ , ‡ \n† Departament de Química, Universitat Autònoma de Barcelona, Bellaterra, 08193, Catalonia, Spain \n‡ Dipartimento di Chimica and Nanostructured Interfaces and Surfaces (NIS) Centre, Università degli Studi di Torino, via P. Giuria 7, 10125, Torino, Italy. \n```\nE-mail: [email protected]; [email protected] Phone: +34-935813723; +39-0116702439\n```', 'Abstract': 'Understanding the interaction between hydrogen cyanide (HCN) and silicate surfaces is crucial for elucidating the prebiotic processes occurring on interstellar grain cores, as well as in cometary and meteoritic matrices. In this study, we characterized the adsorption features of HCN on crystalline forsterite (Mg 2 SiO 4 ) surfaces, one of the most abundant cosmic silicates, by combining experimental infrared spectra at low temperatures (100-150 K) with periodic DFT simulations. Results showed the coexistence of both molecular and dissociative HCN adsorption complexes as a function of the considered forsterite crystalline face. Molecular adsorptions dominate on the \nmost stable surfaces, while dissociative adsorptions occur predominantly on surfaces of lower stability, catalyzed by the enhanced Lewis acid-base behavior of surface-exposed Mg 2+ -O 2 -ion pairs. On the whole set of adsorption cases, harmonic frequency calculations were carried out and compared with the experimental infrared bands. To disentangle each vibrational mode contributing to the experimental broad bands, we run a best non-linear fit between the predicted set of frequencies and the experimental bands. The outcome of this procedure allowed us to: i) deconvolute the experimental IR spectrum by assigning computed normal modes of vibrations to the main features of each band; ii) reveal which crystal faces are responsible of the largest contribution to the adsorbate vibrational bands, giving information about the morphology of the samples. The present straigthforward procedure is quite general and of broad interest in the fine characterization of the infrared spectra of adsorbates on complex inorganic material surfaces.', 'Introduction': "Hydrogen cyanide (HCN) is recognized as an ubiquitous molecule in various astrophysical environments, ranging from diffuse and dense clouds, 1,2 protostellar hot cores, 3Ð5 cometary comas 6Ð8 and meteorites, 9 to planetary atmospheres, 10,11 including Pluto and Saturn's moon Titan. 12Ð14 \nSeveral studies have highlighted the importance of HCN in the synthesis of organic compounds relevant to the emergence of life, prompting experimental investigations into its reactivity under different conditions mimicking both interstellar and prebiotic environments, 15Ð19 which more recently have been extended to a theoretical side. 20Ð24 Within this context, the significance of HCN lays in its ability to polymerize, leading to the formation of complex organic molecules, among which the adenine nucleobase, a fundamental biomolecular building block for life. 25 The gas-phase polymerization involving only HCN has been reported to be hindered by high kinetic barriers, hard to overcome at the interstellar conditions. 26 \nExplorations on HCN and HNC reactive pathways have demonstrated that the introduction of basic or acidic catalysts, and/or other molecules, leads to more feasible mechanisms. 24,27 In liquid-phase, pathways related to HCN polymerization seem to be more promising, 23 due to the presence of basic catalysts. 15,20 Among the few studies on the chemistry of HCN in the solid phase, water ices have been investigated as possible concentrator 28,29 of gaseous HCN and as catalysts for its dimerization. 30 On the other hand, the effectiveness of surface siloxyl radicals (SiO · ) as initiators for the polycyclization of HCN has been theoretically described, 31 thus introducing the potential role of silicate materials as heterogeneous catalysts for HCN prebiotic reactions. \nSilicates represent the most abundant minerals in the Universe as they are the major refractory material in interstellar grains, cometary nuclei, meteoritic chondrites and rocky planets; as an emblematic example, they represent the most abundant and important inorganic material in Earth's crust. Silicates, moreover, possess well-established catalytic properties towards chemical reactions of interstellar 32,33 and prebiotic interest. 34 Therefore, investigating the activation of HCN upon silicate adsorption and the subsequent catalytic reactivity (particularly in the absence of external energy inputs) is of interest in the astrochemistry, cosmochemistry and prebiotic chemistry fields. \nIn a previous work by some of us, spectroscopic experimental measurements of HCN in interaction with synthetic Mg-rich silicate surfaces (both amorphous and crystalline) were performed, with the aim of investigating their catalytic properties in the HCN polymerization. 36 The chosen temperature range (i.e., 100-300 K) aligned well with the conditions found in common prebiotic environments, including interplanetary regions, meteorites, and comets. The study employed innovative methods for the safe production and controlled dosing of pure HCN onto solid samples, followed by Fourier-transformed infrared (FT-IR) spectroscopy and high-resolution mass spectrometry (HR-MS) analysis to characterize the reaction products. For what concerns the adsorption of gaseous HCN, results showed that above 150 K the HCN/silicates complexes undergo an intricate chemistry mainly due to the \nFigure 1: Representations of the optimized geometries (focusing on the binding region). All distances are in Å. Adapted with permission from 35. Copyright 2023 Royal Society of Chemistry. \n<!-- image --> \npresence of surface-exposed Mg 2+ -O 2 -ion pairs, as they present an enhanced Lewis acidbase behavior 37 that enable the deprotonation of adsorbed HCN, which in turn triggers their polymerization. However, the first steps of the HCN adsorption process were not investigated, and accordingly the study lacked a detailed and a thorough atomistic description of the molecule-surface interactions. \nDue to that, a recent computational work 35 on the HCN adsorbed on Mg 2 SiO 4 (forsterite) surfaces was published, being dedicated to characterize the structures and energetics of all possible HCN adsorption complexes (see Figure 1), while vibrational frequencies were only marginally discussed. Indeed, the lack of an experimental counterpart made impossible the assignment of the contribution of each adsorption mode. At the same time, the experimental spectra were rather intricate, with probably overlapping bands difficult to be assigned. \nThe present work aims at overcoming the above-mentioned drawbacks, by combining in \na more thorough way the experimental and theoretically predicted infrared spectra involving the adsorbate modes. The main outcome allows a fine interpretation of the experimental bands, providing a band deconvolution as least biased as possible, and also some hints about the crystal morphology of the forsterite samples and its affectation in the HCN adsorption.", 'Materials': 'Crystalline forsterite was obtained by heating the pristine amorphous magnesium silicate at 1073 K for 24 h in air. The specific surface area was of ≈ 26 m 2 /g. Before HCN dosage, the samples were outgassed at 673 K for 1 hour under high vacuum (residual pressure 5 × 10 -4 mbar) to obtain a dehydrated/dehydroxylated surface. More information about the properties and the synthesis of the two materials are available in ref. 37. The HCN used in the experiment was synthesized using a safe and efficient procedure developed by our group, which is detailed in the supporting information of ref. 36.', 'Experimental methodology': 'For the transmission IR spectroscopic measurements, the powder sample was pressed in the form of self-supporting pellet and placed into a quartz cell equipped with IR-transparent KBr windows and a valve for connection to a vacuum line (residual pressure 5 × 10 -4 mbar). This setup allows in situ adsorption/desorption experiments to be conducted under controlled atmosphere. Additionally, the cell is designed for low-temperature experiments (approximately 100 K) by cooling the sample with a reservoir of liquid nitrogen. In the experiment, HCN (5 mbar equilibrium pressure) was dosed into the cooled cell. Under these conditions, the gas condensed on the walls of the cell. Subsequently, the system temperature was allowed to increase freely, causing HCN to be slowly released and reach the sample as its temperature approached ≈ 150 K. The initial spectra sequence, discussed in this study, can be essentially \ndescribed as the dosing of increasing amounts of HCN at a constant temperature (150 K). The HCN adsorption was monitored in situ by continuously acquiring IR spectra using a Bruker EQUINOX 55 FTIR spectrometer equipped with a DTGS detector. Typically, 128 interferograms (at a resolution of 2 cm -1 ) were acquired for each spectrum to ensure a good signal-to-noise ratio. Further details are reported in ref. 36.', 'Computational methodology': 'All geometry optimizations and vibrational frequency computations were performed with the CRYSTAL17 periodic quantum mechanical code. 38 The DFT-PBE level of theory was used, complemented with the revised version of the Grimme\'s D2 empirical term (namely, the DN correction, specifically customized for periodic systems containing Mg 2+ cations) 39Ð41 to account for dispersion interactions. The Ahlrichs-VTZ 42 basis set augmented with polarization functions for describing the atoms of HCN and the smaller basis set proposed by Bruno et al. 43 for forsterite atoms (8-511G For Mg, 8-6311G* for Si, and 8-411G* for O) were used. The same methodology as that of ref. 35 was used, so that we strongly refer to that work for further computational details. \nVibrational frequencies were computed numerically within the harmonic approximation and at the Γ point, thus including vibrations in the unit cell, only. Each Hessian matrix element was calculated through the second derivatives of the potential energy of the stationary points by displacing each atom from its equilibrium geometry along each cartesian coordinate by ± 0 . 003 Å along each cartesian axis (i.e., central difference formula). To speed up the calculations, only the HCN atoms and those belonging to the first layer of the surface models were allowed to displace. \nComputed vibrational frequencies suffer from a systematic error intrinsic to the DFT method and to the harmonic approximation. As the main purpose of this work is to compare experimental spectra with the computed ones, each HCN computed frequency was corrected by two scaling factors s , one for the C-H and another for the C ≡ N stretching vibrations, by \ncomparing the gas-phase HCN simulated and experimental frequencies. This allowed us to rescale the whole gas phase HCN spectrum by considering that the systematic errors are normal mode dependent, allowing a more sensible and direct comparison with the experimental spectra. Therefore, the following scaling factors were obtained: \nS !" = 𝜈 #$% 𝜈 &\'(% = 3311 3370 = 0.9825 \nS !) = 𝜈 #$% 𝜈 &\'(% = 2097 2127 = 0.9859 \nAdditionally, since in some adsorption complexes HCN dissociation takes place, silanol (SiOH) surface groups are generated. For the OH stretching vibration of these surface SiOH groups, the scaling factor was obtained as the ratio between the experimental frequency of surface isolated silanols and the corresponding computed ones for those surfaces models exhibiting isolated SiOH groups, that is: \nS " = 𝜈 #$% 𝜈 &\'(% = 3744 3753 = 0.9976', 'Simulation of a global IR spectrum': 'i In our previous computational work, 35 we identified 16 adsorption complexes considering 6 different crystalline forsterite surfaces. Thus, in order to simulate a single global spectrum taking into account the contribution of each adsorption complex j , we first associated each computed frequency ν i with a Lorentzian function L j ( ν ) , defined as: \nL + , (𝜈) = I + , Γ + , [𝜋(𝜈-𝜈 + ) -+(Γ + , 2 / ) -] : \ni i where I j is the PBE-DN computed infrared intensity of the stretching mode corresponding to the ν i frequency and Γ j is the full width at half maximum (FWHM) of the peak. \nThe experimental IR signals undergo a broadening in the 2500-3700 cm -1 zone, due \nFigure 2: Example of the effect of the Huggins-Pimentel correction 44 on the band-width of the computed O-H stretching bands. Top: simulated bands with a fixed FWHM = 15 cm -1 . For each signal, the corresponding frequency in cm -1 is also reported. Bottom: peaks computed with a FWHM given by the Huggins-Pimentel formula, as adopted in this work. For each signal, the corresponding FWHM in cm -1 is also reported. \n<!-- image --> \ni to the H-bond interactions occurring between the exposed silanols and either the surfaceexposed O 2 -anions or the CN -adsorbates. In order to better reproduce this broadening of the peaks, we computed Γ j for the O-H groups through the formula proposed by Huggins and Pimentel for the correction of the bandwidth as a function of the OH frequency shift with respect to the free SiOH group 44 (see Figure 2): \ni In order to better reproduce this broadening of the peaks, we computed Γ j for the O-H groups through the empirical formula, parametrized on OH and NH H-bonded systems, proposed by Huggins and Pimentel for the correction of the bandwidth as a function of the OH frequency shift with respect to the free SiOH group 44 (see Figure 2): \ni Γ j = 0 . 72 ∆ i + 2 . 5 cm -1 \nwhere ∆ i represents the bathochromic shift of the stretching frequency of the i -silanol with \ni respect to a free silanol, so that ∆ i = 3744 cm -1 -ν i . For the peaks corresponding to the C-H and C ≡ N stretchings, we instead adopted a fixed Γ j = 15 cm -1 . Accordingly, we computed the IR spectrum of each single j -adsorption complex as a sum of the Lorentzian functions associated with its computed signals: \nS , (𝜈) = ;L + , + (𝜈) \nand generate the total spectrum T ( ν ) through a linear combination of each S j ( ν ) : \nT(𝜈) = ;c , S , , (𝜈) \nwhere c j are the weight coefficients of each spectrum, normalized for the case with the best match between experiment and theory. \nWhen HCN deprotonates upon adsorption, the proton is transferred to the surface forming a SiOH group. The remaining CN -anion interacts with multiple surface-exposed Mg 2+ cations, forming different CN --Mg 2+ interactions of variable strength. 35 Simulations show that the freshly formed SiOH can engage H-bond interactions either with the CN -or nearby surface-exposed O 2 -anions, thus experiencing different degrees of perturbation. 35,36 For these reasons, in this work, we treated the C ≡ N and O-H stretching vibrations separately. That is, we defined two independent S j ( ν ) spectra for C ≡ N and O-H. \nBy proceeding this way, we generated a global spectrum T ( ν ) by linearly combining 23 single S j ( ν ) spectra, which include: 8 spectra given by the sum of C-H and C ≡ N signals (molecular adsorptions), 8 spectra given by the C ≡ N signals alone (dissociative adsorptions) and 7 spectra by the O-H signals (dissociative adsorptions); the remaining one represents the free silanol used to compute the OH stretching scale factor. To match the experimental spectrum, we performed a non-linear best fit between T ( ν ) and the experimental one in the 3700-1300 cm -1 range, thus excluding any contribution given by the vibrations of the surface. The best fit is performed by changing the relative weights c j of each spectrum \nS j . Therefore, only the relative intensities are changed, while keeping the frequencies of the bands fixed at the scaled computed values. From now on, we will refer to the fitted simulated spectrum as T fit ( ν ) . Therefore, the value of each S j ( ν ) contribution in T fit ( ν ) weighs the relevance of each type of HCN/forsterite adsorption complex contributing to the description of the experimental spectrum. In addition, this approach allows us to gauge the contribution of each forsterite crystal face in contributing to the final spectrum, disentangling the experimental bands, as discussed in the next section.', 'Results and Discussion': "Figure 3: Experimental IR spectra of HCN gas adsorbed at 150 K on crystalline Mg-silicate (forsterite) previously outgassed at 673 K. The series shows the effect of gradually increasing the HCN equilibrium up to 5 mbar (from black to red spectrum). The spectrum of the bare activated material has been subtracted from all spectra. Negative peaks represent species that are consumed after the interaction between HCN and the solid, while positive peaks represent species that are formed. \n<!-- image --> \nThe experimental IR spectra obtained at 150 K at increasing HCN equilibrium pressure (from black to red) on dehydrated forsterite are reported in Figure 3, while Figure 4 shows the global computed IR spectrum T fit ( ν ) superimposed to the low-coverage experimental \nFigure 4: Experimental (blue) and fitted computed (red) IR spectra. The letters correspond to the adsorption complexes which mainly contribute to the experimental spectrum. In the assignments, the corresponding stretching mode, namely HC, CN and OH (silanol), is reported in superscript. Atom color legend: H white, C ochre, N blue, O red, Mg cyan, Si yellow. \n<!-- image --> \none (blue line of Figures 3 and 4), as it better represents the isolated adsorptions modelled in the simulations. Only 14 out of the 23 starting S j ( ν ) spectra contribute to T fit ( ν ) , of which here we report the best 10 cases in Figure 4 and in Table 1. In Table S1 all the adsorption modes are reported with the corresponding vibrational frequencies and weight coefficients to the fitted spectrum. \nUpon dosing HCN, there is a noticeable progressive reduction in the intensity of the band at ca. 3744 (3753 computed) cm -1 which represents the isolated silanols, accompanied by the simultaneous growth of two broad absorption bands at ca. 3500 cm -1 and 3200 cm -1 . \nThis transformation is marked by the appearance of a distinct isosbestic point at ca. 3690 cm -1 . Previous studies concerning forsterite characterization and the interaction between HCN gas and SiO 2 36 allow for an interpretation of these features, suggesting the formation of SiOH · · · NCH adducts involving the silanol groups present in the non-completely crystallized portions of the forsterite sample. 37 In our simulations we do not have this specific interaction, as the formation of silanol groups is associated with the HCN deprotonation. Nevertheless, we modelled different degrees of perturbation of the SiOH, thus correctly reproducing the experimental bands in the OH region. \nMoreover, according to our simulations, the band at 3200-3300 cm -1 is ascribed to ν (CH) modes (in addition to highly red-shifted ν (O-H)) coupled with the band at 2095 cm -1 of the ν (C ≡ N). This assignments fit well with the computed cases a and b (c i = 0.5408 and 0.2173) which belong to the second most stable surface (120) and represent a slightly perturbed molecular HCN, ( ν (C-H) = 3305-3276 cm -1 , ν (C ≡ N) = 2099 cm -1 ). The c case on the (010) surface ( ν (C-H) = 3195 cm -1 , ν (C ≡ N) = 2064 cm -1 , c i = 0.1688) undergoes a larger bathochromic shift of both features with respect to a and b , because of the stronger interaction of the N atom with a Mg 2+ cation and the formation of a H-bond with a silicate group Mg 2+ · · · NCH · · · OSiO 3 . This case well explains the C ≡ N feature below 2095 cm -1 while, for the C-H, the presence of a shoulder below 3000 cm -1 suggests that many structures similar to c could contribute to the observation i.e. differently perturbed C-H by H-bond donation to the surfaces. The above mentioned cases are those contributing to the most intense bands of the experimental spectrum and, unsurprisingly, they represent the most stable surfaces of forsterite, i.e. those having the largest surface areas in crystalline nanoparticles (see Figure 5). This means that our simulated IR features of the adsorption complexes, joint which the calculation of surface energies of the bare material, can provide important hints on the morphology obtained during the crystallization of forsterite. \nThe broad band at 3500 cm -1 can be ascribed to interacting silanols, originated from the deprotonation of HCN. This can happen on defective and/or reactive sites, such as surface- \nexposed Mg 2+ -O 2 -ion pairs. From the computational standpoint, this situation can be somehow simulated by high energy forsterite surfaces, i.e. (111) and (021), where indeed the HCN deprotonation easily occurs, generating cases g and h (c i = 0.7301 and 0.2083). \nHowever, we cannot exclude that strong interactions between silanols and deprotonated CN -anions can produce very large OH bathochromic shift that fall into the CH stretching zone (from 3200 cm -1 downwards), as shown by the case f ( ν (C ≡ N) = 3056 cm -1 , c i = 1.0000). \nAs regards the CN -anion , only case i has an appreciable weight in the fitted spectrum ( ν (C ≡ N) = 2151 cm -1 , c i = 0.2016), while all the others contributions (from j to n ) are around 1% or less. In this case the CN -is bridged between two Mg 2+ cations forming a η 2 (C,N) adsorption mode (Mg-C ≡ N-Mg) resulting in a hypsochromic shift of the CN bond, which explains the experimental band at 2137 cm -1 . \nTable 1: Summary of all the contributions to the simulated IR spectrum after the best fit to the experimental one, grouped depending on the adsorption type, and ordered according to decreasing weight in the fitted linear combination. For each item we report the corresponding type of adsorption (molecular (M) or dissociative (D)), the involved stretching modes, the corresponding surface, the normalized weight in the linear combination (c i ) and the computed stretching frequency (in cm -1 .) \nFinally, it is worth mentioning case e ( ν (C=N) = 1526 cm -1 ), which, even if it does not present a good fit with the experimental IR (c i = 0.0074), we believe represents an interesting case as well, possibly explaining the band at 1600-1700 cm -1 . This is the region of the double bond stretching of C=N, among other moieties, and was assigned to a bridged \n<!-- image --> \nFigure 5: Left: surface energy of the six slab models adopted in this work calculated at the PBE-DN level. Right: Wulff's construction of a forsterite nano-crystal at 0 K. Adapted with permission from 35. Copyright 2023 Royal Society of Chemistry. \n<!-- image --> \nspecies of non-deprotonated HCN where the bond order of C ≡ N decreases and a new covalent bond is formed between the C atom of HCN and one of the O atoms of the surface. As for the HCN deprotonation, this particular chemisorption can occur on reactive/defective sites, i.e. surface-exposed Mg 2+ -O 2 -ion pairs, and high energy surfaces.", 'Conclusions': 'This combined experimental and computational study provides a comprehensive understanding of the adsorption behavior of hydrogen cyanide (HCN) on crystalline forsterite (Mg 2 SiO 4 ) silicate surfaces. Through infrared (IR) spectroscopic measurements and periodic density functional theory (DFT) simulations, the intricate HCN/forsterite interactions occurring during the HCN adsorption on the crystalline forsterite surfaces at low temperatures (100150 K) were elucidated at an atomistic detail. \nThe main achievement of this synergy is twofold: on one hand, we remove many biases related to the deconvolution of a complex spectrum by making a weighted convolution of all \nthe simulated IRs, fitted on the experimental spectrum. On the other hand, the exploration of all forsterite surface models allowed us to gauge the role of each specific crystal face in contributing to the vibrational features of the experimental spectrum which, in turn, allows to elucidate the crystal morphology of the forsterite sample. \nResults reveal the prevalence of both molecular and dissociative adsorption complexes, the former dominating on stable surfaces, while the latter occurring predominantly on less stable surfaces. HCN dissociation occurs due to the enhanced Lewis acid-base behaviour of surface-exposed Mg 2+ -O 2 -ion pairs, but also to reactive silicate groups belonging to high energy surfaces, as it has been demonstrated by DFT simulations, which therefore present a non-negligible contribution to the exposed facets of experimental forsterite samples. \nThese findings are useful to deepen our understanding on the interaction of HCN with silicate surfaces and, by extension, to the catalytic properties of silicates towards HCN activation, thereby contributing to improve our know-how of prebiotic chemistry in astrophysical environments. Our combined experimental-theoretical approach can also be used to explore the adsorption properties of other molecules on different inorganic material surfaces, thereby further advancing our knowledge on surface chemical processes such as adsorption and heterogeneous catalysis relevant to fundamental (like astrochemistry and prebiotic chemistry) and more applied fields.', 'Acknowledgement': "This project has received funding within the European Union's Horizon 2020 research and innovation program from the European Research Council (ERC) for the project 'Quantum Chemistry on Interstellar Grains' (Quantumgrain), grant agreement No. 865657. The Italian Space Agency for co-funding the Life in Space Project (ASI N. 2019-3-U.O), the Italian MUR (PRIN 2020, Astrochemistry beyond the second period elements, Prot. 2020AFB3FX) are also acknowledged for financial support. This research is also funded by the Spanish \nMICINN (projects PID2021-126427NB-I00 and CNS2023-144902). This research acknowledges support from the Project CH4.0 under the MIUR program 'Dipartimento di Eccellenza 2023-2027'. The authors thankfully acknowledge RES resources provided by UMA for the use of Picasso (activity QHS-2023-3-0032) and by IAC for the use of LaPalma (activity QHS2023-1-0020). The supercomputational facilities provided by CSUC and CINECA (ISCRAB projects) are also acknowledged. The EuroHPC Joint Undertaking through the Regular Access call project no. 2023R01-112, hosted by the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRA CZ (ID: 90254) is also acknowledged.", 'Supporting Information Available': 'Electronic Supplementary Information (ESI) available: graphical representation of the forsterite slab models adopted in this work, IR spectrum with the isolated contributions to the fitting procedure and table of all the structures contributing to the fitting.', 'References': "- (1) Liszt, H.; Lucas, R. 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From gaseous HCN to nucleobases at the cosmic silicate dust surface: An experimental insight into the onset of prebiotic chemistry in space. Phys. Chem. Chem. Phys. 2022 , 24 , 7224-7230.\n- (37) Signorile, M.; Zamirri, L.; Tsuchiyama, A.; Ugliengo, P.; Bonino, F.; Martra, G. On the Surface Acid-Base Properties of Amorphous and Crystalline Mg 2 SiO 4 as Probed by Adsorbed CO, CO 2 , and CD 3 CN. ACS Earth Space Chem. 2020 , 4 , 345-354.\n- (38) Dovesi, R.; Erba, A.; Orlando, R.; Zicovich-Wilson, C. M.; Civalleri, B.; Maschio, L.; Rérat, M.; Casassa, S.; Baima, J.; Salustro, S.; others Quantum-mechanical condensed matter simulations with CRYSTAL. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2018 , 8 , e1360.\n- (39) Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem 2006 , 27 , 1787-1799.\n- (40) Civalleri, B.; Zicovich-Wilson, C. M.; Valenzano, L.; Ugliengo, P. 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2024A&A...690A.148A	We propose a novel approach for determining the orbital inclination of lowmass Xray binary systems by modelling the H and H line profiles emitted by the accretion disc with a Newtonian i.e. nonrelativistic version of DISKLINE. We applied the model to two sample sources Swift J1357.20933 and MAXI J1305704 which are both transient black hole systems and analyse two observations that were collected during a quiescent state and one observation of an outburst. The line profile is well described by the DISKLINE model although we had to add a Gaussian line to describe the deep inner core of the doublepeaked profile which the DISKLINE model was unable to reproduce. The H emission lines in the spectrum of Swift 1357.20933 and the H emission lines in that of MAXI J1305704 during the quiescent state are consistent with a scenario in which these lines originate from a disc ring between 9.6 5710SUP3SUP RSUBgSUB and 1.94 2010SUP4SUP RSUBgSUB respectively. We estimate an inclination angle of 81 5 degrees for Swift J1357.20933 and an angle of 73 4 degrees for MAXI J1305704. This is entirely consistent with the values reported in the literature. In agreement with the recent literature our analysis of the outburst spectrum of MAXI J1305704 revealed that the radius of the emission region deviates from expected values. It is larger than the orbital separation of the system. This outcome implies several potential scenarios including line profile contamination an alternative disc configuration that deviates from the Keplerian model or even the possibility of a circumbinary disc. We caution that these results were derived from a simplistic model that may not fully describe the complicated physics of accretion discs. Despite these limitations our results for the inclination angles are remarkably consistent with recent complementary studies and the proposed description of the emitting region remains entirely plausible.	2024-10-01T00:00:00Z	['2024A&A...690A.148A', '2024arXiv240911988A', '10.48550/arXiv.2409.11988', 'arXiv:2409.11988', '10.1051/0004-6361/202348907']	['accretion', 'accretion disks', 'line: profiles', 'stars: black holes', 'X-rays: binaries', 'X-rays: individuals: Swift J1357.2–0933', 'X-rays: individuals: MAXI J1305–704', 'Astrophysics - High Energy Astrophysical Phenomena']	Xray view of emission lines in optical spectra Spectral analysis of the two lowmass Xray binary systems Swift J1357.20933 and MAXI J1305704	2,024	197	0.54	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.11988.pdf	{'No Header': '8 \n9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18', 'X-ray view of emission lines in optical spectra: Spectral analysis of the two low-mass X-ray binary systems Swift J1357.2-0933 and MAXI J1305-704': 'A. Anitra 1 , C. Miceli 1 , 2 , 3 , T. Di Salvo 1 , R. Iaria 1 , N. Degenaar 6 , Jon M. Miller 7 , F. Barra 1 , W. Leone 1 , 4 , L. Burderi 5 \n- 1 Università degli Studi di Palermo, Dipartimento di Fisica e Chimica, via Archirafi 36, I-90123 Palermo, Italy\n- 2 INAF / IASF Palermo, via Ugo La Malfa 153, I-90146 Palermo, Italy\n- 3 IRAP, Universitè de Toulouse, CNRS, UPS, CNES, 9, avenue du Colonel Roche BP 44346 F-31028 Toulouse, Cedex 4,France\n- 4 Department of Physics, University of Trento, Via Sommarive 14, 38122 Povo (TN), Italy\n- 5 Dipartimento di Fisica, Università degli Studi di Cagliari, SP Monserrato-Sestu, KM 0.7, Monserrato, 09042 Italy\n- 6 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Postbus 94249, 1090 GE Amsterdam, The Netherlands\n- 7 Department of Astronomy, The University of Michigan, 1085 South University Avenue, Ann Arbor, MI, 48109, USA \nSeptember 19, 2024', 'ABSTRACT': 'We propose a novel approach for determining the orbital inclination of low-mass X-ray binary systems by modelling the H α and H β line profiles emitted by the accretion disc, with a Newtonian (i.e. non-relativistic) version of diskline . We applied the model to two sample sources, Swift J1357.2-0933 and MAXI J1305-704, which are both transient black hole systems, and analyse two observations that were collected during a quiescent state and one observation of an outburst. The line profile is well described by the diskline model, although we had to add a Gaussian line to describe the deep inner core of the double-peaked profile, which the diskline model was unable to reproduce. The H β emission lines in the spectrum of Swift J1357.2-0933 and the H α emission lines in that of MAXI J1305-704 during the quiescent state are consistent with a scenario in which these lines originate from a disc ring between (9 . 6 -57) × 10 3 , Rg and (1 . 94 -20) × 10 4 , Rg, respectively. \nWe estimate an inclination angle of 81 ± 5 degrees for Swift J1357.2-0933 and an angle of 73 ± 4 degrees for MAXI J1305-704. This is entirely consistent with the values reported in the literature. \nIn agreement with the recent literature, our analysis of the outburst spectrum of MAXI J1305-704 revealed that the radius of the emission region deviates from expected values. It is larger than the orbital separation of the system. This outcome implies several potential scenarios, including line profile contamination, an alternative disc configuration that deviates from the Keplerian model, or even the possibility of a circumbinary disc. \nWe caution that these results were derived from a simplistic model that may not fully describe the complicated physics of accretion discs. Despite these limitations, our results for the inclination angles are remarkably consistent with recent complementary studies, and the proposed description of the emitting region remains entirely plausible. \nKey words. accretion, accretion discs - stars: Black hole - stars: individual: Swift J1357.2- stars: individual: MAXI J1305-704 X-rays: binaries - X-rays: general - eclipses', '1. Introduction': 'Low-mass X-ray binaries (LMXBs) are systems that are composed of a compact object, such as a black hole (BH) or neutron star (NS), and a donor star with a mass lower than 1 M ⊙ . The compact object in these systems accretes matter from its companion via Roche-lobe overflow, which leads to the formation of an accretion disc around it. \nThe study of the double-peaked emission line profiles that are often present in the optical spectra of these sources is a powerful tool that allows us to determine the orbital parameters of the system. By examining parameters such as the orbital period and radial velocity semi-amplitude we can dynamically confirm the nature of the compact objects in these systems. \nIt has been acknowledged that the central core within the double-peaked lines that are emitted by an accretion disc become deeper with increasing inclination. This is particularly noticeable for inclinations greater than 67 degrees (Horne & Marsh 1986). Casares et al. (2022) discovered a direct correlation between the \ndepth of the inner core in the double-peaked H α emission line and the inclination angle in a group of quiescent BH LMXBs. By applying this method, the authors were able to determine the inclination angle of di ff erent sources with a high confidence level. \nWang et al. (2009) examined the optical spectrum of the millisecond radio pulsar binary SDSS J102347.6 + 003841. They focused on the analysis of double-peaked emission lines to determine the accretion disc geometry. By adopting two Lorentz functions to fit the lines and calculating the flux, they were able to estimate key properties such as the disc temperature range, the outer and inner emission radii, and the overall mass of the disc. \nIn this work, we investigate whether an estimate of the system inclination angle can be derived by applying the diskline model for the first time to fit the double-peaked emission lines related to the H-series transition that appears in the optical spectra of two candidate transient BH systems. The diskline model can describe simple pure Keplerian fully symmetric disc profiles, where the emissivity is a pure power law. This may not be the case for optical lines that are emitted in the outer disc, \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 \n66 \n67 \n68 \n69 \n70 \n71 \n72 \n73 \n74 \n75 \n76 \n77 \n78 \n79 \n80 \n81 \n82 \n83 \n84 \n85 \n86 \n87 \n88 \n89 \n90 \n91 \n92 \n93 \nwhich is most a ff ected by asymmetries, non-uniform emissivity patterns, strong outer disc features such as hot spots and bulges, departures from Keplerian flow due to tidal interactions, and so on. However, we show that the line profiles observed during Xray quiescence of the BH candidates Swift J1357.2-0933 and Maxi J1305-704 appear to be simple enough to provide useful constraints on their inclination angle.', '2. X-ray transient system samples': 'To ensure the accuracy of the inclination measurements, we chose two transient BH X-ray binaries that have been intensely studied in the literature and for which several indications of a high-inclination angle have been reported.', '2.1. Swift J1357.2-0933': 'Swift J1357.2-0933 (hereafter J1357) is an LMXB X-ray transient that was originally detected during its outburst in 2011. The distance to J1357 is estimated to be greater than 2.29 kpc, which places the source within the thick Galactic disc (Mata Sánchez et al. 2015). Studies of the modulation of the H α double-peak emission line during 14 months of outbursts have led to an orbital period of 2 . 5673 ± 0 . 0006 h (Casares et al. 2022). An estimate of the compact object mass was provided by Mata Sánchez et al. (2015) based on the correlation between the full width at half maximum (FWHM) and the radial velocity of the donor star (K2) (Casares 2015). This revealed that this source is one of the most massive BHs in our galaxy (MBH > 9 . 3 M ⊙ ). The optical light curve of the source displays periodic dips with a consistent pattern during the outburst (Armas Padilla et al. 2014). These dips are thought to be linked to a toroidal structure within the inner accretion disc that gradually moves outward as the outburst proceeds (Corral-Santana et al. 2013). \nThe frequent dipping episodes in the system suggest a highinclination angle i larger than 70 degrees (see Corral-Santana et al. 2013; Torres et al. 2015). Anitra et al. (2023) performed phase-resolved spectroscopy during a quiescent state. They focused on the analysis of the H β line profile. The authors estimated the systemic velocity and the radial velocity semiamplitude of the black hole and detected compelling evidence of a narrow and variable inner core within the double-peaked H β emission line profile. These features were observed in highinclination cataclysmic variables (Schoembs & Hartmann 1983) and are probably associated with an occultation of the inner-disc emission by the outer rim bulge. \nHigh-inclination systems typically exhibit eclipses in their light curves (Anitra et al. 2021). The absence of eclipses in this system appears to contradict a high inclination (see Iaria et al. 2018; Armas Padilla et al. 2014). Corral-Santana et al. (2013) justified this absence by considering an inclination angle of 78 degrees and a low mass ratio, proposing that the radius of the donor star might be comparable to or even smaller than the outer disc rim. Based on these indicators of a high-inclination nature, J1357 is an ideal candidate for our analysis.', '2.2. MAXI J1305-704': 'MAXI J1305-704 (hereafter J1305) was proposed to be a highinclination (BH) X-ray binary because a dipping behaviour was detected during its outburst (Suwa et al. 2012, Shidatsu et al. 2013, Morihana et al. 2013, Kennea et al. 2012). The source has \nan estimated distance of d = 7 . 5 l -1 . 4 + 1 . 8 kpc, which places it in the thick Galactic disc (Mata Sánchez et al. 2021). \nJ1305 was studied in detail in the X-rays, but follow-up at other wavelengths has been scarce (see Shaw et al. 2017; Miller et al. 2014; Shidatsu et al. 2013). The first optical spectroscopic analysis of this source during its 2012 outburst has been performed by Miceli et al. (2024). It focused on the double-peaked H α emission line, which revealed no conclusive evidence of the outflow features within the system. \nMata Sánchez et al. (2021) inferred an orbital period of 9 . 456 ± 0 . 096 h and a MBH = 8 . 9 + 1 . 6 -1 . 0 M ⊙ by analysing the orbital modulation in the optical light curve in quiescence. The same authors obtained a constraint on the orbital inclination of i = 72 + 5 -8 deg, which supports the high-inclination scenario. The number of observations during outburst and quiescence and the well-constrained values of the inclination angle and other orbital parameters (e.g. orbital period and mass of the two objects) make J1305 one of the best candidates for our analysis.', '3. Observations': 'We analysed di ff erent sets of observations for the two sample sources. \nJ1357 was observed by the 10.4-meter Gran Telescopio Canarias (GTC) at the Observatorio del Roque de los Muchachos (ORM) on the island of La Palma (Spain), using the Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy (OSIRIS). Eight observations were collected during a quiescence state. They were focused on the wavelength range from 3950 to 5700 Å. The R2000B grism was set alongside a 0.8" slit during the campaign. This configuration allowed us to obtain a spectral resolution of R = 1903 and a dispersion of D = 0.86 Å / pix. The dispersion is evaluated at λ c = 4755 Å , and the resolution was obtained by the FWHM of the skylines in the background spectra. Data were collected on 5 March 2016 between 03:32:11.4 and 06:19:5. Each observation was 1235 s long for a total exposure of 2.75 h and covered a full orbit (Anitra et al. 2023). \nJ1305 was observed both during an outburst and in a quiescent state. The initial observation took place during its discovery outburst in 2012 and was made with the 6.5-meter Magellan Clay telescope located at Las Campanas Observatory in Chile. This observation involved the use of the Low Dispersion Survey Spectrograph (LDSS-3) and the VPH-ALL grism setup. The observations covered two consecutive nights, 2 May (02:18:34 UTC to 02:56:41 UTC) and 3 May (00:29:36 UTC to 01:09:03 UTC) with a 0.75\' slit. The exposure time for each individual spectrum was set at 300 s for a total of six spectra per night (Miceli et al. 2024). The setup configuration allowed us to focus on a wavelength range between 4500 - 6950 Åwith a resolution power of R = 826. \nThe source was observed during quiescence on 31 March 2016 with the Very Large Telescope Unit Telescope 1 (VLTUT1; Paranal Observatory, Chile) using the focal reducer / lowdispersion spectrograph 2 (FORS2, Appenzeller et al. 1998) in long-slit mode. The data sets were composed of 16 spectra, with an exposure time of 1800 s each. They were collected consecutively for a total exposure of ∼ 9 h, that is, they covered almost one orbital period (Mata Sánchez et al. 2021). The spectra cover a wavelength range between 5800 - 7300 Å with a spectral resolution of R ∼ 2140 and a dispersion of 0.76 Å / pix. This was obtained by measuring the FWHM of the skylines in the background spectra. \n94 \n95 \n96 \n97 \n98 \n99 \n100 \n101 \n102 \n103 \n104 \n105 \n106 \n107 \n108 \n109 \n110 \n111 \n112 \n113 \n114 \n115 \n116 \n117 \n118 \n119 \n120 \n121 \n122 \n123 \n124 \n125 \n126 \n127 \n128 \n129 \n130 \n131 \n132 \n133 \n134 \n135 \n136 \n137 \n138 \n139 \n140 \n141 \n142 \n143 \n144 \n145 \n146 \n147 \n148 \n149 \n150 \n151 \n152 \n153 \n154 \n155 \n156 \n157 \n158 \n159 \n160 \n161 \n162 \n163 \n164 \n165 \n166 \n167 \n168 \n169 \n170 \n171 \n172 \n173 \n174 \n175 \n176 \n177 \n178 \n179 \n180 \n181 \n182 \n183 \n184 \n185 \n186 \n187 \n188 \n189 \n190 \n191 \n192 \n193 \n194 \n195 \n196 \n197 \n198 \n199 \n200 \n201 \n202 \n203 \n204 \n205 \n206 \n207 \n208 \n209 \n210 \n211 \nWe reduced the data using standard procedures based on the iraf 1 software, molly tasks, and python packages from astropy and pyastronomy (Astropy Collaboration et al. 2022). These tools allowed us to correct the observed spectra for bias and flats and to calibrate the data set. We note that cosmic rays contaminated the spectra of J1357, and we corrected for them using the L.A. Cosmic task (van Dokkum et al. 2012). We adopted the optimal extraction technique (Naylor 1998) to extract the twodimensional images. The quiescence spectra of J1305 might be contaminated by the companion emission. However, as claimed by Mata Sánchez et al. (2021), the contamination is probably lower than 10% of the total flux and is therefore negligible.', '4.1. Diskline model': "The emission lines of the Balmer series are associated with the atomic hydrogen transition. They appear as broad symmetrical double-horned lines when observed from accretion discs at highinclination angles. In these systems, the emission is mainly influenced by the Doppler shift, which is caused by the orbital motion of matter in the accretion disc around the compact object. \nIn a disc ring at a certain radius RD, matter moves at the corresponding Keplerian velocity Vkep. For a distant observer, some of the matter within the disc moves toward the observer, while it simultaneously moves away from the observer on the far side of the disc. This dual motion causes the observer to perceive the emission line as simultaneously blue- and red-shifted, and it causes the distinctive symmetrical double-horn profile. Because the emission comes from di ff erent regions of the disc surface, the overall line profile is additionally broadened by the velocity distribution in the disc. As first discussed by Smak (1981) and later by Horne & Marsh (1986), the final line profile is significantly a ff ected by the inclination angle of the system relative to the line of sight because the component of the Keplerian velocity Vkep along the line of sight is V D = ± V Kep ( R D) sin i . Furthermore, the separation of the peaks is determined by the outer radius of the emitting region, but the wings are shaped by the emissivity profile. \nThe X-ray spectrum-fitting software xspec (Arnaud 1996) provides several models for describing an emission line that is shaped by Doppler e ff ects from a Keplerian accretion disc. One of the most commonly adopted models for fitting double-horn emission lines is the diskline model (Fabian et al. 1989). It is usually adopted to describe the iron K α emission line that characterises the X-ray reflection features observed in LMXBs (see e.g. di Salvo et al. 2009). \nThe shape of the broad iron fluorescence line observed at X-ray wavelengths, similar to the hydrogen Balmer series lines in optical spectra, is a ff ected by Newtonian Doppler shift. Its profile appears to be quite di ff erent, however. The Fe K α line originates in the innermost part of the disc, where the velocity of matter reaches relativistic values. Consequently, the relativistic beaming a ff ects the emission and intensifies the blue peak of the line while weakening the red peak. Furthermore, the line is gravitationally redshifted by the strong gravitational pull near the compact object, causing it to shift to lower energies in a way that depends on the distance from the compact object. As a result, the emission line appears to be broadened and asymmetric. \nThe diskline model is composed of the following free parameters that shape the emission line profile: the inner (Rin) and outer radius (Rout) of the emission region in the disc (expressed in units of gravitational radii, Rg = GM / c 2 ), the orbital inclination of the system, and the index, β , which describes the dependence of the disc emissivity on the distance from the compact object (E ∝ R β ). This model was used several times to obtain constraints on the inclination angle of LMXBs (see e.g. Anitra et al. 2021; Iaria et al. 2009; Cackett et al. 2008) when relativistic e ff ects were dominant. Although this model is usually adopted to describe a (relativistic) emission line, it can be adapted to model the hydrogen line profiles as well. When the value of the inner and outer radius is high enough, meaning that the emission region of the line is far from the compact object, the relativistic beaming and the gravitational redshift indeed become negligible, and only the Newtonian Doppler shift dominates (see Fig. 1). Under these conditions, the model is equivalent to the optically thin solution presented in Horne & Marsh (1986). The outer radius of the disc is also most strongly a ff ected by asymmetries, non-uniform emission line emissivity patterns, and strong outer disc features. For instance, self-absorption by an optically thick disc atmosphere can contribute to deepening the inner core of the line. Simultaneously, disc precession may play a significant role in determining the orbit-averaged velocity shifts that are observed in the emission line centroid (Torres et al. 2002), while a hot spot can enhance one of the two peaks of the line (Marsh et al. 1994). For these reasons, it is important to exercise caution with respect to the constraints derived from the analysis. \nIn order to isolate the emission line profile on which we focused our analysis, we normalised the spectra of J1357 and J1305 by dividing them by their polynomial best-fit functions. Specifically, we applied a third-order polynomial fit for the first source and a fifth-order polynomial fit for the second source. We thus adapted our optical data sets for an analysis in the xspec environment by creating a unit diagonal response matrix of the data size. We adhered to the following naming convention: spectra linked to J1357 are denoted src1-Q, and those associated with J1305 are labelled src2-O during outburst and src2-Q during quiescence. \nWe find it convenient to express the constrained best-fit radii in units of the expected tidal radius of the system, that is, the outer edge of the disc truncated by the tidal torque of the companion star (Frank et al. 2002). The tidal radius depends on several binary parameters, but its value can be shown to be close to RT = 0 . 9 R1 (Frank et al. 2002), where R1 is the Roche-lobe radius of the compact object. This can be calculated using the Eggleton (1983) equation \nR 1 a = 0 . 49 q -2 / 3 0 . 6 q -2 / 3 + ln (1 + q -1 / 3 ) , (1) \nwhere q is the mass ratio of the two bodies M1 and M2, and a is the orbital separation of the system, which can be calculated using Kepler's third law. Moreover, adopting black hole masses of about 8.9 solar masses for J1305 (Mata Sánchez et al. 2021) and 9.3 solar masses for J1357 (Mata Sánchez et al. 2015), we can define the gravitational radius Rg for the two systems as 13 km and 14 km, respectively. Consequently, we obtain an orbital separation and tidal radius of a = (1 . 0259 ± 0 . 0002) × 10 6 Rg and RT = (5 . 9 ± 0 . 004) × 10 4 Rg for J1357, and a = (2 . 524 ± 0 . 002) × 10 5 Rg, and RT = (1 . 45 ± 0 . 06) × 10 5 Rg for J1305. \n212 \n213 \n214 \n215 \n216 \n217 \n218 \n219 \n220 \n221 \n222 \n223 \n224 \n225 \n226 \n227 \n228 \n229 \n230 \n231 \n232 \n233 \n234 \n235 \n236 \n237 \n238 \n239 \n240 \n241 \n242 \n243 \n244 \n245 \n246 \n247 \n248 \n249 \n250 \n251 \n252 \n253 \n254 \n255 \n256 \n257 \n258 \n259 \n260 \n261 \n262 \n263 \n264 \n265 \n266 \n267 \n268 \n269 \n270 \n271 \n272 \n273 \n274 \n275 \n276 \n277 \n278 \n279 \n280 \n281 \n282 \n283 \n284 \n285 \n286 \n287 \n288 \n289 \n290 \n291 \n292 \n293 \n294 \n295 \n296 \nFig. 1. Series of examples showing the diskline model profiles with fixed inclinations ( i = 83 degrees), emissivity index ( β = 2.5), and energy centroid (E = 4857.522 Å) for all models. The inner and outer radius were systematically varied within ranges of (100 -10 4 ) Rg and (10 4 -6 × 10 4 ) Rg, respectively, to highlight the variations in the line profiles. The model normalisations are scaled for visual clarity. \n<!-- image -->", '4.2. J1357 spectroscopy': 'The wavelength coverage of our spectra allowed us to focus on the H β line region with a su ffi ciently high signal-to-noise ratio (S / N) 2 . \nWe fitted the eight spectra simultaneously using a power-law model to account for the normalised continuum (the associated parameters are not physically relevant), to which we added the diskline model to fit the H β line profile. The observations were obtained within a time interval of 2.75 h (one orbital period), and we therefore expect very little spectral variation in the di ff erent spectra. As a result, we constrained specific parameters, such as the inclination angle, emissivity index, and inner and outer radii, to be the same for all the spectra. The model, called model 1, achieved a poor fit to the data, with a χ 2 / d . o . f . = 4089.1 / 2811. Nonetheless, the values for the inner and outer radii of the line emission region within the disc, (9 . 6 ± 0 . 2) × 10 3 Rg ( ∼ 0 . 16 RT ) and 5 . 4 + 0 . 1 -0 . 2 × 10 4 Rg ( ∼ 0 . 92 RT ), respectively, agree with expectations. Additionally, the emissivity index characterises a region of the disc that is located farther from the compact object. However, as shown in Fig. 2 (second panel), residuals lie in the region between the two peaks. \nTo improve the fit without modifying the diskline model, we attempted two di ff erent approaches. First, we masked out the core of the line and excluded these data from the fit. The resulting best-fit parameters are consistent with those presented previously, with a χ 2 / dof = 4071 . 72 / 2805. Second, we introduced a Gaussian absorption line for which the centroid and FWHM parameters were linked for all the spectra, but we al- \nalized flux \nNor \nFig. 2. J1357 spectra collected during a quiescent phase and residuals in units of sigma with respect to the diskline model described in the test (top panel). The middle and bottom panel shows the residuals obtained by using only the diskline model ( model 1) and those obtained with the same model plus a Gaussian absorption line ( model 2), respectively. \n<!-- image --> \nlowed an independent variable depth to determine whether each spectrum significantly required this feature (see Sect. 5). This model, called model 2, achieved a better fit to the data with a χ 2 / dof = 3849 . 0 / 2794. We tested the improvement of the fit using the statistical test Ftest and obtained a probability of a chance improvement of ∼ 1 . 1 × 10 -28 . This means that including the Gaussian line in absorption improves the quality of the fit with a confidence level (c.l.) higher than 7 σ . The new model provides a more solid constraint on the inclination angle (i = 83 + 5 -3 degrees), and the H β emission line is consistent with being emitted by a ring in the disc with a radius between 9 . 6 + 0 . 2 -0 . 1 × 10 3 Rg ( ∼ 0 . 16 RT ) and 5 . 7 + 0 . 1 -0 . 2 × 10 4 Rg ( ∼ 0 . 91 RT ). However, for this high number of degrees of freedom, a reduced χ 2 of 1.38 is unacceptable because it implies a very low null-hypothesis probability. In order to obtain reliable error bars for the best-fit spectral parameters, we therefore scaled the errors by the q χ 2 red to achieve a reduced χ 2 of approximately 1 and repeated the fit. Using this approach, we obtained the same values for the inner and outer radii of the emitting region, while the fit was able to provide just a lower limit for the inclination angle of 80 degrees. Notably, this finding only holds true for errors calculated at the 90% c.l. Conversely, when errors are calculated at the 68% c.l, the inclination angle is constrained to be 83 . 8 + 5 . 4 -1 . 8 degrees (see Table A.1).', '4.3. J1305 spectroscopy': 'The wavelength coverage and spectral resolution provided by the telescope set-up allowed us to perform a high-resolution analysis focused on the H α emission line. In the spectra collected during the outburst, we identify H α and H β emission lines, corresponding to the Balmer series and He II 4686 Å as the Bowen blend (a mixture of N iii and C iii emission lines; see e.g. Steeghs \n297 \n298 \n299 \n300 \n301 \n302 \n303 \n304 \n305 \n306 \n307 \n308 \n309 \n310 \n311 \n312 \n313 \n314 \n315 \n316 \n317 \n318 \n319 \n320 \n321 \n322 \n323 \n324 \n325 \n326 \n327 \n328 \n329 \n330 \n331 \n332 \n333 \n334 \n335 \n336 \n337 \n338 \n339 \n340 \n341 \n342 \n343 \n344 \n345 \n346 \n347 \n348 \n349 \n350 \n351 \n352 \n353 \n354 \n355 \n356 \n357 \n358 \n359 \n360 \n361 \n362 \n363 \n364 \n365 \n366 \n367 \n368 \n369 \n370 \n371 \nTable 1. Ratios of the blue and red peaks of the J1305 spectra collected during quiescence. \n& Casares 2002). We decided to focus our analysis on the H α line, which is present in all the spectra in the two days and is the strongest isolated feature (Miceli et al. 2024). We separately analysed the two days of optical observations. For the first day, we used four of the six spectra excluding, in particular, src2O1 because of its strong absorption component and src2-O3 because it does not show a double-peak profile, but a flat top profile (which recalls outflow features; Miceli et al. (2024), see also Cúneo et al. 2020). \nWe applied model 1 to the data and linked the parameters of each spectrum as described in the previous section (see Fig. 3). On the second day of observations, a distinct absorption component is evident in all spectra. It is located at wavelengths longer than those associated with the H α line (about 6563 Å) (see Fig. 4). To take this feature into account, we added to the previous model a Gaussian absorption line for which the centroid and FWHM parameters we linked for all the spectra, but we allowed an independently variable depth. For both days, the adopted models achieve a reasonably good fit to the data, with a χ 2 / dof = 139 . 4 / 153 (first day) and χ 2 / dof = 149 . 0 / 222 (second day). The best-fit parameters are reported in Table A.1. On both days, the inclination angle is 70 ± 4 and 71 ± 4 degrees, which is consistent with the angle reported in the literature ( i = 72 + 5 -8 degrees; Mata Sánchez et al. 2021). Furthermore, for the first day, we derived inner and outer radii values of 1 . 20 + 0 . 09 -0 . 07 × 10 5 Rg ( ∼ 0 . 83 RT ) and 1 . 96 + 0 . 32 -0 . 27 × 10 6 Rg ( ∼ 13 RT ), respectively. For the second day, these values were 1 . 66 + 0 . 10 -0 . 08 × 10 5 Rg ( ∼ 1 . 1 RT ) and 2 . 73 + 0 . 9 -0 . 6 × 10 6 Rg ( ∼ 19 RT ) . A more detailed discussion of these parameters is presented in the following section.', '4.3.1. Quiescence': 'We chose 10 of the 16 spectra for the quiescent phase that showed a more regular line profiles. Our preference lay with line profiles with a symmetrical double peak because the asymmetry in the intensity of the two peaks can be caused by various effects, for instance a hot spot (Shafter et al. 1986), and this might complicate the fitting. However, the exclusion of the 6 spectra does not compromise the credibility of the model. As a test, we individually analysed the excluded spectra and obtained parameter values that were entirely consistent with those reported in the main analysis (see Sect. 4.4). The symmetry of a profile can be estimated by examining the ratios of the two peaks, as reported in Table 1. Following the same procedure as applied to J1357, we first fitted the spectra applying model 1. However, \nFig. 3. J1305 spectra collected during an outburst phase and residuals in units of sigma with respect to the diskline ( model 1), related to the first day of observation. \n<!-- image --> \nmalized flux \nNor \nFig. 4. J1305 spectra collected during an outburst phase and residuals in units of sigma with respect to the diskline model plus an absorption Gaussian line, related to the second day of observation. \n<!-- image --> \nthe latter did not reach a good fit to the data, with a χ 2 / dof of 6432.32 / 2650, because residuals were clearly present in the core region (see the first panel of Fig. 5). Therefore, we applied model 2 and linked the parameters of di ff erent spectra as described for the J1357 spectra. The model achieved a better fit to the data, ensuring a χ 2 / dof of 4161.1 / 2620. The χ 2 value had to be rejected here as well. Therefore, as discussed in the previous section, we multiplied the errors by the square root of the reduced χ 2 and repeated the fitting. We obtain a constraint on the inclination angle of 72 . 6 + 1 . 4 -1 . 3 degrees at 90% c.l., as well as best-fit values for the inner and outer radii of the emitting region associated with the H α emission line of 1 . 94 + 0 . 07 -0 . 09 × 10 4 Rg ( ∼ 0 . 1 RT ) and 2 . 01 ± 0 . 04 × 10 5 Rg ( ∼ 1 . 3 RT ), respectively. \n372 \n373 \n374 \n375 \n376 \n377 \n378 \n379 \n380 \n381 \n382 \n383 \n384 \n385 \n386 \n387 \n388 \n389 \n390 \n391 \n392 \n393 \n394 \n395 \n396 \n397 \n398 \n399 \n400 \n401 \n402 \n403 \n404 \n405 \n406 \n407 \n408 \n409 \n410 \n411 \n412 \n413 \n414 \n415 \n416 \n417 \nFig. 5. J1305 spectra collected during a quiescent phase and residuals in units of sigma with respect to the diskline model described in the text (top panel). The middle and bottom panels show the residuals obtained with the diskline model alone ( model 1) and those obtained with the same model plus a Gaussian absorption line ( model 2). \n<!-- image -->', '4.4. Analysis of the parameter degeneration': 'The line profiles can easily be a ff ected by external factors, such as hot spots or disc precession (e.g.Orosz et al. (1994); Mason et al. (2000); Marsh et al. (1994)). These potential pitfalls for a model might prompt concerns about the reliability of the approach. We therefore explore degeneracies and systematics quantitatively here to ensure that the previously derived constraints are statistically reliable. \nWhen multiple parameter combinations yield similar or indistinguishable results during the model-fitting process, this indicates that the best-fit values of the individual parameters cannot be determined based on the data alone. To investigate this degeneracy, we employed the steppar command in xspec to perform a fit while stepping the value of the parameters through a given range. This approach allowed us to evaluate all possible pairings of the parameters of interest, measuring how the χ 2 value changes in relation to the best fit for each combination. This allowed us to identify any value combinations that might result in an equivalent spectral fit. We performed the steppar on data sets collected during quiescence from the two sources and focussed on parameters such as inclination, outer radius, and inner radius, which significantly a ff ect the line profile. In Fig. 6, we present contour plots obtained from the analysis. \nThe contours of the regions are clearly defined and concentric around the best-fit value, indicating robustness in the parameter estimates. In addition, the contours lack extended overlapping or elongated areas that stretch along the diagonals, which would typically indicate degeneration. It is important to observe that even in regions with a diagonal trend (panels a and d), the contours delineate parameter values that lie within the error limits reported in the text. This shows that the same χ 2 values cannot be achieved with parameter pairs that deviate from those established by the best fits. \nTable 2. Inclination values obtained by separately fitting the spectra related to the two sources in quiescence. \nBecause the physics within an accretion disc is complex and the approach of the diskline model is simplistic, the high precision of the obtained inclination constraints (especially from the quiescence spectrum of J1305) may cast doubt on the reliability of the results. An insightful way to re-evaluating and discussing the error associated with the best-fit parameters, especially the inclination angle, involves fitting each spectrum individually and considering the dispersion of the resulting values. \nWe therefore fitted the spectra acquired during quiescence for J1357 and J1305 separately, allowing all parameters to vary freely 3 . In Table 2 we list the inclination values for each fit. Moreover, we decided to also include the spectra that were rejected in the analysis presented in the text to avoid any bias in the estimation of the inclination angle error. We ensured through a steppar analysis throughout the entire range of possible inclination values that the fit did not yield a value corresponding to a local minimum of the χ 2 . Subsequently, we determined the best fit as the average of the obtained values, with the associated error calculated as the semi-dispersion because it represents the maximum dispersion around the average value. This yielded an inclination of 81 ± 5 degrees for J1357 and 73 ± 4 degrees for J1305, and these inclinations agree with the values derived from the analysis. Moreover, the error bars obtained with this method are consistently similar to those constrained by the diskline model.', '5. Discussion': 'We analysed GTC observations of J1357 collected during quiescence, along with VLT and Magellan observations of J1305, taken during quiescence and outburst phases, respectively. Our study focused on the double-peaked profiles of the H β and H α emission lines. With the aim of gaining insights into the system \n418 \n419 \n420 \n421 \n422 \n423 \n424 \n425 \n426 \n427 \n428 \n429 \n430 \n431 \n432 \n433 \n434 \n435 \n436 \n437 \n438 \n439 \n440 \n441 \n442 \n443 \n444 \n445 \n446 \n447 \n448 \n449 \n450 \n451 \n452 \n453 \n454 \n455 \n456 \n457 \n458 \n459 \nFig. 6. Contour plots for inclination and the outer and inner radii of the spectra related to J1357 (first column) and J1305 (second column). The contours represent the 1 σ , 2 σ , and 3 σ confidence levels, and the cross marks the best-fit values obtained from the best fit. \n<!-- image --> \ngeometry, we employed the diskline model, which is complementary to traditional analysis approaches. As discussed in the previous section, the model including the diskline component along with a (Gaussian) absorption line provides a good fit to the data of both systems. This significantly improved the quality of the fit with respect to the model without the absorption line. \nThe addition of this feature may be necessary due to the potential limitations in the assumptions made by the diskline model. It describes a Gaussian emission line profile, modified \nby the Doppler e ff ect induced by a Keplerian velocity distribution, in an accretion disc. It is clear that the model does not take self-absorption or other features due to the optically thick nature of the disc into account, which was also noted by Orosz et al. (1994). This e ff ect is particularly strong for high-inclination angles, as appears to be the case of J1357 and J1305. This agrees with our results, for which an additional Gaussian absorption enabled a better description of the observed spectra. \n460 \n461 \n462 \n463 \n464 \n465 \n466 \n467 \n468 \n469 \n470 \n471 \n472 \n473 \n474 \n475 \n476 \n477 \n478 \n479 \n480 \n481 \n482 \n483 \n484 \n485 \n486 \n487 \n488 \n489 \n490 \n491 \n492 \n493 \n494 \n495 \n496 \n497 \n498 \n499 \n500 \n501 \n502 \n503 \n504 \n505 \n506 \n507 \n508 \n509 \n510 \n511 \n512 \n513 \n514 \n515 \n516 \n517 \n518 \n519 \n520 \n521 \n522 \n523 \n524 \n525 \n526 \n527 \n528 \nUpon visual examination of the individual spectra, it becomes clear that the central core of the line displays narrow and variable absorption that changes in depth and even reaches (sometimes going below) the continuum average level. Features like this were mainly detected in cataclysmic variables that were observed at a high-inclination angle (with i > 75 degrees, Schoembs & Hartmann 1983), and they are thought to be caused by occultation of the inner regions of the accretion disc. Di ff erently from the e ff ects caused by self-absorption, narrow cores are variable with the orbital phase, and they can reach much deeper than the optically thick absorption e ff ects. They can reach even below the normalised continuum (Rayne & Whelan 1981). Narrow variable cores in the J1357 spectra were first claimed by Mata Sánchez et al. (2015), who noted a variation in the normalised flux of the H α core. This was later confirmed by Anitra et al. (2023), who analysed high-resolution data focused on the H β emission line. Mata Sánchez et al. (2021) performed dynamical studies of the same VLT data of J1305 as we presented here. They reported narrow cores in the spectra. \nAs further proof of this, we calculated the significance of the absorption feature in each spectrum in units of sigma by comparing the normalisation of the line with its uncertainty at the 68% c.l. We found that the intensity of the line in J1357 src1Q2, src1-Q3, and src1-Q6 is ≤ 3 σ , meaning that during these orbital phases, the absorption core of the line is negligible, but it is stronger in src1-Q1, src1-Q7 and src1-Q8. In the quiescent spectra of J1305, all the absorption lines are statistically significant at more than 7 σ , except for the line associated with src2-Q15, where this feature is not necessary. This result is very similar to the result reported by Marsh et al. (1987) in the dwarf nova system Z Cha, which exhibited a core depth variability along the orbit in every line of the Balmer series in the spectrum. \nAs reported in Sect. 4.3, spectra collected during the outburst do not show narrow variable cores. However, the spectra collected on the second day of observation display absorption features at higher wavelengths compared to the H β line, which may resemble an inverted P-Cygni profile, which is associated with inflows (Cúneo et al. 2020). Miceli et al. (2024) have extensively discussed the nature of this phenomenon and proposed that a broad and variable absorption component is observed in all spectra, which can a ff ect the shape of the line. Nevertheless, the origin of this feature remains the subject of ongoing debate.', '5.1.1. Inclination angle': 'The best-fit parameters of both sources provide a description of the geometry of the system in line with the hypothesis of a high inclination. We obtain an angle of 72 . 6 + 1 . 4 -1 . 3 degrees (73 ± 4 degrees considering the semi-dispersion around the average value) for J1305 during quiescence and angles of 70 ± 4 and 71 ± 4 degrees for the two days of outburst, respectively. These results are consistent with the previous constraint reported by Mata Sánchez et al. (2021), i = 72 + 5 -8 degrees. \nRecently, Casares et al. (2022) studied the inner core depth in J1357 spectra and derived an inclination angle of 87 . 4 + 2 . 6 -5 . 6 degrees, which matches the estimate we find by applying the diskline model (i > 80 degrees at 90% c.l., i = 80 + 2 -6 degrees at 68 %c.l. and 81 ± 5 considering the average values). However, the lack of eclipses in both the X-ray and optical light curves of this source (Corral-Santana et al. 2013) contradicts the expected behaviour of an edge-on configuration. Corral-Santana et al. (2013) provided an explanation for this phenomenon, which may be due \nFig. 7. Inclination angle obtained with Eq. 2 by varying the mass ratio q in a range of 0 to 0.5. The dashed black lines represent the lower and upper bounds of the inclination values determined for J1357 with model 2. Consequently, the data points highlighted with a red cross symbolise the lower and upper limits of the mass ratio q allowed for the system in order to preserve the lack of eclipses at this specific inclination. \n<!-- image --> \nto the low mass ratio of the system, implying that the radius of the donor star is either comparable to or smaller than that of the outer rim of the disc. Consequently, even in an edge-on configuration, the central disc region is not obstructed by eclipses of the companion star. It is possible to give quantitative credibility to this hypothesis by evaluating whether the mass ratio q given in the literature allows the occurrence or absence of eclipses at a particular inclination angle. As stated by Iaria et al. (2018), the angle θ between the line of sight and the equatorial plane can be described by the following equation: \ntan θ = R 2 2 -x 2 a 2 -GLYPH<16> R 2 2 -x 2 GLYPH<17> 1 / 2 , (2) \nwhere R 2 is the Roche-lobe radius of the companion star, and x is the obscured region during the eclipse. Without an eclipse, x can be assumed to be zero, while the Roche-lobe radius can be calculated using Eq. 1 by substituting q -1 with q . By substituting R 2 into the previous equation, the angle θ is clearly only a function of the mass ratio q . In order to investigate the combinations of q and θ that preserve the condition without an eclipse, we varied the q parameter within the range of 0 to 0.5 and determined the corresponding values of θ . The results are shown in Fig. 7. \nFor an inclination of 83 . 8 + 5 . 4 -1 . 8 degrees obtained at 68% c.l, we can establish a q ratio range between 0.0006 and 0.0275, which is consistent with the value reported by Mata Sánchez et al. (2015). Moreover, with the mass function for M 1, f(M1) = 11 . 0 ± 2 . 1 M ⊙ (Mata Sánchez et al. 2015), we can analytically estimate the mass of the companion star M 2. We establish a range that falls between 0 . 0006 ± 0 . 0001 M ⊙ and 0 . 32 ± 0 . 06 M ⊙ (note that Mata Sánchez et al. 2015 reported a lower limit of M 2 > 0 . 4 M ⊙ ). In other words, the inclination estimated for this system together with the limits on the mass ratio are in line \n529 \n530 \n531 \n532 \n533 \n534 \n535 \n536 \n537 \n538 \n539 \n540 \n541 \n542 \n543 \n544 \n545 \n546 \n547 \n548 \n549 \n550 \n551 \n552 \n553 \n554 \n555 \n556 \n557 \n558 \n559 \n560 \n561 \n562 \n563 \n564 \n565 \n566 \n567 \n568 \n569 \n570 \n571 \n572 \n573 \n574 \n575 \n576 \n577 \n578 \n579 \n580 \n581 \n582 \n583 \n584 \n585 \n586 \n587 \n588 \n589 \n590 \n591 \n592 \n593 \n594 \n595 \n596 \n597 \n598 \n599 \n600 \n601 \n602 \n603 \n604 \n605 \n606 \n607 \n608 \n609 \n610 \n611 \n612 \n613 \n614 \n615 \nwith the literature, and the mass range for the companion star M 2 characterises it as a low-mass star that fills its Roche lobe.', '5.1.2. Inner and outer radius of the emitting region': 'We found that the H β line in J1357 is emitted from a region bounded between (0 . 16 -0 . 91) RT , while the H α emission line observed in the J1305 spectrum describes two di ff erent emitting regions for outburst and quiescence. During the quiescent phase, the emission line is located between (0 . 1 -1 . 3) RT , while during the outburst, the emitting region extends to radii between (0 . 83 -13) RT and (1 . 1 -19) RT . \nTable A.1 shows that the outer radius of J1357 is smaller than the expected tidal radius, indicating that either the accretion disc does not extend to the tidal radius or that the H β emission line does not originate in the outermost region of the accretion disc. For J1305, the best-fit value for the outer radius obtained during quiescence is consistent with the expected geometry of the system (although the best-fit range includes values that exceed the tidal radius). On the other hand, the outer radius obtained for the outburst spectra is entirely out of scale. A value of about 1319 RT is higher by up to an order of magnitude than the orbital separation. This deviation is not due to a miscalculation or to a limitation of the application of diskline under certain circumstances. Miceli et al. (2024) reported a Doppler-shift velocity of the peaks during the outburst of approximately 400 km s -1 . Considering a Keplerian disc, we can apply the third Keplerian law to calculate the corresponding radius, and even in this case, the derived radius is four times the tidal radius. This result could be due to the e ff ects during the outburst on the line profile. It is possible that the velocity distribution of the emitting region is not Keplerian. In standard discs (Shakura & Sunyaev 1973), the in-flowing material reaches a stable equilibrium in which the radial component of the velocity is negligible compared to the azimuthal component. However, if there is a surge in viscosity within that region, the radial velocity may cease to be negligible, and the way in which matter accretes may deviate from the way in the Keplerian regime. \nAnalternative explanation might be linked to the existence of a circumbinary disc around the binary system. Due to the highaccretion rate, outflows of matter can occur, and the ejected material may subsequently remain around the system, causing the formation of a viscous toroidal disc around the binary system (Chen & Podsiadlowski 2019). Although these discs are typically quite cold, the intense emission produced by the central source (Lx ∼ 10 37 erg s -1 for J1305 during outburst, see Miller et al. 2014) might irradiate these regions, thus providing the energy to trigger the atomic transitions that give rise to the H α emission line. This would explain an emitting region greater than the orbital separation itself. However, these are only hypothetical answers to a question that requires further analysis to be understood.', '5.2. Temperatures and hydrogen ionisation': 'Although our results seem to delineate an emitting region that is consistent with the geometry of the disc, at least during quiescence periods, it is crucial to consider the temperatures at these radii. It is indeed true that the emission lines H β and H α are associated with the energy transitions of neutral hydrogen. This means that when the temperatures are too high, the amount of neutral hydrogen will be negligible. \nShakura & Sunyaev (1973) described the structure of the accretion disc around a BH under the assumption that the disc is optically thick and geometrically thin. The authors divided the disc into three ideal regions, depending on the predominant type of pressure and cross-section, and derived a system of equations for each of them through which the physics of the accretion disc can be completely described. We analysed the outer part of the disc, and the equation that describes the variation in the central temperature of the disc with the radius therefore is \nT = 8 . 6 · 10 7 α -1 / 5 ˙ m 3 / 10 m -1 / 5 r -3 / 4 GLYPH<16> 1 -r -1 / 2 GLYPH<17> 3 / 10 , (3) \nwhere m , ˙ m , andr are non-dimensional parameters that contain the dependence on the BH mass, the accretion rate, and the radius, respectively (see Shakura & Sunyaev (1973) for the explicit dependence), while α is the accretion disc viscosity parameter, which usually lies in the range 0.1- 0.4 (King et al. 2007). In this case, we assumed α = 0 . 1. As the density does not vary significantly across the vertical height ( z ) at these large radii, it can reasonably be assumed that the surface temperature of the disc is approximately equal to the central temperature (Shakura &Sunyaev 1973). \nConsidering the values of Rin and Rout from our analysis, we estimated the temperature ranges for the H β and H α emitting regions during the quiescent phases of J1357 and J1305, as we have already discussed the reliability of the measurements during the outburst. We obtained temperature ranges of 2081 ± 255 K to 353 ± 40 K for J1305 and 2203 ± 317 K to 580 ± 81 K for J1357. This is consistent with a higher quiescent luminosity of LX ∼ 1 . 6 × 10 32 , erg , s -1 for J1305 (Mata Sánchez et al. 2021) compared to J1357, for which LX ∼ 1 . 3 × 10 31 , erg , s -1 (Armas Padilla et al. 2014). \nThe degree of ionisation for the H-atom, that is, the fraction of ionised hydrogen with respect to neutral hydrogen at the estimated temperatures, can be computed using the Saha equation (Payne 1924), \nlog H + H = log u + u + log 2 + 5 2 log T -5040 χ ion T -log Pe -0 . 48 . (4) \nHere, Pe = nekT is the electron pressure, where ne is the electron density, which is typically assumed to be equal to 10 18 cm -3 (Sincell & Krolik 1998). χ ion is the ionization energy (13.6 eV for the H atom), and u is the partition function of the atom (u(H) = 2 and u(H + ) = 1 (see Böhm-Vitense 1992). Assuming the temperatures we inferred above, we obtain a range of H + / H between 10 -25 and 10 -114 for J1357, and between 10 -45 and 10 -186 for J1305. This implies that the amount of ionised hydrogen with respect to neutral hydrogen is negligible, and therefore, the radii estimated for the emitting regions of the H α and H β lines for the two systems appear to be physically plausible.', '6. Conclusion': 'We presented a new approach for analysing the emission lines found in the optical spectra of binary systems that involves the use of the diskline model in order to obtain constraints on the system geometry. \nWe analysed two observations in quiescence of the X-ray binary BH candidates J1357 and J1305 and an observation collected during an outburst of the latter source. The best-fit parameters allowed us to provide a reasonable description of the geometry of these systems. The H β and H α emission lines in the quiescent spectra of J1357 and J1305 are emitted by a ring in the \n616 \n617 \n618 \n619 \n620 \n621 \n622 \n623 \n624 \n625 \n626 \n627 \n628 \n629 \n630 \n631 \n632 \n633 \n634 \n635 \n636 \n637 \n638 \n639 \n640 \n641 \n642 \n643 \n644 \n645 \n646 \n647 \n648 \n649 \n650 \n651 \n652 \n653 \n654 \n655 \n656 \n657 \n658 \n659 \n660 \n661 \n662 \n663 \n664 \n665 \n666 \n667 \n668 \n669 \n670 \n671 \n672 \n673 \n674 \n675 \n676 \n677 \n678 \n679 \n680 \n681 \n682 \n683 \n684 \n685 \n686 \n687 \n688 \n689 \n690 \n691 \n692 \n693 \n694 \n695 \n696 \n697 \n698 \n699 \n700 \n701 \n702 \n703 \n704 \n705 \n706 \n707 \n708 \n709 \n710 \n711 \n712 \n713 \n714 \n715 \n716 \n717 \n718 \n719 \n720 \n721 \n722 \n723 \n724 \n725 \n726 \n727 \n728 \n729 \n730 \n731 \n732 \n733 \n734 \n735 \n736 \ndisc between (0 . 16 -0 . 91) RT and (0 . 1 -1 . 3) RT , respectively. The analysis of the outburst spectra yielded an emission region that is not aligned with the expected system geometry, suggesting a range for the outer radius from the tidal radius to values exceeding the orbital separation of the binary system. We can put forward some hypotheses to account for this behaviour, including a non-Keplerian flow in the outer disc or the existence of a circumbinary disc. \nOur analysis reveals that the inclination angles of the two systems closely match the expected values. This confirms their high-inclination nature. The proposed method requires further investigation because the physics of emission lines from accretion discs is far more complex, and the diskline model provides a simplified description of the emission profile modified by Doppler e ff ects in a Keplerian flow, without any consideration of self-absorption e ff ects in the optically thick atmosphere of accretion discs. However, the application of the diskline model to the disc emission lines in the optical band can be a powerful tool for providing a reliable geometrical description of these sources, allowing us to give precise estimates of the inclination angle, and also to gather information on the expected temperature and ionisation level of the emitting region. Further analyses similar to the analysis we presented here will be useful to provide additional evidence regarding the reliability of this method and to fully explore its capabilities. For example, applying the method to an eclipsing source would represent an optimal approach to test its validity. It would also be interesting to ascertain whether di ff erent emission lines within the same spectrum yield di ff erent insights. 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2024MNRAS.535.1778E	PLATO will begin observing stars in its Southern Field LOPS2 after its launch in late 2026. By this time TESS will have observed the stars in LOPS2 for at least four years. We find that by 2025 on average each star in the PLATO field will have been monitored for 330 d by TESS with a subset of stars in the TESS continuous viewing zone having over 1000 d of monitoring. There are currently 101 known transiting exoplanets in the LOPS2 field with 36 of these residing in multiplanet systems. The LOPS2 field also contains more than 500 TESS planet candidate systems 64 exoplanets discovered by radial velocity only over 1000 bright Vlt13 eclipsing binary systems 7 transiting brown dwarf systems and 2 bright white dwarfs Glt13. We calculate TESS and PLATO sensitivities to detecting transits for the bright FGK stars that make up the PLATO LOPS2 P1 sample. We find that TESS should have discovered almost all transiting giant planets out to approximately 30 d within the LOPS2 field and out to approximately 100 d for the regions of the LOPS2 field within the TESS CVZ inlineformulatexmath idTM0003 notationLaTeXsim 20texmathinlineformula per cent of the LOPS2 field. However we find that for smaller radius planets in the range 1 4 Rinlineformulatexmath idTM0004 notationLaTeXoplus texmathinlineformulaPLATO will have significantly better sensitivity and these are likely to make up the bulk of new PLATO discoveries.	2024-12-01T00:00:00Z	['2024MNRAS.tmp.2387E', '2024MNRAS.535.1778E', '10.48550/arXiv.2409.13039', '10.1093/mnras/stae2427', 'arXiv:2409.13039', '2024arXiv240913039E']	['Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Solar and Stellar Astrophysics']	Viewing the PLATO LOPS2 field through the lenses of TESS	2,024	198	0.58	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.13039.pdf	{'Viewing the PLATO LOPS2 Field Through the Lenses of TESS': "Yoshi Nike Emilia Eschen, 1 Daniel Bayliss, 1 Thomas G. Wilson, 1 Michelle Kunimoto, 2 Ingrid Pelisoli, 1 Toby Rodel 3 \n- 1 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK,\n- 2 Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada,\n- 3 Astrophysics Research Centre, School of Mathematics and Physics, Queen's University Belfast, Belfast, BT7 1NN, UK \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': 'PLATO will begin observing stars in its Southern Field (LOPS2) after its launch in late 2026. By this time, TESS will have observed the stars in LOPS2 for at least four years. We find that by 2025, on average each star in the PLATO field will have been monitored for 330 days by TESS , with a subset of stars in the TESS continuous viewing zone having over 1000 days of monitoring. There are currently 101 known transiting exoplanets in the LOPS2 field, with 36 of these residing in multiplanet systems. The LOPS2 field also contains more than 500 TESS planet candidate systems, 64 exoplanets discovered by radial velocity only, over 1000 bright (V < 13) eclipsing binary systems, 7 transiting brown dwarf systems, and 2 bright white dwarfs (G < 13). We calculate TESS and PLATO sensitivities to detecting transits for the bright FGK stars that make up the PLATO LOPS2 P1 sample. We find that TESS should have discovered almost all transiting giant planets out to approximately 30 d within the LOPS2 field, and out to approximately 100 d for the regions of the LOPS2 field within the TESS CVZ ( ∼ 20 per cent of the LOPS2 field). However, we find that for smaller radius planets in the range 1 - 4 R ⊕ PLATO will have significantly better sensitivity, and these are likely to make up the bulk of new PLATO discoveries. \nKey words: exoplanets - techniques: photometric - planets and satellites: detection - (stars:) binaries: eclipsing', '1 INTRODUCTION': "Following the discovery of the first exoplanets in the 1990s (Wolszczan & Frail 1992; Mayor & Queloz 1995), over 5000 exoplanets have been discovered using various techniques (NASA Exoplanet Archive 2024, accessed on 9 October 2024). The transit method (Sackett 1999; Seager & Mallén-Ornelas 2003; Winn 2010) is, to-date, the most successful technique for discovering exoplanets, currently accounting for 4274 of the known exoplanets (NASA Exoplanet Archive 2024, accessed on 31 July 2024). A large fraction of this success has been due to wide-field space-based photometric surveys such as 𝐶𝑜𝑅𝑜𝑇 (Auvergne et al. 2009), 𝐾𝑒𝑝𝑙𝑒𝑟 (Borucki et al. 2010), 𝐾 2 (Howell et al. 2014), and TESS (Ricker et al. 2015). A new European Space Agency (ESA) mission searching for transiting exoplanets is scheduled for launch in late 2026: Planetary Transits and Oscillations of Stars ( PLATO ; Rauer et al. 2024). One of the main goals of PLATO is to discover terrestrial planets in the habitable zones of solar-like stars. To do this PLATO will observe stars with multiple cameras (between 6 and 24 depending on the location of the star in the field), and is estimated to achieve a precision of ∼ 50 ppm in one hour for a star at V=11 mag (Börner et al. 2024). The initial field to be observed by PLATO has now been confirmed, and is a field in the Southern Ecliptic Hemisphere known as the 'LOPS2' field (Nascimbeni et al. 2022). PLATO will observe the LOPS2 field for at least the first two years of the mission (Rauer et al. 2024). \nThe Transiting Exoplanet Survey Satellite ( TESS ; Ricker et al. 2015) has been conducting an all-sky photometric survey for transiting \nexoplanets since it was launched in 2018. Since TESS has now observed the Southern Ecliptic Hemisphere three times, and will continue to observe it in the future, all of the stars in the PLATO LOPS2 field will have a significant amount of photometric data from the TESS mission prior to the launch of PLATO . In this work, we investigate what this TESS data can tell us about the stars in the LOPS2 field, and what impact that will have for discovering exoplanets with the PLATO mission. \nIn this paper we outline the key aspects of the TESS and PLATO missions in section 2. We then describe our methodology for determining the sensitivities of TESS and PLATO to discovering transiting exoplanets in the LOPS2 field in section 3. In section 4 we present and discuss our results including transiting planets and planet candidates, planets only detected by radial velocity as well as eclipsing binaries and bright white dwarf systems in the PLATO LOPS2 field. This section also includes our results from the precision calculations and sensitivity maps. We summarise our conclusions in section 5.", '2.1 TESS': "TESS (Ricker et al. 2015) is a NASA Astrophysics Explorer mission led and operated by MIT in Cambridge, Massachusetts, and managed by NASA's Goddard Space Flight Center. TESS is in a 13.7 d elliptical orbit in a 2:1 resonance with the moon. It is performing an all-sky \nsurvey by observing sectors of 2300 deg 2 for 27.4 d each (two orbits). These sectors typically tile the ecliptic hemispheres, with overlapping regions that result in areas of longer duration coverage (see Figure 2). Around the ecliptic poles this includes the continuous viewing zones (CVZ) which are monitored for one year in each hemisphere. Temporal data gaps occur between TESS orbits and sectors, and during any periods of technical problems with the cameras or spacecraft. For a typical star in the CVZ this results in approximately 20 per cent less duration than a truly continuous coverage (Rodel et al. 2024). \nTESS consists of four f/1.4 lenses (10 cm effective aperture), each coupled to an array of four 2K × 4K pixel CCDs (with 2K × 2K of imaging pixels) with a pixel scale of 21 arcsec/pixel. Data is read out every 2 seconds and summed to 20 seconds or 2 minutes for postage stamps which are images (11 × 11 pixels) of selected stars containing the star and the pixels around it, to 200 seconds (in Extended Mission 2), 10 minutes (in Extended Mission 1) or 30 minutes (in Primary Mission) for Full-Frame Images. Data processing is used to mitigate the effects of cosmic rays and stray light from Earth or the Moon (Vanderspek et al. 2018). Information on cosmic rays and stray light are made publicly available for each sector. TESS data is then further processed on the ground by two main pipelines: the Science Processing and Operations Centre (SPOC; Jenkins et al. 2016; Caldwell et al. 2020) and the Quick Look Pipeline (QLP; Huang et al. 2020a,b; Kunimoto et al. 2021, 2022a). SPOC processes the sample of preselected stars at the 2-minute cadence (Ricker et al. 2015) as well as up to 160,000 FFI lightcurves per sector using a selection function outlined in Caldwell et al. (2020) using simple aperture photometry. QLPprocesses around 1,000,000 light curves per sector including all stars down to T=13.5 mag and M dwarfs as faint as T=15 mag using multi-aperture photometry. \nTESS has been extremely successful in discovering transiting exoplanets, beginning with pi Mensae c, a Neptune-sized exoplanet transiting a very bright (V=5.7) star in a 6.3 d orbit (Gandolfi et al. 2018; Huang et al. 2018). So far TESS has discovered 543 confirmed planets (NASA Exoplanet Archive 2024, accessed on 1 August 2024) among 7,204 TESS Objects of Interest (TOIs; Guerrero et al. 2021, accessed on 1 August 2024) out of which 5,068 are currently flagged as planet candidates (PCs; NExScI 2024, accessed on 1 August 2024). Notable discoveries to date from TESS include complex multiplanet systems e.g. TOI-178 (Leleu et al. 2021), TOI-561 (Lacedelli et al. 2021), HD 23472 (Trifonov et al. 2019; Barros, S. C. C. et al. 2022), habitable-zone super-Earths e.g TOI-700 (Gilbert et al. 2020), TOI-715 (Dransfield et al. 2024), TOI-2095 (Murgas et al. 2023), circumbinary planets e.g. TOI-1338 (Kostov et al. 2020) and TIC 172900988 (Kostov et al. 2021), gas giants orbiting M Dwarfs (e.g. Bryant et al. 2023; Kanodia et al. 2023; Eschen & Kunimoto 2024) and a planet transiting a white dwarf (WD 1856+534 b; Vanderburg et al. 2020).", '2.2 PLATO': "Planetary Transits and Oscillations of Stars ( PLATO ; Rauer et al. 2024) is an ESA M class mission which is planned to be launched towards L2 at the end of 2026. In order to discover terrestrial planets in the habitable zone of Sun-like stars, PLATO will observe a fixed field in the Southern hemisphere for at least 2 years; this is described in more detail in subsection 2.3. PLATO focuses on observing mainly FGK stars within the field. Depending on their location in the field, stars are monitored by either 6, 12, 18 or 24 cameras. In order to optimise the solar irradiation falling on the solar panels PLATO will rotate by 90 ° every three months and use this data gap to downlink data. \nPLATO has 26 cameras. 24 of these, combined in groups of 6 with a bandpass between 500 and 1000 nm and observing at a cadence of 25 s, are called normal cameras (NCAMs). The remaining 2 so called fast cameras (FCAMs) will observe only bright targets ( ∼ 300 targets of V < 8 . 5 mag in the centre of LOPS2) at a cadence of 2 . 5 s, one with a bandpass between 505 and 700 nm and one with a bandpass between 665 and 1000 nm (Rauer et al. 2024). Each normal camera has four CCDs with detector dimensions of 81.18 mm × 81.18 mm, pixel sizes of 18 𝜇 m x 18 𝜇 m and a pixel scale of 15 arcsec/pixel. Normal cameras have a field of view of 1037 deg 2 . Due to arranging the cameras in four groups of six, overlapping in the centre of the field, PLATO 's overall field of view is 2132 deg 2 . Due to the larger target sample of the normal cameras, we will only focus on the stars that they will observe in this work. \nPLATO will downlink unprocessed imagettes, which are cut-out images of selected stars containing the target and the pixels around it (typically 6 × 6 pixels) and pre-processed lightcurves every 3 months, which will be further processed on the ground and complemented with ground-based follow-up data. Final data products will be summarised into catalogues of candidates and confirmed planetary systems (Rauer et al. 2024).", '2.3 The PLATO LOPS2 Field': "In contrast to the TESS observing strategy, PLATO will focus on the single Long-duration Observation Phase South (LOPS2, see Figure 1) field in the Southern hemisphere for at least the first two years of the PLATO mission (Rauer et al. 2024). The LOPS2 field, and its selection process, is described in full in Pertenais et al. (2021) and Nascimbeni et al. (2022). LOPS2 is centered at RA= 6 h 21 m 14 . 5 s and Dec= -47 ° 53 ' 13 '' (l=255.9375 deg and b=-24.62432 deg in galactic coordinates; Rauer et al. 2024), corresponding to ∼ 5% of the sky due to the field of view of PLATO . The LOPS2 field overlaps with TESS 's southern continuous viewing zone (see Figure 2). The LOPS2 field runs all the way from the galactic plane (galactic latitude 𝑏 =-0.25 ° ) down to a galactic latitude of 𝑏 =-49 ° . The centre of the field is observed by 24 cameras, the corners of the field will be observed by 6 cameras, and intermediate overlap zones are covered by either 12 or 18 cameras.", '2.4 The PLATO Input Catalog': 'The all-sky PLATO Input Catalogue (PIC; Montalto et al. 2021) consists of 2,675,539 stars, and the stellar parameters are derived from the Gaia survey (Gaia Collaboration et al. 2016, 2023). The PIC is divided into four samples (P1, P2, P4, and P5) based on different criteria (Montalto et al. 2021; Rauer et al. 2024). P1 and P2 contain the brightest and most quiet FGK dwarfs and subgiants (V ≤ 11 mag for P1 and V ≤ 8.5 mag for P2; with estimated noise of ≤ 50 ppm/h). P4 contains M dwarfs brighter than V=16 mag, and P5 is a statistical sample covering FGK dwarfs and subgiants brighter than V=13 mag. Hereafter we will refer to samples of stars within the PIC (and each PIC subset) in the LOPS2 field by adding the prefix "LOPS2" (e.g. "LOPS2 PIC", "LOPS2 P1"). The LOPS2 PIC (v.2.0.0) contains 179 , 566 stars. The number of stars in each of the four PIC samples in LOPS2 are shown in Table 1. For the P1, P2 and P4 samples data will be available in the form of imagettes monitored at a cadence of 25 seconds, while stars in the P5 sample will have a mix of imagettes and light curves with cadences of 25, 50 or 600 seconds depending on each star (Rauer et al. 2024). \nRelying on Gaia DR3 data (Gaia Collaboration et al. 2023), the PIC \nFigure 1. The PLATO LOPS2 field. Different shades of blue represent the different number of cameras the respective area will be observed with. Left: In galactic coordinates. Right: In equatorial coordinates \n<!-- image --> \ncatalogues parameters such as magnitude, radius, mass, effective temperature of stars of interest to PLATO . The PIC also contains the Gaia DR3 flags for non single stars, dividing them between photometric, spectroscopic and astrometric binaries. The expected number of PLATO cameras observing each star, as well as the expected systematic and random noise for each star, is recorded in the PIC (Börner et al. 2024).', '3.1 TESS Monitoring of PLATO LOPS2 field': "Weuse TOPCAT (Taylor 2011), an interactive graphical viewer used to edit tabular data, to cross-match the LOPS2 PIC with the target lists of the Science Processing Operations Centre (SPOC; Jenkins et al. 2016) and the Quick Look Pipeline (QLP; Huang et al. 2020a,b; Kunimoto et al. 2021, 2022a) on Exact Values using their TIC IDs. We find that the majority of the stars will be monitored in PLATO 's LOPS2 PIC already have TESS lightcurves created by SPOC and QLP. \nWe find that bright stars within the LOPS2 PIC (the LOPS2 P1 and LOPS2 P2 sample) are covered nearly completely by SPOC and QLP. In the statistical sample (LOPS2 P5) ∼ 70% of the stars are covered by SPOC due to its limit in producing no more than 160,000 FFI light curves per sector, while QLP covers LOPS2 P5 nearly completely. The biggest difference between the two pipelines is found in the coverage of the M dwarf sample (LOPS2 P4). Due to SPOC's selection function's cutoff, only ∼ 30% of P4 are covered, while QLP covers more than 85%. However due to the P4 sample being as faint as V=16 mag several stars are also not covered by QLP. See Table 1 for the breakdown of the TESS data products for each of the LOPS2 PIC samples. We are hence able to analyse P1 and P2 in more detail by using SPOC data. P1 contains ∼ 60% F dwarfs and subgiants, ∼ 30% G dwarfs and subgiants, ∼ 10% K dwarfs and subgiants \nSo far TESS has observed the PLATO field in three of its years. Using TESS-Point (Burke et al. 2020), we determined how many sectors each star in the PIC was monitored by TESS up to year 7 \nof the mission as shown in Figure 3. The PLATO field partially overlaps with TESS 's continuous viewing zone and hence a peak of stars being observed in more than 40 sectors can be found. For these stars more than 1,000 days of data are available. Based on this bimodal distribution we divide the LOPS2 PIC stars into two samples; one with ≤ 20 sectors of TESS monitoring which we will refer to as non-continuous viewing zone (non-CVZ) and stars with > 20 sectors of TESS which we will refer to as near or within the continuous viewing zone (CVZ). We note that approximately 80% of the LOPS2 PIC stars are in this non-CVZ sample, while 20% are in the CVZ sample.", '3.2 Photometric Precision of TESS and PLATO': "Variability in light curves can be of instrumental or astrophysical origin. Both can impact transit detections. While instrumental noise is dependent on the telescope, astrophysical variability could arise from high variability and eclipsing binaries. The Combined Differential Photometric Precision (CDPP; Christiansen et al. 2012), introduced during the Kepler mission, is a metric that quantifies the photometric variability in a light curve over a particular timescale. \nFor TESS lightcurves, the 2-hour CDPP is calculated by the SPOC pipeline and made available in the header of the fits file (keyword: CDPP2\\_0 ). The CDPP was first calculated for the Kepler pipeline and is defined as the root mean square (RMS) photometric noise on transit timescales (Christiansen et al. 2012), which is 2 hours for TESS . QLP data products do not contain a photometric precision metric. In Figure 4 we plot the 2-hour CDPP for all of the 9,244 PLATO P1 stars in the LOPS2 field. As PLATO is yet to launch we rely on predictions for the photometric noise. We use the prediction lines from the PLATO Instrument Noise Estimator (PINE; Börner et al. 2024). The noise estimate curves from PINE are presented in terms of the 𝑉 band magnitude of the stars. In order to compare directly to the TESS data and curve (Kunimoto et al. 2022b), we convert these to 𝑇 band magnitude by assuming a colour term of ( 𝑉 -𝑇 =0.6), typical for solar-like stars in the PLATO P1 sample. PINE also presents photometric noise on a 1-hour timescale. To compare with TESS data we convert the PINE noise estimates to a 2-hour timescale by assuming white noise and dividing by √ 2. \nn \no \ni \nt \na \nn \ni \nl \nc \ne \nD \nRight Ascension \n<!-- image --> \n<!-- image --> \nFigure 2. Overlap of the PLATO field with TESS 's all-sky monitoring. Top: Map showing the number of sectors of TESS observing the sky in equatorial coordinates. Dark purple represents the continuous viewing zone, where stars have more than 30 sectors of data. The sky position of the PLATO LOPS2 field is shown (white line). The PLATO LOPS2 field has a large overlap with the TESS continuous and near-continuous viewing zones. Bottom Left: Number of TESS sectors within the PLATO field, plotted in galactic coordinates. TESS 's continuous viewing zone is coloured in dark purple at the left of the field. Bottom Right: Number of TESS sectors within the PLATO field, plotted in equatorial coordinates. TESS 's continuous viewing zone is coloured in dark purple at the bottom of the field. \n<!-- image --> \nTable 1. Number of stars within each PIC sample and their coverage by SPOC and QLP. \nFigure 3. Histogram showing the number of TESS Sectors observed for each of the 179,566 Target PIC stars in the LOPS2 field based on Years 1-7 of the TESS mission. The peak near 40 Sectors is due to stars in the TESS southern CVZ. \n<!-- image --> \nFigure 4. The 2-hour photometric precision as a function of 𝑇 magnitude. TESS 2h-CDPP precision for the 9,244 stars in the PLATO P1 sample in the LOPS2 field covered by SPOC (grey points). Overplotted is the noise model fit of the TESS CDPP(black line) and the estimated PLATO precision (Börner et al. 2024) for 6, 12, 18, and 24 cameras (pink, yellow, blue and green lines respectively) \n<!-- image --> \n. \nSince the PLATO noise curves do not include noise arising from stellar activity, we fit a model to the minimum CDPP of the TESS data. We use the noise model by Kunimoto et al. (2022b) and fit it to the lowest CDPP value in each magnitude bin after removing outliers with a 5 iteration 2.5 𝜎 clip. This is shown by the black line in Figure 4. We compare the measured TESS photometric precision to the estimated PLATO precision for P1 stars in LOPS2 in Figure 4. Weplot four different scenarios for PLATO based on whether the star is observed in 6, 12, 18, or 24 PLATO cameras (see subsection 2.2).", '3.3 High Photometric Variability Stars': "In order to identify stars in the PLATO P1 sample in the LOPS2 field that may have excess photometric astrophysical noise (caused by binary variability, pulsations, stellar activity variability), we calculate \nFigure 5. TESS 2h-CDPP precision for the 9244 stars in the PLATO P1 sample. Colours show the number of sigmas each star is away from the CDPP mean of the respective magnitude bin. \n<!-- image --> \nthe excess photometric noise in the TESS light curve. To do this we divide the P1 sample into bins of one magnitude in width. We calculate the mean and standard deviation of the CDPP distribution in each bin ignoring CDPP values that are above the 90th quantile of the distribution. For each star within this bin we calculate its CDPP offset from the bin's mean and record this value in multiples of the bin's standard deviation (see Figure 5).", '3.4 Sensitivity Maps': "To explore the parameter space where PLATO will have the greatest potential for new discoveries of transiting exoplanets, we create sensitivity maps for TESS and PLATO . These maps are set out in terms of planetary radius and orbital period space. They are a function of each survey's photometric precision and duration of monitoring. It is important to stress that they show only the sensitivity of the survey to transiting planets, and are agnostic as to the occurrence rate of planets at any particular radius or orbital period. \nTo calculate the sensitivity maps for TESS and PLATO we use the Transit Investigation and Recovery Application (TIaRA; Rodel et al. 2024). TIaRA works on a star by star basis. It reads in the timestamps of each star, the photometric noise in terms of a CDPP value, and the stellar radius which is obtained from the PIC. It then calculates the signal-to-noise for a large number of exoplanet transits at randomly generated radii, orbital periods, impact parameters and phases. Exoplanet transits are deemed detected with a probability based on the signal-to-noise of the transit and a gamma-function selected based on the number of transit events in the data. The details of TIaRA are fully described in Rodel et al. (2024). \nTo calculate sensitivity maps for TESS , we apply TIaRA to each star in LOPS2 P1. TIaRA reads in the available SPOC FFI lightcurve for each sector in which the star was monitored by TESS and extracts the timestamps with good photometric data (quality flags = 0). From the FITS header of the SPOC lightcurve, TIaRA reads in the 2-hour CDPP noise, the crowding metric, the stellar radius, the effective temperature and the TESS magnitude. \nWemodified TIaRA to generate 1,000 transiting planets per star uniformly distributed in log 2 space following occurrence rate studies of Hsu et al. (2019) for periods between 0.5 and 400 days and radii between 0.3 and 16 R ⊕ . We also modified TIaRA to no longer per- \nrm a minimum detectable radius cut-off. Since we are interested here in sensitivities rather than yields, we did not apply the TIaRA functions for including the geometric probability of transit or the planet occurrence rate. For the TESS sensitivity maps, we only use the available data. Hence the sensitivities will improve with more TESS observations of each star. \nAlthough we do not yet have real PLATO light curves, we can simulate PLATO data based on the estimated noise performance by PINE (Matuszewski et al. 2023; Börner et al. 2024) and the plans for the first two years of the PLATO mission (Rauer et al. 2024). Other PLATO tools like the PLATO Solar-like Light-curve Simulator (PSLS; Samadi et al. 2019) or PlatoSim (Jannsen et al. 2024) also provide similar simulations. We use the stellar parameters (radius, mass, temperature) recorded in the PIC (Montalto et al. 2021). The PIC also records an estimate for the beginning-of-life random and systematic noise (BOLrandomsysNSR; Börner et al. 2024). This noise value takes into account how many PLATO cameras will observe each star. We generate timestamps assuming an observation duration of two years, with a cadence of 25 s and one day data gaps every three months for spacecraft rotation and data downlink. This last assumption is based on the operation of the Kepler spacecraft, which also performed a single-field stare from a sun-centred orbit (Borucki et al. 2010) and had a downlink time of ∼ 0.9 days on average (García et al. 2014). We bin the timestamps into bins of 10 minutes, and scale noise accordingly, for computational efficiency. We then run TIaRA with these timestamps and noise properties in the same manner as we ran it for the TESS data. \nTIaRA outputs detection probabilities for TESS and PLATO for transiting planets over a range of orbital periods and radii for each star in the LOPS2 P1 sample. We bin this data into 12 bins in radius and 10 bins in orbital period. From this data we create the sensitivity plots for each star in the LOPS2 P1 sample. Examples of our sensitivity maps for P1 stars in the PLATO LOPS2 field are set out in Figure 6 and Figure 7.", '3.5 Known Systems': 'In addition to new discoveries, the PLATO mission will be monitoring known systems that lie within the LOPS2 field. These include known transiting exoplanet systems, unconfirmed transiting exoplanet candidate systems, and non-transiting exoplanet systems. Within LOPS2 there are also transiting brown dwarf systems, eclipsing binaries, and nearby white dwarfs that will all be of interest to the astrophysics community. \nIn order to find confirmed transiting planets in the LOPS2 field we cross-match the PIC to the transiting exoplanets in the NASA Exoplanet Archive (NASA Exoplanet Archive 2024) and TEPCat (Southworth 2011). Since the PIC only covers FGK stars brighter than V=13 mag (and M Dwarfs brighter than V=16 mag), we also perform a cone search in TOPCAT to find all remaining transiting exoplanet systems in the NASA Exoplanet Archive and TEPCat that lie within the LOPS2 field of view, irrespective of magnitude or spectral type. \nWe perform a similar cross-match for non-transiting exoplanet systems (subsection 4.5), transiting brown dwarfs (subsection 4.4), eclipsing binaries (subsection 4.6), and white dwarfs (subsection 4.7) that lie within the LOPS2 field.', '4.1 Photometric Precision': 'For the P1 sample of stars in the LOPS2 field, the TESS photometric precision is set out in Figure 4. As expected, the estimated PLATO photometric precision is significantly better than the TESS precision over all magnitudes. At the bright end ( 𝑇 =4), we estimate that the PLATO light curves should nominally improve on the TESS precision by a factor of approximately three, from 20 ppm to 7 ppm. For fainter stars the precision is more dependent on the number of PLATO cameras that will monitor the stars. For stars that are monitored by six cameras, the improvement for a 𝑇 =9 magnitude star is 43.1 ppm, from 76.3 ppm to 33.2 ppm. However for the stars monitored with 24 cameras, this improvement over TESS is 58.8 ppm, from 76.3 ppm to 17.5 ppm.', '4.2 High Photometric Variability Stars': 'The TESS CDPP values show that while most (96%) of stars within the LOPS2 P1 lie within 5 𝜎 of the precison distribution (see Figure 4 and Figure 5), there are a number of stars that have much higher CDPP noise than we would expect given their magnitudes. It is important to understand why this is occurring, as detecting transiting planets around these stars will be much more difficult than around photometrically quiet stars. Some fraction of this photometric noise could be due to systematic noise unique to TESS , such as scattered light from Earth and the Moon or other spacecraft specific noise sources (Hattori et al. 2022). As such this will not be relevant for assessing the likely precision of these stars in PLATO . However, some of the stars with high noise may exhibit true astrophysical variability, in which case we would expect such variability to also be present in the PLATO data. \nTo investigate this effect, we take the stars with a 5-sigma increase in TESS noise from the mean noise within the P1 LOPS2 sample. These are the stars plotted in yellow in Figure 5. We find that SPOC calculates a CDPP value for 9,107 of the 9,244 stars it produces lightcurves for in LOPS2 P1 sample. Out of these, 376 stars are within this higher 5-sigma noise band. We inspected these light curves in order to determine the cause of the high CDPP value. We determined that approximately 35% are due to TESS systematic noise, while the remaining 65% are due to true astrophysical variability. More detailed analysis of stellar variability in TESS is beyond the scope of this work but discussed by e.g. Audenaert et al. (2021); Fetherolf et al. (2023). The definition of the P1 sample has the requirement to contain stars with random noise below 50 ppm per hour. Using the available TESS data and the precisions for TESS and PLATO , we scale the actual CDPP values of the stars in P1 from TESS to PLATO noise for the different numbers of cameras (see Figure 8) assuming the noise is photometric and there is no systematic astrophysical noise. In Kepler data, Gilliland et al. (2011) found more astrophysical noise arising from stars than expected. PLATO will detect astrophysical noise at a higher precision than TESS can, which is not taken into consideration in the PLATO noise values. However it may be possible to use the higher precision of PLATO to model and correct for some types of stellar activity. \nAs expected from the TESS CDPP in Figure 5, there are several stars in P1 that have more noise in real data than predicted by PINE. The percentage of stars of higher noise decreases with more cameras. We find that 40.9% of the stars predicted to be monitored by 24 cameras are above the 50 ppm limit, 56.9% for 18 cameras, 60.2% for 12 cameras and 67.4% for 6 cameras. This noise is caused by astrophysical', 'TESS': 'Figure 9 also shows the radius-period space that cannot be covered by TESS or PLATO . This covers radii of < 1 R ⊕ and periods longer than 100 days. \n<!-- image --> \nPLATOFigure 6. Sensitivity maps of the multiplanetary system HD 23472 (VMag=9.73, R 𝑆 =0.7 R ⊙ , Teff=4684 K from Barros, S. C. C. et al. (2022) and a minimum 2-hour CDPP value of 84.8 ppm / h in sector 3 of TESS observations). The positions of HD 23472 b, c, d, e and f are plotted in white. Left : The sensitivity for the twelve sectors (Sectors 1, 2, 3, 4, 11, 29, 30, 31, 34, 64, 68, 69 observed; sectors 95 and 96 to be observed within the next year) observed in Year 1, Year 3 and Year 5 of the TESS mission. Right : Simulated sensitivity for PLATO data assuming two years of PLATO data with gaps of 24 hours per quarter and BOLrandomNSR noise value from the PIC. \n<!-- image --> \ninstrumental effects. These stars are required to be studied further to determine that they meet the 50 ppm noise requirement. We note the different magnitude cutoff for the different cameras in order to achieve the photometric precision of 50 ppm.', '4.3 Sensitivities': "In order to quantify the sensitivity to detect transiting exoplanets for each of the LOPS2 P1 stars, we used the sensitivity maps (subsection 3.4) to determine the smallest planet radius for which we expect to have a 50% probability of detection for different orbital periods in TESS for each star. We select orbital periods of 30 and 100 days and denote these as R\\_min,30d and R\\_min,100d respectively as listed in Table 2. \nSince transit sensitivity is strongly dependent on stellar radius, we split our sample into F, G, and K spectral types. We average all the individual TESS and PLATO sensitivity maps for each spectral type to create a combined transit sensitivity map for populations of stars in the LOPS2 P1 sample to provide an overview of the sensitivities . We also differentiate between stars in the TESS non-CVZ ( ≤ 20 sectors), > \nthe TESS CVZ ( 20 sectors). The results are shown in Figure 7. The averaged sensitivity maps for TESS and PLATO for each spectral type give insights as to where the discovery potential of PLATO lies. Confirmed planets and planet candidates to date lie within the expected discovery space of TESS as shown in Figure 7. PLATO 's sensitivity predictions open up a new discovery space for smaller planets than the ones detected by TESS around FGK dwarfs and subgiants, down to 1 R ⊕ and for planets of longer period up to 400 days due to its higher precision and its continuous observations for 2 years. \nFromaqualitative comparison, these PLATO sensitivity maps are not inconsistent with the expected planet yields by Heller et al. (2022); Matuszewski et al. (2023) and Cabrera et al. (in prep.). \nIn order to highlight the discovery space that PLATO will explore, beyond what TESS has already reached, we map out the sensitivity \ndifferences between the two missions in Figure 9. The sensitivity difference is computed by subtracting the averaged PLATO sensitivity from the averaged TESS sensitivity from Figure 7. Hence the sensitivity difference rages from -1 (blue in the plots) where PLATO is most sensitive to finding planets in comparison to TESS , to 1 (red in the plots) where TESS is most sensitive. 0 sensitivity difference (white in the plots) shows where both telescopes have similar sensitivities. We can see there are two major regions where PLATO will have greater sensitivity than TESS . \nFirstly, for stars of TESS 's non-CVZ, PLATO is predicted to find planets smaller than those TESS is typically sensitive to (below 2-4 R ⊕ depending on spectral type) as well as planets of periods longer than 30 days. For stars in the TESS CVZ,planets of longer periods are already well covered by TESS alone. Here PLATO 's most significant contribution will be in discovering planets smaller than 2 R ⊕ that are difficult for TESS to detect. \nFrom the combined transit sensitivity maps we fit a spline curve to calculate the threshold at which we expect a 50% transit detection at a given planetary radius and orbital period. We again calculate this 50% transit sensitivity for F, G, and K spectral types. The 50% transit sensitivities are calculated for the TESS lightcurves (CVZ and non-CVZ) and for the expected PLATO lightcurves. The results are shown in Figure 10. \nThe interpolation curves show the decrease in detectable radii between the TESS non-continuous viewing zone, the TESS continuous viewing zone, and PLATO for the three different spectral types. We findthe space between the TESS non-continuous zone, the TESS CVZ and PLATO to align with the blue regions in the difference plots (see Figure 9). The interpolation curves don't show a significant difference in long orbital periods which is found in the difference plots. This is due to TESS 's sensitivity being around 0.5 for long period \nFigure 7. Combined sensitivity maps for the FGK dwarfs of the P1 LOPS2 sample. Yellow and light green regions show where planets can get detected with a high sensitivity, while dark blue regions show areas where the respective telescope is not sensitive to detect planets. Confirmed planets detected by TESS in the LOPS2 field are plotted in red, while multiplanetary systems with at least one transiting planet found by TESS are marked with a red star. Planet candidates identified by TESS in the LOPS2 field are plotted in gray. Top: Sensitivity for stars in the TESS non-continuous viewing zone ( ≤ 20 sectors). Middle: Sensitivity for stars in the TESS continuous viewing zone ( > 20 sectors). Bottom: Expected sensitivity for PLATO . Left: ∼ 60% F dwarfs and subgiants. Middle: ∼ 30% G dwarfs and subgiants. Right: ∼ 10% K dwarfs and subgiants. \n<!-- image --> \nTable 2. TESS sensitivities for LOPS2 P1 stars. Only the first 10 rows are shown here; the full table is available online in machine-readable ASCII form. \nFigure 8. TESS CDPP values for the P1 sample scaled to PLATO 's precision for each camera. Due to the P1 noise cutoff at 50 ppm the different numbers of PLATO cameras, which are represented by the different colours. have different magnitude cutoffs. Scaled CDPP values above the P1 noise requirement of 50 ppm are coloured black. This affects 40.9% stars monitored by 24 cameras, 56.9% for 18 cameras, 60.2% for 12 cameras and 67.4% for 6 cameras. \n<!-- image --> \nplanets due to monotransits. Although PLATO will be doing better for long period planets as shown in the difference plots, TESS is still able to detect some planets of longer periods as shown by TOI-4562 (Heitzmann et al. 2023), a gas giant planet with an orbital period of 225 days, in Figure 7 in the LOPS2 field, and even a planet with an orbital period of 483 days, which does not lie in PLATO 's LOPS2 field and had 20 sectors of TESS data (TOI-4600 c Mireles et al. 2023). \nThe K dwarf sample in LOPS2 P1 is very small with only 859 stars, hence all three interpolation curves and sensitivity maps are not representative of the larger K dwarf sample that is contained in P5. This is especially the case for the continuous viewing zone which only includes of 175 K dwarfs and subgiants in P1. In order to obtain a more representative sensitivity curve for K dwarfs, we would require a larger sample and could include the K dwarfs from the P5 sample. Some multiplanetary systems (e.g. HD 23472) are detected although the planets lie very close to TESS 's detection limits. Since this is a K dwarf system where the sensitivity map and interpolation is not highly representative as discussed above, we look at the sensitivity of the star individually and find that the planets lie right at the boundary of the detectable space (see Figure 6). The confirmed multiplanetary systems may contain further transiting planets that are beyond the detection limits of TESS and have great potential to be detected by PLATO . The same applies to planets that have so far only been measured by radial velocity but not by transit. In particular, the lowmass and small-radius or long orbital period planets might be beyond TESS 's detection limit but within PLATO 's. \nKnowing this region where PLATO is most sensitive to discover new planets will guide ground-based follow-up surveys to achieve precisions required to detect the respective planets. PLATO has good sensitivity even below 1 R ⊕ , however the occurrence rates for this population of small radius exoplanets is currently unknown.", '4.4 Known Transiting Planets, Candidates, and Brown Dwarfs': "We find there are currently 101 confirmed transiting planets in the LOPS2 field, orbiting 80 host stars. 65 of these are single-planet systems, and we list these along with the key stellar and planetary \nproperties in Table A1. Additionally, 15 systems are multiplanet systems, and these are set out in Table A2. We note there are 25 transiting planet host stars that are currently not in the PIC since they are either too faint or a different spectral type than FGKM. A number of the known transiting planet system in LOPS2 were discovered over the past decade by ground-based transit surveys such as WASP (15 systems; Pollacco et al. 2006), HATS (11 systems; Bakos et al. 2013), NGTS (8 systems; Wheatley et al. 2017), and KELT (2 systems; Pepper et al. 2007). The remaining 44 systems are discoveries from the TESS mission, and of these 20 are in the TESS CVZ. \nThe radii and orbital periods of the known transiting exoplanets in the LOPS2 field are shown in Figure 11. There is a cluster of hot Jupiter type planets with orbital periods between 1-5 d, and radii between 10-20 R ⊕ . The remaining planets have radii spanning all the way down from Neptune radii to sub-Earth radii. We note that the majority of the currently known transiting planets in the LOPS2 field have periods less than 100 days. Only one transiting planets has a longer period, TOI-4562 (Heitzmann et al. 2023), at 225 days. We also see that most of the longer period planets (P > 20 d) are found around brighter stars that have around 1000 d of TESS monitoring and thus are in or close to TESS 's CVZ. \nThere are 15 known multiplanetary transiting systems in the LOPS2 field (see Table A2). We present these system in terms of their orbital periods in Figure 13. Again we can see the majority of these discoveries are at orbital periods < 100 d. Since it is known that many multiplanet systems are highly co-planar (Figueira et al. 2012), PLATO has the potential for discovering additional transiting planets in these systems, at either longer orbital periods or smaller radii which are challenging to detect in the TESS data. \nWe plot the sky distribution of the known transiting planets over the LOPS2 field in Figure 12. We find they are following two distributions. One is that the majority of planets discovered by TESS are in or close to TESS 's continuous viewing zone. Planets discovered further north are mainly found by other surveys such as NGTS (Wheatley et al. 2017), HATS (Bakos et al. 2013), KELT (Pepper et al. 2007) and WASP (Pollacco et al. 2006). They are also monitored by TESS and are detected by TESS , but most of these planets were already known before the launch of TESS . The planets that are primarily found by instruments other than TESS are distributed further to smaller galactic longitudes in the PLATO field. This is due to the fact that there are not many ground-based observatories at very southerly latitudes. Finally we notice a lack of planets at high galactic longitude and close to 𝑏 = 0 in galactic latitude. This is due to the field approaching the galactic plane. The more crowded regions towards the galactic plane cause more contamination for wide-field transit surveys with large pixel scales. This makes it more challenging to detect transiting planets in these crowded regions. \nWe find ∼ 500 TESS transiting planet candidates (PC) in the LOPS2 field by cross-matching from the TOI catalogue (Guerrero et al. 2021) from ExoFOP (NExScI 2024) and are flagged as PC by TFOPWG. We also plot the sky distribution of these TESS planet candidates in Figure 12. The planet candidates show a gradient towards the TESS CVZandtowardsthegalactic plane, which is simply due to the higher density of stars in the galactic plane. \nMany of these TESS planet candidates will be confirmed or ruled out before the launch of PLATO , although some may remain candidates and new candidates will be added as the TESS mission continues. Several planet candidates flagged by TESS are blended due to the large plate-scale of the TESS cameras (21 '' pix -1 ). Since PLATO's pixel scale is smaller (15 '' pix -1 ), some of these cases may be \nFigure 9. Sensitivity difference between the TESS and PLATO missions for FGK stars of the P1 LOPS2 sample. Sensitivity difference values < 0 (blue) show the regions where PLATO has a higher sensitivity than TESS . Right: F dwarfs. Middle: G dwarfs. Left: K dwarfs. Top: Sensitivity difference between stars in the TESS non-continuous viewing zone ( ≤ 20 sectors) and PLATO predictions for all stars of the respective spectral type in P1. Bottom: Sensitivity difference between stars in the TESS continuous viewing zone ( > 20 sectors) and PLATO predictions for all stars of the respective spectral type in P1. \n<!-- image --> \nFigure 10. 50% transit sensitivities for F (left) G (middle) and K (right) dwarfs in the P1 LOPS2 sample. 50% transit sensitivities are shown for TESS in the non-CVZ (blue), TESS in the CVZ (orange) and PLATO (green). The number of stars of each sample are shown in brackets. \n<!-- image --> \nresolved by PLATO photometry. Currently ruling out blended scenarios is performed with ground-based follow-up (see SG1 (Collins et al. 2018)). \nThere are also a number of known transiting brown dwarfs in the LOPS2 field that we found through manual inspection of the known systems in the LOPS2 field listed on NASA Exoplanet Archive (2024) and TEPCat (Southworth 2011). These typically have masses higher than 13 M 𝐽 , the deuterium burning limit (Burrows et al. 2001), and lower than 80 M 𝐽 , the hydrogen burning limit (Chabrier et al. 2023). Due to their similar radii, radial velocity follow-up is required to distinguish transiting brown dwarfs from transiting gas giant planets. \nWe find there are currently 7 confirmed transiting brown dwarfs in the LOPS2 field, and these are set out in Table A3.", '4.5 Known Radial Velocity Only Planets': "We also conduct a search for known non-transiting planets (detected by radial velocity alone) in the LOPS2 field using the NASA Exoplanet Archive (NASA Exoplanet Archive 2024). \nPLATO data for these systems could be interesting for many reasons. The systems could have additional planets that happen to transit, and hence could be detected by PLATO . Several of the radial velocity planets also have periods longer than 100 days and hence a transit \n<!-- image --> \nFigure 11. The radius-period distribution of the known transiting planets in the LOPS2 field. Top: Colour-coded to show the V-band apparent magnitude of the transiting planet host star. Bottom: Colour-coded to show the number of days each planet host star has been monitored by TESS . \n<!-- image --> \ngeometry may not have been ruled-out yet by TESS monitoring. If these planets do transit, PLATO may be able to detect them. Even if there are no transits in the known radial velocity systems, photometric monitoring with PLATO will provide valuable insights into the host stars' stellar parameters and stellar activity. \nThe LOPS2 field contains 43 stars hosting at least one planet that has only been confirmed by radial velocity alone. 24 of these host stars have only one planet, and we detail these in Table A4. A further 19 host stars host multiplanetary systems, and these are listed in Table A5. We plot the sky distribution of these radial velocity only host stars in Figure 12. Similar to transiting planets, we notice a lack of planets at high galactic longitude and close to 𝑏 = 0 in galactic latitude. This is due to the field approaching the galactic plane. The orbital period of these known planets range from just over 1 d to over 10,000 d, while the minimum masses range from approximately 3M ⊕ to several M J (see Figure 14). Due to magnitude limits of radial velocity detections, the majority of radial velocity planets are found around stars brighter than V=12 mag (see Figure 14). Eight radial velocity planet host stars are not included in the LOPS2 PIC \nsample due to their radius being above the cutoff or falling to close to the edge of the field. The 43 confirmed radial velocity systems in the LOPS2 field (see Table A4) show a much wider distribution of orbital periods than the transiting planets, as seen in Figure 14. In fact 27 of these radial velocity systems have planets with orbital periods longer than 100 days. Some of these radial velocity planets may not have yet received sufficient photometric monitoring to rule out transits (e.g. Kane & von Braun 2009), and PLATO will thus provide useful data for that task.", '4.6 Known Eclipsing Binaries': 'The Gaia Eclipsing Binaries Catalogue (Rimoldini et al. 2023; Mowlavi et al. 2023) identifies ∼ 115,000 eclipsing binary candidates in the PLATO LOPS2 field based on the Gaia 𝐺 -band stellar light-curves. Most of these are very faint stars, with the distribution peaking at 𝐺 =19 (Mowlavi et al. 2023). If we limit ourselves to the LOPS2 PIC sample, this is reduced to 1028 eclipsing binary candidates. \nThe TESS 𝑇 -band lightcurves have also been searched for eclipsing binaries. Prša et al. (2022) presents a catalogue of 4580 eclipsing binary stars found in the first two years (Sectors 1-26) of TESS 2-minute cadence SPOC light curves. The ephemerides for these eclipsing binaries are estimated using the Quasiperiodic Automated Transit Search (QATS; Carter & Agol 2013; Kruse et al. 2019), Eclipse Candidates in Light curves and Inference of Period at a Speedy Rate (ECLIPSR; Ĳspeert et al. 2021) and box least squares periodogram (BLS; Kovács et al. 2002) algorithms. An update to this catalogue is currently in progress (Kruse et al. 2021, Prapotnik Brdnik et al. 2025, in prep), which will include full-frame image TESS data, hence many more lightcurves. Accessing this new catalogue (Prša, priv comm), we find 1023 eclipsing binaries in the LOPS2 PIC sample. Cross-matching these with the 1028 LOPS2 PIC eclipsing binaries identified by Gaia, we find an overlap of 229 targets and 794 eclipsing binaries from TESS that are 𝑛𝑜𝑡 identified by Gaia. This is not surprising given Gaia lightcurves may be very sparsely sampled - between 16 and 259 photometric measurements (Mowlavi et al. 2023) - compared with TESS lightcurves which will have thousands of 30-minute candence photometric measurements. We plot the sky distribution of the known eclipsing binaries found by TESS and Gaia in Figure 12 and find as expected a gradient in distribution towards the galactic plane. This is also analysed in Bray et al. (2023) which notes more false positives to occur towards the \ngalactic plane.', '4.7 White Dwarfs': "Polluted white dwarfs have long hinted that exoplanets orbit white dwarfs (Zuckerman et al. 2003; Jura & Young 2014; Wilson et al. 2019). In more recent years, the detection of transiting planetary debris has provided further evidence to that (Vanderburg et al. 2015; Vanderbosch et al. 2020; Guidry et al. 2021). With the launch of TESS the first candidate transiting exoplanet was discovered orbiting a white dwarf (WD 1856+534 b; Vanderburg et al. 2020). Unfortunately, this remains the only transiting exoplanet candidate orbiting a white dwarf and it is not in the LOPS2 field. However, Jupiter-sized exoplanet candidates around white dwarfs have been directly imaged by JWST (Mullally et al. 2024). With PLATO 's bluer bandpass and fast cadence, PLATO could be well-placed to discover further \nFigure 12. Top Left: Stars hosting at least one transiting planet in the LOPS2 field. Planets detected by TESS are coloured red, Planets detected by other ground-based facilities are yellow. Multiplanetary systems are marked with a diamond, single planet systems with a circle. Top Right: Stars hosting planets detected by radial velocity in the LOPS2 field. Stars with only one detected planet are marked with a circle, stars hosting more than one confirmed planet are marked with a diamond. Bottom Left: TESS planet candidates (TFOPWG Disposition 'PC') in the LOPS2 field. Bottom Right: Eclipsing binaries in the PIC. Eclipsing binaries found by TESS (Prapotnik Brdnik et al. 2025, in prep) are marked in red, eclipsing binaries flagged by GAIA's NSS flag (Mowlavi et al. 2023) ( NSSflag =4 in PIC) are marked yellow. \n<!-- image --> \ntransiting exoplanets around white dwarfs if any are selected for monitoring. Additionally, photometric observations of white dwarfs are important for studies of pulsating white dwarfs(Córsico et al. 2019) and white dwarf - white dwarf eclipsing binaries (Burdge et al. 2019; Munday et al. 2023) . Currently, there are no white dwarf stars in the LOPS2 PIC sample as they do not match the spectral type cuts used to produce that sample (Montalto et al. 2021). However these targets could be proposed under the Guest Observer Program (GO; Rauer et al. 2024) \nWe perform a cone search of the catalogue of known white dwarfs from GAIA EDR3 (Gentile Fusillo et al. 2021) brighter than G=13mag within the LOPS2 field. We find RX J0623.2-3741 and 𝜖 Ret B lie in the LOPS2 field. RX J0623.2-3741 is a metal-polluted white dwarf (Preval et al. 2019) and hence an excellent candidate to \nhost planets. 𝜖 Ret B is within a binary system with a K subgiant (HD 27442 Mugrauer et al. 2007) that is hosting a planet (HD 27442 b) detected by radial velocity (Butler et al. 2001) as listed in Table A4. White dwarfs would have photometric precision similar to many of the stars in the LOPS2 PIC. Since exoplanet transits around white dwarfs are deeper than around FGK stars, they require less precision, hence also fainter white dwarfs in the LOPS2 field have the potential of exoplanet detection.", '5 CONCLUSIONS': "Wehave explored the potential for PLATO discoveries in the LOPS2 field, in particular focusing on the P1 sample of bright FGK stars. We have studied the existing data from the TESS mission in order \nFigure 13. Multiplanetary systems with at least one transiting planet in the LOPS2 field. The size of the star in the plot represents its stellar radius, the colour its effective temperature. Transiting planet are shown in blue circles, the circle size representing their radius. Their distance from the star on the plot corresponds to the planet's orbital period. Systems that also contain a planet that has no transit detection, but by Radial Velocity or TTV are marked by a triangle or square respectively. \n<!-- image --> \nFigure 14. The distribution of minimum mass and orbital period for radial velocity only planets in the LOPS2 field. The colour scale give the V-band apparent magnitude of the host star. \n<!-- image --> \nto understand the LOPS2 P1 sample more fully, and to work out the planets that have already been discovered within the LOPS2 field. We find that there are currently 101 known transiting exoplanets in the LOPS2 field, along with ∼ 500 TESS planet candidates. Several studies by Heller et al. (2022), Matuszewski et al. (2023), and Cabrera et. al. (in prep.), which are summarised in Rauer et al. (2024), show an expected yield of up to 5,000 planets in the field. For all spectral types, TESS has very high sensitivity to discovering hot Jupiter type planets, and therefore we expect PLATO to discover very few if any \nnew transiting hot Jupiters. For spectral types of F, G and K TESS is less sensitive to discovering exoplanets < 2 R ⊕ at periods of more than 30 days, whereas PLATO should have a higher sensitivity to detect such planets. \nFor the 20% of stars in the LOPS2 P1 sample that lie within the TESS CVZ, the TESS mission has good sensitivity all the way out to orbital periods of 300 d. For these stars, we expect TESS will discover the long period giant planets prior to the launch of PLATO . PLATO 's long period contribution for these stars will therefore be limited to planets with radii < 4 R ⊕ . However for the remaining 80% of stars in the LOPS2 P1 sample that are in the TESS non-CVZ, we expect PLATO will discover long period planets down to Earth-size planets which TESS was not able to detect due to limited duration monitoring. Combining TESS and PLATO data could also improve sensitivities, especially towards long periods. Our work shows the contribution of the TESS mission has opened up the long period and small radius space of exoplanets which will be advanced by PLATO .", 'ACKNOWLEDGEMENTS': "The authors gratefully acknowledge the European Space Agency and the PLATO Mission Consortium, whose outstanding efforts have made these results possible. \nThis paper includes data collected by the TESS mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST). Funding for the TESS mission is provided by NASA's Science Mission directorate. We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. This research has made use of the Exoplanet Followup Observation Program website, which is \noperated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. IP acknowledges support from the UK's Science and Technology Facilities Council (STFC), grant ST/T000406/1.", 'DATA AVAILABILITY': 'We make our created catalogue available in a machine readable format. The light curves used for our CDPP calculations are already available as high-level science products (HLSPs) on MAST, from SPOC.', 'REFERENCES': 'APPENDIX A: CONFIRMED TRANSITING AND RADIAL VELOCITY PLANETARY SYSTEMS AND TRANSITING BROWNDWARFS IN THE PLATO LOPS2 FIELD \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \nTable A1. Confirmed transiting planet host stars with only one confirmed planet to date in the PLATO LOPS2 Field. Stellar parameters and the planet radius and the orbital period of each confirmed planet are obtained from the Exoplanet Archive (NASA Exoplanet Archive 2024, accessed on 9 October 2024) and the number of PLATO cameras the stars will be observed with are obtained from the LOPS2 PIC. For planets that are currently not in the LOPS2 PIC the number of cameras is marked with a minus. MNRAS 000 , 1-16 (2024) \nTable A3. Transiting Brown Dwarfs in the LOPS2 Field. \nTable A2. Confirmed multiplanetary systems with at least one transiting planet in the PLATO LOPS2 Field. Stellar parameters and the planet radius and the orbital period of each confirmed planet are obtained from the Exoplanet Archive (NASA Exoplanet Archive 2024, accessed on 9 October 2024) and the number of PLATO cameras the stars will be observed with are obtained from the LOPS2 PIC. For planets that are currently not in the LOPS2 PIC the number of cameras is marked with a minus. \n- 1 TOI-1338 A c with a period of 215.5 days was only detected through radial velocity but not transit.\n- 2 TOI-199 c with a period 273.69 days was detected through TTV but not transit.\n- 3 TOI-4562 c with a period 3390 days was detected through TTV but not transit. \nTable A4. Confirmed radial velocity planet host stars with only one confirmed planet to date in the PLATO LOPS2 Field. Stellar parameters and the planet radius and the orbital period of each confirmed planet are obtained from the Exoplanet Archive (NASA Exoplanet Archive 2024, accessed on 24. May 2024) and the number of PLATO cameras the stars will be observed with are obtained from the LOPS2 PIC. For planets that are currently not in the LOPS2 PIC the number of cameras is marked with a minus. \nTable A5. Confirmed multiplanetary systems in the LOPS2 field with at least one planet that has only been detected through radial velocity. Stellar parameters and the planet radius and the orbital period of each confirmed planet are obtained from the Exoplanet Archive (NASA Exoplanet Archive 2024, accessed on 24. May 2024) and the number of PLATO cameras the stars will be observed with are obtained from the LOPS2 PIC. For planets that are currently not in the LOPS2 PIC the number of cameras is marked with a minus.'}
2024arXiv240811536D	Recent observations have revealed rare previously unknown flashes of cosmic radio waves lasting from milliseconds to minutes and with periodicity of minutes to an hour 14. These transient radio signals must originate from sources in the Milky Way and from coherent emission processes in astrophysical plasma. They are theorised to be produced in the extreme and highly magnetised environments around white dwarfs or neutron stars 58. However the astrophysical origin of these signals remains contested and multiple progenitor models may be needed to explain their diverse properties. Here we present the discovery of a transient radio source ILT J11015521 whose roughly minutelong pulses arrive with a periodicity of 125.5 minutes. We find that ILT J11015521 is an M dwarf white dwarf binary system with an orbital period that matches the period of the radio pulses which are observed when the two stars are in conjunction. The binary nature of ILT J11015521 establishes that some longperiod radio transients originate from orbital motion modulating the observed emission as opposed to an isolated rotating star. We conclude that ILT J11015521 is likely a polar system where magnetic interaction has synchronised the rotational and orbital periods of the white dwarf 9. Magnetic interaction and plasma exchange between the two stars may generate the sporadic radio emission. Such mechanisms have been previously theorised 1013 but not observationally established.	2024-08-01T00:00:00Z	['arXiv:2408.11536', '10.48550/arXiv.2408.11536', '2024arXiv240811536D']	['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Solar and Stellar Astrophysics']	A white dwarf binary showing sporadic radio pulses at the orbital period	2,024	198	0.64	['EPRINT_HTML', 'EPRINT_PDF']	8	https://arxiv.org/pdf/2408.11536.pdf	{'No Header': '5', 'A white dwarf binary showing sporadic radio pulses at the orbital period': "I. de Ruiter 1* , K.M. Rajwade 2 , C.G. Bassa 3 , A. Rowlinson 1,3 , R.A.M.J. Wijers 1 , C.D. Kilpatrick 4 , G. Stefansson 1 , J.R. Callingham 3,5 , J.W.T. Hessels 1,3,6,7 , T.E. Clarke 8 , W. Peters 8 , R.A.D. Wijnands 1 , T.W. Shimwell 3,5 , S. ter Veen 3 , V. Morello 10 , G.R. Zeimann 9 , S. Mahadevan 11,12 \n- 1* Anton Pannekoek Institute for Astronomy, University of Amsterdam, Amsterdam, 1098 XH, The Netherlands.\n- 2 Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK. \n3 \nASTRON, Netherlands Institute for Radio Astronomy, Oude \nHoogeveensedijk 4, Dwingeloo, 7991 PD, The Netherlands. \n4 \nCenter for Interdisciplinary Exploration and Research in \nAstrophysics/Northwestern University, Evanston, 60201, USA. \nLeiden Observatory, Leiden University, PO Box 9513, Leiden, 2300 RA, \nThe Netherlands. \n6 Trottier Space Institute, McGill University, 3550 rue University, Montr'eal, QC H3A 2A7, Canada. \n7 Department of Physics, McGill University, 3600 rue University, Montr'eal, QC H3A 2T8, Canada. \n8 U.S. Naval Research Laboratory, Remote Sensing Division, 4555 Overlook Ave SW, Washington, DC 20375, USA. \n9 Hobby Eberly Telescope, University of Texas, Austin, Austin, TX, 78712, USA. \n10 SKA Observatory, Jodrell Bank, Lower Withington, Macclesfield, SK11 9FT, UK. \n- 11 \nDepartment of Astronomy & Astrophysics, 525 Davey Laboratory, Penn State, University Park, PA, 16802, USA. 12 Center for Exoplanets and Habitable Worlds, 525 Davey Laboratory, Penn State, University Park, PA, 16802, USA. \nCorresponding author(s). E-mail(s): [email protected]; Contributing authors: [email protected];", 'Abstract': 'Recent observations have revealed rare, previously unknown flashes of cosmic radio waves lasting from milliseconds to minutes, and with periodicity of minutes to an hour [1-4]. These transient radio signals must originate from sources in the Milky Way, and from coherent emission processes in astrophysical plasma. They are theorised to be produced in the extreme and highly magnetised environments around white dwarfs or neutron stars [5-8]. However, the astrophysical origin of these signals remains contested, and multiple progenitor models may be needed to explain their diverse properties. Here we present the discovery of a transient radio source, ILT J1101 + 5521 , whose roughly minute-long pulses arrive with a periodicity of 125.5 minutes. We find that ILT J1101 + 5521 is an M dwarf white dwarf binary system with an orbital period that matches the period of the radio pulses, which are observed when the two stars are in conjunction. The binary nature of ILT J1101 + 5521 establishes that some long-period radio transients originate from orbital motion modulating the observed emission, as opposed to an isolated rotating star. We conclude that ILT J1101 + 5521 is likely a polar system where magnetic interaction has synchronised the rotational and orbital periods of the white dwarf [9]. Magnetic interaction and plasma exchange between the two stars may generate the sporadic radio emission. Such mechanisms have been previously theorised [10-13], but not observationally established.', '1 Main': "ILTJ110160 . 52+552119 . 62 (ILTJ1101+5521 hereafter) was discovered in a commensal transient search of Low-Frequency Array (LOFAR) all-sky survey data (LOFAR Two-Metre Sky Survey, LoTSS [14]), to detect radio transients on timescales of seconds to hours using radio images [15]. A single bright radio pulse from ILT J1101+5521 was detected in data from February 8th, 2015, using 8-second snapshot images. We detected six additional pulses in other archival LOFAR data. ILT J1101 + 5521 is localised to RA (ICRS) = 11 h 01 m 50 . s 5 ± 1 . '' 9 and Dec (ICRS) = +55 · 21 ' 19 . '' 6 ± 0 . '' 39, equivalently ( l , b )= (150 . 4551 · ± 0 . 0004 , 55 . 5200 · ± 0 . 0001) in Galactic coordinates. In total, we discovered seven radio pulses lasting between 30 to 90 seconds, with peak flux densities ranging from 41 ± 6 to 256 ± 10 mJy/beam in five LoTSS observations, lasting eight hours each and spanning the years 2015 to 2020. A new 16-hour LOFAR monitoring campaign at the end of 2023 yielded no additional detections. A summary of all LOFAR observations and detected pulses is given in Extended Data Table 1. The Very Large Array's VLITE archive was also searched for pulses from ILTJ1101 + 5521 (Methods), but no additional pulses were found. \nFigure 1a shows the light curves from the five observations in which we detected radio pulses (see Extended Data Table 1). We use the times of arrival (ToAs) of the pulses to determine a phase-connected timing solution (Methods) with a period of \n125 . 51950 ± 0 . 00004 minutes. Furthermore, we are able to obtain a 3 σ upper limit on the period derivative of 3.04 × 10 -11 s s -1 . Figure 1a shows both the pulses and the non-detections at times when we expect to find pulses based on the periodicity. The pulses have a 2% duty cycle, and the intermittency of the pulses combined with the non-detections in the 2023 follow-up observations (not shown in Figure 1a), indicate that the source is intrinsically highly variable in nature: we detect a pulse in 7 out of 26 observed periods. \nThe pulses are visible across the observed range of radio frequencies (120 -168 MHz). For the brightest pulse in our sample, we find that the spectral index of the pulse is extraordinarily steep and we determine the spectral index to be α = -4 . 1 ± 1 . 1 (Methods), with S ν ∝ ν α , where S ν is the flux density and ν the observing frequency. Additionally, for the brightest pulse in our sample, we determine the linear polarization fraction to be 51 ± 6% for a Faraday rotation measure (RM) of 4 . 72 ± 0 . 14 rad m -2 . No circularly polarized emission is detected in the pulses, with an upper limit of < 1 . 6% on the circular polarization fraction for the brightest pulse. For the brightest pulse we reprocessed the raw imaging data with a 1-second time resolution (Figure 1b), which reveals multiple sub-components in the pulse. Additionally, the high-time-resolution data allows us to constrain the dispersion measure (DM) to 16 ± 6 pc cm -3 . \nSimultaneous Swift X-ray Telescope (XRT) observations were carried out during the 2023 LOFAR observations, yielding an upper limit on the quiescent X-ray luminosity of L < (1 -1 . 6) · 10 30 · [ d 504 pc ] 2 erg s -1 (Methods). \nA search in archival multi-wavelength datasets on the coordinates of ILT J1101 + 5521 resulted in a match with a star catalogued in the Sloan Digital Sky Survey, SDSSJ110150.52+552119.9 [16], with an r-band magnitude r = 20 . 86 ± 0 . 05, whose Gaia Data Release 3 (DR3) position [17] is 0 . '' 44 offset from ILT J1101 + 5521 , but within the astrometric uncertainty of the radio-derived position of ILT J1101 + 5521 (error in (RA,Dec)= ± (1 . 9 '' , 0 . 39 '' )). The probability of the optical source aligning with ILT J1101 + 5521 by chance is extremely small ( p < 0 . 00014; Methods) due to the low stellar density at high Galactic latitudes. The geometric distance to this star based on Gaia Early Data Release 3 (EDR3) data is 504 +148 -109 pc [18]. \nSpectroscopic follow-up determined the spectral type of the star to be M4.5V and showed that the star has a significant radial velocity variation of ∼ 200 kms -1 . The radial velocity as a function of time is shown in Figure 2. We fit two sinusoids to the data, one where all fit parameters are unbound and one where we fix the period to the 125-minute period of the radio pulses. A simple sinusoid describes the data well, indicating a close-to-circular binary orbit. There is no significant difference in goodness-of-fit between the two fits (Bayesian Information Criterion, BIC= 51.3 versus BIC=51.9), clearly showing that the period of the radio pulses is tied to the binary period. Figure 2 additionally shows the predicted radio pulse arrival time, according to the phase-connected timing solution. We find that the radio pulses are all emitted when the M dwarf is at superior conjunction with respect to its companion; that is, the M dwarf is seen to be behind and in line with the companion star from the perspective of an observer on Earth. Given the small chance alignment probability, the agreement between the dispersion measure of the radio pulses and the distance to the star (Methods), and the periodicity of the radio pulses being equal to the orbital \n<!-- image --> \nFig. 1 : a) Light curves (flux density versus time) of the LOFAR-detected pulses (blue) and the non-detections (grey) at times when we expect to find pulses based on their 125-minute periodicity. The y-axis shows the flux density in arbitrary units and the x-axis shows time in seconds (bottom) and duty cycle (top). The pulses are aligned according to the measured period, with period derivative assumed to be zero. The orange-shaded region indicates a 2% duty cycle window, where the pulses have been observed from 2015 to 2020. The relative flux scaling between pulses is consistent. b) Temporal profile (top) and the dynamic spectrum (bottom) of the brightest pulse in our sample, shown in dark blue in a. The time resolution is 1 second. Note that this dynamic spectrum is not dedispersed. 4 \n<!-- image --> \nFig. 2 : Radial velocity of the M dwarf associated with ILT J1101+5521 , as calculated from MMT-Binospec (black points) and HET-LRS2-R (grey point) observations. The orange line shows the best fit to the data, leaving all parameters free (BIC=51.3). The blue line shows the best fit to the data if we fix the period to be the 125-minute period of the radio pulses (BIC=51.9). The red dashed vertical lines show the predicted pulse arrival time, according to the phase-connected timing solution. \n<!-- image --> \nperiod of this star, we conclude that ILT J1101 + 5521 is a binary system where one of the components is an M4.5V star. \nThe SDSS ugriz [16] and Pan-STARSS PS1 griz magnitudes [19] yield color-color diagrams in which the candidate optical counterpart is consistently offset towards the blue with respect to the main locus of stars, suggesting that the binary companion to the optical star could be a white dwarf. We note that the blue excess could, in principle, alternatively originate from a star that is highly irradiated by a companion neutron star, as seen in some neutron star - M dwarf binary systems [20]. However, when compared to the spectrum of such systems, we do not see any evidence that would suggest significant irradiation of the M dwarf. Figure 3 shows a simple leastsquares fit to the broadband photometry with a model that allows for a main-sequence star only (top panel), and a model that allows for a main-sequence star and a white dwarf (bottom panel). The difference in Bayesian Information Criterion for both fits, BIC = 88 and BIC = 65, respectively, is very strong evidence that the latter model, including a main-sequence star and a white dwarf, is preferred. We constrain the M dwarf mass, white dwarf mass, temperature, and distance through a Markov-Chain Monte-Carlo likelihood analysis [21] (Methods). We show that an alternative scenario in which the binary consists of two M dwarfs is excluded by the lack of photometric variability (Methods) and conclude that the M dwarf companion is a white dwarf. \nUsing the radial velocity amplitude measurement presented in Figure 2 we apply the binary mass function to constrain the mass of the white dwarf as a function of orbital inclination, as shown in Figure 4. Here we assume a mass of 0 . 188 M ⊙ for the M dwarf (which is a result of the broadband photometry fit, 0 . 188 ± 0 . 01 M ⊙ , see Figure 3), a period of 125.52 minutes, and a radial velocity amplitude of 103 ± 13 km / s. \nFig. 3 : Broadband photometry (SDSS ugriz and 2MASS JHK s ) for ILT J1101 + 5521 fit to a single M dwarf model and a combination of an M dwarf and a white dwarf model. An M dwarf with a mass of M = 0 . 22 M ⊙ at a distance of d = 410pc produces the fit in the top panel. An M dwarf with a M = 0 . 18 M ⊙ and a white dwarf with a mass of M = 0 . 63 M ⊙ , and effective temperature of T eff = 5156K at a distance of d = 322pc produces the fit in the bottom panel. The Bayesian Information Criterion (BIC) for both fits is shown in the top-left of the plot. The residuals are defined as ( O i -C i ) 2 /σ 2 i , where O i is the observed value, C i is the model value, and σ i is the error on the observed value. The reduced chi-squared ( χ 2 ν ) is the sum of the residuals divided by the degrees of freedom (number of data points minus number of fit parameters). \n<!-- image --> \nWe find that even for a white dwarf companion as light as 0 . 2 M ⊙ , the inclination of the system has to be smaller than 40 · . In the known sample of white dwarf - M dwarf binaries, the white dwarf typically has a mass larger than 0 . 6 M ⊙ [25]. Extended Data Figure 8 is a more detailed version of Figure 4, showing that for white dwarf masses above 0 . 2 M ⊙ the point of gravitational equipotential between the two stars (Roche-Lobe radius) would equal the stellar radius of the M dwarf. \nThe radio pulses are incompatible with typical low-frequency stellar radio emission from M dwarfs in terms of luminosity (by five orders of magnitude) and polarimetric properties [26, 27] (Methods). Therefore, we conclude that the radio emission originates from the white dwarf or the interaction between the white dwarf and the M dwarf. The high linear polarization fraction indicates the presence of strongly ordered \nFig. 4 : Binary mass function, with the black line showing the allowed companion mass as a function of system inclination (the shaded region around the black line indicates the 1 σ error). A radial velocity amplitude of 103 ± 13 km/s and an M dwarf mass of 0 . 188 M ⊙ are assumed. In the background we show typical mass ranges of isolated white dwarfs in dark red (0 . 5 -0 . 7 M ⊙ [22]), and the full range of known white dwarf masses (0 . 17 -1 . 35 M ⊙ [23, 24]) in the lighter shade of red. \n<!-- image --> \nmagnetic fields, often found around white dwarfs [28]. Magnetic white dwarfs are the only systems next to neutron stars that are confirmed to emit coherent and highly linearly polarized radio pulses. AR Scorpii [29] and J1912 -4410 [30] are examples of white dwarf binaries that show periodic radio emission, with pulse periods on the order of minutes and orbital periods of about four hours (Methods). \nThe origin and evolution of magnetic white dwarfs in close binary stars are described in detail in [9]. The high magnetic field of the white dwarf originates from a crystallization- and rotation-driven dynamo. A strong white dwarf magnetic field can connect with the field of the M dwarf and provide a synchronising torque on the white dwarf spin. AR Scorpii and J1912 -4410 are thought to be in the beginning stages of the synchronisation process. The shorter orbital period of ILT J1101 + 5521 indicates that the binary system is in the polar stage, where the synchronisation process is complete and the M dwarf fills its Roche-Lobe again. Additionally, the pulses from ILTJ1101 + 5521 show a periodicity that is consistent with the orbital period. This is in strong contrast to AR Scorpii and J1912 -4410, where the pulse and orbital periods differ by a factor of more than 40. For polars, the magnetic field strength of the white dwarf has increased to around B ∼ 1 MG. The formation of an accretion disk is suppressed, but accretion does occur directly onto the magnetic pole of the white dwarf. \nPolars enter states with little to no accretion, and during these times the system appears as a practically detached white dwarf plus M dwarf system[25, 28]. No sustained accretion seems to occur for ILT J1101 + 5521 based on the lack of X-ray emission [31]. Additionally, the presence of radio pulses indicates little to no accretion, \nwhich would likely disturb the creation of coherent radio emission, similar to statetransitioning millisecond pulsar systems (e.g. [32]). A 125-minute period with a M4.5V spectral type donor as observed for ILT J1101 + 5521 sits well within the population of polars; see Figure 7 in [33]. Typical polars have a white dwarf mass ( ∼ 0 . 6 M ⊙ ) and white dwarf effective temperatures of less than 11000 K [25]. The temperature of the white dwarf in ILT J1101 + 5521 is likely to be lower ( T eff between 4500 and 7500 K, see Extended Data Figure 9) indicating a more evolved system compared to the known sample from optical surveys, which are observationally biased to find hotter systems. \nAlthough the energy budget is such that the observed radio luminosity could originate from the rotation of the white dwarf (Methods), we argue that the more likely scenario is that the radio emission comes from the interaction of the M dwarf with the white dwarf magnetic field, as magnetic coupling between the M dwarf and white dwarf is confirmed to occur for polar systems [9, 28]. The exact mechanism that produces the radio emission is unknown but, given the polar configuration, it seems most natural that we observe pulsed radio emission due to beaming effects. Here we observe the system in a certain geometry (superior conjunction, when the M dwarf is seen to be in line with and behind the white dwarf) once per orbital cycle, effectively looking down a beam of radio emission. In this case, the highly intermittent nature of ILT J1101 + 5521 (pulses are seen in one quarter of the observed orbits) could be explained by a strong variation in the brightness of the coherent radio emission, similar to what has been proposed for Rotating Radio Transients (RRATs) [34]. The steep spectral index for ILT J1101 + 5521 (brightest pulse α = -4 . 1 ± 1 . 1) seems consistent with what was found for and J1912 -4410 [30], where the in-band spectra suggest a steep negative spectral index of ∼ -3. These spectral indices are extreme and indicate that the spectrum could be similar to the drastic cutoff seen for electron cyclotron maser instability (ECMI) emission [35]. The radio emission could be triggered by the reconfiguration of the magnetic field (see e.g. [10, 11, 36-38] or in a more exotic scenario, the radio emission could come from accretion of material onto the magnetic pole of the white dwarf (at very low accretion rates), in a similar fashion to [12] or [13]. \nWe know of a small population of long-period transient radio sources (periods longer than 10 minutes) [1-4]. None of the previously detected long-period radio sources have a known binary companion and the periodicity of the pulses is argued to originate from the spin period of either a neutron star or a white dwarf. ILTJ1101+5521 is the only long-period radio source that is confirmed to be a binary and to feature a white dwarf. Furthermore, the radio pulses from ILT J1101+5521 have been shown to occur at the orbital period, and at the time of stellar conjunction. We provide a more detailed comparison of ILT J1101+5521 to the other long-period radio sources in the Methods. ILT J1101 + 5521 reveals that there are likely multiple progenitors that can produce long-period radio pulses. More speculatively, the existence of ILT J1101+5521 may provide an analogy for understanding periodically active fast radio burst (FRB) sources [39], which could originate from highly magnetised neutron stars interacting with a massive stellar companion [40]. Lastly, the high Galactic latitude of ILT J1101 + 5521 makes it easier to study via multi-wavelength observations. This will allow us to further study the exact geometry of this binary system, the properties of the two stars, as well as the detailed emission mechanisms at play. \nAcknowledgements. We appreciate the input from Silvia Toonen's research group at the Anton Pannekoek Institute for Astronomy, and thank Manisha Caleb, Natasha Hurley-Walker, and Zorawar Wadiasingh for valuable discussions. \nThis paper is based in part on data obtained with the International LOFAR Telescope (ILT) under project codes LC3 008 , LT14 003 and DDT20 005 . LOFAR [41] is the Low-Frequency Array designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientific policy. The ILT resources have benefitted from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Universit'e d'Orl'eans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK; Ministry of Science and Higher Education, Poland. \nThis work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France 10.26093/cds/vizier . The original description of the VizieR service was published in [42]. \nThis work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/ consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. \nThis work is based, in part, on observations obtained with the Hobby-Eberly Telescope (HET), which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Ludwig-Maximillians-Universitaet Muenchen, and Georg-August Universitaet Goettingen. The HET is named in honor of its principal benefactors, William P. Hobby and Robert E. Eberly. The Low Resolution Spectrograph 2 (LRS2) was developed and funded by the University of Texas at Austin McDonald Observatory and Department of Astronomy, and by Pennsylvania State University. We thank the Leibniz-Institut fur Astrophysik Potsdam (AIP) and the Institut fur Astrophysik Goettingen (IAG) for their contributions to the construction of the integral field units.", 'Declarations': "- · Funding I.d.R. acknowledges support through the project CORTEX (NWA.1160.18.316) of the research programme NWA-ORC which is (partly) financed by the Dutch Research Council (NWO). AR acknowledges funding from the NWO Aspasia grant (number: 015.016.033). J.W.T.H. and the AstroFlash research group at McGill University, University of Amsterdam, ASTRON, and JIVE are supported by: a Canada Excellence Research Chair in Transient Astrophysics (CERC-2022-00009); the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme ('EuroFlash'; Grant agreement No. 101098079); and an NWO-Vici grant ('AstroFlash'; VI.C.192.045). \nBasic research in radio astronomy at the U.S. Naval Research Laboratory is supported by 6.1 Base Funding. Construction and installation of VLITE was supported by the NRL Sustainment Restoration and Maintenance fund. SM, GS, EF acknowledge funding from NASA XRP Grant 80NSSC24K0155 \n- · Conflict of interest/Competing interests The authors declare no competing interests.\n- · Ethics approval and consent to participate Not applicable\n- · Consent for publication Not applicable\n- · Data availability The data that support the findings of this study will be made available on Zenodo.\n- · Materials availability Not applicable\n- · Code availability The data that support the findings of this study will be made available on Zenodo.\n- · Author contribution I.d.R performed the transient search on LoTSS data and led the writing of the paper with suggestions from all co-authors. I.d.R. is the PI of the LOFAR follow-up observations and performed the analysis of the radio data and optical spectra. K.M.R. undertook the timing and dispersion measure analysis. C.B. developed the methods to fit the broadband photometry. A.R., R.A.M.J.W. and T.W.S. contributed to the development of the transient pipeline. A.R., R.A.M.J.W. and J.W.T.H. helped steer the project and plan follow-up observations.\n- C.B., K.M.R. and S.t.V. commissioned the simultaneous beamformed and interferometric observations for the LOFAR follow-up proposal. C.D.K. performed the MMT spectroscopic observations. G.S., S.M., E.F. acquired the HET LRS2 data and G.Z. performed the reductions. J.R.C. contributed to the analysis of the spectral data and writing of the sections on stellar activity. G.S. analysed the ZTF data. W.P. and T.E.C. led the search through archival VLITE data. R.A.D.W. performed the analysis of the UVOT data. V.M. allowed the use of the periodicity search code Altris .", 'LOFAR observations and pulse search analysis': "The Low-Frequency Array (LOFAR) [41] is a radio telescope that is comprised of many thousands of dipole antennas grouped into stations. The LOFAR Two-Metre Sky Survey (LoTSS; [14]) aims to image the whole northern sky using 3168 pointings. The survey has had two major data releases so far, Data Release 1 (DR1) [43] covering 58 pointings and Data Release 2 (DR2) [44] covering 814 pointings, spanning 5635 deg 2 . LoTSS observes between 120 and 168 MHz. The flux densities of detected sources are referenced to a central frequency of 144 MHz. \nIn [15] a transient detection pipeline is described that looks for transient sources in 8-second to 1-hour image snapshots. The transient detection pipeline uses the Live Pulse Finder ( LPF ) [45]. Transient candidates are marked for further inspection if their position in the snapshot image does not correspond to the position of a known source from the LoTSS catalogue, indicating that the source is only visible for a small part of the full 8-hour integration. This transient detection pipeline was tested on a small subset of 58 LoTSS pointings, corresponding to the DR1 fields [15]. For each of \nExtended Data Figure 1 : Gallery of ILT J1101+5521 pulse profiles. For each pulse both the light curve and the SNR as a function of time are shown. The crosses indicate the initial values for the flux and SNR that triggered the detection of this pulse. These detections are done with a quick source finder ( LPF ) on lower-quality (but 'quick') images. The blue light curves indicate the full pulse profile determined with a sophisticated source finder ( PySE ) using images that are of higher quality. \n<!-- image --> \nthese pointings, only the most sensitive part of the primary beam is considered (an inner circular region with a radius of 1.5 · ). \nILTJ1101 + 5521 was detected in the P164+55 pointing in two 8-second snapshots and the 2-minute snapshot that encompasses this interval. Ref. [15] lists all the steps that were undertaken to test whether ILT J1101 + 5521 is a genuine astrophysical source or whether the signal is an imaging artefact. Once ILT J1101 + 5521 was firmly established as an astrophysical source, additional imaging steps were undertaken to characterise it. This included re-imaging the raw visibility data (without subtracting the sky model) with WSClean [46] with increased padding, and deconvolution cycles to obtain snapshot images with the lowest-possible rms noise. The most important imaging parameter settings are: -minuv-l 50.0 , -maxuv-l 60000 , -weight briggs -0.2 , -auto-mask 3 , -auto-thresholding 0.3 . Additionally, in these new images we use PySE , a more commonly used source finder that fully captures the properties of point sources in radio images [47]. The Extended Data Figure 1a shows the light curve corresponding to the pulse detected in the 2015 dataset. The two black crosses show the flux density (top panel) and signal-to-noise ratio (bottom \nExtended Data Table 1 : LOFAR observations of ILT J1101 + 5521 . The final column shows the individual pulse peak flux entities for observations with one or more detections, whereas '-' indicates observations without detections. : Peak flux density from the 1-second time slices is 431 mJy/beam. \npanel) of the initial detection in the transient detection pipeline (with LPF ). It is clear that the improved imaging settings optimise the signal-to-noise ratio of the pulse. \nAfter discovery of this initial pulse, we searched the LOFAR archive for data where the position of ILT J1101+5521 lies within the observed field-of-view. Four additional observations of the P164+55 field were found in the LOFAR Long Term Archive. Again, these observations had an averaged integration time of 8 seconds, the total observations were 8 hours, and the direction-independent calibrated data products were readily available. The same methods as in Ref. [15] were applied, and six additional pulses were found. Extended Data Figure 1 shows the flux and signal-to-noise ratio profiles for these pulses. The two pulses identified in the 2020-12-29 dataset are just slightly above the signal-to-noise ratio threshold of 6 used to identify candidate pulses. After the re-imaging procedure, one of the pulses turns out to be spurious. For all archival data, we tried pushing the signal-to-noise ratio down to 5 to see whether any additional pulses could be identified. The few candidates that we found turned out to disappear after re-imaging (similar to the first pulse detected on 2020-12-19, top-right of Extended Data Figure 1). An overview of the observations and detected pulses is shown in Extended Data Table 1. After establishing that we had robustly identified 7 bright pulses from this source, we obtained additional LOFAR observations through Director's Discretionary Time (project DDT20 005 ). Here we conducted simultaneous beamformed and imaging observations, in order to obtain both high spatial resolution and sub-second time resolution for any new pulses. Unfortunately, during these observations, no additional pulses were identified. The few candidates that were initially identified after dropping the detection threshold to a signal-to-noise of 5, turned out to disappear after properly re-imaging (see bottom column of Extended Data Figure 1).", 'Periodicity search and timing of the radio pulses': 'Despite the limited number of radio pulses, we performed a search for any potential underlying periodicity in their arrival times. First we computed the time of arrival (ToA) of each pulse. We then barycentred the ToAs before doing any further analysis. Typically, a template is used to compute ToAs but due to the dearth of pulses to \nExtended Data Figure 2 : Number of matched time differences between pulse ToAs as a function trial period in minutes. The red star shows the preferred period of 125.52 minutes, while the cyan and green star show the (sub)harmonics of the preferred period. \n<!-- image --> \ncreate a stable radio profile for template creation, we decided to use every pulse for the ToA calculation individually by fitting a Gaussian to each pulse to compute the ToA based on the estimated peak of the Gaussian. To get a first estimate of the period, we run the barycentered ToAs through a simple folding algorithm 1 that is often used for highly intermittent sources, such as rotating radio transients (RRATs). For ILTJ1101 + 5521 we have seven ToAs, implying 21 unique time differences. Extended Data Figure 2 shows the number of time differences (between pulse ToAs) that are matched given a certain period, trialling periods from 1 second to 360 minutes with a timestep δt = 0 . 01 seconds and maximum 2% tolerance in phase to count as a match. We find that a period of 125 . 52 minutes, indicated by the red star, lines up all the pulse ToAs to within 2% of a full cycle, and we find that this period is preferred over its harmonics. \nAdditionally, we used Altris ( 2 ) a brute-force search algorithm that fits the integer number of rotations within all the gaps between consecutive ToAs. The algorithm attempts to recursively discover the rotation counts under the assumption that a tentative solution for some n shortest gaps is known. Then, a phase model is fitted to the ToA gaps, from which a range of possibilities for the rotation count n + 1 is calculated. If no integer rotation counts are possible, the solution is discarded; otherwise, the algorithm attempts to further expand the set of new tentative solutions for the first n +1 gaps.Using this technique, we found the best period of 125.34 minutes \nFigure 1a shows the light curves of the LOFAR detections folded with a period of 125.52 minutes. The smallest time window that fully captures all pulses implies a duty cycle of ∼ 2%, as well as a jitter in the pulse arrive times of ∼ 1% of the pulse period. \n<!-- image --> \nMJD \nExtended Data Figure 3 : Timing residuals (the difference between the ToA and the ToA predicted by the timing model) divided by the period as a function of MJD. \nwith a Q-factor of 225.6 where the Q-factor is the goodness of fit metric for the period (Q= 1 σ ph where σ ph is the rms of the offset of time of the peak of the pulse from the predicted arrival time of the peak). The Q-factor is a measure of the signal-to-noise ratio of the detected period. Then, we used TEMPO2 [48] to fit a simple timing model to the barycentred ToAs. We only used the spin frequency (F0), here also a proxy for the orbital period, as a fitting parameter, keeping the best-known position (RA, Dec) as fixed parameters. We first phase-connected the 6 ToAs that were close in time to each other (spanning 10 days) to obtain an rms of 23.476 seconds. Projecting that solution back to the first ToA about 4 years before, we obtain a 1σ timing rms of 46.577 seconds. Figure 3 shows the timing residuals after subtracting the best-fit phase-connected timing solution from the ToAs. By phase-connected we mean that we can predict the exact number of rotations/orbits across the entire time span over which the pulses were observed. Since the rms is less than 1% of the period, we believe this to be a phase-connected solution. We now use this rms to obtain an upper limit of 3.04 × 10 -11 s s -1 for the period derivative regardless of the origin of the period (spin or orbital origins).', 'Astrometry of the radio pulses': "Based on the 5 pulses that were identified with peak flux densities of > 68 mJy/beam, we perform an astrometric correction on the images to get the most accurate position possible. Using three bright ( > 200 mJy) point sources close to ILT J1101 + 5521 (within 10 ' ) we define the astrometric offset as the distance between the position of these bright point sources in the snapshot images and their LoTSS catalogue position. Subsequently, we shift ILT J1101 + 5521 in each snapshot according to this \nsystematic offset. The position is then determined again in each snapshot and the final position is determined by averaging the positions weighted by the signal-to-noise ratio of ILT J1101 + 5521 in the specific snapshot. This yields (RA, Dec) (ICRS) = 165 . 4605 · ± 1 . 9 '' , 55 . 35545 · ± 0 . 39 '' , or RA (ICRS) = 11 h 01 m 50 . 5 s ± 1 . 9 '' and Dec (ICRS) = +55 d 21 ' 19 . 6 '' ± 0 . 39 '' . Additionally, we check our procedure by applying FitsWarp [49] to our images. This software is also designed to de-distort ionospheric effects in the image plane. We find a similar correction, that (within errors) matches the result from the aforementioned procedure, but with larger error bars ( ± 10 '' ) on the position.", 'Pulse polarization and Faraday rotation measure': 'We estimate the Faraday rotation measure (RM) using RM-Tools [50] from the Canadian Initiative for Radio Astronomy Data Analysis (CIRADA). For the three brightest pulses we take the 8-second time slices with the highest total intensity and create full Stokes images with 12 frequency channels. Performing the QU-fit we find rotation measures of 4 . 503 ± 0 . 015 rad m -2 for the L801324 pulse, 4 . 72 ± 0 . 14 rad m -2 for the L801366 pulse, and 4 . 457 ± 0 . 033 rad m -2 for the L801380 pulse. We note that a large polarization position angle swing over the duration of the pulse would distort this result. The RM measurements have not been corrected for the time-variable ionospheric RM contribution, which is the most likely explanation for their apparent variability. These RMs are consistent with the contribution from the smoothed Galactic foreground [51], precluding the presence of significant local RM contribution from the corona or wind of a star in the system. We find peak linear polarization fractions of 51 +6 -6 % for the L801324 pulse, 13 +5 -3 % for the L801366 pulse, and 42 +7 -5 % for the L801380 pulse. No significant circular polarization is detected at the position of ILT J1101 + 5521 in the Stokes V images for any of the pulses in our sample. Performing a forced-flux extraction in the Stokes V images, we find circular polarization fractions of < 1 . 6% for the L801324 pulse, < 1 . 6% for the L801366 pulse, and < 4 . 2% for the L801380 pulse. These estimates are limited by the local noise in the Stokes V images.', 'Dynamic spectrum and dispersion measure': 'The average time resolution of 8 seconds does not allow for a sensitive dispersion analysis. We thus extract the original data products for the observations of the brightest pulse from the LOFAR Long Term Archive (project code LT14\\_004 ), and reprocess at the highest-possible time resolution of 1 second. Figure 1b shows the temporal profile and the dynamic spectrum of the reprocessed data for the bright L801324 pulse (see Extended Data Table 1). The higher time resolution reveals that the pulse consists of at least two components. The temporal profile was obtained by running the LPF source finder on the 1-second Stokes I images with a signal-to-noise ratio threshold of three. No clear dispersion track is visible in the dynamic spectrum. We used the LORDS software suite to estimate a DM value of 16 ± 6 pc cm -3 (see Extended Data Figure 4). The Galactic DM in the direction of ILT J1101+5521 to a distance of 504 pc \nExtended Data Figure 4 : Frequency averaged, peak signal-to-noise ratio of the bright pulse presented in Figure 1b as a function of assumed dispersion measure in blue. The red dashed curve shows a Gaussian fit to the data. \n<!-- image --> \n(estimated distance to the M dwarf star) is ∼ 10 pc cm -3 based on the NE2001 electron density model [52, 53]. Within errors, the DM is consistent with the distance to the star. We note that the electron density models in directions well off of the Galactic plane ( b = 55 . 5200 · ± 0 . 0001 for ILT J1101 + 5521 ) are not constrained as well as directions closer to the Galactic plane. \nThe best estimate for the spectral index is obtained from the in-band spectrum of the brightest pulse in L801324, as presented in Figure 1b. The best fit for the first component is α = -4 . 5 ± 1 . 0, and α = -4 . 8 ± 1 . 0 for the second component, assuming S ∝ ν α , where S ν is the flux density and ν the observing frequency. Averaging over the full pulse we obtain a very steep spectral index of α = -4 . 1 ± 1 . 1. However, we caution that the spectrum is not necessarily well described as a power law.', 'VLITE observations and analysis': "The VLA Low-band Ionosphere and Transient Experiment (VLITE 3 ) [54, 55] is a commensal system capable of continuously accessing 64 MHz of bandwidth (centered on 352 MHz) from the 236 -492 MHz Low-Band system deployed on the National Radio Astronomy Observatory's Karl G. Jansky Very Large Array (NRAO, VLA). VLITE has been operational since November 2014 recording data from up to 18 antennas during nearly all regular VLA observations. Since VLITE accumulates a large amount of data, an automated calibration and imaging pipeline has been developed [54] that \ncalibrates the visibilities and produces self-calibrated images. The VLITE Database Pipeline [56] then takes these images and creates a source database using the PyBDSF [57] source finding algorithm. \nWe searched the VLITE Database at the position of ILT J1101+5521 , but found no catalogued source. We re-processed all archival VLITE observations within two degrees of ILT J1101+5521 where the VLA was in the high resolution A or B configuration. The images were broken into 60-second snapshots to match the pulse length of ILT J1101+ 5521 and maximise the VLITE sensitivity given the measured steep spectral index. The 225 resulting images were searched by PyBDSF for any source above 5 σ , and then visually inspected for emission at lower levels for the target position. In the same manner we searched all 103 VLITE Commensal Sky Survey [VCSS, 58] 28-second snapshot images from six separate days between 2017 and 2023 that cover the source position. Emission was not detected in either the VLITE or VCSS snapshots. \nILTJ1101+5521 falls between 0.9 · and 1.9 · , with an average offset of 1.3 · , from the correlation centre of the archival VLITE data, resulting in significant sensitivity loss due to the instrumental response. The VCSS snapshots are observed in highly overlapping, continuous declination scans across the sky. Similar to the targeted images, the average source position is 1.2 · from the VCSS correlation centre, although it ranges from 0.5 · to 1.7 · . Due to the telescope movement during each of the 28 second VCSS frames, there is a slight additional loss of sensitivity. In Extended Data Figure 5, we plot the 3 σ upper limits from VLITE and VCSS snapshots together with the LOFAR detections and predicted 340-MHz flux based on the measured full pulse spectral index and its uncertainty. Finally, we re-imaged the VLITE data between November 2017 and September 2023 with snapshot cadences of 8 and 2 seconds to search for signatures of very bright pulses. No emission was detected, and we place 5 σ upper limits on bright pulses at 350 and 700 mJy/beam for the 8- and 2-second cadences, respectively. Archival VLITE and VCSS observations are not sensitive to pulses as steep as measured for the brightest ILT J1101 + 5521 burst ( α = -4 . 1 ± 1 . 1) as indicated by the lower panel in Extended Data Figure 5. VLITE and VCSS would be sensitive to flatter spectrum or brighter pulses, but unfortunately the temporal coverage in the archive is sparse for this position, making it difficult to rule out such possibilities.", 'Chance alignment probability with optical counterpart': "We performed a cone search through the available optical and near-infrared catalogues using Vizier [42] around the position of ILT J1101 + 5521 . This search yields an optical source at RA (ICRS) = 11 h 01 m 50 . '' 5698 and Dec (ICRS) = +55 · 21 ' 19 . '' 738, which is 0 . '' 44 offset from ILT J1101 + 5521 , but within the astrometric uncertainty of the radio-derived position of ILT J1101 + 5521 (error in (RA,Dec)= ± (1 . 9 '' , 0 . 39 '' )). To identify the probability of chance association we extract all the sources that lie within a radius of 3 degrees around the position of ILT J1101 + 5521 in Gaia EDR3 [59, 60]. We do not apply any other filters to the Gaia catalogue. Subsequently, 10 6 random positions within this patch are drawn and we determine the number of times one of these random positions falls within half an arcsecond of a Gaia source. We find that this only happens in 143 cases, and therefore conclude that the probability of chance alignment is p < 0 . 00014. \nExtended Data Figure 5 : Top panel: 3 σ upper limits from VLITE 1-minute snapshot images (blue triangles) and the peak flux densities of the LOFAR detections (orange crosses) as a function of time. Bottom panel: distribution of 3 σ upper limits from VLITE 1-minute snapshot images (blue) and the extrapolated LOFAR peak flux densities at 340 MHz (VLITE frequency) for a spectral index of α = -4 . 1 (orange lines) and α = -3 . 0 (green dashed dotted lines). \n<!-- image -->", 'Spectroscopic observations of the M dwarf': "We observed the M dwarf associated with ILT J1101+5521 using the Binospec optical spectrograph [61] on the 6.5-m Multiple Mirror Telescope (MMT) located at the MMT Observatory on Mt. Hopkins, Arizona. The observation occurred on 4 April 2024 starting at 02:42:49 UTC. We observed the M dwarf with 6 × 900 s exposures, using a 1 . 0 '' longslit, a 270 l/mm grating, set to a central wavelength of 6560 ˚ A, and using the LP3800 blocking filter. This resulted in spectral coverage from ≈ 3900-9300 ˚ A with an average spectral resolution of R ≈ 1530. Conditions were near photometric at the time of observation, with an average seeing of 1.2 '' , and the airmass ranged from ≈ 1 . 1-1.3 throughout the observation. \nWe reduced all observations using pypeit [62], with dome flat and arc lamp calibration files obtained on the same night and instrumental configuration. We used standard pypeit parameters for Binospec, resulting in a signal-to-noise averaged over \neach optimally extracted spectrum of ≈ 2-3. We then flux calibrated each spectrum with a sensitivity function derived from the standard star G191-B2B obtained in January 2024 and corrected the spectra for telluric absorption using an atmospheric model for the MMT Observatory. \nAdditionally, we observed an optical R = 2 , 500 spectrum of the M dwarf associated with ILT J1101+5521 using the LRS2 [63] instrument on the Hobby Eberly Telescope [64, 65]. We used the LRS2-R unit on the 'red' channel from 6450 -8420 ˚ A. We reduced the LRS2 spectrum using the publicly available Panacea pipeline [66], which performs basic CCD reduction tasks, wavelength calibration, fiber extraction, sky subtraction, and flux calibration. \nExtended Data Figure 6 shows the LRS2-R spectrum and a coadded spectrum of the 6 epochs obtained with Binospec. The highlighted regions are used for radial velocity fitting below. To determine the spectral type of the M dwarf we fit the obtained spectra to a range of spectral templates, including the ATLAS-T spectra [67] obtained with the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, a suite of Keck LRIS spectra of late-M dwarfs [68], UVES/VLT high-resolution spectra of late type sub-dwarfs [69], and a complete M spectral type sequence for a sample of confirm young sources [70] 4 . We find that an M4.5V spectrum fits our data best, but we cannot confidently rule out an M4.0V star. \nNext, we use the individual epochs to look for a radial velocity signature. In the following, we opt to use the [68] M4.5V model, as the resolution of 1.9 ˚ A is similar to the binned resolution of our observations, and the wavelength range overlaps our data. To obtain the Doppler shift of the individual epochs, we select a particular wavelength range. We avoid the noisy edges of the LRS2-R spectrum and disregard data above 8300 ˚ A (see Extended Data Figure 6). For consistency, we therefore also only consider the spectral range between 6500 and 8300 ˚ A for the Binospec data. Additionally, the signal-to-noise ratio (before binning) is below 3 for Binospec data below 6500 ˚ A. In this wavelength range the Binospec data has a resolution of 1 . 3 ˚ A. We bin the data by a factor of 2 and fit the template to the data while applying a Doppler shift. \nWe note that excluding the H α line (6562 . 8 ˚ A) from the radial velocity analysis does not affect the outcome in a way that would alter our conclusions. The signalto-noise ratio in the blue part of the Binospec spectra is insufficient to observe the absorption lines we might expect from the white dwarf (e.g. [71]).", 'Swift X-ray Telescope observations and analysis': 'To search for any flaring activity of ILT J1101 + 5521 , we proposed for Swift observations during the L203* observing runs (see Extended Data Table 1). We obtained four individual visits using the X-ray Telescope (XRT; [72]) in December 2023 for a total time of 13.3 ks on source. We do not find any flares, as no photons are detected at the position of ILT J1101 + 5521 . The 3 σ upper limit that we find by stacking all observations is a count-rate limit of 1 . 43 · 10 -3 counts s -1 (0 . 3 -10 keV). Based on the position of ILT J1101 + 5521 we predict an absorbing Galactic column density of N H ≈ 8 . 7 · 10 19 cm -2 . This estimate was obtained using the web version of the nH tool [73] in HEASOFT [74]. To convert the count rate upper limit to unabsorbed flux \nExtended Data Figure 6 : Spectrum of the M dwarf associated with ILT J1101 + 5521 . The top panel shows the LRS2-R spectra obtained with the HET. The bottom panel shows a coadded spectrum of the 6 epochs obtained by the Binospec instrument on the MMT. The highlighted parts of the spectrum are used for radial velocity fitting. Both spectra have been binned by a factor of two to increase signal-to-noise. \n<!-- image --> \nlimits we use WebPIMMS 5 and assume two spectral models for a pulsar/magnetar, following Ref. [6]. In the first scenario we assume thermal emission from a young pulsar, implying a blackbody spectrum with kT = 0 . 3 keV. In the second scenario, we assume non-thermal emission from an energetic pulsar, for which the corresponding spectral scenario is a power law with index Γ = 2. These two scenarios yield unabsorbed flux limits of 3 . 3 · 10 -14 erg cm -2 s -1 and 5 . 1 · 10 -14 erg cm -2 s -1 , respectively. This translates to L < (1 -1 . 6) · 10 30 · [ d 504 pc ] 2 erg s -1 .', 'Swift Ultra-violet Optical Telescope observations and analysis': "During the Swift XRT observations described above, ILT J1101+5521 was also simultaneously observed using the Swift UV/Optical Telescope (UVOT; [75]). We use several HEASOFT [74] tasks to obtain an upper limit on the UV flux by combining all observations. uvotsource was used to combine extensions within one visit and extract source and background information, uvotimsum was used to combine the visits, and uvotsource was used to determine the flux upper limit at the ILT J1101 + 5521 position. We use an extraction radius of 15 '' on the source location and to determine the background flux. The UVOT flux limit is < 1 . 0 · 10 -17 erg s -1 cm -2 ˚ A -1 in the uvm2 filter at 2246 ˚ A. We note that the data quality of these UVOT observations \nis low. All point sources show up as double point sources which is caused by the attitude control problems of the Swift satellite during the period of early August 2023 until the beginning of April 2024 (due to problems with one of the three on-board gyroscopes; for more details see [76, 77]). Unfortunately, all our UVOT data (taken in December 2023) are affected by this issue. We take this into account by increasing our source extraction radius to 15 '' .", 'Zwicky Transient Facility photometry and analysis': 'Photometric variability might be expected from a close binary like ILT J1101 + 5521 , either at the orbital period due to irradiation of the M dwarf by the white dwarf [78] or at half the orbital period by ellipsoidal variations from the (close to) RocheLobe-filling M dwarf [79]. We analyse the z i , z r and z g photometry from the Zwicky Transient Facility (ZTF, [80]) for ILT J1101 + 5521 . Overall there are 592 datapoints publicly available in the three filters, where we rejected any points that were obtained in non-optimal conditions (non 0 quality flags) following the recommendations in the ZTF pipeline. This left a total of 464 photometric points with 37 in the z g filter, 337 in the z r filter, and 90 in the z i filter, which we used for our analysis. From the ZTF photometry, we find no evidence for modulations at the orbital period or half the orbital period in the periodograms of the ZTF photometry (Figure 7). We note that there are hints of two peaks at 70.3 min and 71.4 min seen in the z r and z i bands, respectively (highlighted in red Figure 7). However, although these two peaks do not correspond to any peaks in the window functions of the datasets, as the two periods are sufficiently different from each other and correspond to false alarm probabilities higher than 0.1%, we conclude that the data do not show significant detections of photometric variability. \nTo probe the sensitivity of the ZTF data to detecting signatures of photometric variability, we performed injection tests where we injected sinusoidal variations with a period of 125.52 minutes with various amplitudes in the data. Doing so, we place an upper limit of 0.1 mag in the z r data and 0.05 mag in the z i data. The expected ellipsoidal variations using the equations in Ref. [81] for a white dwarf - M dwarf system are expected to have ellipsoidal variation amplitudes at or below the detectability threshold of the ZTF filters, in agreement with the non-detections in the ZTF filters. \nAssuming an M dwarf radius of R MD = 0 . 217 R ⊙ and an orbital separation of a = 0 . 76 R ⊙ (for an M dwarf with mass 0 . 188 M ⊙ and a white dwarf with mass 0 . 6 M ⊙ ), we calculate the maximum inclination angle based on the fact that we do not see eclipses in the ZTF data. We find i < 74 · . Similarly, we estimate the irradiation luminosity from the white dwarf on the M dwarf. L irr = L WD · πR 2 MD 4 πa 2 = 0 . 02 L WD . We thus expect the white dwarf to at most contribute to the luminosity of the M dwarf at the level of ∼ 2%. This is consistent with the ZTF data where we do not find any evidence for photometric variability due to irradiation. Finally, we can rule out an M dwarf - M dwarf binary scenario from the lack of photometric variability in the ZTF data (at twice the orbital period). From the binary mass function, we know that a M4.5V dwarf with a sub M4.5V dwarf companion ( M < 0 . 188 M ⊙ ) needs to have an inclination i > 50 · to create the observed radial velocity amplitude. However, for short-period double M dwarf binaries with such inclinations strong photometric \nExtended Data Figure 7 : ZTF photometry of ILT J1101 + 5521 . Data are shown for three filters: (a) z g , (b) z r , and (c) z i . Error bars denote 1 σ uncertainties. (d-f) Generalized Lomb-Scargle periodograms for each filter. The red vertical line denotes the P orb = 125 . 52min orbital period of the system, and the orange vertical line denotes half the orbital period ( P orb / 2) expected from ellipsoidal variations. There are no peaks above the 0.1% false alarm probability threshold (grey horizontal dotted lines) at the orbital period or half the orbital period. In the z r and z i filters the maximum peaks correspond to periods of 70.3min and 71.4min, but are below the 0.1% false alarm probability line. The power in the periodogram is normalized following [82]. \n<!-- image --> \nvariability due to ellipsoidal variations from the (close to) Roche-Lobe-filling M dwarfs are expected and observed [79]. We expect the M dwarf to be close to Roche-Lobe filling for white dwarf masses M WD > 0 . 2 M ⊙ , see Extended Data Figure 8.', 'Broadband photometry fits': 'To constrain the parameters of a possible binary consisting of a white dwarf and a main-sequence star, we compare the observed ugriz magnitudes from the Sloan Digital Sky Survey (SDSS) Data Release 16 (DR16) [16] and the Two Micron All-Sky Survey (2MASS) JHK s [83] to absolute magnitude predictions in the same bands from white dwarf cooling models 6 [84-86] and stellar evolutionary tracks [87-89]. The models are interpolated over the stellar mass, white dwarf mass and white dwarf effective temperature, and the summed flux is corrected for the distance and absorption using an E g -r = 0 . 011 reddening value [90] and R V = 3 . 1 extinction values [91]. \nExtended Data Figure 8 : Constraints on the white dwarf mass given the observed mass function of the M-dwarf from optical spectroscopy. Constraints from orbital inclination are indicated in black with dotted, dashed, solid and dashed-dotted lines indicating an edge-on orbit ( i = 90 · ), and orbits at i = 60 · , i = 30 · and i = 20 · , respectively, with the light grey shaded area being excluded. The orange lines and shaded area indicate constraints provided by the Roche lobe around the M dwarf, indicated in fractions of the M dwarf radius of R 1 = 0 . 217 R ⊙ , where a Roche lobe radius of R L1 = 0 . 9 R 1 indicates Roche lobe overflow. The blue vertical line indicates the M dwarf mass of M 1 = 0 . 188 ± 0 . 010 M ⊙ determined from photometry. \n<!-- image -->', 'White dwarf parameters via MCMC fitting': 'Through a Markov-Chain Monte-Carlo likelihood analysis [21] the observed magnitudes are well matched for a stellar mass of M MD = 0 . 188 ± 0 . 010 M ⊙ at a distance of d = 333 ± 25 pc (see Extended Data Figure 9). The white dwarf mass and the white dwarf effective temperature range are degenerate, with T eff ranging from 4500 to 7500 K for masses of M WD = 0 . 2 to 1.3 M ⊙ . The UVOT-upper limit ( < 1 . 0 · 10 -17 erg s -1 cm -2 ˚ A -1 at 2246 ˚ A, see Methods) is consistent with white dwarfs in this temperature and mass range. The distance following from the MarkovChain Monte-Carlo likelihood analysis is smaller than the Gaia geometric distance of 504 +148 -109 pc, but the distance estimates are consistent within 2 σ . \nExtended Data Figure 9 shows the one and two-dimensional projections of the posterior probability distributions of the parameters used to fit the broadband photometry data presented in Figure 3. We use flat priors T eff ϵ { 3000 , 12000 } K, M WD ϵ { 0 . 2 , 1 . 3 } M ⊙ and M MS ϵ { 0 . 09 , 0 . 5 } M ⊙ and do not put a prior on the distance. The distance of d = 333 ± 25 pc is offset from the Gaia geometric distance estimate of 504 +148 -109 pc. We try setting a Gaussian prior on the distance ( µ = 504pc and σ = 109pc) but find \nExtended Data Figure 9 : One and two-dimensional projections of the posterior probability distributions of the parameters used to fit the broadband photometry data presented in Figure 3. This Figure was created using [92]. \n<!-- image --> \nthat the model quickly converges to a shorter distance again, with a preferred distance of d = 337 ± 25 pc.', 'Chromospheric activity from the M dwarf': 'We use the X-ray observations to investigate whether the M dwarf is chromospherically active. Usually, the X-ray luminosity of chromospherically active M dwarfs in the range 0 . 2 -2 keV varies between 2 · 10 26 -3 . 6 · 10 29 erg s -1 (see supplementary material in [27]). \nFor ILT J1101 + 5521 , assuming a soft X-ray upper limit of ∼ 1 · 10 -14 erg cm -2 s -1 and a distance of 504 pc this translates to L < 3 · 10 29 erg s -1 . Based on the X-ray limits we cannot exclude chromospheric activity. Another indicator for chromospheric activity are the chromospheric emission lines Hα and CaII which are the result from magnetic heating of the stellar atmosphere. The spectra obtained in this work show a clear Hα emission line (see Extended Data Figure 6). This emission line follows the same radial velocity pattern as the M dwarf, thus the Hα line originates from the M dwarf. Following [93] we estimate the equivalent width (EW) of the Hα emission line in the HET LRS2-R spectrum, as the signal-to-noise and spectral resolution are higher than for the combined MMT spectrum. We find the EW of Hα to be -9 . 1, clearly indicating strong chromospheric activity when compared to the sample of [93] (see their Figure 2).', 'Radio emission from the M dwarf': 'Coherent radio emission from stellar systems can be produced by two mechanisms: plasma emission that occurs at the ambient plasma frequency and its harmonics, and cyclotron emission that occurs at the ambient cyclotron frequency and its harmonics [94]. Both plasma and cyclotron emission are expected to be highly circularly polarised [95]. \nWe therefore conclude that ILT J1101+5521 cannot originate from the M dwarf, as the circular polarisation fraction of the radio pulses is less than a few percent at most. In [27] it was found that even the weakest circularly polarised M dwarf still showed a circular polarisation fraction of 38%. Furthermore, the brightest radio luminosity (assuming the distance to the star to be 504 pc) is roughly L = 7 . 8 · 10 19 erg s -1 Hz -1 , which is 5 orders-of-magnitude brighter than the population of M dwarfs that has been observed at low radio frequencies [27]. Stellar flares with significant linear polarisation fractions have been observed [26, 96, 97], but these sources were always significantly circularly polarised as well. We do not know of any physical mechanism by which the M dwarf could produce the observed radio emission. Therefore, this radio emission must originate from the interaction between the companion and the M dwarf.', 'Chromospherically active binaries': 'Close stellar binaries, such as RS Canum Venaticorum (RS CVn) variables are the most luminous stellar radio systems and show heightened levels of chromospheric activities. The typical X-ray luminosity of chromospherically active binaries is > 10 30 ergs/s [98100]. However, RS CVn systems with X-ray luminosities as low as 10 29 ergs/s have also been found [99]. For ILT J1101 + 5521 , the X-ray limit is L < (1 -1 . 6) · 10 30 · [ d 504 pc ] 2 erg s -1 . A stronger argument against a RS CVn origin for ILT J1101+5521 is the relation between the X-ray and radio luminosity of such systems, the Gudel-Benz relation: L X = 9 . 48 · 10 18 L 0 . 73 ν,rad [101]. For the radio luminosities of the detected pulses ( L ν,rad = 1 . 4 -7 . 8 · 10 19 [ d 504 pc ] 2 erg s -1 Hz -1 ), this implies X-ray luminosities of L X = 0 . 9 -3 . 2 · 10 33 erg/s, which is three orders of magnitude above our X-ray \nupper limit. Deviations from the Gudel-Benz relation have been observed [99], but no systems with over an order-of-magnitude discrepancy between X-ray and radio luminosities have been found. Additionally, the radio emission from RS CVn systems is thought to be produced by the electron cyclotron maser instability, which is expected to yield a circularly polarised radio signature [95, 98]. Finally, the RV signature and optical photometry imply a mass that is consistent with a compact object - not with a similar mass object. Therefore, we conclude that ILT J1101 + 5521 is not a close chromospherically active stellar binary.', 'Comparison to AR Scorpii and J1912 -4410': 'AR Scorpii [29] and J1912 -4410 [30] are examples of white dwarf binaries that show periodic radio emission. These systems are both in a binary with an M dwarf, with orbital periods of 3.56 and 4.03 hours, respectively. Both sources show radio pulses with periods of 1.97 and 5.3 minutes, respectively, which is the beat frequency between the spin period of the white dwarf and the orbital period. The pulse period in these systems is more than 23 times shorter than for ILT J1101 + 5521 . The pulse duration is only a fraction of the spin period, and therefore much shorter (roughly an order-ofmagnitude) than the pulses observed for ILT J1101 + 5521 . For AR Scorpii there are two emission components with different polarisation properties that contribute to the radio emission. These two components are likely to be the spin-down from the rapidly rotating, highly magnetised (up to 500 MG) white dwarf and the magnetic interactions between the M dwarf and the white dwarf [102]. Although both ILT J1101 + 5521 and the aforementioned systems are M dwarf - white dwarf binaries, ILT J1101 + 5521 is likely to be in a different evolutionary state than AR Scorpii and J1912 -4410 because of the shorter orbital period and because the radio pulses are seen at the orbital period instead of a beat period between the rotational and orbital period.', 'White dwarf spin-down luminosity': 'Spin-down luminosity is defined as the maximum luminosity that can be derived from the rotation of a magnetised white dwarf or neutron star. We consider whether the radio emission from ILT J1101 + 5521 can be powered by the spin-down of the white dwarf. Assuming that the spin period of the white dwarf has synchronised to the orbital period of the binary (as is the case for polars), we calculate the spin-down luminosity of a pulsar using L spin -down = 4 π 2 I ˙ P P 3 , where P = 7531 seconds and ˙ P ≤ 3 . 04 × 10 -11 . Assuming I ≈ 10 50 g cm 2 for a typical white dwarf we find that L spin -down < 2 . 8 · 10 29 erg s -1 . The peak radio luminosity (256 mJy at 144 MHz) is L ν,rad = 7 . 8 · 10 19 [ d 504 pc ] 2 erg s -1 Hz -1 . Integrating over the LOFAR bandwidth of 48 MHz and assuming a flat spectrum over this bandwidth, we find L rad = 3 . 7 · 10 27 [ d 504 pc ] 2 erg s -1 . This implies that for a radio efficiency of ξ ∼ 10 -2 , the spin-down luminosity could power the observed radio emission. For neutron stars radio efficiencies up to 10 -2 have been observed [103], but typical radio efficiencies for pulsars are much lower ( < 10 -2 ).', 'Comparison to other known long-period transient radio sources': "The first instance of a long-period radio source was the 'Galactic Centre Radio Transient' (GCRT), which initially showed pulses with a 10-minute duration on a period of 77 minutes [1]. Follow-up observations revealed fainter and shorter 2-minute pulses [104]. Two more recent discoveries are GLEAM-X J1627 -52 [2] and GPM J1839 -10 [3], which were found to have pulse periods of 18 and 21 minutes, respectively, with pulses that last for half a minute to 5 minutes. These aforementioned long-period sources are all thought to feature a neutron star or white dwarf, as a coherent emission mechanism is required to create radio emission with the observed brightness temperatures. Finally, ASKAP J1935+2148 was recently reported with a 54-minute period [4]. For ASKAP J1935+2148, two types of radio pulses are observed, one with extremely bright tens of seconds wide, highly linearly polarised pulses, and one with weaker pulses that last only hundreds of milliseconds and are highly circularly polarised. The different emission states are similar to changes in emission that have been observed for pulsars, and the authors argue that for ASKAP J1935+2148 a neutron star scenario is most likely. In contrast to ILT J1101 + 5521 , none of the aforementioned long-period radio sources are known to be in a binary system. \nThe radio pulses from ILT J1101 + 5521 appear to be similar to the radio emission observed from other long-period sources in terms of timing, polarization and spectral properties. This is shown by the clustering of these sources in terms of radio luminosity and duration (see Extended Data Figure 10). Additionally, we find ILT J1101+5521 to be active for over 5 years, which again fits with the other long-period sources as GPM J1839 -10 was found in archival data spanning 30 years. Finally, the lack of Xray emission from ILT J1101+5521 ( L < (1 -1 . 6) · 10 30 · [ d 504 pc ] 2 erg s -1 ) is consistent with the lack of X-ray emission from GLEAM-X J1627 -52 and GPM J1839 -10 ( ≤ 10 32 -33 erg s -1 [2, 3]) and ASKAP J1935+2148 ( < 4 · 10 30 erg s -1 [4]). \nThe four known long-period sources are all located towards the Galactic Plane (Galactic latitudes between -3 · and +1 · ), complicating follow-up at optical and infrared wavelengths. None of the previously known long-period sources has a confirmed optical counterpart. We note that there might be a bias to finding long-period radio sources towards the Galactic Plane because of the nature of the surveys that were used to find them. For GLEAM-X J1627 -52, the lack of an optical/infrared counterpart at the estimated distance of 1.3 kpc does not rule out a lower-mass, mainsequence companion [2], and an AR Scorpii-like system could pass unnoticed if it has a relatively large extinction. [6]. For GPM J1839 -10, the optical limits are not strong enough to rule out a binary companion given its larger distance ( d = 5 . 7 ± 2 . 9 kpc)[3], and a similar situation holds for ASKAP J1935+2148 [4] ( d = 4 . 85 kpc). \nFast radio bursts (FRBs) are coherent radio flashes seen from other galaxies [106]. One repeating FRB source, FRB 20180916B, has a well-established 16.33-day activity period, where the observed bursts systematically occur at lower radio frequencies at later times during the activity window [39, 107]. Another repeating FRB source, FRB 20121102A, has a candidate activity period of about 160 days [108, 109]. The periodic activity of these FRB sources is theorised to originate from orbital motion, rotation, or precession (Ref. [40] and references therein). The discovery of \nExtended Data Figure 10 : The luminosity of different types of radio transients as a function of their width (W) and frequency ( ν ), adapted from [4]. Diagonal lines represent constant brightness temperatures. The brightness temperature of 10 12 K roughly separates coherent emitters from the incoherent ones, with the shaded region (lower-right triangle) encapsulating the incoherent emitters. The long-period sources presented in [1, 3, 4, 105] shown in the legend appear to cluster together as indicated by the box, which is merely to highlight the sources and does not have a physical significance. The pulses from ILT J1101 + 5521 are indicated with the green squares. The green squares with a white face colour are the two sub-pulses as resolved in the higher time resolution data, see Figure 1b. The pulses from ILT J1101 + 5521 fit with the population of long-period sources. \n<!-- image --> \nILTJ1101+5521 presents a possible analogy for periodically active FRBs, which could originate from highly magnetised neutron stars (potentially 'magnetars') interacting with a massive stellar companion [40], and which would have greatly scaled-up energetics compared with a white dwarf system. FRB 20180916B is at a luminosity distance of about 150 Mpc, and its millisecond-duration bursts are at least trillions of times more luminous than ILT J1101 + 5521 [110]. The 16.33-day activity period of FRB 20180916B is comparable to the orbital periods of high-mass binaries known to harbour a neutron star. For instance, LS I +61 · 303 has an orbital period of about 26.5 days and has recently been shown to harbour a neutron star producing sporadic pulsations with a period of 269 ms [111]. In contrast to LS I +61 · 303, however, the periodic burst activity of FRB 20180916B could be related to the orbital period of the \nsystem and share some of the viewing geometry and magnetic interaction effects we propose to explain ILT J1101 + 5521 .", 'References': '- [21] Foreman-Mackey, D., Hogg, D.W., Lang, D., Goodman, J.: emcee: the mcmc \n- [32] Stappers, B., Archibald, A., Hessels, J., Bassa, C., Bogdanov, S., Janssen, G., \n[111] Weng, S.-S., Qian, L., Wang, B.-J., Torres, D.F., Papitto, A., Jiang, P., Xu, R., Li, J., Yan, J.-Z., Liu, Q.-Z., et al. : Radio pulsations from a neutron star within the gamma-ray binary ls i+ 61 303. Nature Astronomy 6 (6), 698-702 (2022)'}
2024ApJ...973L..47B	SN 2023ixf was discovered in M101 within a day of the explosion and rapidly classified as a Type II supernova with flash features. Here we present ultraviolet UV spectra obtained with the Hubble Space Telescope 14 19 24 and 66 days after the explosion. Interaction between the supernova ejecta and circumstellar material CSM is seen in the UV throughout our observations in the flux of the first three epochs and asymmetric Mg II emission on day 66. We compare our observations to CMFGEN supernova models that include CSM interaction inlineformula mmlmath overflowscrollmmlmover accenttruemmlmrowmmlmiMmmlmimmlmrowmmlmrowmmlmommlmommlmrowmmlmovermmlmoltmmlmommlmsupmmlmrowmmlmn10mmlmnmmlmrowmmlmrowmmlmommlmommlmn3mmlmnmmlmrowmmlmsupmmlmath inlineformula M SUBSUB yrSUP1SUP and find that the power from CSM interaction is decreasing with time from L SUBshSUB 5 10SUP42SUP erg sSUP1SUP to L SUBshSUB 1 10SUP40SUP erg sSUP1SUP between days 14 and 66. We examine the contribution of individual atomic species to the spectra on days 14 and 19 showing that the majority of the features are dominated by iron nickel magnesium and chromium absorption in the ejecta. The UV spectral energy distribution of SN 2023ixf sits between that of supernovae which show no definitive signs of CSM interaction and those with persistent signatures assuming the same progenitor radius and metallicity. Finally we show that the evolution and asymmetric shape of the Mg II 2796 2802 emission are not unique to SN 2023ixf. These observations add to the early measurements of dense confined CSM interaction tracing the massloss history of SN 2023ixf to 33 yr prior to the explosion and the density profile to a radius of 5.7 10SUP15SUP cm. They show the relatively short evolution from a quiescent red supergiant wind to high mass loss.	2024-10-01T00:00:00Z	['arXiv:2408.03993', '2024arXiv240803993B', '2024ApJ...973L..47B', '10.3847/2041-8213/ad7855', '10.48550/arXiv.2408.03993']	['Type II supernovae', 'Red supergiant stars', 'Stellar mass loss', 'Ultraviolet transient sources', 'Ultraviolet spectroscopy', 'Hubble Space Telescope', '1731', '1375', '1613', '1854', '2284', '761', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Solar and Stellar Astrophysics']	Circumstellar Interaction in the Ultraviolet Spectra of SN 2023ixf 1466 Days After Explosion	2,024	198	0.62	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	9	https://arxiv.org/pdf/2408.03993.pdf	{'Circumstellar Interaction in the Ultraviolet Spectra of SN 2023ixf 14-66 Days After Explosion': "K. Azalee Bostroem, 1, ∗ David J. Sand, 1 Luc Dessart, 2 Nathan Smith, 1 Saurabh W. Jha, 3 Stefano Valenti, 4 Jennifer E. Andrews, 5 Yize Dong ( 董 - 泽 ) , 4 Alexei V. Filippenko, 6 Sebastian Gomez, 7 Daichi Hiramatsu, 8, 9 Emily T. Hoang, 4 Griffin Hosseinzadeh, 10 D. Andrew Howell, 11, 12 Jacob E. Jencson, 13 Michael Lundquist, 14 Curtis McCully, 11, 12 Darshana Mehta, 4 Nicolas E. Meza Retamal, 4 Jeniveve Pearson, 1 Aravind P. Ravi, 4 Manisha Shrestha, 1 and Samuel Wyatt 15 \n1 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA \n2 Institut d'Astrophysique de Paris, CNRS-Sorbonne Universit'e, 98 bis boulevard Arago, 75014 Paris, France \n3 Department of Physics and Astronomy, Rutgers, the State University of New Jersey, \n136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA \n4 Department of Physics and Astronomy, University of California, Davis, 1 Shields Avenue, Davis, CA 95616-5270, USA 5 Gemini Observatory, 670 North A'ohoku Place, Hilo, HI 96720-2700, USA \n6 Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA \n7 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218-2410, USA \n8 Center for Astrophysics \| Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA \n9 The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, USA \n10 Department of Astronomy & Astrophysics, University of California, San Diego, 9500 Gilman Drive, MC 0424, La Jolla, CA 92093-0424, USA \n11 Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA \n12 Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA \n13 IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125 \n14 W. M. Keck Observatory, 65-1120 M¯amalahoa Highway, Kamuela, HI 96743-8431, USA \n15 Astrophysics Science Division, NASA Goddard Space Flight Center, Mail Code 661, Greenbelt, MD 20771, USA", 'ABSTRACT': 'SN 2023ixf was discovered in M101 within a day of explosion and rapidly classified as a Type II supernova with flash features. Here we present ultraviolet (UV) spectra obtained with the Hubble Space Telescope 14, 19, 24, and 66 days after explosion. Interaction between the supernova ejecta and circumstellar material (CSM) is seen in the UV throughout our observations in the flux of the first three epochs and asymmetric Mg II emission on day 66. We compare our observations to CMFGEN supernova models that include CSM interaction ( ˙ M < 10 -3 M ⊙ yr -1 ) and find that the power from CSM interaction is decreasing with time, from L sh ≈ 5 × 10 42 erg s -1 to L sh ≈ 1 × 10 40 erg s -1 between days 14 and 66. We examine the contribution of individual atomic species to the spectra on days 14 and 19, showing that the majority of the features are dominated by iron, nickel, magnesium, and chromium absorption in the ejecta. The UV spectral energy distribution of SN 2023ixf sits between that of supernovae which show no definitive signs of CSM interaction and those with persistent signatures assuming the same progenitor radius and metallicity. Finally, we show that the evolution and asymmetric shape of the Mg II λλ 2796, 2802 emission are not unique to SN 2023ixf. These observations add to the early measurements of dense, confined CSM interaction, tracing the mass-loss history of SN 2023ixf to ∼ 33 yr prior to the explosion and the density profile to a radius of ∼ 5 . 7 × 10 15 cm. They show the relatively short evolution from a quiescent red supergiant wind to high mass loss. \nKeywords: Type II supernovae; Red supergiant stars; Stellar mass loss; Ultraviolet transient source; Ultraviolet spectroscopy; Hubble Space Telescope \[email protected]', '1. INTRODUCTION': "Hydrogen-rich supernovae (Type II) are thought to come from red supergiants (RSGs) with zero-age main sequence masses between ∼ 8 and ∼ 25 M ⊙ . While it is accepted that all RSGs lose mass, there are no quantitative mass-loss rates derived from stellar evolution theory. Adding to this, there is no consensus about the empirical mass-loss rates for RSGs; Massey et al. (2023) infer mass-loss rates higher than the canonical rates of de Jager et al. (1988) from the RSG luminosity function, while other studies (e.g. Beasor et al. 2020), measure lower mass-loss rates from the infrared (IR) excess of RSGs. Different mass-loss rates affect the final star that explodes, impacting the final hydrogen envelope mass as well as the density structure of the circumstellar material (CSM) into which the supernova explodes. \nSignatures of mass loss can also be observed in supernovae, allowing us to constrain late-stage RSG mass loss using post-explosion observations. One definitive sign of CSM is the presence of narrow emission lines. As the number of young Type II supernovae discovered has increased with the prevalence of high-cadence wide-field surveys, it has become clear that a significant fraction of them have confined CSM (Khazov et al. 2016; Bruch et al. 2023; Jacobson-Gal'an et al. 2024). Early-time spectra of these supernovae display flash features: narrow emission lines with Lorentzian wings formed by the ionization of surrounding CSM by shock breakout and the interaction of the ejecta with CSM. These narrow features disappear a few days to weeks after explosion and the evolution then proceeds as a typical Type IIP/L supernovae. \nWhile these lines disappear when the CSM is swept up, the influence of this CSM on the remaining photometric and spectroscopic evolution of the supernova is uncertain. Observations and recent theoretical modeling of the effects of the overall dynamics and CSM properties on the light curve and spectral evolution suggest that, even in the absence of narrow emission lines, the CSMejecta interaction will convert some of the kinetic energy of the supernova into thermal energy, boosting the supernova luminosity (Smith et al. 2015; Dessart et al. 2017; Dessart & Hillier 2022). Furthermore, this interaction will produce a bluer spectral energy distribution (SED) after a few days. The discrepancy between the U -V color curves of a supernova with and without CSM grows with time as the noninteracting supernova ejecta expand and cool (Dessart & Hillier 2022). Spectroscop- \nically, high-velocity absorption features, from the cool dense shell that forms at the interface between ejecta and CSM, are predicted to be visible in H α , H β (Chugai et al. 2007), Na I D λλ 5890, 5896, and the Ca II nearinfrared (NIR) triplet λλλ 8498, 8542, 8662, and produce strong emission lines in the ultraviolet (Dessart & Hillier 2022). In particular, as the ejecta cool, Mg II λλ 2796, 2802 emerges in emission with increasing strength as the level of CSM interaction increases. Thus, the characterization of the long-term evolution of supernovae weeks to years after explosion probes RSG mass loss centuries to millennia prior to explosion. \nObservational evidence supporting these predictions is unclear. Sample analyses found contradictory results about whether supernovae with flash features are systematically bluer and more luminous than noninteracitng supernovae (Khazov et al. 2016; Bruch et al. 2023). Additional contradictions are found in individual objects (e.g. SN 2018zd, Hiramatsu et al. 2021; SN 2020pni, Terreran et al. 2022; various, Jacobson-Gal'an et al. 2024; PTF11iqb, Smith et al. 2015; SN 2013fs, Bullivant et al. 2018). Spectroscopically, while an absorption feature is often observed blueward of H α (Guti'errez et al. 2017) and has been interpreted as high-velocity hydrogen due to CSM interaction, the relationship between the sample of supernovae with this feature and those with flash features is unclear. UV observations of Type II supernovae are especially sparse, with early-time data limited by the availability of sensitive rapid-response UV telescopes, and with late-time observations limited by the fading of the UV when CSM interaction is not present (Panagia et al. 1980; Pettini et al. 1982; Cassatella 1987; Baron et al. 2000; Fransson et al. 2005; Vasylyev et al. 2022, 2023a). \nThe study of Type II supernovae with dense, confined CSM was recently enriched by the explosion of SN 2023ixf in the nearby galaxy M101. The proximity, early discovery and classification (Itagaki 2023; Perley & Gal-Yam 2023), and identification of narrow emission lines have led to this being the best-observed supernova with flash features to date. Of particular note is the periodic pre-explosion light curve of the RSG progenitor in the optical and IR (Jencson et al. 2023; Kilpatrick et al. 2023; Niu et al. 2023; Qin et al. 2023; Soraisam et al. 2023; Xiang et al. 2024), the incredibly deep multiband pre-discovery data which characterize the shock breakout and light-curve rise starting at an absolute magnitude of V ≈ -11 together with high-cadence photometry through the end of the plateau phase (JacobsonGal'an et al. 2023; Hiramatsu et al. 2023; Hosseinzadeh et al. 2023; Sgro et al. 2023; Teja et al. 2023; Yamanaka \net al. 2023; Zimmerman et al. 2024; Bersten et al. 2024; Li et al. 2024; Martinez et al. 2024), intranight optical spectroscopy for more than a week after discovery (Bostroem et al. 2023a; Jacobson-Gal'an et al. 2023; Teja et al. 2023; Hiramatsu et al. 2023; Yamanaka et al. 2023; Zhang et al. 2023), early-time optical spectropolarimetry (Vasylyev et al. 2023b; Singh et al. 2024), highresolution spectroscopy of the first 17 days of evolution (Smith et al. 2023; Hiramatsu et al. 2023; Zimmerman et al. 2024), X-ray and radio detections (Berger et al. 2023; Grefenstette et al. 2023; Matthews et al. 2023; Chandra et al. 2024), and the first early-time UV spectra of a supernova with flash features (Zimmerman et al. 2024). While a coherent picture of these observations is still being explored, they point toward a confined, dense CSM with a radius R ≲ 10 15 cm and a mass-loss rate ˙ M ≈ 10 -2 -10 -4 M ⊙ yr -1 . \nIn this paper, we describe UV spectroscopy of SN 2023ixf obtained with the Hubble Space Telescope (HST) at 14, 19, 24, and 66 days past explosion, all after the flash features had faded. Throughout the paper we use UTC 2023-05-18 18:00:00 (MJD 60082.75 ± 0.10; Hosseinzadeh et al. 2023) as the explosion epoch, E ( B -V ) = 0 . 0077 ± 0 . 0002 mag (Schlafly & Finkbeiner 2011) as the Milky Way (MW) extinction, and E ( B -V ) = 0 . 031 mag (Smith et al. 2023) as the host-galaxy extinction. In Section 2 we describe the data reduction and in Section 3 the evolution of the spectra, focusing on the UV. SN 2023ixf is compared to models with CSM interaction in Section 4 and to other supernovae with UV observations in Section 5. We close with a discussion of CSM interaction through day 66 in Section 6 and summarize our results in Section 7.", '2. OBSERVATIONS AND DATA REDUCTION': "HST observations were obtained on days 14, 19, 24, and 66 with the CCD detector of the Space Telescope Imaging Spectrograph (STIS) using the 52 '' × 0. '' 2 slit at the E1 position to mitigate flux loss to charge transfer efficiency loss and the G230LB, G430L, and G750L gratings (DD GO-17313; Bostroem et al. 2023b). Unfortunately, our planned visit on day 50 failed to acquire guide stars. However, the observation was successfully repeated on day 66. We obtained at least four exposures with each grating to automatically remove cosmic rays. A list of observations is given in Table 1. \nBias, flat-fielding, and cosmic-ray rejection were automatically performed prior to download from the Mikulski Archive for Space Telescopes. Additionally, onedimensional (1D) spectra were automatically extracted, wavelength calibrated, and flux calibrated, prior to download for all G230LB and G430L observations ex- \ncept the G230LB observation on day 66. At this phase, the UV flux was very low at the blue end, preventing the automatic identification of the spectrum location by the pipeline. These observations were manually extracted with the stistools.calstis pipeline, using the red end of the spectrum to identify the location of the trace. \nThe G750L grating suffers fringing at the reddest wavelengths, which can be corrected using a contemporaneous fringe flat. We obtained a fringe flat with each visit using the 0 . '' 3 × 0 . '' 09 aperture, as recommended for the E1 aperture position. The fringe flat was applied to the G750L observations using the module stistools.defringe and 1D spectra were extracted using the stistools.x1d routine. \nWe utilized data-quality flags 16 (high dark rate) and 512 (bad reference pixel) to eliminate bad pixels. In particular, we note that narrow emission features that appear in all spectra are flagged as high dark rate pixels and therefore removed from the analysis. We confirm that these are isolated pixels with high count rates in the dark files by examining the extraction location in the two-dimensional dark images themselves. Nevertheless, it is possible that there is a real unresolved emission line at the same pixel as at least one of these features. However, this would be contaminated by the high dark rate and therefore unable to provide any constraint on the CSM. Thus, given the characteristics described above, we treat these features as artifacts and do not consider them further. \nWe add to this dataset the HST observations taken between days 3 and 11 as part of GO-17205 (Zimmerman et al. 2022), which have been presented by Zimmerman et al. (2024). 1D-extracted and flux-calibrated files were downloaded from the Mikulski Archive for Space Telescopes. As noted by Zimmerman et al. (2024), some of these exposures suffered from pointing, acquisition, and/or saturation issues, which led to unreliable flux due to the supernova not being aligned in the slit and the nonlinearity of the CCD response near the saturation point. To mitigate these issues, we scale all observations with each grating in a given visit to the spectrum with the most flux (assuming this is the best-centered observation) using a first-degree polynomial fit to the ratio of the spectra as the scale factor. A constant multiple of a few percent was then applied to align the G430L and G750L observations to the G230LB observation, and then the fluxes were combined by taking the median value at each wavelength, excluding pixels with data-quality flags of 16 and 512. We note that the majority of pixels are saturated in the range ∼ 3200-5000 ˚ A \nTable 1. HST /STIS observations of SN 2023ixf from GO-17313 \nin visit 1, ∼ 2100-7600 ˚ A in visits 2 and 3, and ∼ 29007900 ˚ A in visit 4, making these wavelength ranges the most unreliable in terms of flux calibration. However, in this paper, we are primarily analyzing the later epochs (14 -66 days) which are not saturated and only use the early epochs ( < 10 days) to identify the presence and timing of relevant spectral features. Thus, we do not attempt to further correct for this effect.", '3. SPECTRAL EVOLUTION': "The near-UV (NUV) through NIR evolution of SN 2023ixf from day 3 to 66 is shown in Figure 1. No Type II supernovae (excluding Type IIn supernovae) since SN 1987A have been observed this early in the UV and with this cadence, revealing unprecedented details of the evolution. As has been noted by other authors, narrow emission lines are present in the early-time spectra, including notably a narrow P Cygni line from N IV λ 1718 which is present in the first spectrum and fades by day 8, and C III λ 2297 which is present from day 4 to day 8 (Yamanaka et al. 2023; Jacobson-Gal'an et al. 2023; Bostroem et al. 2023a; Teja et al. 2023; Hiramatsu et al. 2023; Zimmerman et al. 2024). These early narrow lines indicate dense CSM surrounding the supernova that has not been shocked. By day 14, these lines disappear and the optical spectrum is that of a young Type II supernova. Although slow to develop as a result of early interaction, a typical Type II supernova optical spectrum emerges with prominent Doppler-broadened P Cygni profiles in H I , He I , and other metals. \nA more detailed view of the evolution from day 14 through day 66 is shown in Figure 2. Although still dominated by iron line blanketing (the broad absorption of flux due to a forest of iron lines that blend together), the UV continuum flux is clearly present in the day 14 spectrum. This is unusual for classical Type II supernovae in which the UV flux fades rapidly (Baron et al. 2000; Brown et al. 2007; Vasylyev et al. 2023a; Bostroem et al. 2023c). \nWhile the flux blueward of 2500 ˚ A fades significantly by day 20, prominent absorption features can still be seen in the range 2500-3500 ˚ A, including Mg II , and just blueward Fe II /Fe III . In contrast with the fading UV flux, a prominent 'emission' line arises around 1925 ˚ A which reaches peak strength at 25 days before decreasing in strength. \nBy day 66, there is very little UV flux as the supernova ejecta cool causing the SED to shift to longer wavelengths and iron absorption to increase in strength. However, the Mg II λλ 2796, 2802 doublet is clearly present as a boxy broad emission line with a slanted top. Also, at this time H α has fully developed a broad P Cygni profile, and other metal lines typical of Type II supernovae are present in the optical, including the Ca II NIR triplet λλλ 8498, 8542, 8662. \nAcomparison of the Mg II and H α profiles provides insight into the origin of the Mg II emission (see Figure 3). When CSM is present, a cool dense shell forms at the interface between the ejecta and CSM. While the H α emission originates from the ejecta and the cool dense shell, the Mg II is thought to originate only from the \nFigure 1. Evolution of the NUV to NIR spectra of SN 2023ixf from 3.5 to 66 days after explosion. Prominent features (N IV : 1718, 7109, 7122 ˚ A; C III : 2297 ˚ A; Mg II : 2796, 2802 ˚ A; He II : 3203, 4685.5, 4860 ˚ A; C IV : 5801, 5812 ˚ A; H α : 6563 ˚ A; He I : 6678.1 ˚ A; Ca II : 8498, 8542, 8662 ˚ A) are marked with vertical lines at their rest wavelength and labeled at the top of the figure. Narrow ISM absorption lines have been removed to highlight the supernova spectra. \n<!-- image --> \nshell and should thus be at a similar velocity as the outer edges of the ejecta. We compare Mg II and H α in Figure 3, finding that the profiles extend to similar velocities on both the red and blue sides, confirming that the Mg II is emitted from the cool dense shell at the outer edge of the ejecta. However, H α sits in the optical where there is continuum and forms throughout the ejecta and cool dense shell, leading to a P Cygni profile, while Mg II , without UV continuum and forming in the outer ejecta, is seen solely in emission. There is a shallow, narrow dip at the blue edge of the H α profile that could be absorption from the cool dense shell at ∼ 9200 km s -1 , although it is not clear if this feature is real given the signal-to-noise ratio (S/N) of the spectrum. \nTo characterize the Mg II feature further, we fit three different profiles in Figure 3, representing different physical configurations. A spherical distribution of material with a Gaussian emitting profile will produce a Gaussian profile. Although the profile is clearly not Gaussian, it could be coming from a spherical distribution of material that is being attenuated. To explore the nonattenuated spherical distribution, we fit a Gaussian to the blue side of the Mg II profile from -10 , 000 to -7000 km s -1 . This fit is shown as a pink line in Figure 3; it has a mean of µ = 0 km s -1 and a FWHM of 7780 km s -1 . The observed blue side of Mg II is well described by a Gaussian profile. An alternate emitting region could \nbe a sphere with a Gaussian profile that has had the center removed to form a thick shell. This configuration will have a flat-topped profile with Gaussian wings (Jerkstrand 2017) if the cool dense shell and ejecta are optically thin. Given our slanted top profile, we use the maximum of our observed flux as the flat-topped maximum of the model profile. This defines the inner radius velocity of the shell as v in = -6800 km s -1 , and the profile is shown as a dotted-blue line in the right panel of Figure 3. \nTo model the slated top of the profile, we model the line as if it were generated by an asymmetric shell of material (i.e., a shell in which there is more flux from one side than the other reaching the observer). Following Kwok et al. (2023, 2024), we model this asymmetric shell as a Gaussian profile with an off-center hole cut out, which results in a slanted top instead of a flat-topped profile. We parameterize this model by the FWHM and mean of the Gaussian, the velocity of the center of the hole ( v c ), and the velocity of the radius of the hole ( v in ). We find FWHM = 7580 km s -1 , µ = -2995 km s -1 , v in = 5000 km s -1 , and v c = 1750 km s -1 , where v c and v in are measured relative to µ . This model fits the data exceptionally well and is shown as a green dashed line in the right panel of Figure 3. While the CSM is likely asymmetric, we caution that this parameterization is more likely to indicate different optical depths along the \nFe II \nFigure 2. Acomparison of the spectra of SN 2023ixf from 14.25 to 66.25 days post-explosion (solid lines) with the best-matching CSM interaction model (semi-transparent). The zero flux levels for each epoch are drawn as a dot-dashed line in the same color as the observed spectrum at that epoch. Over time, the UV emission declines and P Cygni features develop in the optical, until day 66, when only Mg II λλ 2796, 2802 is visible in the UV. This corresponds to a sequence of models with decreasing interaction power. Even though the model spectra do not match the SED of the observations, especially in the first two spectra where they appear to be cooler (resulting in lower-ionization species and more UV line blanketing), the match is impressive given the 1D nature and the ad-hoc implementation of the mass, density, and clumping of the cool dense shell. Prominent spectral features have been marked with dotted lines at 7000 km s -1 , the approximate location of the absorption throughout the spectroscopic sequence. The 'emission' feature at 1925 ˚ A is marked at rest with a dashed line. Narrow ISM absorption lines have been removed to better highlight the supernova spectra. Observed and model spectra at each epoch have been placed at the same distance with the same offset applied. \n<!-- image --> \nline of sight through the ejecta and cool dense shell, and should not be taken as a literal description of the ejecta. It does, however, demonstrate that this profile can be created by a shell of material in which a higher flux of photons from the blue side of the ejecta is reaching us, than photons from the red side. Additionally, it defines the inner edge of the blue side of the shell to be at velocity -6245 km s -1 (where the profile peaks), close to our earlier estimate of -6800 km s -1 . From these two values, we adopt v in = -6500 km s -1 . From these fits we conclude that the emitting region of the Mg II is well described by a Gaussian profile with a FWHM ≈ 7700 km s -1 and an inner radius of v in = -6500 km s -1 that has been attenuated on the red side due to the opacity of the ejecta and cool dense shell. We note that as the profile approaches the peak flux from the blue side, the rise is shallower than the Gaussian profile, possibly indicating asymmetry. Finally, we note that if the CSM is asymmetric, inner and outer shell velocities would represent projected velocities. \nNarrow absorption lines are present in the spectra from Fe II λ 2344, λ 2374, λ 2383, λ 2587, λ 2600 and Mg II λλ 2796 , 2802, λ 2852 . 96. These lines are centered at the host redshift and there is no evolution in their equivalent widths with time. From this, we conclude that these lines are either from the interstellar medium (ISM) in the host galaxy or distant CSM and remove them from the remainder of the analysis.", '4. COMPARISON WITH LITERATURE MODELS': "Although by day 14 the narrow emission lines typically associated with strong CSM interaction have faded, it is possible that some energy from the interaction of the supernova ejecta with CSM is still contributing to the observed supernova spectrum. When supernova ejecta encounter CSM, a cool dense shell is formed between the forward and reverse shocks. The cool dense shell absorbs and reemits some of the shock power from CSM interaction. Dessart & Hillier (2022) published 1D (spherically symmetric) models of the interaction of supernova ejecta and cool dense shell with a low-density wind ( ˙ M < 10 -3 \nFigure 3. Left: H α and Mg II line profiles on day 66 in velocity space of SN 2023ixf compared to the CSM interaction models of Dessart & Hillier (2022) with L sh = 1 × 10 40 erg s -1 , scaled to the distance of SN 2023ixf. The shapes of both lines are well captured by the models, given that the cool dense shell is at a higher velocity in the models than in SN 2023ixf. Right: Mg II profile (black) fit with a pure Gaussian (pink) modeling spherical ejecta, a symmetric shell (blue dotted) modeling emission from the cool dense shell in the absence of optical-depth effects, and an asymmetric shell (green dashed) modeling the cool dense shell with optical depth. \n<!-- image --> \nM ⊙ yr -1 ). Unlike those of Dessart et al. (2017), which model interaction in a spherically symmetric dense, confined CSM, these models ignore the hydrodynamics of the CSM interaction in exchange for more accurate radiative transfer. While Dessart & Hillier (2022) compared their models to optical observations, they show that the signatures of this interaction are strongest in the UV, and our spectral series allows us to test these model predictions. \nBriefly, these models start with a 15 M ⊙ progenitor of solar metallicity evolved with Modules for Experiments in Stellar Astrophysics (Paxton et al. 2011, 2013) which has been exploded with the radiation-hydrodynamics code V1D (Livne 1993; Dessart et al. 2010) and evolved to 10 days post explosion. On day 10, these observations are mapped to the nonlocal thermodynamic equilibrium radiation transport code CMFGEN (Hillier & Miller 1998; Hillier & Dessart 2012; Dessart et al. 2013; Hillier & Dessart 2019) assuming homologous expansion after this point. A cool dense shell is simulated by placing 0.1 M ⊙ of material at 11,700 km s -1 into the density profile. This velocity was chosen as the shock and cool dense shell are not expected to slow down much in the presence of a low-density CSM over a time scale of weeks (Dessart et al. 2017). The power from the CSM-ejecta \ninteraction is then deposited at a constant rate into this region, mimicking the conversion of kinetic energy from the ejecta into radiative energy. Implicit in this implementation is the assumption that the cool dense shell is fully formed during the first few days after the explosion and does not grow in mass. \nWith this setup, they produce models with no power deposited and continuous depositions of 1 × 10 40 , 1 × 10 41 , 5 × 10 41 , 1 × 10 42 , 2 . 5 × 10 42 , 5 × 10 42 , and 1 × 10 43 erg s -1 from day 15 through at least day 120 (some models are run longer). We emphasize that these models were not created for this specific supernova (or any supernova with early narrow emission lines). One difference between the model setup and SN 2023ixf is the presence of confined dense CSM, which produces the flash features observed in the early spectra. Dessart et al. (2017) find that in the presence of dense CSM, the maximum ejecta velocity decreased from 11,500 km s -1 in the low-density scenario to 7200 km s -1 in the highdensity scenario (see also Dessart & Jacobson-Gal'an 2023). We thus expect the velocity of the outer ejecta of SN 2023ixf to be slower than the models of Dessart & Hillier (2022). Another consequence of the dense CSM is that the initial conditions of the interaction region of the model may not be representative of the physi- \nitions of the outer ejecta of SN 2023ixf, leading to a larger mismatch between the observations and the models in the earliest epoch than at later times. \nWe visually compare each model at a given epoch to our observed flux, starting from the day 14.25 spectrum, and identify the model that best matches the observed flux. Although each of the models represents a steadystate wind, the cool, dense shell is optically thin to continuum radiation and radiates efficiently. For this reason, it will react promptly to a change in shock power. Thus, we expect that a decreasing mass-loss rate with radius would effectively have the same result as taking a different steady-state model for different epochs. The spectra of SN 2023ixf and best-matched models are shown in Figure 2 and the model properties are listed in Table 2. As expected, the model velocity is faster than the observed ejecta velocity leading to blueshifted and broader features. We find that the observations of SN 2023ixf indicate a progressive drop in interaction power over time, which translates into a drop in progenitor mass loss with increasing distance. \nConsidering that these models were not produced for this specific supernova, it is impressive that no scaling beyond correcting for distance is required to match the observations and many of the features are reproduced, although at varying strengths. The first two epochs of models are too faint in the NUV and too bright in the optical, a discrepancy that improves with time. Nevertheless, we consider the best-matched model to be indicative of the CSM interaction level as the selected model most closely matches the observed flux: the model with more CSM interaction is too bright at all wavelengths and the model with less CSM interaction is even more discrepant for the majority of the spectrum. This consideration is further validated by the custom modeling in Section 6. One possible reason for the difference between the models and observations is that the temperature is too low in the emitting region in the early models. This is corroborated by the lack of He I λ 5875 in the models, whose formation requires higher temperatures (Dessart & Hillier 2005) \nThe UV is dominated by a forest of iron-groupelement absorption lines that blend together, making it challenging to associate features with the emission or P Cygni profiles of individual species (Bostroem et al. 2023c; Dessart & Hillier 2005). To facilitate the identification of regions with contributions from different elements, we post-process the CMFGEN models to calculate the spectra on day 14 and day 19, omitting the bound-bound transitions of individual species. The full \nspectrum is then divided by these spectra to obtain the relative flux contribution of each species. We perform this calculation for H, He, C, N, O, Ne, Na, Mg, Al, Si, S, K, Ca, Sc, Ti, Cr, Fe, Co, and Ni. Comparing the omitted spectrum to the full spectrum for each species, regions were identified as at least partially resulting from an individual species. These are marked in the top panel of each column of Figure 4, which shows the full model spectrum compared to the observed spectrum at the same epoch. Elements that have absorption features with a depth of > 2% of the total flux are shown in the bottom panel of Figure 4. In these panels, the strong influence of the iron-group elements (specifically Fe and Ni on day 14 and Fe, Ni, and Cr on day 19) on the overall spectrum is clear. In particular, we point out that the apparent developing 'emission line' at 1924 ˚ A is in fact a window of lower absorption between two prominent iron absorption regions, similar to the 'emission line' at 2970 ˚ A identified in the NUV spectra of SN 2022acko on day ∼ 20 (Bostroem et al. 2023c). Additionally, we show the continuum emission if no absorption is present in the top panel of each column of Figure 4, demonstrating how strongly the spectrum deviates from blackbody emission. \nAnother emission line of particular note is Mg II in the day 66 spectrum, which is compared to the CMFGEN models in Figure 5. To account for the higher velocity of the models, we scale the models by the ratio of the blue edge of the feature in the observation to the blue edge in the model which preserves the morphology of the line. The Mg II feature is only present in models with CSM interaction after day ∼ 50 and rapidly increases in strength for greater interaction power. Our observation on day 66 falls between the models with a shock power of L sh = 1 × 10 40 erg s -1 and L sh = 1 × 10 41 erg s -1 and looks clearly distinct from models with no interaction. \nIn the left panel of Figure 3, we compare the Mg II and H α profiles of our observation and the L sh = 1 × 10 40 erg s -1 model (without any shift in wavelength) on day 66. We find that both profiles are similar, although the model extends to bluer wavelengths due to the lack of dense CSM in the early supernova evolution, which caused the cool dense shell to form at lower velocities in SN 2023ixf. While the models have a large narrow absorption feature on the blue side of H α from the blue side of the cool dense shell, no such prominent feature is present in the observed H α profile. There is a hint of an absorption feature around 9200 km s -1 , although this is near the noise threshold of the spectrum and thus we cannot confidently associate it with this feature. On the red edge of H α , the models show a subtle red shelf. \n) \n1 \nÅ \n1 \ns \n2 \nm \nc \ng \nr \ne \n( \ny \nt \ni \ns \nn \ne \nD \nx \nu \nl \nF \n<!-- image --> \nFigure 4. Top: UV spectra of SN 2023ixf on day 14.25 (left) and 19.25 (right) compared to radiative-transfer models. Absorption lines from individual species are marked below the spectra. In the day 19.25 spectrum, the asterisk on Fe at 1924 ˚ A indicates that this is a window of minimal iron absorption rather than an emission line due to any atomic transition. The individual features are reproduced quite well, albeit at higher velocities in the models. Bottom: decomposition of the model spectrum at each epoch into species that contribute to the spectrum. Although the individual spectra are offset, the scales of the spectra are consistent, showing the relative intensity of each species. The dominance of iron-group elements (and in particular iron) in the spectra at both epochs is clear. \n<!-- image --> \nTable 2. Characteristics of the CSM around the progenitor of SN 2023ixf assuming v shock = 10000 km s -1 and v wind = 55 km s -1 . \nNote -The last two rows bracket the true shock luminosity as the Mg II λλ 2796, 2802 emission lies between L sh = 10 40 and L sh = 10 41 erg s -1 . \nFigure 5. Mg II line profiles of SN 2023ixf on day 66 in velocity space (black) compared to models with no power from CSM interaction (cyan) and the two lowest levels of CSM interaction modeled ( L sh = 1 × 10 40 erg s -1 ; blue, L sh = 1 × 10 41 erg s -1 ; magenta). This feature is present only in those models with CSM interaction, indicating that SN 2023ixf is interacting with CSM on day 66. The velocity scale of the models has been scaled to match the observations. \n<!-- image --> \nWith the S/N of our data, we cannot verify whether this feature is present. \nThe model Mg II is very boxy, rising sharply at the blue edge of the H α profile to a plateau before falling off just after 0 km s -1 . This shape arises from a shell of material with the back side obscured by the cool dense shell and ejecta, leading to what appears to be blueshifted emission. Compared to our observations, the Mg II feature in the models is boxier and more blueshifted. Additionally, the Mg II emission in the observations rises more gradually to a peak and begins to fall off immediately, rather than plateau. Nevertheless, the observations have the same basic structure as the models, indicating a similar emitting mechanism and structure.", '5. COMPARISON WITH OTHER SUPERNOVAE': "We focus our comparison of SN 2023ixf to other supernovae on the small number of Type II supernovae with UV observations, defining a sample of objects with a range of CSM properties and supernova types (see Table 3). Although many of these objects continued to exhibit signs of CSM interaction later in their evolution (e.g., Pooley et al. 2002; Mauerhan & Smith 2012; Smith et al. 2017), we limit the discussion to their plateauphase evolution, which corresponds to our observations of SN 2023ixf. \nSN 1993J was a Type IIb supernova, with the hydrogen lines fading and helium lines increasing in strength \nsoon after explosion (Filippenko et al. 1993). Many lines of evidence indicated interaction with CSM including narrow emission lines (Benetti et al. 1994) and X-ray (Zimmermann et al. 1994) and radio (Pooley & Green 1993) detections. SN 1979C and SN 1998S are both Type II supernovae with steeply declining light curves and above-average photospheric phase luminosities. They also showed narrow emission lines for weeks after explosion; however, these did eventually disappear (Panagia et al. 1980; Leonard et al. 2000). SN 1979C was also detected at X-ray and radio wavelengths indicating interaction with CSM beyond the nominal RSG wind (Panagia et al. 1980). SN 1980K had a decline rate similar to that of SN 1979C, although it was about a magnitude fainter in the V band. While it did not exhibit narrow emission lines, it was detected in X-rays (Canizares et al. 1982) and radio (Weiler et al. 1992). SN 1987A was a peculiar Type II supernova, originating from a compact, blue supergiant progenitor. No CSM interaction was initially detected; however, months after explosion narrow emission lines appeared as high-energy photons ionized a ring of CSM. SN 1999em, SN 2021yja, and SN 2022wsp are considered fairly normal Type II supernovae (Leonard et al. 2000; Baron et al. 2000; GalYam et al. 2008; Hosseinzadeh et al. 2022; Vasylyev et al. 2022, 2023a), although Hosseinzadeh et al. (2022) note that SN 2021yja was exceptionally blue at early times and Vasylyev et al. (2023b) find that the absorption of H α and H β in SN 2022wsp was suppressed in the P Cygni profiles which they ascribe to CSM interaction. Finally, SN 2005cs and SN 2022acko were low-luminosity supernovae (Brown et al. 2007; Bostroem et al. 2023c). The only possible sign of CSM interaction in these supernovae is a 'ledge' feature in the early spectra, which has been attributed to CSM, although other possibilities exist (see Pearson et al. 2023, for a detailed discussion). SN 2021yja also showed the 'ledge' feature. Details on the explosion epochs, extinction values, and spectra considered in this analysis are given in Table 3. We note that UV data also exist for SN 2010jl, a superluminous Type IIn supernova. Throughout its evolution, the UV spectrum is dominated by the cool dense shell with narrow spectral features and enhanced flux, in contrast to the other supernovae in this sample where the spectra are dominated by the supernova photosphere. We therefore exclude it from our analysis. \nIn Figure 6 and Figure 7 we show a comparison of the spectra of SN 2023ixf to this sample of UV spectra, scaled to the distance of SN 2023ixf, on day ∼ 10 and day ∼ 20, respectively. For this comparison, we exclude SN 1979C and SN 1998S, as their UV flux is significantly higher than that of SN 2023ixf and all mod- \nconsidered in this paper. On day 10, SN 2023ixf is similar in brightness to SN 2021yja. Although it has a lower flux, SN 2022wsp is spectroscopically similar. The Mg II λλ 2796, 2802 doublet is present in absorption in SN 1980K, SN 2021yja, SN 2022wsp, and SN 2022acko, although it is blueshifted to higher velocities in SN 2021yja and SN 2022acko. Additionally, blueward of Mg II , the Fe absorption feature identified by Vasylyev et al. (2023a) in SN 2022wsp is clearly present in SN 2023ixf and SN 1980K. Although not a pronounced absorption trough, it is also possibly present in SN 2021yja and SN 2022acko and blended with Mg II in SN 1999em. Like SN 2023ixf, the UV spectra of SN 1980K, SN 2021yja, and SN 2022wsp all show an absorption complex of Fe and Ni between 1750 ˚ A and 2050 ˚ A. The Fe absorption complex between 2050 ˚ A and 2550 ˚ A, which is notably shallower in SN 2023ixf and SN 1980K than models predict, is fairly well reproduced in SN 2021yja, although all of the observed profiles have less prominent individual troughs than the model. \nOne of the most distinctive features of CSM interaction is excess UV flux. On day 10, we see that SN 1980K has a higher flux than SN 2023ixf, SN 2021yja has a very similar flux while SN 2022wsp is a bit lower, SN 2022acko is significantly lower, and the remaining supernovae have virtually no flux on day 10. We note, however, that both excess UV flux and the slope of the UV SED are highly dependent on the assumed distance, extinction, and explosion epoch of the supernovae. In particular, the redshift-independent distance of NGC 4269 ranges from 4 to 7.8 Mpc (see Van Dyk et al. 2019, for a thorough discussion), significantly affecting the luminosity of SN 1980K. While the distances and thus overall luminosities are uncertain, individual features can indicate different levels of CSM interaction. For example, the development of an Fe absorption complex redward of 3000 ˚ A is not present in SN 1980K, SN 2021yja, and SN 2023ixf. This feature only appears in models with lower levels of CSM interaction. This line does appear to be developing in SN 2022wsp, consistent with the interpretation that the lower flux level is indicative of less CSM interaction. Comparing with the models, this suggests that SN 2021yja and SN 2022wsp have close to 1 × 10 43 erg s -1 interaction power added to the ejecta luminosity on day 10. \nBostroem et al. (2023c) noted that SN 1999em has an extremely low UV flux relative to comparable sequences. It is interesting to consider both the anomalous SED and features of SN 1999em relative to the other supernovae in this sample, as SN 1999em is a relatively prototypical Type IIP supernova (although a high-velocity feature was detected in both H α and H β during the photo- \nheric phase, interpreted as a sign of CSM interaction; Chugai et al. 2007). The explosion epoch used is halfway between the last nondetection and first detection. It is possible that, at discovery, SN 1999em was up to 5 days older, as suggested by Dessart & Hillier (2006), which would make the UV flux less unusual; however, even at a later phase, the shape of the UV flux is different from that of the other supernovae - either more suppressed below 2750 ˚ A or with increasing flux redward of that. The presence of an'emission' feature owing to the window of absorption at 1924 ˚ A further supports a later explosion epoch. Given this uncertainty, SN 1999em is plotted in both Figure 6 and Figure 7. \nOn day 20, the UV flux has decreased in all supernovae. As was observed in SN 2022wsp (Vasylyev et al. 2023a), the Fe feature blueward of Mg II has faded in SN 2023ixf. Interestingly, SN 2021yja shows an extended feature that encompasses both the Fe and Mg II features, possibly indicating that these are blended. The slope and features of SN 2021yja and SN 2022wsp are very similar to the model with L sh = 2 . 5 × 10 42 erg s -1 , although they all have a larger flux. The slope of SN 2023ixf and SN 1999em is not reproduced by any model, even if an overall flux offset is applied. It is possible that models customized for each supernova with more sophisticated physics would provide a better fit. For example, a compelling time-dependent model for SN 1999em is presented in Dessart & Hillier (2006). The flux of SN 2021yja has fallen to a level similar to that of SN 2022wsp, indicating a lower level of power from CSM interaction. The fact that the flux of SN 2021yja fell relative to SN 2023ixf and SN 2022wsp from the flux levels on day 10 gives us confidence that this is not a distance effect. The window of absorption at 1924 ˚ A creates a clear 'emission line' in SN 2021yja, SN 2022wsp, and SN 2023ixf. \nIn interacting supernovae, the Mg II emission is attributed to emission from the cool dense shell, which has been excited by radiation from the reverse shock (Chevalier & Fransson 1994; Dessart & Hillier 2022). Originating from a shell of material, the emission-line profile is expected to be broad, boxy, and blue shifted (Dessart & Hillier 2022). It has been noted in SN 1979C 14 yr after explosion (Fesen et al. 1999), in SN 1993J on day 670 (Fransson et al. 2005), in SN 1995N on day 943 (Fransson et al. 2002), in SN 1998S on days 28-485 (Fransson et al. 2005), and in SN 2010jl on days 34-543 (Fransson et al. 2014). However, these supernovae represent the more strongly interacting objects in our sample. It remains to be seen if this feature is present in supernovae that do not have any period of strong interaction with dense CSM. \nFigure 6. Top : comparison of the UV spectrum of SN 2023ixf on day 9 with that of other Type II supernovae at a similar epoch (phase for each supernova given in the panel legend; all spectra scaled to the distance of SN 2023ixf): SN 1980K, SN 1987A, SN 1999em, SN 2005cs, SN 2021yja, SN 2022wsp, and SN 2022acko. At NUV wavelengths, SN 2023ixf is among the brightest in the sample. Bottom : day 9 (black) and 14 (gray) spectra of SN 2023ixf compared with CMFGEN models having varying degrees of shock power on day 14. Although the majority of UV observations have phases around day 10, day 14 is the first epoch modeled by Dessart & Hillier (2022). We show both SN 2023ixf spectra to demonstrate possible evolution from the spectra in the top panel and the models in the bottom. The day 14 spectrum best matches the model with L sh = 5 × 10 42 erg s -1 below ∼ 3000 ˚ A but is brighter than the model at shorter wavelengths. Elements are marked in both panels in black at -7000 km s -1 to match the SN 2023ixf absorption. \n<!-- image --> \nWe searched all available NUV spectra of our comparison sample of Type II supernovae for the broad, boxy Mg II emission that was observed on day 66 in SN 2023ixf. As Mg II is present in absorption on day 25 and then in emission on day 66, we do not know the exact timing of this transition. Additionally, this timing is related to the ejecta and CSM properties and thus may not be a universal property of Type II supernovae. We therefore consider all spectra with phases greater than 30 days. One complicating factor is the presence of Mg II absorption from both the MW and the host galaxy, which contaminates the supernova emission and can make it challenging to assess the shape of the emission profile. Whenever possible, we remove this narrow absorption from our data. \nWe do not detect this feature in SN 1987A, although this is perhaps not surprising given its blue supergiant progenitor and lack of UV flux. The feature is clearly detected in the remaining supernovae as a broad, boxy, blue-shifted profile. In SN 1980K, we marginally detect it. However, the S/N of the spectra is fairly low and the feature is not as strong as in other supernovae. In SN 1979C we detect it developing already on day 24, although it could be as old as 32 days if the supernova explosion occurred immediately after the last nondetection. From this date, Mg II evolves from a blue-shifted, flat-topped profile into one with a slanted profile (higher blue edge) starting around day 60. Similarly, Mg II evolves in SN 1998S from a boxy, blue-shifted, symmetric profile on day 72 to an asymmetric profile on days 237 and 485. In SN 1993J, the Mg II emission is fairly constant with a boxy, asymmetric profile from day 173 to 669. This is the earliest epoch of data for SN 1993J after the UV continuum has faded. While the feature is clearly present as broad emission in SN 1995N and SN 2010jl (days 30-572), the ISM absorption from both the Milky Way and the host galaxy obscure the shape of the profile and we do not include these in our analysis. \nThe Mg II emission features of SN 1979C, SN 1980K, SN 1993J, and SN 1998S are compared to that of SN 2023ixf in Figure 8. We choose the closest available epoch to day 66, and for SN 1998S we also include the first epoch at which the profile is asymmetric. From these comparisons, we find that if a change in the shape of the Mg II emission is observed, the emission begins as a symmetric, boxy, blue-shifted profile and always evolves to the same asymmetric, boxy profile, with more flux on the blue side. For supernovae that, like SN 2023ixf, always show an asymmetric profile, the asymmetry is always higher on the blue side. The consistency of this asymmetry tells us that this is not an effect of asymmetric CSM that we would expect \nto observe from a different viewing angle, but rather an optical-depth effect. This is corroborated by the Dessart &Hillier (2022) models, which show a blue-shifted, boxy emission profile with no emission from ejecta at velocities below 7000 km s -1 (i.e. the emission is isolated to the cool dense shell). Additional validation is derived from a custom model that reproduces the models of Dessart & Hillier (2022) but with the cool dense shell at 8500 km s -1 and shows the same slanted profile as we observed in the Mg II feature. While optical-depth effects are responsible for the blue-shifted and slated profile at early times, at later times dust formation likely attenuates the red side of the feature. This sample shows a variety in maximum velocity of the blue side of these features, indicating diversity in the location of the cool dense shell, as well as other supernova parameters such as the ejecta mass, hydrogen envelope mass, explosion energy, and CSM mass. However, they uniformly rise to maximum more slowly than the models, indicating a different cool dense shell density profile than is implemented in the models. \nOverall, this comparison confirms that UV spectra are sensitive to different levels of CSM interaction, and the Mg II feature can reveal weak CSM interaction even when it is not clearly seen in the optical. The sparse time series of the other supernovae that we have assembled indicates a diversity in features and their strength, velocity, and evolution, but a larger sample is required to evaluate further. Additionally, while the models reproduce many features, they are not able to reproduce the slope and possibly the overall flux of the observations, indicating room for further detailed modeling.", '6.1. Testing Model Predictions': "With our sequence of observations, we can test the following predictions of signatures of CSM interaction made by Dessart & Hillier (2022) through day 66. \n- 1. Higher luminosity and flux in all filters for supernovae having more interaction, with this effect being more prominent in the UV and at later times when the ejecta luminosity has faded.\n- 2. Emergence of the Mg II doublet as broad, boxy emission ∼ 50 days after explosion, increasing in luminosity until day ∼ 175.\n- 3. Weakened absorption features as the photospheric spectrum of the supernova is filled in by emission from the cool dense shell.\n- 4. A sharp-edged red shelf extending beyond the photospheric emission in H α originating from the cool \nFigure 7. Top : comparison of the spectrum of SN 2023ixf on day 19 with that of other Type II supernovae at a comparable epoch (phase for each spectrum given in the panel legend; all spectra scaled to the distance of SN 2023ixf): SN 1980K, SN 1987A, SN 1993J, SN 2021yja, SN 2022wsp, and SN 2022acko. SN 2023ixf remains UV bright, especially when compared with SN 2021yja, which has faded to a flux similar to that of SN 2022wsp. The explosion epoch of SN 1999em is uncertain and thus we show the spectrum here, too, to demonstrate its similarity to the other spectra at these epochs. Bottom : UV spectrum of SN 2023ixf on 19 days (black) compared with the CMFGEN models having varying degrees of shock power on day 20 (colors). Although at optical wavelengths the spectrum best matches the model with L sh = 2 . 5 × 10 42 , below ∼ 2750 ˚ A the spectrum best matches the model with L sh = 5 × 10 42 . Elements are marked in both panels in black at -7000 km s -1 to match the SN 2023ixf absorption. The asterisk denotes the feature caused by a window between two strong iron absorption troughs. \n<!-- image --> \nFigure 8. Comparison of the Mg II emission line in SN 2023ixf (day 66; black) to that of other supernovae that show this feature (SN 1979C, brown; SN 1993J, yellow; and SN 1998S, magenta). All lines have a blueshifted peak. In all but the first SN 1998S spectrum, the profiles are asymmetric indicating that the red side of the line is attenuated. The gap in some spectra is due to the removal of narrow Galactic and host ISM absorption. \n<!-- image --> \ndense shell, although this feature is only starting to become visible on day 50 \n- 5. A strong, narrow absorption feature at the velocity of the dense shell in H α , H β , Na I D λλ 5890, 5896, and the Ca II NIR triplet λλλ 8498, 8542, 8662 increasing in strength with time starting around day 30. \nFor the first point, we find that the UV luminosity is consistent with models with CSM interaction at all epochs, and not just limited to the early epochs when the narrow emission features are present. The fact that the shock power in the best-matched model decreases with the lookback time indicates that the ejecta are sweeping through CSM created from decreasing massloss rates. This is consistent with the light-curve modeling results of Singh et al. (2024) which require extended CSM ( ˙ M=10 -4 M ⊙ yr -1 to R = 10 16 cm) to match the late-time UV excess. We detect a boxy Mg II emission feature at 66 days, indicating CSM interaction even without the presence of a UV continuum, confirming the second expectation. As described in the third point, the absorption components in the P Cygni profiles of the observed spectra are very shallow and slow to develop, especially in H α which is only clearly identified in the final epoch. This is also notes by Singh et al. (2024). The fourth prediction of a sharp red shelf in H α is subtle on day 66 and not visible prior to this date. At the S/N of our data, we cannot confidently identify it nor \nrule out its presence. Similarly, the narrow high-velocity absorption component, if present, is significantly weaker than predicted in the models. This could be due to a lower density CSM, asymmetric CSM where the bulk of the CSM is offset from the line of sight, and/or the fragmenting of the cool dense shell due to Rayleigh Taylor instabilities which would spread this feature out in the velocity space. \nTo explore whether clumping and lowering the velocity of the cool dense shell would provide a better fit to our observations, we calculate a custom CMFGEN model. We place the cool dense shell at 8500 km s -1 . At this velocity, the ejecta are optically thick. This has two effects: first, the material has not been able to cool, so the temperature is higher; and second, it responds much more quickly to the power from the CSM interaction. However, we found that this change alone was not enough to reproduce our observations and continued to further tune the model. To better approximate the physical conditions that create the emission from the cool dense shell (i.e., the reprocessing of X-rays from the forward and reverse shock), we add a uniform field of X-rays starting at 8000 km s -1 with a total power of 5 × 10 42 erg s -1 instead of inserting power into the cool dense shell directly. Given the energy of these X-rays, we add higher energy ions to our model, namely, C III , N III , O III , Mg III , and Ne III . We also increase the clumping from 1% to 10% volume filling factor outside of 8000 km s -1 , keeping the total mass the same. \nIn Figure 9 we compare this model on day 14.6 to our day 14.2 observed spectrum and the best-matching model from Dessart & Hillier (2022) on day 14.6. Reducing the velocity of the cool dense shell aligns the absorption and emission. However, the resulting increase in the temperature of the cool dense shell shifts flux from the optical to the UV, causing the UV to overshoot the observed spectrum and the optical to underestimate the true flux. The change in power deposition from direct injection into the cool dense shell to an Xray field reduces the UV flux to align with the observations but does not resolve the low optical flux issue. The increase in clumping from a volume filling factor of 1%10% increases the optical flux to match the full SED. The observed spectrum is very well matched by our custom model, demonstrating the success of this approach. However, further improvements could be made to better match the strength of features in the UV, most notably the Fe II absorption complex in the range 2250-2760 ˚ A, which is too strong in the models. \n6.2. CSM Density Profile and Progenitor Mass-loss History \nFigure 9. Observed spectrum on day 14 compared to the L sh = 5 × 10 42 erg s -1 model from Dessart & Hillier (2022, blue) and our new model with the cool dense shell at lower velocity, increased clumping, and a more complete atomic model (pink). The new model is able to match the velocity of most of the features and provides a significantly better match to the SED. \n<!-- image --> \nUsing the best models at each epoch, we derive the density profile of the RSG progenitor of SN 2023ixf, which is given in Table 2 and plotted in Figure 10. We use the shock luminosity of each model, assuming a shock velocity of v sh = 10 , 000 km s -1 to find the CSM density at each epoch from ρ ( r ) = L sh / (2 πv 3 sh r 2 ) g cm -3 . Note that this is a simplified model that assumes both spherical symmetry and a shock velocity. From this, we find that the density decreases more rapidly than a constant wind, going from ρ = 5 . 2 × 10 -16 g cm -3 at 14 days after explosion to ρ = 4 . 8 × 10 -20 g cm -3 on day 66. \nIf we further use the RSG wind velocity of v wind = 55 km s -1 measured by Zhang et al. (2023), we can derive a mass-loss rate from the density and a time of mass loss from the radius (see Figure 11). We find ˙ M = 8 . 7 × 10 -4 M ⊙ yr -1 14 days after the explosion, decreasing to 4 . 4 × 10 -4 M ⊙ yr -1 19 and 24 days after the explosion, and finally arriving at a canonical RSG wind of 1 . 7 × 10 -6 M ⊙ yr -1 at 66 days after the explosion. This traces the mass loss from elevated rates ∼ 7 yr before the explosion down to nominal RSG rates ∼ 35 yr before explosion. The clear presence of Mg II emission on day 66 indicates that we can detect ejectaCSM interaction, even with canonical RSG winds of 10 -6 M ⊙ yr -1 . This mass-loss rate and timing are dependent on the assumed wind velocity. If Zhang et al. (2023) measured a wind that had already been radiatively accelerated, the velocity could be lower, which would increase the period of time prior to the explosion \nFigure 10. CSM density of SN 2023ixf as a function of the radius assuming a shock velocity of 10,000 km s -1 . The CSM density constraints from this paper are plotted with star symbols. The filled symbols are constrained by the overall UV and optical flux. The open stars represent the value from the Mg II line, which lies between L sh = 10 40 erg s -1 (black) and L sh = 10 41 erg s -1 (pink). Density constraints from flash spectroscopy are shown at the epoch of the first spectrum, day 4, when the majority of the narrow features disappear, and day 7, when all of the narrow features disappear (Bostroem et al. 2023a; Jacobson-Gal'an et al. 2023; Zhang et al. 2023). Mustard triangles represent a mass-loss rate of 10 -2 M ⊙ yr -1 and green inverted triangles represent a mass-loss rate of 10 -3 M ⊙ yr -1 . Values derived from the H α luminosity from Zhang et al. (2023) are shown as light-blue solid squares and the value from Bostroem et al. (2023a) is a light-blue open square. CSM density from NuSTAR X-ray observations (Grefenstette et al. 2023) are plotted as filled blue circles and X-ray observations from Chandra (Chandra et al. 2024) are shown as open blue circles. Lines of constant density corresponding to mass-loss rates of 10 -1 , 10 -2 , 10 -3 , 10 -4 , 10 -5 , and 10 -6 M ⊙ yr -1 are shown as solid gray lines. While there is significant scatter in the density, it is clear that density is decreasing more rapidly than would be expected from constant mass-loss rates down to the density expected for canonical RSG winds. \n<!-- image --> \nprobed by these observations and decrease the mass-loss rate. \nThe CSM density and mass-loss rates derived in this paper can be combined with other published values to paint a more complete picture of the CSM around the progenitor of SN 2023ixf. We plot these densities and mass-loss rates in Figure 10 and Figure 11. Grefenstette et al. (2023) use NuSTAR X-ray obser- \nvations on day 4.4 and day 11 to measure a column density of N H = 26 +5 -7 × 10 22 atoms cm -2 and N H = 5 . 6 ± 2 . 7 × 10 22 atoms cm -2 , respectively. Additionally, from Chandra observations on days 13 and 86, Chandra et al. (2024) find a column density of N H = 2 . 5 +0 . 40 -0 . 34 × 10 22 atoms cm -2 and N H = 0 . 36 +0 . 22 -0 . 17 × 10 22 atoms cm -2 , respectively. From the column density, the CSM density can be found assuming a density profile from a constant wind and that N H = ∫ ∞ R sh ρ ( r ) /m p dr . With these assumptions, the density at each epoch is ρ = ( N H m P ) /R sh . From the best-matching models of the observed flash features between 1 and 7 days, Bostroem et al. (2023a), JacobsonGal'an et al. (2023), and Zhang et al. 2023 infer a massloss rate of 10 -3 -10 -2 M ⊙ yr -1 from the CMFGEN models of Dessart et al. (2017) and Jacobson-Gal'an et al. (2023). As the line emission in these models is dependent on the density and mass loss is a parameterization of this, we use the model wind velocity of v wind = 50 km s -1 to find the density rather than the measured wind velocity. We plot the density derived from flash features at the time of the first spectrum (day 1.1), day 4 when many of the narrow features disappear, and day 7 when the H α narrow component disappears. Bostroem et al. (2023a) and Zhang et al. (2023) use the H α luminosity to calculate the CSM radius and density. Bostroem et al. (2023a) find R CSM ≳ 8 . 7 × 10 13 cm and ρ CSM = 3 . 4 × 10 -14 g cm -3 . Zhang et al. (2023) determine R CSM = 2 . 33, 2.59, and 2 . 74 × 10 14 cm and ρ CSM = 7 . 41, 7.26, and 7 . 31 × 10 -15 g cm -3 . \nThese combined measurements show that the massloss rate of the progenitor of SN 2023ixf was relatively low until quite close to the time of core collapse ( R ≲ 5 × 10 15 cm or t ≲ 33 yr prior to explosion), at which point it increased to the large densities inferred from flash-feature observations. However, this figure also highlights that the systematics, uncertainties, and simplifying assumptions in these different techniques lead to 1 order of magnitude uncertainties in mass-loss rates and densities.", '7. SUMMARY': "We present UV observations of SN 2023ixf 14-66 days after the explosion. SN 2023ixf is UV bright, with continuum emission through day 24. Its UV spectrum shows weaker metal absorption than other Type II supernovae that do not exhibit definitive signs of CSM interaction. In the optical, the H α P Cygni profile is slow to develop and has a shallow absorption component, which we infer is due to the additional emission from CSM interaction. On day 66, the Mg II profile and H α emission component have a similar extent in veloc- \nFigure 11. Same as Figure 10 for the mass-loss history of SN 2023ixf assuming a shock velocity ( v sh = 10 , 000 km s -1 ) and the CSM wind velocity ( v wind = 55 km s -1 ). In its last ∼ 30 yr of life, the mass-loss rate increased from a nominal RSG wind of ∼ 10 -6 M ⊙ yr -1 to ∼ 10 -2 M ⊙ yr -1 . \n<!-- image --> \ny space, but extremely different profile shapes, resulting from the lack of UV continuum from the supernova ejecta. We examine the contribution of different species to the UV spectrum of SN 2023ixf on days 14 and 19, finding that the majority of the absorption in the UV spectrum is due to iron, nickel, and magnesium at both epochs, with additional contributions by chromium at the later epoch. \nWe compare SN 2023ixf to CMFGEN models of supernova ejecta with power from CSM interaction (Dessart & Hillier 2022). Impressively, even though these models were not made for a supernova like SN 2023ixf, which interacted with dense CSM during its early evolution, they match the overall evolution and the presence of individual features. It is likely that the cool dense shell in SN 2023ixf is broader and centered at a lower velocity (8500 vs. 11,700 km s -1 ) and higher temperature than the models. This is confirmed by a custom model that places the cool dense shell at 8500 km s -1 . Additionally, the predicted presence of the Mg II emission after 50 days is confirmed, demonstrating that SN 2023ixf is interacting with CSM at all epochs through day 66. \nWe visually identify the model that best matches our observed spectrum at each epoch, finding that the power from CSM interaction decreases with time. From this, we conclude that at larger radii, there is lower density CSM, suggesting that the mass-loss rate increased \nrapidly just prior to the explosion. While the flash features favor a confined CSM, these disappear after a week, limiting the radius out to which the mass-loss history can be traced by this method. Our UV observations are sensitive to much lower CSM densities, and we use them to trace the CSM density to significantly larger radii and the mass-loss history to earlier epochs. With a shock velocity of 10,000 km s -1 , our observations probe the CSM density between a radius of 1 . 2 × 10 15 cm to 5 . 2 × 10 15 cm and show the density evolving from 5 . 2 × 10 -16 g cm -3 to 4 . 8 × 10 -20 g cm -3 over this range. Using the wind velocity of 55 km s -1 , we find that the mass-loss rate decreases from 8 . 7 × 10 -4 M ⊙ yr -1 in our first observation to 1 . 7 × 10 -6 M ⊙ yr -1 in our last observation, showing the dramatic change in mass-loss rates 33-7 yr before the explosion. Additionally, the mass-loss rate derived from the day 66 spectrum shows that 33 yr before the explosion, the RSG progenitor's mass-loss rate was consistent with a quiescent RSG wind (Beasor et al. 2020). \nWe also compare SN 2023ixf with other supernovae having UV spectra around days 10 and 20. These objects cover a range of UV fluxes, similar to the range spanned by the CMFGEN models with varying levels of CSM interaction. This suggests that we can use the UV to sensitively distinguish varying levels of CSM interaction. Around day 10, we see that Mg II is clearly present in absorption in the spectra with the most CSM interaction. Often appearing next to it is Fe II ; this can be seen as a distinct feature in SN 2023ixf and SN 2022wsp, and it may be present in the other supernovae either blended with Mg II owing to its formation at higher velocities or as a weaker feature. Mg II and Fe II are weaker around day 20 in all supernovae. Additionally, the apparent emission line at 1924 ˚ A is caused by a window of lower iron absorption, rather than emission from an individual species, demonstrating the complexity of associating a single species with an individual feature in UV spectra. \nBased on the shape and width of the Mg II emission in the day 66 spectrum, we conclude that this emission is from the cool dense shell, which is well described by a thick shell with a Gaussian density profile: FWHM ≈ 7700 km s -1 and an inner radius velocity of ∼ -6500 km s -1 . We model the emission with an asymmetric Gaussian shell, finding that the profile is consistent with more of the observed emission coming from the blue side than the red side. Additionally, in other supernovae, this feature is either symmetric or asymmetric with a higher blue side, leading us to conclude that the profile shape is due to attenuation of the far side of the shell, likely from opacity in the shell and ejecta at early times and dust \nattenuation in the later-time spectra (e.g. SN 1998S at 240d). \nFinally, we compile the density and mass-loss measurements of SN 2023ixf from the literature using a consistent shock velocity of 10,000 km s -1 and a wind velocity of 55 km s -1 . We show that the density profile from 9 . 5 × 10 13 cm to 7 . 5 × 10 15 cm (0.5-42 yr before explosion) decreases from 1 . 1 × 10 -12 g cm -3 to 4 . 9 × 10 -20 g cm -3 . With a wind velocity of 55 km s -1 , this corresponds to a change in mass-loss rate of ∼ 10 -2 M ⊙ yr -1 to ∼ 10 -6 M ⊙ yr -1 . This shows that over the final ∼ 40 yr of the progenitor's life, the mass-loss rate went from that of a quiescent RSG to an extremely high mass-loss rate. The densities and mass-loss rates of these different techniques span about an order of magnitude, highlighting the uncertainties and systematic errors in our methods, even when some consistent assumptions are made. \nSN 2023ixf is the best-observed Type II supernova since SN 1987A, providing us with a Rosetta Stone to interpret other observations and against which to benchmark theoretical predictions. Although relatively few UV observations of Type II supernovae exist (and even fewer time series), it is clear that this wavelength range is rich in information about the temperature, composition, and dynamics of the CSM and the ejecta. It also probes the pre-supernova mass-loss history, tracing it back to quiescent RSG winds 40 years before explosion and can provide insights into the role of CSM interaction in the diversity of optical properties that we observe in Type II supernovae. In particular, time series which look for the presence of Mg II emission for all Type II supernovae and trace its evolution to even later epochs are warranted inform our understanding of mass loss in RSGs, especially in the years leading up to explosion. The observations of SN 2023ixf provide a critical link between early- and late-time UV observations, and also between 'normal' Type II supernovae and stronglyinteracting Type II supernovae.", 'T able 3 . Prop erties of Sup erno v ae with UV Sp ectra Analyzed in This P ap er': 'Note -+ SN 1980K do es not h a v e a reliable explosion ep o c h. W e adopt an explosion ep o c h 7 da ys prior to an observ ed sp ectrum on 1980 No v em b er 1 whic h sho ws broad H α emission. † See also Dessart & H illier 2006 , ∗ See also Dessart et al. 2008 . (A): Sc hlafly & Finkb einer 2011 . (B): Gall et al. 2015 . (C): F errarese et al. 1996 . (D): Barb on et al. 1982b . (E): P anagia et al. 1980 . (F): P a lum b o 1982 . (G): Ben v en uti et al. 1982 . (H): F esen et al. 1999 . (I): V an Dyk et al. 2019 . (J): Barb on et al. 1982a . (K): P ettini et al. 1982 . (L): Roma niello et al. 2000 . (M): Castagnoli et al. 1987 . (N): Scuderi et al. 1996 . (O): W amstek er et al. 1987 . (P): Cassatella et al. 1987 . (Q): Cassatella 1987 . (R): Kirshner et al. 1987 . (S): P anagia 1988 . (T): Pun et al. 1995 . (U): F reedman et al. 1994 . (V): Le wis et al. 1994 . (W): Ergon et al. 2014 . (X): Jeffery et a l. 1994 . (Y): F ransson et al. 2005 . (Z): Willic k et al. 1997 . (AA): B os tro em et al. 2023a . (BB ): Leonard et al. 2000 . (CC): Len tz e t al. 2001 . (DD): F ransson et al. 2005 . (EE): Leonard et al. 2003 . (FF): Elmhamdi et al. 2003 . (GG): Silv erman et al. 2017 . (HH): Baron et al. ( 2000 ). (I I): T ully 1988 . (JJ): Tsv etk o v et al. 2006 . (KK): Gal-Y am et al. 2008 . (LL): Sabbi et al. 2018 . (MM): P astorello et al. 2006 . (NN): Baron et al. 2007 . (OO): Bro wn et al. 2007 . (PP): Zhang et al. 2012 . (QQ): Stoll et al. 2011 . (RR): F ransson et al. 2014 . (SS): Hosseinzadeh et al. 2022 . (TT): V asyly ev et al. 2022 . (UU): V asyly ev et al. 2023a . (VV): Anand et al. 2021 . (WW): Bostro em et al. 2023c . \n- We thank the referee for their thoughtful comments. 1\n- This research is based on observations made with the 2\n- NASA/ESA Hubble Space Telescope obtained from the 3\n- Space Telescope Science Institute, which is operated by 4\n- the Association of Universities for Research in Astron5\n- omy, Inc., under NASA contract NAS 5-26555. These 6 \n7 \nobservations are associated with programs GO-17313 \n- and GO-17205. Huge thanks to our HST program coor8\n- dinator Alison Vick and our contact scientist Dan Welty 9\n- for assistance in planning and executing the observations. The data described here may be obtained from the MAST archive at doi:10.17909/787k-c897. 10 11 12 \n13 \n14 \nK.A.B. is supported by an LSST-DA Catalyst Fellow- \nship; this publication was thus made possible through \n- the support of grant 62192 from the John Templeton 15\n- Foundation to LSST-DA. A.V.F. is grateful for financial 16\n- assistance from the Christopher R. Redlich Fund and 17\n- many other donors. Time domain research by D.J.S. 18 \n19 \nand team is supported by NSF grants AST-1908972 \n- and 2108032, and by the Heising-Simons Foundation 20\n- under grant No. 20201864. Research by Y.D., S.V., 21\n- N.M.R, E.H., and D.M. is supported by NSF grant AST22\n- 2008108. 23', 'REFERENCES': 'Facilities: \nHST(STIS) \nSoftware: astropy (Astropy Collaboration 2013; 2018; 2022), CMFGEN (Hillier & Miller 1998; Dessart et al. 2013; Hillier & Dessart 2019), MatPLOTLIB (Hunter 2007), NumPy (Harris et al. 2020), Scipy (Virtanen et al. 2020), stistools (Sohn 2019) \nZhang, J., Lin, H., Wang, X., et al. 2023, Science Bulletin, 68, 2548, doi: 10.1016/j.scib.2023.09.015 \nZhang, T., Wang, X., Wu, C., et al. 2012, AJ, 144, 131, doi: 10.1088/0004-6256/144/5/131 \nZimmerman, E. A., Gal-Yam, A., Bruch, R. J., et al. 2022, Explosions in Real-Time: Rapid UV Supernova Flash Spectroscopy, HST Proposal. Cycle 30, ID. #17205 Zimmerman, E. A., Irani, I., Chen, P., et al. 2024, Nature, 627, 759, doi: 10.1038/s41586-024-07116-6 Zimmermann, H. U., Lewin, W., Predehl, P., et al. 1994, Nature, 367, 621, doi: 10.1038/367621a0'}
2024arXiv240902807C	It is a common belief that a theory of quantum gravity should ultimately cure curvature singularities which are inevitable within General Relativity and plague for instance the Schwarzschild and Kerr metrics usually considered as prototypes for primordial black holes PBHs as dark matter DM candidates. We continue our study initiated in a companion paper of nonsingular objects as PBHs considering three regular nontr nontimeradialsymmetric metrics all of which are oneparameter extensions of the Schwarzschild spacetime the SimpsonVisser PeltolaKunstatter and DAmbrosioRovelli spacetimes with the latter two motivated by loop quantum gravity. We study evaporation constraints on PBHs described by these regular metrics deriving upper limits on ftextpbh the fraction of DM in the form of PBHs. Compared to their Schwarzschild counterparts these limits are weaker and result in a larger asteroid mass window where all the DM can be in the form of PBHs with the lower edge moving potentially more than an order of magnitude. Our work demonstrates as a proofofprinciple that quantum gravityinspired spacetimes can simultaneously play an important role in the resolution of singularities and in the DM problem.	2024-09-01T00:00:00Z	['arXiv:2409.02807', '2024arXiv240902807C', '10.48550/arXiv.2409.02807']	['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory']	Primordial regular black holes as all the dark matter. II. Nontimeradialsymmetric and loop quantum gravityinspired metrics	2,024	198	0.32	['EPRINT_HTML', 'EPRINT_PDF']	9	https://arxiv.org/pdf/2409.02807.pdf	{'Primordial regular black holes as all the dark matter. II. Non-time-radial-symmetric and loop quantum gravity-inspired metrics': "Marco Calz'a, 1, 2, ∗ Davide Pedrotti, 1, 2, † and Sunny Vagnozzi 1, 2, ‡ \n1 Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy § 2 Trento Institute for Fundamental Physics and Applications (TIFPA)-INFN, Via Sommarive 14, 38123 Povo (TN), Italy (Dated: December 9, 2024) \nIt is a common belief that a theory of quantum gravity should ultimately cure curvature singularities which are inevitable within General Relativity, and plague for instance the Schwarzschild and Kerr metrics, usually considered as prototypes for primordial black holes (PBHs) as dark matter (DM) candidates. We continue our study, initiated in a companion paper, of non-singular objects as PBHs, considering three regular nontr (non-time-radial)-symmetric metrics, all of which are oneparameter extensions of the Schwarzschild space-time: the Simpson-Visser, Peltola-Kunstatter, and D'Ambrosio-Rovelli space-times, with the latter two motivated by loop quantum gravity. We study evaporation constraints on PBHs described by these regular metrics, deriving upper limits on f pbh , the fraction of DM in the form of PBHs. Compared to their Schwarzschild counterparts, these limits are weaker, and result in a larger asteroid mass window where all the DM can be in the form of PBHs, with the lower edge moving potentially more than an order of magnitude. Our work demonstrates as a proof-of-principle that quantum gravity-inspired space-times can simultaneously play an important role in the resolution of singularities and in the DM problem.", 'I. INTRODUCTION': "Once regarded as objects of pure mathematical interest, over the past decade black holes (BHs) have gone on to become some of the most fascinating objects in the Universe [1]. At the time of writing, observational effects associated to astrophysical BHs are detected on a regular basis allowing us to use these extreme regions of space-time as unique laboratories for testing fundamental physics in the strong-field regime [2-52]. On the more theoretical end of the spectrum, a widespread hope is that BHs may hold the key towards the unification of quantum mechanics and gravity, although a somewhat more humble goal could be that of using BH observations to test candidate theories of quantum gravity (QG). On the more phenomenological side, the possible role of BHs in accounting for the dark matter (DM) which makes up ≃ 25% of the energy budget of the Universe [53, 54] is now widely acknowledged. In our work, these two aspects - DM and candidate theories of QG - will naturally meet, with BHs being the common denominator, and astrophysical observations the playing ground. \nThe collapse of large density perturbations upon horizon re-entry in the early Universe can lead to the formation of hypothetical relics known as primordial BHs (PBHs), whose role as potential DM candidates has long been recognized [55-154] (for recent reviews on the topic, see Refs. [155-164]). A wide range of observations (mainly of astrophysical nature) severely limit the ability of PBHs to account for the entire DM component: in \npractice, this is potentially possible (although this possibility is not completely free of debates) only in the so-called ' asteroid mass window ', i.e. 10 17 g ≲ M pbh ≲ 10 23 g, with lighter and heavier PBHs being tightly constrained by observational signatures of their evaporation and microlensing respectively [165-180]. However, it is important to note that virtually all constraints on PBHs, including those determining the existence and extension of the asteroid mass window, are subject to the underlying assumption about these object being either Schwarzschild or Kerr BHs [155-164]. This assumption is perfectly reasonable from the phenomenological and observational point of view, but at the same time may be cause of some apprehensiveness on the more theoretical side. In fact, these metrics feature pathological curvature singularities, whose existence is virtually inevitable in General Relativity (GR), and is at the essence of the well-known singularity problem [181-185]. \nGiven that significant efforts are being devoted to the study of so-called regular space-times, free of curvature singularities, a relevant question is therefore what happens if PBHs are regular. This is a question we started to systematically address in a companion paper focused on tr (time-radius)-symmetric metrics, i.e. where the product of the coefficients of the dt 2 and dr 2 terms in the line element in four-dimensional Boyer-Lindquist coordinates is equal to -1 [186]: these metrics include, for instance, the well-known Bardeen [187] and Hayward regular BHs [188]. As we show in our companion paper, the phenomenology of the resulting primordial regular BHs (PRBHs) can be very rich, and can result in the asteroid mass window opening by up to an extra decade in mass [186]. The choice of studying tr -symmetric metrics was adopted to make the equations simpler to handle, but is certainly not exhaustive. Indeed, as we shall soon see, such a choice does not cover a number of well-known and well-motivated metrics, potentially including space- \ntimes rooted into candidate theories of QG. In this sense, it is worth recalling that the metrics considered in our companion paper [186] are purely phenomenological in nature. It is therefore our goal in the present work to extend our earlier study of PRBHs to nontr -symmetric metrics, some of which carry very strong theoretical motivation and can arise within candidate theories of QG. \nTo be concrete, in what follows we will consider three regular, static spherically symmetric space-times, characterized by an additional regularizing parameter ℓ , and recovering the Schwarzschild space-time in the ℓ → 0 limit. All three space-times enjoy quite different properties compared to the phenomenological ones considered in our companion paper [186]. The first metric we consider is the so-called Simpson-Visser metric: this is arguably one of the best known black-bounce space-times, and interpolates between the Schwarzschild metric, regular BHs, and traversable wormholes. 1 The other two metrics are instead deeply rooted within Loop Quantum Gravity (LQG) [190-192]: arguably one of the leading QG approaches, LQG is a fully non-perturbative and manifestly background-independent approach towards a consistent theory of QG, wherein space-time is fundamentally discrete (see e.g. Refs. [193-222] for various follow-up studies and applications). More specifically, the two regular LQG-motivated metrics we analyze as candidates for DM in the form of PRBHs are the PeltolaKunstatter [223, 224] and D'Ambrosio-Rovelli spacetimes [225, 226]. As a cautionary note, we remark that ours is to be intended as a pilot study in this direction, and that much more follow-up work is needed before primordial regular BHs as DM candidates are characterized to the same extent as their Schwarzschild counterparts. 2 \nThe rest of this paper is then organized as follows. We briefly introduce the regular space-times studied in our work in Sec. II. Theoretical aspects of the Hawking evaporation process are presented in the next two sections, with Sec. III A devoted to the derivation of the greybody factors, Sec. III B to the calculation of the resulting photon spectra, and Sec. III C to the derivation of constraints on the fraction of DM which may be in the form of PRBHs. The resulting limits are discussed in Sec. IV. Finally, in Sec. V we draw concluding remarks. Technical issues regarding the asymptotic solutions of the \nradial Teukolsky equation which may be of interest to some readers are discussed in Appendix A. Unless otherwise specified, we adopt units where G = c = ℏ = 1. We recall once again that a related study focusing on tr -symmetric, phenomenological metrics is presented in our companion paper [186]. If time allows our recommendation is that the interested reader consult our companion paper [186] prior to reading the present work.", 'II. REGULAR BLACK HOLES': "As is well known, GR predicts the almost unavoidable existence of essential space-time singularities, where curvature invariants diverge. Nevertheless, it is a commonly held belief that these unwanted features are merely a reflection of our ignorance of a more fundamental theory of QG, which would ultimately cure these singularities (see, however, Refs. [234-236]). Various regular BH (RBH) metrics, free of singularities in the entire spacetime, have in fact been studied in recent years, both from a more phenomenological standpoint [237-287] as well as from a first-principles theoretical basis [288-302]. 3 These RBHs are usually controlled by an extra regularizing parameter (which we denote by ℓ ), and typically (but not necessarily) recover the Schwarzschild metric as ℓ → 0. In what follows, similarly to our companion paper [186] we will entertain the possibility that DM may be in the form of primordial RBHs. \nThe line element of the space-times we consider can all be written in the following general form: \nds 2 = -f (˜ r ) dt 2 + 1 f (˜ r ) [1 -g ℓ (˜ r )] d ˜ r 2 + ˜ r 2 d Ω 2 , (1) \nwhere d Ω 2 = dθ 2 + sin 2 ( θ ) dϕ 2 is the metric on the 2sphere and ˜ r is manifestly the areal radius. On the other the function g ℓ , which depends on the regularizing parameter ℓ , goes to g ℓ (˜ r ) → 0 for both ℓ → 0 and ˜ r →∞ . Such a space-time possesses horizon-like structures located at radial coordinates ˜ r such that: \nf (˜ r ) [1 -g ℓ (˜ r )] = 0 , (2) \ni.e. at ˜ r = ˜ r H , ˜ r 0 such that f (˜ r H ) = 0 and/or g ℓ (˜ r 0 ) = 1. If ˜ r 0 > ˜ r H , the value ˜ r 0 determines the location of a wormhole (WH) throat, whereas no event on the manifold is associated to the location of ˜ r H . Note that, in general ˜ r 0 depends on the regularizing parameter, i.e. ˜ r 0 = ˜ r 0 ( ℓ ). On the other hand, when ˜ r H > ˜ r 0 , the value ˜ r H characterizes the event horizon of a BH (in \nthis case the WH throat is located within the BH event horizon and is therefore causally disconnected from the relevant BH exterior space-time). Typically, in regions where these space-times are regular, the above geometry describes a bounce into a future incarnation of the universe [309-311]. Due to this peculiar characteristic, geometries of this type are sometimes referred to as blackbounce space-times. \nFor these types of metric, it generally proves advantageous to perform a change of variable for what concerns the radial coordinate, going from an extrinsic description to an intrinsic one through the coordinate transformation ˜ r = √ r 2 + ℓ 2 . The metric in Eq. (1) can then be expressed in the following form: \nds 2 = -f ( r ) dt 2 + g ( r ) -1 dr 2 + h ( r ) d Ω 2 , (3) \nwhere h ( r ) = r 2 + ℓ 2 . When expressed in the above form, the Petrov-D nature of this class of metrics is manifest. We additionally require asymptotic flatness, in other words that f (˜ r ) → 1 for ˜ r → ∞ , from which it follows that: \nf ( r ) r →∞ ---→ 1 , g ( r ) r →∞ ---→ 1 , h ( r ) r →∞ ---→ r 2 . (4) \n̸ \n̸ \nFinally, we note that the metrics in question are nontr -symmetric, since in general f ( r ) = g ( r ) and h ( r ) = r 2 . The tr -symmetric case is treated in our companion paper [186], whereas we have chosen to deal with the nontr -symmetric case in a separate work both because it introduces non-trivial complications on the mathematical side, and at the same time allows us to treat metrics which are strongly motivated from first-principles theoretical considerations (unlike those considered in our companion paper, which are introduced on purely phenomenological grounds), such as LQG. \nA key quantity characterizing the RBHs we are considering is their temperature T , since this directly controls the strength of the radiation emitted from Hawking evaporation. Assuming that the temperature is the usual Gibbons-Hawking one, which in turn tacitly implies that we are assuming the standard Boltzmann-Gibbs distribution (see our companion paper for a slightly more detailed discussion on this point [186]), the temperature is given by the following: \nT = √ g ( r ) f ( r ) f ' ( r ) 4 π \| r H , (5) \nwhere the prime indicates a derivative with respect to r , and r H denotes the location of the event horizon. In Fig. 1 we show the evolution of the temperatures, normalized to the temperature of Schwarzschild BHs T Sch = 1 / 8 πM , of the three RBHs we will discuss shortly, as a function of the regularizing parameter ℓ normalized to the event horizon radius r H . As we see, for all three metrics the temperature is a monotonically decreasing function of the regularizing parameter. One may therefore qualitatively expect that the intensity of the Hawking evaporation radiation should decrease relative to that \nFIG. 1. Evolution of the temperatures (normalized to the temperature of Schwarzschild black holes, T Sch = 1 / 8 πM ) as a function of the regularizing parameter ℓ (normalized to the event horizon radius r H ) for the three regular space-times studied in the work: the Simpson-Visser (blue solid curve, Sec. II A), Peltola-Kunstatter (red dashed curve, Sec. II B), and D'Ambrosio-Rovelli (green dotted curve, Sec. II C) regular space-times. \n<!-- image --> \nof Schwarzschild BHs of the same mass: this expectation in fact turns out to be correct, as we will explicitly show later, with important consequences for f pbh limits.", 'A. Simpson-Visser space-time': "The Simpson-Visser (SV) metric is a one-parameter extension of the Schwarzschild space-time and easily one of the best known black-bounce metrics. In the words of Simpson and Visser, in some sense this metric 'represents the minimal violence to the standard Schwarzschild solution' needed to enforce regularity [312]. The line element analytically interpolates between black holes and traversable wormholes according to the value of the regularizing parameter. In the notation of Eq. (1), the SV space-time is characterized by the following functions: 4 \nf (˜ r ) = 1 -2 M ˜ r , g ℓ (˜ r ) = ℓ 2 ˜ r 2 , (6) \nwhereas, in the notation of Eq. (3), the line element of the SV space-time is given by the following: \nds 2 = -( 1 -2 M √ r 2 + ℓ 2 ) dt 2 + dr 2 1 -2 M √ r 2 + ℓ 2 +( r 2 + ℓ 2 ) d Ω 2 . (7) \nThe SV space-time encompasses a rich phenomenology, as it interpolates between the Schwarzschild BH ( ℓ = 0), a regular BH with a one-way space-like throat (0 < ℓ/M < 2), a one-way WH with an extremal null throat ( ℓ/M = 2), and a traversable WH with a two-way timelike throat ( ℓ/M > 2). 5 This metric has been the subject of several follow-up studies (see e.g. Refs. [313323]) and, while originally introduced on phenomenological grounds, can potentially originate as a solution of GR coupled to non-linear electrodynamics in the presence of a minimally coupled phantom scalar field [324].", 'B. Peltola-Kunstatter space-time': 'The Peltola-Kunstatter (PK) space-time is a LQGmotivated metric obtained upon applying effective polymerization techniques to 4D Schwarzschild BHs. Although there are indications that LQG may be capable of resolving the singularities which plague GR, the inherent difficulty in solving the complete system has led to the development of semi-classical polymer quantization techniques, which provide an unitarily inequivalent alternative to Schrodinger quantization while maintaining the key aspect of space-time discreteness. The PK space-time is obtained polymerizing only area but not the conformal mode, and results in a space-time whose singularity is replaced by a complete and regular bounce, where the space-time reaches a minimum radius before expanding into a Kantowski-Sachs metric [223, 224]. In the notation of Eq. (1), the SV space-time is characterized by the following functions: \nf (˜ r ) = √ 1 -ℓ 2 ˜ r 2 -2 M ˜ r , g ℓ (˜ r ) = ℓ 2 ˜ r 2 (8) \nwhereas, in the notation of Eq. (3), the line element of the PK space-time is given by the following: \nds 2 = -( r -2 M √ r 2 + ℓ 2 ) dt 2 + dr 2 r -2 M √ r 2 + ℓ 2 +( r 2 + ℓ 2 ) d Ω 2 . (9) \nIn what follows, we shall take the PK space-time as an example of regular metric motivated by first-principles quantum gravity considerations, unlike the other phenomenological metrics considered earlier. \n̸', "C. D'Ambrosio-Rovelli space-time": "The D'Ambrosio-Rovelli (DR) space-time was originally developed with motivations other than singularity avoidance, and is in fact also motivated by LQG considerations. This space-time represents a natural extension of the Schwarzschild space-time which crosses the r = 0 singularity smoothly into the interior of a white hole, and one can see it as the ℏ → 0 limit of an effective QG metric [225, 226]. It has been argued that this black hole-to-white hole tunneling mechanism can shed light on possible solutions to the information paradox. Of interest to us is the fact that the DR metric is regular, as a result of the curvature of the effective metric being bound at the Planck scale. The ansatz for the effective metric written by D'Ambrosio and Rovelli is similar to that of the SV metric, but differs in the form of the g function - specifically, the two relevant functions are given by the following: \nf (˜ r ) = 1 -2 M ˜ r , g ℓ (˜ r ) = ℓ ˜ r , (10) \nwhereas, in the notation of Eq. (3), the line element of the DR space-time is given by the following: \nds 2 = -( 1 -2 M √ r 2 + ℓ 2 ) dt 2 + dr 2 1 -2 M √ r 2 + ℓ 2 ( 1 + ℓ √ r 2 + ℓ 2 ) +( r 2 + ℓ 2 ) d Ω 2 . (11) \nMuch like the PK space-time, we will take the DR spacetime as another well-motivated example of QG-inspired metric. We note that the assumption of primordial DR BHs inevitably leads to the existence of long-lived primordial DR white holes from quantum transitions near the would-be singularity. This can potentially lead to an interesting phenomenology whose exploration, however, is well beyond the scope of this work.", 'A. Greybody factors': "We now discuss the computation of the greybody factors (GBFs), functions of energy and angular momentum which characterize the shape of the emitted Hawking radiation (and in particular its deviation from a blackbody) and therefore play a key role in determining the resulting evaporation constraints [325-327]. It is worth noting that, in the notation of Eq. (3), both the SV and PK metrics share the fact that f ( r ) = g ( r ), which makes the calculations somewhat easier. This is not the case, however, for the DR space-time. Therefore, in what follows, we consider the more general case where f ( r ) = g ( r ), \n̸ \nwhich differs from the much simpler tr -symmetric case considered in our companion paper [186]. \nWe adopt the Newman-Penrose (NP) formalism, denoting by Υ s a general perturbation of spin s (defined by the appropriate NP scalars) and dropping the l and m indices to lighten the notation. Then, the Teukolsky equation for the evolution of massless perturbations of given spin upon a background metric characterized by functions f ( r ), g ( r ), h ( r ) as in Eq. (3) reduces to the following master equation [328]: \n-h g ∂ 2 t Υ s + s √ f g ( hg ' g -h ' ) ∂ t Υ s + fh∂ 2 r Υ s + ( hf ' 2 +( s +1 / 2) fhg ' g +( s +1) fh ' ) ∂ r Υ s + ( 1 sin θ ∂ θ (sin θ ∂ θ ) + 1 sin 2 θ ∂ 2 ϕ + 2 is cot θ sin θ ∂ ϕ -s 2 cot 2 θ -s ) Υ s ( s hfg '' g + 3 s -2 s 2 4 (2 fh '' + f ' h ' ) + s 2 ( hf ' g ' g -fhg ' 2 g 2 ) + 2 s 2 -s 4 fh ' 2 h + 2 s 2 +5 s 4 fg ' h ' g ) Υ s = 0 , (12) \nwhich is separable with the following ansatz: \nΥ s = ∑ l,m e -iωt e imϕ S l s ( θ ) R s ( r ) , (13) \nwith ω , l , and m being the perturbation frequency, angular node number, and azimuthal node number respectively, whereas S l s ( θ ) are related to the so-called spin-weighted spherical harmonics S s l,m ( θ, ϕ ) through S s l,m ( θ, ϕ ) = ∑ S l s ( θ ) e imϕ . \nWe now define the functions A s , B s , and C s as follows: \nA s = √ f g 1 ( gh ) s , (14) \nB s = √ fg ( gh ) s h, (15) \nC s = s fhg '' g + s 2 ( hf ' g ' g -fhg ' 2 g 2 ) + s (3 -2 s ) 4 (2 fh '' + f ' h ' ) + s (2 s -1) 4 fh ' 2 h . (16) \nWith these definitions, the decoupled radial Teukolsky equation reduces to the following general form [329]: \nA s ( B s R ' s ) ' + [ h g ω 2 + iωs √ f g ( h ' -hg ' g ) + C s ] R s = 0 . \nWe further define the tortoise coordinate r ⋆ as follows: \ndr ⋆ dr = 1 √ f ( r ) g ( r ) , (18) \nnoting that, being our space-times asymptotically flat, r ⋆ → r for large r . In order to compute the GBFs for the metrics in question, we set purely ingoing boundary conditions. In addition, we need to know the asymptotic behaviour of R s as at infinity and close to the horizon. These asymptotic behaviours are given as follows: \nR s ∼ R in s e -iωr ⋆ r + R out s e iωr ⋆ r 2 s +1 ( r →∞ ) (19) \nR s ∼ R hor s A s e -iωr ⋆ ( r → r H ) , (20) \nas proven in more detail in Appendix A (the case of the DR metric is actually far from trivial). \nTo compute the GBFs, we make use of the shooting method, widely used earlier in similar contexts (see e.g. Refs. [330-337]), including in our companion paper [186]. We begin by defining the rescaled coordinate x : \nx ≡ r -r H r H , (21) \nwhere r H is the largest real root of the equation f ( r ) = 0. In order to further simplify our notation, in what follows we work in units of horizon radius, setting r H = 1, so that r = x +1. The decoupled radial Teukolsky equation, Eq. (17), then takes the following form: \nA s R s + B s ˙ R s + C s R s = 0 , (22) \nwhere the functions A , B , and C are defined as follows: \nA s = f 2 h, (23) \nB s = (( s + 1 2 ) f 2 h ˙ g g + hf 2 +(1 + s ) f 2 ˙ h ) , (24) \nC s = f 4 ( s (2 s -1) f ˙ h 2 h +2 s (3 -2 s ) f h -2 s fh ˙ g 2 g 2 + 1 ( s (5 + 2 s ) f ˙ g ˙ h +2 h ( 2 ω 2 +2 sf g (25) \ns ˙ g ( 2 iω √ f g + ˙ f ))) + sh ( (3 -2 s ) ˙ f +4 iω √ f g )) , \ng - \n(17) where the dot denotes a derivative with respect to the rescaled coordinate x . For completeness, we note that f and g as as function of x for the metrics considered here \nare given by the following: \nh SV ( x ) = h PK ( x ) = h DR ( x ) = ( x +1) 2 + ℓ 2 , g SV ( x ) = f SV ( x ) = 1 -√ 1 + ℓ 2 √ ℓ 2 +( x +1) 2 , g PK ( x ) = f PK ( x ) = x √ ℓ 2 +( x +1) 2 , g DR ( x ) = 1 -√ 1 + ℓ 2 √ ℓ 2 +( x +1) 2 , -. (26) \nf DR ( x ) = g DR ( x ) ( 1 + ℓ √ ℓ 2 +( x +1) 2 ) 1 \nWe express the solution to Eq. (22) as a Taylor expansion as follows [330, 331, 338-341]: \nR s ( x ) = x -s -iω τ ∞ ∑ n =0 a n x n . (27) \nHere, τ is a function of the field's spin and regularizing parameter ℓ , and varies with the metric being considered. For further details, we refer the reader to the Appendix of our companion paper [186], where the issue is discussed in more depth. We then determine the coefficients a n iteratively, by repeatedly substituting Eq. (27) into Eq. (22). \nOnce we have the near-horizon solution, we treat it as a boundary condition from which we numerically integrate outwards, where the solution takes the following form: \nR ( x ) r →∞ ---→ R in s e -iωx x + R out s e iωx x 2 s +1 , (28) \nwith the GBFs then given by the following: \nΓ s lm ( ω ) = δ s \| s R lm in ( ω ) \| -2 , (29) \nwhere δ s is given by: \nδ s = ατ ie iπs (2 ω ) 2 s -1 Γ(1 -s -2 iω τ ) Γ( s -2 iω τ ) . (30) \nwith α and τ depending on the metric considered. More specifically, for what concerns α , we find that within the SV and PK metrics for which f = g the following holds: \nα SV = α PK = 1 + ℓ 2 , (31) \nwhereas for the DR metric we find the following: \nα DR = 1 + ℓ ( ℓ + √ 1 + ℓ 2 ) . (32) \nFinally, for the three metrics τ is given by the following: \nτ SV = 1 1 + ℓ 2 , τ PK = 1 √ 1 + ℓ 2 , τ DR = √ 1 + ℓ ( ℓ -√ 1 + ℓ 2 ) 1 + ℓ 2 . (33) \nFIG. 2. Greybody factors Γ s =1 l =1 as a function of ω/M for Schwarzschild BHs (black curve), as well as the SimpsonVisser (blue solid curve), Peltola-Kunstatter (red dashed curve), and D'Ambrosio-Rovelli (green dotted curve) regular space-times. For illustrative purposes we only plot Γ s =1 l =1 , since we are interested in photons ( s = 1) and the dominant emission mode is the l = 1 one. We have fixed the regularizing parameter to ℓ = 0 . 3 r H for all three regular space-times. We see that in all three cases the GBFs are consistently (slightly) higher than their Schwarzschild counterparts. \n<!-- image --> \nAs can be seen in Eqs. (31,32), deviations from α = 1 are associated to the nontr -symmetric nature of the metrics. As a general consideration, we note that the computation of GBFs for the metrics under consideration is significantly more involved compared to those considered in our companion paper [186]. \nIn Fig. 2 we plot the Γ s =1 l =1 GBFs for the Schwarzschild BH and the three regular space-times we study, focusing for illustrative purposes on the Γ s =1 l =1 GBF and fixing ℓ = 0 . 3 r H for all three regular space-times. We observe that the GBFs are slightly higher than their Schwarzschild counterparts (by ≲ 10% at most), and asymptote to the latter for both ω/M ≲ 0 . 3 and ω/M ≳ 0 . 7.", 'B. Evaporation spectra': "As in our companion paper [186], we only consider the primary photon spectrum resulting from Hawking evaporation, while also checking that within the mass region of interest the secondary spectrum will provide a negligible contribution. For a particle species i of spin s (characterized by n i degrees of freedom), the rate of emission (particles per unit time per unit energy) through Hawking radiation is given by the following [342-346]: \nd 2 N i dtdE i = 1 2 π ∑ l,m n i Γ s l,m ( ω ) e ω/T ± 1 , (34) \nFIG. 3. Primary photon spectra resulting from the evaporation of Simpson-Visser black holes of mass 10 16 g for different values of the regularizing parameter ℓ (normalized by the horizon radius r H ): ℓ/r H = 0 . 3 (red dotted curve), 0 . 6 (green dashed curve), and 0 . 9 (magenta dash-dotted curve). The blue solid curve corresponds to the case ℓ/r H = 0, which recovers the Schwarzschild black hole. \n<!-- image --> \nFIG. 4. As in Fig. 3, but for Peltola-Kunstatter black holes, with identical values of the regularizing parameter ℓ/r H and identical color coding. \n<!-- image --> \nwith ω = E i being the mode frequency, whereas the positive (negative) sign in the denominator corresponds to fermions (bosons). To compute the (photon) GBFs Γ s l,m ( ω ) we adopt the methodology discussed earlier, going up to node number l = 4, but verifying that including higher l modes leads to negligible corrections. \nExamples of the resulting evaporation spectra are provided in Figs. 3, 4, and 5 for a representative PBH of mass M pbh = 10 16 g, located somewhat halfway in the mass range of interest (although the features we discuss shortly do not depend on the chosen mass). For the SV and DR PRBHs, increasing ℓ leads to the intensity of the \nFIG. 5. As in Fig. 3, but for D'Ambrosio-Rovelli black holes, with identical values of the regularizing parameter ℓ/r H and identical color coding. \n<!-- image --> \nspectra decreasing at all energies. This feature confirms the intuition raised when we discussed the temperatures of these BHs (see Fig. 1), which we observed to decrease with increasing ℓ . However, we remark that a decrease in T alone is not sufficient to draw this conclusion, since the GBFs also enter in Eq. (34). Nevertheless, one can expect the temperature to play a more important role, as it enters exponentially into Eq. (34), unlike the GBFs which enter linearly. \nThe role of the GBFs can be noticed in the case of the PK space-time, whose temperature evolution as a function of ℓ is identical to that of the SV space-time (see Fig. 1). However, as ℓ is increased, the evaporation spectra of PK PBHs (Fig. 4) decreases in intensity only at energies approximately above the peak (located roughly between 5 MeV and 10MeV), while conversely increasing in intensity for lower energies, albeit to a lesser extent compared to the decrease at higher energies. From Fig. 2 we can see that the GBFs for the SV and PK space-times are close to each other for ℓ = 0 . 3 r H , explaining why there is little difference between the red dotted curves in Fig. 3 and Fig. 4. The difference between the two become more important as ℓ increases, and are particularly noticeable at ℓ = 0 . 9 r H . We have explicitly checked that, in this case (not shown in Fig. 2), the PK GBFs start increasing at lower values of ω/M compared to their SV counterparts, explaining the observed trends. At any rate already at a qualitative level, inspecting the spectra just discussed, we can expect that the upper limits on f pbh obtained assuming Schwarzschild PBHs should loosen (thereby opening the asteroid mass window) when considering the PRBHs in question, at the very least for the SV and DR metrics - as we shall see shortly, the expectation is in fact confirmed for all three space-times.", 'C. Evaporation constraints': 'In the mass range 10 13 g ≲ M pbh ≲ 10 18 g, the dominant constraints on the PBH abundance come from measurements of the extragalactic photon background [347], and more precisely of the diffuse extragalactic γ -ray background (EGRB) in the energy range 100 keV ≲ E γ ≲ 5 GeV, which can be directly compared against theoretical expectations for the PBH Hawking evaporation spectra. Of particular interest to us is the fact that the lower edge of the asteroid mass window, where PBHs could make up the entire DM, is precisely set by evaporation constraints. In what follows, we set evaporation constraints on the fraction of DM in the form of PBHs f pbh ( M ) ≡ Ω pbh / Ω dm , assuming that PRBHs are described by the three metrics discussed so far. We work under the same set of approximations adopted in our companion paper [186]. Namely, we assume that PRBHs are isotropically distributed on sufficiently large scales and cluster in the galactic halo in the same way as other forms as DM, we only compute the primary photon spectrum, and finally we assume a monochromatic mass distribution (see e.g. Refs. [348-362] for studies on the effects of an extended mass distribution). We refer the reader to Sec. IIIC of our companion paper [186] for a more detailed discussion of why these, which are clearly all approximations, are appropriate for the scope of our work (while allowing for a more direct comparison to earlier works, including our companion paper). \nWe therefore focus on a population of PRBHs which all share the same mass M pbh . Following Ref. [363], the number of emitted photons in the logarithmic energy bin ∆ E γ ≃ E γ is approximated as ˙ N γ ( E γ ) ≃ E γ ( d ˙ N γ /dE γ ). The rate of emitted photons with present-day energy E γ 0 per unit time per unit area per unit solid angle is then obtained by integrating over the entire cosmological time, accounting for the redshift scaling of the photon energy and density, and is given by the following: \nI ( E γ 0 ) = A I ∫ z ⋆ 0 dz H ( z ) d 2 N γ dtdE γ ( M pbh , (1 + z ) E γ 0 ) , (35) \nwhere the normalization factor A I is given by: \nA I = c 4 π n pbh ( t 0 ) E γ 0 . (36) \nIn Eqs. (35,36), d 2 N γ /dtdE γ is computed via Eq. (34), whereas z ⋆ is the redshift of recombination, H ( z ) is the expansion rate, and n pbh ( t 0 ) is the present-day PRBH number density, itself related to the parameter of interest f pbh via the following: \nf pbh ( M pbh ) ≡ Ω pbh Ω dm = n pbh ( t 0 ) M pbh ρ crit , 0 Ω dm , (37) \nwith ρ crit , 0 = 3 H 2 0 / 8 πG being the present-day critical density, H 0 the Hubble constant, and Ω dm the presentday DM density parameter. In what follows, in order \nto specify H ( z ), ρ crit , and Ω dm , we adopt the same spatially flat ΛCDM cosmological model used by the seminal Ref. [363]. This allows us to have a reliable reference against which we can cross-check our limits on f pbh in the Schwarzschild case ( ℓ → 0), although we stress that the choice of underlying cosmology does not play a significant role in determining our constraints. Analogously to our companion paper, we focus on the mass range M pbh > 10 15 g. A Schwarzschild PBH of this mass has a lifetime which is much longer than the age of the Universe, and is therefore far from having fully evaporated today. Importantly, it has only lost a negligible fraction of its mass from formation until today, and it is therefore safe to approximate these PBHs as being quasi-static throughout the lifetime of the Universe, while denoting by M pbh the values of the PBH mass both at formation and today. These considerations hold even more strongly for the regular BHs we consider, as these are colder and hence longer lived compared to their Schwarzschild counterparts at a given mass. We refer the reader to Appendix B of our companion paper [186] for further discussions on this point. \nWith the cosmological model specified, the only unknown parameter in Eqs. (35,36) is the present-day PRBH number density n pbh ( t 0 ), or equivalently, through Eq. (37), the PRBH fraction f pbh . For each value of M pbh , we set upper limits on the only free parameter f pbh using EGRB flux measurements, and more specifically measurements from the HEAO-1 X-ray telescope in the 3-500 keV range [364], the COMPTEL imaging Compton γ -ray telescope in the 0 . 8-30 MeV range [365], and the EGRET γ -ray telescope [366]. To do so, for given values of M pbh and ℓ/r H , the maximum allowed value of f pbh is determined by the requirement that the theoretical prediction for the photon flux given in Eq. (35) does not overshoot any of the ERGB measurements by more than 1 σ (see e.g. Fig. 6 in our companion paper [186], and note that different datapoints are first overshot when changing M pbh ). 6 For each of the three metrics, we use this method to set upper limits on f pbh as a function of M pbh for fixed, representative values of ℓ/r H = 0 . 3, 0 . 6, and 0 . 9. Finally, we note that while the origin of the EGRB is not fully understood [373], our approach is conservative in this sense given that we remain agnostic as to the level of astrophysical (non-PBH) contribution to the EGRB. \nFIG. 6. Upper limits on f pbh , the fraction of dark matter in the form of primordial regular Simpson-Visser black holes, as a function of the black hole mass M pbh . The limits are derived for different values of the regularizing parameter ℓ (normalized to the horizon radius r H ), with the shaded regions excluded: ℓ/r H = 0 . 3 (red dotted curve), 0 . 6 (green dashed curve), and 0 . 9 (magenta dash-dotted curve). Note that the blue solid curve corresponds to the case ℓ/r H = 0, which recovers the Schwarzschild black hole, whereas the value of M pbh corresponding to the upper right edge of the f pbh constraints marks the lower edge of the asteroid mass window. \n<!-- image --> \nFIG. 7. As in Fig. 6, but for primordial regular PeltolaKunstatter black holes, with identical values of the regularizing parameter ℓ/r H and identical color coding. \n<!-- image -->', 'IV. RESULTS': "The resulting upper limits on f pbh as a function of PRBH mass M pbh , for different values of ℓ , are shown in Figs. 6, 7, and 8 for the SV, PK, and DR space-times respectively. In each figure, shown as blue solid curves are the corresponding constraints in the Schwarzschild PBH case ( ℓ → 0), which we have verified to recover the \nFIG. 8. As in Fig. 6, but for primordial regular D'AmbrosioRovelli black holes, with identical values of the regularizing parameter ℓ/r H and identical color coding. \n<!-- image --> \nresults of Ref. [363]. We stress that for a given value of ℓ , the value of M pbh where the overclosure limit f pbh < 1 is saturated sets the lower edge of the modified asteroid mass window (potentially enlarged or contracted). \nFor all three cases, we see that increasing the regularizing parameter ℓ results (at a given value of M pbh ) in weaker limits on f pbh . This confirms the expectation raised at the end of Sec. III B upon inspection of the resulting evaporation spectra, all of which decrease in intensity relative to the Schwarzschild case (except for the slight increase in the PK PRBH case for energies below the peak, which we recall reflects the different behaviour of the GBFs). We see that, for ℓ/r H = 0 . 9, the upper limits on f pbh weaken by up to an order of magnitude at a given M pbh relative to the Schwarzschild limit for the SV and DR PRBHs, whereas for PK PRBHs the extent to which the f pbh limits weaken is more limited again unsurprisingly, given that the enhanced intensity of the evaporation spectrum at low energies counteracts the decrease at higher energies in the integral of Eq. (35). \nThe aforementioned shifts result in the asteroid mass window being enlarged for all three metrics considered, because the lower edge of the window (lying roughly at M pbh ≃ 10 17 g in the Schwarzschild case) moves towards lower masses. In general, we observe that the asteroid mass window further opens up by about half a decade in mass or more (an increase which is less dramatic than what we observed for the phenomenological metrics in our companion paper [186]). As a result, there is a wider available range of parameter space where PRBHs of the type we are considering could make up all the DM. We note that our constraints assume that all PRBHs in the Universe carry the same value of 'hair parameter' ℓ/r H (in the language of Ref. [374], we are treating it as an 'universal hair'). Whether or not this is a reasonable assumption requires a deeper investigation of the theo- \ncal underpinning of the adopted metrics (in particular the LQG-inspired one) which, in the spirit of the present work being a pilot study, we defer to follow-up work. \nFinally, in our companion paper [186] we extensively commented on a few caveats concerning the extension of the asteroid mass window and, more generally, on other existing constraints on f pbh , which bear repeating here, albeit in a more condensed form (we refer the reader to Sec. IV of our companion paper for a significantly more detailed discussion). While evaporation constraints set the lower edge of the asteroid mass window, the upper edge thereof is instead set by lensing constraints. Our claim that the asteroid mass window is enlarged because the lower edge moves towards even lower values is therefore contingent upon the upper edge remaining the standard Schwarzschild one, even within the adopted metrics. We do, in fact, expect this to be the case since, at fixed M pbh , lensing constraints depend only on the mass of the lensing object. We can therefore assert that the asteroid mass window is indeed enlarged when considering the three PBH metrics introduced here. Finally, a variety of other constraints on f pbh exist, including dynamical, accretion, and CMB constraints (see e.g. Ref. [160] for a recent summary): however, with the exception of a few debated constraints [375-378], we expect these to be relevant within significantly different mass ranges (unless accretion dynamics are significantly different around the RBHs under consideration), although we reserve a detailed study to follow-up work.", 'V. CONCLUSIONS': "It is a commonly held belief that the singularities which plague General Relativity, and represent one of the most important open problems in theoretical physics, will eventually be solved once the long sought after theory of quantum gravity is unveiled. While a consensus theory of quantum gravity remains elusive, progress on the singularity problem can still be made by considering ansatze for singularity-free space-times, either introduced phenomenologically or somewhat motivated from candidate quantum gravity frameworks (such as LQG). These regular black holes, if produced early on in the Universe from the collapse of large density perturbations (thus being primordial regular BHs), could also have a role to play in the dark matter problem. Our work is a pilot study which goes precisely in this direction, examining what are the consequences of PBHs being described by non-singular metrics. In fact, it bears reminding that the usual constraints on the fraction of DM in the form of PBHs, f pbh , are derived under the assumption of PBHs being described by the Schwarzschild metric, which is well-known to be plagued by the r = 0 singularity. In the present work, we have explored three so-called nontr -symmetric metrics as candidates for describing PBHs: the Simpson-Visser black-bounce, Peltola-Kunstatter, and D'Ambrosio-Rovelli black-to-w \nite-hole-bounce space-times, with the latter two enjoying strong theoretical motivation from LQG (we note that the mathematically simpler tr -symmetric case is covered in our companion paper [186], and includes well-known phenomenological regular BHs such as the Bardeen and Hayward BHs). \nAfter discussing the impact of the regularizing parameter ℓ (with the Schwarzschild BH corresponding to the ℓ → 0 limit) on the resulting evaporation spectra, we find that as ℓ increases, at a fixed PRBH mass M pbh the corresponding upper limit on f pbh from observations of the EGRB weakens for all three metrics considered. This results in the lower edge of the asteroid mass window shifting down by up to approximately half a decade in M pbh parameter space (down from M pbh ≃ 10 17 g to M pbh ≃ 3 × 10 16 g). As a result, there is a larger range of available parameter space where the PBHs in question could make up the entire DM in the Universe, which could be targeted by proposed probes of the asteroid mass window [169, 171, 173-175, 177, 178, 379, 380]. \nOur work (alongside our companion paper [186]) demonstrates, as a proof-of-principle, that the intersection of the DM and singularity problems is a fertile terrain worthy of further studies. The most important avenue for immediate follow-up work would be to systematically revisit, within the metrics under consideration, all other non-evaporation constraints which have been extensively discussed for Schwarzschild PBHs (including lensing, accretion, and dynamical constraints). Moreover, since the Peltola-Kunstatter and D'AmbrosioRovelli space-times are rooted within an underlying quantum gravity theoretical framework, a first-principles study of their formation mechanism (which is otherwise not possible for metrics introduced at a phenomenological level, such as the Bardeen and Hayward BHs) and whether this leads to additional interesting complementary signatures is definitely worth pursuing. For instance, quantum transitions near the would-be singularity of the D'Ambrosio-Rovelli space-time should lead to the existence of long-lived (primordial) white holes, which in turn could potentially lead to a wide range of exotic signatures one could hope to search for. In continuing our exciting program at the interface of the DM and singularity problems, it is our intention to return to these and related issues in future follow-up work.", 'ACKNOWLEDGMENTS': "We acknowledge support from the Istituto Nazionale di Fisica Nucleare (INFN) through the Commissione Scientifica Nazionale 4 (CSN4) Iniziativa Specifica 'Quantum Fields in Gravity, Cosmology and Black Holes' (FLAG). M.C. and S.V. acknowledge support from the University of Trento and the Provincia Autonoma di Trento (PAT, Autonomous Province of Trento) through the UniTrento Internal Call for Research 2023 grant 'Searching for Dark Energy off the beaten track' (DARKTRACK, grant \nagreement no. E63C22000500003). This publication is based upon work from the COST Action CA21136 'Addressing observational tensions in cosmology with systematics and fundamental physics' (CosmoVerse), supported by COST (European Cooperation in Science and Technology).", 'Appendix A: Asymptotic solutions to the radial Teukolsky equation': "We recall that, in order to compute the GBFs for the regular BHs studied in this work, we need to know the asymptotic behaviour of the function R s , introduced in the ansatz of Eq. (13) and solution to the radial Teukolsky equation given by Eq. (17), both at infinity and close to the horizon. In the main text we reported these limits as being given by Eqs. (19,20). We now set out to prove this more formally. \nThe radial Teukolsky equation, Eq. (17), simplifies considerably if we make the following substitution [329]: \ndr ⋆ dr = 1 √ f ( r ) g ( r ) , Y s = √ B s √ fg R s . (A1) \nThis allows one to get rid of first derivatives of Y s , with the radial Teukolsky equation as a function of the latter now taking the following form: \nY s , ⋆⋆ + [ ω 2 + iωs √ f g ( h ' -hg ' g ) g h + C s g h -√ β, ⋆⋆ √ β ] Y s = 0 , (A2) \nwhere β ≡ ( hg 2 ) h = B s / √ fg , and , ⋆ denotes differentiation with respect to the tortoise coordinate r ⋆ . We now consider the r → + ∞ and r → r H limits separately.", '1. Asymptotic behaviour at infinity': 'In this limit, we trivially see that Eq. (A2) reduces to the following: \nY s , ⋆⋆ + ( ω 2 + 2 iωs r ) Y s = 0 , (A3) \nwhose asymptotic solutions are given by: \nY s ∼ r ± s e ∓ iωr ⋆ . (A4) \nwhich implies that R s scales as follows: \nR s ∼ e -iωr ⋆ r and R s ∼ e iωr ⋆ r (2 s +1) , (A5) \nconfirming the asymptotic scaling quoted in Eq. (19).', '2. Asymptotic behaviour near the horizon': "In the vicinity of the event horizon, Eq. (A2) reduces to the following: \nY s , ⋆⋆ + ω -is √ f g g ' 4 2 + g ' s 4 ( f ' -fg ' g ) Y s = 0 , (A6) \nwhere we kept only terms scaling as ∼ f 0 , g 0 . It is easy to see that in the cases of the SV and PK space-times, for which f = g , the second term in square brackets vanishes. With some effort, one can show that this is true for the DR metric as well. This then leaves us with the following equation for Y s : \nY s , ⋆⋆ + ω -is √ f g g ' 4 2 Y s = 0 , (A7) \nwhose asymptotic solutions are given by: \nY s ∼ exp [ ± i ( ω -isg ' 2 √ f g ) r ⋆ ] . (A8) \nBy definition, the tortoise coordinate r ⋆ is determined by the following integral: \nr ⋆ = ∫ dr √ f ( r ) g ( r ) r → r H ----→ Kln( r -r H ) , (A9) \nwhere r H is the radial coordinate of the event horizon and K is a coefficient to be determined. Combining Eqs. (A8,A9) we reach the following expression for the asymptotic scaling of Y s : \nY s ∼ e ± iωr ⋆ exp [ ± sg ' 2 √ f g Kln( r -r H ) ] . (A10) \nIt is straightforward to check that in the case of Schwarzschild BH, Eq. (A9) reduces to the following: \nr ⋆ = ∫ dr 1 -2 M r r → r H ----→ 2 M ln( r -2 M ) , (A11) \nfrom which we recover the asymptotic behaviour found in Ref. [328]: \nY s ∼ e ± iωr ⋆ ∆ ± s/ 2 = ⇒ R s ∼ e iωr ⋆ or R s ∼ ∆ -s e -iωr ⋆ , (A12) \nsince it is easy to show that the following holds asymptotically: \nln( r -r H ) ∼ ln(∆) , (A13) \nwhere ∆ ≡ r 2 -2 Mr = r 2 g ( r ). Note that the R s ∼ e iωr ⋆ solution in Eq. (A12) is then discarded on the basis \nof the purely ingoing boundary conditions we fix at the horizon when setting up the scattering problem, thereby confirming the asymptotic scaling quoted in Eq. (20). The above results hinged upon the r ⋆ ∝ ln( r -r H ) scaling in Eq. (A9). We now check that this scaling does indeed hold for the three space-times we consider in our work.", 'a. Simpson-Visser space-time': "We recall that the functions f ( r ) and g ( r ) are given by the following: \nf ( r ) = g ( r ) = 1 -2 M √ r 2 + ℓ 2 , (A14) \nimplying that the tortoise coordinate is given by the following: \nr ⋆ = ∫ dr 1 -2 M √ r 2 + ℓ 2 r → r SV H ----→ 4 M 2 √ 4 M 2 -ℓ 2 ln ( r -r SV H ) , (A15) \nwhere r SV H = √ 4 M 2 -ℓ 2 . For this metric g ' ( r ) = 2 Mr/ ( r 2 + ℓ 2 ) 3 / 2 , from which one easily finds that Eq. (A10) can be expressed as follows: \nY s ∼ e ± iωr ⋆ e ± s 2 ln( r -r H ) ∼ e ± iωr ⋆ [ h ( r ) g ( r )] ± s 2 , (A16) \nwhere in the last step, in light of the discussion in Ref. [183], we generalized ∆ to the following: \n∆ ≡ r 2 g ( r ) -→ D ≡ ( h ( r ) g ( r )) . (A17) \nFinally, returning to R s , it is easy to show that the asymptotic behaviour of the latter is given by: \nR s ∼ e iωr ⋆ , and R s ∼ A s e -iωr ⋆ (A18) \nwhere A s is defined according to Eq. (14), and confirming the asymptotic scaling quoted in Eq. (20).", 'b. 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2024ApJ...976...96P	JWST has revealed a population of lowluminosity active galactic nuclei at z gt 4 in compact red hosts the Little Red Dots or LRDs which are largely undetected in Xrays. We investigate this phenomenon using General Relativistic Radiation Magnetohydrodynamics simulations of superEddington accretion onto a supermassive black hole SMBH with M SUBSUB 10SUP7SUP M SUBSUB at z 6 representing the median population the spectral energy distributions SEDs that we obtain are intrinsically Xray weak. The highest levels of Xray weakness occur in SMBHs accreting at mildly superEddington rates 1.4 lt f SUBEddSUB lt 4 with zero spin viewed at angles gt30 from the pole. Xray bolometric corrections in the observed 210 keV band reach 10SUP4SUP at z 6 5 times higher than the highest constraint from Xray stacking. Most SEDs are extraordinarily steep and soft in the Xrays median photon index 3.1 mode of 4.4. SEDs strong in the Xrays have harder spectra with a highenergy bump when viewed near the hot gt10SUP8SUP K and highly relativistic jet whereas Xray weak SEDs lack this feature. Viewing an SMBH within 10 of its pole where beaming enhances the Xray emission has a 1.5 probability matching the LRD Xray detection rate. Nextgeneration observatories like AXIS will detect Xrayweak LRDs at z 6 from any viewing angle. Although many SMBHs in the LRDs are already estimated to accrete at superEddington rates our model explains 50 of their population by requiring that their masses are overestimated by a mere factor of 3. In summary we suggest that LRDs host slowly spinning SMBHs accreting at mildly superEddington rates with large covering factors and broad emission lines enhanced by strong winds providing a selfconsistent explanation for their Xray weakness and complementing other models.	2024-11-01T00:00:00Z	['10.48550/arXiv.2407.15915', '2024arXiv240715915P', '10.3847/1538-4357/ad84f7', '2024ApJ...976...96P', 'arXiv:2407.15915']	['Active galaxies', 'Supermassive black holes', 'Black holes', 'Accretion', 'Spectral energy distribution', '17', '1663', '162', '14', '2129', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies']	Mildly SuperEddington Accretion onto Slowly Spinning Black Holes Explains the XRay Weakness of the Little Red Dots	2,024	198	0.66	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	26	https://arxiv.org/pdf/2407.15915.pdf	{'Mildly Super-Eddington Accretion Onto Slowly-Spinning Black Holes Explains the X-Ray Weakness of the Little Red Dots': 'Fabio Pacucci 1, 2 and Ramesh Narayan 1, 2 \n1 Center for Astrophysics \| Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA 2 Black Hole Initiative, Harvard University, 20 Garden St, Cambridge, MA 02138, USA', 'ABSTRACT': "JWST has revealed a population of low-luminosity AGN at z > 4 in compact, red hosts (the 'Little Red Dots', or LRDs), which are largely undetected in X-rays. We investigate this phenomenon using GRRMHD simulations of super-Eddington accretion onto a SMBH with M · = 10 7 M ⊙ at z ∼ 6 , representing the median population; the SEDs that we obtain are intrinsically X-ray weak. The highest levels of X-ray weakness occur in SMBHs accreting at mildly super-Eddington rates ( 1 . 4 < f Edd < 4 ) with zero spin, viewed at angles > 30 · from the pole. X-ray bolometric corrections in the observed 2 -10 keV band reach ∼ 10 4 at z = 6 , ∼ 5 times higher than the highest constraint from X-ray stacking. Most SEDs are extraordinarily steep and soft in the X-rays (median photon index Γ = 3 . 1 , mode of Γ = 4 . 4 ). SEDs strong in the X-rays have harder spectra with a high-energy bump when viewed near the hot ( > 10 8 K) and highly-relativistic jet, whereas X-ray weak SEDs lack this feature. Viewing a SMBH within 10 · of its pole, where beaming enhances the X-ray emission, has a ∼ 1 . 5% probability, matching the LRD X-ray detection rate. Next-generation observatories like AXIS will detect X-ray weak LRDs at z ∼ 6 from any viewing angle. Although many SMBHs in the LRDs are already estimated to accrete at super-Eddington rates, our model explains 50% of their population by requiring that their masses are overestimated by a mere factor of ∼ 3 . In summary, we suggest that LRDs host slowly spinning SMBHs accreting at mildly super-Eddington rates, with large covering factors and broad emission lines enhanced by strong winds, providing a self-consistent explanation for their X-ray weakness and complementing other models. \nKeywords: Active galaxies (17) - Supermassive black holes (1663) - Accretion (14) - Spectral energy distribution (2129) - Black holes (162)", '1. INTRODUCTION': "X-ray photons have been foundational to studying compact objects. For example, the first X-ray source detected outside the Solar System (Sco X-1, Giacconi et al. 1962, a highly accreting neutron star, Shklovsky 1967), the first confirmed stellar-mass black hole (Cygnus X1, Bowyer et al. 1965) and the first confirmed supermassive black hole (SMBH, 3C 273, Schmidt 1963) are all powerful X-ray sources. Searching for black holes in the high-energy sky is a fruitful method because X-ray sources are sparse ( ∼ 500 deg -2 , in the Chandra Source Catalog, Evans et al. 2024). Hence, an X-ray detection of a centrally located source within a galaxy has been the 'gold standard' to assess the presence of a SMBH. \nA newly discovered, enigmatic population of galaxies in the highz Universe is challenging this fact: the 'Little Red Dots' (LRDs). LRDs are a population of high- \nz sources detected in many JWST deep fields (Labbé et al. 2023; Kocevski et al. 2023; Harikane et al. 2023; Maiolino et al. 2023; Matthee et al. 2024), with the bulk of their distribution in the redshift range 4 < z < 8 , and a median of z ∼ 6 (Kocevski et al. 2024; Akins et al. 2024; Kokorev et al. 2024). \nA few properties of the LRDs are apparent. They have very red near-infrared colors (Matthee et al. 2024) and are very compact, with a median effective radius of r e ∼ 150 pc (Baggen et al. 2023; Labbé et al. 2023), while some are even smaller with r e < 35 pc (Furtak et al. 2023). This property would lead to extreme stellar densities in their galactic cores (Pacucci & Loeb 2024). Additionally, these sources are abundant compared to the previously known population of highz AGN. With a number density of 10 -4 -10 -5 Mpc -3 mag -1 at z ∼ 5 , they are 10 -100 times more numerous than the faint end of quasar luminosity function (Greene et al. 2024). \nHence, if they contain central SMBHs, they belong to a previously unexplored and fainter population of AGN. \nGiven the properties of the LRDs, their physical interpretation bifurcates into two paths. The LRDs are either very massive, star-forming, and compact galaxies (see, e.g., Labbé et al. 2023), or they contain a central SMBH, typically in the mass range M · ∼ 10 6 -10 8 M ⊙ (see, e.g., Harikane et al. 2023; Maiolino et al. 2023). \nA majority of these sources do show telltale signs of the presence of an active SMBH. Greene et al. (2024) found that ∼ 60% of the LRDs investigated display definitive evidence of broad emission lines, with H α FWHMs of > 2000 kms -1 : a signature of Type-1 AGN, for which an estimate of the mass can be derived (Greene & Ho 2005). The absence of broad forbidden lines (e.g., the [OIII]5007) suggests that such emission cannot be associated with outflows (Maiolino et al. 2024). If the LRDs contain a central SMBH, their SEDs are assembled by a dust-obscured AGN and a young, blue galaxy (Kocevski et al. 2023). Pérez-González et al. (2024) argue that many LRDs are extremely intense and compact starburst galaxies, which produce a significant amount of dust; their global energy output is dominated by emission from OB stars and an obscured AGN. \nSurprisingly, most LRDs observed thus far are undetected in the X-rays, leading to the 'X-ray weakness problem', which starkly contrasts with other AGNrelated properties of the LRDs. For example, in a comprehensive sample of 341 LRDs spanning several JWST fields, Kocevski et al. (2024) found only two Xray detected LRDs, which are moderately obscured with log 10 ( N H / cm 2 ) of 22 . 72 and 23 . 3 . Although the LRDs account for only ∼ 10% of the AGN population discovered by JWST (Maiolino et al. 2024), their lack of X-ray emission is compelling, as it adds to their already puzzling properties. \nPrevious studies systematically searched for X-ray emissions in LRDs. Yue et al. (2024) used archival Chandra data on 19 spectroscopically selected LRDs with broad emission lines and detected none. The X-ray stacking analysis also led to non-detection, with an inferred soft-band ( 0 . 5 -2 keV) upper limit that is ∼ 1 dex lower than what is expected from typical Type-1 AGN. They also suggest that extremely large absorbing column densities, i.e., log 10 ( N H / cm 2 ) > 24 , would be required to entirely suppress the X-ray emission, especially since Chandra is probing energies of ∼ 2 . 5 -50 keV at a representative redshift of z = 5 . Although the X-ray obscuration does increase with redshift (Peca et al. 2023), it is unlikely that very broad H α lines are associated with such substantial absorbing column densities. As pointed out by Yue et al. (2024), some broad absorption \nline (BAL) AGN show values of log 10 ( N H / cm 2 ) > 24 (Blustin et al. 2008); however, the spectra of LRDs do not show typical BAL features (Greene et al. 2024). Additionally, Type 1 AGN with log 10 ( N H / cm 2 ) > 24 are very rare in the local Universe, i.e., ≲ 5% of the whole population (Ananna et al. 2022). Considering these constraints, Yue et al. (2024) suggest that LRDs may be intrinsically X-ray weak, as also pointed out by Inayoshi & Ichikawa (2024). \nMaiolino et al. (2024) also executed an X-ray stacking analysis of a large JWST sample of Type-1 and Type2 AGN, leading to widespread non-detections. Their stacking analysis led to upper limits of 1 -2 dex lower than expected from standard AGN emission. Contrarily to Yue et al. (2024), they suggest that dust-poor clouds from the broad-line region (BLR) with sufficiently large column densities and abnormally large covering factors can explain the X-ray weakness. They also indicate that steep X-ray spectra, such as those observed in Narrow Line Seyfert 1 (NLSy1) galaxies, can contribute to explaining an intrinsic X-ray weakness (Vasudevan & Fabian 2007). \nAnanna et al. (2024) performed a stacking analysis of a sample of 21 LRDs detected behind the lensing galaxy cluster Abell 2744. This study also led to widespread non-detections and an upper limit on the X-ray emission, with constraints similar to those reported by Yue et al. (2024). \nTo conclude, King (2024) suggested that LRDs may be highly-beamed sources, fed at or above the Eddington rate, similar to the case of ultraluminous X-ray sources (ULXs). The resulting gas outflow velocities of ∼ 0 . 1 -0 . 2 c would make virial mass estimates difficult and lead to an overestimate. Interestingly, the Galactic X-ray binary Cyg X-3, which may be viewed as a ULX from observers located along the axis of the funnel, shows signs of X-ray weakness (Veledina et al. 2024). Additionally, based on semi-empirical models of the emission in different accretion regimes, Lupi et al. (2024) also suggested that the masses of the SMBHs in the LRDs may be overestimated due to accretion at largely super-Eddington rates. \nIn this Letter, we demonstrate that the LRDs' Xray weakness can be explained by SMBHs accreting at mildly super-Eddington rates, with SEDs significantly different from those of radiatively efficient, 'standard' AGN. Intrinsically soft X-ray spectra, redshifted out of Chandra's range, can elegantly explain the lack of highenergy detections while supporting the other hallmarks of the black hole hypothesis (e.g., broad lines and significant covering factors). For the first time, we support this interpretation with a suite of GRRMHD (General \nRelativistic Radiation Magnetohydrodynamics) simulations of SMBHs accreting above the Eddington rate, with various values of the spin parameter and viewing angles. In Sec. 2, we provide a primer on different accretion regimes for black holes. In Sec. 3, we describe our numerical codes for the GRRMHD simulations and raytracing. Then, we present our results in Sec. 4. Finally, in Sec. 5, we summarize our findings and conclude.", '2. A PRIMER ON ACCRETION REGIMES': "The Eddington ratio f Edd is defined as the ratio between the mass accretion rate ˙ M · onto the black hole and the corresponding Eddington rate: \nf Edd = ˙ M · ˙ M Edd , (1) \nwhere ˙ M Edd is the mass accretion rate at which a relativistic thin accretion disk (Novikov & Thorne 1973) would radiate at the Eddington luminosity. f Edd is the main parameter that defines the accretion regime and, ultimately, the characteristics of the accretion disk, jet, and the shape of the emerging spectral luminosity. \nFor 10 -2 ≲ f Edd ≲ 1 , a geometrically-thin, optically thick, radiatively efficient accretion disk is present (Shakura & Sunyaev 1973; Novikov & Thorne 1973) 1 . This accretion mode is often referred to as the 'standard disk' model and is the most widely studied for black holes spanning the entire mass range. The 'bias' towards this accretion regime is likely caused by the combination of two effects: black holes accreting in this regime are sufficiently bright to be detected but, at the same time, do not require extremely large (thus, infrequent at z = 0 ) reservoirs of cold, accretable gas. In the radiatively efficient thin disk regime, typical matter-toenergy conversion factors ϵ are of the order ϵ ∼ 0 . 1 (depending on the spin, Bardeen 1970; Novikov & Thorne 1973), where the emitted luminosity is L = ϵ ˙ Mc 2 . \nThe standard thin disk regime is itself sub-divided into two sub-regimes. For f Edd ≲ 0 . 1 , the disk is gas-pressure dominated and is thermally stable. However, for 0 . 1 ≲ f Edd ≲ 1 , radiation pressure dominates, and the disk becomes viscously and thermally unstable (Lightman & Eardley 1974; Shakura & Sunyaev 1976; Jiang et al. 2013). Unfortunately, the unstable regime overlaps with the classic luminosity range of quasars, which prevents self-consistent modeling of these astrophysically important systems. \nFor f Edd ≲ 10 -2 and f Edd ≳ 1 , the disk becomes radiatively inefficient. In the lower range ( f Edd ≲ 10 -2 ), an extremely hot, two-temperature, opticallythin, advection-dominated accretion flow (ADAF) is formed, which is geometrically thick and has typical values of ϵ ≪ 0 . 1 (Narayan & Yi 1994, 1995; Abramowicz et al. 1995; Narayan & McClintock 2008; Yuan & Narayan 2014). What little radiation is present in this accretion state has hardly any effect on the dynamics of the optically thin gas. Sgr A* is a classic example of a SMBH accreting in ADAF mode (Yuan et al. 2003). Low f Edd systems often accumulate a strong magnetic field at small radii around the black hole to form a Magnetically Arrested Disk (MAD, Igumenshchev et al. 2003; Narayan et al. 2003). In this state, they produce relativistic jets extending to thousands of gravitational radii r g = 2 GM · /c 2 if the black hole is spinning (Tchekhovskoy et al. 2011). It is likely that most black holes, at least in the local Universe, accrete in the ADAF mode and are in the MAD state (Narayan et al. 2022). The accretion mode alternative to the MAD state is aptly named SANE (Standard And Normal Evolution). \nFor f Edd ≳ 1 , the accretion disk is radiatively inefficient because of photon trapping (Begelman 1979; Abramowicz et al. 1988). The disk is geometrically thick, prompting the name 'slim disk' for these systems. Crucially, the slim disk regime is thermally stable (Abramowicz et al. 1995; Narayan & Yi 1995; Chen et al. 1995), which facilitates theoretical studies, especially using numerical simulations (Sądowski et al. 2014; McKinney et al. 2014). Jets are prominent if the disks reach the MAD state and if the black holes are spinning (Narayan et al. 2017; Curd & Narayan 2023; Ricarte et al. 2023). Such systems will be extremely X-ray-bright for poleon observers because of relativistically beamed radiation from the hot disk and jet (Curd & Narayan 2019). \nRadiation plays a critical role in both the dynamics and thermodynamics of standard and slim disks (i.e., for the entire range f Edd ≳ 0 . 01 ). There is a strong coupling between the gas and the radiation, while radiative pressure inflates the disk in the vertical direction as f Edd approaches or exceeds unity (Abramowicz et al. 1988). Additionally, jets and winds driven only by radiation can occur in super-Eddington systems (Sądowski et al. 2014). For these reasons, a GRRMHD code, i.e., a GRMHD code with a self-consistent treatment of the radiation field, is required. The work reported in this paper uses the GRRMHD code KORAL (Sądowski et al. 2013, 2014). \nThis primer has highlighted the physical differences characterizing the three fundamental accretion regimes. \nGRRMHD-derived spectral models are available in the ADAF regime ( f Edd ≲ 0 . 01 ) and, albeit more scarcely, in the super-Eddington regime ( f Edd ≳ 1 ). However, accretion rates in between are challenging to simulate. For f Edd ≲ 0 . 1 , disks are geometrically very thin and require extremely high resolution, making them challenging for GRRMHD codes. On the other hand, for f Edd ≳ 0 . 1 and especially as f Edd → 1 , although disks are geometrically thicker and potentially accessible to GRRMHD codes, they are thermally unstable, as already mentioned (Shakura & Sunyaev 1976). Only a handful of full GRRMHD simulations have been successfully run in this accretion regime; in these few simulations, strong magnetic fields stabilize the accretion flow (Sądowski 2016) and a curious 'puffy disk' structure is observed (Lančová et al. 2019; Wielgus et al. 2022). The models so far are limited to non-spinning black holes. \nIn the present work, we focus on the super-Eddington regime ( f Edd ≳ 1 ), where the accretion disk is stable, and we have both GRRMHD codes to simulate the gas magnetohydrodynamics and radiation field of the disk (e.g., Sądowski et al. 2013, 2014) as well as postprocessing codes to compute spectra (e.g., Zhu et al. 2015; Narayan et al. 2016). The results we report here are, however, valid only for the super-Eddington regime and cannot be extended to thermally unstable f Edd ≲ 1 systems. The latter regime, though of great interest for modeling traditional quasars, is beyond the reach of our codes. Simulating them would require entirely different simulation techniques (that do not yet exist) and theoretical assumptions.", '3. NUMERICAL METHODS': 'Calculating the spectral luminosity emitted by SMBHs accreting in the super-Eddington regime involves a two-stage process. First, we conducted GRRMHD simulations of the accretion flow for specific model parameters; then, we solved for the radiation field using a post-processing code. The two steps are described in their generalities in the following subsections; the interested reader is referred to the original studies describing the GRRMHD code KORAL (Sądowski et al. 2013, 2014), the radiation post-processing code HEROIC (Zhu et al. 2015; Narayan et al. 2016), as well as their application to the case of ultra-luminous X-ray sources (Narayan et al. 2017) and tidal disruption events (Curd & Narayan 2019, 2023).', '3.1. GRRMHD Simulations with KORAL': 'The super-Eddington accretion process onto SMBHs is simulated using the GRRMHD code KORAL (Sądowski et al. 2013, 2014), which incorporates the effects of gas \ndynamics, magnetic fields, and radiation within a fixed gravitational field. These simulations are performed with the Kerr metric in Kerr-Schild coordinates and employ the M1 closure method (Levermore 1984) to handle radiative transfer, along with a radiative viscosity term to address limitations inherent in the M1 scheme (Sądowski et al. 2015). The simulations include radiative processes such as synchrotron emission, free-free and bound-free (taken from Sutherland & Dopita 1993) emission and absorption, and Compton scattering. All these simulations are performed in 3D to correctly resolve the magnetorotational instability (Balbus & Hawley 1991). \nSimulations begin with weakly magnetized gas in an equilibrium torus orbiting the SMBH; the parameters are adjusted to achieve the desired mass accretion rate (Narayan et al. 2022), which, in our case, varies between mildly to strongly super-Eddington (see Sec. 3.3). The topology of the initial magnetic field is appropriate for MAD models (Tchekhovskoy et al. 2011). MAD-type accretion disk models around SMBHs are characterized by strong magnetic fields, significantly affecting the accretion process. These disks occur when the magnetic field becomes so intense that it disrupts the inflow of gas onto the SMBH, creating a region where magnetic forces dominate the dynamics of the disk. \nOur simulations are run on a grid with 256 × 192 × 32 cells in ( r, θ, ϕ ) , where ϕ extends over only a π/ 2 wedge (hence the effective resolution is 128 cells over 2 π ). Each simulation is run up to a time of 30000 t g ≈ 410 hrs, where the gravitational time scale for a black hole of mass M · is t g = GM · /c 3 ≈ 50 s for M · = 10 7 M ⊙ . By the end of the simulation, the disk is in a steady state out to roughly ≈ 50 r g (see Narayan et al. 2017; Ricarte et al. 2023 for additional details). Once the system has reached a steady state, the spectra are calculated via ray tracing from the last 5000 t g . \nThe standard procedure of a σ = 1 cut was applied to the GRRMHD runs. This technique addresses problematic regions in GRRMHD simulations where the quantity σ = B 2 / (4 πρc 2 ) , the ratio of magnetic stress to rest mass energy density, exceeds unity. In these regions, which commonly coincide with the highly magnetized funnel, the density becomes very low, causing issues as numerical errors accumulate (Gammie et al. 2003); hence, cutting out these areas makes the simulation reliable and more accurate. This method is commonly used in various GRMHD simulations, including those used for interpreting the Event Horizon Telescope observations (see, e.g., Event Horizon Telescope Collaboration et al. 2019; Tsunetoe et al. 2021).', '3.2. Radiation Post-Processing with HEROIC': 'Post-processing of the radiation field is carried out using the general relativistic HEROIC code (Zhu et al. 2015; Narayan et al. 2016), which solves the detailed radiation field based on data from the GRRMHD simulations. HEROIC accounts for the angular distribution of radiation in the local fluid frame and spans a wide range of frequencies, from radio to gamma rays. Since its development, HEROIC has been enhanced to better handle bremsstrahlung in the relativistic regime, utilizing methods developed from Narayan & Yi (1995) and spectral distribution from Gould (1980). Additionally, it uses opacity tables from the CHIANTI database (Dere et al. 1997; Landi et al. 2013; Del Zanna et al. 2015) as well as from Sutherland & Dopita (1993), for temperatures below 10 8 K, and includes a relativistic Comptonization module for high temperatures (Jones 1968; Coppi & Blandford 1990). HEROIC also incorporates thermal synchrotron emission and absorption (Narayan & Yi 1995; Mahadevan et al. 1996) and can handle two-temperature plasmas (Sądowski et al. 2017). \nIn the present work, post-processing involves two stages. First, HEROIC takes the time-averaged (over the final 5000 t g ) and axisymmetrized data from KORAL for the gas density, temperature, and four-velocity, as well as the magnetic field strength. To be conservative, it caps the gas temperature 2 at 10 9 K, which is the typical coronal temperature in Seyfert galaxies (see, e.g., Akylas & Georgantopoulos 2021). Using this information and the appropriate Kerr spacetime metric, the code iteratively solves for the frequency-dependent radiation field at each grid point, including all the emission, absorption, and scattering opacities and the effect of relativistic Comptonization. At the conclusion of this stage, the code generates a source function at each position on the grid. Using this source function, in the second stage HEROIC performs general relativistic ray-tracing, again with all opacities and scattering included, to compute the observed spectra from different viewing angles. Note that HEROIC only calculates the continuum spectrum. It does not model relativistically broadened X-ray iron lines or related reflection phenomena (e.g., Reynolds & Nowak 2003). \nThe observed SEDs are re-normalized so that their bolometric luminosity matches the value expected from the Eddington ratio of each run, given a fixed black hole \nmass of 10 7 M ⊙ . This task required a consistent renormalization across all runs of a factor of 10 for the radiation field, suggesting that KORAL overestimated the temperature field by a factor of 10 1 / 4 ≈ 1 . 8 . This procedure is standard practice to correct discrepancies in calculating the radiation and temperature fields performed by KORAL , using the M1 closure method. A similar radiation and temperature field matching was also applied to other studies based on KORAL and HEROIC , e.g., Narayan et al. (2017). We also note that the fact that a consistent re-normalization was required for all runs suggests that our simulations are consistent with each other.', '3.3. Description of the Simulations': 'Our study involves 12 GRRMHD simulations, all assuming a SMBH mass of 10 7 M ⊙ . This mass is representative of the sample of SMBHs discovered thus far in LRDs at z > 4 (Harikane et al. 2023; Maiolino et al. 2023). For example, the sample of LRDs recently analyzed in Maiolino et al. (2024) is characterized by a median mass of M · = 3 × 10 7 M ⊙ . \nThe GRRMHD simulations vary the Eddington ratio, ranging from mildly super-Eddington (i.e., f Edd = 1 . 4 ) to strongly super-Eddington (i.e., f Edd = 13 . 4 ). Three values of the dimensionless SMBH spin parameter a are considered: 0, 0.68, 0.9. By definition, a is required to lie in the range 0 ≤ \| a \| ≤ 1 , where a = 0 is a Schwarzschild (i.e., non-rotating) black hole, and a = 1 is a maximally rotating black hole (although the limiting value a = 1 is not achievable in practice, Thorne 1974). \nThese 12 simulations are then post-processed to obtain the emerging spectral luminosity for 8 different viewing angles i of the observer relative to the pole. The inclination angle varies between i = 10 · to i = 80 · , in intervals of 10 · . Hence, a grid of 12 simulations × 8 angles leads to a total of 96 SEDs, which are used to investigate the physical parameter spaces relevant to our problem. A summary of the physical properties of the 12 simulations performed is presented in Table 1.', '4. RESULTS': 'In this Section, we present the results of our study. We first summarize the X-ray properties of our superEddington SEDs and show how their observed X-ray emission depends on four parameters: Eddington ratio, spin, inclination, and redshift. Then, we apply these findings to explain the X-ray weakness of the population of SMBHs detected in the LRDs and how a slight overestimate of a factor of ∼ 3 in their mass measurement could lead the majority of them to the mildly superEddington regime investigated here. Super-Eddington accretion leads naturally to X-ray weakness, lower spin, \nTable 1. Properties of the 12 GRRMHD simulations performed. The SED emerging from each simulated system is analyzed from 8 different viewing angles: 10 · to 80 · from the pole. \nlarge covering factors, and broad emission lines due to strong winds.', '4.1. X-ray Bolometric Corrections': 'We begin by investigating what fraction of the bolometric luminosity of a given SED is emitted in the 2 -10 keV X-ray band. Hence, we calculate the X-ray bolometric correction in the rest frame of the SMBH: \nk X = L bol L z =0 2 -10 keV . (2) \nValues of k X for standard AGN are in the range ∼ 10 -50 , except for very bright quasars (i.e., with L bol > 10 47 erg s -1 ) for which it can reach values of ∼ 100 (Duras et al. 2020). Note that a larger value of k X indicates a weaker X-ray source. \nIf the emitted SED is soft in the X-ray, we expect a portion of the spectrum to be redshifted out of the observed 2 -10 keV energy band for z ≫ 0 , corresponding to a restframe band [2 -10](1 + z ) keV. Hence, we are also interested in the observed X-ray bolometric correction calculated at z , which we define as: \nk z =6 X = L bol L z =6 2 -10 keV . (3) \nWechose a representative redshift of z = 6 as the median value computed from the most comprehensive catalogs of LRDs available to date (Kocevski et al. 2024, where the median redshift is z = 6 . 1 , Akins et al. 2024, where it is z = 6 . 5 , and Kokorev et al. 2024, where it is z = 5 . 6 ). We note that using a value of z ∼ 5 , which is closer to the median redshift of the sources presented in Harikane \net al. (2023) and Maiolino et al. (2023), does not change our results in any significant way. \nOur results for the restframe (left) and z = 6 (right) bolometric corrections are displayed in Fig. 1, as a function of the Eddington ratio, spin value and inclination angle from the pole. This analysis suggests the following general properties: \n- · Mildly super Eddington accretion, with 1 . 4 < f Edd < 4 , leads to the highest bolometric corrections (i.e., they are the most X-ray weak).\n- · SEDs generated by zero spin SMBHs have higher bolometric corrections than those with larger spins 3 .\n- · Large inclinations from the pole lead to higher bolometric corrections. On the contrary, observing the accretion from a viewing angle close to the pole increases the X-ray flux. \nAs the LRDs are highz sources, we are primarily interested in the observed bolometric corrections calculated for a source at the median redshift of z = 6 ; they are displayed in the right panel of Fig. 1. While the general trends with Eddington ratio, spin, and inclination angle do not change, the values of the bolometric corrections are substantially larger, indicating an astounding observed X-ray weakness. In particular, 2 / 3 of all the SEDs with mildly super-Eddington rates (i.e., f Edd < 4 ) are characterized by k z =6 X > 10 3 , with some of them even reaching k z =6 X > 10 4 . These values can explain even GN28074 (Juodžbalis et al. 2024a), the most extreme X-ray weak AGN found to date by JWST (see Fig. 7). \nThe significant variations in the X-ray bolometric corrections (at a fixed redshift) are due to a combination of different factors. Generally, X-rays are produced by hot gas with temperature T > 10 7 K. In the present simulations, such gas is found exclusively in the jet, both at the jet base and along the walls of the funnel. Higher spin values lead to a more prominent jet (Blandford et al. 2019); correspondingly, non-spinning black holes are the most X-ray-quiet. Jet emission is beamed, hence the observed X-ray emission is strongly dependent on the inclination angle. The Eddington ratio determines the properties of the accretion disk, with higher temperatures being associated with higher values of f Edd , which leads to a higher level of X-ray emission. \nFigure 2 displays that most of our SEDs are characterized by values of the α ox that are significantly different from the standard α ox ( L 2500 ) relation presented in Lusso et al. (2010). In particular, ∼ 55% of our SEDs are outside the dispersion region for typical lowz sources. In extreme cases, the (restframe) luminosity at 2 keV is ∼ 20 times weaker than expected from a standard AGN with the same optical-UV luminosity. Interestingly, our \n<!-- image --> \nFigure 1. Comparison of the 2 -10 keV bolometric correction as a function of the logarithm of the Eddington ratio in the restframe (left panel) and in the observed 2 -10 keV band when the source placed at the median redshift of z = 6 (right panel). Different colors indicate varying inclination angles from the pole (degrees), and different markers represent different spin values: triangles for a = 0 . 9 , circles for a = 0 . 68 , and stars for a = 0 . Mildly super-Eddington accretion rates ( 1 . 4 < f Edd < 4 ) onto slowly spinning black holes, if observed at large inclination angles from the pole, lead to extremely large values of the bolometric correction, especially if the SMBH is highly redshifted. \n<!-- image --> \nSection 4.5 describes how the structure of the accretion flow determines the observed X-ray emission. \nIn summary, GRRMHD simulations of superEddington accreting SMBHs show that the observed SEDs are soft in the X-rays, especially for mildly superEddington accretion ( 1 . 4 < f Edd < 4 ) on slowly spinning black holes when observed at large inclination angles from the pole. If these SEDs, which are already intrinsically X-ray weak , are observed at z ∼ 6 , they reach staggering values of the observed X-ray bolometric correction, > 2 orders of magnitude higher than (restframe) reference values for standard AGN. \nA note on terminology is warranted. In this work, an "intrinsically X-ray weak" system is defined as one where reduced X-ray emission arises from the intrinsic physical properties of the accretion disk, jet, or corona, or due to the influence of structures close to the black hole (within 100 r g ) on the emitted radiation. This definition excludes the effects of obscuration from structures at larger radii, such as a torus. Thus, we consider the system within 100 r g as a whole, where few high-energy photons are produced, leading to an intrinsically weak X-ray spectrum. Relativistic beaming (see Sec. 4.4) is not included in this definition; while beaming can enhance the observed X-ray flux from a weak system, it cannot fundamentally alter the intrinsic emission to produce an X-ray strong system. \nWe conclude by pointing out that the behavior of super-Eddington accretion in the SANE regime is generally similar to that of MAD systems with zero spin, both leading to X-ray-weak outcomes. This is consistent with the results from Curd & Narayan (2019), where both SANE ( a = 0 ) and MAD ( a = 0 ) models exhibit \nX-ray weakness, while only the MAD ( a = 0 . 9 ) model is X-ray strong. Therefore, the MAD state is not critical for explaining the X-ray weakness observed in the LRDs, and a SANE accretion state would yield similar results. In the MAD case, a low spin must be invoked to reduce X-ray luminosity, but SANE systems do not require this additional constraint.', '4.2. Optical-UV to X-ray Ratios: the α ox': 'We expand on the study of the X-ray weakness by calculating the values of the α ox (Tananbaum et al. 1979). While the bolometric corrections compare one specific energy range to the entire spectrum, the α ox is a pointby-point comparison between the optical-UV and the X-ray emission. \nFor our purposes, the α ox is a valuable tool to investigate how much our super-Eddington SEDs are X-ray weak (in the soft band) compared to other AGN with the same optical-UV luminosity. We use the following definition from Lusso et al. (2010): \nα ox = -log 10 ( L 2 keV /L 2500 Å ) 2 . 605 , (4) \nwhich is characterized by a mean value of α ox ≈ 1 . 5 for standard high-luminosity AGN (Lusso et al. 2010). \nFigure 2. The relationship between α ox and log 10 L 2500 for our super-Eddington SEDs. The dashed red line (with its intrinsic scatter) represents the standard α ox ( L 2500 ) relation from Lusso et al. (2010). Many of our super-Eddington SEDs are characterized by values of the α ox that are significantly higher, indicating a severe lack of X-ray emission. Our SEDs become X-ray stronger, instead of weaker, with increasing optical-UV luminosity. \n<!-- image --> \nmodels show increasing X-ray emission, instead of X-ray weakening, for increasing UV luminosity.', '4.3. Characterizing the X-ray Slope: the Photon Index': 'The photon index, Γ , characterizes the spectral slope of the power-law component of the SED, and it is a valuable measure to investigate how soft (or hard) an SED is at high energy. Assuming a power-law component F ν ∝ ν -α , where α is the spectral index, the photon index Γ is usually defined as Γ = α +1 , which leads to: \nνF ν ∝ ν 2 -Γ . (5) \nTypical values are 1 . 5 < Γ < 2 . 5 (Piconcelli et al. 2005). \nIt is crucial to note that Γ < 2 increases νF ν with frequency, while Γ > 2 leads to a decrease. Larger values of Γ > 2 lead to a rapidly steepening SED, which can become extremely soft in the X-rays. \nIn Fig. 3, we display the distribution of photon indexes Γ for our 96 SEDs. The photon index is calculated in the observed frame at 2 -10 keV (i.e., 14 -70 keV restframe at z = 6 ). The median value is Γ = 3 . 1 , while the most common is Γ = 4 . 4 . About 86% of our SEDs are characterized by Γ > 2 , which means that νF ν ∝ ν 2 -Γ declines with higher frequency. These values of Γ indicate an extremely steep, X-ray soft SED, declining rapidly with increasing frequency. \nOur values of Γ resemble the highest among those recently measured in NLSy1 galaxies by eROSITA (Grün- \nld et al. 2023). Maiolino et al. (2024) correctly suggested that an intrinsic X-ray weakness in the LRDs could be caused by a steep, extremely X-ray soft SED, similar to that of NLSy1. Our study confirms that this is the case. Care needs to be taken when choosing the photon index Γ to characterize the (restframe) bolometric corrections of the LRDs. Choosing too low values, especially if Γ < 2 , will lead to overestimating the restframe bolometric correction. Values in the range Γ = 1 . 7 -1 . 9 , which are typical of sub-Eddington accretion, have been thus far used in the literature to characterize the X-ray weakness of the LRDs (Yue et al. 2024; Maiolino et al. 2024; Ananna et al. 2024). \nIt is important to note that the spectral descriptors, i.e., k X , α ox , and Γ , derived from our GRRMHD simulations are compatible with those recently observed in super-Eddington accreting SMBHs, both in the local and in the highz Universe. For example, Laurenti et al. (2022) studied a sample of lowz AGN ( 0 . 4 ≤ z ≤ 0 . 75 ) with bolometric luminosities of ∼ 10 46 erg s -1 and selected for accretion levels close to Eddington, with 0 . 9 ≤ f Edd ≤ 1 . 2 ; they found substantial values of the bolometric correction, ranging from k X = 50 to k X = 5100 , and with a median value of k X = 175 . They observed, however, smaller values of the photon index, i.e., Γ ∼ 2 , in agreement with the values found by another study of local, super-Eddington accreting black holes (Maithil et al. 2024). However, the values of Γ for super-Eddington accreting SMBHs are higher than those for sub-Eddington ones. In the highz Universe, Zappacosta et al. (2023) selected highly-accreting AGN at z > 6 in the HYPERION sample, finding steep values of the photon index Γ = 2 . 4 ± 0 . 1 , and up to a value of Γ = 3 . 03 +1 . 08 -0 . 89 , which is compatible with the median value for our SEDs.', '4.4. The Cause of X-ray Weakness: Mildly Super-Eddington, Low Spin, and Beaming': "Thus far, we described how super-Eddington SEDs can be extremely X-ray weak globally (with k X ), and compared to their optical-UV emission (with α ox ); we have also described how they can be exceptionally soft and steep in the X-rays (with Γ ). This Section provides a closer look at specific, representative SEDs to detail (i) what parameters cause the X-ray weakness and (ii) how the SEDs are affected by the typical redshift of the LRDs. \nFigure 4 compares two representative superEddington SEDs, which are named the 'reference SEDs' for the remainder of this study: \n- · X-ray Weak: SED for a spin a = 0 SMBHaccreting at a mildly super-Eddington rate ( f Edd = 2 . 4 ). \nFigure 3. The photon index distribution for our 96 superEddington SEDs. These values of Γ indicate extremely steep, X-ray soft SEDs, declining rapidly with increasing energy. \n<!-- image --> \n- · X-ray Strong: SED for a spin a = 0 . 9 SMBH accreting at a mildly super-Eddington rate ( f Edd = 2 . 8 ). \nDespite a similar Eddington ratio, the SEDs are markedly different. In particular, the X-ray strong SED is characterized by a much harder X-ray spectrum, with a significant bump extending at (restframe) frequencies of 10 19 -10 20 Hz for viewing angles close to the pole (i.e., < 30 · ). This high-energy bump is caused by Compton scattering and Lorentz boosting the photons originating from the hottest parts of the accretion flow, as shown in Sec. 4.5. Even the SED averaged over the solid angle (shown in red) is hard in the X-ray. On the contrary, the X-ray weak SED does not show any high-energy bump, except, minimally, for a viewing angle of 10 · . Hence, the emission is much softer in the X-rays. The principal difference between the two models is the black hole spin. The a = 0 . 9 model has a powerful jet and correspondingly has a large X-ray luminosity. The a = 0 model has a much weaker jet and is much more X-ray-quiet. \nTo fully understand how observable these SEDs are with, e.g., the Chandra X-ray observatory, it is necessary to redshift them to a representative redshift for the LRDs, e.g., z = 6 . This exercise is performed in the bottom panels of Fig. 4. A source characterized by an X-ray strong SED can be detectable by Chandra in the observed 2 -10 keV range, assuming a very deep field with a flux limit of 10 -17 erg s -1 cm -2 . In particular, Chandra can detect the mean SED, which is averaged over the solid angle. \nThe situation is radically different for a source characterized by an X-ray weak SED. Chandra can barely \ndetect only sources observed near the pole, and it is ∼ 2 orders of magnitude in flux away from detecting the mean SED. At the flux limit of 10 -17 erg s -1 cm -2 , the mean SED is too soft, rendering these sources impossible to detect with Chandra. \nThe fact that Chandra may detect such sources, at the median redshift of z = 6 , only for viewing angles very close to the jet (i.e., < 10 · ) leads to a significant corollary. Assuming a SMBH with a collimated jet ejecting from its pole and a random orientation in space, we can calculate the probability that Chandra observes it within an inclination of 10 · from the pole. The solid angle subtended by a cone with a half-angle of 10 · is given by Ω = 2 π (1 -cos i ) , where i is the inclination in radians. The probability is then the ratio of this solid angle (multiplied by 2 , to account for the two jets) to the total solid angle of a sphere, i.e., 4 π steradians, resulting in a probability of ∼ 1 . 5% . Interestingly, Kocevski et al. (2024) reported the X-ray detection of only 2 LRDs out of a sample of 341 , leading to a similar detection frequency of ∼ 0 . 6% . Thus, it is plausible that Chandra is detecting only the LRDs viewed from minimal angles from the jet of the central SMBH. \nNext-generation X-ray observatories, such as AXIS (Reynolds et al. 2023; Cappelluti et al. 2024), can detect such weak sources. Assuming an AXIS deep field with a flux limit of 10 -18 erg s -1 cm -2 (Marchesi et al. 2020; Cappelluti et al. 2024), and an energy range of 0 . 2 -10 keV, AXIS will detect the mean SED of a typical X-ray weak source at z ∼ 6 easily and, most likely, from any viewing angle. Note that the SEDs shown in Fig. 4 belong to sources that are easily detectable by JWST/NIRSpec, assuming a flux limit of ≈ 5 . 5 × 10 -19 erg s -1 cm -2 for a 3 σ detection at ∼ 1 µm ≈ 3 × 10 14 Hz (Pacucci et al. 2023). \nThe effect of the viewing angle is essential and warrants a deeper analysis. Figure 5 zooms in on the reference X-ray weak SED (at z = 6 ) and clarifies the socalled 'beaming effect' due to observing the emission from lines-of-sight progressively closer to the pole. In particular, the same SMBH, with the same Eddington ratio and spin, produces SEDs whose peaks differ by 1 . 5 dex in flux when the viewing angle changes from 80 · to 10 · . \nExtending the analysis of the SED to lower frequencies, we note that recent studies (e.g., Mazzolari et al. 2024) point out that LRDs that are X-ray weak are also radio-weak. Regarding the predicted radio luminosity, our simulations do not account for non-thermal jet emission, as both KORAL and HEROIC assume thermal gas and focus on thermal synchrotron and Bremsstrahlung radiation. Nevertheless, assuming that the radio emission of \nFigure 4. Top Row: Comparison (at z = 0 ) between typical X-ray weak (left, mildly super-Eddington with zero spin) and X-ray strong (right, mildly super-Eddington with high spin) SEDs. Bottom Row: Same as in the top row, but at z = 6 . The 8 gray lines represent the spectrum for lines of sight ranging from 80 · (bottom) to 10 · (top) from the pole. The red line indicates the spectrum averaged over 4 π steradians. In the bottom panel, the arrows indicate typical flux limits for deep-field X-ray surveys with Chandra ( 2 -10 keV) and AXIS ( 0 . 2 -10 keV). Some of these SEDs extend up to ∼ 1 MeV. \n<!-- image --> \nthe jet is proportional to the jet power, P jet , we estimate that the X-ray strong model, with P jet ≈ 0 . 9 ˙ Mc 2 , would be more radio luminous than the X-ray weak model, which has P jet ≈ 0 . 08 ˙ Mc 2 .", '4.5. X-ray Weak and Strong Accretion Structures': 'What are the physical causes of such extreme differences in the X-ray emission? A detailed view of the structure of the accretion flows in the models corresponding to the two reference SEDs (see Sec. 4.4 for \ntheir definition) is provided in Fig. 6. The density distribution in the central panel shows that the accretion disks of X-ray weak and X-ray strong SEDs are similar, with a \'puffed up\' structure typical of super-Eddington accretion. \nIn general, for sub-Eddington accretion rates, the disk is thinner (Novikov & Thorne 1973; Shakura & Sunyaev 1973), and a more significant fraction of the hot inner region is visible, leading to a more substantial X-ray component in the observed SED, especially if there is \nFigure 5. Effect on the spectrum of a changing line of sight from the pole. The strongest X-ray emission is achieved by looking from a direction close to the pole (i.e., from small inclination angles). The red line indicates the spectrum averaged over 4 π . \n<!-- image --> \nan optically thin hot corona above the disk. At superEddington accretion rates, although the total luminosity increases, the X-ray contribution diminishes because the hot regions are more obscured by the thicker disk (in Fig. 6, notice that extreme values of the optical depth occupy a larger fraction of the solid angle in the X-ray weak case). This angle dependence implies that the apparent fraction of AGNs with intrinsic strong Xray emission may be underestimated, as many may be viewed at angles where X-rays are suppressed. This is consistent with the explanation of the X-ray weakness provided by Maiolino et al. (2024), which requires significant covering factors. Additionally, the third row of panels in Fig. 6 shows the scattering optical depth out to z = 100 r g , illustrating the transverse optical depth (integrated over θ from the pole at a constant radius). These results indicate that X-rays cannot escape laterally, suggesting significant opacity in the transverse direction of the jet. \nThe main differences in our super-Eddington simulations are shown in the temperature field (top row) and the βγ field (bottom row), where β = v/c and γ is the Lorentz factor. The X-ray strong accretion structures display significantly higher values for the temperature and βγ , indicating that the jet is relativistic and more powerful. The combination of these two effects leads to \nenhanced X-ray emission. Higher spin values generally lead to more efficient jet production, as the spin influences the inner accretion disk\'s structure and energy extraction efficiency (see Ricarte et al. 2023 for detailed results on jet efficiencies as a function of black hole spin for super-Eddington accretion). \nThe temperature distribution within the accretion disk is particularly revealing: the hottest regions, with T > 10 8 K , are located at the base of the jet and on the wall of the funnel and are the primary sources of X-ray emission. Hence, observers aligned with the jet would predominantly see high-energy X-rays. Conversely, observers at larger inclination angles are screened from the hottest gas by the geometrically thick disk and see cooler regions, resulting in spectra dominated by UV and optical emissions. \nIn our GRRMHD simulations, the X-rays come from the base of the jet and the wall of the funnel, which is consistent with the lamppost interpretation of the Xray \'coronae" in AGN (see, e.g., Matt et al. 1991; Martocchia & Matt 1996; Miniutti & Fabian 2004). Our simulations (similar to other studies using GRRMHD simulations) do not include an explicit treatment of the X-ray reflection component from an extended corona in the polar region.', '4.6. Explaining the X-ray Weakness of the LRDs': 'We can now apply our super-Eddington SEDs to the case of the LRDs. Figure 7 shows the bolometric correction k z =6 X in the observed 2 -10 keV frame for our sample of super-Eddington SEDs, calculated at z = 6 . Note that the bolometric luminosities of our SEDs are compatible with those observed. We display the entire sample of 12 simulations × 8 viewing angles, totaling 96 values of the bolometric correction. These are compared to (i) lower limits on the bolometric corrections for LRDs at 4 < z < 11 derived from Maiolino et al. (2024) and Yue et al. (2024); (ii) the X-ray stacking of the Maiolino et al. (2024) sample, which provides the most robust constraint; (iii) the standard (restframe) relation between the bolometric luminosity and the 2 -10 keV bolometric correction presented in Duras et al. (2020); (iv) a density distribution summarizing the (restframe) location in the L bol -k X plane of standard, lowz Type 1 AGN from Lusso et al. (2020). Note that the data points from Maiolino et al. (2024) and Yue et al. (2024) have been shifted to the observed 2 -10 keV range to be comparable to our SEDs, using the redshift of each source, and Γ = 1 . 7 and Γ = 1 . 8 , respectively, as indicated in their study. \nAlso note that the LRDs are, in principle, a subset of the collection of Type 1 AGN at 4 < z < 11 presented \nFigure 6. Comparison between accretion structures in the poloidal plane (the black hole is at (0 , 0) and the jets are oriented in the vertical direction) for our reference cases of X-ray weak (left) and X-ray strong (right) SEDs (see Sec. 4.4). The rows are, from top to bottom: temperature, gas density, scattering optical depth, and βγ . X-ray strong accretion structures are significantly hotter and more relativistic. \n<!-- image --> \nin Maiolino et al. (2024). The X-ray weakness is not exclusively associated with the LRDs but with a broader collection of JWST-discovered sources (Maiolino et al. 2024). However, most sources in Maiolino et al. (2024) are also classified as LRDs (Maiolino et al. 2023; Matthee et al. 2024). In addition, some of the LRDs presented in Yue et al. (2024) are also included in Maiolino et al. (2024); for those not included, we calculated the bolometric luminosity from the provided H α broad component luminosity, following the standard relation in Richards et al. (2006). \nFigure 7 demonstrates the following: \n- · Many SEDs (i.e., ∼ 40% ) are characterized by Xray bolometric corrections higher than the lower limits on the single LRDs.\n- · Many SEDs (i.e., ∼ 35% ) are characterized by Xray bolometric corrections higher than even the lower limit on the stacked sample.\n- · Most SEDs (i.e., ∼ 88% ) are characterized by values of the (observed) X-ray bolometric correction significantly higher than those of lowz Type 1 AGN. Some SEDs have values of k z =6 X that are 2 orders of magnitude higher than the highest (restframe) bolometric corrections measured by Duras et al. (2020), despite being associated with significantly (bolometrically) fainter objects. \nThe single SEDs shown in Fig. 7 are not equally probable. In Sec. 4.4, we showed how most X-ray emission is associated with SEDs seen from a small (i.e., < 30 · ) angle from the pole. Assuming random orientations (and two jets), the probability of observing such geometrical configuration is ∼ 13% . Hence, it is most likely to observe such systems along a line of sight that leads to an X-ray weak SED. \nIt is then apparent that the X-ray weakness observed in the LRDs can be explained by SEDs for mildly superEddington accreting SMBHs, typically characterized by lower or zero spin and observed away from the pole, which is the most likely observing condition. \nRemarkably, our models predict that the restframe optical and UV spectra would still exhibit BLRs, potentially with higher velocities and implied line widths due to strong winds from the super-Eddington accreting SMBH. There is recent observational evidence of strong winds launched from super-Eddington accreting AGN (Vietri et al. 2020, 2022; Ding et al. 2022), supported by robust investigations on the theoretical side (Yang et al. 2014; Sądowski et al. 2015; Sądowski 2016; Yang et al. 2023; Zhang et al. 2024). This implies that BLRs could appear artificially broadened by these winds, lead- \ng to an overestimation of the black hole mass and an underestimation of the Eddington ratio. \nOf course, explaining the X-ray weakness of one specific source requires a detailed study involving GRRMHD simulations tailored for the particular (estimated) black hole mass and Eddington ratio. A study of a compilation of such SEDs, as a function of the spin parameter and viewing angle, will then associate a precise probability for that specific source to be Xray weak. However, our analysis, based on the median SMBH mass estimated ( M · = 10 7 M ⊙ ) and the median redshift ( z ∼ 6 ), has demonstrated that this is a valid explanation for the X-ray weakness of the LRDs. Thanks to the thick accretion disk produced by the super-Eddington accretion regime, our solution supports the requirement of large covering factors proposed by Maiolino et al. (2024). \nTwo final questions remain: (i) Are the LRDs possibly accreting at super-Eddington rates? (ii) Are the superEddington rates and the low spin required compatible?', '4.7. Eddington Ratios of the LRDs': 'LRDs have a median Eddington ratio of f Edd = 0 . 4 , (Harikane et al. 2023; Maiolino et al. 2023; Larson et al. 2023; Kokorev et al. 2023). Some LRDs are already estimated to be accreting at super-Eddington rates (Maiolino et al. 2023, 2024; Harikane et al. 2023), while some Type 1 AGN seem to be well below the limit (Juodžbalis et al. 2024b). Remarkably, some collections of LRDs, such as the one presented in Harikane et al. (2023), have a median Eddington ratio already superEddington: f Edd = 1 . 1 . \nAs the bolometric luminosity is measured (or, better, inferred from bolometric corrections, such as the ones in Richards et al. 2006), a decrease in the SMBH mass would automatically increase the Eddington ratio. Hence, assuming that the SMBH masses estimated for the LRDs are overestimated by a mere factor of ∼ 3 would bring the median value of the Eddington ratio well inside the mildly super-Eddington regime, characterized by extreme X-ray weakness. We note that a factor of ∼ 3 is well within the uncertainty in the SMBH mass estimates via single-epoch virial relations even in the local Universe (Kaspi et al. 2000; Greene & Ho 2005). \nHence, if the SMBH masses are systematically overestimated by a mere factor of ∼ 3 , our analysis for the median mass ∼ 10 7 M ⊙ at the median redshift of 6 can explain the median population, i.e., 50% of the sources.', '4.8. Spin-down Effect for Super-Eddington SMBHs': "We have shown that mildly super-Eddington ( 1 . 4 < f Edd < 4 ) accreting SMBHs when slowly spinning ( a ∼ \nFigure 7. In the plane log 10 L bol vs. X-ray bolometric correction (in the observed 2 -10 keV range, for z = 6 ), we compare the X-ray weakness of our super-Eddington SEDs with that of highz LRD populations, and some well-known lowz samples. Teal dots represent the bolometric corrections derived in this study from super-Eddington SEDs at z = 6 (i.e., the median redshift from LRDs catalogs). Note that the bolometric luminosities of our SEDs are compatible with those observed. Orange circles indicate LRD populations at 4 < z < 11 from Maiolino et al. (2024) and Yue et al. (2024), with arrows showing lower limits, while the red star marks the stacked data point (for sources at z > 4 ) from Maiolino et al. (2024), which provides the most robust constraint. The dark orange circle displays the location of GN-28074, which is the most X-ray weak among all AGN discovered by JWST ( z = 2 . 26 , Juodžbalis et al. 2024a). Note that each data point has been shifted to the observed 2 -10 keV range to be comparable to our SEDs. The solid black line (with its spread) represents the restframe k X ( L bol ) relation from Duras et al. (2020). The grey distribution in the background shows lowz Type 1 AGN data from Lusso et al. (2020). \n<!-- image --> \n0 ) and viewed far from the pole are extremely X-ray weak. \nThe combination of super-Eddington accretion rates and low spin is self-consistent: super-Eddington rates naturally lead to a spin-down of the black hole (see, e.g., recent studies by Curd & Narayan 2019; Ricarte et al. 2023; Lowell et al. 2024; Jacquemin-Ide et al. 2024). In particular, these studies have shown that the spin evolution and jet efficiency of an accreting SMBH are linked to the accretion rate. \nA black hole accreting from a disk with a strong poloidal magnetic field (i.e., a MAD accretion disk) is characterized by a jet, which is powered by the spin energy of the black hole. The spin-down effect becomes crucial as the accretion rate surpasses the Eddington \nlimit. Detailed GRRMHD simulations indicate a dramatic reduction in the black hole's spin in the superEddington regime, driving it toward a near-zero spin state (Ricarte et al. 2023). In particular, the spin-down process, for the range of Eddington ratios investigated here (i.e., 1 . 4 < f Edd < 13 . 4 ), occurs on cosmologicallynegligible timescales of 3 -30 Myr (Ricarte et al. 2023). \nHence, if the SMBHs hosted by the LRDs are accreting at a super-Eddington rate due to the large availability of cold gas (Pacucci & Loeb 2020), their spin is expected to reach near-zero equilibrium levels quickly, further enhancing their X-ray weakness. Thus, with the LRDs, we may witness, for the first time, a ubiquitous occurrence of slowly spinning SMBHs accreting at mildly superEddington rates.", '5. SUMMARY AND CONCLUSIONS': "This study was motivated by the widespread JWST detection at z > 4 of sources characterized by solid evidence for the presence of low-luminosity AGN, with SMBHmasses of M · = 10 6 -8 M ⊙ . These SMBHs, found in compact and red hosts (the LRDs), are largely undetected in X-rays, leading to the 'X-ray weakness' problem. \nUsing a suite of GRRMHD simulations, we investigated the super-Eddington accretion process onto a SMBH with mass M · = 10 7 M ⊙ at z ∼ 6 ; these values represent the medians for the LRDs population in the largest catalogs thus far available (Harikane et al. 2023; Maiolino et al. 2023; Kocevski et al. 2024; Akins et al. 2024; Kokorev et al. 2024). Our key findings are summarized here. \n- · The highest levels of X-ray weakness occur in SMBHs accreting at mildly super Eddington rates ( 1 . 4 < f Edd < 4 ), with low or zero spin ( a ∼ 0) , viewed at angles > 30 · from the pole. Viewing angles aligned with the pole increase the observed X-ray emission.\n- · X-ray bolometric corrections in the observed 2 -10 keV energy band reach values of ∼ 10 4 at z ∼ 6 , which is ∼ 5 times higher than the highest constraint from X-ray stacking. These X-ray corrections are compatible even with the most extreme X-ray weak sources detected. About 35% of our SEDs have X-ray bolometric corrections higher than limits imposed by X-ray stacking.\n- · About 55% of our super-Eddington SEDs have optical-UV to X-ray ratios ( α ox ) outside the typical range for standard Type 1 AGN. Contrary to the standard, our SEDs show increasing X-ray emission for increasing optical-UV luminosity.\n- · Our super-Eddington SEDs are extraordinarily steep and soft in the X-rays. Their median photon index is Γ = 3 . 1 , while the most common value is Γ = 4 . 4 . In 86% of our SEDs, νF ν declines rapidly with increasing frequency. These values resemble the highest measured in NLSy1 galaxies.\n- · X-ray strong SEDs have a harder X-ray spectrum with a bump at 10 19 -10 20 Hz (restframe) when viewed close to the pole (i.e., < 30 · ), while X-ray weak SEDs lack this bump except, minimally, for a viewing angle of ∼ 10 · .\n- · If X-ray weak spectra characterize the population of LRDs, Chandra may detect only those viewed \nfrom minimal angles from the jet. The probability of observing a SMBH within 10 · from the jets is ∼ 1 . 5% , which is similar to the X-ray detection rate ( ∼ 0 . 6% ) in the most extensive catalog of LRDs with X-ray coverage available (Kocevski et al. 2024). \n- · Next-generation X-ray observatories, such as AXIS, can detect typical X-ray weak sources at z ∼ 6 from any viewing angle.\n- · The effect of the viewing angle, or beaming, is crucial. The same SMBH, with the same Eddington ratio and spin, produces SEDs whose peaks differ by 1 . 5 dex in flux when the viewing angle changes from 80 · to 10 · . In general, observers aligned with the jet axis are most likely to see X-rays. Observers at larger and more probable viewing angles see cooler regions, resulting in spectra dominated by optical-UV emission.\n- · The inner structure of the accretion flow reveals the physical difference between X-ray weak and strong SEDs. Enhanced X-ray emission is linked to higher temperatures ( T > 10 8 K) close to the SMBH and a highly relativistic jet, with higher spin values leading to more powerful jets and harder X-ray spectra.\n- · In the super-Eddington regime, while the total luminosity increases, the X-ray contribution diminishes because the hot regions are more obscured by the thicker disk. This picture is consistent with the model by Maiolino et al. (2024), which requires significant covering factors. Thus, our proposal consistently complements other models to explain the lack of X-rays.\n- · Super-Eddington accretion rates naturally lead to strong winds, artificially broadening BLRs and potentially overestimating the black hole masses.\n- · Our picture is fully self-consistent: superEddington rates lead naturally to lower spin, large covering factors, and broad emission lines, leading to an intrinsic X-ray weakness. The spin-down process occurs over 3 -30 Myr for the range of Eddington ratios investigated.\n- · Many SMBHs in the LRDs are already estimated to accrete at super-Eddington rates. If the SMBH masses are systematically overestimated by a mere factor of ∼ 3 (which is compatible with even local measurements), our analysis for the median mass ∼ 10 7 M ⊙ at the median redshift of 6 can explain the median population, i.e., 50% of the sources. \n- · The phenomenon of the LRDs is mainly observed in the redshift range 4 < z < 8 , where large availability of accretable gas would render widespread super-Eddington accretion feasible. An older Universe, with far less free gas available, would make this phenomenon disappear, as observed. \nA comprehensive understanding of the masses, accretion rates, and spins of the SMBHs in the LRDs is crucial for enlightening their role in the early formation and evolution of their hosts. Specifically, accurate measurements of the black hole and the host's stellar mass are essential. Recent analyses indicate that the central SMBHs in LRDs appear to be overmassive by a factor of 10 -100 relative to the stellar content of their hosts (Pacucci et al. 2023; Pacucci & Loeb 2024; Durodola et al. 2024), according to standard local relations (e.g., Reines & Volonteri 2015). If confirmed, this discrepancy could provide the most substantial evidence yet for the occurrence of heavy seed formation in the highz Universe (Agarwal et al. 2013; Natarajan et al. 2017; Scoggins et al. 2023; Pacucci & Loeb 2024). For this and other reasons, it is essential to constrain the mass \nand accretion properties of the SMBHs detected in the LRDs. \nIn summary, our study suggests that with the LRDs we are witnessing, for the first time, the ubiquitous occurrence of slowly spinning SMBHs accreting at mildly super-Eddington rates. \nAcknowledgments: F.P. acknowledges fruitful discussions with Nico Cappelluti, Andrea Ferrara, Erin Kara, Vasily Kokorev, Roberto Maiolino, Joseph Silk, and Minghao Yue. F.P. also acknowledges support from a Clay Fellowship administered by the Smithsonian Astrophysical Observatory. This work was also supported by the Black Hole Initiative at Harvard University, funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation. This work used the Anvil supercomputer at Purdue University through allocation AST-080028 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296.", 'REFERENCES': '```\nAbramowicz, M. A., Chen, X., Kato, S., Lasota, J.-P., & Regev, O. 1995, ApJL, 438, L37, doi: 10.1086/187709 Abramowicz, M. A., Czerny, B., Lasota, J. P., & Szuszkiewicz, E. 1988, ApJ, 332, 646, doi: 10.1086/166683 Agarwal, B., Davis, A. J., Khochfar, S., Natarajan, P., & Dunlop, J. S. 2013, MNRAS, 432, 3438, doi: 10.1093/mnras/stt696 Akins, H. B., Casey, C. M., Lambrides, E., et al. 2024, arXiv e-prints, arXiv:2406.10341, doi: 10.48550/arXiv.2406.10341 Akylas, A., & Georgantopoulos, I. 2021, A&A, 655, A60, doi: 10.1051/0004-6361/202141186 Ananna, T. T., Bogdán, Á., Kovács, O. E., Natarajan, P., & Hickox, R. 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2024arXiv240907511A	The Radar Echo Telescope for Cosmic Rays RETCR a pathfinder instrument for the radar echo method of ultrahigh energy UHE neutrino detection was initially deployed near Summit Station Greenland in May 2023. After a 4 week commissioning period 9 days of data were taken before the instrument went offline. In this article we describe the instrument as it was deployed and the initial performance of the detector. We show that the technical aspects of running a radar based particle cascade detector in the ice have been demonstrated. Analysis of the 2023 data informed improvements that were incorporated into the MayAugust 2024 deployment which has just concluded at time of writing. Results from the 2024 run will be presented in forthcoming publications.	2024-09-01T00:00:00Z	['arXiv:2409.07511', '10.48550/arXiv.2409.07511', '2024arXiv240907511A']	['High Energy Physics - Experiment', 'Astrophysics - High Energy Astrophysical Phenomena']	Initial performance of the Radar Echo Telescope for Cosmic Rays RETCR	2,024	199	0.33	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.07511.pdf	{'Initial performance of the Radar Echo Telescope for Cosmic Rays, RET-CR': 'P. Allison, 1 J. Beatty, 1 D. Besson, 2 A. Connolly, 1 A. Cummings, 3 C. Deaconu, 4 S. De Kockere, 5 K.D. de Vries, 5 D. Frikken, 1 C. Hast, 6 E. Huesca Santiago, 5 C.-Y. Kuo, 7 A. Kyriacou, 2 U.A. Latif, 5 J. Loonen, 5 I. Loudon, 8, 9 V. Lukic, 5 C. McLennan, 2 K. Mulrey, 9 J. Nam, 7 K. Nivedita, 9 A. Nozdrina, 2 E. Oberla, 4 S. Prohira, 2 J.P. Ralston, 2 M.F.H. Seikh, 2 R.S. Stanley, 5 S. Toscano, 8 D. Van den Broeck, 5, 10 N. van Eijndhoven, 5 and S. Wissel 3 \n(Radar Echo Telescope) \n1 Department of Physics, Center for Cosmology and AstroParticle Physics (CCAPP), \nThe Ohio State University, Columbus OH 43210, USA \n2 University of Kansas, Lawrence, KS 66045, USA \n3 Departments of Physics and Astronomy & Astrophysics, Institute for Gravitation and the Cosmos, \nPennsylvania State University, University Park, PA 16802, USA \n4 Enrico Fermi Institute, Kavli Institute for Cosmological Physics, \nDepartment of Physics, University of Chicago, Chicago, IL 60637, USA \n5 Vrije Universiteit Brussel, Dienst ELEM, IIHE, Brussels, Belgium \n6 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA \n7 National Taiwan University, Taipei, Taiwan \n8 Universit´e Libre de Bruxelles, Brussels, Belgium \n9 Department of Astrophysics/IMAPP, Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands 10 Vrije Universiteit Brussel, Astrophysical Institute, Brussels, Belgium \nThe Radar Echo Telescope for Cosmic Rays (RET-CR), a pathfinder instrument for the radar echo method of ultrahigh energy (UHE) neutrino detection, was initially deployed near Summit Station, Greenland, in May 2023. After a 4 week commissioning period, 9 days of data were taken before the instrument went offline. In this article, we describe the instrument as it was deployed, and the initial performance of the detector. We show that the technical aspects of running a radar based particle cascade detector in the ice have been demonstrated. Analysis of the 2023 data informed improvements that were incorporated into the May-August 2024 deployment, which has just concluded at time of writing. Results from the 2024 run will be presented in forthcoming publications.', 'I. INTRODUCTION': 'The Radar Echo Telescope for Cosmic Rays (RET-CR) [1] is a pathfinder instrument for the radar echo method of ultrahigh energy (UHE) neutrino detection. When a UHE particle initiates a cascade within a dense material, a short lived cloud of ionization is created that can reflect incident radio waves. With the radar echo method, a transmitting antenna and receiving antenna(s) are deployed within a large target volume, such as the large ice sheets found in the Arctic and Antarctic, to detect the radar reflections from neutrino initiated cascades. For more information on the history, theory, and status of the method, see Refs. [1-5]. \nRET-CR was developed to test the radar echo method in nature and demonstrate the technology needed to detect neutrino initiated cascades. RET-CR targets the in-ice component of a UHE cosmic ray extensive air shower (EAS) [6]. At zenith angles less than ∼ 30 degrees and energies above ∼ 10 PeV a significant fraction, > 10%, of the primary energy is deposited into the ground within a few cm of the EAS axis [7]. This produces a secondary, dense cascade that mimics a neutrino initiated cascade, but at a far higher event rate. Targeting UHE cosmic rays is therefore a bridge between the successful laboratory detection [5, 8] and an eventual neutrino detector [9]. \nIn this article we detail the detector as deployed at Summit Station, Greenland in May 2023, the deployment itself, and the initial performance of the instrument. We begin by discussing the differences between what was described in Ref. [1] and what was actually deployed; primarily a differ- \nFIG. 1. RET-CR layout as deployed in May 2023. Different system components (surface stations, data acquisition system [DAQ], transmitter [TX] and receiver [RX]) are indicated on the figure, and described in the main text. \n<!-- image --> \nence in layout geometry and number of stations. We then describe the deployment itself, as well as the commissioning phase data run. We then describe the performance of the various subsystems, and conclude with a discussion of the upgrades and improvements that were realized for the full data run in 2024.', 'II. DESCRIPTION OF THE INSTRUMENT': 'RET-CR can be divided into two main systems, the surface component and the radar component. The surface component consists of autonomous, solar powered stations, each with a pair of scintillator detectors and a dual polarized radio antenna. The radar component consists of a solar power system, a communications system, a central data acquisiton system (DAQ), a radar transmitter, and radar receivers. The principle of operation of the instrument is unchanged from what was presented in Ref. [1], though some technical aspects are detailed below. The entire instrument was comprised of three surface stations, three downhole receiver strings, and one downhole phased transmitter string, in a layout shown in Fig. 1. \nRET-CR operates via triggers from the surface system. A surface station will send a trigger to the radar system when it registers a coincident minimally ionizing particle (MIP) in both panels. The window for coincidence is 1 µ s. The radar system triggers an event readout when ≥ N surface stations trigger within a 1 µ s window, with N being set in software. The radar system can also be triggered via software, GPS, or via a selectable edge threshold in the receiver (RX) channels. \nFIG. 2. The magnitude of the change in phase required to cancel the TX signal correlates with wind speed. \n<!-- image -->', 'A. Radar System': 'The radar system is an 8 channel phased transmitter with 4 receive channels featuring active transmitter cancellation (TC). The heart of the DAQ is a xilinx RFSoC, featuring an 8 channel analog-to-digital converter (ADC) and an 8 channel digital-to-analog converter (DAC). The RFSoC was clocked at ∼ 3GHz. A custom-made board breaks out all 8 DAC channels and all 8 ADC channels. The DAC channels have a software controlled variable attenuator providing up to 32 dB of \nattenuation per channel. Four of the DAC channels are used to actively cancel the TX signal in the RX channels by injecting a scaled, time-delayed copy of the transmitted signal into the RX signal path. Overall attenuation of the transmitted signal in the RX of > 80 dB is achieved. The TX and RX antennas are all identical wide-cylinder dipoles with a 6.35 cm radius. For the 8-channel TX phased array, each antenna is fed by a single 20 W power amplifier, and each pair of power amplifiers is fed by a single channel of the DAC, allowing for pairwise phasing of the transmitted polar angle. The 8 TX antennas are separated by ∼ 0.3 m 1 , deployed such that the midpoint between the fourth and fifth antennas is approximately 10 m deep. Each RX channel consists of an antenna, a combiner for the transmitter cancellation, a ∼ 60 dB low-noise amplifier, and a bandpass filter from 100-300 MHz. Each RX is lowered down a borehole to a depth of 10 m a distance of 30 m from the transmitter (TX) borehole. LMR-400 coaxial cable was used for all RF lines, except for the TX cancellation line, which was LMR-240. The radar system is powered by a triangular solar array with 1.2 kW nominal power per face, and a 200 Ah, 24 V battery buffer. GPS provides accurate timestamping of events, and a WLAN bridge provides telemetry and commanding from Summit Station. \nThe FPGA firmware included a software interface for debugging and diagnostic information as well as commanding. The interface allows us to: change the TX frequency and modulation, steer the polar angle of the phased transmitter, send software triggers, run the TX cancellation routine, change the attenuation setting on the DAC channels, enable/disable the RF trigger and set the threshold, select the number of surface stations required to form a trigger, set the delay time between a trigger and an event readout (to ensure enough data has been buffered before readout) and request different levels of debug and diagnostic data. Changing the TX frequency and modulation allows us to observe the reflected signal in different bands. The carrier cancellation finds the optimal amplitude and timing offset to cancel the transmitter in each receive channel simultaneously. A single event record is a header with useful housekeeping information and 4 records of 32,768 bytes, corresponding to 16,384 samples of data, one per RX channel. Data is readout to a raspberry pi single board computer (SBC) via ethernet for fast transfer and recorded redundantly on two disks before telemetry over WLAN to Summit Station, where it is redundantly recorded once more and transmitted to servers in the US. A power distribution system allows for on/off control of the individual power amplifiers via software.', 'B. Surface System': 'Asingle surface station consists of 2 scintillator panels[10], a cross-pol log periodic dipole antenna[11], a power system \nconsisting of an omnidirectional 4x20 W solar array with a 10 Ah, 12 V battery buffer, a raspberry pi SBC with redundant storage, and electronics for readout of the scintillators and radio antenna. The SBC controls the readout electronics for the scintillators and radio antenna, the latter of which is repurposed from the CODALEMA experiment [12]. There are 3 such stations on a 40 m radius from the TX. Communication with the surface stations is achieved via ethernet on a cat-7 cable, with trigger signals sent along a separate cat-7 cable. The scintillator panels were deployed on the surface, and the radio antenna was deployed atop bamboo poles to elevate it approximately 1 m off the surface. Every cosmic ray, RF, and forced/software trigger is sent to every surface station, triggering an event readout of the surface radio.', 'III. DEPLOYMENT': 'The deployment of RET-CR took place in May 2023 with a team of five people. RET-CR is located approximately 6 km N/NE of Summit Station, Greenland. Summit Station is ideally suited to RET-CR, situated at 3,216 m above sea level at roughly the center of the Greenland Ice Sheet (72.579583, 38.459186). This high elevation, uniform ice facilitates the deposition of a significant fraction of the primary cosmic ray energy into the ice, ideal for maximizing chances of measuring a radar echo. \nStation staff assisted in setting up a shelter/work tent at the site. The four boreholes were drilled using a Kovacs MarkV coring system. The surface stations, including antennas, were assembled and deployed. The radar power system was assembled on site and raised, using guylines set to dead-men anchors to protect from the wind. The WLAN link was established by facing one point of the PV triangle directly toward Summit Station and mounting the WLAN antenna at this point. The DAQ, a single Nanuk 975 enclosure with custom bulkhead panels and inlet/outlet vents for thermal regulation, was buried in a 2 x 2 x 1 m snow pit vault covered with plywood, to protect it from blowing snow. A photograph of the deployed instrument is shown in Fig. 3, with the PV array visible at center right, and one of the surface stations in the middle distance. Flags indicate buried elements such as the receiver strings (far right) and cables. The large snow disturbance in the foreground was drifting against our deployment tent, which Summit Station staff had removed at the end of our deployment, just before this photo was taken.', 'IV. INITIAL PERFORMANCE': 'The instrument worked nominally for several hours as soon as it was powered on. By the next day, we noticed that two of the surface stations had powered off. It was concluded that the surface radio electronics were perhaps drawing too much power, and so these were unplugged from the systems for the duration of the deployment. During the commissioning phase one of the surface station trigger lines went offline, though we still had contact with that station via ethernet, and could verify \nthat it was still triggering nominally. Therefore, all of our data was with N=2 surface station logic, though the timestamped triggers for the 3rd station were also recorded and stored. Additionally, only 4 transmitter channels were powered on at any time, as we noticed very quickly that the DAQ became far hotter than expected with all 8 transmitter power amplifiers active. \nAfter a 4 week commissioning period, during which the various subsystems were tested and firmware upgrades were written and pushed, a data run commenced. Communication with RET-CR was lost 9 days into this data taking run. Subsequent debugging efforts concluded that the system was still powered on, but that the WLAN link was down, and the system was not fully operational, as evidenced by an inability to detect the TX signal from the surface. The DAQ vault was then uncovered to reveal that overheating resulted in the DAQ box melting down approximately one meter into the snow, and severing the WLAN cable, as well as one of the surface station trigger lines, which had become unresponsive some days before. The DAQ itself had reset into a safe mode during the 9th day of the data run and was undamaged. \nOur 220 hours of livetime demonstrated several important technical successes. The first and most important of these was the transmitter cancellation. Without active cancellation, the receiver amplifiers would be fully saturated by the direct transmitter signal. To remedy this, a time delayed and scaled version of the transmitted signal is fed directly into the RX channels before the front-end amplifier to cancel out the incoming TX signal. A firmware/software routine periodically runs to find the optimal cancellation phase and amplitude. The phase and amplitude that are selected are those which optimally cancel out the received transmitter signal. This is the sum total of the direct TX signal, plus all reflected paths from internal reflecting layers, and most prominently, the surface. Therefore, the cancellation phase will vary as the surface profile changes through wind or precipitation. Fig. 2 shows this qualitatively. Here the magnitude of the change in cancellation phase \| ∆ ϕ \| between hourly runs of the TX cancellation in one receiver is plotted against the local windspeed, as measured at Summit Station. Higher windspeed results in increased drifting, which changes the reflected path lengths for the TX signal, thus requiring a more significant change in cancellation phase. \nEven at the relatively modest power of 20 W per channel, the signal received directly by the RX was far higher than the dynamic range input of the front end amplifiers. We were able to use the TC to nearly completely eliminate the TX signal from the RX, even with the TX at full power. The TC happens in two stages. First, the transmitter is set to fractional power, so that the RX amplifiers do not saturate, and the attenuators on the TC DACs are set to a high value. A first stage cancellation is run, to find an approximate value for the cancellation phase and amplitude. Second, the transmitter is increased to full power and the attenuation on the TC DACs are reduced appropriately to best match the amplitude of the incoming signal. Then the second stage TC routine is run, to fully eliminate the signal. Consequently, a plot comparing TC on versus off cannot be shown, as turning the TC off would damage the receivers. We can approximate the cancellation \nFIG. 3. RET-CR, as deployed for the 2023 run. Solar panels are evident in the center, and flags denote the buried receivers and transmitter. In the middle distance are smaller solar trees and a surface radio antenna. \n<!-- image --> \nFIG. 4. An example forced trigger event. Data from all 4 receiver channels are overlaid, the smallest trace being that of an unamplified surface antenna, the other three being the dedicated radar receiver channels. Evident in the data are broadband RF noise spikes from inadequately shielded RF power supplies, which were remedied for the 2024 data run. \n<!-- image --> \nby a simple calculation: a 20 W output, assuming zero gain to the TX and RX, results in a direct TX signal of ∼ 100 mV at the RX. When the TC is run, the TX is removed to the level of noise, which is nominally ∼ 10 µ V thermal for our band, equating to roughly 80 dB of rejection in power. \nSteering of the beam was demonstrated to work as expected by varying the relative phase of the transmitter antennas and observing the changing amplitude in the receiver strings. Fig. 5 shows the arrival time difference between pairs of surface system triggers for a period during commissioning when N=3. The straight-line distance between stations \nFIG. 5. The time difference in trigger arrival times between pairs of surface systems (each line style/color represents a surface system pair), as recorded on the central radar DAQ. Direct line separation of approximately 70 m sets the maximum light crossing time. \n<!-- image --> \nis roughly 70 m, corresponding to a maximum light crossing time of approximately 230 ns. After debugging during commissioning, the data readout was also nominal for the data run, with readout taking ∼ 50 ms per event. \nAnother issue that became evident during the run was regular broadband radio frequency interference (RFI) that contaminated the data, visible in Fig. 4. We subsequently identified the source of this RFI to be the high current switching power supplies feeding the transmitter amplifiers, which were removed entirely for the subsequent 2024 run.', 'V. IMPROVEMENTS FOR 2024': 'The most critical improvement made after the 2023 run was a redesign of the thermal regulation system. Our design of this initial system was predicated upon a surface deployment, and it allowed us to shed heat sufficiently in such a scenario. However, on arrival at Summit we realized that the filtration system we had designed to keep snow out of the inlet/outlet ports would be insufficient for the local conditions: the inlet filter was ineffective against the finest snow, which could damage/destroy the electronics, and the outlet filter could clog/freeze without regular maintenance. We therefore elected to bury the system, which inhibited cooling, and ultimately caused the system to melt down into the snow. To mitigate this, the 2024 enclosure has an improved, passive thermal regulation system to meet the local conditions. The power amplifiers are bolted to the inside of a large heat sink that forms the lid of the revised electronics enclosure, which was deployed on a platform above the snow surface for 2024. \nOther improvements include reconnecting the surface system radio receivers for improved reconstructability of the primary cosmic rays. To achieve this, we doubled the capacity of the surface system power supply. The 2024 system also has two additional surface stations, for a total of 5, with the two additional stations at longer baselines. As deployed in 2023, 3 RX channels had the full front end chain, with a 4th channel connected, but not buried, and without any front end electronics. The 2024 system included an additional amplified, buried channel, opting to break the symmetry of the layout and deploy one RX at a longer baseline. Finally, the 2024 system has different power regulation to completely remove the main sources of local RFI. \nThe 2024 data season from May-August has just concluded. The improvements allowed for the system to operate all through the summer, capturing O (10 5 ) cosmic ray triggered events. Each event record in the 2024 system contains the data from 4 amplified channels (three with active carrier \n- [1] S. Prohira et al. , (2021), arXiv:2104.00459 [astro-ph.IM].\n- [2] M. Chiba et al. , SUSY07 (2007), arXiv:0710.4186v1.\n- [3] K. D. de Vries, K. Hanson, and T. Meures, Astropart. Phys. 60 , 25 (2015), arXiv:1312.4331 [astro-ph.HE].\n- [4] S. Prohira and D. Besson, Nucl. Instrum. Meth. A922 , 161 (2019), arXiv:1710.02883 [physics.ins-det].\n- [5] S. Prohira et al. , Phys. Rev. Lett. 124 , 091101 (2020), arXiv:1910.12830 [astro-ph.HE].\n- [6] P. Auger, Review of Modern Physics 11 , 288 (1939).\n- [7] S. De Kockere, K. D. de Vries, N. van Eijndhoven, and U. A. Latif, Phys. Rev. D 106 , 043023 (2022), arXiv:2202.09211 [astro-ph.HE]. \ncancellation), one unamplified monitoring channel, and 5 surface stations including the data from surface radio antennas and two scintillator panels per station. Analysis of these data, and details of the improved hardware and firmware, will be presented in forthcoming publications.', 'VI. CONCLUSIONS': 'Wehave presented the first deployment of RET-CR, and the initial performance of the instrument. RET-CR was deployed for the first time in May 2023, and took 9 days of data after a four week commissioning period. The instrument performed well overall during the data taking period, before going offline, though the data was contaminated by broadband RFI. All problems that were identified during the run were remedied in advance of the data run from May-August 2024, which has just concluded.', 'VII. ACKNOWLEDGMENTS': "We sincerely thank the support staff at Summit Station for their tremendous efforts in getting RET-CR into the ice safely and successfully. We also thank K. Hughes and the RNOG collaboration for their assistance. We thank M. Kauer, C. Wendt, D. Tosi, and the IceCube collaboration for providing our scintillator panels and associated technical support. We thank the CODALEMA experiment for providing our surface radio electronics. We recognize support from The National Science Foundation under grant numbers 2012980, 2012989, 2306424, and 2019597 and the Office of Polar Programs, the Flemish Foundation for Scientific Research FWOG085820N, the European Research Council under the European Unions Horizon 2020 research and innovation programme (grant agreement No 805486), the Belgian Funds for Scientific Research (FRS-FNRS), IOP, and the John D. and Catherine T. MacArthur Foundation. \n- [8] S. Prohira et al. , Phys. Rev. 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2024arXiv240905807F	Quantum angular moment transport schemes are an important avenue toward describing neutrino flavor mixing phenomena in dense astrophysical environments such as supernovae and merging neutron stars. Successful implementation will require new closure relations that go beyond those used in classical transport. In this paper we derive the first analytic expression for a quantum M1 closure valid in the limit of small flavor coherence based on the maximum entropy principle. We verify that the resulting closure relation has the appropriate limits and characteristic speeds in the diffusive and freestreaming regimes. We then use this new closure in a moment linear stability analysis to search for fast flavor instabilities in a binary neutron star merger simulation and find better results as compared with previously designed ad hoc semiclassical closures.	2024-09-01T00:00:00Z	['arXiv:2409.05807', '10.48550/arXiv.2409.05807', '2024arXiv240905807F']	['High Energy Physics - Phenomenology', 'Astrophysics - High Energy Astrophysical Phenomena']	Quantum maximum entropy closure for small flavor coherence	2,024	199	0.37	['EPRINT_HTML', 'EPRINT_PDF']	2	https://arxiv.org/pdf/2409.05807.pdf	{'Quantum Maximum Entropy Closure for Small Flavor Coherence': 'Julien Froustey ID , 1,2, ∗ James P. Kneller ID , 2 and Gail C. 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Raffelt, Fast flavor conversions at the edge of instability, arXiv:2403.12189 [hep-ph] (2024).", 'Quantum Fermi-Dirac Maximum Entropy Closure': "We derive here the fermionic version of our quantum maximum entropy closure, first derived in the classical case in [13]. \nTaking into account the fermionic nature of neutrinos, the functional (4) becomes: \nS [ ϱ ] = ∫ 1 -1 d µ Tr[ ϱ ln( ϱ ) + ( I -ϱ ) ln( I -ϱ )] -〈 α, ∫ 1 -1 d µϱ -N 2 πϱ 0 〉 -〈 Z, ∫ 1 -1 d µµϱ -F z 2 πϱ 0 〉 , (S.1) \nwhere ϱ 0 is a normalization factor, needed to ensure that ϱ aa is a distribution function with 0 ≤ ϱ aa ≤ 1. The additional term compared to the Maxwell-Boltzmann limit (4) gives a contribution identical to (11) with ϱ aa → 1 -ϱ aa , that is, \nTr[( I -ϱ ) ln( I -ϱ )] = (1 -ϱ ee ) ln(1 -ϱ ee ) + (1 -ϱ xx ) ln(1 -ϱ xx ) -ln(1 -ϱ ee ) -ln(1 -ϱ xx ) ϱ ee -ϱ xx \| ϱ ex \| 2 + · · · (S.2) \nFor completeness, we quote the same formulae, but written for any number of flavors: † \nTr[ ϱ ln( ϱ )] = ∑ a ϱ aa ln( ϱ aa ) + 1 2 ∑ a = b ln( ϱ aa ) -ln( ϱ bb ) ϱ aa -ϱ bb \| ϱ ab \| 2 + · · · , (S.3) \n̸ \nTr[( I -ϱ ) ln( I -ϱ )] = ∑ a (1 -ϱ aa ) ln(1 -ϱ aa ) -1 2 ∑ a = b ln(1 -ϱ aa ) -ln(1 -ϱ bb ) ϱ aa -ϱ bb \| ϱ ab \| 2 + · · · . (S.4) \n̸ \nFlavor-diagonal elements Maximizing (S.1) over ϱ aa leads at leading order to: \nln( ϱ aa ) -ln(1 -ϱ aa ) -α aa ϱ aa -Z aa µϱ aa = 0 , (S.5) \nsuch that \nThe constraints read: \n2 π ∫ 1 -1 d µ [ 1 µ ] ϱ 0 e -α aa e -Z aa µ +1 = [ N aa F z aa ] . (S.7) \nThey were historically numerically inverted or considered in a Pad'e approximation, before Cernohorsky and Bludman [14] found an approximate closed-form for the Eddington factor P zz aa /N aa (see also [9, 10]). \nϱ aa 1 -ϱ aa = e α aa e Z aa µ , or equivalently ϱ aa = 1 e -α aa e -Z aa µ +1 . (S.6) \nFlavor off-diagonal elements We now maximize (S.1) with respect to Re( ϱ ex ) and Im( ϱ ex ). The result is straightforward given the Maxwell-Boltzmann derivation in the main text, as the term in S [ ϱ ] involving ϱ ex that changes because of Fermi-Dirac statistics reads: \nS [ ϱ ] ⊃ ln ( ϱ ee 1 -ϱ ee ) -ln ( ϱ xx 1 -ϱ xx ) ϱ ee -ϱ xx \| ϱ ex \| 2 , (S.8) \nsuch that Eq. (13) becomes: \nϱ ex ( µ ) = ϱ ee -ϱ xx ln ( ϱ ee 1 -ϱ ee ) -ln ( ϱ xx 1 -ϱ xx ) ( α ex + µZ ex ) . (S.9) \nTherefore, Eqs. (15)-(17) keep the same form, where the I ( n ) integrals are modified via ϱ aa → ϱ aa / (1 -ϱ aa ) inside the logarithms. Notably, the angular dependence of the denominator of I ( n ) remains linear in µ , just like in the Maxwell-Boltzmann case.", 'Closure Parameters in the Free-Streaming Regime': 'In this section, we provide a derivation of the limits of the closure coefficients κ and η when the flux factors for ν e and ν x approach 1. \nWhen f ee → 1, Z ee → + ∞ according to Eq. (3), such that ϱ ee ( µ ) ∼ ( N ee / 2 π ) Z ee e Z ee ( µ -1) . This is a function that, for Z ee → + ∞ , is non-zero only in a small neighborhood of µ = 1. The same arguments apply to ϱ xx . Recalling that \nI ( n ) = 2 π ∫ 1 -1 d µµ n ϱ ee -ϱ xx ln( ϱ ee ) -ln( ϱ xx ) ≡ 2 π ∫ 1 -1 d µµ n h ( µ ) , (S.10) \nwe distinguish different cases. \n̸ \n̸ \nFirst, if ϱ ee ( µ ) = ϱ xx ( µ ), since the limit of x ↦→ ( x -1) / ln( x ) for x → 1 is 1, then h ( µ ) = ϱ ee ( µ ). Otherwise, we have N ee = N xx and/or Z ee = Z xx , such that the denominator of h ( µ ) reads, in the f aa → 1 limit, ln( N ee /N xx ) + ( Z ee -Z xx ) µ , which is a slowly varying function of µ compared to the exponentials in the numerator. Since the support of I ( n ) is very localized at µ ≲ 1, we can take the denominator as constant. Assuming, without loss of generality, that Z xx > Z ee ≫ 1, we have \nϱ ee -ϱ xx = ϱ ee ( 1 -ϱ xx ϱ ee ) ∼ ϱ ee , (S.11) \nas ∀ µ < 1, ϱ xx /ϱ ee ∼ ( N ee Z ee /N xx Z xx ) e -( Z xx -Z ee )(1 -µ ) ≪ 1. \nConsequently, in any case, I ( n ) is given, in the free-streaming limit, by I ( n ) = C × J ( n ) (1 /Z ee ), with C a constant independent of n , and \nJ ( n ) ( ε ) ≡ ∫ 1 -1 d µµ n 1 /ε 2 sinh(1 /ε ) e µ/ε . (S.12) \nIn particular, this means that the limits of κ and η , defined by Eq. (17), are given by: \nlim f ee ,f xx → 1 κ = lim ε → 0 J 2 (2) -J (1) J (3) J (0) J (2) -J 2 (1) , lim f ee ,f xx → 1 η = lim ε → 0 J (0) J (3) -J (1) J (2) J (0) J (2) -J 2 (1) . (S.13) \nWe have, for Z ee →∞ and thus ε → 0, \nJ (0) = 1 , J (2) ∼ 1 -2 ε +2 ε 2 , J (1) ∼ 1 -ε , J (3) ∼ 1 -3 ε +6 ε 2 -6 ε 3 . (S.14) \nAs a consequence, \nJ (0) J (2) -J 2 (1) ∼ ε 2 , J 2 (2) -J (1) J (3) ∼ -ε 2 , J (0) J (3) -J (1) J (2) ∼ 2 ε 2 . (S.15) \nInserting those asymptotic equivalents in Eq. (S.13), we proved that κ →-1 and η → 2. If the flux factors tend to -1, the support of J ( n ) is around -1 and there is an extra minus sign in front of all the odd J ( n ) , such that η →-2.', 'Numerical Study of λ ( ± ) ex': 'In order to verify that the additional characteristic speeds (23) satisfy the causality requirements \| λ ( ± ) ex \| ≤ 1, we perform a numerical scan over the parameter space that can be spanned by the Minerbo distributions entering Eq. (17). First, assuming without loss of generality that N ee ≥ N xx , we rewrite the I ( n ) integrals as: \nI ( n ) = N xx 2 ∫ 1 -1 d µµ n ˜ N Z ee sinh( Z ee ) e Z ee µ -Z xx sinh( Z xx ) e Z xx µ ln [ ˜ N Z ee sinh( Z xx ) Z xx sinh( Z ee ) ] +( Z ee -Z xx ) µ , (S.16) \nwith ˜ N = N ee /N xx . Since ratios of I ( n ) appear in the functions κ and η , these quantities (and the related characteristic speeds λ ( ± ) ex ) are functions of three parameters: ˜ N ∈ [1 , + ∞ ), Z ee ∈ R , and Z xx ∈ R (note that a negative Z aa corresponds to F z aa ≤ 0). We plot in Fig. A the maximum quantum characteristic speed, scanning over the parameter ranges. When \| Z aa \| ≫ 1, meaning that ν a is free-streaming, the maximum characteristic speed goes to 1, as expected by causality requirements [51]. When both \| Z ee \| , \| Z xx \| ≪ 1, this is the isotropic limit for which η → 0 and κ → 1 / 3, so that \| λ ( ± ) ex \| → 1 / √ 3 ≃ 0 . 577. \nFIG. A. Maximum λ ex characteristic speed across the range of values of { Z ee , Z xx } , for three relative number densities ˜ N ∈ { 1 . 01 , 10 , 100 } . \n<!-- image --> \nWe thus explicitly show that characteristic speeds are everywhere physical, an important requirement for a closure to be used in large-scale hydrodynamic simulations. \nAs an additional illustration, we show on Fig. B the largest quantum characteristic speed at each point in the transverse slice of the 5 ms post-merger snapshot from [58], studied in the main text (see Fig. 1). As proved above, the characteristic speeds are always below unity, approaching the isotropic limit in the center of the slice ( λ ( ± ) ex → ± 1 / √ 3 ≃ 0 . 577). The characteristic speeds do not get significantly close to 1 in this snapshot, as the simulation box is rather close to the remnant. \nFIG. B. Largest characteristic speed \| λ ( ± ) ex \| in a transverse slice of the 5 ms post-merger snapshot from the NSM simulation [58]. \n<!-- image -->'}
2024PhRvD.110f4058W	In the EinsteinMaxwell theory with nonlinear electrodynamics NED fields the singularity problem in general relativity is potentially resolved leading to regular black hole solutions. In NED theories photons follow null geodesics of an effective geometry that differs from the spacetime geometry itself. This raises an important question Do NED fields produce unique observational signatures in the electromagnetic spectrum that can test regular black holes and NED theories We analyze the shadows of two NEDcharged regular black holes and their horizonless ultracompact objects HUCOs under two accretion models comparing them with Schwarzschild black holes focusing on shadow size central brightness depression and photon ring characteristics. Our results identify distinctive NED signatures that could be observable by the EHT providing empirical evidence of NED fields and potentially ruling out previously considered viable candidates for astrophysical black holes models based on shadow measurements. Notably NEDcharged HUCOs generally exhibit only one unstable photon ring thus avoiding the dynamical instability associated with stable photon rings and challenging the idea that objects with photon rings must be black holes.	2024-09-01T00:00:00Z	['10.1103/PhysRevD.110.064058', '2024arXiv240913290W', '10.48550/arXiv.2409.13290', 'arXiv:2409.13290', '2024PhRvD.110f4058W']	['General relativity', 'alternative theories of gravity', 'General Relativity and Quantum Cosmology']	Exploring nonlinear electrodynamics theories Shadows of regular black holes and horizonless ultracompact objects	2,024	199	0.32	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	2	https://arxiv.org/pdf/2409.13290.pdf	{'Exploring Nonlinear Electrodynamics Theories: Shadows of Regular Black Holes and Horizonless Ultra-Compact Objects': 'Rahul Kumar Walia ∗ \nDepartment of Physics, University of Arizona, 1118 E 4th Street, Tucson, AZ, USA (Dated: September 23, 2024) \nIn the Einstein-Maxwell theory with nonlinear electrodynamics (NED) fields, the singularity problem in general relativity is potentially resolved, leading to regular black hole solutions. In NED theories, photons follow null geodesics of an e ff ective geometry that di ff ers from the spacetime geometry itself. This raises an important question: Do NED fields produce unique observational signatures in the electromagnetic spectrum that can test regular black holes and NED theories? We analyze the shadows of two NED-charged regular black holes and their horizonless ultracompact objects (HUCOs) under two accretion models, comparing them with Schwarzschild black holes, focusing on shadow size, central brightness depression, and photon ring characteristics. Our results identify distinctive NED signatures that could be observable by the EHT, providing empirical evidence of NED fields and potentially ruling out previously considered viable candidates for astrophysical black holes models based on shadow measurements. Notably, NED-charged HUCOs generally exhibit only one unstable photon ring, thus avoiding the dynamical instability associated with stable photon rings and challenging the idea that objects with photon rings must be black holes.', 'I. INTRODUCTION': "The presence of curvature singularities and geodesic incompleteness at the center of black holes remains one of the most intriguing and longstanding problems in general relativity (GR). Resolving this issue has been a primary motivation for various developments in both classical and quantum gravity. While reasonable classical modified gravity theories have failed to provide vacuum singularity-free asymptotically flat black hole solutions in four dimensions, various quantum gravity theories have successfully predicted such models, notable examples include loop quantum gravity [1, 2], asymptotic safe gravity [3, 4], string theory [5, 6], and noncommutative gravity [7]. However, in this conventional top-down approach to quantum gravity, regular black holes do not usually appear as solutions of the gravitational field equations, but rather as ad hoc models based on quantum gravity arguments. The introduction of the first model of a regular black hole by Bardeen in 1968 [8] prompted the bottom-up approach to derive the regular black hole solutions within the framework of GR, before resorting to the quantum gravity. This approach utilized classical fields to violate certain energy conditions or bypass some other assumptions of singularity theorems in order to obtain regular black hole solutions within GR. \nNonlinear electrodynamics (NED) fields were anticipated as one such possible solution to spacetime singularities [9]. This approach stems from the observation that, in classical Maxwell electrodynamics theory, the electric field and selfenergy of a point-like charged particle diverge at its location, and this infinity could be resolved by nonlinear modifications of Maxwell's theory in strong-field regimes. In the same spirit, one might expect that substituting the Maxwell field with a suitable NED field in Einstein-Maxwell theory could replace the singular Reissner-Nordstrom (RN) black hole so-l \ntion with a regular counterpart. It turns out that this is indeed the case: NED fields not only resolve field singularities in electrodynamics but also, under certain reasonable conditions [10, 11], regularize spacetime singularities in both black holes and early universe cosmology. Ay'on-Beato and Garc'ıa [12] derived the first regular black hole solution from the EinsteinMaxwell field equations sourced by a NED field from an electric charge, and later showed that the Bardeen regular black hole model can also be explained as an exact solution of the field equations with a NED field from a magnetic monopole charge [13]. \nBuilding upon the foundational works of Born-Infeld [14] and Euler-Heisenberg [15] on NED field models, several static and spherically symmetric regular black hole solutions within various well-motivated NED theories are now known [1621]. Some interesting and important no-go theorems for the existences of regular black holes with the NED fields have also been formulated [10, 11, 22-24]. For a recent review and progress on the topic see [25-29] and references therein. \nNED theories exhibit a long list of model-independent interesting predictions, including: (i) regularization of electromagnetic fields; (ii) vacuum polarization of virtual e -e + pair; (ii) pair production; (iii) photon-photon scattering; (iii) birefringence, wherein an electromagnetic wave splits into two normal modes with mutually orthogonal linear polarization propagating at distinct velocities (Born-Infeld NED model is an exception for the birefringence [30]); (iv) photons propagating along the null geodesics of an 'e ff ective metric' modified by the NED field instead of the background metric; (v) significant enhancement of light polarization; (vi) bending of light purely due to the NED field in vacuum. Since these e ff ects are absent in the Einstein-Maxwell theory, they o ff er exciting opportunities to explore distinct observational signatures of NED charged regular black holes. \nOf course, it is still unclear whether ordinary astrophysical conditions allow for the existence of such black holes. Electrically charged black holes can, in principle, be produced in gravitational collapse [31], but they are normally expected to be rapidly neutralized by the abundance of free charges in \nastrophysical plasma. Formation scenarios for black holes with magnetic monopole charges are not available at present and are objectively hard to justify. However, hereafter, we will simply assume that such a black hole has somehow been produced (eternal or primordial black hole sourced by GUT predicted monopole), and we will explore the consequences of this assumption. In particular, we examine photon geodesics influenced by the e ff ective metric of these black holes, analyze the resulting shadows, and propose further tests of NED fields in astrophysical contexts. Specifically, we address the following key questions: (i) How do the shadows of NED charged regular black holes, as determined from the e ff ective metric, di ff er from those dictated by the background metric? (ii) What are the similarities and di ff erences between the shadow of a NED charged regular black hole and that of a singular black hole, such as a Schwarzschild black hole? (iii) Can a regular horizonless ultracompact object (HUCO) mimic a regular black hole as far as its shadow size is concerned within the current Event Horizon Telescope (EHT) angular resolutions? (iv) Do the EHT's shadow size measurements of M87* and Sgr A* support the predictions of regular black holes from the e ff ective metric or the background metric? (v) Could the EHT observations of M87* and Sgr A* black holes be attributed to regular HUCO rather than black holes? (vi) How does the interior geometry of regular black holes and HUCOs influence their shadows? \nThese questions are important from both the perspective of EHT shadow observations and for understanding the NED field e ff ects in curved spacetimes. Our approach to address these questions is rather simple. We consider two static and spherically symmetric NED charged regular spacetimes: Bardeen [8] and the Ghosh-Culetu (GC) [32, 33] models. Both metrics smoothly interpolate between regular black holes ( k ≤ k E) and regular HUCO spacetimes ( k > k E) depending on the value of the regularization parameter k that enters the metric; values of k E are given in Secs. VI A and VI B.To distinguish two cases, we use Bardeen-BH and GC-BH to represent black hole spacetimes, and BardeenHUCO and GC-HUCO for regular HUCO spacetimes. While both Bardeen and GC spacetimes are globally regular and can be attributed to distinct NED fields originating from magnetic monopole charges k , they are not particularly unique among other known regular black hole models. However, they exhibit significant di ff erences in interior geometry and in the weak-field limits, representing two distinct classes of regular black holes, thus making them suitable candidates for addressing previously raised inquiries. Additionally, we consider two simple astrophysical scenarios: a spherically symmetric and radially infalling Bondi-Michel flow, and a Novikov-Thorne thin disk. Note this sets up an interesting challenge for developing a numerical ray-tracing technique involving two spacetime metrics: (i) the accreting matter follows timelike geodesics of the background metric; (ii) the emitted photons follow null geodesics of the e ff ective metric; (iii) the observations are conducted within the background metric. Comparing the shadows of two black holes and their regular HUCO counterparts, both internally and against that of the Schwarzschild black hole under identical astrophysical \nconditions, we address the posed questions. \nThe paper is structured as follows: We begin by providing an executive summary of the main results in Sec. II, and addressing the motivation for NED fields for regular black holes, along with their current experimental and observational evidence in Sec. III. Because the EHT shadow size bounds ruled out the Bardeen-BH and Bardeen-HUCO based on the shadows deduced from the e ff ective metric, in the rest of the paper we focus on the GC spacetimes shadows. In Sec. IV, we derive the e ff ective metric for photon geodesics. Section V covers the setup for the black hole shadow and two accretion models. We present the shadows of GC-BH and GC-HUCO, using the two accretion scenarios in Sec. VI. Some general results for the NED fields in regular black hole spacetime are discussed in Sec. VII. Finally, Sec. VIII summarizes our main results. We present the shadows of the GC-BH and GC-HUCO deduced from the background metric in the Appendix A and those of Schwarzschild black hole under identical accretion models in the Appendix B.", 'II. EXECUTIVE SUMMARY': 'Here, we highlight the important findings of this paper. These results have significant implications for both EHT theory and observation, o ff ering novel perspectives on NEDcharged regular black holes. \n- 1. The analytical expression for the shadow radius of a general static and spherically symmetric magnetically charged black hole in NED theory is presented in Eq. (35).\n- 2. Regular HUCOs exhibit null circular orbits from the background metric only within a limited parameter range of k , whereas those from the e ff ective metric exist for arbitrarily large values of k (cf. Figs. 3 and 4).\n- 3. While the background metric is singularity-free and geodesically complete, the e ff ective metrics for Bardeen and GC spacetimes exhibit curvature singularities that are observable only by photons (see Secs. V A and V B). These singularities become particularly relevant for HUCOs, where they influence the resulting shadows. Refer to Sec. VII for detail discussion on general magnetically charged NED spacetimes.\n- 4. HUCOs cast shadows with a central brightness depression and shadow size comparable to that of a black hole (see Sec. V B).\n- 5. The e ff ects of magnetic monopole charge are more evident in the shadows derived from the e ff ective metric than in those predicted by the background metric. Bardeen black holes can be ruled out by the EHT shadow size measurements of the Sgr A* and M87* black holes (see Sec. III).\n- 6. Absence of stable photon rings renders NED charged regular HUCOs as viable alternatives to astrophysical black holes.', 'III. EXPERIMENTAL AND OBSERVATIONAL INTERESTS FOR NED': "Although initially purely theoretical, interest in NED models has grown over time, as evidenced by reports [28, 34]. The main obstacle in obtaining direct evidence of the NED field is its exceedingly high Schwinger limit, E cric ∼ 10 18 V / m and B cric ∼ 10 13 G. Despite this challenge, recent advancement in experimental facilities o ff er promising opportunities to reach the Schwinger limit. For instance, laser facilities like Extreme Light Infrastructure (ELI) [35] are actively searching for evidence of photon-photon scattering and e + e -pair creation processes. The X-ray Free Electron Lasers (XFEL) [36] facility is dedicated to investigating the nonlinear response of matter to extremely high-intensity electromagnetic fields and has recently observed multi-photon nonlinear Compton scattering and second harmonic generation at x-ray wavelengths for the first time. The pair production process, such as the Breit-Wheeler process 2 γ = e + + e -, conceptually modifies the linearity of Maxwell's theory and naturally necessitates a transition to a NED theory. Thus, observing the BreitWheeler process would experimentally confirm the NED theory. The LUXE experiment [37] proposed at DESY aims to investigate nonperturbative e ff ects of QED and measure the pair production rate from the QED vacuum through collisions between a 16.5 GeV electron beam and a 40 TW optical laser. Relativistic electrons in LUXE experience electric fields approaching or exceeding the Schwinger limit in their frame of reference compared to the actual laser electric field in the laboratory frame of reference. Another approach to probe the NED scale is through relativistic heavy-ion collisions Z ≥ 1 α , where ions scatter quasielastically with an impact parameter larger than the sum of their radii. The recent direct experimental detection of γγ → γγ scattering in ultrarelativistic Lead ions collisions in the ATLAS detector at the LHC stands as one of the most cleanest and strongest confirmations of the nonlinear extension of the Maxwell's electrodynamics theory [38, 39]. Electromagnetic vacuum birefringence and dichroism have been searched for in a number of experiments, one of which is PVLAS [40]. Although the experimentally measured value did not match the predicted value from QED, constraints could be placed on the Euler-Heisenberg NED parameter. At B = 2 . 5 T ∼ 10 -10 B cric, the di ff erence in the refractive index for two di ff erent light polarization mode is tiny, ∆ n ∼ α B 2 / B 2 c ∼ 10 -23 , which can be significantly increased by the strong background magnetic field [40]. \nAs experimental facilities continue to improve to achieve the Schwinger limit, some astrophysical environments such as neutron stars, magnetars and black holes naturally o ff er electromagnetic fields exceeding this limit. Magnetars, in particular, have been spotted with magnetic fields stronger than 10 13 G. The Imaging X-ray Polarimetry Explorer (IXPE) [41] is one of the space observational facilities designed to measure light polarization from black holes, neutron stars, pulsars, and galactic centers. In May 2022, IXPE detected linearly polarized x-ray emission from the pulsar 4U 0142 + 61, with an estimated dipole magnetic field of 10 14 G [42], marking the first-ever detection of x-rays polarization in a magnetar \nsource. The observed polarization values in two competing polarization modes lend support for the vacuum birefringence and, consequently, the NED theories. Additionally, vacuum birefringence has been confirmed through optical-polarimetry measurements of the isolated neutron star RX J1856.5-3754, showing a polarization degree of 16.43 ± 5.26 % [43]. A recent addition to the light polarization space observational facility is X-ray Polarimeter Satellite (XPoSat), launched in 2024, by ISRO. \nHowever, these existing data does not favor any specific NED model, necessitating to gather diverse observational signatures of the NED field. Of particular importance is the calculation of potential observables within black hole spacetimes, given the theoretical significance of NED fields in mitigating curvature singularities. \nNED charged regular black holes, considering their e ff ects solely within the background metric, have already been tested through various observations including EHT and LIGO / Virgo observations [44-50], and with X-ray data from the Cygnus X-1 [51]. However, only recently have the e ff ects of the e ff ective geometry on observational features been considered [52-56], led by Stuchl'ık and Schee [57]. Given that EHT observations directly involve the observation of light photons from synchrotron emission, accounting for the e ff ective metric ˜ g µν becomes imperative to calculate accurate predictions for the shadows. \nThe EHT, with an observing frequency of 230 GHz, captured total-intensity and polarized images of the horizonscaled radio emission regions of the supermassive black holes, Sgr A* [58-61] and M87* [62-64]. For testing the observational predictions of NED theories and our black hole models, and to address the queries raised in the introduction section, our focus lies solely on the observed angular sizes of Sgr A* and M87* shadows. The bounds, within 1 σ uncertainty region, on the shadow angular diameters of the M87* and the Sgr A* black holes, respectively, are translated to bounds on the shadow radius as follows [58, 65] \n4 . 31 M ≤ R sh ≤ 6 . 08 M (1) \n4 . 54 M ≤ R sh ≤ 5 . 22 M . (2) \nFigure 1 illustrates the sizes of black hole shadows for Bardeen and GC spacetimes as a function of NED charge k , inferred from photons following the null geodesics of their e ff ective metrics. Few comments are in order. Contrary to expectations, in Bardeen spacetime, R sh does not converge to the Schwarzschild value even in the vanishing charge limit as k → 0. A relevant question arises: Is this discontinuous limit at k → 0 specific to only the Bardeen spacetime? No, this holds true for a class of magnetically charged NED black holes, as we shall see in the following Section VII. GC spacetime is one such example, where the shadow size predicted by the e ff ective metric has a smooth Schwarzschild limit as k → 0. The shadow sizes R sh for both Bardeen and GC spacetimes decreases slowly with k , reaches a minimum value, and thereafter monotonically increases with k , indicating that now HUCOs also cast shadows. The smallest shadow appears for a HUCO with k = 0 . 97 M and R sh = 6 . 18519 M for Bardeen spacetime, and k = 1 . 4856 M and R sh = 3 . 3061 M \nFIG. 1. The shadow radii of Bardeen spacetime (red curve) and GC spacetime (blue curve) predicted by the e ff ective metric are shown. The darker green and lighter green regions indicate the EHT bounds for the Sgr A* and M87* black hole shadow radii, respectively. Bardeen black hole and HUCO can be ruled out solely from the EHT shadow-size measurement. \n<!-- image --> \nfor GC spacetime. Shadow radii R sh of Bardeen-BHs and Bardeen-HUCOs fall outside of the EHT bounds (cf. Fig. 1): Shadows of Bardeen-BHs and Bardeen-HUCOs, as predicted by the e ff ective metric, fail to meet the shadow size bounds set by Sgr A* and M87. On the other hand, the shadow sizes of GC-BHs within 0 ≤ k ≤ 0 . 986 M and 0 ≤ k ≤ 1 . 119 M are consistent, respectively, with the Sgr A and M87* shadow sizes within 1 σ bounds. Additionally, GC-HUCOs within 2 . 059 M ≤ k ≤ 2 . 257 M and 1 . 987 M ≤ k ≤ 2 . 476 M are also consistent, respectively, with the observed shadows of Sgr A* and M87. This results in a degeneracy between GC-BHs and GC-HUCOs, where a shadow of a given size could correspond to either a black hole or a HUCO. Furthermore, the observed shadows of M87 and Sgr A* can be explained not only by a GC-BH but also by a GC-HUCO. Thus, further observables are required to distinguish a black hole from a HUCO, such as photon rings as discussed in this paper. In summary, as predicted by the null geodesics of the e ff ective metric, unlike the Bardeen model, where both black holes and the HUCOs were ruled out by the EHT shadow size observations, GCBHs and GC-HUCOs satisfy the observed shadow size. This finding is one of the important results of this work. \nIn light of the conclusive evidence against Bardeen spacetime provided by the EHT shadow size measurement, our focus in the subsequent sections of this paper will shift exclusively to the examination of the GC spacetime shadows as derived from the e ff ective metric.", 'IV. EFFECTIVE SPACETIME GEOMETRY FOR PHOTON PROPAGATION IN NED FIELDS': "The main object of our interest in this paper is the e ff ective metric, so we briefly outline the formulation of the e ff ective metric ˜ g µν for photon propagation in the NED charged regular \nblack hole spacetime. We start with the Einstein-Hilbert action minimally coupled with a generic NED field featuring a smoothly varying Lagrangian density L ( F ) [13]: \nS = c 4 16 π G Z d 4 x √ -g ( R - L ( F )) , (3) \nwhere R is the Ricci scalar. The choice of L ( F ) is guided by the specific NED model phenomenology under investigation. One might expect that under weak electromagnetic fields, the action would reduce to that of Einstein-Maxwell theory. However, as we will see later, there exist interesting NED models that defy these assumptions. Likewise Maxwell field, the NED field L ( F ) respects U (1) gauge invariance and Lorentz symmetry group, making it a function of the invariant F = F µν F µν , where F µν is the Faraday tensor associated with the four-potential A µ through F µν = ∂ [ µ A ν ]. L ( F ) is a nonlinear function of the single Lorentz invariant F = 2( ⃗ E 2 -⃗ B 2 ); ⃗ E and ⃗ B are the radial electric and magnetic fields. On varying the action (3) with the metric field tensor g µν and gauge field A µ , the Einstein-Maxwell field equations read [13] \nG µν = T µν ≡ 2 GLYPH<16> L ' F α µ F να -1 4 g µν L GLYPH<17> , (4) \n∇ µ GLYPH<16> L ' F µν GLYPH<17> = 0 , (5) \n∇ µ GLYPH<16> ∗ F µν GLYPH<17> = 0 , (6) \nwhere L ' : = ∂ L ( F ) /∂ F , L '' : = ∂ 2 L ( F ) /∂ F 2 and ∗ F µν is the Faraday dual tensor. \nContrary to the Einstein-Maxwell theory, where the action is quadratic and the field equations are linear in F µν , NED theories exhibit nonlinear behavior, as both the action (3) and the field equations (4) and (6) are nonlinear functions of F µν . The NED field stress tensor is diagonal but not trace-free and violates the strong energy condition \nT µ ν = -diag h 1 2 L , 1 2 L , 1 2 L-FL ' , 1 2 L-FL ' i . (7) \nEquation (5) characterizes the nonlinear interaction between electromagnetic fields and Eq. (6) is the consequence of the Bianchi identity. \nBecause we are interested in static and spherically symmetric black hole spacetimes, we opt for a general metric ansatz in Schwarzschild coordinates { x µ } = ( t , r , θ, ϕ ) \nds 2 = g µν dx µ dx ν = -A ( r ) dt 2 + B ( r ) dr 2 + C ( r ) d Ω 2 2 , (8) \nwhere d Ω 2 2 = ( d θ 2 + sin 2 θ d ϕ 2 ) is metric on unit 2-sphere, the Faraday tensor is F µν = 2 δ θ [ µ δ ϕ ν ] k sin θ , F = 2 k C 2 ( r ) , and k is the magnetic-monopole charge. In spherically symmetric spacetime, electromagnetic fields exhibit two primary configurations: radial electric fields and radial magnetic fields arising from charged monopoles. Here, we focus only to the magnetic monopole fields. Making a suitable choice of NED Lagrangian density L ( F ), one can solve the field equations above to get the metric functions in (8). Bronnikov [10] demonstrated that simply requiring finite values for the electromagnetic fields ( ⃗ E , ⃗ B ) and the self-energy of a charged \npoint particle in NED models is not enough to produce regular black hole solutions from the gravitational field equations; and that additional conditions are required. For instance, although the Born-Infeld NED model [14] regularizes the ⃗ E field and self-energy of a test charged particle, it fails to produce regular black hole solutions when coupled with gravity. Intriguingly, regular black holes naturally result from Eqs. (4) and (6), if in the weak-field limit, L ( F ) → 0 as F → 0, and in the strongfield limit, L ( F ) → finite as F → ∞ [10, 11, 23]. \nUsing Hadamard's approach of light propagation [66] where the electromagnetic field is continuous, but its derivatives are discontinuous across the light wavefront - Novello et al. [67, 68] have shown that photons no longer propagate along the null geodesics of a fixed background geometry g µν ; instead, they follow the null geodesics of an 'e ff ective' Riemannian geometry ˜ g µν modified by the NED e ff ects. This results in the fact that the photon four-momentum, p µ , is a geodesic vector satisfying the following equations [67-71] \np µ ∇ µ p ν = 0 , (9) \ng µν p µ p ν , 0 , rather ˜ g µν p µ p ν = 0 . (10) \nThe e ff ective metric ˜ g µν for photons depends on the background metric g µν and the NED field L ( F ), such that 1 \n˜ g µν = L ' g µν -4 L '' F µα F ν α , (11) \nwhere \n˜ g µν ˜ g µσ = δ σ ν . (12) \nThis e ff ective metric for photon propagation in NED backgrounds is an example of nonlinear medium induced modification on light propagation which is generic for any nonlinear field theory and known for a quite long time now [67, 72-75]. \nFor the background geometry g µν given in Eq. (8), the e ff ective metric ˜ g µν reads [67, 70] \n˜ ds 2 = ˜ g µν dx ν dx ν = -A ( r ) L ' dt 2 + B ( r ) L ' dr 2 + C ( r ) Φ d Ω 2 2 , (13) \nwith \nΦ : = L ' + 2 FL '' . (14) \nThe e ff ective metric (13) mirrors the static and spherically symmetric nature of the background metric (8). Note that the e ff ective metric ˜ g µν is relevant only for photon motion; other uncharged, massless (or massive) particles remain una ff ected by the nonlinearity of the NED field and adhere to the null (or timelike) geodesics of the background metric g µν . Additionally, under a linear Maxwell's field ( L ( F ) ∼ F ), photons also traverse along the null geodesics of the background \nmetric, indicating that the conformal modification outlined in Eq. (13) does not alter their trajectory. Furthermore, contrary to expectations, the metric ˜ g µν does not necessarily match the g µν metric in the larger limit. This will be emphasized in Section (VI), where we discuss two black hole models with di ff erent weak-field limits of ˜ g µν . \nIn summary, in the presence of NED fields, the spacetime geometry seen by the photons is more complex than the geometry seen by other relativistic particles yielding some unexpected and wide-ranging consequences. \nNotably, Bardeen metric is also treated as quantumcorrected Schwarzschild spacetime [76, 77], with parameter k controlling the quantum correction. Now, instead of a NED field, stringy e ff ects create a de Sitter core at the origin, stabilizing the matter configuration against collapse. This o ff ers an alternative explanation for the most of regular black holes, wherein photons continue to follow the null geodesics of the background metric. However, due to lack of a welldefined action principle, these alternative explanations are not widely accepted. Nevertheless, considering the EinsteinNED theory resulting in regular black holes, our focus is to determine the qualitative e ff ects of the NED fields on black hole's observational features . For this, in the rest of the paper, we compare the shadows predicted by the null geodesics of the e ff ective metric ˜ g µν , presented in section VI B, with those predicted by the null geodesics of the background metric g µν , presented in appendix A. This comparative analysis serves as the focal point of our investigation, shedding light on the distinct outcomes stemming from the inclusion of NED fields within the black hole spacetimes.", 'V. BLACK HOLE SHADOW': 'Isometries of the e ff ective metric (13) enable the formulation of the photon geodesic equations; the four-velocity of photon reads as follows: \n{ ˙ x µ } = n E L \' A ( r ) , ± L \' B ( r ) s B ( r ) A ( r ) E 2 -( K + L 2 ) B ( r ) Φ C ( r ) L \' , ± Φ C ( r ) p K L 2 cot 2 θ, L Φ C ( r ) sin 2 θ o , (15) \nwhere ˙ x µ = dx µ / d τ , and τ is the a ffi ne parameter along the geodesics. E and L represent the energy and angular momentum of a photon, respectively, while K stands for the Carter constant. For photons confined to geodesics along the equatorial plane θ = π/ 2, K = 0. With this, the radial motion equation takes the following simpler form at θ = π/ 2 \n-˜ gtt ˜ grr ˙ r 2 + V = E 2 , (16) \nsuch that the radial potential V for the null geodesics of the e ff ective metric \nV = -˜ gtt ˜ g ϕϕ L 2 = Φ A ( r ) L \' C ( r ) L 2 , (17) \nand for the null geodesics of the background metric, it simplifies as \nV = A ( r ) C ( r ) L 2 . (18) \nThe radial potential vanishes at the black hole horizon A ( r ) = 0 and at Φ ( r ) = 0. Radial motion is possible only when ˙ r 2 ≥ 0 and any turning point r = r tp in trajectory can be obtained by solving ˙ r = 0 as a function of the impact parameter b that is defined as L / E [see Eqs. (15) and (18)] \nb = L E = s ˜ g ϕϕ -˜ gtt GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = r tp = s L \' C ( r ) Φ A ( r ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = r tp . (19) \nWe will focus on the unstable circular photon orbits, which correspond to the local extrema of the radial potential. The radius r p of a circular orbit is determined by simultaneously solving the following equations for r \np r = 0 , dp r dr = 0 , (20) \nwhich can be cast in a single equation as follows \nd dr GLYPH<16> ˜ gtt ˜ g ϕϕ GLYPH<17> = 0 , (21) \nor equivalently \nH ( r ) : = A \' ( r ) A ( r ) -C \' ( r ) C ( r ) ! -GLYPH<16> L \'\' L \' -Φ \' Φ GLYPH<17> = 0 , (22) \nsuch that H ( r = r p) = 0 gives circular photon orbit radius r p . Equation (22) for Maxwell electrodynamics reduces to the simpler form \nA \' ( r ) C ( r ) -A ( r ) C \' ( r ) = 0 . (23) \nReal positive roots of Eq. (22) and (23) give radii of null circular orbits from the e ff ective metric and the background metric, respectively. The critical value of the impact parameter, b cr, is related with r p as following \nb cr = s ˜ g ϕϕ -˜ gtt GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = r p = s L \' C ( r ) Φ A ( r ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = r p . (24) \nIn other words, as the impact parameter reaches its critical value b → b cr, the turning radius approaches the null unstable circular orbit radius, r tp → r p [cf. Eqs. (19) and (24)]. Determining the shadow of a black hole requires careful use of both the ˜ g µν and g µν metrics: light rays forming the shadow boundary follow ˜ g µν metric, whereas the observer receives them in the metric g µν . To illustrate this, let us consider a static observer located at ( ro , θ o ) shooting light rays into the past toward the black hole, as shown schematically in Fig. 2. Light rays emanating within a cone with semiopening angle Ψ sh, the angle between a light ray tangent and the radial direction, at the observer\'s position falls into the black hole, forming the \ndark region on the image plane-\' shadow \' [78]. Whereas, following the past-oriented light rays which asymptotically approach the unstable circular photon orbit with radius rp , one can determine the shadow boundary curve, known as the \'critical curve\' or \'light ring\' that is characterized by the angular radius Ψ sh. \nTo locate the shadow boundary, we project the fourmomentum p µ of these light rays onto the observer\'s tetrad frame defined by bases e ( ρ ) determined from g µν . These bases satisfy \ng µν e µ ( ρ ) e ν ( σ ) = η ( ρ )( σ ) , (25) \nwhere η ( a )( b ) is the Minkowski metric in the observer\'s local frame. The basis vectors are defined such that e ( t ) is timelike, while e ( r ), e ( θ ) and e ( ϕ ) are spacelike and mutually orthonormal. Consider in the observer\'s frame, a light ray with locally measured four-momentum components p ( a ) = e µ ( a ) p µ forms an angle X from the e ( r ) -e ( ϕ ) plane, while its projection onto the e ( r ) -e ( ϕ ) plane forms an angle Ψ from the e ( r ) direction, such that \np ( r ) = \| P \| cos X cos Ψ , p ( θ ) = \| P \| sin X , p ( ϕ ) = \| P \| cos X sin Ψ , \nwith \| P \| 2 ≡ \| p ( r ) \| 2 + \| p ( θ ) \| 2 + \| p ( ϕ ) \| 2 and \np ( a ) p ( a ) = -\| p ( t ) \| 2 + \| P \| 2 , (26) \nwhere ( X , Ψ ) defines the celestial coordinates in the observer frame. From the normalization condition \np ( a ) p ( a ) = η ( a )( b ) p ( a ) p ( b ) , = η ( a )( b ) e µ ( a ) e ν ( b ) p µ p ν, = g µν p µ p ν , 0 , (27) \ntherefore, in a local intertial frame of the observer, the norm of light four-momentum does not vanish, implying that the light does not propagate along the null geodesics in the observer\'s frame. \nWe establish an image plane centered at the black hole, and defined by Cartesian coordinate ( α, β ) \nα : = -r p ( ϕ ) p ( r ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> ( ro ,θ o ) , β : = r p ( θ ) p ( r ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> ( ro ,θ o ) . (28) \nThis gives a relation between image plane coordinates ( α, β ) and celestial coordinates ( X , Ψ ) as follows \nα = -ro tan Ψ , β = ro tan X sec Ψ , (29) \nsuch that a light ray starting with local four-momentum p ( a ) or in the direction ( X , Ψ ) intersects the image plane on a point with coordinate ( α, β ). Using Eqs. (26)-(29), \ntan Ψ = p ( ϕ ) p ( r ) = e ( ϕ ) ϕ p ϕ e ( r ) r p r = √ g ϕϕ √ grr d ϕ dr , (30) \nwhere d ϕ/ dr is determined from the e ff ective metric. Because the shadow boundary is circular for a static black hole, its radius on the image plane is defined as \nR sh = q α 2 + β 2 ; (31) \nand this is the only quantity characterizing the shadow. On the other hand, additional features appear in the shadow in the case of stationary black holes and these could be generically captured via a Fourier expansion of the polar curve describing the shadow (see, [79]). Furthermore, the light trajectories are planar, i.e. p ( θ ) = 0, this simplifies the shadow radius expression as \nR sh = ro tan Ψ sh . (32) \nFrom the null geodesic equations, we get \ndr d ϕ ! 2 = ˜ g ϕϕ ˜ grr ˜ g ϕϕ ˜ gtt 1 b 2 -1 ! , (33) \nand using this in the Eq. (30), yields the shadow angular radius Ψ sh from \ncot Ψ sh = √ grr √ g ϕϕ s ˜ g ϕϕ ˜ grr ˜ g ϕϕ ˜ gtt 1 b 2 cr -1 ! , (34) \nand the shadow radius as follows \nR sh = ro tan Ψ sh , = ro p g ϕϕ ( ro ) p grr ( ro ) " s ˜ g ϕϕ ( ro ) ˜ grr ( ro ) ˜ g ϕϕ ( ro ) ˜ gtt ( ro ) ˜ gtt ( r p) ˜ g ϕϕ ( r p) -1 !# -1 / 2 , (35) \nwhere we have used Eq. (24) to substitute b cr. Equation (35) gives shadow radius for arbitrary static spherically symmetric NEDblack hole, where photons follow e ff ective metric. Note, shadow radius is not the same as the impact parameter. In the Maxwell electrodynamics, ˜ g µν → g µν , and consequently the shadow radius simplifies to \nR sh = ro " g ϕϕ ( ro ) gtt ( ro ) gtt ( r p) g ϕϕ ( r p) -1 # -1 / 2 , (36) \nwhich, for an asymptotically flat spacetime and a far away observer ro ≫ M , further simplifies to \nR sh = s g ϕϕ ( r p) -gtt ( r p) = b cr . (37) \nOnly if the e ff ective metric is asymptotically flat, i.e., lim ro →∞ { ˜ gtt ( ro ) , ˜ grr ( ro ) , ˜ g ϕϕ ( ro ) } → {-1 , 1 , r 2 o sin 2 θ } , and the observer is in the asymptotically flat region, then shadow radius is \nR sh = s ˜ g ϕϕ ( r p) -˜ gtt ( r p) = s L \' C ( r ) Φ A ( r ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = r p = b cr . (38) \nFIG. 2. A schematic diagram for the black hole shadow. The observer, represented by an orange dot, is situated at a distance ro from the black hole. A light ray with an impact parameter b experiences a turning point at radial distance r tp and makes an angle Ψ on the observer\'s screen. See text in Section V for more details. \n<!-- image --> \nHowever, the e ff ective metric is not asymptotically flat in most of the non-Maxwellian magnetically charged NED spacetimes, and thus the shadow radius cannot be calculated from Eq. (38); even though it is sometime used for NED black holes [57]. Hereafter, we will use expression given in Eq. (35) to calculate the shadow size of black holes and HUCOs. \nTwo features of the spacetime are of particular importance: photon orbit radius r p and shadow radius R sh. While the shadow radius solely depends on the black hole parameters and observer\'s location, the overall intensity profile additionally depends on the accretion details. In this paper, we consider two very simple accretion models: a spherically symmetric radially infalling Bondi-Michel accretion and a Novikov-Thorne type optically and geometrically thin accretion disk on the equatorial plane. Although these accretion models are simplistic representations and do not fully capture the complexity of supermassive black hole accretion [59, 63], they su ffi ce to address the objectives outlined in the introduction. In particular, since we are interested in quantifying the e ff ects of a NED e ff ective metric on black hole shadows and comparing the images of regular black holes with those of the regular HUCOs, these two accretion models serve the purpose well. Additionally, these models are simple enough to be analytically tractable, allowing us to compute the intensity profile semi-analytically. Moreover, we argue that the major findings are independent of the choice of accretion physics.', 'A. Radially Infalling Spherical Accretion Model': 'For our first accretion model, we consider that the regular black holes or the regular HUCOs are surrounded by a spherically symmetric, optically thin, and isotropically radiating matter [80, 81]. This matter, lacking azimuthal angular momentum, is radially infalling and following timelike geodesics of g µν . For further insights into the interplay between the gravitational physics of black holes and the emission profile for spherically symmetric accretion in resulting shadows, refer to [82-84]. \nThe matter is emitting radiation, which follow null \ngeodesics of the metric ˜ g µν . The shift in photon\'s frequency from the point of emission re to the point of observation ro is defined by the redshift factor \nz : = ν o ν e = p ρ u ρ o p σ u σ e , (39) \nwhich includes both the gravitational and Doppler redshift from ν e → ν o . Notice that the asymptotic observer is static and moves along a timelike word-line of the background metric, i.e., g ρσ u ρ o u σ o = -1, with four-velocity u ρ o = ˙ x ρ = 1 √ -gtt ( r = ro ) δ ρ t . Similarly, the timelike unit normalization fixes the radial four-velocity of the emitter \nu σ e = -1 gtt , -s -1 -gtt gtt grr , 0 , 0 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = re . (40) \nUsing the photon geodesics equations and the normalization condition with respect to the metric ˜ g µν p µ p ν = 0, the redshift factor becomes \nz = s -1 gtt GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = ro -1 gtt -s -1 -gtt gttgrr pr pt -1 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = re , z ϵ = A ( re ) √ A ( ro ) " 1 -ϵ h (1 -A ( r ))(1 -b 2 Φ L \' A ( r ) C ( r ) ) i 1 / 2 # -1 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> r = re (41) \nwith \npr = ϵ pt s -1 ˜ g rr GLYPH<0> ˜ g tt + b 2 ˜ g ϕϕ GLYPH<1> , (42) \nand we set ϵ to either -1 or + 1, depending on whether the photon velocity is directed locally toward or away from the black hole, respectively. \nAn infinitesimal path length for photons along emission point to the observer is given by \nd ℓ = -p µ u µ e d λ = -s -1 gtt ( ro ) pt z ϵ p r dr . (43) \nAdditionally, we assume isotropic and monochromatic radiation emission in the rest frame of the accreting matter. While the matter may possess a finite absorption coe ffi cient, for the purposes of EHT observing frequencies, we consider an optically-transparent accretion flow with zero absorption. For emissivity j ( ν ), we consider frequency-independent, corresponding to emission close to the peak of the synchrotron emissivity, spherically symmetric profile which scales as j ( ν ) ∝ 1 / r 2 [80, 85]. The observed specific intensity I obs ν at the photon frequency ν o can be obtained by integrating the emissivity along the photon path Γ as follows [80, 86] \nI obs ν = Z Γ z 3 j ( ν e ) d ℓ. (44) \nIntegrating Eq. (44) over all the observed frequencies, we obtain the observed photon intensity [80] \nI obs ∝ Z Γ z 4 ϵ r 2 d ℓ ∝ -Z Γ z 3 ϵ r 2 s -1 gtt ( ro ) pt p r dr , (45) \nwhere for a far away observer -gtt ( ro ) = 1, and we transform an integral over the a ffi ne parameter, ℓ , to an integral over the radial coordinate r . \nFor b < b c, light rays are traced backward in time from the observer to the central object. Photons along these rays experience a monotonic redshift forward in time. Conversely, for b > b c, light rays are first traced backward from ro to a turning point r tp along their trajectory, where photons are redshifted, and then from r tp to re , where photons experience a blueshift, so that \nI obs = R ro r + z 3 + r 2 pt p r dr if b < b c -R r tp re z 3 -r 2 pt p r dr + R ro r tp z 3 + r 2 pt p r dr if b ≥ b c (46) \nOverall, the e ff ects of the NED field are most evident in the redshift factor given by Eq. (41) and in Eq. (42). Thereafter, we will use Eq. (46) to calculate the images of regular black holes and their corresponding HUCOs within the e ff ective metric. For the background metric, we simply use ˜ g µν → g µν .', 'B. Thin Disk Model': "In our second model of accretion, we adopt a NovikovThorne like geometry [87] for an optically thin accretion disk, neglecting absorptivity. The disk is made up of massive particles rotating along nearly Keplerian timelike circular orbits on the equatorial plane defined by the background metric. The emission region is assumed to be optically thin, allowing light to traverse it multiple times; otherwise, only a direct image of the disk is observable. We set the outer edge of the disk at a fixed radius of r out = 20 M . While horizons exist only for k ≤ k E, timelike circular orbits persist even for larger values k ≤ k TL c , where k TL c > k E. For k ≤ k TL c , disk inner edge is at the r in = r ISCO, where ISCO is the innermost stable circular orbit, whereas for k > k TL c , it is determined by solving d Ω o dr GLYPH<12> GLYPH<12> GLYPH<12> r = rd = 0, where Ω o is disk angular velocity defined as follows [88] \nΩ o = r -gtt , r g ϕϕ, r . (47) \nFor e ffi cient accretion to happen, Ω o should increase with decreasing r , therefore the inner edge is defined where Ω o is globally maximum. Thus disk inner edge is fixed as [88, 89] \nr in = r ISCO if k ≤ k TL c rd ; Ω ' o GLYPH<12> GLYPH<12> GLYPH<12> ( r = rd ) = 0 if k > k TL c . (48) \nKeplarian velocity Ω o monotonically increases as we radially move inward along the accretion disk. Consequently, matter \nlocated closer to the disk's center orbits faster than matter farther out. This di ff erential rotation induces tension within the disk, compelling matter in the inner regions to decelerate, thus leading to a loss of angular momentum in the outward direction and subsequent inward migration of matter toward lower orbits. However, if Ω o were to decrease with r , it would be impossible for angular momentum to be transported outward, making the accretion process impossible. \nNo emission is coming from r > 20 M and within r < r in. The flux of electromagnetic radiation emitted from a radial position re within the accretion disk follows the standard formula [87, 88]: \nF ( r ) = -˙ M 4 π p -g (3) Ω ' o ( Eo -Ω oLo ) 2 Z re r in ( Eo -Ω oLo ) L ' o dr , (49) \nwhere ˙ M is the mass accretion rate which is assumed to be time independent and fixed. Eo , Lo and Ω o are the energy, specific angular momentum and the orbital angular velocity of massive particles moving along the circular orbits in the accretion disk \nEo = -gtt p -gtt -Ω 2 o g ϕϕ , Lo = Ω o g ϕϕ p -gtt -Ω 2 o g ϕϕ . (50) \np -g (3) is the determinant of the induced 3 × 3 metric on the accretion disk plane. \nThe redshift factor is given in Eq. (39), but now the fourvelocity of the emitter is \nu σ e = u t e (1 , 0 , 0 , Ω o ) GLYPH<12> GLYPH<12> GLYPH<12> re , (51) \nand g µν u µ e u ν e = -1 fixes the normalization coe ffi cient as \nu t e = s -1 gtt + Ω 2 o g ϕϕ . \nRedshift factor reads [52, 53] \nz = 1 ˜ gtt GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> ro p -gtt -Ω 2 o g ϕϕ -gtt ˜ ggtt + b Ω o g ϕϕ ˜ g ϕϕ GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> re , (52) \nwhich in the background metric simplifies to [52, 90] \nz = p -gtt -Ω 2 o g ϕϕ (1 -b Ω o ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> re . (53) \nz determines how much the frequency of photons changes when they are emitted from a point re on the disk and then observed far away at a distance ro . The photon flux detected by the distant observer is quantified by \nI obs ( r ) = z 4 F ( r ) . (54) \nTo determine the observed flux at specific image screen coordinates ( α i , β i ), we start by solving the null geodesic equations derived from the metric ˜ g µν , with initial conditions fixed in terms of the coordinates ( α i , β i ). Backtracking this \nlight ray helps us determine the point of intersection or the emission point on the accretion disk. Then, we calculate the corresponding redshift to be used in Eq. (54). On the other hand, if for some other specific values of ( α i , β i ), null geodesics do not intersects the accretion disk, we assign a flux value of zero to those points on the image plane. We systematically vary the ( α i , β i ) values to cover the entire image plane, thus generating an intensity map of the accretion disk's images on the observer's image plane. For this, we consider that the observer is placed at the radial coordinate ro = 10 4 M , which corresponds e ff ectively to the asymptotic infinity. In the optically thin limit, each pixel represents the cumulative emission along an entire null geodesic", 'VI. REGULAR BLACK HOLE SPACETIMES': "It is important to declare here that we refer 'regular black hole' as one that is free from the curvature singularity, i.e., black holes with globally finite spacetime curvature, while being aware that curvature singularities might not be synonymous with geodesic incompleteness in general; see, for instance, [96, 97]. However, for spherically symmetric black holes with one shape function in four-dimensional GR, the finiteness of curvature invariants and the geodesics completeness are equivalent [98]. Nevertheless, the possibility of black holes with finite curvature scalars and extendable geodesics, or those with curvature singularity and inextendable geodesics, remains open. \nWhile regular black holes mimic the Schwarzschild metric in the exterior weak-gravitational field limit and are asymptotically flat, their interiors can manifest a variety of geometries near the core, for a in-depth exploration, refer to the reviews by Sebastiani and Zerbini [26] and CarballoRubio et al. [27]. Noteworthy examples include: (i) a deSitter interior geometry with an even number of horizons, (ii) a Minkowski core with an even number of horizons, (iii) a spacelike wormhole throat hidden inside a trapping horizon or a black-bounce core, and (iv) discrete spacetime structure or the presence of a fundamental length. Table I presents some static and spherically symmetric NED charged regular black holes, along with their key geometric features including interior geometry nature. Our selection of models is by no means complete and represent only a subset of NED regular spacetimes. \nFor the purposes of this study, we focus only on two particular models: Bardeen [8] and Ghosh-Culetu [32, 33]. The spacetime curvature remains globally finite for r ≥ 0, rendering both black holes geodesically complete and asymptotically flat. Both Bardeen and GC metrics are characterized by a charge k in addition to mass M , such that in the limit k → 0, both metrics converge to the Schwarzschild metric. Furthermore, within a limiting parameter space k < k E, these metrics depict a regular black hole with two distinct horizons r ± , where r + > r -; when k = k E, the horizons coincide r + = r -≡ r E, representing an extremal regular black hole with degenerate horizons at r E. In the parameter space k > k E, the horizon disappears, and the metrics describe the \nTABLE I. A nonexhaustive list of NED-charged regular black holes. Table columns include: (i) NED spacetimes, (ii) magnetic (M) or electric (E) monopole charges of NED, (iii) black hole's core geometry, (iv) ✓ for Maxwell weak-field limit of NED model or × otherwise, (v) ✓ for Schwarzschild limit as k → 0 for the e ff ective metric or otherwise × . \nexterior of regular HUCOs, which are similar to the Gravastar model of Mazur-Mottola [99]. Thus, Bardeen and GC metrics describe both black holes for k ≤ k E and the regular HUCOs spacetimes for k > k E. These black hole models are further examined in detail in the subsequent sections, exploring both the background metric and the e ff ective metric descriptions to address the questions raised in the introduction.", 'A. Bardeen Spacetime': "Solving field equations (4) and (6) for the following NED Lagrangian density [13] \nL ( F ) = 6 sk 2 GLYPH<16> p k 2 F / 2 1 + p k 2 F / 2 GLYPH<17> 5 / 2 , (55) \nwith F = 2 k 2 r 4 , we obtained the metric functions \nA ( r ) = 1 B ( r ) = 1 -2 M r GLYPH<16> r 2 r 2 + k 2 GLYPH<17> 3 / 2 , C ( r ) = r 2 , (56) \nwhich described both Bardeen-BH and Bardeen-HUCO for [8]. Here, s is a constant determined by the field equations to be s = k 2 M . Utilizing Eqs. (13) and (14), and substituting L ( F ) from Eq. (55) and the metric functions from Eq. (56), we obtained the e ff ective metric ˜ g µν for the Bardeen model. Some key properties of interest are listed as follow \n- · The Bardeen spacetime center exhibits a de-Sitter geometry with an e ff ective cosmological constant lim r → 0 T µ ν = \n6 M k 3 δ µ ν . \n- · The strong and dominant energy conditions are satisfied only within, respectively, r ≥ q 2 3 k and r ≤ 2 k , whereas the weak and null energy conditions are satisfied globally r ≥ 0.\n- · In the limit k → 0, while g µν recovers the Schwarzschild metric, ˜ g µν does not match.\n- · For r ≫ M , g µν does not reduce to the RN metric.\n- · For r ≫ M , ˜ g µν does not reduce to g µν . \nFIG. 3. The radial coordinates of key features in the Bardeen spacetime are depicted in this figure: the event (Cauchy) horizon by the black solid (dashed) curve, curvature singularity r sing by red solid curve, unstable (stable) null circular orbits from the background metric by the green solid (dashed) curve, and null (photon) circular orbits from the e ff ective metric by the blue solid curve. Three vertical orange dashed lines, from left to right, represent the extremal black hole case k = k E, extremal null circular orbits from the background metric case k = k c, and the minimum photon orbit radius case k = k p. While the e ff ective metric allows for at least one photon circular orbit for all values of k , null circular orbits are only possible for k ≤ 0 . 85865 M = k c in the background metric. The background metric is free of singularities, but a curvature singularity emerges in the e ff ective metric followed by photons. \n<!-- image --> \nVarious limits of the NED source and their repercussions on the g µν and ˜ g µν are further discussed in detail in Sec. VII. Some interesting geometrical features of the Bardeen spacetime are shown in Fig. 3. Both metrics, g µν and ˜ g µν , predict identical black hole horizons radii. Extremal black hole exists for k E = 0 . 7698 M with horizons at rE = 1 . 0405 M . The metric ˜ g µν is singular at L ' = 0 and Φ = 0. Here, L ' = 0 ( ⇒ r = 0) is merely a coordinate singularity, while Φ = 0 gives a true curvature singularity at \nr = 2 √ 3 k ≡ r sing . (57) \nIt is important to recall here that the Bardeen metric g µν is regular with globally finite curvature scalars, while this singularity manifests solely within the metric ˜ g µν seen only by photons. The singularity resides on the surface of a twosphere with radius r sing, which is always enclosed by the event horizon when k ≤ k E, but becomes globally naked for k > k E. Novello et. al. [68] have identified similar curvature singularity in the e ff ective metric of a electrically charged NED regular black hole. Curvature singularity in ˜ g µν metric is still a subtle issue being further discussed in detail in Sec. VII. For the present discussion, we consider that the range accessible to photon motion is within r ≥ r sing. \nNull circular orbits of the metric g µν are compared with those of the metric ˜ g µν followed by photons. Within the background metric, the Bardeen-BH spacetimes accommodate two distinct null circular orbits, with the outer one being unstable outside the event horizon and the inner one being stable between the event and Cauchy horizons. Even within Bardeen-HUCOs spacetimes ( k > k E), null circular orbits continue to exist (shown as green curves in Fig. 3), converging at r = 1 . 717966 M ≡ r c for k = 0 . 85865 M ≡ k c. For k > k c, no null circular orbits exist. The existence of null circular orbits in regular HUCOs spacetimes leads to some interesting observational e ff ects, as presented in Ref. [100]. HUCOs have recently grabbed significant attention and are now being actively investigated in various settings [101-105]. On the other hand, the e ff ective metric ˜ g µν supports multiple null circular orbits for photons. For k < 0 . 781099 M ≡ ˜ k c, three null circular orbits exist, whereas for k > ˜ k c, only one null circular orbit exist that is unstable under radial perturbations. Interestingly, ˜ g µν ensures that the HUCOs always possess at least one null (photon) circular orbit. This is not surprising; indeed, a su ffi ciently strong NED field even in vacuum can compel photons to traverse circular orbits [71]. The radial coordinate, r p, of the outermost photon circular orbit corresponding to the photon sphere initially decreases slowly with k , reaching a minimum value of r p = 2 . 32506 M at k = 1 . 07663 M ≡ k p, and then monotonically increases with k . The minimum radius of photon orbits can be determined by simultaneously solving H ( r ) = 0 and ∂ H /∂ k = 0 in Eq. (22). It is important to note that, for HUCOs, the photon orbits always remain outside the curvature singularity, i.e., r p > r sing (cf. Fig. 3). In summary, even though Bardeen-BH horizons exist only for k ≤ 0 . 7698 M = k E and null circular orbits for the background metric exist for k ≤ 0 . 85865 M = k c, timelike circular orbits for massive particles continue to exist in HUCO spacetime for even larger values of k , specifically, within k ≤ 0 . 95629 M = k TL c and atleast one null circular orbit for the e ff ective metric exists for all values of k . \nThe shadows size depicted in Fig. 1 for Bardeen-BH and Bardeen-HUCO spacetimes do not align with the size constraints of M87* and Sgr A. Consequently, we will not pursue further investigation into the ray-traced images of these spacetimes. Instead, our focus will be solely on discussing the GC-spacetime in detail. \nFIG. 4. The radial coordinates of key features in the GC spacetime are depicted in this figure: the event (Cauchy) horizon by the black solid (dashed) curve, curvature singularities r sing by red solid curves, unstable (stable) null circular orbit from the background metric by the green solid (dashed) curve, and null (photon) circular orbits from the e ff ective metric by the blue solid curve. Three vertical orange dashed lines, from left to right, represent the extremal black hole case k = k E, extremal null circular orbits of the background metric case k = k c, and the minimum photon orbit radius case k = k p. While the background metric is singularity-free, these curvature singularities appear only in the e ff ective metric followed by photons. \n<!-- image -->", 'B. GC Spacetime': "For GC model, we considered the following form of the NED field Lagrangian density: \nL ( F ) = F e -s GLYPH<16> k 2 F 2 GLYPH<17> 1 / 4 , (58) \nwhich upon solving the field equations (4) and (6), give the Ghosh-Culetu black hole metric functions [32, 33] \nA ( r ) = 1 B ( r ) = 1 -2 M r e -k 2 2 Mr , C ( r ) = r 2 . (59) \nHere, we defined s = k 2 M and F = 2 k 2 r 4 . From Eqs. (13) and (14), we obtained the e ff ective metric ˜ g µν for photon propagation in GC spacetime. Some notable features of interest for this model are listed as follow \n- · The GC spacetime center exhibits a Minkowski geometry such that lim r → 0 ( G µ ν = T µ ν ) = 0.\n- · All classical energy conditions are violated in deep core within 0 < r < k 2 / 8 M .\n- · In the limit k → 0, both the metrics g µν and ˜ g µν recover the Schwarzschild metric.\n- · For r ≫ M , g µν metric reduces to the RN metric.\n- · For r ≫ M , ˜ g µν metric reduces to g µν . \n<!-- image --> \n<!-- image --> \nFIG. 5. This figure illustrates shadows of GC-BH with k = 0 . 5 M and GC-HUCOs with k = 0 . 80 M and k = 2 . 0 M determined from the e ff ective metric under the radially infalling spherical accretion. \n<!-- image --> \n<!-- image --> \nFIG. 6. Photon orbits from the e ff ective metric are shown for GC-BH with k = 0 . 7 M (left) and GC-HUCO with k = 2 . 0 M (right). The black dashed circle, red solid circle, and black disk represent the photon sphere, curvature singularity, and black hole event horizon, respectively. \n<!-- image --> \nIn the limit r ≫ M , unlike the Bardeen background metric, the GC metric approaches the RN metric. This convergence was expected, as the NED source for the GC spacetime exactly satisfies a correspondence with the Maxwell linear electrodynamics in the weak-field regime. Even within the e ff ective metric ˜ g µν , the radii of black hole horizons remain unchanged. The GC-BH exhibits two distinct horizons r ± for k < 1 . 2130 M ≡ k E, with degenerate horizons at r -= r + = 0 . 73576 M ≡ r E for the extremal case k = k E. For L ' = 0 and Φ = 0, the e ff ective metric functions turn singular, respectively, at \nr = k 2 8 M ≡ r 2 , r = (7 ± √ 17) k 2 32 M ≡ r 1 , 3 , (60) \nwhich give three curvature singularities r 1 > r 2 > r 3 only for \nthe photons. These singularities are on the surface of twosphere with increasing radii r sing ∈ { r 1 , r 2 , r 3 } , scaling proportionally to k 2 / M , where curvature scalars diverge. While for k < k E, these singularities remain hidden behind the horizon, emerging as globally naked singularity for k > k E. The horizon radii, curvature singularities, and the null circular orbits of the background and the e ff ective metric are shown in Fig. 4. GC background spacetime exhibits two null circular orbits for k < 1 . 2645 M ≡ k c: an outer unstable orbit and an inner stable one, which merge together for k = k c at r = 1 . 1467 M ≡ r c. These null circular orbits within GC-HUCO spacetimes were first reported in Ref. [100]. Conversely, the metric ˜ g µν facilitates three null (photon) circular orbits when k < 1 . 2651 M ≡ ˜ k c, whereas for k > ˜ k c, only one such orbit exists. The outermost photon circular orbit that \nFIG. 7. A schematic illustrating the identification of photon rings around a spherically symmetric static black hole. The black disk at the center represents the black hole, while the brown thick vertical line depicts the edge-on view of the accretion disk at θ = π/ 2 viewed from an observer located to the right at θ o = π/ 2. Black dots mark the inner edge of the disk, defined as in Eq. (48). Photon rings are characterized by an order number -n , representing the number of half-orbits around the black hole. Orbits defining the inner diameter of the n = 0 image (green orbit), n = 1 ring (orange orbit), n = 2 ring (magenta orbit), and n = 3 ring (red orbit) are depicted. Black dashed circle represent the photon sphere around the black hole, \n<!-- image --> \nexists for all values of k is radially unstable, and its radius initially decreases with k , reaching a minimum value of r p = 0 . 99129 M at k = 1 . 356402 M ≡ k p, and then monotonically increases with k . This outermost photon orbit is always outside the curvature singularity, i.e., r p > r 1 , 2 , 3. (cf. Fig. 4). In numerical ray-tracing, we consider the photon motion only outside the outermost singularity, i.e, in the region limited to r > r 1. \nHere, we employed our accretion models to investigate the GC spacetime shadows predicted from the e ff ective metric. We first considered the optically thin, radiating, radially infalling spherical accretion flow, as elaborated in the Sec. V A. Shadows of GC-BHs and GC-HUCOs are shown in Fig. 5. Di ff erent colors correspond to di ff erent values of the observed intensity, and we used one color function for all shadow plots, where the greater (smaller) intensity means the brighter (darker) color. All observed intensities are normalized to one. With increasing k , photons redshift from the emission region to the observer decrease, and eventually, the central brightness depression in the image increases. This explains a brighter emission ring outside the shadow boundary for k = 2 . 5 in Fig. 5. The observed intensity at the image plane exhibits circular symmetry due to the spherical symmetry of the emission flow in the bulk. \nFor a GC-BH with k = 0 . 50 M , the shadow exhibits an expected intensity peak at the lensed position of the photon orbit, resulting in the shadow critical curve at α 2 + β 2 = R 2 sh on \nthe image plane. Within α 2 + β 2 ≤ R 2 sh , a prominent intensity depression represents the shadow feature. The observed intensity I obs peaks immediately outside the shadow boundary, diminishing gradually with radial distance from it. However, the shadow inner region within α 2 + β 2 < R 2 sh exhibits a finite intensity and is not entirely dark, as would be observed if the accreting flow were entirely behind the black hole. This arises because the accreting flow is also present along the lines of sight intersecting the surface of the black hole. A fraction of radiation within this region can escape to the distant observer, albeit highly redshifted, contributing to the non-zero intensity within shadow inner region. Interestingly, HUCOs, for e.g. GC-HUCO with k = 1 . 22 M and k = 2 . 50 M , also display a similar intensity depression at the screen center, casting black hole-like shadows. This occurs because for k > k E, light rays with b < b cr intersect the singularity at r = r 1 and do not experience any radial turning point outside of r > r 1. These light rays intersecting the singularity account for the dark patch at the image screen center. Similar features have been reported for the Janis-Newman-Winicour [106] naked singularity, where a naked curvature singularity appears at the 2-sphere boundary of radial coordinate r sing = 2 M γ , and a photon sphere form at r p = M γ (2 γ + 1); γ is the scalar charge. For 0 . 5 < γ ≤ 1, JNW naked singularity exhibit photon orbit outside the curvature singularity, thus casting a shadow reminiscent of a black hole [107-109]. In the framework of the electrically charged NED e ff ective metric, photons approaching the singularity undergo infinite blueshift, potentially destabilizing the spacetime [25, 68]. Conversely, singularities in the magnetically charged NED e ff ective metric have not been previously explored in detail. Interestingly, in this scenario, photons reaching the singularity at r = r 1 undergo only a finite blueshift in frequency. \nFor our second accretion model, we simulate images of thin equatorial disk accreting onto a GC spacetime as discussed in Sec. V B. Photon orbits around the GC-BH and GCHUCO are shown in Fig. 6. Identifying photon orbits with multiple intersections of the accretion disk is further useful for determining the gravitationally lensed image of an accretion disk. Let the thin accretion disk is along the vertical line passing through the black hole center, with the observer positioned far away from it to the right. When tracing light rays backward in time, from the observer to the central object, our focus is on identifying the point of intersection on the accretion disk to project it onto the observer's screen. In summary, light rays intersecting the θ = π/ 2 plane n + 1 times complete n half-orbits around the central object, generating the n th order image of the accretion disk on the observer's screen. n = 0 and n → ∞ , respectively, correspond to the direct image of the disk and the black hole's critical curve. As shown in Fig. 6, di ff erent colors distinguish orbits based on the number of intersections of the θ = π/ 2 plane or the number of the half-orbits around the central object. Light rays of the black color intersect the θ = π/ 2 plane only once without making any loop around the central object, directly imaging the front side of the accretion disk. Orange-colored light rays intersect the plane twice, completing one half-orbit, thus forming the first-order or primary image of the backside \nFigure 8 displays the resulting accretion disk images for \n<!-- image --> \nFIG. 8. Shadows of GC-BH (left) and GC-HUCO (right) cast by photons tracing null geodesics of the e ff ective metric ˜ g µν under an equatorial accretion disk model. Observed flux I obs is represented on a logarithmic scale. \n<!-- image --> \nof the accretion disk. Red-colored light rays intersect the θ = π/ 2 plane thrice, completing two half-orbits, resulting in the second-order image of the accretion disk, and so forth. Light rays coming with the critical value of impact parameter b = b cr asymptotically approach the photon sphere, shown by the black dashed circle. The optically thin accretion flow enables such multiple crossing of disk, such that, the direct image is generically accompanied by a series of higher-order images. However, only when the emission is non-spherical do these successive higher-order images manifest as a discrete series of concentric exponentially demagnified ' photon rings ,' labeled by their half-orbit number n , converging toward the 'critical curve' [110]. The higher the ring order, the less it depends on the accretion disk features and more on the black hole geometry. Nevertheless, these higher-order orbits help identify the photon ring structures discussed subsequently. Our convention for identifying the n = 0 , 1 , 2 , 3 photon rings around a black hole is illustrated in Fig. 7. For simplicity, we have chosen the observer to be at θ o = π/ 2, but a similar convention holds for all inclination angles. For k = 2 . 0 M , we traced the photon geodesics only outside the curvature singularity. \nTo generate the images of the accretion disk, we defined an screen with coordinates α ∈ [ -25 M , 25 M ] and β ∈ [ -15 M , 15 M ]. This screen is perpendicular to the line of sight of the observer to the black hole in bulk, positioned at ro = 5000 M , which, for practical purposes, can be treated as asymptotic infinity. The screen was further divided into 500 × 300 pixels ( α i , β i ). While the outer radius of the accretion disk remained fixed at r out = 20 M , the inner edge is at r in determined from Eq. (48). The observer inclination angle was set to θ o = 80 · for all accretion disk images. We traced null geodesics backward in time from each pixel ( α i , β i ) on the screen until they either: (i) intersected the horizon in the case of a black hole, or reached the singularity of the e ff ective metric, which is visible only to photons in HUCOs spacetimes; or (ii) approached the asymptotic source. Along these trajectories, we tracked the point of intersection re on the accretion disk, where r in ≤ re ≤ r out, to use in Eq. (49) for calculating the observed intensity. The resulting intensity was computed for all pixels ( α i , β i ) on the screen. \nGC-BH with k = 0 . 70 M and GC-HUCO with k = 2 . 0 M . While for GC-BH, photon rings are clearly visible and well within the emission ring, for GC-HUCO, photon rings are immensely thin and are close to the emission ring. The relative positions of the photon and emission rings depend on the inner and outer edges of the accretion disk. Nonetheless, the morphology of the direct and indirect images of the Keplerian disk closely resembles those associated with Schwarzschild black holes. Diameters and widths of the first three photon rings of the Sgr A black hole, as a GC-BH with k = 0 . 50 M , are calculated and summarized in Table III. Precise measurements of radiation emanating from the innermost regions of Keplerian accretion disks near the black hole horizon could facilitate the potential validation or elimination of direct NED e ff ects around supermassive black holes like Sgr A* and M87. \nTo further visualize the gravitational lensing e ff ect, we established a celestial sphere of radius rs = 30 M , centered on the GC spacetime, refer to Fig.9. The observer is situated within this sphere, positioned o ff -centered at a distance of ro = 15 M from the center. This celestial sphere is partitioned into four quadrants, each painted with a distinct color; along the observer's line of sight, the top-left, top-right, bottomleft and bottom-right quadrants, are, respectively, seen as red, green, yellow and blue. We further subdivided the sphere boundary into evenly spaced grids using latitude and longitude lines separated by π/ 18 degrees to aid in the interpretation of the distortion of the resulting images. In the absence of any object along the observer's line of sight, these latitude and longitude lines are nearly straight, resulting in colored grid of identical area. However, in the presence of a black hole or a HUCO, strong gravitational lensing causes multiple copies of a single grid to appear, each with decreasingly smaller area approaching the shadow boundary. To analyze these e ff ects, we partitioned the observer screen into 800 × 800 small pixels and traced the paths of light rays from each pixel, following the ˜ g µν metric, until they intersected either the celestial sphere or the horizon for k ≤ k E or the singularity for k > k E. Redshift e ff ects were disregarded, focusing solely on the spatial distortion of the resulting images. Figure 9 presents lensed images of GC-BH with k = 0 . 70 M and GC-HUCO with k = 2 . 0 M . Dark regions indicate photons captured by the \n<!-- image --> \nFIG. 9. Gravitational lensing of light as predicted by the GC e ff ective metric ˜ g µν for a black hole with k = 0 . 70 M (left figure) and a HUCO object with k = 2 . 0 M (right figure) located at the center of the celestial sphere. The celestial sphere has a radius of 30 M , and the observer is situated at a distance of 15 M from the center of the celestial sphere. The black lines denote lines of constant longitude and latitude, with the central black circular region representing the shadow. \n<!-- image --> \nblack hole or those plunging into the singularity of the HUCO, resulting in the observable shadow. The apparent angular size of the shadow appears larger in the observer's field of view, primarily due to geometric factors arising from the observer's proximity to the black hole. Strong gravitational lensing induced by the compact mass is evident in the warping of the grid lines, particularly pronounced around the shadow boundary. Additionally, the region of the celestial sphere located behind the observer undergoes lensing on the image plane. The lensing images of HUCOs closely resemble those of a black hole, di ff ering primarily in the size of shadow and the width of higher-order photon rings.", 'C. Observational Predictions of NED Spacetimes': "In Sec. IV, we presented the shadow radii of Bardeen and GC spacetimes derived from the e ff ective metric and placed constraints on k using EHT measurements of Sgr A and M87. In this section, we derive additional observables within two frameworks: the background metric, applicable to non-NED alternative theories of regular black holes, and the e ff ective metric, relevant to the NED theory of the same black hole. This allows for a comparative analysis of how di ff erent theoretical models of a given regular black hole influence observable phenomena. \nThe shadow sizes predicted by the null geodesics of the background metric for Bardeen and GC spacetimes are shown in Fig. 10. Firstly, the shadow sizes of both BardeenBHs and GC-BHs precisely match the Schwarzschild value \nR sh = 3 √ 3 M in the limit k → 0, decreasing further with k . The background metrics predict no shadow like features for regular HUCOs. Notably, for a fixed value of k , the e ff ective metrics predict larger radii of photon circular orbits and shadow size compared to that from the background metric (cf. Fig. 1 and Fig. 10). This discrepancy in the shadow size is also reported for electrically charged NED black holes [56]. The predicted R sh values for both Bardeen and GC spacetimes from the background metric are within the 1 σ bounds deduced by the EHT for Sgr A and M87* black holes. Specifically, derived constraints from bounds (2) on (Bardeen and GC) parameters are, respectively, ( k ≤ 0 . 7634 M , k ≤ 0 . 8311 M ) and ( k ≤ k E = 0 . 7698 M , k ≤ 0 . 948 M ). Similar constraints on parameter k , deduced from the background metric, were reported by the EHT Collaboration [60, 65] and independently in Refs [44-46]. Predictions from e ff ective metric, especially for Bardeen spacetime, fail to reconcile with EHT measurements, unlike GC spacetime. \nFurthermore, our numerical ray-tracing code allows us to track the properties of each individual photon rings: radii in the image plane and in bulk, impact parameters, number of half-orbits, and their corresponding fluxes. We modeled the Sgr A* black hole as Bardeen-BH and GC-BH with k = 0 . 50 M , and calculated the diameters and widths of the first three photon rings, both from the background metric and the e ff ective metric, using the accretion disk model with θ o = 80 o , and summarized the results in Table II and III. For the Bardeen-BH, the photon ring diameter from the e ff ective metric is about 25% larger than those derived from the background metric. The width, and subsequently the flux, \nFIG. 10. The shadow radii of Bardeen spacetime (red curve) and GC spacetime (blue curve) predicted by the background metric are shown. The darker green and lighter green regions indicate the EHT bounds for the Sgr A* and M87* black hole shadow radii, respectively. \n<!-- image --> \nTABLE II. We model Sgr A* as Bardeen-BH with k = 0 . 50 M . Average diameter and width of first three photon rings are shown. From the background metric, the diameter of n = 0 inner ring is d = 13 . 014 M (62 . 767 µ as) and critical curve is at 2 R sh = 9 . 920 M (47 . 84 µ as), whereas from the e ff ective metric, diameter of n = 0 inner ring is d = 14 . 48 M (69 . 842 µ as) and critical curve is at 2 R sh = 12 . 609 M (60 . 819 µ as). \nTABLE III. We model Sgr A* as GC-BH with k = 0 . 50 M . Average diameter d and width w of first three photon rings are shown. From the background metric, the diameter of n = 0 inner ring is d = 13 . 084 M (63 . 1053 µ as) and critical curve 2 R sh = 9 . 9463 M (47 . 972 µ as), whereas from the e ff ective metric, the diameter of n = 0 inner ring is d = 13 . 174 M (63 . 54 µ as) and critical curve 2 R sh = 10 . 087 M (48 . 65 µ as). \nof these rings exponentially falls with order number n , making them challenging to observe. For instance, the n = 3 ring itself is positioned already very close to the critical curve and has a width of the order of nano-arc-seconds. While the critical curve is completely independent of the accretion and emission profiles and remains unobservable, our best bet are n = 1 and n = 2 rings for the precision test of gravity [110, 111]. Although these rings have not been observed directly in the shadows of M87* and Sgr A* due to the limited angular resolution of the EHT, preliminary studies suggested that they would produce a clean interferometric signature in the black hole images with a space-based interferometer with higher angular resolutions and / or flux-sensitivities, such as the Black Hole Explorer (BHEX) [112-115]. The 230GHz interferometric signatures of first three photon rings, derived from the e ff ective metric, of diameters d and width w , as presented in Table II and III, are shown in Fig. 11 as a function of baseline length. The visibility amplitude is normalized to have a value of unity at zero baseline length. Increasing the baseline improves angular resolution, enabling the resolution of diameter and width of higher-order photon rings. The important feature of our interest is two distinct periodicities in visibility amplitude for each ring: smaller and larger periodicities, respectively, encode the ring diameter and its width information. Specifically, to resolve image features of size L , baselines of length greater than 1 / L are required. Therefore to resolve a diameter d , and both the diameter d and width w , of a photon ring, we require baseline u given as follows (i) and (ii), respectively [112] \n( i ) : 1 d ≪ u ≪ 1 w , ( ii ) : 1 d ≪ 1 w ≪ u . (61) \nThe ratio of the diameter to the width of a given photon ring is the same as the ratio of the two periodicities in the Fourier domain. While the diameter and width of the n = 1 photon ring can be resolved with a small baseline such as with ngEHT, resolving the width of the n = 2 ring necessitates an exceedingly large baseline as with BHEX. A larger baseline is required to resolve the photon ring of GC-BH compared to that of Bardeen-BH due to its narrower width. \nThe instability of null geodesics is better understood in terms of Lyapunov exponent γ [116], defined such that a nearly-bound photon starting at a radius δ r 0 close to one of the photon orbit ends-up at δ rn after making n -loops around the black hole \nδ rn = e γ n δ r 0 . (62) \nFor su ffi ciently higher-order photon rings, γ solely controls the relative width and essentially the flux between successive rings on the image plane and is an observable. Similar to the shadow boundary curve, γ is determined only by black hole parameters, completely independent of surrounding astrophysical processes. Therefore, a precise measurement of γ gives shadow size independent constrains on black holes' parameters. For instance, for the Schwarzschild black hole, γ = π , so each successive photon ring is e π ∼ 23 . 1 times fainter than the previous one [110, 112, 117]. Space-based \n¨ \n¨ \n¨ \n¨ \n<!-- image --> \n<!-- image --> \n¨ \n¨ \n¨ \n¨ \n¨ \n¨ \n<!-- image --> \n¨ \n¨ \n<!-- image --> \n<!-- image --> \nFIG. 11. Normalized visibility amplitude as a function of EHT baseline length at 1.3mm observing wavelength for n = 1 , 2 , 3 photon rings from the e ff ective metric of Bardeen-BH with k = 0 . 50 M (top panel) and GC-BH with k = 0 . 50 M (bottom panel) for the Sgr A* black hole. The diameter and width of photon rings are presented in the Table II and III. \n<!-- image --> \n@ \nD \n@ \nD \nFIG. 12. Lyapunov exponent for the null circular orbits of the background metric (dashed curve) and those from the e ff ective metric (solid curve) for Bardeen and GC spacetimes. \n<!-- image --> \ninterferometer is expected to measure the Lyapunov exponents of black holes [112]. Figure 12 depicts the Lyapunov exponent of the Bardeen-BHs and GC-BHs determined from the background metric and the e ff ective metric as a function of k . γ is smaller than the Schwarzschild value, resulting in photon rings that are more wider and brighter than those for the Schwarzschild black hole. However, the e ff ective metric predicts a non-monotonic behavior for γ , where HUCO with su ffi ciently large k values have γ > π , resulting in successive photon rings becoming thinner and fainter compared to those \n@ \nD \n@ \nD \nin black holes. Note a discontinuous limit of γ as k → 0 for Bardeen-BH ˜ g µν .", 'VII. DISCUSSION ON NED INDUCED EFFECTIVE SPACETIME': "Westart by investigating the NED field within a Minkowski vacuum background, akin to the analysis conducted for Bardeen and GC NED models but without black holes in background. Our objective is to identify the unique characteristics of the resulting e ff ective metric ˜ g µν for photon propagation in the absence of any external gravitational field. It turns out that the light propagation in Minkowski NED vacuum shows some profound and wide-ranging consequences due to the modification of the Minkowski geometry, e ff ectively simulating an external gravitational field: \n˜ g µν = L ' η µν -4 L '' F µα F ν , (63) \nα \nwhere η µν is the Minkowski metric. \nFor Bardeen model, using L ( F ) from Eq. (55) in (63), the NED vacuum e ff ective metric ˜ g µν reads \n˜ ds 2 = 4( k 2 + r 2 ) 7 / 2 s 15 kr 6 -dt 2 + dr 2 + 2( k 2 + r 2 ) r 2 3 r 2 -4 k 2 d Ω 2 2 ! , (64) \nwhich mimics an 'e ff ective' gravitational field with non-zero spacetime curvature guiding the photons trajectories. While the metric (64) does not replicate a black hole spacetime, i.e., lacking an event horizon, it does demonstrate a photon's unstable circular orbit at a radius of r p = q 2 3 (2 + √ 7) k , \n@ \nD \n@ \nD \n<!-- image --> \nFIG. 13. E ff ective metric ˜ g µν for NED fields even in purely Minkowski background, without black hole, lead to some interesting consequences. Both the Bardeen NED model (left figure) and the GC NED model (right figure) show that ˜ g µν exhibits a curvature singularity (red curve), a photon circular orbit (blue curve), and a shadow radius (green curve). \n<!-- image --> \nalongside a curvature singularity at r sing = 2 √ 3 k . Both the curvature singularity and the photon orbit disappear as k → 0. \nSimilarly, for GC NED model (58), the vacuum e ff ective metric ˜ g µν \n˜ ds 2 = 4 re ks / r 4 r -ks -dt 2 + dr 2 + 2 r 3 (4 r -ks ) 8 r 2 -7 krs + k 2 s 2 d Ω 2 2 ! , \nyields three curvature singularities located at r sing = { ks 4 , 1 16 (7 ± √ 17) ks } , along with one photon circular orbit at r p = 1 . 02469 ks . This illuminates intriguing aspects of solely NED fields and their impact on photon trajectories through ˜ g µν metric. \nIf we compare the ˜ g µν metric predictions of a NED field in vacuum from Eq. (63) and the same NED field with a black hole background from Eq. (11), the findings are surprising. In both scenarios, the metric ˜ g µν exhibits photon circular orbits, curvature singularities, and yields a finite photon-captured cross-section for all k values as shown in Fig. 13. Despite sharing the identical radial coordinates for the curvature singularities, the photon circular orbit and resulting shadow are notably smaller in NED vacuum compared to those in the presence of a black hole, particularly for k ≤ k E. However, as k increases significantly ( k ≫ k E), the photon circular orbit radius and shadow size in the presence of a black hole converge to their values in vacuum background. This convergence occurs because, at high charge values of k , the NED field dominates over gravitational e ff ects in the metric ˜ g µν , influencing photon trajectories and shadows primarily through the NED field rather than gravitational mass. In summary, the curvature singularities in the metric ˜ g µν for both Bardeen and GC models are attributed to their NED fields rather than black hole geometry. Furthermore, the circular photon orbits in Bardeen-HUCO and GC-HUCO spacetimes for k > k c are not due to non-trivial features of HUCOs gravity, but rather a consequence of the NED field. These e ff ects highlight the important impact of the NED field on photon propagation. Notably, in electrically charged NED vacuum spacetime, Novello et al. have demonstrated the formation of trapped surfaces by NED fields, resulting in an electromagnetic analogue of gravitational black holes [118], and the imposition of closed photon orbits [119]. \nFor Bardeen NED model (55), the weak-field expansion of Lagrangian density is as follows \nL ( F ) ≃ 3 × 2 3 / 4 M √ k F 5 / 4 -15 √ kM 2 3 / 4 F 7 / 4 + O ( F 11 / 4 ) , (65) \nwith some interesting limits \nlim r → 0 {L ( F ) , L ' ( F ) , Φ ( F ) } → ( 12 M k 3 , 0 , 0 ) \nlim k → 0 {L ( F ) , L ' ( F ) , Φ ( F ) } → ( 0 , 15 M 2 r , 45 M 4 r ) . \n, lim r →∞ {L ( F ) , L ' ( F ) , Φ ( F ) } → { 0 , 0 , 0 } , (66) \nand the e ff ective metric \nlim k → 0 ˜ ds 2 = 2 r 15 -1 -2 M r ! dt 2 , + 1 -2 M r ! -1 dr 2 + 2 r 2 3 d Ω 2 2 ! . (67) \nBecause the leading-order term of L ( F ) in Eq. (65) does not precisely match with the Maxwell's term, both the Bardeen metrics g µν and ˜ g µν fail to reproduce the RN metric in the weak electromagnetic field regime ( r ≫ k ). In addition, ˜ g µν , neither in the vacuum nor in presence of a black hole, converge to the Minkowski metric or the Schwarzschild metric, respectively, as the limit k → 0 approached. This deviation of ˜ g µν from GR reflects in the discontinuous limits of the black hole shadow size and the Lyapunov exponent as well, where R sh and γ fail to converge to the Schwarzschild black hole values as k → 0 (cf. Fig. 1 and 12). \nFor GC NED model, the weak-field expansion of Lagrangian density is \nL ( F ) ≃ F k 3 / 2 2 5 / 4 M F 5 / 4 + k 3 8 √ 2 M 2 F 3 / 2 + O ( F 2 ) (68) \nwith some interesting limits \nlim r → 0 {L ( F ) , L ' ( F ) , Φ ( F ) } → { 0 , 0 , 0 } , lim r →∞ {L ( F ) , L ' ( F ) , Φ ( F ) } → { 0 , 1 , 1 } , lim k → 0 {L ( F ) , L ' ( F ) , Φ ( F ) } → { 0 , 1 , 1 } , (69) \nand the e ff ective metric \nlim k → 0 ˜ ds 2 = -1 -2 M r ! dt 2 + 1 -2 M r ! -1 dr 2 , + r 2 d Ω 2 2 . (70) \nThe leading-order term of L ( F ) in Eq. (68) is exactly Maxwell, and thereby the GC metrics g µν and ˜ g µν converge, to the RN metric in the limit r ≫ k , to the Minkowski metric as r → ∞ , and to the Schwarzschild metric as k → 0. This limiting behavior of the ˜ g µν is apparent in the continuous limits of R sh and γ , where they converge to the Schwarzschild black hole values as k → 0. \nFor a general magnetically charged NED model, its e ff ects in the metric ˜ g µν directly appear through two terms: L ' and Φ . This means that we can predict certain features of the resulting metric ˜ g µν by analyzing L , L ' , and Φ , independent of the black hole metric g µν . In particular, lim r → 0 L / 2 gives the energy density at the black hole center, revealing the black hole's interior topology. As stated earlier, the de-Sitter and Minkowski natures of Bardeen and GC centers, respectively, are attributed to the lim r → 0 T µ ν values. Null and weak energy conditions hold if L ≥ 0 and L ' ≥ 0 (cf. Eq. (7)). We find that \n- · For a black hole featuring a de-Sitter center and asymptotic flatness, L is a definite positive, monotonically decreasing function of r without an extremum, ensuring L ' > 0 for r > 0 (e.g., Bardeen-BH).\n- · Conversely, for a black hole with a Minkowski core and asymptotic flatness, L vanishes as r → 0 and r → ∞ , remaining nonzero in between, implying the existence of at least one extremum at r > 0 where L ' = 0 (e.g., GC-BH).\n- · However, Φ always vanishes for magnetically charged NED black holes somewhere between the black hole center and the spatial infinity [25]. Notably, Φ = 0 results in a divergence in the angular part of the metric ˜ g µν (13).\n- · L ' = 0 or Φ = 0 generally occurs at distinct radii and introduce curvature singularities to the e ff ective metric ˜ g µν . \nAlthough at spacetime center, L ' , Φ → 0, curvature scalars from the e ff ective metric vanish. While, Bardeen ˜ g µν exhibits curvature singularity only from Φ = 0, GC exhibits from both L ' = 0 and Φ = 0. Our analysis suggests that the e ff ective metric of a magnetically charged NED regular black hole exhibits at least one curvature singularity, which depending on the charge might appear inside or outside the horizon. Similar singularities in the NED e ff ective metric for the electrically charged ABG black hole were reported in Ref. [68], where both the curvature scalar and the photon radial potential diverge. \nAt the singularities, photon geodesics equations take the following form \nlim L ' → 0 { ˙ x µ } = n 0 , 0 , ± Φ C ( r ) p K L 2 cot 2 θ, L Φ C ( r ) sin 2 θ o , lim Φ → 0 { ˙ x µ } = n E L ' A ( r ) , ± E L ' √ A ( r ) B ( r ) , 0 , 0 o . (71) \nThe geodesic equations are integrable across the Φ = 0 singular surface. Particularly for Bardeen e ff ective metric, ingoing null geodesics directed toward r = 0 smoothly traverse this curvature singularity, crossing it within a finite a ffi ne parameter and finite radial velocity. Once past the singularity, these photons do not encounter any radial turning point and approach the center r → 0 asymptotically in infinite a ffi ne time. However, across the L ' = 0 singularity, geodesic \nequations lose integrability; ˙ r 2 < 0 for r < r 2 where L ' ( r 2) = 0. Equations (16) and (17) illustrate that the singularity at r 2 acts like a potential wall for the photons, where ˙ r → 0 as r → r 2. Consequently, for BH-II, infalling photons from the exterior region r > r + , except those with zero angular momentum, cannot approach the center [10]. These photons, in principle, can either bounce back to the same spacetime or traverse to a distinct copy of the spacetime. Additionally, the ˜ g µν metric signature switches from ( -, + , + , + ) to ( -, + , -, -) within the outermost singularity, where the angular coordinates ( θ, ϕ ) become timelike, facilitating compact and closed orbits. From the geodesic equations (71), the angular velocity vanish when Φ = 0 (i.e., at the outermost singularity) even for photons with non-zero angular momentum L . Therefore, in HUCO spacetimes, a given photon with L > 0 have positive (negative) angular velocity outside (inside) the singularity, such that photons always cross the outermost singular surface orthogonally. The physical relevance of these curvature singularities in metric ˜ g µν and photon trajectories within it are not clear and require further investigation. The causal structure of the metric ˜ g µν might help to better understand this. However, addressing this issue is beyond the scope of this paper and will be explored in future work. Nevertheless, for black holes, these singularities are covered by the event horizon; thus, once photon trajectories intersect the event horizons, further tracing them to the singularity becomes irrelevant for black hole observations. However, for HUCOs, we exclusively consider photon motion only outside the outermost singularity. Similar assumptions were made in the papers [56, 120]. An interesting study on the photon geodesics in the metric ˜ g µν spacetime is presented in Ref. [25]. \nNow, let us examine a general spherically symmetric model of a single-invariant F dependent magnetically charged NED field L ( F ). In the weak-field ( k ≪ M ) regime, the Lagrangian density and its derivatives approximately take the following forms \nL ( F ) ≃ w F s + 1 + ...., L ' ≃ w ( s + 1) F s + ...., L '' ≃ ws ( s + 1) F s -1 + ..... (72) \nHere, we consider that the NED field lacks a correct Maxwell weak-field limit, i.e., s > 0, with w being a positive constant. Considering a two-parameter ( M , k ) background metric g µν , such that lim k → 0 g µν → g Schw µν , the e ff ective metric in weak-field reads as: \n˜ g µν = L ' g µν + 2 s r 2 δ µ θ δ ν θ + 1 sin 2 θ δ µ ϕ δ µ ϕ !! . (73) \nOne immediate consequence is that, in general, the metric ˜ g µν is not conformal to the metric g µν unless s = 0; otherwise, both the background and the e ff ective metrics would give identical null geodesics, resulting in identical shadows. In contrary to GC ( s = 0), the Bardeen ( s , 0) metric ˜ g µν is not conformal to the Schwarzschild metric, even in the limit k → 0, explaining the discontinuous limits of the BardeenBH shadow size and the Lyapunov exponent. This establishes \na general result: 'For spherically symmetric magnetically charged single Lorentz-invariant NED models L ( F ) with the correct Maxwell limit in the weak-field and a background metric g µν , the e ff ective metric ˜ g µν is conformal to the metric lim k → 0 g µν in the limit of vanishing magnetic charge.' Additionally, for such NED black holes, the lensing observables smoothly approach the Schwarzschild value as the magnetic charge tends to vanish. This result is independent of the black hole interior geometry and solely depends on the weak-field limit in r ≫ k . \nNevertheless, the ( t , r )-part of a generic ˜ g µν , for a magnetically charged NED field, is always conformal to the part of g µν . Consequently, the radial null geodesics of ˜ g µν and g µν are the same; radially moving photons follow the same null geodesics in both metrics. However, for ˙ θ , 0 and / or ˙ ϕ , 0, the null geodesics of the metric ˜ g µν are either timelike or spacelike trajectories of the metric g µν . From the null geodesics normalization Eq. (27), one gets \ng µν p µ p ν = -4 L '' L ' g θθ g ϕϕ ( F θϕ ) 2 ( g θθ p 2 θ + g ϕϕ p 2 ϕ ) . (74) \nThis rea ffi rms that a radially directed light ray, with p θ = p ϕ = 0, moves along the null directions of the background metric. Noticing that the background NED field is also radial, only photons propagating orthogonal to the NED field are subject to follow the e ff ective metric. Utilizing Eq. (74), we establish the causality conditions, ensuring that although light rays follow the null geodesics of the e ff ective metric, they remain causal curves of the background metric. \ng µν p µ p ν ≤ 0 ⇒ L '' L ' ≥ 0 . (75) \nIn the Bardeen spacetime, light follows timelike geodesics of the metric g µν within r ≥ √ 6 k and spacelike for r < √ 6 k . Conversely, in the GC spacetimes, light follows timelike geodesics of the metric g µν only within the range k 2 / 10 < r < k 2 / 8. Bronnikov [25] obtained similar results for a generically magnetically charged NED black hole, indicating a violation of causality near the black hole's center. Additionally, Ref. [56] demonstrated that photons follow spacelike geodesics of the background spacetime in the presence of an electrically charged NED field. The net e ff ect of the NED field's force on photons resembles that of gravity [71]. In this context, from the background metric perspective, photons move under a radially attractive force thus causing larger shadows than that for the null geodesics of the background metric [56, 71]. \nRegarding stability, while the NED charged regular black holes demonstrate stability against gravitational and electromagnetic perturbations [121], regular HUCOs lack conclusive stability assessment due to limited dynamics understanding. In the context of the background metric, these regular HUCOs have pair of null circular orbits for k < k c, degenerate orbits for k = k c, and no orbits for k > k c, where k c > k E (cf. Fig. 3). Indeed, it is now an established fact in GR that HUCOs spacetime, if su ffi ciently compact enough to develop null circular orbits, always accompanied orbits in pairs: a \nstable circular orbit along with an unstable orbit [122-124]. Notably, similar stable circular null orbit exists also for RN and regular black holes, however, being invariably located inside the event horizon, it does not raise concerns for black hole stability or observations. However, in HUCOs spacetimes, their presence has profound implications as slow decay of linearized fluctuations suggests potential nonlinear instability, potentially leading to collapse into a black hole [123, 124]. This leads to an interesting result that 'HUCOs with null (photon) circular orbits are black hole' [122, 123]. On the other hand, in the e ff ective metric, Bardeen-HUCO and GCHUCOexhibit an odd number of null (photon) circular orbits, specifically, three photon orbits for k E < k < ˜ k c, comprising two unstable orbits and one stable orbit, and only one unstable photon orbit for k > ˜ k c. For k E < k < ˜ k c, while the presence of a stable photon orbit might lead to nonlinear instability due to photons accumulation and subsequent back-reaction on the spacetime, a comprehensive stability understanding requires modeling the matter interaction with this photon orbit and calculating its reflectivity or absorption properties. This is certainly a feature worth investigating for NED charged spacetimes, but it is beyond the scope of this paper, and we will not consider the e ff ects of the stable photon orbits in the HUCOs spacetimes. Nevertheless, HUCOs with k > ˜ k c possess only one photon circular orbit, and that too unstable (cf. Fig. 3). This is in clear contrast with the GR predictions. Consequently, the HUCOs spacetime instabilities associated with stable photon orbits are absent in these e ff ective metric spacetimes. As a result, unlike in GR, where stable photon orbits always accompanied unstable orbits and yield spacetime instability on astrophysically short time scales and potentially destroy HUCOs that could otherwise be plausible candidates for astrophysical black holes, the e ff ective metric description suggests that regular NED charged HUCOs lack stable photon orbits (except for a small parameter space k E < k < ˜ k c beyond the extremal limit) and are thereby free from associated instability. Therefore, if we observe photon orbits around astrophysical HUCOs, then one possible alternative explanation to black holes is NED regular HUCOs. \nIn summary, these features of the NED fields and the distinctive behavior of photon geodesics indicate that the NED description of regular black holes and the e ff ective metric formalism for photon geodesics are not completely understood, and warrant further investigation into these topics.", 'VIII. SUMMARY': "NED fields significantly impact spacetime geometry, with important implications in astrophysics and cosmology. In black holes, NED fields, with a suitable Lagrangian-density L ( F ), can eliminate central curvature singularities, resulting in globally regular black hole geometries. Another relatively less explored consequence is that, in the presence of NED fields, photons do not follow the null geodesics of the background metric g µν but instead propagate along those determined by an 'e ff ective metric' ˜ g µν ( g µν, L ( F )). Notably, this e ff ective metric is specific to only photons, while the \ntrajectories of other particles remain governed by g µν . \nIn this paper, we focused on these two primary aspects of NED fields: singularity-free regular black holes and the e ff ective metric descriptions of photon propagation. We examined two well-studied NED charged regular black hole metrics, namely the Bardeen and Ghosh-Culetu (GC) models, both featuring two parameters-mass M and magnetic charge k . Although both metrics interpolate between regular black holes ( k ≤ k E) and regular HUCOs ( k > k E), they manifest considerable geometrical distinctions at their cores ( r ∼ 0) and in weak-field regimes ( r ≫ M ). The e ff ective metric ˜ g µν introduces intriguing e ff ects in observational features, ranging from strong gravitational lensing to changes in the emitted photon frequency and variations in the size and shape of shadows. We analyzed these e ff ects using two accretion models: spherically symmetric and radially infalling flow, and Novikov-Thorne type optically and geometrically thin disk. Additionally, to assess NED e ff ects, we compared null circular orbits and resulting shadows predicted by photons following the null geodesics of ˜ g µν with those predicted by the null geodesics of g µν . In both Bardeen and GC spacetimes, null circular orbits derived from the e ff ective metric have larger radii than those from the background metric. While null circular orbits from the e ff ective metric persist for arbitrarily large values of magnetic charge k , those from the background metric exist only within the parameter space k ≤ k c. \nWe found that, from the background metric, HUCOs shadows are markedly di ff erent from black holes shadows (see appendix A). Under the spherical accretion model, HUCOs shadows typically lack intensity depression at the image center, instead, the intensity grows toward the image center, resulting in 'full-moon' like shadows, unlike black hole shadows. However, HUCOs with null circular orbits exhibit an additional weak signature-a faint circular ring along with steady rise in central intensity. These di ff erences in black holes and HUCOs shadows were anticipated because, in the absence of a horizon, radially ingoing null geodesics with small impact parameters always have a turning point at r ∼ 0 that contribute to the intensity at the image center. \nConversely, the e ff ective metric predicted completely different shadows. The shadow sizes are larger than those predicted by the background metric, Additionally, the shadows of HUCOs displayed intensity depression at the image center resembling those of black holes. This di ff erence arises from the fact that, now in HUCOs spacetimes, the ingoing null geodesics intersect a singularity at r = r sing > 0 with no turning point outside it, resulting in a dark patch at the image center. This outcome holds true for all magnetically charged NED spacetimes with a de-Sitter geometry at the center. \nAccretion disk images have revealed additional e ff ects of NED field on gravitational lensing. In the limit of large r or small NED charge k ≪ M , the NED field for Bardeen spacetime is stronger than the Maxwell's field, unlike for GC spacetime, where it correctly matches with the Maxwell's field. This residual NED field in Bardeen spacetime resulted in stronger gravitational lensing of photons in larger regime, in comparison to that for GC spacetime. This subsequently reflected as bulging in accretion disk images around Bardeen- \nBHs, as shown in [52], but not around GC-BHs, as shown in 8. Additionally, images of HUCOs resemble those of black holes. Unlike the spherical accretion model, accretion disk images exhibit a sequence of discrete photon rings. For both Bardeen-BHs and GC-BHs, the diameter of photon rings from the e ff ective metric is larger than those rings of the same order from the background metric. \nWe assessed the impact of NED field e ff ects on photon propagation through the e ff ective metric ˜ g µν , determining shadow sizes and establishing new constraints on the NED charge parameter k for Bardeen and GC spacetimes using EHT measurements of the Sgr A* and M87* black holes. Although shadow sizes predicted by the background metric and the e ff ective metric both decrease with k for k < k E, the e ff ective metric predicts larger shadows than the background metric for the same charge k . This confirms that the gravitational lensing of null geodesics in the e ff ective metric is stronger than those in the background metric. BardeenBH shadow sizes from the background metric fall within EHT bounds, while those from the e ff ective metric exceed 1 σ bounds for all k . Therefore, with the assertion that the shadows are cast by photons following the ˜ g µν , Bardeen NED model can be completely ruled out for Sgr A* and M87* black hole candidates. In contrary, shadows of GCBHs from both metrics are consistent with the 1 σ bound. Interestingly, GC-HUCOs with 1 . 987 M ≤ k ≤ 2 . 476 M and 2 . 059 M ≤ k ≤ 2 . 257 M also cast shadows that are consistent with the EHT shadow size measurements for M87* and Sgr A, respectively. Indeed, a shadow of a given angular size can correspond to either a GC-BH or a GC-HUCO. \nIn GR, it is known that any HUCOs, if compact enough to posses null circular orbits, exhibit stable photon rings along with the unstable rings. These stable rings possess longlived perturbations, leading to nonlinear instability of these objects. This leads to a beautiful and very strong result in favor of astrophysical black holes over HUCOs 'objects with a light ring are black holes' [122, 125]. However, we have shown that NED charged regular HUCOs, except for a small parameter space beyond the extremal limit k E < k < ˜ k c, do not posses any stable null (photon) circular orbits, but only unstable null (photon) circular orbits. Consequently, the absence of stable photon rings renders NED charged regular black holes and HUCOs as viable alternatives to astrophysical black holes. In this case, 'objects with a light ring do not rule out the NED charged regular HUCOs'. However, a conclusive answer can be made after investigating the gravitational and electromagnetic field stability on model-by-model basis of NED spacetimes. \nBased on these results, we aim to further explore the full general-relativistic magnetohydrodynamic (GRMHD) simulation of plasma flow and the subsequent general-relativistic radiative-transfer (GRRT) images around NED charged regular black holes and HUCOs. The NED field is expected to significantly enhance synchrotron light polarization. Estimating the e ff ect of the NED field on GRMHD simulated images and light polarization is the focus of our future research. Furthermore, vacuum birefringence induced by the NED field leads to time delays in light signals polarized orthogonally, \nproviding an avenue to independently constrain NED models in astrophysical contexts. \nWhile this work was in preparation, some theoretical aspects of photon propagation in NED-charged regular black hole spacetimes were recently explored in Ref. [126].", 'ACKNOWLEDGMENTS': "RKW expresses gratitude to Luciano Rezzolla for the invitation to visit Goethe University, where this project was conceived, and for continuous suggestions and feedback on the project. RKW also acknowledges discussions with Rajibul Shaikh at various stages of the project, Prashant Kocherlakota for the invitation to visit BHI, Harvard University, and for insightful discussions on e ff ective metrics for photon propagation and their causal structure, Ramesh Narayan for insights into the NED field and to Sam Gralla for discussions on ray tracing. RKW's research is supported by the Fulbright-Nehru Postdoctoral Research Fellowship (award 2847 / FNPDR / 2022) from the United States-India Educational Foundation.", 'Appendix A: GC spacetime shadows from the background metric': "As discussed in the Sec. IV, regular black holes can be motivated from various theories, such as from the quantum gravity with a minimum length parameter or a classical Einstein's gravity with some non-NED source of finite energymomentum tensor. However, the Lagrangian formalism for both alternative theories is mostly missing. Within these alternate explanation of the regular black holes, photons continue to follow the null geodesics of the background metric g µν . For a comparative study, we determine the shadows of GC spacetimes, originating from the non-NED theory, within the background spacetime. \nFigure 14 illustrates the shadows of GC-BHs and GCHUCOs under spherical accretion. For GC-BH with k < k E, null geodesics lead to a strong intensity depression within α 2 + β 2 = b 2 cr , which is a signature of black hole shadow. Note that the shadow radius is di ff erent from that determined for the e ff ective metric, as explained previously. Conversely, for HUCOs with k > k E, the absence of a horizon allows accreting matter to exist up to r = 0, resulting in a rising intensity toward the observer's screen center, akin to a 'full moon-like' image, attributed to the repulsive radial potential near r → 0. Ingoing null geodesics experience a radial turning point r tp > 0, reflecting back toward the observer, contributing to image intensity. These new image features emerge because null geodesics can now pass through the spacetime region that was previously inside the horizon. Although radiation emitted from the close to r = 0 experiences high redshift, this is compensated by the high emission rate of radiation ( j ∝ 1 / r 2 ) from the accreting matter near r = 0, thereby contributing to the intensity. For further analysis, we looked at two cases: GC-HUCOs with ( k E < k < k c) and without ( k > k c) null \ncircular orbits. For k E < k < k c, ( k = 1 . 22 M in Fig. 14) we observed a secondary subdominant peak in the observed intensity, or a faint circular ring in the image, at the lensed position of null circular orbits, which disappeared for k > k c ( k = 2 . 50 M in Fig. 14). Consequently, as predicted by the null geodesics of the background metric, shadows of GC-HUCOs markedly di ff er from those of GC-BHs. This feature is in stark contrast with the behavior observed with null geodesics of the e ff ective metric, where a 'full-moon' like shadow was absent. However, null geodesics in the e ff ective metric undergo more redshift compared to those in the background metric, leading to a more pronounced intensity contrast in shadows from ˜ g µν than those from the g µν . Such 'full moon' images generically appear for RN naked singularity, and were recently reported for the JMN-1 and JMN-2 naked singularities by Shaikh et. al. [127]. \nIn Fig. 15, the accretion disk images cast by null geodesics of the background metric of GC-BH and GC-HUCO are presented. In the case of a GC-BH, besides a central dark region, a bright and narrow ring appears that can be decomposed into a series of exponentially stacked n = 1 , 2 , 3 , ... subrings, identified by the image order, converging to the black hole's critical curve. These subrings were absent for first accretion model discussed earlier. For k = 0 . 70 M , the inner edge of the disk is located at r in = 5 . 227 M , and its direct image delineates the inner boundary of the emission ring on the image plane as shown in the top-left panel of Fig. 15. For example, along the α = 0 axis, the direct image of the disk's inner edge is mapped onto the screen coordinate at β = 6 . 084 M , and the null circular orbits at r p = 2 . 651 M are mapped onto β = β c = 4 . 7438 M . Backtracing null geodesics from 0 < β < β c end at the horizon without intersecting the accretion disk and therefore do not contribute to the intensity. However, for GC-HUCO, geodesics from 0 < β < β c also play a role and yield distinctive additional signatures in the images. The shadows of GC-HUCO, both with ( k = 1 . 22 M < k c) and without ( k = 2 . 0 M > k c) null circular orbits, are shown in the bottom panel of Fig. 15. \nFor GC-HUCO with k = 1 . 22 M , along the α = 0 axis on the screen, β = 4 . 385 M and β = 3 . 45 M map, respectively, the direct image of the inner edge of the disk at r in = 3 . 124 M and null circular orbits at r p = 1 . 577 M . Interestingly, unlike the black hole case, null geodesics starting from β < 3 . 45 M still exhibit deflection angles exceeding 2 π and intersect the accretion disk for discrete values of β , manifesting as an additional series of concentric circular rings at the center of the image plane. Consequently, two series of circular rings emerge for HUCO with k < k c: one for b > b cr resembling those found for black holes, and another for b < b cr, representing a novel feature of HUCO. For k = 1 . 22 M and for a very small values of β , the deflection angle is less than π/ 2, thus these geodesics neither make loops around the central object nor intersect the accretion disk, resulting in a dark region at the center of the image plane, as shown in the bottom-left panel of Fig. 15. However, these inner rings disappear in the images for GC-HUCO without null circular orbits, as shown for k = 2 . 0 M in the bottom-right panel of Fig. 15. This happens because light deflection angle decreases \nFIG. 14. This figure illustrates shadows of GC-BH with k = 0 . 5 M and GC-HUCOs with k = 0 . 80 M and k = 2 . 0 M from the background metric under the radially infalling spherical accretion. \n<!-- image --> \nwith increasing k for a fixed value of b . For instance, for k = 2 . 0 M , the light deflection angle is less than even π/ 2, and for observer's inclination angle θ o = 80 · , light rays intersect the accretion disk at most twice, constructing only the direct image and the first-order image of the backside of the disk with no higher-order rings, as illustrated in the bottom-right panel of Fig. 15. The inner boundary of the emission ring on the screen corresponds to the inner edge of the disk in bulk. In summary, the intensity dip or the dark region at the image center results from the fact that the accretion disk could be extended only up to r in > 0. If r in → 0, then this dark region at the image center should disappear. Although the angular frequency of Keplerian orbits becomes imaginary for r < r in, accreting matter can radially fall into black holes within this region. GC-HUCO with and without null circular orbits manifest significantly distinct shadows compared to those of GC-BH. 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2024ApJ...975...10S	Oscillatory reconnection is a specific type of timedependent reconnection which involves periodic changes in the magnetic topology of a null point. The mechanism has been reported for a variety of magnetic field strengths and configurations background temperatures and densities. All these studies report an oscillation in the current density at the null point but also report a variety of periods amplitudes and overall behaviors. We conduct a parametric study for equilibrium magnetic field strength and initial background temperature solving twodimensional resistive magnetohydrodynamic equations around a magnetic Xpoint. We introduce a parameter space for the ratio of internal to magnetic energy and find selfsimilar solutions for simulations where this ratio is below 0.1 which represents a magnetically dominated environment or equivalently a lowbeta plasma. Selfsimilarity can be seen in oscillations in the current density at the null including amplitude and period ohmic heating and the temperature generated via reconnection jets. The parameter space of energy ratios also allows us to contextualize previous studies of the oscillatory reconnection mechanism and bring those different studies together into a single unified understanding.	2024-11-01T00:00:00Z	['10.3847/1538-4357/ad7600', '2024arXiv240912130S', '10.48550/arXiv.2409.12130', 'arXiv:2409.12130', '2024ApJ...975...10S']	['Solar magnetic reconnection', 'Solar physics', 'Solar coronal transients', 'Solar coronal heating', 'Magnetohydrodynamics', '1504', '1476', '312', '1989', '1964', 'Astrophysics - Solar and Stellar Astrophysics', 'Physics - Plasma Physics', 'Physics - Space Physics']	Selfsimilar Solutions of Oscillatory Reconnection Parameter Study of Magnetic Field Strength and Background Temperature	2,024	199	0.44	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.12130.pdf	{'No Header': 'Self-similar solutions of oscillatory reconnection: parameter study of magnetic field strength and background temperature \n<!-- image --> \n1 Northumbria University, Newcastle upon Tyne, NE1 8ST, UK', 'ABSTRACT': 'Oscillatory reconnection is a specific type of time-dependent reconnection which involves periodic changes in the magnetic topology of a null point. The mechanism has been reported for a variety of magnetic field strengths and configurations, background temperatures and densities. All these studies report an oscillation in the current density at the null point, but also report a variety of periods, amplitudes and overall behaviors. We conduct a parametric study for equilibrium magnetic field strength and initial background temperature, solving 2D resistive MHD equations around a magnetic X-point. We introduce a parameter space for the ratio of internal-to-magnetic energy and find self-similar solutions for simulations where this ratio is below 0.1 (which represents a magnetically-dominated environment or, equivalently, a low-beta plasma). Self-similarity can be seen in oscillations in the current density at the null (including amplitude and period), Ohmic heating and the temperature generated via reconnection jets. The parameter space of energy ratios also allows us to contextualize previous studies of the oscillatory reconnection mechanism and bring those different studies together into a single unified understanding. \nKeywords: Solar magnetic reconnection(1504) - Solar physics(1476) - Solar coronal transients(312) - Solar coronal heating(1989) - Magnetohydrodynamics(1964)', '1. INTRODUCTION': "Magnetic reconnection is a fundamental plasma process allowing stored magnetic energy to be released into thermal and kinetic energy, as well as accelerate particles and allow a change in magnetic connectivity (Pontin & Priest 2022; Browning et al. 2024). Reconnection is understood to be at the heart of several fundamental processes, including coronal mass ejections (e.g. Webb & Howard 2012) and solar flares (e.g. Benz 2017). For example, observations of chromospheric anemone jets have provided evidence of reconnection events occurring at smaller spatial scales in the chromosphere, suggesting a potential link between the heating of the solar chromosphere and corona and small-scale reconnection (Shibata et al. 2007). \nOscillatory reconnection is a specific type of timedependent reconnection which involves periodic changes in the magnetic connectivity and topology of the field. The concept of oscillatory reconnection was first identified by Craig & McClymont (1991) during their investigation of the relaxation of a two-dimensional (2D) Xpoint configuration magnetic field. One distinguishing feature of oscillatory reconnection is its intrinsic periodicity, which arises naturally from the relaxation process \nitself rather than being externally imposed (i.e. the generation of periodic outputs even from aperiodic drivers McLaughlin et al. 2012a). \nMcLaughlin et al. (2009) investigated a 2D X-point configuration in a cold plasma simulating the resistive magnetohydrodynamic (MHD) equations for a ideal fully-ionized plasma. In this study, oscillatory reconnection was initiated by perturbing the magnetic X-point using an external fast magnetoacoustic pulse. The research identified several key properties of this mechanism, including periodic changes in the orientation of the resulting current sheet corresponding to alterations in magnetic connectivity. Additionally, the formation of both fast and slow oblique magnetic shocks was observed as part of the oscillatory reconnection process. Studies have also highlighted the role of oscillatory reconnection in generating quasi-periodic waves and flows in the solar atmosphere, providing a physical explanation for high-speed, quasi-periodic, transverse outflows and jets (McLaughlin et al. 2012b). \nInvestigations into the three-dimensional (3D) nature of oscillatory reconnection have revealed that reconnection at fully 3D nulls can occur naturally in a timedependent and periodic fashion (Thurgood et al. 2017). The periodicity of oscillatory reconnection has been \nfound to be independent of the initial pulse in 2D Xpoint simulations (Karampelas et al. 2022a). More recently, Talbot et al. (2024) investigated the impact of resistivity on oscillatory reconnection, and discovered that the reconnection period is independent of background resistivity. Additionally, simulations of emerging flux in a coronal hole have demonstrated the initiation of oscillatory reconnection in a self-consistent manner, with signatures comparable to observations of select flux emergence events and solar and stellar flares (Murray et al. 2008). Moreover, oscillatory reconnection has been proposed as a possible mechanism underlying various periodic phenomena in the solar atmosphere, including quasi-periodic pulsations (QPPs, e.g. McLaughlin et al. 2018; Zimovets et al. 2021). \nOscillatory reconnection phenomena have been investigated across various plasma configurations. Among these studies, Prokopyszyn et al. (2019) delved into the dynamics of a null point perturbed by a continuous driver perpendicular to a 2D plane that contains an X-point using 2.5D simulations, revealing phase-mixing due to the magnetic field inhomogeneities. Santamaria & Van Doorsselaere (2018) examined oscillatory reconnection within a 2D arcade configuration, considering a stratified atmosphere, finding that the null point behaves as a resonant cavity generating waves at certain frequencies that depend upon the equilibrium parameters. Tarr et al. (2017) studied a null point in an arcade configuration, modeling a stratified atmosphere from the photosphere to the lower corona. In their simulation, they added a wave packet driver in the photosphere. They analyzed the energy conversion of this incident wave packet at the null point. They reported that 70% of the energy incident on a null point is converted to slow magnetoacoustic waves, 7% into fast magnetoacoustic waves, and 23% remains at the null until dissipated. Stewart et al. (2022) found that, during flux rope coalescence, oscillatory reconnection can occur intrinsically without an external oscillatory driver, resulting in both a periodic signal and the generation of radiallypropagating nonlinear waves. In these investigations (Prokopyszyn et al. 2019; Santamaria & Van Doorsselaere 2018; Stewart et al. 2022; Tarr et al. 2017), the plasma was either analyzed under fixed atmospheric conditions or within a single magnetic field configuration. \nKarampelas et al. (2022b) investigated the periodicity and decay rate in the oscillatory reconnection pattern observed at the current sheet at a null point. In their study, the effect of temperature was evaluated from 0 K up to 1 MK for a 1.44 G magnetic field. Their findings show that the oscillatory reconnection signal is only affected by temperatures above 10,000 K. This \nstudy's parametric exploration of temperature and magnetic field variations was conducted within a limited parameter space. Karampelas et al. (2023) performed a parametric study for evaluating the impact of solar atmospheric conditions and magnetic field on the oscillatory reconnection period. They evaluated a temperature range from 3 MK up to 10 MK for a magnetic field range from 10 G to 30 G and proposed an empirical formula to describe the oscillatory reconnection period. \nThus, oscillatory reconnection has been studied in a variety of magnetic topologies, albeit all containing a null point, and for a variety of coronal conditions, including different initial temperature profiles and varying equilibrium magnetic field strengths. All these studies report an oscillating signal in the current density at the null point; a tell-tale sign of oscillatory reconnection. However, these studies also report a variety of periods, amplitudes and behaviors for such a signal, and it is currently unclear how to bring these different studies together into a single unified understanding. This paper aims to do just that: to perform a parametric study involving variations in magnetic field strength from B = 5 G to 100 G and plasma temperature between 0 K to 10 MK, with the aim of exploring a wider parameter space than previous explored, in order to consolidate these different results under a single explanation. \nThis paper has the following structure: the numerical model, initial conditions and boundary conditions are detailed in § 2; the results are presented in § 3, including the dependence on the equilibrium magnetic field ( § 3.2), the influence of the initial background temperature ( § 3.3) and the unification of these results into an energy map ( § 3.4). The conclusions are presented in § 4.", '2.1. Governing equations': 'In our investigation, we solve the 2D resistive MHD equations through the utilization of the LARE2D code (Arber et al. 2001). The equations are solved in Lagrangian form, employing a Lagrangian-Eulerian remap procedure and can be expressed in dimensionless form as follows: \nDρ Dt = -ρ ∇· v , D v Dt = 1 ρ ( ∇× B ) × B -1 ρ ∇ p, D B Dt =( B · ∇ ) v -B ( ∇· v ) -∇× ( η ∇× B ) , Dϵ Dt = -p ρ ∇· v + η ρ \| j \| 2 , p = ρϵ ( γ -1) . \nTable 1. Initial conditions employed in the parametric study, with values given at 1 Mm from the null point. \nHere, v denotes the velocity vector, B represents the magnetic field, j is the current density, ρ signifies plasma density, p corresponds to plasma thermal pressure, ϵ represents specific internal energy, η characterizes the resistivity, and γ is the ratio of specific heats, set to 5/3 for a hydrogen plasma. To accurately accommodate steep gradients like shocks and address numerical instabilities, LARE2D utilizes a numerical viscosity (Caramana et al. 1998; Arber et al. 2001). \nThe model assumes full ionization of the plasma and non-dimensionalizes the governing equations with respect to length-scale L 0 , magnetic field B 0 , and density ρ 0 . These constants define non-dimensionalization for velocity v 0 = B 0 / √ µ 0 ρ 0 , thermal pressure p ∗ = B 2 0 /µ 0 , time t 0 = L 0 /v 0 , current density j 0 = B 0 /µ 0 L 0 , specific internal energy ϵ 0 = v 2 0 , temperature T ∗ = ϵ 0 m/k B and resistivity η 0 = µ 0 L 0 v 0 , where µ 0 is the vacuum magnetic permeability, k B is the Boltzmann constant and m the average mass of ions. Simulation results can be scaled with appropriate reference scales, with typical values for the solar corona being L 0 = 1 Mm and ρ 0 = 1 . 67 × 10 -12 kg/m 3 . We set the resistivity as η = 10 -4 η 0 . Our investigation explores a variety of B 0 values and initial background temperatures, and the physical values of these are presented in Table 1. There is no physical viscosity in our system and the numerical dissipation is negligible. It is important to notice that \nTable 2. Non-dimensionalization scales showing the influence of magnetic field strength B 0 on time t 0 , temperature T ∗ and current densities j 0 . \nsome normalization scales depend on B 0 , as shown in Table 2. \nWe will conduct two types of parametric studies: simulation set A, where the temperature and thermal pressure are initially constant while varying the magnetic field intensity, and simulation sets B, C and D, where we choose and fix an equilibrium magnetic field strength B 0 , while exploring increasing the initial plasma temperature from a cold state ( T = 0) up to 10 MK, for cases at B 0 = 10 G, 50 G and 100 G. Details of each simulation set is shown in Table 1. \nIn our analysis, we consider a fully-ionized, pure hydrogen plasma, wherein the average mass of ions can be approximated to the proton mass, m p . However, for conditions typical of the solar corona, where the plasma composition includes various elements, one can account for an average ion mass by setting m = 1 . 2 m p . Due to the non-dimensionalization in our system, it is straightforward to consider either a pure hydrogen plasma ( m = m p , where these are the results presented in this paper) or the temperature derived from our simulations can be divided by 1.2 to obtain the corresponding temperature for an average ion mass of m = 1 . 2 m p (where this adjustment would then account for heavier ions in the plasma composition).', '2.2. Equilibrium magnetic field and initial condition in velocity': "For the equilibrium magnetic field, we consider a 2D X-point defined by: \nB = B 0 L 0 ( y, x, 0) . (1) \nWe take the equilibrium density, ρ 0 , and initial background temperature, T 0 , to be uniform, and the magnetic Reynolds number was set as R m = 10 4 . The initial temperature profile and equilibrium magnetic field strength are detailed in Table 1. Table 1 also shows information on the ratio between internal energy E i = ρϵ , and magnetic energy E B = \| B \| 2 / 2 µ 0 , per unit of volume. V s = √ γp/ρ is the speed of sound and V a = \n\| B \| / √ µ 0 ρ is the Alfv'en speed. Note that the equilibrium magnetic field given in Equation (1) is highly inhomogenous and is scale-free. Thus, our choice of B 0 is only the initial value of magnetic field strength at t = 0 and r = L 0 , where r = √ x 2 + y 2 , i.e., \| B ( r = L 0 , t = 0) \| = B 0 . Similarly, V a ( r = L 0 , t = 0) = V a 0 , E i ( r = L 0 , t = 0) = E i 0 and E B ( r = L 0 , t = 0) = E B 0 denote the initial Alfv'en speed, internal energy per unit of volume, and magnetic energy per unit of volume, respectively, at r = L 0 and t = 0. In the same way, T ( t = 0) = T 0 , p ( t = 0) = p 0 and V s ( r = L 0 , t = 0) = V s 0 denote the initial background temperature, pressure and sound speed, respectively, which we take to be constant when t = 0 (and thus there is no need to specify r = L 0 ). All these parameter choices are detailed in Table 1. \nThe plasma β denotes the ratio of thermal pressure ( p ) to magnetic pressure ( p magnetic = \| B \| 2 / 2 µ 0 ) and is given by β = 2 µ 0 p/ \| B \| 2 . Initially and at r = L 0 , we define β 0 as: \nβ 0 = β \| r = L 0 ,t =0 = 2 µ 0 p \| B \| 2 ∣ ∣ ∣ ∣ r = L 0 ,t =0 = 2 µ 0 p 0 B 2 0 . (2) \nThe initial velocity field is computed as it was in McLaughlin et al. (2009), where it is imposed based in two variables v ⊥ = v × B · ˆ z and v ∥ = v · B that is related to propagation perpendicular and parallel to the magnetic field lines. The initial velocity pulse is given by: \nv ⊥ ( x, y ) = 2 C sin[ π ( r -4 . 5)] 4 . 5 ≤ r ≤ 5 . 5 , (3) \nv ∥ ( x, y ) = 0 , (4) \nwhere 2 C is our initial amplitude. The expression describe a circular, sinusoidal pulse shown in Figure 1a. When the simulation begins, this initial pulse will naturally split into two waves, each of amplitude C , traveling in different directions: a radially-outgoing wave and a radially-incoming wave. The incoming wave, i.e. the wave traveling towards the null point, is the wave we are primarily interested in since it is the wave that is responsible for triggering the oscillatory reconnection. The Cartesian velocity field, equivalent to Equations (3) and (4), can be obtained via: \nv x = v ∥ B x + v ⊥ B y \| B \| 2 and v y = v ∥ B y -v ⊥ B x \| B \| 2 .", '2.3. Boundary conditions and domain setup': 'We adopt a Neumann boundary condition imposing zero gradient at the boundaries for velocities, magnetic field and thermodynamic variables. We also employ a stretched grid characterized by finer resolution closer to the null point, i.e. the region of primary interest, \nFigure 1. Evolution of the plasma flow for case D2, where B 0 = 10 G and T 0 = 1 MK. Panel (a) shows contour of v ⊥ and separatrices, panels (b) to (d) present contours of ∆ T = T -T 0 and black lines represent the (evolving) separatrices. \n<!-- image --> \nand coarser resolution in the outer regions. The grid is equally spaced and highly refined around the null point and initial pulse at -5 ≤ x, y ≤ 5, and it employs a stretching at x, y > 5. We adopt a hyperbolic tangent stretching function that smoothly changes the growth rate of grid spacing from 0 to 7% for x, y > 5. The mesh stretching in the outer regions also creates some numerical dissipation, which is useful in terms of dissipating away the outgoing waves and thus reducing the impact of reflected waves which could then go on to further perturb our null point, which is undesirable. We also employed a damping region at r > 6, such that this damping region removes kinetic energy from the outgoing waves in the outer region, again so it does not reflect back and perturb the null. The details of this kineticenergy damping condition are described in Talbot et al. (2024). Our total grid has 1700 × 1700 points, and the total domain box extends to -93 ≤ x, y ≤ 93.', '3.1. Overall behavior and temperature evolution': 'The numerical set-up, choice of equilibrium magnetic field and initial velocity perturbation closely follow the work of previous authors (such as McLaughlin et al. 2009; Karampelas et al. 2022b, 2023) and readers are referred to these works for a detailed explanation of the system evolution. Instead, this section will focus on previously unexplored details, such as the analysis of the evolution of temperature perturbation. \nFigure 1 depicts the evolution for simulation D2 in Table 1, where B 0 = 10 G and T 0 = 1 MK (There \nFigure 2. Time evolution of the current density at the null point for case D2, j z (0 , 0 , t ) /j 0 , with the × symbol denotes the roots of the function. \n<!-- image --> \nis nothing special about the choice of case D2; it just represents a typical simulation across all our cases). In \nFigure 3. Integrated temperature perturbation ⟨ ∆ T ⟩ as function of non-dimensionalized time for simulation sets A1 to A5 in Table 1. The circles indicate the maximum of each time series. \n<!-- image --> \nthe Figure 1a, the initial condition is shown by a circular pulse represented by v ⊥ contours, which generates two fast magnetoacoustic waves, one propagating inward and the other outward. The inward wave perturbs the null point and initiates oscillatory reconnection. Note that, in order to focus attention towards the area of primary interest (the null) the numerical domain presented here is a subset of the full domain. \nSubsequent Figures 1b -1d illustrate temperature perturbation contours ∆ T , with black lines representing the magnetic field lines separatrices. We define ∆ T as the temperature difference between the evolving temperature field and the initial background temperature (which is initially uniform): \n∆ T ( x, y, t ) = T ( x, y, t ) -T ( x, y, 0) = T ( x, y, t ) -T 0 . (5) \nIn Figure 1b, at t = 2 t 0 , the fast magnetoacoustic wave approaches the null point and develops into a shock wave. This initial shock wave elevates plasma temperature and gives rise to two jet streams along the x -axis, emanating from the null point. The plasma attains a highly-localized temperature of 12 MK at the jets. \nIn Figure 1c, ∆ T is presented at t = 4 t 0 , after the shock wave reaches the null point. The X-point is highly deformed from its equilibrium profile (due to the passage of the fast oblique magnetic shocks) and the magnetic field lines assume a new configuration featuring a horizontal current sheet aligned with the x -axis, marking the first cycle of oscillatory reconnection. Additionally, the temperature peak of 12 MK decreases and spreads near the jets, heating the plasma locally to 2.5 MK. \nFigure 1d, at t = 6 t 0 , presents the plasma after the first horizontal current sheet has formed, force imbalance has then peeled the horizontal current sheet apart, overshoot the original magnetic configuration, and so a new current sheet has formed (the first vertical current sheet), parallel to the y -axis. This process characterizes an oscillation cycle, where the reconnection and reorientation will repeat at the points where the roots of the oscillations of j z (0 , 0 , t ) are displayed in Figure 2. In Figure 1d, we observe local heating at the ends of the vertical current sheet, albeit the localized heating is still strongest close to x ≈ ± 1 , y = 0. In other words, the strongest localized heating comes from that first horizontal current sheet.', '3.2. Sensitivity to choice of equilibrium magnetic field strength': 'In this section, we shall analyze the influence of magnetic field strength on heating for a given initial background temperature. To achieve this, we utilize the integrated temperature perturbation ⟨ ∆ T ⟩ , to measure the \nFigure 4. (a) maximum integrated temperature ⟨ ∆ T ⟩ max as function of equilibrium magnetic field strength B 0 . The brown dot-dash line denotes a fitted quadratic dependence on the magnetic field strength following ⟨ ∆ T ⟩ max ∼ B 2 0 , where the slope is offset artificially for ease of comparison. (b) maximum integrated non-dimensional temperature ⟨ ∆ T ⟩ max as function of internalto-magnetic energy ratio. The purple line represents the maximum obtained for simulation set A seen in Fig. 3. The red line represents the maximum from simulations B2, C2 and D2, the green line corresponds to simulations B3, C3 and D3, the orange line cases B4, C4 and D4 and the blue B5, C5 and D5. \n<!-- image --> \nheating effect near the null point. The symbol ⟨ ⟩ denotes a spatial average over an area S , defined as: \n⟨ f ⟩ = 1 S ∫∫ S f ( x, y, t ) dS, (6) \nwhere f is some arbitrary function, and S the integration surface we defined as -2 < x,y < 2 Mm. The integration area was selected to enclose the magnetic field lines, jet stream and the heated region, as depicted in Figure 1. The integrated temperature perturbation serves as a metric for assessing the heating effect in the vicinity of the null point. The circles in Figure 3 represent the maximum heating observed in the time series. \nFigure 3 illustrates the values of ⟨ ∆ T ⟩ for simulation sets A1-A5, as outlined in Table 1, wherein the plasma is maintained at 10 MK and we vary the equilibrium magnetic field strength, B 0 . As mentioned in Section 2.2, Equation (1) is scale-free and so our choice of B 0 is the value magnetic field strength at r = L 0 (and then B varies throughout the domain). \nIn Figure 3, initially a sharp increase in plasma temperature is observed, coinciding with the passage of the shock wave propagating towards the null point. This initial heating phase concludes approximately before t = 2 t 0 . Subsequently, oscillatory reconnection processes sustain the heating by converting magnetic energy into internal energy. \nIn cases A1, A2 and A3, corresponding to B 0 = 5 G, B 0 = 10 G and 20 G respectively, a spike in heating is observed around t =1.5 t 0 , attributed to the initial shock \nwave. Following this, the plasma undergoes cooling before the onset of oscillatory reconnection, heating up to 0.19 MK, 0.48 MK and 1.06 MK, respectively. These cases are characterized by a high β 0 regime, as indicated by the initial internal-to-magnetic energy ratio of 4.16 for case A1, 1.04 for case A2, and 0.26 for case A3 (at a distance of 1 Mm from the null point). Consequently, the magnetic field strength in cases A1 and A2 is insufficient to heat the plasma to temperatures exceeding 1.0 MK and never above 1.1 MK for case A3. \nConversely, in cases A4 and A5, corresponding to B 0 = 50 G and 100 G respectively, the plasma is in a low β 0 regime, resulting in more pronounced heating. Maximum temperatures reach up to 15 MK and 63 MK, respectively, indicating a more significant heating effect due to the stronger magnetic fields. \nFigure 4 illustrates the maximum heating obtained from all the different simulation sets in Table 1, namely A1-to-A5, B1-to-B5, C1-to-C5 and D1-to-D5 (note that B1=A5, C1=A4 and D1=A2) cases. Observing Figure 4a, it becomes apparent that the heating is directly proportional to the choice of equilibrium magnetic field strength at 1 Mm from the null point. The maximum heating follows a power law and exhibits a quadratic dependence on the magnetic field strength following a ⟨ ∆ T ⟩ max ∼ B 2 0 line slope. \nWe performed a fitting on the curves shown in Figure 4a and we were able to obtain an expression to quantify the heating and the average temperature as: \n⟨ ∆ T ⟩ max = aB b 0 , (7) \nwhere in this expression B 0 is the initial magnetic field in Gauss at 1 Mm from the null point and the initial output temperatures are given in MK. The constants a and b are presented in Table 3 together with respective standard deviations, σ a and σ b . Under the assumption that the covariance σ ab between the coefficients a and b is negligible, we can derive the following expression to describe the standard deviation for the fitted expression using uncertainty propagation: \nσ ∆ T ≈ aB b 0 √ ( σ a a ) 2 + B 2 b 0 (ln( B 0 ) σ b ) 4 . \nTable 3 shows that there is no variation in the index b , or amplitude a , for cases with temperatures lower than 1 MK. For cases at 1 MK, there is a negligible variation in the coefficients, and for the hotter case at 10 MK, we can observe a slight variation. The curve at 10 MKcomprises the simulation set A1 to A5 (representing the peaks observed in Figure 3), which has a significant variation in β 0 , ranging from 0.016 to 6.24. The other curves have low β 0 values, as shown in Table 1, which can explain the self-similar behavior. \nIn addition, in Figure 4b, we analyzed the maximum integrated-temperature perturbation nondimensionalized by T ∗ as a function of the energy ratio for multiple initial background temperatures. We observe self-similar behavior for background temperatures up to 0.1 MK. The simulation cases of 1 MK and 10 MK show a decrease in heating when E i 0 /E B 0 > 0.01.', '3.3.1. Integrated temperature perturbation ⟨ ∆ T ⟩': 'Figure 5 displays the results of the integrated temperature perturbation ⟨ ∆ T ⟩ and integrated Ohmic heating ⟨ ηj 2 ⟩ non-dimensionalized by η 0 j 2 0 for simulation sets B, C and D. Firstly, analyzing the integrated temperature perturbation Figure 5a, significant heating is found in simulation set B, which is attributable to the high equilibrium magnetic field strength of 100 G. The curves exhibit a self-similar pattern during the initial transient, \nTable 3. Fitting coefficients and standard deviation for each curve from Figure 4a, coefficients a and b are presented in Equation (7). \nheating the plasma to a localized maximum of 65 MK in simulation set B. \nOf particular interest, the heating in the simulation set at B 0 =100 G does not seem to be influenced by the initial temperature. This observation is significant since the initial temperature from dataset B1 and B5 differs by seven orders of magnitude (0 to 10 MK). \nIn the Figure 5c, simulation set C (50 G) exhibits maximum heating nearly four times smaller than the results found in simulation set B (100 G). This difference highlights the significant importance of choice of equilibrium magnetic field strength on plasma heating. Additionally, a self-similar solution is observed in temperatures between T 0 = 0 to 1 MK, whereas the simulation at T 0 = 10 MK achieves maximum heating faster and with a 1.5 MK reduction compared to simulations at lower temperatures. \nThe Figure 5e shows results for simulation set D (10 G). Here, the initial background temperature significantly impacts simulations, and the heating is only selfsimilar for simulations with temperatures ranging from 0 to 0.1 MK. The simulation at 10 MK display a spike in t ≈ t 0 generated by the initial shock wave; similar behavior can also be observed in B 0 = 5 G in Figure 3. \nOhmic heating (Figures 5b, 5d and 5f) increases quickly, reaching its peak by t =2 t 0 , then decreasing significantly by t = 4 t 0 . This increase corresponds with the sharp rise observed in the integrated temperature disturbance (Figure 5 left column), indicating that magnetic energy is being converted to internal energy. The initial background temperature does not affect the Ohmic heating patterns observed at magnetic field strengths of 100 G and 50 G. However, simulations at 10 G reveal a decrease in the magnitude of Ohmic heating for temperatures of 1 and 10 MK, which aligns with the temperature disturbance plots in Figures 5a, 5c and 5e. The peak at t = 2 t 0 suggests that a substantial portion of Ohmic heating comes from the initial jets shown in Figure 1b.', '3.3.2. Evolution of j z (0 , 0 , t )': 'The evolution of the current density at the null point, j z (0 , 0 , t ), is a key aspect of oscillatory reconnection. In our simulations the null point is stationary and located at the origin ( x, y ) = (0 , 0). Figure 6 left column displays the j z (0 , 0 , t ) evolution (normalized against j 0 ) for datasets B, C and D in Table 1 and the right column displays the power spectral density (PSD) of j z (0 , 0 , t ) non-dimensionalized. \nThe observed oscillations in j z (0 , 0 , t ) are characteristic of signals associated with oscillatory reconnection, indicating periodic changes in the configuration of mag- \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n(d) 50 G \n0 \n<!-- image --> \nFigure 5. Left column: Integrated temperature perturbation ⟨ ∆ T ⟩ as function of normalized time. Right column: Integrated non-dimensionalized Ohmic heating as function of non-dimensionalized time. Panels represent simulations with B 0 = 100 G (top), 50 G (middle) and 10 G (bottom), respectively in Table 1. \n<!-- image --> \nFigure 6. Left column displays variations of the current density at the null point, j z (0 , 0 , t ) /j 0 , displays for different equilibrium magnetic field strengths and initial background temperature configurations. Right column shows the power spectral density (PSD) of j z (0 , 0 , t ) /j 0 of the same simulations. \n<!-- image --> \nTable 4. Periods and amplitudes extracted from j z (0 , 0 , t ) /j 0 power spectral density (PSD) for the dominant period ( P ). ∆ P and ∆ A represent the period decrease and amplitude decrease from a cold case to a hot case for the same magnetic field strength. \nnetic field lines over time (McLaughlin et al. 2009). In the Figures 6a and 6c, we see that variations in initial background temperature profile do not significantly affect the amplitude and period of the j z (0 , 0 , t ) signal for equilibrium magnetic field strengths of B 0 = 100 G (cases B) and 50 G (cases C), and these cases exhibit a self-similar solution in non-dimensional units. However, for simulations with equilibrium magnetic field strength B 0 = 10 G (cases D), we observe variations in amplitude and period for temperatures of 1 MK (case D2) and 10 MK (case D1), while temperatures below 0.1 MK demonstrate a self-similar solution (and are thus comparable to B 0 = 100 G and 50 G). \nIn Figures 6a, 6c and 6e, we can also observe that j z (0 , 0 , t ) /j 0 oscillates around a non-zero value. This is due to asymmetric heating in the plasma around the neutral point, which itself is due to the sequence of reconnection jets: at the end of the simulation, the plasma to the left and right of the neutral point is fractionally hotter than the plasma above and below it, as a result of the iteration of hot reconnection jets that formed and heated the plasma (each subsequent current sheet and corresponding heating event is shorter/weaker than the last. Consequently, the plasma pressure is very slightly higher on the left and right of the null point at the very end of the simulation, making it easier for the system to form a very slight vertical current sheet). This leads to j z (0 , 0 , t ) tending towards a small positive value. This phenomenon was originally reported in McLaughlin et al. (2009). \nThe power spectral densities (PSD) exhibit uniform behavior for all cases at magnetic field strengths of 100 \nG (cases B) and 50 G (cases C). We observe a constant oscillation period for the dominant period of 9.001 t 0 for all cases except for case D1 ( B 0 =10 G and T 0 =1MK). Case D1 exhibits a dominant period of 7.508 t 0 , representing a 16.7% decrease from the other cases. We also calculated the amplitude difference for the dominant period between a cold case and a hot case with the same magnetic field strength. We only observed a difference larger than 5% when E i 0 /E B 0 > 10%, which represents cases D1 and D2. These results are presented in Table 4.', '3.4. Energy map': "An explanation for the apparent independence on initial background temperature for some cases may be found in Table 1, where we observe that simulation set B operates in a low β 0 regime. The internal-to-magnetic energy ratio in simulation set B is below 0.01, indicating that magnetic energy is approximately 100 times larger than internal energy. This dominance of magnetic energy leads to self-similar solutions between a cold plasma and a plasma at 10 MK. \nLet us now further explore the dependence of the internal-to-magnetic 'energy ratio' on the equilibrium magnetic field strength and the initial background temperature profile. The internal energy per unit volume, E i , for an ideal fully-ionized hydrogen plasma is given by: \nE i = ρϵ = 2 ρk b T m p ( γ -1) = p γ -1 , (8) \nFigure 7. Contour of the ratio of internal energy non-dimensionalized by magnetic energy for the initial condition (1 Mm away from the null point). Color bar shows the energy ratio as function of plasma β 0 and 100 E i 0 /E B 0 . \n<!-- image --> \nand the magnetic energy per unit volume, E B , is given by: \nE B = \| B \| 2 2 µ 0 , (9) \nand thus the ratio between energies can be written as: \nE i E B = 4 µ 0 ρk b T m p ( γ -1) \| B \| 2 = 1 γ -1 p p magnetic = β γ -1 , (10) \nwhere p magnetic is the magnetic pressure. \nAlternatively, it can be formulated for the initial condition at r = L 0 = 1 Mm of the null point: \nE i 0 E B 0 = 4 µ 0 ρ 0 k b T 0 m p ( γ -1) B 2 0 = 2 µ 0 γ -1 p 0 B 2 0 = β 0 γ -1 . (11) \nNotice that E i 0 /E B 0 is inversely proportional to the equilibrium magnetic field strength (at r = 1 Mm), whereas it is directly proportional to density and temperature. These three variables can provide insights into \nwhether the system is more dependent on magnetic energy or not. Notice also that in Equation (11) E i 0 /E B 0 is directly proportional to β 0 . \nFigure 7 denotes the internal-to-magnetic energy ratio (Equation 11, presented as a percentage) as a function of equilibrium magnetic field strength and initial background pressure at a distance of 1 Mm from the null point. Alternatively, since the internal-to-magnetic energy ratio is directly proportional to β 0 , Figure 7 also denotes contours of β 0 . Thus, we refer to Figure 7 as an energy map of the parameter space. \nThe energy map is divided into several regions delineated by isolines of E i 0 /E B 0 in percentage at 1%, 5%, 10%, 50%, and 100%. The contour lines provide insights into the relative dominance of internal energy compared to magnetic energy. Regions below the 10% line suggest a dominance of magnetic energy, indicating that the system's behavior may resemble that of cold plasma, with self-similar solutions for plasma heating profiles. Solu- \ntions above the 50% line indicate a significant contribution from internal energy, leading to a decrease in the maximum heating. Beyond the 100% threshold, hydrodynamic effects become dominant, resulting in further reduction in heating. \nIn the solar corona, we can estimate the equilibrium pressure of a plasma at 1 MK and a density of ρ 0 = 1 . 67 × 10 -12 kg/m ³ to be around 0.0276 Pa and 0.276 Pa during a 10 MK flare. Regions with pressure below this threshold are typical of the solar corona. \nThe solution in the lower-right-corner of the map indicate regions of low β 0 and are thus magneticallydominated. The upper-left-corner indicates regions of high β 0 and so simulations here are more hydrodynamically dominated. Note that increases in thermal pressure on the y -axis can be obtained via increasing the initial background temperature profile and/or simulations that have higher initial background densities, according to Equation (8). \nThe energy map allows us to contextualize previous studies by placing them at specific points within the 2D parameter space: \n- · Karampelas et al. (2022b) investigated the influence of temperature and heat conduction on plasma conditions. Their simulations focused on a single magnetic field with varying background temperatures. They observed self-similar behavior in plasma temperature profiles at low temperatures, which is consistent with our findings.\n- · Karampelas et al. (2023) conducted a parametric study covering a subset of both axes of the energy map. Their research focused on modeling j z oscillations at null points and developing empirical formulas based on their simulations. Their work primarily lies in a region above the 10% line, which means that based on the energy map we can infer that temperature profiles may not exhibit selfsimilar behavior in their studied conditions.\n- · The simulation from Prokopyszyn et al. (2019) is expected to display the exact solution of a cold plasma since it is placed below the E i 0 /E B 0 =10% line; the same is expected from the Stewart et al. (2022) simulation. On the other hand, it is expected that Prokopyszyn et al. (2019) and Stewart et al. (2022) display a distinct behavior in terms of heating, since ⟨ ∆ T ⟩ max ∼ B 2 0 as discussed in Section 3.2.\n- · The simulation case from Santamaria & Van Doorsselaere (2018) is placed above the 10%, \nwhere the magnetic field strength is weak, meaning that magnetic and internal energies are comparable in magnitude. Their heating profile, j z oscillation period and amplitude are expected to be more sensitive to initial temperature and pressure variations. In their study, they considered an arcade configuration in a stratified atmosphere. The reference values were extracted from their plots and interpolated at the height of the null point at a 1 Mm distance for placement on the energy map. \n- · Tarr et al. (2017) considered a stratified atmospheric condition and modeled the effect of heavier ions where m = 1 . 25 m p . Their energy ratio at the height of the null point is E i 0 /E B 0 =83%, which means that they are in a regime where plasma heating is smaller than in a cold plasma, and also it is expected to present a smaller maximum amplitude for j z signal at the null point.\n- · Previous studies considering a cold plasma, T = 0 K, such as McLaughlin et al. (2009) and Talbot et al. (2024) do not appear as dots on the energy map since the initial condition considers a zero pressure. Instead, they would appear towards the bottom of the y -axis, clearly within the low β /magnetically-dominated regime. Thurgood et al. (2017) did not provide the nondimensionalization scales to place it into the energy map precisely. However, their energy ratio in their simulations was 1.25%, which would place them between the 1% and 5% lines, which means that their simulation is in a magneticallydominated regime. \nDetails about the initial configuration of these previous studies in the literature can be found in Section 1. As discussed, the energy map can be used to estimate the heating and j z sensitivity of a 2D X-point to different magnetic field strengths and atmospheric conditions.", '4. CONCLUSIONS': 'We conducted 2D MHD simulations of a magnetic field with an X-point configuration perturbed by an initial condition in velocity for a fully-ionized, resistive plasma. Through a parametric study involving adjustments in equilibrium magnetic field strength and initial background plasma temperature, we investigated their influence on plasma heating around the null point and the oscillatory reconnection signal j z (0 , 0 , t ). \nFirstly, we performed a parameter study for different values of equilibrium magnetic field strength, B 0 . Our \nequilibrium magnetic field (Equation (1)) is scale-free and so our choice of B 0 is the value of the magnetic field strength at r = L 0 (and then B varies throughout the domain). We found that the choice of B 0 has a significant effect on the evolution, with the maximum temperature generated by the initial reconnection jets exhibiting a quadratic dependence ⟨ ∆ T ⟩ max ∼ B 2 0 . For example, our simulation for B 0 = 50 G and T = 10 MK (case C1) generated reconnection jets with maximum temperatures of around 15 MK. We also obtained an expression to quantify the heating and the average temperature as ⟨ T ⟩ max = aB b 0 + T ( t = 0), with Table 3 detailing the amplitude a and index b for all our simulations. \nThis behavior aligns with expectations, as the magnetic energy increases quadratically with the magnetic field strength, resulting in a larger reservoir of magnetic energy available for conversion into internal energy, creating a self-similar behavior of Figure 4b. \nWe also analyzed the maximum integratedtemperature perturbation (non-dimensionalized by T ∗ ) as a function of the internal-to-magnetic energy ratio for multiple initial background temperatures. We found self-similar behavior for all cases where E i 0 /E B 0 < 0.01, including self-similar behavior for all background temperatures up to 0.1 MK, and a decrease in heating for simulation cases of 10 MK (case D1) and 1 MK (case D2). \nSecondly, we performed a parameter study for different values of initial background temperature T 0 . We found that the integrated temperature perturbation ⟨ T ( x, y, t ) -T 0 ⟩ displayed different behavior depending upon the value of B 0 . For B 0 = 100 G (cases B) and B 0 = 50 G (cases C), the integrated temperature perturbation did not seem to be influenced by the initial background temperature; all the curves exhibit a selfsimilar pattern, over seven orders of magnitude (0 to 10 MK) for cases B and six orders of magnitude (0 to 1 MK) for cases C: there was only a slight departure from self-similarity for case C1 ( T 0 = 10 MK). \nFor B 0 = 10 G simulations (cases D), we found that the evolution of the integrated temperature perturbation is only self-similar for simulations with temperatures ranging from 0 to 0.1 MK (cases D3 to D5), whereas for 10 MK (case D1) and 1 MK (case D2) the initial background temperature does significantly impact the evolution (a departure from self-similarity). \nAlthough the heating effect is more pronounced in simulations at lower temperature, where β 0 is smaller, the actual temperature is larger in simulations when the initial background plasma is hotter. This is expected \nsince final temperature is still dependent on the initial temperature in the system, as seen in Equation (5). \nWe also investigated the amount of Ohmic heating in the system, finding that the initial background temperature does not affect the integrated Ohmic heating patterns observed at magnetic field strengths of 100 G (Cases B) and 50 G (Cases C). However, simulations at 10 G reveal a decrease in the magnitude of Ohmic heating for temperatures of 10 MK (case D1) and 1 MK (case D2), in agreement with the temperature disturbance plots. \nThirdly, we also investigated the evolution of the current density at the null point, j z (0 , 0 , t ) for different values of initial background temperature T ( t = 0). We found that variations in initial background temperature profile do not significantly affect the amplitude and period of the j z (0 , 0 , t ) signal for equilibrium magnetic field strengths of B 0 = 100 G (cases B) nor 50 G (cases C), and that all these cases exhibit a self-similar solution in non-dimensional units. However, for simulations with B 0 = 10 G (cases D) we observe variations in amplitude and period for temperatures of 10 MK (case D1) and 1 MK (case D2), while temperatures below 0.1 MK demonstrate a self-similar solution (and are thus comparable to j z (0 , 0 , t ) for B 0 = 100 G and 50 G), i.e. a similar result as that for the integrated temperature perturbation sensitivity. \nThe power spectral densities (PSD) exhibit uniform behavior for all cases at magnetic field strengths of 100 G (cases B) and 50 G (cases C) and for 10G Cases D2D5, where we observed a constant oscillation period for the dominant period of 9.001 t 0 for all cases except for case D1 ( B 0 = 10 G and T 0 = 1 MK). Case D1 exhibits a dominant period of 7.508 t 0 , representing a 16.7% decrease from the other cases. \nAdditionally, we calculated the amplitude difference for the dominant period between a cold case and a hot case with the same magnetic field strength. We only observed a difference larger than 5% when E i 0 /E B 0 > 10%, which represents cases D1 and D2. \nFourthly, the results for the parameter studies across B 0 and T ( t = 0) let us visualize the 2D parameter space as an energy map (Figure 7). This energy map presents the ratio of internal-to-magnetic energy E i 0 /E B 0 or, equivalently, plasma β 0 , since E i 0 /E B 0 = β 0 / ( γ -1). \nThe energy map is divided into several regions delineated by isolines of E i 0 /E B 0 , where contour lines provide insights into the relative dominance of internal energy compared to magnetic energy or, equivalently, in terms of β 0 : \n- · Regions below the 10% line suggest a dominance of magnetic energy, with self-similar solutions for \nplasma heating profiles and for j z (0 , 0 , t ). Equivalently, the lower-right-corner of the map indicates regions of low β 0 and are thus magneticallydominated; this is where cold plasma simulations exist. \n- · Solutions above the 50% line indicate a considerable contribution from internal energy, leading to a decrease in the maximum heating.\n- · Beyond the 100% threshold, hydrodynamic effects become dominant, resulting in further reduction in heating. Equivalently, the upper-left-corner indicates regions of high β 0 and so simulations here are more hydrodynamically dominated. There is a significant departure from the self-similar solutions of the magnetically dominated regimes. \nThe energy map also allows us to contextualize previous studies by placing them at specific points within the 2D parameter space (see Figure 7). All these previous studies reported an oscillating signal in j z (0 , 0 , t ) but also reported a variety of periods, amplitudes and behaviors for such a signal, and the energy map now brings these different studies together into a single unified understanding (i.e. there exists a 2D parameter space, dependent upon choices of B 0 and T ( t = 0), that can affect resultant behavior and departure from selfsimilar solutions in extreme high β 0 situations). Thus, this energy map serves as a valuable tool for interpreting plasma behavior within a broader parameter space. \nIt is important to mention that these findings rely on the hypothesis that the plasma is fully ionized, and ther- \nmal conduction and radiation effects were not considered in this study (Karampelas et al. 2022b investigated the effect of thermal conduction of the oscillatory reconnection system, finding that its inclusion has a small effect on the resultant period).', 'ACKNOWLEDGMENTS': 'We would like to thank the anonymous referee for the constructive comments that improved the quality of the paper. All authors acknowledge the UK Research and Innovation (UKRI) Science and Technology Facilities Council (STFC) for support from grant ST/X001008/1 and for IDL support. This work used the DiRAC Data Intensive service (CSD3) at the University of Cambridge, managed by the University of Cambridge University Information Services on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC component of CSD3 at Cambridge was funded by BEIS, UKRI and STFC capital funding and STFC operations grants. DiRAC is part of the UKRI Digital Research Infrastructure. Numerical simulations were conducted with LARE2D which is available at https://github.com/Warwick-Plasma/Lare2d. The data behind Figure 1, and a python script to reproduce it, are available in the following repository, DOI: 10.25398/rd.northumbria.26310742. \nSoftware: LARE2D (Arber et al. 2001), NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), matplotlib (Hunter 2007).', 'REFERENCES': "Arber, T. D., Longbottom, A. W., Gerrard, C. L., & Milne, A. M. 2001, Journal of Computational Physics, 171, 151, \ndoi: 10.1006/jcph.2001.6780 Benz, A. O. 2017, Living Reviews in Solar Physics, 14, 2, doi: 10.1007/s41116-016-0004-3 Browning, P. K., Gordovskyy, M., Schiavo, L. A., & Stewart, J. 2024, Fundamental Plasma Physics, 10, 100049, doi: https://doi.org/10.1016/j.fpp.2024.100049 Caramana, E. J., Shashkov, M. J., & Whalen, P. P. 1998, \nJournal of Computational Physics, 144, 70 \nCraig, I. J., & McClymont, A. N. 1991, Astrophysical Journal; (United States), 371, doi: 10.1086/185997 \nHarris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2 \nHunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55 \nKarampelas, K., McLaughlin, J. A., J. Botha, G. J. J., & R'egnier, S. 2022a, The Astrophysical Journal, 933, 142, doi: 10.3847/1538-4357/ac746a \n-. 2022b, The Astrophysical Journal, 925, 195, \ndoi: 10.3847/1538-4357/ac3b53 \n-. 2023, The Astrophysical Journal, 943, 131, \ndoi: 10.3847/1538-4357/acac90 \nMcLaughlin, J. A., De Moortel, I.and Hood, A. W., & Brady, C. S. 2009, A&A, 493, 227, doi: 10.1051/0004-6361:200810465 \nMcLaughlin, J. A., Nakariakov, V. M., Dominique, M., Jel'ınek, P., & Takasao, S. 2018, Space Science Reviews, doi: 10.1007/s11214-018-0478-5 \nMcLaughlin, J. A., Thurgood, J., & MacTaggart, D. 2012a, Astronomy and Astrophysics, doi: 10.1051/0004-6361/201220234"}
2023NIMPA105268253A	The HighAltitude Water Cherenkov HAWC observatory is a secondgeneration continuously operated wide fieldofview TeV gammaray observatory. The HAWC observatory and its analysis techniques build on experience of the Milagro experiment in using groundbased water Cherenkov detectors for gammaray astronomy. HAWC is located on the Sierra Negra volcano in Mxico at an elevation of 4100 meters above sea level. The completed HAWC observatory principal detector HAWC consists of 300 closely spaced water Cherenkov detectors each equipped with four photomultiplier tubes to provide timing and charge information to reconstruct the extensive air shower energy and arrival direction. The HAWC observatory has been optimized to observe transient and steady emission from sources of gamma rays within an energy range from several hundred GeV to several hundred TeV. However most of the air showers detected are initiated by cosmic rays allowing studies of cosmic rays also to be performed. This paper describes the characteristics of the HAWC main array and its hardware.	2023-07-01T00:00:00Z	['10.1016/j.nima.2023.168253', '2023arXiv230400730T', 'arXiv:2304.00730', '10.48550/arXiv.2304.00730', '2023arXiv230400730A', '2023NIMPA105268253A']	['Physics - instrumentation and detectors', 'Water Cherenkov Detectors', 'Astrophysics', 'High energy physics - experiment', 'Nuclear experiment', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Instrumentation and Methods for Astrophysics']	The HighAltitude Water Cherenkov HAWC observatory in Mxico The primary detector	2,023	199	0.51	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	25	https://arxiv.org/pdf/2304.00730.pdf	{"The High-Altitude Water Cherenkov (HAWC) Observatory in M'exico: The Primary Detector 1": "A.U. Abeysekara a , A. Albert b , R. Alfaro c , C. Alvarez d , J.D. ' Alvarez e , M. Araya f , J.C. Arteaga-Vel'azquez e , K.P. Arunbabu g,h , D. Avila Rojas c , H.A. Ayala Solares i , R. Babu j , A.S. Barber a , A. Becerril c , E. Belmont-Moreno c , S.Y. BenZvi k , O. Blanco n , J. Braun o , C. Brisbois p , K.S. Caballero-Mora d , J.I. Cabrera Mart'ınez c,l,m , T. Capistr'an q , A. Carrami˜nana r , S. Casanova s , M. Castillo t,e , O. Chaparro-Amaro u , U. Cotti e , J. Cotzomi t , S. Couti˜no de Le'on o , E. de la Fuente 2 3 , n,v,w , C. de Le'on e , T. De Young x , R. Diaz Hernandez r , B.L. Dingus b , M.A. DuVernois o , M. Durocher b , J.C. D'ıaz-V'elez v , R.W. Ellsworth p , K. Engel p , C. Espinoza c , K.L. Fan p , K. Fang o , B. Fick j , H. Fleischhack y , J. L. Flores z , N. Fraija q , J.A. Garc'ıa-Gonz'alez aa , G. Garcia-Torales z , F. Garfias q , G. Giacinti aw , H. Goksu ab , M.M. Gonz'alez q , A. Gonz'alez-Mu˜noz c,ac , J.A. Goodman p , J.P. Harding b , E. Hernandez t , S. Hernandez c , J. Hinton ab , B. Hona a , D. Huang j , F. Hueyotl-Zahuantitla d , C.M. Hui ax , T.B. Humensky p , P. Huntemeyer j , A. Iriarte q , A. Imran o , A. Jardin-Blicq ai,aj,ab , V. Joshi ak , S. Kaufmann ad , D. Kieda a , G.J. Kunde b , A. Lara g , R. Lauer ah , W.H. Lee q , D. Lennarz al , H. Le'on Vargas c , J.T. Linnemann 4 , x , A.L. Longinotti q , G. Luis-Raya ad , J. Lundeen x , K. Malone am , V. Marandon ab , A. Marinelli c,ae,af,ag , O. Martinez t , I. Mart'ınez-Castellanos an , J. Mart'ınez-Castro u , H. Mart'ınez-Huerta ao , J.A. Matthews ah , P. Miranda-Romagnoli aq , T. Montaruli o,ar , J.A. Morales-Soto e , E. Moreno t , M. Mostaf'a i , A. Nayerhoda s , L. Nellen ap , M. Newbold a , M.U. Nisa x , R. Noriega-Papaqui aq , T. Oceguera-Becerra w , L. Olivera-Nieto ab , N. Omodei as , A. Peisker x , Y. P'erez Araujo q , E.G. P'erez-P'erez ad , E. Ponce t , J. Pretz i , C.D. Rho av , D. Rosa-Gonz'alez r , E. Ruiz-Velasco ab , H. Salazar t , D. Salazar-Gallegos x , F. Salesa Greus at,s , A. Sandoval c , M. Schneider p , H. Schoorlemmer au,ab , J. Serna-Franco c , G. Sinnis b , A.J. Smith p , Y. Son av , K. Sparks Woodle i , R.W. Springer 4 , a , I. Taboada al , A. Tepe al , O. Tibolla ad , K. Tollefson x , I. Torres 4 , r , R. Torres-Escobedo aw , R. Turner j , F. Ure˜na-Mena r , T.N. Ukwatta b , E. Varela t , M. Vargas-Maga˜na c , L. Villase˜nor t , X. Wang j , I.J. Watson av , F. Werner ab , S. Westerhoff o , E. Willox p , I. Wisher o , J. Wood ax , G.B. Yodh ay , D. Zaborov i,az , A. Zepeda ba , H. Zhou 4 , aw", 'The historical and present HAWC Collaboration 3 , 4': "a Department of Physics and Astronomy, University of Utah, Salt Lake City, UT, USA b Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA c Instituto de F'ısica, Universidad Nacional Aut'onoma de M'exico, Ciudad de M'exico, M'exico d Universidad Aut'onoma de Chiapas, Tuxtla Guti'errez, Chiapas, M'exico e Universidad Michoacana de San Nicol'as de Hidalgo, Morelia, M'exico f Escuela de F'ısica, Universidad de Costa Rica, San Jos'e, Costa Rica g Instituto de Geof'ısica, Universidad Nacional Aut'onoma de M'exico, Ciudad de M'exico, M'exico h Department of Physics, St. Albert ' s College (Autonomous), Cochin, 682018 Kerala, India \n- i Department of Physics, Pennsylvania State University, University Park, PA, USA j Department of Physics, Michigan Technological University, Houghton, MI, USA\n- k\n- Department of Physics & Astronomy, University of Rochester, Rochester, NY , USA l Facultad de Ciencias, Universidad Nacional Aut'onoma de M'exico, 04510, Ciudad de M'exico, M'exico\n- m Colegio de Ciencias y Humanidades Plantel Sur, Universidad Nacional Aut'onoma de M'exico, Ciudad de M'exico, M'exico\n- n Departamento de F'ısica, CUCEI, Universidad de Guadalajara, Guadalajara, M'exico \no \nDepartment of Physics, University of Wisconsin-Madison, Madison, WI, USA \np \nDepartment of Physics, University of Maryland, College Park, MD, USA \n- q Instituto de Astronom'ıa, Universidad Nacional Aut'onoma de M'exico, Ciudad de M'exico, M'exico r Instituto Nacional de Astrof'ısica, ' Optica y Electr'onica, Puebla, M'exico\n- s\n- Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 IFJ-PAN, Krakow, Poland\n- t Facultad de Ciencias F'ısico Matem'aticas, Benem'erita Universidad Aut'onoma de Puebla, Puebla, M'exico\n- u Centro de Investigaci'on en Computaci'on, Instituto Polit'ecnico Nacional, M'exico City, M'exico. v Doctorado en F'ısica-Matem'aticas, CUValles, Universidad de Guadalajara, Guadalajara, M'exico\n- w Doctorado en Tecnolog'ıas de la informaci'on, CUCEA, Universidad de Guadalajara, Guadalajara, M'exico\n- x Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA y NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA\n- z Departamento de Bioingenier'ıa Traslacional, CUCEI, Universidad de Guadalajara, Guadalajara, M'exico\n- aa Tecnologico de Monterrey, Escuela de Ingenier'ıa y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L., M'exico, 64849\n- ab Max-Planck Institute for Nuclear Physics, 69117 Heidelberg, Germany\n- Departamento de Ciencias B'asicas, Tecnol'ogico Nacional de M'exico Campus Oaxaca, Oaxaca,\n- ac M'exico\n- ad Universidad Politecnica de Pachuca, Pachuca, Hgo, M'exico\n- ae Dipartimento di Fisica 'Ettore Pancini', Universit'a degli studi di Napoli 'Federico II', Complesso Univ. Monte S. Angelo, Napoli, Italy\n- af INFN - Sezione di Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy \nag \nINAF-Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, Napoli, Italy \nah \nDept of Physics and Astronomy, University of New Mexico, Albuquerque, NM, USA \nai \nChulalongkorn University, 254 Phayathai Road, Pathumwan, Bangkok, Thailand \n- aj National Astronomical Research Institute of Thailand (Public Organization), Don Kaeo, MaeRim, Chiang Mai, Thailand\n- ak Erlangen Centre for Astroparticle Physics, Friedrih-Alexander-Universitat, Erlangen-Nurnberg, Erlangen, Germany\n- al Center for Relativistic Astrophysics School of Physics, Georgia Institute of Technology, Atlanta GA, USA\n- am Space Science and Applications Group, Los Alamos National Laboratory, Los Alamos, NM, USA an NASA Goddard Space Flight Center, Greenbelt, MD,USA\n- ao\n- Departamento de F'ısica y Matem'aticas, Universidad de Monterrey, Monterrey, NL, M'exico ap Instituto de Ciencias Nucleares, Universidad Nacional Aut'onoma de M'exico, Ciudad de M'exico, M'exico\n- aq Universidad Aut'onoma del Estado de Hidalgo, Pachuca, M'exico\n- ar D'epartement de Physique Nucl'eaire et Corpusculaire, Facult'e de Sciences de l'Universit'e de Gen'eve, CH-1205 Gen'eve\n- as Department of Physics, Stanford University: Stanford, CA 94305-4060, USA at 'Instituto de F'ısica Corpuscular, CSIC, Universitat de Val'encia, E-46980, Paterna, Valencia, Spain au Radboud Universiteit, Nijmegen, Netherlands\n- av Department of Physis, Sungkyunkwan University, Suwon 16419, South Korea aw Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, China ax NASA Marshall Space Flight Center, Astrophysics Office, Huntsville, AL 35812, USA ay University of California Irvine, Irvine, CA 92697, USA\n- az\n- Institute for Nuclear Research of Russian Academy of Sciences, Moscow, Russia ba Centro de Investigaciones y Estudios Avanzados del Instituto Polit'ecnico Nacional, Ciudad de M'exico, M'exico", 'Abstract': "The High-Altitude Water Cherenkov (HAWC) observatory is a second-generation continuously operated, wide field-of-view, TeV gamma-ray observatory. The HAWC observatory and its analysis techniques build on experience of the Milagro experiment in using ground-based water Cherenkov detectors for gamma-ray astronomy. HAWC is located on the Sierra Negra volcano in M'exico at an elevation of 4100 meters above sea level. The completed HAWC observatory principal detector (HAWC) consists of 300 closely spaced water Cherenkov detectors, each equipped with four photomultiplier tubes to provide timing and charge information to reconstruct the extensive air shower energy and arrival direction. The HAWC observatory has been optimized to observe transient and steady emission from sources of gamma rays within an energy range from several hundred GeV to several hundred TeV. However, most of the air showers detected are initiated by cosmic rays, allowing studies of cosmic rays also to be performed. This paper describes the characteristics of the HAWC main array and its hardware.", 'Keywords:': 'Physics - Instrumentation and Detectors; Water Cherenkov Detectors; Astrophysics; High Energy Physics - Experiment; Nuclear Experiment', '1. Introduction': "Very-high-energy (VHE) gamma rays (E > 100 GeV) probe the non-thermal universe, tracing the sites of particle acceleration around black holes, neutron stars, astrophysical jets, massive star formation regions, and other objects where strong shocks are present. While human-made particle accelerators can currently accelerate particles to a few tens of TeV, nature can accelerate particles to at least 10 8 TeV (e.g. [1, 2]). Gamma rays have been observed to energies above 100 TeV (e.g. [3]). VHE gamma rays can be produced by inverse Compton scattering, synchrotron and bremsstrahlung radiation, and the decay of neutral pions created by collisions of accelerated hadrons with an ambient medium or radiation fields (e.g. [1, 4] and references therein). VHE gamma-ray observations shed light on environments where charged particles are accelerated, thereby gaining insight into some of the most extreme regions of our universe. From radio to VHE gamma rays, multi-wavelength observations create a detailed picture of these acceleration environments. More recently, multi-messenger observations of gamma rays, neutrinos, and gravitational radiation have been used to improve our understanding of mergers of black holes and neutron stars [5]. \nThe Earth's atmosphere is opaque to X-rays and gamma rays. At sufficiently high energies, gamma and cosmic rays can generate extensive air showers (EAS), with a pancakeshaped wavefront of relativistic particles (mostly electrons, positrons, and lower-energy photons from primary gamma rays, and pions and muons from cosmic rays). The EAS generates Cherenkov light in the atmosphere (detected by atmospheric Cherenkov telescopes; e.g. [6, 7] and references therein), and if the initial gamma ray has sufficiently high energy, particles in the EAS will survive to ground level and can be detected by an Preprint submitted to Nuclear Instruments and Methods in Physics Research Section A April 12, 2023 \nFigure 1: The HAWC observatory, seen looking north from the Sierra Negra volcano. The Citlaltepetl volcano is visible in the background. The primary detector (HAWC) is at the center, surrounded by the smaller plastic tanks (outriggers) not discussed in this paper (see Section 7). Credit: The HAWC Collaboration. \n<!-- image --> \narray of detectors on the ground. Thus, there are three (overlapping) gamma-ray energy regimes depending on the detection techniques. At low energies, space-based instruments (e.g. Fermi-LAT; [8]) are used to detect the primary gamma rays from ∼ 50 MeV to ∼ 100 GeV, atmospheric Cherenkov telescopes work from ∼ 50 GeV to ∼ 10 TeV and particle detection arrays operate from ∼ 1 -∼ 1000 TeV [3, 9]. In this paper, we discuss the High Altitude Water Cherenkov (HAWC) TeV gamma-ray observatory, which uses water Cherenkov detectors (WCDs) to detect the particles within an EAS that survive to ground level at high altitude. The HAWC observatory was designed to extend the capabilities of the first-generation water Cherenkov TeV gamma-ray observatory, Milagro ([10-12] and references therein). HAWC became the first EAS gamma-ray detector to observe dozens of sources at energies from one TeV to above 100 TeV (e.g [13] and references therein). For a review of WCDs for gamma-ray astronomy, EAS, and Cherenkov radiation, see [2] (and references therein). \nThe HAWC observatory is located in the state of Puebla, M'exico in the Pico de Orizaba National Park at an altitude of 4100 m.a.s.l. (meters above sea level), in the saddle region between the Sierra Negra volcano, or Tliltepetl (altitude 4582 m.a.s.l.), and Pico de Orizaba, or Citlaltepetl ('star mountain' in Nahuatl; altitude 5636 m.a.s.l. Figure 1 shows a picture of the location of the HAWC observatory with Citlaltepetl in the background. The Gran Telescopio Milim'etrico (GTM) Alfonso Serrano or Large Millimeter Telescope 5 (LMT; [14]) site is at the summit of Sierra Negra. The facilities and infrastructure of the nearby LMT have been leveraged for the benefit of the HAWC observatory. The HAWC observatory is comprised of the original primary detector (HAWC hereafter) and an upgrade consisting of 345 outrigger detectors 6 (see Section 7; [15] and references therein). The facilities also include a central Counting House (CH) with electronics and \ncomputers, and a support building, called the HAWC Utility Building (HUB), containing water purification facilities and an office. More details about the development of the site, history, and origin of the HAWC observatory are presented in [16-19]. \nThe latitude of the HAWC observatory is 18.99 · North. Taking 50 · from zenith as the limit of the viewable field, we are able to observe gamma-ray sources to declination to at least 31 · South (including the Galactic center). In a sidereal day, we observe up to 9 sr ( > 70% of the entire sky). Observations at larger zenith angles are possible, but the sensitivity declines, and the energy threshold increases rapidly with further increasing inclination due to increased atmospheric depth and shorter transits. The 97.31 · W longitude of the HAWC observatory is similar to that of major observatories in M'exico, the United States, and South America. This location facilitates prompt multi-wavelength follow-up observations of transients found in HAWC data. \nThe observatory was constructed with financial support led by the Consejo Nacional de Ciencia y Tecnolog'ıa (CONACyT) of M'exico, and the United States National Science Foundation (NSF) and Department of Energy (DoE). \nThis paper focuses on the hardware of the HAWC main array. A forthcoming paper will describe the outrigger array (additional small tanks to improve HAWC performance at high energy). In Section 2, we present an overview of HAWC, including a history of its construction and a summary of HAWC science topics. In Section 3, we present a description of the instrument and its construction. The electronics, calibration, and operations are discussed in Sections 4, 5, and 6, respectively. The future of the HAWC observatory and its recent outrigger upgrade is briefly discussed in Section 7.", '2. The HAWC Primary Detector: Overview, Installation History, and Science': "HAWC is an array of 300 steel water tanks equipped with large-area photomultiplier tubes (PMTs). These WCDs cover an area of over 22,000 m 2 (Figure 2 top), five times that of the top layer of Milagro and about ten times larger than its bottom layer 7 . The combination of increased array area, higher altitude (4100 m.a.s.l. vs. 2630 m.a.s.l.), optical isolation of the HAWC WCDs, and all HAWC PMTs being located at an intermediate depth of 4 m, results in a sensitivity increase of more than one order of magnitude when compared to Milagro [11, 20]. HAWC's vertical atmospheric depth is 637 g cm -2 , equivalent to 17.4 radiation lengths. The reduced overburden compared to Milagro (20.8 radiation lengths) allows HAWC to sample air showers more than 3 radiation lengths earlier (nearer to shower maximum) allowing more particles to arrive at the detector. This lowers HAWC's energy threshold and improves reconstruction accuracy and hadron rejection. \nWe flattened the HAWC platform, with a slope of approximately 1.0%, enough for the drainage of precipitation. The downward direction of the slope points 15.8 ° West of North along the direction of the tank columns. Consequently, the normal to the platform is tilted from the zenith along the direction of this slope by approximately 0.7 ° . \nThe WCDs (see Figure 2 center) are cylindrical steel structures ( ∼ 7.3 m in diameter and ∼ 5.4 m in height that support a light-tight bladder containing ∼ 180,000 liters of \npurified water (see Section 3.2). The water depth is 4.5 m. The minimum separation between two adjacent WCDs is 60 cm, sufficient to service the entire exterior of a tank. There is an aisle 2.6 m wide between each column pair to allow passage of a standardwidth truck. A 4.3m access road, known as 10th Avenue, runs diagonally across the array. Cable trenches with conduits and pull-boxes were placed in 10th Avenue as well as the narrower diagonal paths visible in Figure 2. \nHAWC reuses 900 Hamamatsu R5912 eight-inch PMTs [21] and 1200 front-end electronic channels from Milagro, with a newly developed Data Acquisition (DAQ) System. An additional 300 larger Hamamatsu R7081 ten-inch PMTs [21] with higher quantum efficiency (30%) were added to increase sensitivity to lower-energy gamma rays (see Section 3.9). Thus, every WCD has four upward facing PMTs anchored by a mounting system. The ten-inch PMT at the center is surrounded by three eight-inch PMTs in an equilateral triangle configuration. The PMTs are ∼ 0.5 m above bottom (one radiation length = 0.4 m). The distance from the eight-inch PMTs to the central ten-inch PMT is 1.83 m, midway between the center and the outer wall (see Figure 2 down). \nThe ∼ 200 m × 150 m HAWC layout shown in the top panel of Figure 2 was simulated using HAWCSim. HAWCSim contains a detailed description of the HAWC array and propagates the air shower particles provided by CORSIKA [22] at the top of the WCDs. It uses GEANT 4 [23], including the generation of Cherenkov photons and the response of the PMT's down to a single photoelectron. A detailed description of the HAWC design can be found in [24]. We chose HAWC's final layout by balancing cost, site restrictions, and performance. Given the constraints and insights from simulation, the design of HAWCensures that it functions effectively as a calorimeter for secondary electromagnetic particles. The water depth above the PMTs ( ∼ 4 m ≈ 10 radiation lengths) optimizes sensitivity to Cherenkov emission from secondary electromagnetic particles. \nIf the water column above the PMT is too large, the sensitivity to the number of photoelectrons (PEs) per energy will be too low. If the water above the PMTs is too shallow, the response due to the proximity of the secondary electromagnetic particles from the EAS to the photocathode would be too large and not proportional to the deposited energy. For the chosen water depth, HAWC detects ∼ 40 PEs/GeV of deposited energy 8 for electromagnetic particles. More details of the configuration and detector design are presented in [2, 24-34] and references therein. \nThe use of individual tanks (as opposed to the large single water volume of Milagro) enabled HAWC to be constructed using a staged approach, which allowed for operating with a subset of the final array. We used these first observations to perform system-wide testing of components and the data acquisition system, ensuring a smooth transition to full operations. Even with a fraction of the complete array, HAWC was the most sensitive all-sky observatory in the VHE regime. The total time to complete construction was 3.5 years, beginning in 2011. HAWC-30 (with 30 WCDs), completed in September 2012, was used mainly as an engineering array. As we installed additional WCDs, they were incorporated into the data acquisition system. HAWC-111 operated from August 2, 2013, through July 8, 2014, with three to five times greater sensitivity than Milagro ([34]). The complete 300 WCD (HAWC) array began operations on March 20, 2015. More details \nabout HAWC construction and its engineering prototype Verification and Assessment Measurement of Observatory Subsystem (VAMOS) are presented in [35]. \nHAWC has detected over 100 sources of VHE gamma rays and has contributed significantly to our comprehension of the high-energy universe. The third HAWC catalog of very-high-energy gamma-ray sources is presented in [13]. The science topics (with representative references) can be summarized as: \n- · Discovery and studies of the TeV gamma-ray sky, including extragalactic sources such as active galaxies [36], gamma-ray bursts and transients [37], as well as Galactic gamma-ray sources such as PeVatrons [38-40], pulsar-wind nebulae and the Crab [41], TeV Halo Objects [42], microquasars and binary systems [43], Fermi bubbles [44], and molecular clouds [45].\n- · Cosmic-ray studies of the anisotropy of arrival directions, all-particle energy spectrum, and composition studies [46].\n- · Fundamental physics such as probing beyond the standard model of particle physics by searching for dark matter [47] and violations of the Lorentz invariance [48] in the high energy regime. \nOther topics include searches for primordial black holes evaporation [49] and studies of the Sun, the interplanetary medium [50] and space weather [51]. The synergy of HAWC with other observatories such as IceCube, VERITAS, Fermi -LAT, and H.E.S.S. is exemplified in [52].", '3. Components of the HAWC Detector Hardware': 'We describe the steel tanks, bladders, water, electrical infrastructure, cabling, and photomultiplier tubes that comprise the hardware elements of the HAWC observatory in the following sub-sections.', '3.1. Steel Water Tanks': 'Each of the 300 water tanks is made of corrugated, hot-dipped galvanized steel panels which are bolted together. The WCDs are covered with a dome-shaped, UV-hardened, sand-colored, scrimmed, vinyl polyester fabric roof to prevent rain and snow accumulation on the bladder and to serve as an additional light barrier. The ground inside the steel structure is flattened and covered with a layer of sand to prevent rocks from penetrating the bladder. Geo-textile felt on the ground and walls prevents sharp objects such as bolts from puncturing the bladder. \nWe constructed the steel tanks top-down. We assembled the top ring of steel panels first, at ground level, and then raised it with jacks to allow the assembly of the next ring of panels. Then we raised these two rings again to allow another ring to be added below. This procedure continued until the bottom ring. Finally, we lowered the entire tank into a .6m trench and back-filled it with rammed earth, serving as an anchor that prevents tank upheaval during an earthquake (HAWC tanks met 2011 standards for International Building Code Risk Category III [53] and Seismic Design Category D [54]). Thus, we assembled the tank safely from the ground without working on ladders, scaffolding or a crane.', '3.2. Tank Bladder': 'The cylindrical tank bladders are made of flexible, low-density polyethylene. This is shown in Fig. 3a. Bladders enclose the water volume and act as a light barrier. The bladder material ( ∼ 0.4 mm thick) is composed of two layers of three-substrate film fused/bonded during a co-extrusion process. \nAlthough the metal tank sides also provide a light barrier, the bladders act as the primary protection against external light reaching the PMTs. We tested to ensure that the laminate and seals were entirely opaque at single-photon levels for light with wavelengths between 260 nm and 600 nm. In addition, over the top of the bladder, we put a black film (agriculture foil) as an extra light blocker, covering the excess fiber to avoid light leaks. Each bladder has a PVC hatch on the top for access and installation of the PMTs and their cabling. The hatch has a film cover over it to block light passage through the hatch and the penetrations of the cables and optical fibers. The bottom of the bladder has an integrated mounting fixture for each PMT. \nThe part of each mount outside the bladder bottom is attached to a stake in the ground, which was surveyed prior to tank construction, to ensure that the position of the PMT is known to be within 0.16 cm. \nIn Fig. 3 b, c, and d, we show the pulley system to install the PMTs after the tanks are filled with water. This system decouples PMT deployment from water delivery and allows easy replacement of PMTs during operations. A loop of Kevlar string connects the hatch of the bladder (at the top of the tanks) with the PMT mounts at the bottom of the bladders. The bottom of each PMT is attached to a small plastic ball which locks the PMT to the mounting fixture. The PMTs point upward due to their buoyancy of about 80 N. PMTs can be easily removed by pulling from the other side of the Kevlar loop and disengaging the ball from the mount.', '3.3. Water Filtration System': "We soften, sterilize, and filter well and surface water to fill the WCDs. HAWC water filtration is a multi-step process to provide clear, pure water, that starts at the water source and continues at the HAWC site. \nAt the well (located in the valley at 2640 m.a.s.l.), the water goes through a 30-micron pre-filter before it enters the softener system, where the hardness is reduced to below 2 grains. We do the softening at the well because the softener backwash water cannot be dealt with at the HAWC site since it is in a national park. Trucks of 15 kiloliter capacity transport water from the well to HAWC. At HAWC, the water is pre-filtered again at 15 microns before it is transferred to one of five 'dirty' water storage tanks. \nThe filtration system is designed to run 8 hrs/day and can fill at least one tank (13 trucks) per workday. Trucks ran up to 7 days a week when recovering from bad weather or transportation problems. \nFigure 4 shows the filtration system components, including one-micron filters, an Aquafine Model TSG 253 UV sterilizer [55], charcoal filters, and a second set of onemicron filters. The throughput is limited by the internal impedance and the flow capacity of the UV sterilizer. \nThe filtered water is stored in two 'clean' HAWC water tanks. A pump connected to a fixed underground PVC pipe and flex pipes fills a WCD in seven hours. If water needs further filtration, it can be pumped back through the filtration system.", '3.4. Attenuation length and water transparency': "The attenuation length λ describes the water transparency and is computed from the light-path length l and fractional transparency f by inverting f = exp( -l/λ ). A sufficiently large attenuation length is necessary for the successful long-term operation of a WCD, to avoid loss of Cherenkov light due to the presence of organic material (e.g., bacteria and algae). Our instruments measure the fraction of light arriving at a photodetector, which is dominated by apparent losses due to scattering. For detector purposes, scattering matters less than actual light absorption, but our measurement of the combined scattering and attenuation length is sufficient to monitor relative water quality. We chose a minimum λ of 15 m in the filtration system and 10 m inside the WCD. We determined these values by considering the 4.5 m water column inside the WCD, the sensitivity of PMTs, and the duration of HAWC operation ( ∼ 10+ years). \nTo monitor the transparency and water quality of HAWC, we perform several measurements both on-site and in a laboratory. We obtain these water samples from: water collection at the HAWC site (melting snow and natural springs), the truck delivering the water from the well, after the water filtering processes, when filling each WCD with water, and from inside the WCDs over time. \nWe use a convenient commercial transmission meter (C-star [56]) for water quality monitoring. Typical attenuation measurements do not consider the UV regime. The transmission meter operates with a 470 nm blue light laser diode as the emitter, which travels a path length of 0.25 m until it reaches the receiver. In each measurement, the fractional transparency f r is determined as [57]: \nf = C sample -C darkness C calibration -C darkness , (1) \nwhere C sample is the measurement for the sample, C darkness is the respective dark current, and C calibration is the measurement for distilled water. The transmission length is determined by \nλ = -l ln( f ) . (2) \nWe also performed more precise calibration measurements in a reference laboratory (see [58] for details). That setup consisted of a 30 mW power 405 nm wavelength laser, a spectrometer, and a longer 1 m cylindrical tube with two quartz windows. Using an empty, dry, and clean tube, we measured the loss per quartz window to be between 5% and 7%. Measurements for the transmittance of the reference distilled water are taken with the same tube and the same orientation. With this system, the corrected fractional transparency f ' for the water alone is determined by: \nf ' = C A + f w , (3) \nwhere C is the measurement of the water sample, A is the intensity of the laser, and f w is the fractional loss per window (with water in the tube, only the two air/quartz interfaces are important). \nThis transmission measurement for the water alone allows comparison of the C-star measurements with the measurements performed at the laboratory, and confirms that \nthe transparency of water reaches at least 15 m in the filtration system and about 12 m in the WCDs. \nTo save money on water transport from the well, we used some water collected from melting snow and natural springs. This water (especially when spring flow is low) has a yellowish color due to lichens, which remains even after all organic matter is filtered out. Processing the discolored water depleted the charcoal and stained the filters. We diluted all spring water with well water, and eventually stopped using it. Some tanks with mixed water had scattering/attenuation length as low as 8m. Less than a quarter of the tanks had λ of 8-10 m; the vertical muon charge deposition in such tanks was less than 3% different from that in tanks with λ > 10m.", '3.5. AC Power and Backup Solar Photovoltaic System': "The Mexican power grid is HAWC's principal electrical power source. We extended the existing medium high voltage (HV) transmission line (installed for the LMT) by one km to the HAWC site. A 225 kVA transformer steps down the 34.5 KV transmission line to 3-phase 220 V for distribution to equipment throughout the HAWC site. \nHAWC uses backup solar power for essential services. The HAWC 4.1kW solar-power system has 18 panels. A deep cycle battery bank, which in normal conditions is charged by the AC line voltage, or in its absence by the solar panels, provides at least a week of power for critical monitoring and communication tasks.", '3.6. Electrical AC grounding': 'HAWC is constructed on the slope of a volcano that has low soil conductivity. The ground impedance is typically around a hundred Ohms measured with a four-point ground monitoring technique and not suitable for a safe electrical ground without special treatment. Therefore, HAWC has three separate electrical grounds to provide a secure electrical system. The first electrical earthing is at the central transformer station where the incoming underground AC high voltage is transformed into three-phase 220 V and 120 V. A second earthing is at the HUB, which is created using the rebar iron in the large concrete pad in addition to long copper stakes - a system referred to as a Ufer ground or a concrete-encased electrode (CEE) by NEC guidelines [59, 60]. The third and most critical earthing is established at the CH substation for electronics. This transformer provides power for all HAWC electronics through 2 UPS (Uninterruptible Power Supply) systems. The 4 m × 10 m ground structure is based on three components: 3 m long copper stakes, large area short copper stakes to increase surface contact, and ground enhancer along the connecting copper wire. This system is about 0.5 m underground to reduce effects of surface drying. Earth resistivity measurements show a typical value of 2.5 Ω to 5.0 Ω depending on the season. This ground is also used by the outdoor spark gaps which protect every copper cable entering the CH against voltage surges; optical fiber connections are used wherever possible.', '3.7. Lightning and Grounding Improvements': "In 2014 lightning struck about 200 m from HAWC and damaged electronics. Although lightning is common at HAWC in the summer, most strikes land nearer the peak of Sierra Negra (close to the LMT). Most of the damage was to the emitter-coupled logic (ECL) interfaces among electronics components, requiring expensive and lengthy repair. Parts \nof the water-level monitoring system were also impacted. The damage was likely caused by different electronics crates floating to different levels, perhaps 10V or more apart, exceeding the allowed range of differential ECL inputs. The damage mechanism was not completely understood. \nWe examined HAWC grounding inside and outside the CH and took many preventive actions: \n- · We ensured by measurement, solid connections of each VME crate to the ground, adding jumpers as needed.\n- · We tied crate grounds to each other more strongly with copper braid to the central point (the transform AC ground), and we connected them to the racks with serrated washers. We also added (for fast transients) copper foil/tape amid crates of a rack, among racks, and below cable, paths running between racks.\n- · We changed the GPS signal entry to the CH from a wire cable to a radio repeater.\n- · We separated solar power from 'clean' electronics power and grounding.\n- · In addition, we connected all the steel WCDs to each other in a large mesh made with 6 AWG (American Wire Gauge [61]) copper wire. This eliminates surface voltage differences between the WCD structures which had previously been observed when low clouds passed over the array. This WCD grounding is not connected to any electrical ground and has no electrical connection to PMTs inside tanks. This change did not, as hoped, much reduce PMT rate changes due to local electric fields. \nWe had operated for 2-3 years before the problem occurred. We have not had lightning damage in the CH for eight years after these preventive actions were taken, despite significant lightning storms and strikes within 20m of the HAWC main array.", '3.8. Cabling': 'Several hundred km of cables and optical fibers (installed about 1m underground) connect to the WCDs of the HAWC observatory. From each WCD, four HV/signal cables (one per PMT), one optical fiber, and one ethernet category five (CAT5) cable run to the CH. Each PMT base is directly connected to an SHV-terminated cable running to a tank-side access box in which these cables connect to the long HV cables that run toward the CH. Junction boxes for the optical fibers and CAT 5 cables are also mounted on each tank. Locating the CH at the center of the array minimizes the cable length to the most distant PMTs. Figure 5 shows the layout of WCDs and cable trenches.', '3.8.1. High-Voltage Cables and their Spark Gap Protection': 'HAWC HV/signal cables are RG-59 Belden 8241 coaxial cables designed for analog video signals. They have an external diameter of 6.15 mm, and a central wire gauge of 23 AWG [62]. This cable type was successfully used (for outrigger detectors) in Milagro. The cables can withstand 3100 V DC, so they are appropriate for carrying both HV and our PMT signals together on the same cable. The PMT signal is isolated from the DC HV through a pick-off capacitor in the HV sector of the front-end electronics (Section 4). \nTo ensure cables have identical electrical properties, we delay-match each HV cable to a specific standard reference Golden Cable . The standard Golden Cable has a length of \n149.3 m with a delay of 761.2 ns. A sample of 585 cables used in HAWC had a measured average time delay of 761.2 ns with a standard deviation of 0.8 ns. A full description of the HV cable manufacture and testing is given in [63]. \nSpark gaps protect the electronics from high voltage bursts induced by lightning, transmitted through the signal/HV cables. Each HV cable from the PMTs (long cable) connects to a spark gap box in a electrical cabinet near the CH (see Figure 6). From these spark-gap boxes, a 10 m length of RG-59 cable (short cable) runs to the CH. Buried boxes near the cabinets store any excess cable length of long cables , as well as of optical or CAT5 cables. \nThe HAWC spark gap has the same design as used in Milagro. The device is a small aluminum box (shown in Figure 6) containing two Teledyne-Reynolds spark gaps and SHV connectors for the incoming and outgoing HV/signal cable. A DKF-3000L spark gap tube is connected between the conductor and the cable shielding braid, limiting their voltage difference to 3 kV. A DFK-0230L spark gap between the cable shielding braid and the HAWC ground structure, limits their voltage difference to 230 V. \nThe spark gap tubes have an initial resistance of over a gigaohm, and a capacitance of less than 10 pF. They do not interfere with the DC HV or PMT signal. HV shrink tubing insulates the spark gaps. \nThe spark gaps are rated for 40,000 amperes (for a wave shape with an 8-microsecond rise and 20-microsecond decay to 50%) and can sustain up to three to five lightning strikes. The aluminum box itself connects the input and output shield braids but is otherwise floating and does not connect to the ground structure or any other device. Individual boxes are grouped and mounted inside a weatherproof outdoor electrical cabinet. To prevent condensation during weather below the dew point, we added heating tape to warm each spark gap cabinet.', '3.8.2. Fiber Optics': 'The HAWC optical fiber system is used to measure the relative PMT time offsets vs. signal amplitude as well as to calibrate the signal amplitudes. The optical fibers run from the calibration system (see Section 5 ) fiber-optic distribution network in the CH to distribution boxes at the base of each pair of neighboring WCDs, where they connect with a local fiber leading to a diffuser ball in each WCD tank.', '3.8.3. Water Depth Monitoring Cables': 'A 12m cable attached to the water depth sensor (see Section 6.2) at the bottom of the tank ends in a junction box at the WCD with an XLR connector. Unshielded Twisted Pair (UTP) CAT5 cables run from junction box toward the CH. These cables have eight conductors, sufficient for signals from 2 WCDs. As with the signal/HV cables, the long CAT5 cable run goes to an electrical cabinet near the CH containing ethernet spark gaps, after which a 12 m cable runs to the CH.', '3.9. Photomultipler Tubes': "As mentioned in Section 2, HAWC has two PMT types: Hamamatsu R5912 and R708102, which have the following specifications: \n- · Spectral response from 300 to 650 nm with a peak response at a wavelength of 420 nm. \n- · Dynode chain of 10 stages\n- · Anode Dark current 100 nA (typical) 1000 nA (max)\n- · Typical Rise Time of 3.6 ns\n- · Typical Transit Time of 62 ns with a spread of 2.4 ns \nThe R7801 has 530 cm 2 of photocathode area, ∼ 40% larger than the R5912 (340cm 2 ). The increase in area and efficiency of the R7081 enhances response, while only increasing the transit time spread from 2.4 ns to 3.4 ns. More details are reported in the respective Hamamatsu manuals [21]. \nThe PMT base consists of a passive resistor chain with high voltage capacitors on the last two dynodes in the chain to prevent any voltage drop from large pulses. To maximize the sensitivity to single photoelectrons and compensate for the cable attenuation (see Figure 7), we operate the PMTs a high gain of ≈ 10 7 . The dispersion in the cable is calibrated to minimize its impact on timing due to threshold discrimination electronics. The bases have generally been stable, but some carbon Megaohm HV resistors have degenerated over time (many of these bases are over 20 years old), causing slow or catastrophic decreases in gain. HV capacitors (perhaps due to component quality issues) have also failed in some newer bases. We have an ongoing campaign to replace broken carbon resistors with metal film resistors. \nWe measured the gain of each individual PMT and grouped PMTs with similar voltage versus gain curves into the same WCD tanks. There are advantages to grouping similar PMTs in the same tank: \n- · The high voltage cables for eight PMTs (a WCD tank pair) are connected to a single HV supply channel. Therefore, matched PMTs have similar gains.\n- · The PMT gain in all tanks can be made similar by selecting an appropriate voltage for each HV channel, simplifying data analysis and simulation. \nWe measured the dependence of response as a function of photon impact position of the R5912 PMTs using a Robotic Characterization System (RCS) [64]. This system automatically measures PMT response at 101 locations distributed over the PMT's spherical active surface. A charge integrating ADC with a 20 ns time window and a conversion of 0.25 pC per channel digitized pulses from PMTs in response to the LED's flashing. We varied the PMT's HV from 1400 V to 1850 V to measure gain curves. With typical operating voltages corresponding to a gain of 1.5 × 10 7 , the charge spectra for the LED mean light output varied from 1 to 5 PE. A detailed description is available in [65]. Subsequent measurements, with varying light intensity, allowed us to distinguish gain variation from charge collection efficiency; we found that PMTs suffer significant collection efficiency loss at the edges the photocathodes.", '4. HAWC Readout Electronics': "Here we describe the components of the readout electronics chain. Figure 8 shows a schematic overview of the HAWC electronics in the CH. Front-end boards (reused from \nMilagro) make digital signals from the PMT inputs, including a time-over-threshold (ToT) measurement of amplitude at two thresholds. Each PMT's ToT signal is recorded by a commercial Time to Digital Converter (TDC). A central custom GPS timing and control system (GTC) provides timing and control of all TDCs. Single-board computers (SBCs) read out the TDCs. The SBCs group data into fixed-length time blocks before sending them to the online reconstruction CPUs. A separate data acquisition system records hardware scaler rates. The main elements of HAWC electronics are described in the following sub-sections. Further details about the electronics can be found in [65]. We found it very beneficial to develop and test software and hardware interfaces at a comfortable integration site before shipping and installation in the HAWC CH. \nFigure 9 shows a sketch of the HAWC DAQ system. The control computer starts and stops data taking runs and interacts with the GTC electronics to manage the TDCs and readout computers. The DAQ and online processing system assemble, store and reconstruct the TDC data. Analysis clients provide real-time event-based monitoring, and analysis clients can produce real-time alerts (see Section 6.7). The DAQ system and software are discussed in more detail in [70].", '4.1. Front-End Board (FEB) Electronics': 'The cable from the PMT (after the spark gap) connects to a FEB. The signal cable supplies HV to the PMT and the analog anode PMT signal is AC-coupled to the RG59 cable. The PMT signal is capacitively picked off at the FEB, amplified and shaped with a 75 ns time constant, then digitized by a two-threshold ToT circuit. The time at which the threshold is passed provides the timing used in the angular reconstruction. \nFigure 10 shows PMT signals (on the left) and associated FEB discriminator outputs (on the right). For a small signal (upper waveforms on the left and right), the resulting output of the FEB is a single digital pulse with two edges at T 0 and T 1 : the times the pulse crosses the Low Th threshold while rising and falling (respectively). The time duration that the PMT signal exceeds the low threshold, LoToT, in this case, is T 1 -T 0 . For the large-signal case (lower waveforms), the output of the FEB is two digital pulses with four edges at T 0 , T 1 , T 2 , and T 3 . These are the times of rising crossings of the Low Th and High Th, and the falling crossing times of the High Th and then the Low Th. The time duration that the PMT signal exceeds the low threshold, LoToT, in this case, is T 3 -T 0 . The time duration that the PMT signal exceeds the high threshold, HiToT, is T 2 -T 1 . \nThe FEB electronics are implemented in ECL logic which is fast but requires high power. FEBs are described in more detail in [68]. FEBs were modified in several ways for HAWC. First, analog FEBs distribute two input HV levels to 8 PMTs, instead of one HV for 16 channels. Second, HAWC PMTs are operated at a lower gain than in Milagro to allow for larger showers without saturation. As a result, the Low Th and High Th thresholds were reduced from 30 and 80 mV to 20 and 50 mV, maintaining thresholds at about 1/4-1/2 photoelectrons and 3-4 photoelectrons Low Th and High Th, respectively. Finally, because the HAWC TDCs (see below) have better two-pulse resolution (5 ns) than Milagro TDCs, timing constants in the edge processing were adjusted to reduce the minimum delay time between Lo and Hi edges. This reduces the chance that two small hits nearby in time could be confused with a single large hit passing both Lo and Hi thresholds, since either scenario produces 4 edges.', '4.2. Power Supplies': 'Wiener PL506 low voltage power supplies provide the +5.2 V, +5 V, and -5 V for the FEBs. Wiener 6023 × 610 VME crates housed the scaler and TDC data acquisition system. A Wiener MPOD mainframe (with five ISEG EHS 32-channel 20125p modules) provides the PMT high voltage. Selecting Wiener for all provides a consistent software control interface. \nThe High Voltage passes through a custom breakout box to split the high-density 32 channel HV cable into individual SHV cables that connect to the front-end boards. PMTs were grouped to ensure that four tubes with consistent HV settings were deployed in each tank, and a single SHV connector fed two tanks with compatible HV requirements. Each front-end board holds 16 channels or four tanks. This arrangement facilitated installation and repair, as only one HV channel has to be turned off to service a tank. The PMTs are operated with positive HV so that the photocathode in the water is at ground.', '4.3. Scaler DAQ': 'The scalar DAQ provides a robust set of scaler rates, allowing the monitoring of rates even under severe weather conditions that overwhelm the TDC system, and independent of how we operate the TDC DAQ system. The hardware scaler system is also used in cosmic ray and solar physics analyses (e.g [51] and references therein). We use Struck SIS3820 VME scalers with rear transition modules with a density of 64 channels per VME slot. One Wiener VME crate records all 1200 PMT signals. Input signals come from the Low Th threshold output of digital FEBs for each PMT, and the monitoring system records rates once per minute.', '4.4. TDC DAQ': 'The TDC DAQ for the HAWC PMTs is based on the 128 channel VME CAEN V1190S2eSST TDC. These TDCs have multiple-hit capability, record times with a granularity of .1 ns and support several readout modes. As discussed in Section 4.1, the input to the TDCs combines arrival time and pulse height information, allowing measurement of ToT for two separate thresholds (a 2-bit nonlinear ADC encoding), whose analysis is described further in Section 5. \nTDCs are located in two VME crates with each backplane subdivided into five independent sub-backplanes. Each independent 4-slot sub-backplane holds a TDC and a GEXVB602 single-board computer by GE/FANUC (SBC) powered by an Intel i7 processor. One SBC reads out each TDC using the 2eSST protocol [69] using a CENTOS5 library kindly provided by Sergey Boiarinov of the Jefferson Laboratory. The ten TDCs read out in parallel at 50 MB/s using this protocol, with a total data volume of 500 MB/s passing through a network switch to the online farm. Rather than selectively reading out TDCs on air shower events, TDCs are periodically read out by a TRG signal generated every 25 µ s, and the entire data stream is transferred to the processing farm. Each data frame covers a 26 µ s time interval, slightly overlapping to avoid complex processing at frame boundaries. The online farm applies a software event triggering criterion of 28 of the 1200 PMTs having hits within 150 ns and saves those hits; this multiplicity threshold is comfortably lower than the requirements of most physics analyses. The multiplicity criterion fires at a 25 kHz event rate and reduces the stored raw data stream to 20 MB/s. Events are fully reconstructed online for real-time analysis and monitoring, but the full \n20MB/s is saved for offline reconstruction. The DAQ system is described in more detail in [70]. \nThe HAWC PMTs exhibit some after-pulses due to ionized residual gas molecules inside the PMT which are accelerated to the photocathode. We veto after-pulses arriving several microseconds after a large hit, so they are not used in the analysis. This and other TDC-specific effects produce an estimated inefficiency of 1-2% level (per PMT channel, not per event). \nPre-pulses also occur but are mitigated by using the High threshold start time as a reference (whenever it is available) rather than the Low threshold start time. \nIn this high-throughput mode, the system has approximately a 2-3% event deadtime fraction, as deduced by fitting the distribution of time differences between events.', '4.5. Timing and Control': "The primary responsibility of the HAWC GPS Timing and Control (GTC) system is to provide the control signals needed by the TDCs and scalers. The TDC design is based on a 40MHz clock. The GTC generates a 40MHz clock from the 10 MHz clock signal provided by the GPS. The GTC supplies TDC control signals which zero counters and clears TDC event data; the signals arrive well separated from the 40 MHz clock edges. These control signals are a crucial part of cleanly starting a run simultaneously in all TDCs and providing the internal event counts, and the counters of the 40MHz clock which label the TDC data headers. The GTC system was capable of handling either an asynchronous event-based trigger or a synchronous trigger. HAWC chose the latter option, with the GTC providing a precise 40 kHz periodic TRG signal to the TDCs derived from the 40MHz clock. The GTC also provides a periodic readout signal to the scaler DAQ crate. \nThe GTC encoded a sub-microsecond global time stamp for sky positioning in the TDC data channels. However, the analysis requirements were sufficiently met with NTP (Network Time Protocol) computer timestamps with an accuracy of approximately one millisecond. The GTC operation is described in a previous publication [70] on the overall HAWC data acquisition system. The design of the GTC is discussed in detail in [71]. \nIn 2018, we replaced the GPS timing portion of the GTC with a White Rabbit (WR) system with a custom WR-ZEN module [72] to synchronize the main HAWC DAQ with the HAWC outrigger DAQ, which has native timing based on the WR. WR implements the IEEE 1588 Precision Time Protocol (see e.g. [73]) in an open hardware framework. With the WR system, the 40 kHz readout TRG signal is synchronized to the GPS one pulse per second signal, and TDC runs are started at the top of a GPS second. Therefore, the time of the event trigger is now determined to the accuracy of the GPS. HAWC's WR system will be discussed in greater detail in a future publication on the outrigger extension of HAWC.", '5. The Calibration of HAWC': 'Accurate reconstruction of the air shower requires precise timing and charge measurements from the PMT signals. To this end, we have a calibrated and monitored laser system that can deposit light in the tanks with known time and consistent amplitude. We use the laser calibration system to determine in-situ PMT charge and timing characteristics. This calibration information is used to perform arrival time corrections of the \nparticles from the EAS and to convert ToT to number of photoelectrons. The calibration system of HAWC, including charge, timing, and its analytical framework is summarized by [74-76], with further details found in [77, 78] (and references therein). \nTwo aspects of the calibration system are particularly worth emphasizing: we designed the hardware and software to operate remotely, and we record laser firing times in the TDC data stream so that we can take calibration data (a few hundred events/second) simultaneously with shower data, without causing significant down time. Below we describe the hardware of the laser calibration system and the analysis software. \nThe charge and timing information for the energy deposited in WCDs of EAS are deduced from the width and leading-edge time of the discriminator pulses from each TDC (described in Section 4.1). Key tasks of the calibration system include determining the electronic time-slewing as a function of ToT and the offset between the PMT measured time and the fitted air shower front expected time (time pedestals) among PMT channels. Figure 11 illustrates the time slewing effect resulting from a pulse leading edge crossing a fixed voltage threshold at a time which depends on the pulse amplitude. \nFigure 12 displays the overall layout of the hardware for the laser calibration system. A laser-based optical system delivers short pulses of light to the WCDs. The PMT response to this calibrated light source is used to determine the relationship between ToT and PE for charge calibration. The measured time between laser firing (as measured by a photodiode and recorded by the TDC) and TDC edges is used to determine the PMT response time and cable delay, and measure the charge-dependent slewing effect and time pedestal for each PMT. \nThe optical system includes a Teem Photonics PNx-M green (532nm) laser [79], four LaserProbe RM-3700 radiometers [80], optical splitting cubes, Spectral Products AB301 filter wheels [81], 1:2, 1:4, 1:19, and 1:37 splitters, and chassis-based optical switches. Pairs of optical fibers of approximately 200 m length connect this system to and from each of the HAWC WCDs, as described in Section 3.8. The laser delivers light in short pulses ( < 1ns). The fact that shower photons arrive over a longer period (up to 10ns, giving longer ToT for the same number of photons), which leads to one of the systematic uncertainties in HAWC calibration [20]. \nThree filter wheels each hold six neutral density filters with different opacities to give 68 distinct filter combinations providing transmittance values ranging from 1 to 10 -6 . 5 . Thorlabs optical diffusers, which hold the end of the optical fiber, are anchored to the bottom of each WCD tank and are submerged approximately 3 m above the central PMT. The intensity of laser light at PMTs can vary from approximately 0.1 PE to more than 1000 PE. \nThe relative intensity of the light before and after the filter wheel is recorded by the LaserProbe RM3700 Control Box radiometer [80] and a RjP-465 Energy Probe [82]. At the lowest light levels, we determine the number of PEs by occupancy in each PMT, assuming Poisson statistics. At higher light levels, we determine the number of PEs by scaling the occupancy-derived PEs by the ratio of radiometer values. \nThe calibration computer controls the laser system devices [83]. The regular TDC DAQ reads out the PMTs signals from the laser pulses. Two Thorlabs DET02AFC photodiodes provide calibration start and stop timing signals to the DAQ TDCs. The time the laser fired is recorded in a channel of the TDC DAQ system; the presence of the laser time in an event tags calibration events for later processing. \nWe do absolute charge calibration based on a physical standard: a clean sample of \nnearly vertical isolated muons passing though the center of the tanks. We select tanks far from the shower core and with no hits in neighboring tanks and then require that the central PMT has a large signal and three outer PMTs hits occur at nearly the same time. We use the observed signals to measure the absolute sensitivity of each PMT and to provide an absolute normalization for the simulated PMT response. \nTiming corrections are done based on EAS data: we select clean air showers from nearly overhead with > 200 hits and use average shower plane fit residuals to measure individual PMT timing offsets to within less than 1ns. Then we incorporate these corrections into the calibration chain. \nAfter these corrections, we use the charge information to determine the EAS core location and the lateral distribution of the EAS at HAWC. We reconstruct the arrival direction of the EAS using the timing information. Combining all this information, we reconstruct the initial energy of the EAS. \nThe analysis software uses both the calibration system (radiometers) and the DAQ system (TDC data) to provide the final calibration products. These products are used in the reconstruction process to correct the different responses of the individual PMT channels. The calibration analysis software determines the charge calibration, which is the relationship between the measured ToT and the number of PEs detected by the PMT photocathode for varying light intensities. The relative time of the PMT responses and the slewing of measured times due to pulse-height variation is determined in the timing calibration. As a final product, calibration constants are provided to the event reconstruction analysis system. The charge and timing calibration results and raw calibration data are archived in a dedicated database.', '6. HAWC Operations': "We designed HAWC for autonomous operation. The key to autonomous operation is comprehensive remote monitoring, automated control, and the absence of consumables. The site is remote, and access has at times been restricted for scientists and even the local maintenance crew. Down periods can become extended if spare equipment is not immediately available on site. The HAWC site has many features which motivate design for remote operation. The volcanic soil is fine and abrasive, so we minimize traffic through the CH and clean it regularly. That same soil makes electrical grounding difficult (see Section 3.6). The site is subject to power outages due to electrical storms, heavy rain and snow, and even hurricanes and earthquakes. Humidity varies widely at the HAWC site and can be low enough to require careful anti-static protections for handling electronics or high enough to require measures to prevent unwanted condensation. \nThe live time fraction for each day since the beginning of engineering operations with 250 tanks on November 27, 2014 up to November 29, 2021 is shown in Figure 13. Over these 7 years, HAWC was operational and collecting data for more than 94% of the time. The main source of downtime is power outages, mainly caused by lightning during the spring and summer seasons. HAWC was twice off for long periods: the first was in April of 2016 and the second in June and July of 2021. In both cases, lightning damaged critical transformers. We recently upgraded the electrical protection of the main transformer. \nThe COVID-19 pandemic tested the efficiency of remote HAWC operations. This pandemic forced us to work with minimal personnel. HAWC data-taking operation \nwas not interrupted, and detector maintenance was adequately addressed during this unprecedented time. \nHAWC also continued operations despite earthquakes which shook the site. On June 23, 2020, a 7.5 Richter magnitude earthquake hit M'exico with an epicenter about 400 km South of Sierra Negra. Effects near the site were at level V on the Modified Mercalli scale. No damage in HAWC was observed, even though the WCDs registered a response to the earthquake in the scaler data and water level sensors. We present an example of the response of a water level sensor showing the sloshing of the water in a WCD in Figure 14. This event confirms the robustness of HAWC construction. \nIn January 2021, an overhead internet optical fiber was damaged between the site and the base camp at the foot of the mountain where our removable disk drives are located. Local data caching in the CH allowed us to continue data taking for three weeks during the repair. HAWC ran successfully without remote monitoring or intervention. We recently added a Starlink [84]) internet ground station to provide a reliable backup path for communications and monitoring. \nThe high altitude also challenges the health and cognitive abilities of humans, especially those coming to the site from low altitude. HAWC provided personnel training on the awareness of the symptoms of altitude sickness and implemented policies and procedures to help ensure safety. Some of these procedures included making safety equipment such as a satellite phone, supplemental oxygen and pulse oximeters available at the site. Safety policies included the use of two-person rules, site visitation logs and required medical waiver forms that informed personnel of their risks.", '6.1. Networking and Site Computers': "The critical site network components run on UPS or solar power backup, particularly the main switches, router, hardware, the environmental monitoring system or EMS (Section 6.2), and the site monitoring computers. The solar backup (Section 3.4) maintains basic connectivity during power outages and enables remote recovery after power is restored, as the EMS computer is employed to reboot other systems and computers. For personnel security on site, our solar photovoltaic system powers the internet, satellite phone, and WiFi. Remote login to the site proceeds by VPN (Virtual Private Network). \nA fiber directly connects the site internet to the base camp at the foot of the mountain. The connection from the base camp to the computer center at the Instituto de F'ısica of the Universidad Nacional Aut'onoma de M'exico (UNAM) initially included a microwave link. However, it was insufficiently reliable for remote monitoring and control. We now have a commercial 40Mb/s fiber link with excellent interactive response. This comfortably handles monitoring data, and we have even updated control room operating systems with this connection, but it is not able to handle raw data transfers (See Section 6.6). \nStandard computer disks do not operate reliably above ∼ 3 km (10,000 ft) because their design requires sufficient air density for cooling and to support the disk heads above the rotating platter. Standard disks are sufficient at the 2689 m altitude base camp (see Section 6.6). The larger HAWC site data storage disks with heavy rewrite cycles are Hefilled disks designed for computer-center use; they function reliably at HAWC altitude. We use fast SSDs to hold the operating system in all HAWC site computers, and for disks smaller than 1TB where cost is not prohibitive. \nThe cooling issues for site computers are dealt with by maintaining low CH temperatures, preferring low-power computers, or auxiliary fans. Site computers include low-power computers for the EMS system, monitoring, DAQ control, and calibration; six computers carry out the reconstruction and a dedicated archive computer stores the raw data. A separate server runs real-time science analysis tasks using reconstructed data.", '6.2. The Environmental Monitoring System (EMS)': 'The EMS system performs low-level environmental monitoring; its components provide inputs to the monitoring system described in the next section. The EMS includes a weather station, an electric field monitor, temperature and voltage monitoring, water level monitoring for the WCDs, and webcams. The weather station is a Campbell Scientific Datalogger CR1000. We record atmospheric pressure, wind speed and direction, temperature, relative humidity, rain, and solar radiation every minute. The logging software was developed by ADVANTECH [85]. \nStrong electric fields, even without lightning, can cause elevated PMT count rates. We monitor the electric field with a Boltek EFM-100C RS485 Electric Field Mill. We installed it above the metal HAWC utility building, which is a good conductor and does not distort measurements. The high humidity and low temperatures at the site initially froze the moving parts of the field mill, so we added a warming circuit. We read out the detector over optical fiber to avoid grounding issues. \nThe hardware for the temperature and voltage monitoring in the CH is based on two Advantech autonomous data acquisition boards, an ADAM-6015-7 channel RTD input module [86], and Balco 1000 RTD thermistors as temperature sensors [87]. These are placed on the top of each crate of high-power FEBs, for which it is essential to track temperature. We also monitor the temperature of the calibration and electronics rooms of the CH. An ADAM 6017 digitizes low voltages from the FEB power supplies. Every 10 seconds, the 6017 also digitizes the external AC voltages through a rectifier circuit. The EMS also records the state of the UPS system and battery charge level. \nWe monitor the water level in tanks, because some WCD bladders developed slow leaks. The water level monitoring system is based on an MPX4250AP pressure sensor [88] placed at the bottom of the WCDs water volume. Each sensor requires three connections: a 5V power supply voltage along with ground and signal connections. In order to prevent lightning-related transient over-voltages, protection diodes are required on the monitoring cables before entering the WCDs. An ethernet cable spark gap and another diode prevent over-voltages where the cables enter the CH. The signal ranges from 1 to 2 V, equivalent to a water depth between 0 to 5 m. Signals, linearly proportional to absolute pressure, are multiplexed by a MEGA Arduino board to a Labjack U3-HV voltmeter. We record water levels with a 10-minute cadence for most tanks and every minute for a few tanks. Care was taken to use USB or RJ45 connectors in this system to avoid searching for exotic connectors at a remote site (see Section 3 for more on the cabling). While much less expensive than commercial sensors, up to 8% of our water level sensors fail per year, usually due to compromised epoxy potting around the pressure sensor. As a final aspect of our monitoring, we periodically store images from webcams inside the CH and outdoors looking at the WCDs.', '6.3. Monitoring': "Maintaining HAWC's 95% duty cycle requires a system specifically designed for remote monitoring. Monitoring data must be transferred off-site in an efficient manner that is robust to potential network outages. Scientists on the monitoring shifts must be able to view diagnostic information in a clear and user-friendly way that updates in real time so that experts can be notified of potential issues. Experts also require readily available tools to obtain more in-depth information and archival monitoring data to check abnormal behavior and validate candidate gamma-ray transients. Two software packages were developed to meet these goals. The Advanced Tracking of HAWC Experiment Notifications and Alerts (ATHENA) system handles data collection. The HAWC Observatory Monitoring for Experiment and Reconstruction (HOMER) system provides the monitoring user interface. \nATHENA is a Python library which polls experimental hardware and stores the information in an SQL database. While the polling routine is different for each monitored device, ATHENA deals with all formatting in the back end, so data source experts can ignore database details. ATHENA automatically structures the data in a standardized way and has a built-in version control system to define a data stream change. This approach allows for new devices by simply adding a polling script that registers a new data stream with ATHENA. Most components are polled every minute, giving an effectively real-time view of the detector's health. ZeroMQ [89] handles communication between processes reading data and inserting the data into the SQL database. We write approximately 8 GB of monitoring data per year. \nATHENA moves monitoring data to the off-site computers. A custom Python database synchronization script compares on-site and off-site versions of tables, and the time column of each row. It then writes an SQL query to copy only new entries. Tables are synchronized one at a time. The synchronization launches at nearly the same cadence as the monitoring polling. Synchronizations time out and re-launch with approximately the same frequency (with a more permissive timeout for the larger tables), preventing a momentary bad connection or a slow table update from interrupting the entire monitoring process. After a prolonged network outage, we run scripts to move data in smaller packets to re-synchronize the monitoring more efficiently. \nThese optimizations mean approximately only 20 KB of data must be sent during each synchronization instance to keep everything up to date, which is well within the tolerance of the network. Unexpected outages can cause the sync routines to hang. A second master script running under Crontab automatically kills and restarts hanging sync scripts. This process has become less critical with the advent of a more reliable internet connection. \nHOMER is a collection of PHP web pages. It uses SQL wrapper libraries to sort the ATHENA database into arrays. These arrays are then either parsed for summary information or passed into a Google Charts API for plotting. While this API introduces an external dependency, it makes plots optimized for readability and provides interactivity to zoom in or out or to get the exact value of a data point by dragging the mouse over it. \nThe top of the HOMER home page summarizes the most critical information and links to sub-pages with more detail. The critical information includes run status, temperatures of the electronics, scaler rates, and high voltage status. The home page is intuitive with \ncolor codes to indicate whether a component is in a normal state. HOMER also colorcodes data if it is stale. \nA specialized local version of HOMER exists at the site with reduced features. It is lighter-weight and independent of the Google APIs, allowing it to run even if the network connection is down. It is only used by experts at the site, and optimized for reliability and conciseness. The home page contains only the critical information and links to onsite troubleshooting data such as scaler rates and electronics status. The site version of HOMER runs on the same machine as the on-site database so it reflects detector information in real-time up to the polling frequency. This system has proven vital during extended network outages. It also contains pages with JavaScript functions to turn high voltage channels on and off which directly and instantly access the high voltage modules to display status. These features are only available in the on-site version of HOMER to prevent HV channels being toggled by an off-site user or by a security breach at off-site computers. \nAn additional monitoring page outside the HOMER framework emphasizes data quality rather than hardware and environmental monitoring. The web page includes static figures and HTML text generated at the site and transferred to the off-site monitoring application using rsync . The plots provide a light-weight snapshot of analysis and detector status updated every 20 minutes and include results from the online analysis, such as the current significance of bright gamma-ray sources. The page also provides more technical diagnostic plots of hardware based on higher-level reconstruction results. \nThe final component of HAWC remote monitoring is the alert system integrated with the chat platform Slack 9 . Python processes poll various DAQ components; abnormal responses send alerts to the HAWC Slack workspace. Alert scripts of this type monitor the high voltage, low voltage, run status, AC voltage, FEB temperatures, output file sizes, and data copying. Each alert sends pertinent details (such as which high voltage channels tripped) if a problem is detected. These messages alert experts to critical issues in real time and allow a quick response to correct problems. To prevent overloading the Slack channels, alerts are temporarily silenced after a certain number of repetitions.", '6.4. Control of Cooling and Power': "The HVAC (Heating Ventilation and Air Conditioning) system has the primary responsibility for maintaining CH temperature. The front-end boards are ECL based and require high power: the FEB crates together dissipate over 5kW, and the entire HAWC counting house consumes approximately 25kW of power. Thus temperature control in the CH is critical. \nSeparate from the monitoring system, a bimetallic safety interlock switch resides on top of each FEB crate; it turns off the DC power when the temperature exceeds 32 · C. This can happen within seven minutes if we lose the AC power to the HVAC (Heating Ventilation and Air Conditioning) system while the crate is powered by UPS. Since overheating can occur quickly, the EMS monitoring of temperature and low voltage power is critical to provide information to the control scripts. \nHAWCis in a challenging environment for air conditioning: the outdoors is often cooler than inside the CH, which stresses the air conditioning and could freeze the condenser. \nThe HVAC system removes most of the heat from the CH, but we supplement the HVAC with a water heat exchanger system described below. This heat exchanger slows the rate at which the HVAC cycles on to half the time, thus extending the HVAC working life. The heat exchanger system also allows the analog part of the front-end boards to operate more stably in a narrower temperature range. \nA dual-loop heat exchange system performs supplemental heat removal. The heat exchanger is a Coolflow System III from Neslab with a maximum cooling capacity of up to 70 kW. One of the working WCDs near the CH provides the large heat sink of 3 · C water. This cool water circulates through one side of the heat exchanger, returning to the WCD slightly warmer water. The secondary water loop is maintained at 6 to 12 · C by controlling the flow rate to 'radiator' units in each of the four high-power electronics racks, where air warmed by the electronics exchanges heat with the water from the secondary loop. Each of the 4 radiators removes 6kW of heat. \nRemote shifters can determine when the cooling system has tripped by viewing a webcam aimed at a LED indicator of the heat exchanger system. However, resetting after a trip is an expert task: it is necessary to first verify that the trip was not caused by a water leak, by checking the water sensors under each heat exchanger and under the pump.", '6.5. Slow Control': 'Slow control refers to operations related to control of the electronics which take place on the time scales (seconds or longer) much longer than the real-time DAQ scale needed for handling data from individual events.', '6.5.1. Automatic Run Restart': 'HAWC runs normally last for 24 hours, after which the run is restarted automatically. However, the DAQ can crash when high electric fields near the site cause elevated PMT rates. Automatic scripts detect crashes (whatever the cause), end a crashed run, reset the system, and restart a run without remote interaction, continuing this cycle until conditions are stable enough that attempted runs no longer crash. This minimizes downtime during stormy weather.', '6.5.2. Power Outage Bridging and Shutdown': "HAWCuses two 30kVA (UPS) to bridge short power interruptions and allow automatic controlled power-down scripts to act for more prolonged interruptions, or in case of climate control failures leading to temperatures outside of the safe range in electronic crates. Under regular operation, the UPS conditions AC input power, thus supplying 'clean' power to the electronics; unconditioned ('dirty') AC power is restricted to lesscritical uses such as running HVAC, fans, soldering stations, lighting, and powering non-DAQ computers for on-site users outside the electronics room. \nWhen the external AC power voltage is less than 100 V, and after HAWC has been powered by UPS power for 2 minutes, scripts detect the loss of AC power. These scripts stop the data-taking run, turn off DAQ power supplies and power distribution units (PDUs) controlling non-internet-controllable items such as cooling fans, then turn off all DAQ computers in a controlled fashion. The scripts leave the monitoring system, internet switches, and the monitoring computer running under the power provided by \nthe solar-power system and its separate UPS. Since the main UPS could run the system for 40 minutes, it has more than sufficient power storage to safely restart the site as needed.", '6.5.3. Power Outage Recovery': 'The Slack Alert system ensures that experts know when we lose AC power. Experts use the remote monitoring to detect when AC power has returned. They run restart scripts to turn HAWC back on after power or HVAC is restored. The SNMP protocol restarts DAQ computers, turns on cooling fans via remote-controlled PDUs and then the DAQ low and high voltage power supplies. Developing and testing the shutdown and restart scripts took over a year to achieve smooth operation, but it has been critical to maintaining HAWC data taking with minimal downtime.', '6.6. Data Transfer': "HAWC records data at a rate of approximately 2 TB/day (20 MB/s), or 730 TB per year. The site's primary disk array storage capacity is 82 TB, which provides storage for about 40 days of data recording. The data transfer from the HAWC observatory to the closest data center located at UNAM is carried out by transporting portable disk arrays (PDAs). Each disk array has a capacity of 29TB or 36TB. HAWC has 12 of these PDAs. We transfer data down the mountain on a 15km 1 Gb optical fiber to the nearby town of Atzitzintla, where our base camp is: an office of the Instituto Nacional de Astrof'ısica, ' Optica, and Electr'onica (INAOE). This office is set up to handle the data transfer to the PDAs and make backup copies of the data. \nWe can store up to 110 days of data and backup copies in the office or at the site. We ship the PDAs to UNAM approximately every six weeks. The local storage ability made it possible for the HAWC experiment to continue operating during the COVID-19 pandemic despite its effects on the regular schedule for data movement. \nOnce the PDAs arrive at UNAM, they are backed up and checked against on-site data. This first identical copy stays in permanent storage at the UNAM data center. Afterwards, the data moves via a 10 Gb/s connection (of which HAWC uses less than 10%) to a second data center located at the University of Maryland. Once the data is checked and archived at both data centers, it is removed from the site. In case of data loss at one facility, data can be recovered from the copy at the second location.", '6.7. Real-time Science Programs': "HAWC carries out several science analyses in real time at the HAWC site, based on the online reconstruction. We search for transients of the Crab, Mrk 421 and Mrk501 daily, search the sky for bright sources on time scales from hours to 3 days, and search for GRBs (independently, or based on satellite alerts) on time scales between 0.3 and 1000 seconds. Automatic email alerts are sent to partners that have an agreement with HAWC. We post Astronomer's Telegrams (ATels) and Gamma-ray Coordinates Network (GCN) circulars for interesting alerts. We send events and alerts to Astrophysical Multimessenger Observatory Network (AMON) [90] from our independent GRB search. Anything below a false-alarm rate of 12 per year is sent as a Notice to GCN. The latency is less than a minute.", '7. The Upgrade of HAWC': 'While the gamma/hadron discrimination method in HAWC works well, it performs better with increasing gamma-ray energy, with the best performance at the highest energies. Additionally, a significant fraction of events passing the multiplicity trigger criterion have their shower core outside the 300-WCD array. While gamma/hadron discrimination is possible for these events, without good knowledge of the position of the shower core, there is ambiguity in the reconstruction of the shower direction and shower size (needed to determine the energy of the primary gamma ray or cosmic ray). \nHAWC, like Milagro before it, has been upgraded by adding outrigger detectors. This expanded array, installed between 2016 and 2018, enhances the sensitivity above 10 TeV by increasing the area over which shower cores can be detected by a factor of 3-4 and improving the angular resolution (see [15] and references therein). The details of this outrigger array will be published in a forthcoming paper.', 'Acknowledgement': "We acknowledge the support from: the US National Science Foundation (NSF); the US Department of Energy Office of High-Energy Physics; the Laboratory Directed Research and Development (LDRD) program of Los Alamos National Laboratory; Consejo Nacional de Ciencia y Tecnolog'ıa (CONACyT), M'exico, grants 271051, 232656, 260378, 179588, 254964, 258865, 243290, 132197, A1-S-46288, A1-S-22784, c'atedras 873, 1563, 341, 323, Red HAWC, M'exico; DGAPA-UNAM grants IG101320, IN111716-3, IN111419, IA102019, IN110621, IN110521, IN102223 ; VIEP-BUAP; PIFI 2012, 2013, PROFOCIE 2014, 2015; the University of Wisconsin Alumni Research Foundation; the Institute of Geophysics, Planetary Physics, and Signatures at Los Alamos National Laboratory; Polish Science Centre grant, DEC-2017/27/B/ST9/02272; Coordinaci'on de la Investigaci'on Cient'ıfica de la Universidad Michoacana; Royal Society - Newton Advanced Fellowship 180385; Generalitat Valenciana, grant CIDEGENT/2018/034; Chulalongkorn University's CUniverse (CUAASC) grant; Coordinaci'on General Acad'emica y de Innovaci'on (CGAI-UdeG); Cuerpo acad'emico PRODEP-SEP UDG-CA-499. H.F. acknowledges support by NASA under award number 80GSFC21M0002. We also acknowledge the significant contributions of former members of the HAWC collaboration: R. Arceo, E. Almaraz, M. Alvarez, J. R. Angeles Camacho, B. M. Baughman, D. Berley, P. Colin Farias, M. A. Diaz Cruz, D. W. Fiorino, D. Garcia, V. Grabski, Z. Hampel-Arias, J. Jablonski, N. Kelley-Hoskins, M. Lamprea, S. S. Marinelli, J. Mart'ınez, C. Rivi'ere, M. Rosenberg, P. Vanegas, X. J. V'azquez, O. V'azquez-Estrada, G. Vianello, D. Warner, T. Weisgarber, T. Yapici, and P. W. Younk. Thanks to Scott Delay, Luciano D'ıaz, Eduardo Murrieta and Janina Nava for technical support.", 'References': "- [3] The LHAASO Collaboration, Z. Cao, et al., Ultrahigh-energy photons up to 1.4 petaelectronvolts from 12 γ -ray Galactic sources, Nature, 594 (2021) 33-36, https://doi.org/10.1038/s41586-02103498-z\n- [4] F. A. 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The mounting fixture is at the bottom of the WCD. One end of the Kevlar string loop runs through the omega clamp (see c) and the other end attaches to the bladder hatch. c) Drawing of the omega clamp attached to the mounting fixture, which locks the acrylic ball from the PMT to the mounting fixture. d) The pulley system before installing a PMT in a WCD. The kevlar string, ball, and mounting fixture are labeled. \n<!-- image --> \nFigure 4: Scheme of the water filtration system \n<!-- image --> \nFigure 5: Cabling layout (grey-violet solid lines). The HAWC platform is outlined in brown. Power lines are shown running from the substation to the CH and the HUB. The spark gaps are located next to the CH, where all cabling converges. \n<!-- image --> \nFigure 6: Top: The spark gap protection device used for HAWC RG-59 signal/HV cables. On an overvoltage the spark-gaps clamp the conducting wire to the HAWC ground at the counting house. Down : A cabinet containing spark gap protection devices. \n<!-- image --> \nFigure 7: Top: Simulated response for a PMT pulse with and without the cable attenuation. down: attenuation due to 100 meters of RG59 cable as a function of frequency from 1 MHz to 1 GHz is shown. This frequency range is relevant for PMT pulses, and shows strong attenuation at the highest frequency values. Based on [65]. \n<!-- image --> \nFigure 8: Top-level diagram of the HAWC electronics showing a summary of the critical subsystems and the interconnections, including HV and optical fiber cabling. Based on [65]. NMEA refers to the National Marine Electric Association format in which GPS presents data [66, 67]; CLR, TRG and RST are control signals for the TDC system. The LoToT and HiToT time over threshold signals are discussed in section 4.1. \n<!-- image --> \nFigure 9: Schematic overview [70] of the HAWC data acquisition and online processing system, as described in the text of section 4. \n<!-- image --> \nFigure 10: The analog PMT signals are split and passed through two paths. In each path, there is an amplifier and discriminator circuit. The ratio of the amplifier gains is 7 to 1. The higher gain circuit has an effectively lower threshold (Low Th). There is a time (T) delay in the high threshold (High Th) path. The 2-edge event is related with the Low Th, while the 4 edge event is related to the High Th. \n<!-- image --> \nFigure 11: Time slewing due to varying pulse height. As illustrated above, the discriminated pulse from a smaller signal starts later than a larger signal due to a delay in crossing the fixed voltage threshold ([77]). \n<!-- image --> \nFigure 12: A schematic of the laser calibration system [77, 83]. See text for details. \n<!-- image --> \nFigure 13: Live time fraction plots of HAWC since the beginning of engineering operations with 250 WCDs. \n<!-- image --> \nFigure 14: Signal of the June 23, 2020 earthquake in M'exico water level measured by the HAWC WCD E12. \n<!-- image -->"}
2023IJMPD..3250030B	We consider the Kepler twobody problem in the presence of a cosmological constant . Several dimensionless parameters characterizing the possible orbit typologies are used to identify open and closed trajectories. The qualitative picture of the twobody motion is described and critical parameters of the problem are found.	2023-04-01T00:00:00Z	['10.1142/S021827182350030X', '2024arXiv240911427B', '2023IJMPD..3250030B', 'arXiv:2409.11427', '10.48550/arXiv.2409.11427']	['Galaxies', 'two-body problem', 'cosmological constant', '95.36.+x', '98.80.Es', '98.65.At', 'Dark energy', 'Observational cosmology', 'Interacting galaxies', 'galaxy pairs and triples', 'General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	Orbital precession and other properties of twobody motion in the presence of dark energy	2,023	199	0.3	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	2	https://arxiv.org/pdf/2409.11427.pdf	{'No Header': 'International Journal of Modern Physics D © World Scientific Publishing Company', 'G.S. Bisnovatyi-Kogan': "Space Research Institute, Profsoyusnaya 84/32, Moscow, Russia 117997. National Research Nuclear University MEPhI, Kashira Highway, 31, Moscow, Russia 115409. Moscow Institute of Physics and Technology MIPT, Institutskiy Pereulok, 9, Dolgoprudny, Moscow region, Russia 141701. Department of Physics, University of Rome 'La Sapienza', Rome, Italy. \nM. Merafina \nDepartment of Physics, University of Rome 'La Sapienza', Rome, Italy \nReceived Day Month Year Revised Day Month Year \nWe consider the Kepler two-body problem in the presence of a cosmological constant Λ. Several dimensionless parameters characterizing the possible orbit typologies are used to identify open and closed trajectories. The qualitative picture of the two-body motion is described and critical parameters of the problem are found. \nKeywords : galaxies; two-body problem; cosmological constant. \nPACS numbers: 95.36.+x; 98.80.Es; 98.65.At", '1. Introduction': "The discovery of dark energy (DE) in the universe is based on observations of the supernova SN Ia at redshift z ≤ 1 1, 2 , and on measurements of the spectrum of fluctuations of the cosmic microwave background radiation (CMB) 3, 4 . These measurements give a value of the cosmological constant Λ ≈ 10 -56 cm -2 . For review articles considering the role of DE in cosmology, see e.g. Refs. 5, 6, 7, 8, 9, 10, 11, 12. \nBefore the observational measurements of the DE density in SN Ia and CMB investigations, the limits on the present value of the cosmological constant Λ which determined the DE density to be ρ DE = Λ c 2 / 8 πG were obtained using precision measurements of the binary pulsar timing and planetary motion in the Solar System. From measurements of the perihelion shift of Mercury 13 an upper limit Λ < 10 -42 cm -2 was obtained that decreased to Λ < 10 -55 cm -2 15 years later. 14 This result was later invalidated in Ref. 15, where the upper limit Λ < 4 · 10 -45 cm -2 was given, see also Refs. 16, 17, 18, 19. A theoretical analysis of the influence of the cosmological constant on the gravitomagnetic clock effect and the gravitational \ntime delay of electromagnetic waves, as well as the effect of Λ on the pericenter precession, had been done in Ref. 21. Ref. 22 showed that the measured value of Λ has a negligible effect on the measurement of the perihelion shift of Mercury. \nLimits on the cosmological constant from effects on Solar and stellar systems as well as on binary pulsars have been discussed in Refs. 23, 24, where it was claimed that the best constraint comes from the perihelion precession of Earth and Mars, Λ < 1 · 10 -46 cm -2 . Various Solar system effects in the Schwarzschild-de Sitter space-time had been calculated in Ref. 25. The behavior of Keplerian orbits due to central-force perturbations and cosmological expansion was studied in Refs. 26, 27, 28, 29. Anthropic constraints on the cosmological constant from the Sun's motion through the Milky Way were discussed in Ref. 30. \nRefs. 31, 32 showed that the outer parts of galaxy clusters (GC) may be strongly influenced by dark energy. If we consider the relative motion of two rich clusters, then we should deal with sizes for which the influence of the DE in the form of the cosmological constant Λ is important. We consider here a simplified problem of the relative motion of two rich clusters represented by two point masses. \nThe analytic solution of the problem of two-body motion in the presence of a nonzero Λ in the quasi-Newtonian approximation was given in Ref. 33, using the table of integrals from Ref. 34. Numerical approximations were used to calculate the elliptical integrals used in this analytic solution. We solve this problem using a different mathematical approach and identify the main critical parameters for the two-body system, which had not been calculated in that article. Our calculations describe orbital precession for non-circular motion in the presence of DE, and evaluate the associated periods and quasi-periods in terms of various parameters of the system.", '2. Equations for two-body problem in the presence of Λ in the quasi-Newtonian approximation': "We use the equations governing Keplerian motion in the presence of nonzero Λ given in Ref. 36. For two masses m 1 and m 2 rotating around each other, these equations describe the behaviour of their separation vector in the plane of the motion characterized by its length r and polar angle φ in a polar coordinate system as in the Kepler problem in Newtonian gravity. 35 Introducing the reduced mass µ , the total mass M , and the conserved value of the angular momentum L by \nµ = m 1 m 2 / ( m 1 + m 2 ) , M = m 1 + m 2 , L = µr 2 ˙ φ, (1) \nthe equations of motion take the form 36 \nr = -GM r 2 + L 2 µ 2 r 3 + Λ c 2 3 r, ˙ φ = L µr 2 . (2) \nOrbital precession and other properties of two-body motion in the presence of dark energy 3 \nIntegrating the first equation in (2), we obtain the expression for the conserved total energy E of the system in the form \nE = 1 2 µ ˙ r 2 + µ ( -GM r + L 2 2 µ 2 r 2 -Λ c 2 6 r 2 ) . (3) \nThere are two characteristic radii in this problem: r 0 , at which gravity of the binary system is balanced by the antigravity of DE (zero-gravity radius), 32 and the Keplerian radius r k of the circular orbit in Newtonian gravity, 35 namely \nr 0 = ( 3 GM Λ c 2 ) 1 / 3 , r k = L 2 GMµ 2 . (4) \nThe equation for ˙ r following from (3) is \n˙ r = ± √ 2 E µ -2 ( -GM r + L 2 2 µ 2 r 2 -Λ c 2 6 r 2 ) , (5) \nwhere the ' ± ' sign distinguishes the increasing and decreasing phases of the radial motion. From (5) and the second equation in (2), we obtain the relation connecting r and φ along the trajectory \ndr dφ = ± r √ 2 Eµ L 2 r 2 +2 GMµ 2 L 2 r -1 + Λ c 2 µ 2 3 L 2 r 4 . (6)", '2.1. Dimensionless variables': 'Introducing the dimensionless radius x and dimensionless time τ , the second equation in (2) becomes \nx = r r k , τ = L µr 2 k t , dφ dτ = 1 x 2 . (7) \nRewriting equations (5) and (6), taking into account (4) and (7), leads to \ndx dτ = ± 1 x √ 2 Er k GMµ x 2 +2 x -1 + ( r k r 0 ) 3 x 4 (8) \nand \ndx dφ = ± x √ 2 Er k GMµ x 2 +2 x -1 + ( r k r 0 ) 3 x 4 . (9) \nAs in the Kepler problem it is convenient to use the reciprocal variable u = 1 /x instead of x and introduce the dimensionless parameter d = ( r k /r 0 ) 3 . From equations (8) and (9), we obtain (the upper sign corresponds to ˙ r > 0) \ndφ dτ = u 2 , du dτ = ∓ u 2 √ -u 2 +2 u + ε + d/u 2 (10) \nand then \ndu dφ = ∓ √ -u 2 +2 u + ε + d/u 2 , (11) \nwhere ε = 2 Er k /GMµ and d = Λ c 2 r 3 k / 3 GM .', '3. Keplerian limit': "In the absence of DE (Λ = 0 implies d = 0), equations (10) and (11) reduce to \ndu dτ = ∓ u 2 √ -u 2 +2 u + ε , du dφ = ∓ √ -u 2 +2 u + ε , (12) \nwhere -1 ≤ ε < 0 for closed orbits. Consider the branch with the sign '+', writing the solution in the form \n∆ τ = ∫ u + u -du u 2 √ -u 2 +2 u + ε , ∆ φ = ∫ u + u -du √ -u 2 +2 u + ε . (13) \nHere the roots u + = 1 + √ 1 + ε and u -= 1 -√ 1 + ε of the expression inside the square root are related to the minimal (pericenter) and maximal (apocenter) separation between the two bodies, respectively. Integration leads to \n∆ φ = ∫ 1+ √ 1+ ε 1 -√ 1+ ε du √ 1 + ε -( u -1) 2 = ∫ + √ 1+ ε -√ 1+ ε dv √ 1 + ε -v 2 = ∫ +1 -1 dz √ 1 -z 2 = π . (14) \nThe first integral in (13) can be evaluated analytically. 34 We have then \n∆ τ = [ -√ -u 2 +2 u + ε εu + 1 ( -ε ) 3 / 2 arcsin u + ε u √ 1 + ε ] u + u -= π ( -ε ) 3 / 2 . (15) \nThe results obtained in (14) and (15) are related to half of the periodic trajectory, which is closed for Keplerian motion, with the change of the angle equal to 2 π during a cycle. The '-' sign in (12) is related to the second half of the closed elliptical trajectory, describing the motion from pericenter to apocenter, with a decreasing u -velocity ( ˙ u < 0, corresponding to ˙ r > 0). \nThe dimensional period of the Keplerian motion P k , taking into account equations (11) and (15), can be rewritten as (see also Ref. 35) \nP k = 2∆ t = 2 π ( -ε ) 3 / 2 µr 2 k L = πGM [ µ 3 2( -E ) 3 ] 1 / 2 , since ∆ t = µr 2 k L ∆ τ . (16) \nThe trajectory of the Keplerian motion is obtained from the indefinite integral \nφ = ∫ du √ -u 2 +2 u + ε = arcsin u -1 √ 1 + ε +const . (17) \nChoosing const = π/ 2, which corresponds to φ = 0 at the apocenter of the trajectory, and returning to the dimensional variables, we obtain finally \nr = r k 1 + √ 1 + ε sin( φ -π/ 2) = r k 1 + e sin( φ -π/ 2) . (18) \nHere, the quantity e = √ 1 + ε is the eccentricity of the elliptical trajectory in the Keplerian motion described by the equation (18). The whole family of Keplerian trajectories in the ( u, Φ) plane is plotted in Fig. 1. The quantity Φ( u ) = du/dφ = ∓ √ -u 2 +2 u + ε is the angular velocity of the u -variable. The main parameters of the Keplerian trajectories ( d = 0) of Fig. 1 are also given in Table I. \n/s100/s61/s48 \nFig. 1. The family of Keplerian orbits described by the quantity Φ( u ) = ∓ √ -u 2 +2 u + ε as a function of u = r k /r at fixed values of d = 0, for various values of the dimensionless total energy of the system ε = 2 Er k /GMµ (see also Table I). The curves correspond to the values a) ε = -1; b) ε = -0 . 85; c) ε = -0 . 5; d) ε = 0; e) ε = 1. The zeros of Φ (roots) are the turning points of the trajectory: left ( u -) and right ( u + ) zeros correspond to the apocenter and pericenter of the trajectory, respectively. Circular orbit (a) corresponds to the black spot at u = 1 and Φ = 0. Dotted curve (d) with a root at u = 0 corresponds to the parabolic motion ( ε = 0). Curves with ε > 0 correspond to unbound systems (hyperbolic trajectories). No solutions exist for ε < -1. \n<!-- image --> \n/s32 \nTable I. Trajectory parameters of two body motion for d = 0 (Keplerian orbits, without DE). The value u M corresponds to the maximum of the positive branch of each curve. Letters in the 'trajectory type' column refer to the labels in Fig. 1.", '4. Trajectories of the two body motion in the presence of DE': "The two-body motion in the presence of DE exhibits qualitatively different behavior with particular features of the motion at certain ranges of the values d and ε . Here \n/s32 \nFig. 2. The family of trajectories described by the quantity Φ( u ) = ± √ -u 2 +2 u + ε + d/u 2 as a function of u = r k /r at the fixed value d = 0 . 02, for selected values of the dimensionless total energy of the system ε = 2 Er k /GMµ (see also Table II). The curves correspond to the values a) ε = -1 . 2; b) ε = -1 . 02043; c) ε = -0 . 9; d) ε = -0 . 731954; e) ε = -0 . 5; f) ε = 0; g) ε = 1. For each value of ε in the range -1 . 02043 < ε < -0 . 731954 we have one hyperbolic (unbound) and one quasi-elliptical (bound) trajectory. The zeros of Φ for bound orbits are the turning points of the trajectory: left ( u -) and right ( u + ) zeros correspond to the apocenter and pericenter of the trajectory, respectively. The circular orbit (b) corresponds to the black spot at u = 0 . 978663 and Φ = 0. The crossing in the dash-dotted curve (d) at u = 0 . 306691 and Φ = 0 identifies an instability point on the trajectory (transition). Zeros of Φ in unbound (infinite) curves refer to the pericenter of pure hyperbolic ( u 0 ) or semi-hyperbolic ( u + ) trajectories. Finally, the dotted curve (f) corresponds to zero system total energy ( ε = 0). \n<!-- image --> \n/s100/s61/s48/s46/s48/s50 \nΦ( u ) = ± √ -u 2 +2 u + ε + d/u 2 , with d > 0. The trajectories in the ( u, Φ) plane are plotted in Figs. 2-5 for selected values of ε at four fixed values of d . Analogously, numerical parameters referring to the figures are given in Tables II-V, respectively. \nIn order to analyze the features of the different curves, we calculate the roots of the equation Φ( u ) = 0 by considering the expression -u 2 Φ 2 ( u ) = 0. We obtain the following fourth degree equation \nu 4 -2 u 3 -εu 2 -d = 0 . (19) \nThis equation has four roots, but one root always lies in the range u < 0 so is irrelevant. The other three roots depend on the values of parameters ϵ and d , and \nOrbital precession and other properties of two-body motion in the presence of dark energy 7 \nTable II. Trajectory parameters of the two body motion for d = 0 . 02 (presence of DE). The values u m and u M correspond to the minimum and maximum of the positive branch of each curve, respectively. Letters in the 'trajectory type' column refer to the labels in Fig. 2. \nsometimes two of them are not real. Therefore we have a maximum of three real roots (or only one) for Φ. These roots are calculated numerically and will be designated by u 0 , u -and u + . The analytic solution of the equation Φ( u ) = 0 was considered in Ref. 33 via elliptical integrals using formulas from Ref. 34, but this does not make the problem easier because the elliptical integrals must be evaluated numerically. \nIt is also useful when calculating Φ ' ( u ) to evaluate the minimum ( u m ) and maximum ( u M ) values of Φ( u ). From the positive branch of Φ, we obtain \nΦ ' ( u ) = -u +1 -d/u 3 √ -u 2 +2 u + ε + d/u 2 . (20) \nTo analyze the extrema we transform the condition Φ ' ( u ) = 0 to \nu 4 -u 3 + d = 0 . (21) \nThe solutions of this equation do not depend on ε . The solutions can be calculated graphically by considering the functions y 1 = -d/u 3 and y 2 = u -1 and varying the parameter d . We have two real solutions (one minimum and one maximum of Φ) for d ≤ 27 / 256 and no real solutions (no extrema) for d > 27 / 256. In particular, for d = d ∗ = 27 / 256 we have a unique solution corresponding to an inflection point \nTable III. Trajectory parameters of the two body motion for d = 0 . 08 (presence of DE). The values u m and u M correspond to the minimum and maximum of the positive branch of each curve, respectively. Letters in the 'trajectory type' column refer to the labels in Fig. 3. \n/s32 \nFig. 3. The family of trajectories described by the quantity Φ( u ) = ± √ -u 2 +2 u + ε + d/u 2 as a function of u = r k /r at fixed value of d = 0 . 08, for selected values of the dimensionless total energy of the system ε = 2 Er k /GMµ (see also Table III). The curves correspond to the values a) ε = -1 . 5; b) ε = -1 . 08892; c) ε = -1 . 08; d) ε = -1 . 06133; e) ε = -0 . 8; f) ε = 0; g) ε = 1. For each single value of ε in the range -1 . 08892 < ε < -1 . 06133 we have one hyperbolic (unbound) and one quasi-elliptical (bound) trajectory. Zeros of Φ (roots) in bound orbits are the turning points of the trajectory: left ( u -) and right ( u + ) zeros correspond to the apocenter and pericenter of the trajectory, respectively. The circular orbit (b) corresponds to the black spot at u = 0 . 884319 and Φ = 0. The crossing in the dash-dotted curve (d) at u = 0 . 571571 and Φ = 0 identifies an instability point on the trajectory (transition). Zeros of Φ in unbound (infinite) curves refer to the pericenter of pure hyperbolic ( u 0 ) or semi-hyperbolic ( u + ) trajectories. Finally, the dotted curve (f) corresponds to zero system total energy ( ε = 0). \n<!-- image --> \n/s100/s61/s48/s46/s48/s56 \nat u = 3 / 4. For ε = -9 / 8 the inflection point lies at Φ( u ) = 0, generating a cusp (see Fig. 4). Moreover, the condition d > 0 in (21) implies that 0 < u < 1. \nThere is another interesting analysis devoted to finding the value of d ∗ corresponding to the last stable circular orbit. Using the equations (21) and (19) we obtain \n2 u 2 -3 u -ε = 0 , (22) \nwhich takes into account both the conditions Φ( u ) = 0 and Φ ' ( u ) = 0. Equation (22) yields two solutions corresponding to the critical point of the circular orbit \nd = d \n= 27/256 \nFig. 4. The family of trajectories described by the quantity Φ( u ) = ± √ -u 2 +2 u + ε + d/u 2 as a function of u = r k /r at fixed value of d = d ∗ = 27 / 256, for selected values of the dimensionless total energy of the system ε = 2 Er k /GMµ (see also Table IV). The curves correspond to the values a) ε = -1 . 5; b) ε = -9 / 8; c) ε = -1; d) ε = -0 . 5; e) ε = 0; f) ε = 1. In this case, there are no values of ε corresponding to bound trajectories. The cusp in the dash-dotted curve (b) at u = 0 . 75 and Φ = 0 identifies an instability point on the trajectory (transition). Zeros of Φ of the unbound (infinite) curves refer to the pericenter of pure hyperbolic ( u 0 ) or semi-hyperbolic ( u + ) trajectories. Finally, the dotted curve (e) corresponds to zero system total energy ( ε = 0). \n<!-- image --> \nTable IV. Trajectory parameters of the two body motion for d = d ∗ = 27 / 256 (presence of DE). The values u m and u M correspond to the minimum and maximum of the positive branch of each curve, respectively. Letters in the 'trajectory type' column refer to the labels in Fig. 4.u circ and the point of disappearance of closed trajectories u lim \nu circ = 3 + √ 9 + 8 ε circ 4 , u lim = 3 -√ 9 + 8 ε lim 4 , (23) \nd=0.3 \nFig. 5. The family of trajectories described by the quantity Φ( u ) = ± √ -u 2 +2 u + ε + d/u 2 as a function of u = r k /r at fixed value of d = 0 . 3, for selected values of the dimensionless total energy of the system ε = 2 Er k /GMµ (see also Table V). The curves correspond to the values a) ε = -1 . 5; b) ε = -1; c) ε = -0 . 5; d) ε = 0; e) ε = 1. There are no values of ε corresponding to bound trajectories. Zeros of Φ of the unbound (infinite) curves refer to the pericenter of semi-hyperbolic ( u + ) trajectories. Finally, the dotted curve (d) corresponds to zero system total energy ( ε = 0). \n<!-- image --> \nTable V. Trajectory parameters of the two body motion for d = 0 . 3 (presence of DE). Letters in the 'trajectory type' column refer to the labels in Fig. 5. \nwhere from (21), \nd circ = u 3 circ -u 4 circ , d lim = u 3 lim -u 4 lim . (24) \nOrbital precession and other properties of two-body motion in the presence of dark energy 11", '5. Limiting parameters': 'The two-body motion in the Kepler problem is characterized by a circular orbit at ε = -1, and elliptical orbits, with a large axis tending to infinity at ε → 0 (see Fig. 1 and Table I). In the presence of DE, the circular orbits exist only in the limiting interval of the parameter d values, 36 and the transition from a finite to infinite trajectories happens abruptly, at a finite value of maximal separation.', '5.1. Circular orbits': 'Consider first circular orbits, which were analyzed in detail in Ref. 36. It follows from the first of (23) that circular orbits in the presence of DE exist only in the interval \n-9 8 < ε circ < -1 . (25) \nHere the left inequality follows from the need for a positive value inside the square root, and the right one is connected with a positive value of d circ . The values ε = -1, u circ = 1, d circ = 0 correspond to the Keplerian motion in the absence of DE. The values \nε = -9 8 , u = 3 4 , d = 27 256 ≈ 0 . 1055 \ncorrespond to the maximum value of d , generating a cusp as previously discussed. Comparing this result with the corresponding one in Ref. 36 where this extremum is characterized by the value b lim = r 3 k / 2 r 3 0 ≈ 0 . 053, we see that it agrees with our result since d max = 2 b lim ≈ 0 . 1055. Note also that the stable part on the right plot in Fig. 3 of Ref. 36, obtained numerically, is now reproduced by the analytic relations (23) and (24). The dependence of ε circ on u circ and d circ is plotted in Figs. 6 and 7.', '5.2. Limiting finite orbits': 'The limiting orbits, separating finite and infinite motion in the presence of DE (transition orbits), exist in the following interval \n-9 8 ≤ ε lim < 0 . (26) \nHere the left inequality follows from the need for a nonnegative value inside the square root, and the right one is connected with the need for a positive value of u lim due to (23). The values ε = 0, u lim = 0, d lim = 0 correspond to the parabolic orbit motion in the absence of DE. The limiting parameters for closed trajectories of the two-body motion in the presence of DE are presented in Figs. 8 and 9, plotted using the solutions (23). Such transition orbits are shown in Figs. 2-4 and the related parameters in Tables II-IV. Therefore, the limiting values for the parameters u lim and d lim are given by \n3 4 ≥ u lim ≥ 0 , 27 / 256 > d lim ≥ 0 , for -9 8 ≤ ε < 0 . (27) \nFig. 6. The dependence of the dimensionless total energy ε circ on the dimensionless inverse radius u circ for circular orbits. \n<!-- image -->', '6. Orbital precession in the presence of DE': 'In the presence of DE the orbits are not closed (excluding circular motion in some cases), 33 but it is convenient to consider a quasi-period for such motion, defined as the angular distance between two subsequent pericenters or apocenters of the trajectory. While the orbit is closed in the purely Keplerian case without DE, the change of this angular distance relative to 2 π may be interpreted as an orbital pericenter precession due to the DE. The angular distance between two subsequent apocenters of the trajectory ϕ tb using (11) is \nϕ tb = 2 ∫ u + u -du √ -u 2 +2 u + ε + d/u 2 , (28) \nwhere u ± are the largest positive roots of the equation -u 2 +2 u + ε + d/u 2 = 0, with u -< u + . \nFig. 7. The dependence of the dimensionless total energy ε circ on the parameter d circ for circular orbits. \n<!-- image -->', '6.1. The case ε = -1': 'For ε = -1 the roots can be found analytically \nu 0 = 1 2 -√ 1 4 -√ d, u -= 1 2 + √ 1 4 -√ d u + = 1 2 + √ 1 4 + √ d, u 4 = 1 2 -√ 1 4 + √ d . (29) \nOnly the roots u + and u -define finite trajectories; u 4 is always negative and physically not relevant; u 0 can only define infinite trajectories. Numerical integration of (28) gives the results presented in Table VI. The precession angle ϕ pr , by definition equals \nϕ pr = ϕ tb -2 π. (30) \nTherefore, the orbital precession in the presence of DE is the most important feature. As follows from the solutions (29), finite trajectories exist only when d < 1 / 16 for ε = -1. At larger d there is only one real positive root ( u + ) defining an infinite trajectory. It is clear that finite motion in the presence of DE is possible only inside the zero gravity radius r 0 defined by the first equation of (4). The presence of the centrifugal force, at finite angular momentum L , decreases the limiting value of the radius of the finite trajectory, so that actually r lim < r 0 . The dimensionless value \n0.0 \nFig. 8. The dependence of the dimensionless total energy ε lim on the dimensionless inverse radius u lim for transition orbits. \n<!-- image --> \nTable VI. Half angular distance 0.5 ϕ tb between two subsequent apocenters ( ε = -1) \nof the zero gravity radius x 0 = r 0 /r k is directly defined by d , according to the definition of the parameter d = ( r k /r 0 ) 3 : \nx 0 = d -1 / 3 . (31) \nFor ε = -1 we have x 0 = (1 / 16) -1 / 3 ≈ 2 . 52. As follows from the Table VI, the value of the limiting radius of finite trajectories corresponds to u -at d = 1 / 16, namely u lim = u -= 0 . 5. This means that x lim = 1 /u lim = 2 < x 0 ≈ 2 . 52. \n0.0 \nFig. 9. The dependence of ε lim (upper curve) and ε circ (lower curve) on the parameter d . The lower curve is the same as in Fig. 6. The finite orbits of the two-body motion in the presence of DE correspond to values of the parameters ( ε, d ) lying between these two curves. \n<!-- image -->', '6.2. Periods in the quasi-periodic two-body motion in the presence of DE': 'The dimensionless quasi-period ˜ P tb of the two-body motion in the presence of DE is \n˜ P tb = 2 ∫ u + u -du u 2 √ -u 2 +2 u + ε + d/u 2 . (32) \nThen the quasi-period P tb in units of the Keplerian period P k at the same ε and d = 0 is given by \nP tb P k = ˜ P tb ( -ε ) 3 / 2 2 π . (33) \nIn the Kepler problem the period is a function of one parameter ε (see Eq. (16)), while in the presence of DE the quasi-period also depends on d .', '7. Trajectories at L = 0': 'As mentioned above, the limiting value of the radius for finite trajectories is less than the zero-gravity radius r 0 , due to additional repulsion from a centrifugal force. Only in the case of zero angular momentum does the limiting radius coincide with the', '16 G.S. Bisnovatyi-Kogan, M. Merafina': "zero-gravity radius. The previous dimensionless considerations cannot be applied to the case with L = 0 because the scaling radius r k vanishes. It is easy to analyze this case in the original dimensional variables. From equations (2) and (3) we have \nE = 1 2 µ ˙ r 2 -µ ( GM r + Λ c 2 6 r 2 ) , ˙ φ = 0 . (34) \nAnalogous to (5) the equation for radial dependence on time is \ndr dt = ± √ 2 E µ +2 ( GM r + Λ c 2 6 r 2 ) , Ψ( r ) = 2 E µ +2 ( GM r + Λ c 2 6 r 2 ) . (35) \nThe zeros of the function Ψ( r ) define the turning points of the linear trajectory, with nonlinear oscillations. a The boundary between finite and infinite trajectories is a saddle point of the trajectory where Ψ and its derivative become zero \nΨ( r ) = 2 E µ +2 ( GM r + Λ c 2 6 r 2 ) = 0 , Ψ ' = -2 GM r 2 + 2Λ c 2 3 r = 0 . (36) \nThis system determines the boundary between finite and infinite trajectories r lim , and value of the energy E lim , at which this boundary is reached (see Eq. (4)) \nr lim = ( 3 GM Λ c 2 ) 1 / 3 = r 0 , E lim = -3 2 µGM r 0 . (37) \nThe finite amplitude oscillations occur only for E < E lim ; at larger E the trajectory goes to infinity. To solve the equation for the linear trajectory, we use the first equation of (35) in dimensionless variables, introducing an arbitrary radius r ∗ \nx = r r ∗ , τ = t t ∗ , t ∗ = r 3 / 2 ∗ √ GM , ε = 2 Er ∗ µGM , Λ c 2 r 3 ∗ 6 GM = 1 2 ( r ∗ r 0 ) 3 , (38) \nin the form \ndτ dx = ± 1 √ ( r ∗ /r 0 ) 3 x 2 +2 /x + ε . (39) \nIt is convenient to use the variable y = 1 /x , and Eq. (39) assumes the form \ndτ dy = ∓ 1 y 2 √ ( r ∗ /r 0 ) 3 /y 2 +2 y + ε . (40) \nHere ε ≤ -3 ( r ∗ /r 0 ), according to (37) and (38). At Λ > 0 we may use r ∗ = r 0 < ∞ , and Eq. (40) takes the simpler form \ndτ dy = ∓ 1 y 2 √ ε +2 y +1 /y 2 . (41) \nOrbital precession and other properties of two-body motion in the presence of dark energy 17 \nIn the presence of a singularity at r = 0, there are two possible interpretations of the oscillations. They may be interpreted as a limiting trajectory of two-body motion as L → 0. At any nonzero L the singularity is avoided, and the period of oscillations is defined as \nP tb 0 = 2 ∫ ∞ y -dy y 2 √ ε +2 y +1 /y 2 . (42) \nIn the second interpretation the two-body motion with zero angular momentum is passing through the singularity and Eq. (41) describes the oscillating motion between the points y = + y -and y = -y -, where two bodies cross through each other and exchange their positions. In this case the period of oscillations is equal to 2 P tb 0 . \nIn absence of DE we have from Ref. 34, and equations (40) and (42), the expression for the limiting Keplerian period P k 0 at L = 0 \ndτ dy = ∓ 1 y 2 √ ε +2 y , P k 0 = 2 ∫ ∞ -ε/ 2 dy y 2 √ ε +2 y = 2 [ √ ε +2 y -εy + 2 ( -ε ) 3 / 2 arctan √ ε +2 y -ε ] ∞ -ε/ 2 = 2 π ( -ε ) 3 / 2 . (43) \nComparing with (40), we see that the Keplerian period oscillations is defined by the same expression at all L ≥ 0. Taking into account the result (43), we can express the period of linear oscillations (42) at L = 0 in units of the Keplerian period ˜ P tb 0 as \n˜ P tb 0 = P tb 0 ( -ε ) 3 / 2 2 π . (44)", '8. Conclusions': 'We considered the Keplerian two-body problem with non-circular orbits, in the presence of dark nergy (identified with the cosmological constant Λ) introduced as a third additional force. The values of dimensionless parameters determining the typology of trajectories for variable Λ (or equivalently d ) and ϵ are determined. It is found that in the presence of a dark energy only two types of trajectories are present. \n- 1. Pure unbound trajectories for a family of parameters, corresponding to very large distance between the two gravitating bodies at a large negative total energy of the pair, or parameters corresponding to positive total energy.\n- 2. Simultaneous existence of bounded and unbounded trajectories for the same combination of parameters, defining the intermediate values of the total energy. \nThe bound trajectories could be represented approximately by precessing quasielliptical trajectories for small values of Λ, treated as small perturbation of the Keplerian elliptical orbit. Critical parameter values are found which determine the points at which the spontaneous transition from bound to unbound orbits can occur.', 'References': "- 1. S. Perlmutter et al., Astrophys. J. 517 (1999) 565.\n- 2. A. G. Riess et al., Astron. J. 116 (1998) 1009.\n- 3. D. N. Spergel et al., Astrophys. J. Suppl. 148 (2003) 175.\n- 4. M. Tegmark et al., Phys. Rev. D. 69 (2004) 103501.\n- 5. S. Weinberg, Rev. Mod. Phys. 61 (1989) 1.\n- 6. S. M. Carroll, W. H. Press and E. L. Turner, Ann. Rev. Astron. Astrophys. 30 (1992) 499.\n- 7. S. M. Carroll, Living Rev. Relativ. 4 (2001) 1.\n- 8. W. 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Sandage and B. Reindl, Nucl. Phys. B, Proc. Suppl. 80 (2000) ( Proc. of the Texas Symposium on Relativistic Astrophysics and Cosmology , Paris, France, 14-18 December, 1998), arXiv:astroph/9904360.\n- 48. H. Frommert and C. Kronberg, http://www.messier.seds.org/more/vir loc.html (1998).\n- 49. H. Frommert and C. Kronberg, http://www.messier.seds.org/more/virgo.html (2006).\n- 50. H. Frommert and C. Kronberg, http://www.messier.seds.org/more/local.html (2019).\n- 51. K. Nagamine and A. Loeb, New Astron. 8 (2003) 439.\n- 52. D. N. Spergel, R. Flauger and R. Hloˇzek, Phys. Rev. D 91 (2015) 023518.", 'Orbit precession, and binary period correction, in a linear approximation, for small values of Λ': 'A study of the orbital precession in linear approximation was studied earlier in Refs. 20, 21, using rather complicated methods of time averaging of equations following from Lagrangian and Hamiltonian functions. Here we obtain the precession frequency, and corrections to the Keplerian period, for small influence of Λ (small values of d ), in linear approximation for small valued of d , using a simple method considered in Ref. 35, which permitted to avoid non-physical singularities. \nWe find the precession frequency ω pr , calculating precession angle, in linear approximation, during one Keplerian period (15), using equations (28) and (30). \nω pr = ϕ pr P k , ϕ pr = ϕ tb -2 π, ϕ tb = 2 ∫ u + u -du √ -u 2 +2 u + ε + d/u 2 = 4 ∂ ∂ε [∫ u + u -√ -u 2 +2 u + ε + d/u 2 du ] = 2 ∫ u + u -du √ -u 2 +2 u + ε +2 d ∂ ∂ε [∫ u + u -du u 2 √ -u 2 +2 u + ε ] = 2 π +2 d ∂ ∂ε [∫ π 0 dφ u 2 ( φ ) ] = 2 π +2 d ∂ ∂ε [∫ π 0 dφ (1 -e cos φ ) 2 ] . (45) \nHere it is taken into account Eq. (18), from which \nu ( φ ) = 1+ e sin( φ -π/ 2) = 1 -e cos φ, e = √ 1 + ε, dφ = du √ -u 2 +2 u + ε (46) \nThe last integral in (45) is present in Ref. 34, from where we have \n∫ π 0 dφ (1 -e cos φ ) 2 = 1 1 -e 2 ∫ π 0 dφ 1 -e cos φ = π (1 -e 2 ) 3 / 2 = π ( -ε ) 3 / 2 . (47)', '20 G.S. Bisnovatyi-Kogan, M. Merafina': 'From these equations, with parameters of the Kepler motion obtained by equations (16) and (4), \nP k = 2 π a 3 / 2 √ GM , a = L 2 µ 2 GM (1 -e 2 ) , ω k = 2 π P k = √ GM a 3 / 2 , t 0 = µr 2 k L , (48) \nwe get the precession angle during one Keplerian period ϕ pr , and precession frequency ω pr in the form in which they had been derived in Refs. 20, 21: \nϕ pr = 3 πd ( -ε ) 5 / 2 , d = ( r k r 0 ) 3 = Λ c 2 L 6 3 µ 6 ( GM ) 4 , ϕ pr = 3 πd (1 -e 2 ) 5 / 2 = π Λ c 2 L 6 µ 6 ( GM ) 4 (1 -e 2 ) 5 / 2 = π Λ c 2 a 3 √ 1 -e 2 GM , ω pr = ϕ pr P k = Λ c 2 √ 1 -e 2 2 ω k = Λ c 2 a 3 / 2 √ 1 -e 2 2 √ GM = 3 d -2 εt 0 = 3 d 2(1 -e 2 ) t 0 . (49) \nTo find corrections to the period between two subsequent apocenter transitions of the given point of the trajectory, we start from (32). We have \n˜ P tb = 2 ∫ u + u -du u 2 √ -u 2 +2 u + ε + d/u 2 = 4 ∂ ∂ε [∫ u + u -du u 2 √ -u 2 +2 u + ε + d/u 2 ] = 2 ∫ u + u -du u 2 √ -u 2 +2 u + ε +2 d ∂ ∂ε [∫ u + u -du u 4 √ -u 2 +2 u + ε ] = 2 π ( -ε ) 3 / 2 +2 d ∂ ∂ε [∫ π 0 dφ u 4 ( φ ) ] = 2 π ( -ε ) 3 / 2 +2 d ∂ ∂ε [∫ π 0 dφ (1 -e cos φ ) 4 ] . (50) \nFor derivation of analytic expression for the binary period in the presence of DE, as a perturbation, we need to calculate the following integrals, using Ref. 34 \n∫ π 0 dφ (1 -e cos φ ) 4 = 1 1 -e 2 ∫ π 0 dφ (1 -e cos φ ) 3 + 2 e 3(1 -e 2 ) ∫ π 0 cos φdφ (1 -e cos φ ) 3 , ∫ π 0 dφ (1 -e cos φ ) 3 = 1 1 -e 2 ∫ π 0 dφ (1 -e cos φ ) 2 + e 2(1 -e 2 ) ∫ π 0 cos φdφ (1 -e cos φ ) 2 . (51) \n∫ π 0 cos φdφ (1 -e cos φ ) 3 = 1 2(1 -e 2 ) ∫ π 0 (2 e +cos φ ) dφ (1 -e cos φ ) 2 , ∫ π 0 cos φdφ (1 -e cos φ ) 2 = e 1 -e 2 ∫ π 0 dφ 1 -e cos φ = πe (1 -e 2 ) 3 / 2 . (52) \nWe have from equations (47) and (50)-(52) \nI 4 = ∫ π 0 dφ (1 -e cos φ ) 4 = π (1 -e 2 ) 7 / 2 ( 1 + 3 2 e 2 ) = π ( -ε ) 7 / 2 ( 5 2 + 3 2 ε ) , (53) \nOrbital precession and other properties of two-body motion in the presence of dark energy 21 \n˜ P tb = 2 π ( -ε ) 3 / 2 +2 d ∂I 4 ∂ε , ∂I 4 ∂ε = 5 π (1 -e 2 ) 9 / 2 [ 7 4 -3 4 (1 -e 2 ) ] = 5 4 π (1 -e 2 ) 9 / 2 (4 + 3 e 2 ) , ∆ P pr = P tb -P k = ( ˜ P tb -˜ P k ) t 0 = 2 dt 0 ∂I 4 ∂ε = 5 π 6 Λ c 2 ω 3 k (4 + 3 e 2 ) . (54)'}
2024A&A...690L..18S	We present the first observations of HCOSUPSUP10 and HCN10 emission in the northern filaments of Centaurus A with ALMA. HCOSUPSUP10 is detected in nine clumps of the Horseshoe complex with similar velocities as the CO10 emission. Conversely HCN10 is not detected and we derive upper limits on the flux. At a resolution of 40 pc the line ratio of the velocityintegrated intensities ISUBHCOSUBISUBCOSUB varies between 0.03 and 0.08 while ISUBHCOSUBISUBHCNSUB is higher than unity with an average lower limit of 1.51. These ratios are significantly higher than what is observed in nearby starforming galaxies. Moreover the ratio ISUBHCOSUBISUBCOSUB decreases with increasing COintegrated intensity contrary to what is observed in the starforming galaxies. This indicates that the HCOSUPSUP emission is enhanced and may not arise from dense gas within the Horseshoe complex. This hypothesis is strengthened by the average line ratio ISUBHCNSUBISUBCOSUB lt 0.03 which suggests that the gas density is rather low. Using nonlocal thermal equilibrium large velocity gradient modelling with RADEX we explored two possible phases of the gas which we call diffuse and dense and are characterised by a significant difference in the HCOSUPSUP abundance relative to CO respectively NSUBHCOSUBNSUBCOSUB 10SUP3SUP and NSUBHCOSUBNSUBCOSUB 3 10SUP5SUP. The average CO10 and HCOSUPSUP10 integrated intensities and the upper limit on HCN10 are compatible with both diffuse nSUBHSUB 10SUP3SUP cmSUP3SUP TSUBkinSUB 15 165 K and dense gas nSUBHSUB 10SUP4SUP cmSUP3SUP TSUBkinSUB gt 65 K. The spectral setup of the present observations also covers SiO21. While undetected the upper limit on SiO21 is not compatible with the RADEX predictions for the dense gas. We conclude that the nine molecular clouds detected in HCOSUPSUP10 are likely dominated by diffuse molecular gas. While the exact origin of the HCOSUPSUP10 emission remains to be investigated it is likely related to the energy injection within the molecular gas that prevents gravitational collapse and star formation.	2024-10-01T00:00:00Z	['2024A&A...690L..18S', 'arXiv:2409.11031', '10.48550/arXiv.2409.11031', '10.1051/0004-6361/202450952', '2024arXiv240911031S']	['methods: data analysis', 'galaxies: ISM', 'galaxies: individual: Centaurus A', 'galaxies: star formation', 'radio lines: galaxies', 'Astrophysics - Astrophysics of Galaxies']	Physical conditions in Centaurus As northern filaments II. Does the HCOSUPSUP emission highlight the presence of shocks	2,024	200	0.51	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.11031.pdf	{"Physical conditions in Centaurus A's northern filaments II. Does the HCO + emission highlight the presence of shocks? ⋆": "Q. Salomé 1 , 2 , P. Salomé 3 , B. Godard 3 , 4 , P. Guillard 5 , A. Gusdorf 4 , 3 \n- 1 Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Vesilinnantie 5, 20014 Turku, Finland email: [email protected]\n- 2 Aalto University Metsähovi Radio Observatory, Metsähovintie 114, 02540 Kylmälä, Finland\n- 3 Observatoire de Paris, LERMA, CNRS, Université PSL, Sorbonne Université, 75014 Paris, France\n- 4 Laboratoire de Physique de l'ENS, Ecole Normale Supérieure, Université PSL, CNRS, Sorbonne Université, 75005 Paris, France\n- 5 Sorbonne Université, CNRS, Institut d'Astrophysique de Paris, 98bis bd Arago, 75014 Paris, France \nReceived 31 May 2024 / Accepted 15 September 2024", 'ABSTRACT': "We present the first observations of HCO + (1-0) and HCN(1-0) emission in the northern filaments of Centaurus A with ALMA. HCO + (1-0) is detected in nine clumps of the Horseshoe complex, with similar velocities as the CO(1-0) emission. Conversely, HCN(1-0) is not detected, and we derive upper limits on the flux. At a resolution of ∼ 40 pc, the line ratio of the velocity-integrated intensities IHCO + / ICO varies between 0.03 and 0.08, while IHCO + / IHCN is higher than unity, with an average lower limit of 1.51. These ratios are significantly higher than what is observed in nearby star-forming galaxies. Moreover, the ratio IHCO + / ICO decreases with increasing CO-integrated intensity, contrary to what is observed in the star-forming galaxies. This indicates that the HCO + emission is enhanced and may not arise from dense gas within the Horseshoe complex. This hypothesis is strengthened by the average line ratio IHCN / ICO < 0 . 03, which suggests that the gas density is rather low. \nUsing non-local thermal equilibrium, large velocity gradient modelling with RADEX, we explored two possible phases of the gas, which we call 'di ff use' and 'dense' and are characterised by a significant di ff erence in the HCO + abundance relative to CO, respectively NHCO + / NCO = 10 -3 and NHCO + / NCO = 3 × 10 -5 . The average CO(1-0) and HCO + (1-0) integrated intensities and the upper limit on HCN(1-0) are compatible with both di ff use (nH = 10 3 cm -3 , Tkin = 15 -165 K) and dense gas (nH = 10 4 cm -3 , Tkin > 65 K). The spectral setup of the present observations also covers SiO(2-1). While undetected, the upper limit on SiO(2-1) is not compatible with the RADEX predictions for the dense gas. We conclude that the nine molecular clouds detected in HCO + (1-0) are likely dominated by di ff use molecular gas. While the exact origin of the HCO + (1-0) emission remains to be investigated, it is likely related to the energy injection within the molecular gas that prevents gravitational collapse and star formation. \nKey words. methods:data analysis - galaxies:individual:Centaurus A - galaxies:ISM - galaxies:star formation - radio lines:galaxies", '1. Introduction': "Active galactic nuclei (AGN) are believed to play a role in regulating and / or quenching star formation in galaxies (Bower et al. 2006; Croton et al. 2006). However, evidence of AGN positive feedback, which enhances star formation, is also seen in radio galaxies, in particular in regions of radio jet-gas interaction at low (Croft et al. 2006; Morganti 2010; Salomé et al. 2015; Zovaro et al. 2020; Capetti et al. 2022) and high redshifts (Nesvadba et al. 2020; Duncan et al. 2023). \nCentaurus A is a large double-lobed radio source that extends over about 600 kpc (see Ne ff et al. 2015 and references therein). About 15 kpc away from the radio core, the radio lobe emission encounters a large HI shell (Schiminovich et al. 1994) associated with recent star formation (Rejkuba et al. 2001; Auld et al. 2012; Joseph et al. 2022). Its proximity (d = 3 . 8 Mpc; Harris et al. 2010) makes Centaurus A the ideal target for studying the impact of an AGN-driven jet on its environment. Salomé et al. (2016b,a) observed a large reservoir of molecular gas at the intersection of the radio continuum and the HI gas, within the northern optical filaments (Blanco et al. 1975; Morganti et al. 1991). \n- ⋆ This paper makes use of the following ALMA data: ADS / JAO.ALMA#2015.1.01019.S and #2016.1.00261.S. \nThe bulk of the molecular gas lies outside the HI shell, suggesting that the jet-gas interaction induced star formation by triggering the atomic-to-molecular gas phase transition (Salomé et al. 2016a). However, the large reservoir of molecular gas is very ine ffi cient at forming stars compared with nearby star-forming galaxies (Salomé et al. 2016b,a). \nThe Atacama Large Millimeter / submillimeter Array (ALMA) resolved the CO emission in the so-called northern filaments of Centaurus A and enabled a collection of giant molecular clouds (GMCs) distributed along a Horseshoe-like structure to be observed (Salomé et al. 2017). These molecular clouds have very similar physical properties (mass, size, and velocity dispersion) as in the inner Milky Way. However, the virial parameter indicates that kinetic energy is injected into the molecular clouds and prevents gravitational collapse. Moreover, the excitation of the ionised gas associated with the Horseshoe complex suggests the presence of shocks (Salomé et al. 2016a, 2017), which could explain the ine ffi cient jet-induced star formation. \nSalomé et al. (2019) used mid-J CO lines observed with the Atacama Pathfinder Experiment (APEX) in position 16 from Salomé et al. (2016a). Using the Paris-Durham shock model (Flower & Pineau des Forêts 2015), they show that the CO line \nTable 1: ALMA and ACA observations during Cycle 3 and Cycle 4. \nFig. 1. Moment 0 map of the CO(1-0) emission at the resolution of the HCO + observations. The red contours are those of the HCO + emission. The solid line and dashed circles correspond to the field of view and the FWHM of the HCO + primary beam, respectively. The numbers are the clump labels. Subpanel: Position of the HCO + field of view with respect to the HI shell (black contours) and the radio continuum (grey contours). A larger spatial overview is provided in Fig. 1 of Oosterloo & Morganti (2005), where the region is labelled 'outer filament'. \n<!-- image --> \nRA (2000) \nratios are compatible with low-velocity shocks in di ff use gas (shock velocities between 4 and 20 km . s -1 and pre-shock density nH = 100 cm -3 ). However, these predictions are an average over the large area covered by the beams ( > 250 pc). The high resolution of ALMA now allows the shocks experienced by the di ff erent GMCs to be studied. \nIn this Letter we present ALMA observations of dense gas tracers (HCN and HCO + ) in the Horseshoe complex and compare them with the cold molecular gas traced by CO. We aim to constrain the properties of the molecular gas when the GMCs are resolved. The data are presented in Sect. 2. We analyse the data in Sects. 3 and 4, and then compare our observations with a grid of RADEX models in Sect. 5.", '2. Observations': "The HCN(1-0) and HCO + (1-0) lines were observed with the ALMA 12m array during Cycle 4 using the Band 3 receivers (project 2016.1.00261.S). The observations consist of one pointing centred on the eastern CO-bright region from Salomé et al. (2016a), which was later identified as the Horseshoe complex by Salomé et al. (2017). The baselines ranged from 15.1 m to 460 m, corresponding to a resolution of 2 . 1 '' × 1 . 6 '' and maximum recoverable scales (MRS) of about 15 . 5 '' ∼ 280 pc. \nTo improve the sensitivity, we decreased the spatial resolution by cutting any baselines larger than 272.6 m (corresponding to the 90th percentile), and we reduced the channel sampling to 3 . 0 km . s -1 . The total integration time of 2.1 h provides a noise level of 0.58-0.59 mJy / beam in a synthesised beam of \n2 . 3 '' × 2 . 0 '' ∼ 42 × 37 pc (PA ∼ 88 · ). Our observations also covered the SiO(2-1) line with a rms of 0.60 mJy / beam. \nFor the CO(1-0) emission, we re-calibrated, re-combined, and re-imaged the observations (Salomé et al. 2017, 2019; project 2015.1.01019.S) because an issue with CASA a ff ected the flux in the Atacama Compact Array (ACA) data (see NAASC Memo117 1 ). We obtained a spatial resolution of 1 . 56 '' × 1 . 15 '' ∼ 29 × 21pc (PA = 89 · ). The noise level is 4.7 mJy / beam at a spectral resolution of 1 . 5 km . s -1 . \nTable 1 summarises the main characteristics of the various data cubes. The combination of ALMA and ACA data for the CO(1-0) emission enabled us to cover a large range of spatial scales and limit the spatial filtering by the interferometer. For HCN, HCO + , and SiO, the maximum recovered scale corresponds to 280-290 pc. The HCN and HCO + emission is commonly used as a tracer of dense gas in galaxies. Moreover, in local GMCs, most of the emission is localised in the densest regions (e.g. in Orion B; Pety et al. 2017). While we cannot rule out a filtering of possible extended HCN and HCO + emission, we expect the emission to be much more compact than the MRS, and the spatial filtering to be negligible. We tested this hypothesis by comparing our results with those obtained when considering the CO(1-0) emission from ALMA alone, which has a MRS of 10 . 9 '' (i.e. ∼ 200 pc) closer to the MRS of HCN, HCO + , and SiO. \nTable 2: Properties of the di ff erent clumps detected with ALMA. \nNotes. The clump numbers are the same as those indicated in Fig. 1. The integrated flux densities were derived by summing the channels where emission is detected, multiplied by the channel width. The velocity (relative to Centaurus A) and FWHM were estimated by fitting the spectrum with a single Gaussian profile. For the HCN(1-0) and SiO(2-1) emission, upper limits at 3 σ were derived assuming the same FWHM and the same area as for HCO + (1-0). The last three columns report the line ratios of the integrated intensities in K . km . s -1 and are therefore corrected by the ratio of the rest frequencies (see Sect. 4).", '3.1. HCO + (1-0) emission': 'We first used the mapping package from the GILDAS software 2 to produce moment 0 maps of the HCN(1-0), HCO + (1-0), and SiO(2-1) lines. We used the velocity range of the CO(10) emission in this region ( -350 < v < -175 kms -1 ; Salomé et al. 2017) and a threshold of 1 . 1 σ . HCO + is detected, but HCN and SiO are not. Using the moment 0 map of HCO + (10) as a guide, we explored the data cube with the viewer tool in mapping to constrain the velocity range of the line emission from each clump. For each clump, we then spectrally averaged the uv table over the corresponding velocity range to produce an image of the emission. \nThe signal-to-noise ratio of the data cube is rather low ( < 5 in each pixel). Therefore, we created a 3D mask to exclude channels that do not contain signal, to limit the impact of noise, and to produce more accurate moment maps of the HCO + (1-0) emission. For each clump, the mask selects the channels within the velocity range of the line emission and the pixels within the 2 σ contours of the spectrally integrated emission. \nFigure 1 shows the moment 0 map of the CO(1-0) emission, along with the contours of the HCO + (1-0) emission. The HCO + emission is distributed into nine clumps and has the same morphology as the CO(1-0) emission, with the Horseshoe-like feature clearly identified. Dense gas tracers have already been detected in molecular outflows (e.g. by Salas et al. 2014 in M82, by Alatalo 2015 and Cicone et al. 2020 in Mrk 231, by Walter et al. 2017 in NGC 253, and by Barcos-Muñoz et al. 2018 in Arp 220). However, this is the first detection of dense gas tracers in a region of jet-gas interaction outside the galactic plane. \nWe extracted the integrated spectrum of each clump within the 2 σ contour of this new moment 0 map. The spectra were analysed using the CLASS package of GILDAS. The spectral resolution was first decreased to 6 km s -1 in order to improve the signal-to-noise ratio without under-sampling the line. The integrated flux density R SHCO + d 3 was derived by integrating the spectra over the velocity ranges used to build the 3D mask. For the peak velocity vpeak and the full width at half maximum (FWHM), we fitted the emission with a single Gaussian. The characteristics of the HCO + (1-0) emission of each clump are reported in Table 2. The spectra are presented in Fig. A.1.', '3.2. HCN(1-0) emission': 'The HCN(1-0) line is not detected in these ALMA observations. Since the HCN and HCO + lines are commonly expected to trace dense gas, we assumed that the HCN(1-0) line would be emitted by the same region as the HCO + (1-0), with similar linewidths (e.g. Pety et al. 2017; Jiménez-Donaire et al. 2019). We extracted the HCN(1-0) spectrum of each clump within the 2 σ contour of the HCO + (1-0) moment 0 map and derived an upper limit at 3 σ : \nZ S HCNd 3 (mJy km s -1 ) < √ 2 π 2 . 354 × 3 σ (mJy) FWHMHCO + (1) \nwhere we assume FWHMHCN = FWHMHCO + (i.e. the FWHM of the HCO + from Table 2). The HCN(1-0) integrated flux density is lower than 13 -90 mJy . km . s -1 , similar to the upper limit of 33 . 5mJy . km . s -1 reported by Salomé et al. (2016b) based on Australia Telescope Compact Array (ATCA) observations within the HI shell. \nThe SiO(2-1) line is not detected either. Using the same method as for HCN(1-0), we derived upper limits of 16 -104 mJy . km . s -1 .', '3.3. CO(1-0) emission': 'We smoothed the CO(1-0) cube to the spatial resolution of the HCO + (1-0) and extracted the CO(1-0) spectra of the clumps within the 2 σ contour of the HCO + (1-0) moment 0 map. The CO emission covers the same range of velocities as the HCO + emission (see Fig. A.1). The CO line profiles were fitted with a Gaussian profile. The integrated fluxes are reported in Table 2. The integrated fluxes obtained when considering the ALMA data alone are 10% to 30% lower (not reported in Table 2), likely due to the di ff use CO(1-0) emission.', '4. Line ratios': "We studied the ratios of the velocity-integrated intensities ICO, IHCO + , and IHCN (in K . km . s -1 ) and compared them to literature values. The line ratios of the integrated intensities of the clumps can be derived from the integrated flux densities by taking the rest frequency of the lines into account: \nI 1 / I 2 = Z S 1 d 3 GLYPH<30> Z S 2 d 3 ! × ( ν 2 /ν 1) 2 , (2) \nwhere Ii are the integrated intensities, R Si d 3 are the integrated flux densities, and ν i are the frequencies. The line ratios of each clump are reported in Table 2. \nThe IHCO + / ICO line ratio varies between 0.03 and 0.08. This is significantly higher than what was observed in nearby star-forming galaxies by the EMPIRE (Jiménez-Donaire et al. 2019) and ALMOND surveys (Neumann et al. 2023). Moreover, IHCO + / ICO decreases with increasing CO integrated intensity, contrary to what is observed in star-forming galaxies (Fig. 2 - left). This suggests that the HCO + (1-0) emission is not tracing the dense gas within the northern filaments of Centaurus A. Instead, it looks as if the HCO + emission is being enhanced by an external process that is not related to the density. \nFor the IHCN / ICO and IHCO + / IHCN line ratios, we derived upper and lower limits, respectively. The line ratio IHCN / ICO is lower than 0.06 and IHCO + / IHCN is higher than unity. The average IHCO + / IHCN of the clumps is > 1 . 51, while the IHCO + / IHCN ratio of the total emission from the clumps is > 1 . 72. In particular, three clumps detected in HCO + have a line ratio IHCO + / IHCN higher than 2, similar to that in the nuclear region of NGC 5128 (McCoy et al. 2017). The IHCO + / IHCN line ratio is significantly higher than the typical ratio observed in star-forming galaxies (Fig. 2 - right; Brouillet et al. 2005; Jiménez-Donaire et al. 2019; Forbrich et al. 2023; García-Rodríguez et al. 2023; Neumann et al. 2023). \nThe IHCO + / IHCN ratio for the total emission is typical of what is observed in starbursts (Imanishi et al. 2007; Salas et al. 2014; Schirm et al. 2016; Walter et al. 2017) or 'composite' AGN associated with a nuclear starburst (Kohno 2003, 2005; Krips et al. 2008; Privon et al. 2015). Such high line ratios can also be associated with a low gas metallicity (e.g. Galametz et al. 2020; Forbrich et al. 2023). However, there is no evidence of recent star formation in the Horseshoe complex, and the gas metallicity is only slightly sub-solar (see Salomé et al. 2016a, 2017 and references within), suggesting that the enhanced HCO + (1-0) emission likely has another origin. \nThe di ff use CO emission represents 10% to 30% of the total emission of the clumps. Therefore, if the HCN, HCO + , and SiO emission is extended and not fully recovered by ALMA, our conclusion does not change since the IHCO + / ICO ratio would be even higher.", '5. Low-velocity gradient modelling': "We used the non-local thermal equilibrium radiative transfer code RADEX (van der Tak et al. 2007) to constrain the physical conditions of the clumps. Given a molecule, X, and the triplet {NX, nH, Tkin}, RADEX models the integrated intensities of the lines of the molecule. We adopted a linewidth ∆ V = 35 km . s -1 , which corresponds to the average FWHM of the clumps. We compared the outputs of the RADEX models with the average integrated intensities over the clumps: ICO = 36 . 0 ± 11 . 9K . km . s -1 , IHCO + = 1 . 56 ± 0 . 12 K . km . s -1 , and IHCN < 1 . 03 K . km . s -1 . \nWe explored the triplet {NX, nH, Tkin} with (i) nH = 50 , 100 , 500 , 10 3 , and 10 4 cm -3 , (ii) NCO between 10 15 and 10 19 cm -2 , and (iii) kinetic temperatures from 10 K to 200 K. We consider two phases of the gas that we call 'di ff use gas' (n ≲ 500 cm -3 ; e.g. Snow & McCall 2006) and 'dense gas' (n ≥ 10 4 cm -3 ; e.g. Snow & McCall 2006). This two phase are characterised by typical and significantly di ff erent relative abundances, NHCO +/ NCO and NHCN / NCO. We therefore derived the column density of HCO + and HCN from their relative abundances. In the following, we adopt the relative abundances NHCO + / NCO = 10 -3 and NHCN / NHCO + = 1 . 9 (Liszt et al. 2010; Godard et al. 2010) \nfor the di ff use molecular gas and NHCO + / NCO = 3 × 10 -5 and NHCN / NHCO + = 1 . 5 (Liszt et al. 2010; Liu et al. 2013) for the dense molecular gas. When running RADEX, we took into account the excitation by collisions with H2, He, and electrons. We adopted the following abundance ratios: nHe / nH = 0 . 1 and ne / nH = 0 , 10 -5 , and 10 -4 . \nThe present grid of models only reproduces the observed average integrated intensities simultaneously for densities of 10 3 and 10 4 cm -3 (presented in Fig. B.1). The models that reproduce the observations are highlighted in red and reported in Table 3. Two regimes of the molecular gas can reproduce the observations: dense gas at nH = 10 4 cm -3 and Tkin > 65 K; and diffuse gas at nH = 10 3 cm -3 and Tkin = 15 -165 K. We get the same predictions regardless of whether the CO emission from ALMA + ACA or from ALMA alone is used. We note that the predictions for the di ff use gas are in agreement with the midJ CO line ratios at lower spatial resolutions from Salomé et al. (2019), which are indicated by the dotted contours in Fig. B.1. \nTable 3: Predictions from RADEX. \nNotes. This table shows the triplets {NCO, nH, Tkin} that reproduce the CO and HCO + intensities and the upper limit on HCN for the di ff use and dense gas. \nWe explored the e ff ects of small variations in relative abundance values around their typical estimates taken here to characterise the dense and di ff use gas. This is illustrated in Appendix C. For the dense gas, varying the abundance does not change the possible values of the gas density, which remain high ( ≥ 10 4 cm -3 ). However, if the abundance is significantly reduced (by 40% or more), then there are no longer any compatible solutions at n = 10 4 cm -3 . On the other hand, if the abundance is increased significantly (by 40% or more), then the possible temperature solutions tend to decrease and to become bounded. For the di ff use gas, varying the abundance does not change the possible values of the gas density either: it remains low ( ≤ 10 3 cm -3 ). If the abundance is significantly reduced (by 40% or more), then there are again no longer any compatible solutions, even at high densities. On the other hand, if the abundance is significantly increased (by 30% or more), then solutions at lower temperatures and lower densities become possible. In conclusion, abundances significantly lower than the typical values used for the dense and di ff use gas here are excluded or would lead to models not compatible with our observations. To the contrary, significantly higher abundances would imply possible lower temperatures for dense gas and lower temperatures and / or lower densities for diffuse gas. It should be noted, however, that these conclusions are based solely on variations around characteristic abundance values for two cases: dense and di ff use gas. A detailed study of the variation in abundance in a less constrained parameter space is not carried out here. \nSiO emission - While SiO(2-1) is not detected, we derive upper limits (see Sect. 3.2 and Table 2). We investigated the e ff ect of considering an upper limit ISiO < 1 . 08 K . km . s -1 when constraining the RADEX. To do so, we used the SiO abundance \n<!-- image --> \nFig. 2: IHCO + / ICO as a function of ICO ( left ) and IHCN / ICO ( right ). The red points are the clumps in Table 2, and the blue and green points come from the EMPIRE (Jiménez-Donaire et al. 2019) and ALMOND surveys (Neumann et al. 2023). The dashed line in the right panel shows the unity relation. \n<!-- image --> \nNSiO / NH 2 = 10 -8 . 5 (from Towner et al. 2024). The upper limit on SiO is indicated by the magenta line in Fig. B.1. This upper limit is not compatible with the predictions for the dense gas, suggesting that the detected HCO + emission is tracing di ff use molecular gas. \nMolecular gas mass - Eight of the nine HCO + clumps seem to be unresolved with the present ALMA observations, for which the synthesised beam has a characteristic radius of 20 pc. The triplets {NX, nH, Tkin} from RADEX are average values of the total emission. Assuming spherical clouds, the average molecular gas mass of the clumps predicted by the models is Mpred = 8 . 3 × 10 5 M ⊙ for the di ff use gas abundances and Mpred = 8 . 3 × 10 6 M ⊙ for the dense gas abundances. Those estimates are larger than the average molecular gas mass derived from the CO emission with a standard CO-to-H2 conversion factor: Mobs = 2 . 5 × 10 5 M ⊙ . This suggests that these eight HCO + clumps are smaller than the beam of ALMA, and likely associated with di ff use gas. \nTotal H 2 luminosity - Cooling of the gas commonly occurs via H2 emission. It is possible to estimate the total H2 luminosity that a molecular cloud of radius R = 20 pc would produce, assuming local thermal equilibrium: \nLH 2 = 4 π D 2 L Ω X 1 4 π h ν i j Ai j Ni ! , (3) \nwhere ν ij and Aij are the frequency and the Einstein A coe ffi cient of the H2 transition i → j, DL is the luminosity distance, Ω is the solid angle of a clump of radius 20 pc, and Ni is the column density of H2 molecules in the excitation level i given by \nNi = Ntot × gi exp -Ei kTex !! GLYPH<30> X gi exp -Ei kTex !! , (4) \nwith gi and Ei the weight and energy of level i, and Tex the excitation temperature. The total column density is estimated assuming a spherical cloud: \nNtot = (4 / 3) π nR 3 4 π R 2 . (5) \nCooling by H 2 - We predict that the total H2 luminosity produced by a molecular cloud of radius 20 pc would be LH 2 = 2 × 10 28 -3 . 7 × 10 37 erg . s -1 if the gas is di ff use (nH = 10 3 cm -3 and Tkin = Tex = 20 -125K), and LH 2 = 9 . 7 × 10 37 -4 . 1 × 10 39 erg . s -1 if the gas is dense (nH = 10 4 cm -3 and Tkin = Tex = 100 -200K). We note that the H2 luminosity is highly dependent on the gas temperature. Better constraints on the gas temperature are thus important, in particular for the di ff use gas solution. Observations with the K-band Multi Object Spectrograph (KMOS) of ro-vibrational lines of H2 will allow the cooling energy radiated by H2 to be measured (Salomé et al., in prep.). If the observed H2 luminosity is lower than 10 38 erg . s -1 , we would be able to eliminate the dense gas solutions and constrain the gas temperature: LH 2 = 3 . 4 × 10 30 erg . s -1 at 25 K, 7 . 1 × 10 34 erg . s -1 at 50 K, 1 . 6 × 10 36 erg . s -1 at 75 K, and 9 . 7 × 10 36 erg . s -1 at 100 K. \nHeating by the radio jet - The northern radio emission of Centaurus is complex, with several structures extended to di ff erent scales. Ne ff et al. (2015) estimated the total power within the di ff erent structures and reported a power PNML ∼ 10 44 erg . s -1 for the 'Northern Middle Lobe' over an area of 425 kpc 2 . If we consider a homogeneously distributed power within the Northern Middle Lobe, the H2 emission produced by the dense gas would be at least 30% of the power of the radio plasma available locally at the scale of the clumps. In particular, this fraction is higher than unity for a gas temperature higher than 120 K. This suggests that the energy of the Northern Middle Lobe is not enough to heat dense gas at nH = 10 4 cm -3 . Conversely, this fraction would be only 3% for di ff use gas at 100 K. The energy injected by the radio jet is thus quantitatively a possible source of excitation of the H2. \nCosmic ray heating - In the case of heating by cosmic rays (Yusef-Zadeh et al. 2007; Ferland et al. 2008), we can estimate the cosmic-ray ionisation rate, ζ , needed to balance the cooling by the H2 line emission. We considered two cases representative of the di ff use and dense gas conditions predicted by RADEX: (i) nH = 10 3 cm -3 and Tkin = Tex = 75 K, and (ii) nH = 10 4 cm -3 and Tkin = Tex = 150 K. Using the molecular gas mass and the \ntotal H2 luminosity predicted by RADEX, the average total line emission per H2 molecule would be L H2 = 3 . 2 × 10 -34 W . H -1 2 for the di ff use gas and L H2 = 2 . 0 × 10 -32 W . H -1 2 for the dense gas (Table 4). \nFollowing the discussion of Ogle et al. (2010), we estimated the cosmic ray ionisation rate using the following equation: \nζ = 1 . 2 × 10 -13 × L H 2 1 . 3 × 10 -31 W ! s -1 . H -1 . (6) \nTo balance the H2 cooling with cosmic ray heating, an ionisation rate ζ = 3 . 0 × 10 -16 s -1 . H -1 and 1 . 9 × 10 -14 s -1 . H -1 is required for the di ff use and dense gas, respectively (Table 4). This is respectively at least a factor of 0.4 and 24 greater than the cosmic ray ionisation rate derived in the centre of Centaurus A by Van der Tak et al. (2016). This suggests that the heating of the molecular gas within the filaments is unlikely dominated by the cosmic rays. \nTable 4: Cosmic ray heating predicted by RADEX, assuming local thermal equilibrium. \nNotes. We considered two sets of parameter characteristics for the diffuse and dense gas. \nPossible origin of the HCO + emission -At densities n ≤ 10 4 cm -3 , high IHCO + / IHCN line ratios can be explained by photodissociated regions (PDRs) or X-ray-dominated regions (XDRs; Meijerink et al. 2007). In particular, the line ratios between dense gas tracers found in the nuclear region of NGC 5128 by McCoy et al. (2017) indicate that the HCO + might come from XDRs. HCO + emission can also be enhanced by shocks in the di ff use gas (Godard et al. 2019), and there is evidence of shocks in the filaments of Centaurus A (Oosterloo & Morganti 2005; Salomé et al. 2016a, 2019). Studying the presence of PDRs, XDRs, or shocks in the northern filaments of Centaurus A is beyond the scope of the present paper and would need additional observational constraints. In particular, H2 ro-vibrational lines can be used to identify the relative contribution of shocks and PDRs to the energy injection within the molecular gas (Villa-Vélez et al. 2024).", '6. Conclusion and discussion': 'We have presented the first observations of the HCN(1-0), HCO + (1-0), and SiO(2-1) emission in the northern filaments of Centaurus A conducted with ALMA. HCO + (1-0) is detected in nine clumps distributed along the Horseshoe complex, but no HCN or SiO is detected. We extracted the spectra of the clumps for the HCO + (1-0) and CO(1-0) emission and computed upper limits of the HCN(1-0) and SiO(2-1) fluxes within the 2 σ contours of the HCO + moment 0 map. We derive relatively high IHCO + / ICO and IHCO + / IHCN line ratios and a relatively low IHCN / ICO ratio compared to that typically observed in nearby star-forming galaxies. Moreover, we find that the IHCO + / ICO ratio decreases with increasing ICO. This indicates an enhanced HCO + emission that is likely not associated with a high dense-gas fraction. \nWe used the average CO(1-0) and HCO + (1-0) integrated intensities, as well as the upper limit on IHCN, to constrain the \ngrid of large velocity gradient models from RADEX. The observations can be explained either by di ff use molecular gas at nH = 10 3 cm -3 or by dense molecular gas at nH = 10 4 cm -3 . However, we note that dense molecular gas is not compatible with the upper limit on ISiO. Moreover, an analysis of the predicted molecular gas mass and H2 luminosity suggests that the HCO + (1-0) likely arises from di ff use gas at n ≤ 10 3 cm -3 . \nAcknowledgements. We thank the referee for his / her comments. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI / NRAO and NAOJ. QSthanks the Observatoire de Paris for the access to the computational facilities. Q.S. acknowledges the financial support from the visitor and mobility program of the Finnish Centre for Astronomy with ESO (FINCA), funded by the Academy \n- of Finland grant nr 306531.', 'References': "Alatalo, K. 2015, ApJ, 801, L17 Auld, R., Smith, M. W. L., Bendo, G., et al. 2012, MNRAS, 420, 1882 Barcos-Muñoz, L., Aalto, S., Thompson, T. A., et al. 2018, ApJL, 853, L28 Blanco, V. M., Graham, J. A., Lasker, B. M., & Osmer, P. S. 1975, ApJ, 198, L63 Bower, R. G., Benson, A. 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The blue and green areas show the range of observed CO(1-0) and HCO + (1-0) integrated intensities, while the black and magenta lines indicate the upper limits of HCN(1-0) and SiO(2-1). The columns correspond to predictions for relative electron abundances of ne / nH = 0 (left), ne / nH = 10 -5 (middle), and ne / nH = 10 -4 (right). The red areas correspond to the models that reproduce the CO and HCO + intensities and the HCN upper limit. The dotted lines indicate the models that reproduce the mid-J CO lines emission from Salomé et al. (2019). \n<!-- image --> \nA & A proofs: manuscript no. Physical\\_conditions\\_CenA\\_filaments\\_II\\_Dense\\_gas\\_tracers \n<!-- image --> \nFig. B.1. Continued. \nFig. C.1: Radiative transfer predictions from RADEX for dense gas at nH = 10 4 with a relative electron abundance ne / nH = 10 -5 . The HCO + / CO abundance ratio is respectively 50, 80, 100, 120, and 150% of the typical value for the dense gas. The blue, green, and red areas and the black line are the same as in Fig. B.1. \n<!-- image --> \nFig. C.2: Same as Fig. C.1 but for di ff use gas at nH = 10 3 . \n<!-- image -->'}
2024MNRAS.534..400B	We studied the ionized gas in the inner region inlineformulatexmath idTM0001 notationLaTeXsimtexmathinlineformulainlineformulatexmath idTM0002 notationLaTeX680times 470texmathinlineformula pcSUP2SUP of the galaxy NGC 6868 using GeminiGMOS Gemini MultiObject Spectrograph integral field unit observations. Channel maps reveal complex kinematics and morphology indicating multiple processes at work in NGC 6868. Through emissionline fitting we identified two ubiquitous components in our data a narrow inlineformulatexmath idTM0003 notationLaTeXsigma sim 110texmathinlineformula km sinlineformulatexmath idTM0004 notationLaTeX1texmathinlineformula tracing an ionized gas disc and a broad component inlineformulatexmath idTM0005 notationLaTeXsigma sim 300texmathinlineformula km sinlineformulatexmath idTM0006 notationLaTeX1texmathinlineformula mainly associated with inflowingoutflowing gas. The derived Vband reddening shows a spatial distribution consistent with that obtained from stellar population synthesis although with generally higher values. For the first time we measured the electron temperature in NGC 6868 finding values ranging from inlineformulatexmath idTM0007 notationLaTeXsimtexmathinlineformula14 000 K in the central region to inlineformulatexmath idTM0008 notationLaTeXgtrsim 20000texmathinlineformula K with an outward increasing temperature gradient. The electron density map exhibits an inverse relationship with central values reaching inlineformulatexmath idTM0009 notationLaTeXNesim 4000texmathinlineformula cmSUP3SUP for the broad component decreasing to inlineformulatexmath idTM0010 notationLaTeXNesim 100texmathinlineformula cmSUP3SUP towards the edges of the field of view. Using BPT diagrams we found that all spaxels are consistent with both active galactic nucleus AGN and shock ionization. However when this information is combined with our kinematic and temperature findings and further supported by the WHAN diagram we argue that an AGN is the dominant ionization mechanism in the central region of NGC 6868 while the extended outer component is ionized by a combination of hot lowmass evolved stars and shocks. According to our findings shocks play a significant role in the ionization balance of this galaxy.	2024-10-01T00:00:00Z	['2024MNRAS.tmp.2034B', '10.1093/mnras/stae2077', '2024arXiv240908047B', 'arXiv:2409.08047', '10.48550/arXiv.2409.08047', '2024MNRAS.534..400B']	['Astrophysics - Astrophysics of Galaxies']	Digging deeper into NGC 6868 II ionized gas and excitation mechanism	2,024	200	0.53	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.08047.pdf	{'Digging deeper into NGC 6868 II: ionized gas and excitation mechanism': 'João P. V. Benedetti, 1 , 2 , 3 ★ Rogério Riffel, 1 † Tiago Ricci, 4 Rogemar A. Riffel, 5 Miriani Pastoriza, 1 Marina Trevisan, 1 Luis G. Dahmer-Hahn, 6 Daniel Ruschel-Dutra, 7 Alberto Rodríguez-Ardila, 8 Anna Ferré-Mateu, 2 Alexandre Vazdekis 2 , 3 and João Steiner 9 ‡ \n- 1 Departamento de Astronomia, Universidade Federal do Rio Grande do Sul. Av. Bento Gonçalves 9500, 91501-970 Porto Alegre, RS, Brazil\n- 2 Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain\n- 3 Departamento de Astrofísica, Universidad de La Laguna, E-38205, Tenerife, Spain\n- 4 Universidade Federal da Fronteira Sul, 97900-000 Campus Cerro Largo, RS, Brazil\n- 5 Departamento de Física, Universidade Federal de Santa Maria, Centro de Ciências Naturais e Exatas, 97105-900 Santa Maria, RS, Brazil\n- 6 Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan road, Shanghai 200030, China\n- 7 Departamento de Física - CFM - Universidade Federal de Santa Catarina, 476, 88040-900 Florianópolis, SC, Brazil\n- 8 Laboratório Nacional de Astrofísica/MCT - Rua dos Estados Unidos 154, Bairro das Nacões. CEP 37504-364 Itajubá, MG, Brazil\n- 9 Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, 05508-900 São Paulo, Brazil \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': 'Westudied the ionized gas in the inner region ( ∼ 680 × 470 pc 2 ) of the galaxy NGC 6868 using Gemini/GMOS integral field unit observations. Channel maps reveal complex kinematics and morphology, indicating multiple processes at work in NGC 6868. Through emission-line fitting, we identified two ubiquitous components in our data: a narrow ( 𝜎 ∼ 110 km s -1 ) tracing an ionized gas disc and a broad component ( 𝜎 ∼ 300 km s -1 ) mainly associated with inflowing/outflowing gas. The derived V-band reddening shows a spatial distribution consistent with that obtained from stellar population synthesis, although with generally higher values. For the first time, we measured the electron temperature in NGC 6868, finding values ranging from ∼ 14000 K in the central region to ≳ 20000 K with an outward increasing temperature gradient. The electron density map exhibits an inverse relationship, with central values reaching 𝑁 𝑒 ∼ 4000 cm -3 for the broad component decreasing to 𝑁 𝑒 ∼ 100 cm -3 towards the edges of the field of view. Using BPT diagrams, we found that all spaxels are consistent with both AGN and shock ionization. However, when this information is combined with our kinematic and temperature findings, and further supported by the WHAN diagram, we argue that an AGN is the dominant ionisation mechanism in the central region of NGC 6868, while the extended outer component is ionized by a combination of hot low-mass evolved stars and shocks. According to our findings, shocks play a significant role in the ionization balance of this galaxy. \nKey words: galaxies: individual (NGC 6868), galaxies: nuclei, galaxies: elliptical and lenticular, cD, galaxies: ISM, galaxies: kinematics and dynamics', '1 INTRODUCTION': 'Since the discovery of tight correlations of the central black-hole (BH) mass with galaxy properties, such as stellar velocity dispersion and bulge mass (Magorrian et al. 1998; Gebhardt et al. 2000; Häring & Rix 2004) and with the ever-growing evidence that the different manifestations of active galactic nuclei (AGN) were the counterpart of BHs, a picture of co-evolution and interaction between the two components has emerged (Fabian 2012; Kormendy & Ho 2013; Heckman & Best 2014). The injection of energy by the most energetic AGN in the interstellar medium (ISM) plays a crucial role in quenching the star formation (SF) mainly in the most massive galaxies in cosmological simulations (e.g. Croton et al. 2006; Segers et al. \n- ★ E-mail: [email protected](JPVB)\n- † E-mail: [email protected] (RR)\n- ‡ In Memorian. \n2016) by heating and expelling the available gas for star-formation. However, observational studies trying to link SF and AGN activity are still inconclusive. \nThe majority of studies trying to establish this link focused on relatively bright objects (e.g. Seyferts and quasars, e.g. Nayakshin & Zubovas 2012) and the effects of low-luminosity AGN (LLAGN) in the circumnuclear stellar population is even less studied. Many LLAGNs are classified as LINERs (Low-Ionization Nuclear Emission Regions) and proved to be numerous in the nearby Universe, being present in 1/3 of all galaxies and corresponding to 2/3 of all different types of AGN (Ho 2008). Hence detailed studies of the circumnuclear region are needed to improve our knowledge of the impact of these sources in their vicinity. \nLINERs were formerly described by Heckman (1980) as having strong lines from low-ionization species (e.g. O /i.pc) and weaker high ionization lines (e.g. O /i.pc/i.pc/i.pc). Over the decades with compiling evidence (Ferland & Netzer 1983; Halpern & Steiner 1983; Ho et al. 1996, \n1997; Constantin & Vogeley 2006) a growing consensus was formed over the picture that they were toned down versions of Seyfert nuclei, meaning the ionized gas features were due to the photoionization of an LLAGN. Detection of parsec-scale radio nuclei in 50% of LINERsandsubparsec jets (Nagar et al. 2005) and, more recently, the detection of ionized gas outflows in approximately ∼ 46% of LINERs in the sample from Hermosa Muñoz et al. (2022) related with the central supermassive black-hole further endorsed this picture. On the other hand, X-ray studies hint that AGN could not be solely responsible for the optical emission-line intensities observed (Flohic et al. 2006). \nWith improved spatial resolution studies, LINER-like signatures were found not only in the nuclear regions of galaxies ( < 1 kpc) but also at greater distances, hence the terms LIER or LI(N)ER adopted by some authors (e.g. Singh et al. 2013; Belfiore et al. 2016). In this scenario, other ionization mechanisms need to be taken into account, such as shocks (galactic or from an outflow, Heckman 1980; Dopita & Sutherland 1995; Ho et al. 2014, 2016), starbursts dominated by Wolf-Rayet stars (Barth & Shields 2000) and post-asymptotic giant branch stars (pAGB, Binette et al. 1994). The latter scenario has been gaining support as the dominant ionization mechanism in objects with LINER-like extended emission (Stasińska et al. 2008; Cid Fernandes et al. 2011; Yan & Blanton 2012; Singh et al. 2013). Compelling evidence has been found in red-and-dead galaxies (Hsieh et al. 2017), such as a correlation between the H 𝛼 surface density and the stellar population. This hints at a stellar origin for the LINER signature and the lack of ionizing photons due to LLAGN to explain the LINER spectra (Eracleous et al. 2010). In this sense, LINERs (or LIERs) can be found in different objects and phenomena in different parts of the spectra and, despite the similarity in spectral signatures, cannot be grouped as a homogeneous class (Herpich et al. 2016). \nIntegral Field Spectroscopy (IFS) has been used in the past decade to improve our understanding of these objects as they are a powerful tool to disentangle the different ionization mechanisms present in objects with LI(N)ER-like emission. Sarzi et al. (2010), using the SAURON (Spectroscopic Areal Unit for Research on Optical Nebulae) IFS survey, found a tight correlation between the stellar mass surface density and H 𝛽 surface density hinting at a stellar origin behind the ionization process. Loubser & Soechting (2013) found that LLAGN can explain the observed ionization, but shocks and pAGB photoionization could not be ruled out. Ricci et al. (2014a,b, 2015a,b) extensively analysed a group of 10 LI(N)ER galaxies from a range of morphological types. In these objects, they found convincing AGN presence in at least 8 of the objects. However, they also report discs of ionized gas in 7 of them, with 3 having pAGB stars as the most probable source of ionizing photons for this disc component. Belfiore et al. (2015) studied 14 galaxies and found extended ionized gas components, consistent with emission from pAGB stars. Hsieh et al. (2017) using a larger subset of MaNGA (Mapping Nearby Galaxies at APO) also found a correlation between the stellar surface density and H 𝛼 surface density, indicating the same scenario as the one derived using SAURON. Studies using CALIFA data (Calar Alto Legacy Integral Field Area Survey, Kehrig et al. 2012; Papaderos et al. 2013; Singh et al. 2013; Gomes et al. 2016) found "ubiquitous hot evolved stars and rare accreting black-holes". Lagos et al. (2022) using MUSE data to study group-dominant early-type galaxies found that the central regions of these objects are more influenced by LLAGNs with outer regions ionized by pAGB stars. A more recent study from Menezes et al. (2022) using the mini-DIVING 3D (Deep IFS View of Nuclei of Galaxies) sample was able to separate the nuclear from the circumnuclear region, thus allowing a different treatment for each component due to the high spatial resolution for their data. From their sample, \nTable 1. Table showing some basic parameters of the galaxy NGC 6868. \nData available in NED 1 \na de Vaucouleurs et al. (1991) \nb Carrasco et al. (2006) \nc Lauberts & Valentĳn (1989) \nd Babyk et al. (2018) \ne Rickes et al. (2008) \nf Healey et al. (2007) \ng Schlafly & Finkbeiner (2011) \nh Ramella et al. (1996) \ni Tully et al. (2013) \n23% present LINER-like emission of which 69% have signs of AGN activity. Ricci et al. (2023) analysed the nuclear region of 56 earlytype galaxies contained in the DIVING 3D and classified 52 ± 7 % of them into LINER/Seyfert, detecting broad components in H 𝛼 in at least 29 ± 7 %. With the addition of multi-wavelength data from other works (Rampazzo et al. 2013; She et al. 2017; Bi et al. 2020), out of the 48 ETGs with emission lines detected, 41 have compelling evidence of an AGN. One remarkable result is that in their sample, they found no transition objects, which are thought to encompass objects with LINER/H /i.pc/i.pc mixture (Ho et al. 1993; Kewley et al. 2006). \nIt is clear that to fully understand the nature behind objects classified as LINERs and the impact of an LLAGN in its host galaxy, one needs to engage in detailed spatially resolved studies, tracing both the properties of the ionized gas and the underlying stellar population focused on the region surrounding the SMBH. Therefore, in this series of papers, we present a detailed GMOS IFU study of the object NGC6868. It is a nearby (27 . 70 Mpc, Tully et al. 2013) elliptical galaxy (E2, de Vaucouleurs et al. 1991), presents LINER-like signatures and a complex ionization profile (Rickes et al. 2008). We notice that, to the best of our knowledge, no dedicated studies of its central region have been made yet. Table 1 shows some basic parameters for NGC6868, while Fig. 1 shows the galaxy at different scales.', '1.1 About NGC6868': 'NGC6868is the brightest member of the Telescopium group and has been already observed in different wavelengths. It was featured in a series of papers investigating the ISM on early-type galaxies through long-slit spectroscopy (e.g. Buson et al. 1993; Zeilinger et al. 1996; Macchetto et al. 1996; Pizzella et al. 1997; Ferrari et al. 1999; Caon et al. 2000; Ferrari et al. 2002). They found that the ionized gas in NGC6868 has a \'regular extended\' morphology with possible small filaments. Also, the stars present a kinematically decoupled core (KDC) as the stars in the central region counter-rotate with respect to \nFigure 1. Images from NGC 6868 in three different scales. (a) Composite DSS image showing NGC 6868 and close neighbours. It is the brightest group member from the Telescopium group (AS0851). (b) Recreation of H 𝛼 +[N/i.pc/i.pc] image present in Macchetto et al. (1996) using NTT+EFOSC2. They describe the galaxy as having a "regular extended" morphology where small filaments may be present. (c) H 𝛼 +[N/i.pc/i.pc] image extracted from the final GMOS data cube after Voronoi binning the data. The (0,0) is kept the same as in Paper I for consistency. \n<!-- image --> \nthe stars in the outer regions. Moreover, they found that NGC 6868 harbours complex ionized gas kinematics with the superposition of two ionized gas discs. The hot dust mass estimated from the mid-IR is 70 M ⊙ (Ferrari et al. 2002). Veron-Cetty & Veron (1988) detected a dust lane in the centre of NGC 6868 further endorsed by Bregman et al. (1998) who detected cold dust using IRAS data. Hansen et al. (1991) examined this galaxy using CCD images and an International Ultraviolet Explorer low-resolution spectrum, detecting a dust lane with spiral features emerging from it. They also report that the Ly 𝛼 distribution follows that of dust, which suggests that NGC6868 has captured a gas-rich galaxy. In fact, Machacek et al. (2010) using X-ray data found strong evidence of a past encounter between NGC 6868 and NGC6861 in the past hundred Myr, displaying tidal tails and shells. Moreover, they found X-ray cavities, indicative of past AGN activity triggered by the interaction. Radio observations (Slee et al. 1994; Mauch et al. 2003; Healey et al. 2007) revealed a low-power flat spectrum radio source in its centre ( 𝛼 ∼ 0 . 07) and the brightness, temperature and spectral slope are inconsistent with HII regions, thus hinting at an AGN as the most likely source of the emission. Rose et al. (2019) analysed molecular gas in the centre of NGC 6868 and concluded it is drifting in non-circular motions. More recently, Ricci et al. (2023), using the same GMOS data presented in this paper, included NGC 6868 in their analysis of the nuclear region of early-type galaxies, finding compelling evidence for an AGN, despite no detection of a broad-line region. They integrated all the spectra from the NGC 6868 data cube within the PSF, centred in the stellar \nphotometric peak. In this work, we explored the whole FoV of this observation. \nIn Benedetti et al. (2023, hereafter Paper I) we already studied the stellar content of this galaxy through stellar population synthesis and indices measurements. We found that this galaxy is dominated by old stars ( ∼ 12 Gyr) with high-metallicity (1.0-1.6 Z ⊙ ) with a shallow contribution of a young ( ∼ 63 Myr) also high-metallicity (1.6 Z ⊙ ). Indices revealed that this object has a complex chemical evolution, further endorsed by the [ α /Fe] map showing regions with very distinct enrichment. The stellar kinematics in the centre of the galaxy is dispersion dominated and no apparent ordered rotation was detected as well as no evidence for a featureless continuum. These findings led us to conclude that this galaxy probably is experiencing only a residual level of star formation. This contribution, however, is not enough to explain the complex enrichment profile which we attribute to past mergers due to the nature of NGC 6868 (brightest galaxy of the Telescopium group). \nIn this paper, we will focus on the ionized gas content of NGC 6868, organised as follows: in § 2, we describe the observations and the reduction procedures; in § 3, we present the methodology; in § 4, the results are presented. Discussion of the results is made in § 5 and the conclusions and summary are made in § 6. Throughout this paper, we assume that solar metallicity corresponds to Z ⊙ = 0 . 019 (Girardi et al. 2000).', '2 OBSERVATIONS AND DATA REDUCTION': 'The acquisition of the observational data, the reduction processes and subsequent data processing steps were already thoroughly described in Paper I. Briefly, NGC 6868 was observed as part of the DIVING 3D survey (Steiner et al. 2022) on 2013 May 04 using the Gemini MultiObject Spectrograph (GMOS) in the IFU mode mounted on the Gemini South Telescope. The configuration was a one-slit setup (resulting in a FOV of 3 . 5 × 5 . 0 arcsec 2 ), with the B600-G5323 grating centred at 5620 Å. This results in a spectral baseline covering 4260 - 6795 Å. The spectral resolution was estimated with the O /i.pc 𝜆 5577 line resulting in 1.8 Å (FWHM). Using the acquisition image from GMOS in the r -band and field stars present, the seeing was measured at 0.77 arcsec. Lastly, the DA white dwarf EG 274 (Hamuy et al. 1992) was observed to perform the spectrophotometric calibrations. Standard IRAF procedures (Tody 1986, 1993) were employed to reduce the data using the G/e.pc/m.pc/i.pc/n.pc/i.pc /i.pc/r.pc/a.pc/f.pc package and the /l.pc/a.pc/c.pc/o.pc/s.pc software (van Dokkum 2001) was used to remove cosmic rays. The final sampling of the data cube was 0.05 arcsec. \nOther data treatments were applied to improve data visualisation as described in Menezes et al. (2019): the removal of high-frequency spatial noise using a Butterworth filter (Gonzalez & Woods 2008; Ricci et al. 2014a); the correction by the differential atmospheric refraction; and the PCA Tomography technique (Steiner et al. 2009, and references therein) was applied to remove instrumental fingerprints. The data cube was then corrected by the Galactic reddening using the CCM law (Cardelli et al. 1989) and assuming 𝐴 𝑉 = 0 . 152 mag (Schlafly & Finkbeiner 2011). Telluric lines were removed and the spectra were brought to rest-frame velocities using the redshift from Ramella et al. (1996) ( 𝑧 = 0 . 00952). To improve the PSF from our data, the Richardson-Lucy deconvolution (Richardson 1972; Lucy 1974) was applied reaching a PSF of 0.71 arcsec. The final spatial coverage of the data cube in the source corresponds to an FoV of ( ∼ 680 × 470 pc 2 ). In Fig. 1, a map from [N /i.pc/i.pc]+H 𝛼 is shown, extracted after the subtraction of the stellar content and the subsequent Voronoi binning of the data.', '3.1 Pre-processing: Voronoi binning and continuum fitting': 'As mentioned, we have already analysed the stellar populations of this galaxy in Paper I. In order to analyse the pure emission line spectrum, we subtracted the modelled stellar continuum from our data. To improve the S/N mainly towards the boarders of the FoV, we used the Voronoi binning technique (Cappellari & Copin 2003) which bins the data preserving the maximum spatial resolution given a minimum S/N to be achieved. We measured the signal as the mean flux density between 6528-6615 Å which encompasses the [N /i.pc/i.pc] 𝜆𝜆 6548 , 6584+H 𝛼 emission. The noise was estimated as the standard deviation between 4800-4845 Å. We opted to use this wavelength range because it is on the bluer side of our data and therefore is noisier, functioning as an upper limit to the noise in the whole spectrum. For this work, we set the threshold to S/N=30 so that no binned spaxels would be greater than the seeing. In Fig. 3, the H 𝛼 flux map after binning can be seen. \nThe stellar synthesis method, despite its precision, can lead to discrepancies between the data and the stellar continuum. This can be worked around by fitting a high-degree polynomial to model a nonphysical continuum that can affect our measurements. We masked the emission lines and fitted the continuum using double the weight in arbitrary continuum bands of 20 Å of width surrounding each line. \nWe took extra care to avoid the regions of absorption or emission lines within these bands. We tested different configurations for the fit, varying the weights in each window or masking certain parts of the spectrum, the degree of the polynomial fitted, and selecting a 13thdegree polynomial as the most adequate for the pseudo-continuum. This pseudo-continuum accounts for template mismatch issues or limitations in the stellar population base used to fit the stellar continuum (e.g. abundance ratio variations). Therefore, at first glance, a 13-degree polynomial seems redundant for a polynomial fit, but considering the number of points in the wavelength range ( > 2000points) coupled with the number of unknowns previously enumerated, we suggest this is an adequate degree. Moreover, when polynomials with smaller degrees were used, the bluest and reddest wavelengths suffered from spurious features created in the pseudo continuum. In Fig. 2, we show an example of a continuum fit.', '3.2 Emission line fitting': 'After this pre-processing, we modelled the emission line using /i.pc/f.pc/s.pc/c.pc/u.pc/b.pc/e.pc package (Ruschel-Dutra & Oliveira 2020) 2 , which fits the different emission lines using Gaussian functions with predefined components. \nThe spectrum from NGC 6868 is rich in emission lines as can be seen in Fig. 2. The properly fitted lines are H /i.pc H 𝛼 , H 𝛽 and H 𝛾 , [N/i.pc/i.pc] 5755 , 6548 , 6583 Å, [O /i.pc/i.pc/i.pc] 4959 , 5007 Å, [O /i.pc] 6300 , 6360 Å, [N/i.pc] 5197 , 5200 Å and [S /i.pc/i.pc] 6716 , 6731 Å. By looking at Fig. 3, the immense diversity in line profiles from this galaxy becomes clear. We carried out a series of tests with different configurations, such as varying the number of components in each line, adding a broad component compatible with what we would expect for a broad-line region in the H /i.pc recombination lines, fitting each section of the spectra separately, removing or adding kinematical constraints (e.g. coupling or not the [O /i.pc/i.pc/i.pc] kinematics with the other lines). Looking at the residuals and using the model with the least number of components, we ended up with a model consisting of two distinct kinematical components for each line: a narrow ( 𝜎 ∼ 80 -130 km s -1 ) and a broad ( 𝜎 ∼ 100 -450 km s -1 ) component. This has been applied for all the spaxels and some example fits can be seen in Fig. 3. \nIn order to reduce the degeneracy from our fit, we made some assumptions and established some constraints to our models. Firstly, we used the following well-known line ratios (Osterbrock & Ferland 2006): f [N/i.pc/i.pc] 𝜆 6583 = 3.06 · f [N/i.pc/i.pc] 𝜆 6548 , f [O/i.pc/i.pc/i.pc] 𝜆 5007 = 2.94 · f [O/i.pc/i.pc/i.pc] 𝜆 4959 , f [O/i.pc] 𝜆 6300 = 3.05 · f [O/i.pc] 𝜆 6360 . On the first try, we only established kinematical groupings with components from lines produced by the same ion. We noticed, however, that all lines ended up with really similar kinematical and flux distributions among different ions. Despite typically behaving differently when compared to the other lines, this is valid even for the [O /i.pc/i.pc/i.pc] lines. Secondly, we constrained that each component from all the lines fitted would be in the same kinematical group e.g. the blueshifted component needs to have the same velocity and velocity dispersion among all the different lines. This helps to disentangle the different components in weaker lines as the kinematical information from stronger lines is used to improve the fit. \nElectron temperature ( 𝑇 𝑒 ) and density ( 𝑛 𝑒 ) were computed using P/y.pcN/e.pc/b.pc (Luridiana et al. 2015) with line-ratios [N /i.pc/i.pc] 𝜆 5755 / 𝜆 6583 and[S /i.pc/i.pc] 𝜆 6716 / 𝜆 6731, respectively. We also calculated the emission line ratios of [N /i.pc/i.pc] 𝜆 6583/H 𝛼 , [S /i.pc/i.pc] 𝜆 6716 , 6731/H 𝛼 , [O /i.pc] 𝜆 6300/H 𝛼 , [O /i.pc/i.pc/i.pc] 𝜆 5007/H 𝛽 . We derived the reddening also using the P/y.pcN/e.pc/b.pc \nFigure 2. In black, Absorption line free emission spectrum extracted at the peak of the continuum for an aperture of 0 . 05 × 0 . 05 arcsec 2 . The fitted pseudocontinuum is displayed in red. The most prominent emission lines are identified. More details of the kinematical nature of the lines can be seen in Fig. 3. \n<!-- image --> \nFigure 3. Different regions and the respective line profiles from each assigned Voronoi bin. In blue, the observed profile of the [O /i.pc/i.pc/i.pc] (left) and H 𝛼 +[N/i.pc/i.pc] (right), in dashed black lines, each modelled Gaussian and, in orange, the total model. The huge diversity among the line profiles becomes evident, showing double peaks, broad, narrow and wing components which our model successfully reproduces. The central bottom panel shows the total H 𝛼 flux estimated through the fitting procedure. \n<!-- image --> \npackage using the ratio between the H 𝛼 and H 𝛽 . Considering the case B recombination and rough estimates of both density and temperature, as a first approximation (100 cm -3 ; 10000 K). Using this parameters, we estimate the theoretical value of F H 𝛼 / F H 𝛽 = 2 . 87. Assuming, a CCM extinction law ( 𝑓 𝜆 ; Cardelli et al. 1989, R V =3.1) \nand following Riffel et al. (2021b), we get \nE ( B -V ) = 𝐸 ( H 𝛽 -H 𝛼 ) 𝑓 𝜆 ( H 𝛽 ) -𝑓 𝜆 ( H 𝛼 ) (1) = 2 . 5 3 . 1 · ( 1 . 164 -0 . 818 ) log ( F H 𝛼 / F H 𝛽 ) obs ( F H 𝛼 / F H 𝛽 ) theo ! A V = 7 . 22 log ( F H 𝛼 / F H 𝛽 ) obs 2 . 87 ! . (2) \nTherefore, using the Balmer recombination lines (H 𝛼 , H 𝛽 ), we \nderived the reddening in the V band (A V ) and deredden all emission lines, following \nF int = F obs 10 0 . 4 · A 𝜆 = F obs 10 0 . 4 · A V · 𝑓 𝜆 . (3)', '4.1.1 Channel Maps': 'Weslice the data cube in steps of 2.0 Å ( ∼ 91km s -1 ) in order to create several H 𝛼 lines maps (Fig. 4). In this way, we can correlate a spatial counterpart from the gas with a particular kinematical signature. We masked all spaxels that had flux values smaller than 3 times the value of the standard deviation as calculated in § 3 for that spaxel. We have applied the same procedure for the [O /i.pc/i.pc/i.pc] 5007 Å line and have not found significant differences between both lines. \nThe central region of NGC 6868 displays several distinct features. In the high-velocity maps ( 𝑣 = + 411 and + 502 km s -1 ), a circular distribution is seen, with its centre coinciding with that estimated on Paper I. In the lowest velocity map ( 𝑣 = -319 km s -1 ), the same centrally concentrated region appears albeit with a slight perturbation to the NW that can be followed in the maps of 𝑣 = -228 and -137 km s -1 . In the maps from 𝑣 = -228 to + 228 km s -1 , a signature of a disc can be traced, going from the SW to NE. In the 𝑣 = + 319 km s -1 a distortion in the central profile can also be seen in the NE region, which appears to be independent of the disc as such a profile does not emerge in the 𝑣 = -319 km s -1 map. Interestingly, a region in the N section does not seem to have any detected ionized gas, which is better seen in the 𝑣 = + 45 km s -1 map. A closer inspection of the upper panel of Fig. 1 reveals that this feature is also seen in the observation of Macchetto et al. (1996), showing a slight decrease in flux right in the N border of our FoV when compared to the surrounding region.', '4.1.2 Velocity and velocity dispersion maps': 'The centroid velocities and velocity dispersion for the narrow and broad components are presented in Fig. 5. It is clear from this figure that the narrow component ranges from ∼ 140 km s -1 to ∼ 90 km s -1 , broadening in the central regions. Looking at the centroid velocities of this component, one can infer that the central area of NGC 6868 hosts an ionized gas disc, with velocity amplitude ranging from ± 150 km s -1 . The centre of the rotation profile appears to coincide with the centre of NGC 6868, estimated in Paper I. The broader component appears to have a more diverse nature, with regions with a similar velocity dispersion as the narrow component (NE region) but higher centroid velocities ( ∼ 250 km s -1 ). In the central part, the velocity dispersion reaches ∼ 400 km s -1 . We also notice that in the NW direction, a blueshifted (approximately -100 km s -1 ) broader ∼ 250 km s -1 wing is apparent. \nCaon et al. (2000) have reported a rotating gas disc in NGC 6868. Our detection agrees with the orientation reported in their study. However, they find profiles peaking at ∼ 180 and 190 km s -1 , slightly larger than reported here. As their modelling relied on only one Gaussian, they are probably having contamination of the high-velocity components that we can detect separately, mainly in PA=120 · , where they report a counter-rotating gas disc which likely comes from the blueshifted broad component in the NW affecting their measurements.', '4.1.3 Flux distributions': 'The flux distributions derived are presented in Fig. 6. We show only the [N /i.pc/i.pc] 6583 Å and [O /i.pc/i.pc/i.pc] 5007 Å emission lines flux distributions because [N /i.pc/i.pc] is the most prominent line in our cube. Moreover, most lines follow its distribution except for [O /i.pc/i.pc/i.pc] 5007 Å which presents its peculiarities, as shown. \nThe narrow component distribution appears in a flat distribution along the SW-NE direction, following the rotation profile already described. In the [O /i.pc/i.pc/i.pc] line, however, it appears that the SW region is enhanced. This does not appear to be a feature from the fitting procedure, but rather some physical process enhancing it at this location. This can be roughly seen in Fig. 3 as it appears that, in region E, [O /i.pc/i.pc/i.pc] has the largest intensity within all the spectra shown. The broader component shows a distribution that departs from the narrow one with a clear centrally concentrated flux distribution with two wings: one at NE and another at NW, detected in both [O /i.pc/i.pc/i.pc] and [N/i.pc/i.pc] lines. This coincides with the regions described also detected in the channel maps and the kinematical maps.', '4.2 Ionized gas physical parameters': 'Using the Balmer recombination lines H 𝛼 and H 𝛽 , we derived the reddening in the V-band (A V ). We decided to sum the fluxes from both components of each line to measure the A V as H 𝛽 is a rather weak emission line in this object. Thus, having a single measurement of this property in each spaxel. We assumed Case B recombination and, as a first approximation, set the temperature and density respectively as 10000 K and 100 cm 3 . Using the CCM extinction law (Cardelli et al. 1989), the obtained A V map is shown in Fig. 7. \nThe A V map reveals a dust lane that has already been reported in previous studies (e.g. Veron-Cetty & Veron 1988; Hansen et al. 1991; Bregman et al. 1998; Ferrari et al. 1999) and also in Paper I, having the same orientation (PA ∼ 120 · ) and roughly the same spatial extension. The reddening, however, is enhanced when compared to these previous studies. In Paper I, we found that the stellar dust distribution peaks at ∼ 0 . 65 mag whereas the gas A V derived here reaches ∼ 1 . 5 mag. This has been found in past studies, such as Riffel et al. (2021b), with a ratio close to the one reported here. In that work, it was suggested that this was due to single extinction laws used in /s.pc/t.pc/a.pc/r.pc/l.pc/i.pc/g.pc/h.pc/t.pc, the fitting code used (Cid Fernandes et al. 2005). Newer generations of stars, however, are typically embedded in the dust reminiscent of the star-formation processes. Thus, the stellar synthesis can underestimate the dust content mainly if old stellar populations dominate the galaxy. Using the same reasoning, the ionising radiation might come from a heavily dust-obscured region, not necessarily young stars, such as an AGN, or the dustier gas ejected during the stellar evolution process, explaining the different line ratios observed. Using the extinction-corrected emission line fluxes, we derived both the electron temperature and density, respectively shown in Fig. 8 and Fig. 9 which the code computes in an interactive fashion using the result from one parameter to estimate the other until they converge. \nWe were not able to find any other measurement in the literature of the electron temperature and density for NGC 6868, making this the first for this object in the literature. Indeed, spatially resolved measurements of the electron temperature in AGN hosts are available only for a few objects, as they require the detection of typically weak emission lines in AGN spectra (Revalski et al. 2018; Riffel et al. 2021a,c; Negus et al. 2023). In Fig. 8, the temperature profile of the ionized gas is co-spatial with the broader component flux distribution. \npc \npc \n60 \n120 \n60 \n60 \nFigure 4. Channel maps for the H 𝛼 emission line using 2.0 Å ( ∼ 91 km s -1 ) as our sampling width. We masked values smaller than three times the standard deviation as calculated in § 3. In the last map we can already see a contamination from the surrounding [N /i.pc/i.pc] 6583 Å. \n<!-- image --> \nv \n+150 \n0 \n150 \n60 \nE \n100 \n0 \n100 \nFigure 5. Kinematical results for the central region of NGC 6868. The top panels show the results for the narrow component and the bottom panels, for the broad component. For consistency, we will maintain this scheme in all figures unless specified. Centroid velocities are displayed on the left and velocity dispersion, on the right. \n<!-- image --> \n300 \n150 \n0 \n+150 +300 \npc \n400 \n300 \n200 \n100 \n400 \n300 \n200 \n100 \n120 \nN \n110 \n260 \n110 \nE \nE \nN \nN \nk \nm \ns \n1 \nk \nm \ns \n1', '[NII] 6583': '<!-- image --> \npc \npc \n<!-- image --> \n<!-- image -->', '[OIII] 5007': 'Figure 6. Flux distribution from narrow (top) and broad components (bottom) and [N /i.pc/i.pc] 6583 Å (left) and [O /i.pc/i.pc/i.pc] 5007 Å (right) emission lines not corrected by dust extinction. The components are fairly similar between the different lines. \n<!-- image --> \nmag \nFigure 7. The reddening in the V band derived through the Balmer recombination lines ratio H 𝛼 /H 𝛽 . A dust lane emerges in the centre, agreeing with past results found through stellar population synthesis. \n<!-- image --> \nAlso, it appears that an outward gradient emerges, ranging from 14000 K to nearly 21000 K at the edges of our detection. \nWe also provide density estimates for the whole FoV separating the narrow and broad components in the [SII] 6716 , 6731 Å. In some regions, the [SII] 6716 , 6731 Å ratio is larger than 1.45 and density estimates are no longer valid. As this implies a low-density environment, when values exceeded 1.45, we set 100 cm -3 as the density in that spaxel. \nFor the narrow component, a ubiquitous low-density component is found, with values ranging from the lower limit 100 cm -3 to less than \nFigure 8. Temperature profile derived from the [N /i.pc/i.pc] 6583 , 5755 Å line ratio corrected by the extinction. The temperature profile seems to display a negative gradient with values ranging from 14000 to 21000 K. \n<!-- image --> \n500 cm -3 . This is not the case for the broader component with values similar to the ones found in the narrow component for the NE region and values in the central region reaching over 4000 cm -3 . Hansen et al. (1991) based on data from Bonatto et al. (1989) has estimated the electron density to be 800 cm -3 . As a kinematical decomposition was not carried out in those studies, the in-between value most likely stems from their treatment of the two components found in this work as one. \nWe estimate the total ionized gas mass using the relation \nM gas ≈ 2 . 4 × 10 5 L 41 ( H 𝛼 ) n 3 M ⊙ (4) \nas in do Nascimento et al. (2019), where L 41 ( H 𝛼 ) is the luminosity \n<!-- image --> \npc \nFigure 9. Electron density profile derived from [S /i.pc/i.pc] 𝜆 6716 / 𝜆 6731 corrected by extinction. Top panel: Density values for the narrow component with values ranging from 100 cm -3 to 550 cm -3 . Bottom panel: Broader component showing a large variation in the derived density values. In the central part of the FoV electron density is over 4000 cm -3 reaching similar values to the narrow component in the NE and NW region. A 500 cm -3 contour line has been added to aid the visualisation \n<!-- image --> \nof H 𝛼 in units of 10 41 erg s -1 and n 3 is the electron density in units of 10 3 cm -3 . Using the extinction corrected H 𝛼 flux, the distance from Table 1 and previously derived electron density, we calculated the ionized gas mass spaxel by spaxel and then summed over the FoV. \nThe total mass in our FoV is ∼ 9 . 1 ± 1 . 2 × 10 5 M ⊙ . This, however, is a lower limit as the ionized gas component stretches beyond our FoV, as can be seen in Fig. 1. Hansen et al. (1991), using narrow band data to measure the flux of H 𝛼 , have evaluated the H /i.pc/i.pc mass as ∼ 2 × 10 4 M ⊙ , bellow what our estimate suggests. Macchetto et al. (1996) estimated the ionized gas mass as ∼ 5 × 10 4 M ⊙ also using H 𝛼 narrow band data. We attribute this difference to several facts. First, each study uses a different value for the distance to NGC 6868: 27.70 Mpc (this work), 36.8 Mpc (Hansen et al. 1991) and 48.6 Mpc (Macchetto et al. 1996). Also, both studies have used narrow band images to estimate H 𝛼 using previously determined [N /i.pc/i.pc] 𝜆 6583/H 𝛼 ratios, instead of having the fine details that spectroscopy can provide. With a data cube, we were able to separate the [N /i.pc/i.pc] and H 𝛼 emission properly, while the narrow band data may suffer some contamination from the [N /i.pc/i.pc] lines. Also, they use a fixed electron density value, which as can be seen in Fig. 9, is a simplification. The reddening correction we apply could also explain the different values as they do not apply such a correction. Also, in Hansen et al. (1991) they report problems with their spectrophotometric standards, leading to a 30% error in their measurements. Of course, our data is also subject to its problems, mainly regarding the flux calibration as standard stars are sometimes observed on different nights, resulting in ∼ 30 % error \nin the absolute flux. Therefore, all these differences might come into play to explain the different results.', '4.3 Emission line diagnostic diagrams': 'In order to assess the ionization mechanism present in the central region of NGC 6868, diagrams discerning the different ionization sources are needed. The most widely used diagrams in the literature are the BPT diagrams and rely on the [N /i.pc/i.pc] 𝜆 6583/H 𝛼 vs. [O/i.pc/i.pc/i.pc] 𝜆 5007/H 𝛽 (Baldwin et al. 1981), [S /i.pc/i.pc] 𝜆 6716 , 6731/H 𝛼 vs. [O/i.pc/i.pc/i.pc] 𝜆 5007/H 𝛽 (Veilleux & Osterbrock 1987) and [O /i.pc] 𝜆 6300/H 𝛼 vs. [O /i.pc/i.pc/i.pc] 𝜆 5007/H 𝛽 (Veilleux & Osterbrock 1987) line ratios to disentangle the possible ionization mechanisms. Kauffmann et al. (2003) and Kewley et al. (2006) have established calibrations to separate AGNs from LINERs and star-forming galaxies. We thus employed these lines and the corrected fluxes previously derived to create the diagnostic diagram. The BPT diagrams for [N /i.pc/i.pc] 𝜆 6583/H 𝛼 vs. [O /i.pc/i.pc/i.pc] 𝜆 5007/H 𝛽 , [O/i.pc] 𝜆 6583/H 𝛼 vs. [O /i.pc/i.pc/i.pc] 𝜆 5007/H 𝛽 and [S /i.pc/i.pc] 𝜆 6716 , 6731/H 𝛼 vs. [O /i.pc/i.pc/i.pc] 𝜆 5007/H 𝛽 are seen respectively in Fig. 10, Fig. 11 and Fig. 12. \nThe three diagrams classify the whole FoV as having LINER-like emission. This was expected as other studies already mentioned had detected the same signatures (e.g. Rickes et al. 2008). As mentioned, shock-heated gas may be present in NGC 6868. Therefore shock models from Allen et al. (2008) were overplotted in the broad lines BPT. We used twice-solar metallicity models and a pre-shock density of 1 cm -3 , creating our grid with models of velocities between 300 and 1000 km s -1 , and magnetic fields between 1.0 and 4.0 𝜇 G. Also to establish a basis of comparison, we employ AGN models from Groves et al. (2004) with 4 𝑍 ⊙ metallicity, ionization parameters log 𝑈 = -3 and log 𝑈 = -2 with power-law indices ( 𝐹 𝜈 ∝ 𝜈 𝛼 ) 𝛼 = -1 . 2, 𝛼 = -1 . 4 and 𝛼 = -1 . 7 that were overplotted in the narrow component. We chose these models because they are the only ones able to explain our data and have been employed in past studies to discern the ionization mechanisms in LINER sources (e.g. Ricci et al. 2023). \nIn order to make a deeper analysis of the nature of the ionization source of NGC 6868, we used the WHAN diagram (Cid Fernandes et al. 2011). It uses the equivalent width of the H 𝛼 (W H 𝛼 ) to measure if the light from hot low-mass evolved stars (HOLMES) is enough to reproduce the ionization levels of the observed emission lines, allowing for the distinction between true AGNs and retired galaxies. Particularly, the 3 Å line in this diagram comes from the bimodal distribution the authors find using SDSS data. This line, therefore, separates galaxies where the dominant ionization source is HOLMES. In Fig. 13, we show the WHAN diagram for each component with the relevant separation lines as discussed. The diagrams show a diverse scenario for the ionization phenomena in NGC 6868 where the centre falls in the weak AGN region, going to lower and lower values of W H 𝛼 moving away from the centre of the galaxy. Looking at the W H 𝛼 maps also presented in Fig. 13, in the case of the narrow component, the region we attribute to the central disc is consistent with AGN ionization as well as the central part of the broad component. The contours until 1 Å values in the broad component stretch towards the regions with the peculiar kinematics and flux distributions as described above. \nnarrow \nFigure 10. [N/i.pc/i.pc] 𝜆 6583/H 𝛼 BPTdiagram for the central region of NGC 6868 for the narrow (top panel) and broad (bottom panel). The points were colourcoded following, respectively, the velocity and the velocity dispersion in each component. The whole FoV of our observation is kept within the LINER region, as expected. AGN models from Groves et al. (2004) were overplotted in the top panel with log 𝑈 = -3 and log 𝑈 = -2 (continuous lines) and power-law indices 𝛼 = -1 . 2, 𝛼 = -1 . 4 and 𝛼 = -1 . 7 (dashed lines), as indicated in the plots. Shock grids from Allen et al. (2008) were overplotted in the bottom panel with models of velocities between 300 and 1000 km s -1 (dashed lines), and magnetic fields between 1.0 and 4.0 𝜇 G(continuous lines). In both panels, we also plot the results obtained in the bins defined in Fig. 3, following the same tags. \n<!-- image -->', '5.1 Detection of an ionized gas disc': 'The kinematics from the ionized gas derived from our emission line fitting procedure can disentangle the different components that dominate the central region of NGC 6868. The narrow component that can be seen in Fig. 5 resembles a rotation profile. Therefore, we used a method within /i.pc/f.pc/s.pc/c.pc/u.pc/b.pc/e.pc to fit the disc model extracted from Bertola et al. (1991). It assumes the gas is orbiting in circular trajectories and follows a velocity field described by \n𝑣 𝑟 = 𝐴 𝑟 ( 𝑟 2 + 𝑐 2 0 ) 𝑝 / 2 . (5) \nnarrow \n1 \nFigure 11. [S /i.pc/i.pc] 𝜆𝜆 6716 , 6731/H 𝛼 BPT diagram for the central region of NGC6868 for the narrow (top panel) and broad (bottom panel). The points were colour-coded following, respectively, the velocity and the velocity dispersion in each component. The immense majority of the FoV of our observation is kept within the LINER region, as expected. AGN models from Groves et al. (2004) were overplotted in the top panel with log 𝑈 = -3 and log 𝑈 = -2 (continuous lines) and power-law indices 𝛼 = -1 . 2, 𝛼 = -1 . 4 and 𝛼 = -1 . 7 (dashed lines), as indicated in the plots. Shock grids from Allen et al. (2008) were overplotted in the bottom panel with models of velocities between 300 and 550 km s -1 (dashed lines), and magnetic fields between 1.0 and 4.0 𝜇 G (continuous lines). In both panels, we also plot the results obtained in the bins defined in Fig. 3, following the same tags. \n<!-- image --> \nThus the projected velocity distribution follows \n𝑣 ( 𝑅, Ψ ) = 𝑣 sys + (6) 𝐴 𝑅 cos ( Ψ -Ψ 0 ) sin Θ cos 𝑝 Θ h . \n𝑅 2 GLYPH<16> sin 2 ( Ψ -Ψ 0 ) + cos 2 Θ cos 2 ( Ψ -Ψ 0 ) GLYPH<17> + 𝑐 2 0 cos 2 Θ i 𝑝 / 2 \nThe resulting fit is shown in Fig. 14 along with the residuals. As can be seen, we can properly model the ionized gas disc with residuals kept within ∼± 50 km s -1 for most of the FoV. \nThis ionized gas disc has already been reported in previous studies (e.g. Buson et al. 1993; Zeilinger et al. 1996; Macchetto et al. 1996; Pizzella et al. 1997; Ferrari et al. 1999; Caon et al. 2000; Ferrari et al. 2002). We observe that the A V profile follows roughly the same spatial distribution as the gas disc, as pointed in Hansen et al. (1991). The molecular gas already reported for NGC 6868 is found in distances compatible with our findings for the dust lane, therefore the dust is likely shielding the molecular gas from the ionising radiation \nnarrow \n1 \nFigure 12. [O/i.pc] 𝜆 6300/H 𝛼 BPT diagram for the central region of NGC 6868 for the narrow (top panel) and broad (bottom panel). The points were colourcoded following, respectively, the velocity and the velocity dispersion in each component. The immense majority of the FoV of our observation is kept within the LINER region, as expected. AGN models from Groves et al. (2004) were overplotted in the top panel with log 𝑈 = -3 and log 𝑈 = -2 (continuous lines) and power-law indices 𝛼 = -1 . 2, 𝛼 = -1 . 4 and 𝛼 = -1 . 7 (dashed lines), as indicated in the plots. Shock grids from Allen et al. (2008) were overplotted in the bottom panel with models of velocities between 300 and 550 km s -1 (dashed lines), and magnetic fields between 1.0 and 4.0 𝜇 G (continuous lines). In both panels, we also plot the results obtained in the bins defined in Fig. 3, following the same tags. \n<!-- image --> \npresent in the region. Rose et al. (2024) investigated the molecular gas kinematics using the CO(2-1) emission line and their results closely match the ones found in this study. \nThe main exception to our best model is the region towards the NE (see Fig. 14) where larger values for the residuals are found to which we provide some hypothesis. The first one is the realisation that the residual profile resembles a rotating disc that could be the remainder of a past merger episode experienced by this galaxy. However, we have not found such evidence in the stellar population analysis of this galaxy (Paper I). We suggest that the more likely scenario involves two main aspects. First, at NE, we see a co-spatial high-velocity redshifted broad component (see Fig. 5) which we will discuss the origins of in § 5.2. The interaction with this component could have disturbed the gas in the disc producing the observed deviation. Second, as already described, towards the north there is the region where the emission line fluxes drop and are barely detectable. This affects our capability to distinguish both components and, there- \nore, constrain their kinematics which could have resulted mainly in the redshifted residuals.', '5.2 What is the origin of the different emission line profiles?': "As can be seen in Fig. 3, one of the characteristics that stand out in NGC6868 is the variation in line profiles found. So if the narrow component traces an ionized gas disc, the broad component has a much more diverse nature. \nIn the NE region, the broad component traces a high-velocity redshifted component ( ∼ 250 km s -1 ). It is also characterised by a similar velocity dispersion when compared to the narrow component ( ∼ 100 km s -1 ). Beyond the kinematics, line fluxes are enhanced in this direction and to a smaller extent, the A 𝑉 . \nHansen et al. (1991) detected gas filaments with spiral shapes using photometric data in a FoV larger than ours (35 × 35 arcsec 2 ). In Fig. 15 we compare Hansen et al. (1991) B-r (Gunn r filter) colour map, which traces the dust in this object, with our H 𝛼 flux map. From this figure, it is clear that the NE structure we found connects to their description of a spiral arm at NE. Following their interpretation of these spiral structures, this is likely material falling towards the centre of NGC 6868 forming the spiral structures observed which could come from various sources. It could indeed be a past gas-rich merger or outskirts material travelling towards the galaxy centre. Apparently, this structure has not experienced any significant increase in turbulence and remains kinematically coherent as the velocity dispersion in this region remains similar to the one in the disc. Also, the low electron density values found in this region (Fig. 9) indicate that the inflowing material did not experience any significant compression. \nAnother of these distinct kinematic components is the blueshifted ( ∼-100 km s -1 ) broad ( ∼ 250 km s -1 ) component (see Fig. 5, bottom panels). This region is even clearer in the flux maps (Fig.6) where a tail emerges towards the NW as a result of the blue wing found in the line profiles. The higher velocity dispersion hints at a different kinematical origin for this component in which more turbulence has been indicated. It is worth noting that the higher density values towards the centre hit at a gas compression. \nTwoscenarioscanexplain our findings, concerning either an inflow or an outflow from the galaxy centre. The scenario where this traces an inflow is possible because, as we just discussed, this galaxy shows signs of inflowing material towards its centre. Moreover, Rose et al. (2019) analysing the molecular gas present in this object described the gas as drifting in non-circular orbits, leading to the resulting infall of gas towards the vicinity of the SMBH. In Rose et al. (2024) they found that the molecular gas is 'predominantly contained in a 0.5 kpc wide and slightly inclined disky structure'. This inflowing material could originate from streams falling towards the centre of the galaxy, and showing different kinematical components (e.g. one moving in the direction of the observer and another away from it), such as those observed in NGC 5044 (a galaxy with similar morphological and physical characteristics as NGC 6868, for details see Diniz et al. 2017). \nSince NGC6868 hosts an LLAGN and presents distinct kinematical characteristics (when compared to the other component at NE) an alternative scenario emerges, e.g. this could be an outflow from the central SMBH. This would also explain the higher velocity dispersion, which indicates that the AGN is inducing turbulence in this gas. A growing number of studies has found that outflows in LINER sources are rather common when their nucleus present AGN signatures (e.g. Ilha et al. 2019; Rodríguez del Pino et al. 2019; Riffel et al. 2019; Ruschel-Dutra et al. 2021; Ilha et al. 2022; Heckler et al. 2022; Deconto-Machado et al. 2022; Hermosa Muñoz et al. \nnarrow \n<!-- image --> \n<!-- image --> \nFigure 13. Left panel: WHAN diagram for the narrow component of our data. The points are colour-coded according to the velocity, as in Figs. 10 and 12. Different regions in this diagram indicate a prevalence of a given ionization source with AGN showing dominating at 𝑊 𝐻𝛼 > 3 Å, further diving the region for strong AGNs following Cid Fernandes et al. (2011) for 𝑊 𝐻𝛼 > 3 Å. Moreover, the results obtained for the specific bins defined in Fig. 3 are overplotted. Centre panel: Same as the left panel but showing the results for the broader component. Top-right panel: the equivalent width of H 𝛼 for the narrow component, showing where the 3 Å line maps into our data. Bottom-right panel: Same as top-right, but for the broad component. \n<!-- image --> \n<!-- image --> \nFigure 14. Top panel: Map containing the resulting fit from the kinematical map of the narrow component Fig. 5. Bottom panel: residuals (fitted disc model discounted from the observed velocity profile) from the fit. We can confidently model the gas disc as our residuals are kept within ∼± 50 km s -1 for most of the FoV. White patches are masked regions in the fit due to low S/N. \n<!-- image --> \nFigure 15. Top panel: Fig. 2 extracted from Hansen et al. (1991) showing B-r (Gunn r filter) colour for the 35' × 35' inner region of NGC 6868. Reddened areas are seen in black. Beyond the central dust lane, diffuse elongated components are seen, mainly in the NE direction where a broader structure turning to S is found. Bottom panel: H 𝛼 flux map for our data cube where the flux towards the NE seems to follow the same direction as the one indicated by Hansen et al. (1991). \n<!-- image --> \n2022; Riffel et al. 2024; Gatto et al. 2024; Falcone et al. 2024; Hermosa Muñoz et al. 2024), finding outflow mass rates ranging from 10 -5 to 1 M ⊙ yr -1 . \nIn this context, we can estimate the outflow rate considering the following relation: \n/ 𝑀 = 𝑀 out 𝑣 out 𝑟 out (7) \nwhere / 𝑀 is the outflow rate, 𝑀 out is the mass of the outflow, 𝑣 out is the velocity of the outflow and 𝑟 out is the spatial extent of the outflow. 𝑀 out can be estimated using Eq. 4 applying it only to the spaxels where a component characterised by 𝑣 < -95 km s -1 and 𝜎 < 260 km s -1 is detected. For 𝑣 out we assume it to be the absolute median velocity of this component ∼ 110 km s -1 , and since it reaches the edge of our FoV we use as 𝑟 out the distance from galaxy centre (the red cross in all the figures) to the upper right corner. Of course, we are limited by our FoV so the adopted 𝑟 out is indeed a lower limit. It is also true that other studies have seen that in-situ acceleration could also produce this turbulent component (Rodríguez del Pino et al. 2019; Falcone et al. 2024, e.g.). In that case, the radius of the outflow would be simply the size of the region, diminishing the 𝑟 out , but making the outflow mass rate get higher (see Eq. 7). This results in an outflow mass rate of 0.04 M ⊙ yr -1 which is well within the values found by Hermosa Muñoz et al. (2024), further supporting the claim that this emission is due to an outflow from the SMBH. Another evidence supporting this is that the emission line ratios are well explained by models considering shocks. As can be seen in Fig. 16, line ratios seem to be enhanced both in the NE direction towards the edge of the FoV and near the centre in the NW direction, where we detect the peculiar kinematics, indicative of shock-heated gas (see § 5.3). \nThe region towards the North of our data where no emission lines are detected, also presents a challenging interpretation. We tried to sum more pixels to get a higher S/N to allow us to improve the fit but nearby regions where emission is strong make dealing with the contamination of this rather difficult. Given that the A V map does not show any trend towards that region nor in Fig. 7 nor in the stellar population fits (Paper I), it is unlikely that this is due to dust extinction. It is more likely that this is a region where devoid of gas. This is further supported by the fact that there is no molecular gas in this same region (Rose et al. 2024). Machacek et al. (2010) using X-ray data, found X-ray cavities in larger scales which they have correlated with an echo of AGN activity. \nIn the SW region, we only need one Gaussian component to fit the data. At first glance, this seems to only be a part of the ionised gas disc which, kinematically is true, but when we look at the flux of [O/i.pc/i.pc/i.pc] 𝜆 5007 it is evident that the SW shows a peculiar behaviour also maintained in the [O /i.pc/i.pc/i.pc] 𝜆 5007/H 𝛽 map with a clear enhancement of the [O /i.pc/i.pc/i.pc] 𝜆 5007 flux. Because this is a high ionization ion some form of added excitation mechanism is necessary to explain this observation. Given that the temperature increases towards this region (Fig. 8) and that Rose et al. (2024) has detected a stream of molecular gas that extends beyond our FoV, we suggest that this enhancement is due to the interaction with this stream.", '5.3 What is driving the gas ionization in NGC 6868?': 'NGC6868 displays an extended ionized gas component as can be seen in Fig. 1 being classified as a LINER object in the BPT diagram. The ionization mechanisms behind these objects are diverse and IFS studies have allowed for the disentanglement of them. In this sense, NGC6868 seems to have a huge variety of excitation mechanisms. \nIt is already known that this galaxy hosts an LLAGN due to the detection of a radio source, an X-ray core as well as [Ne /v.pc] (She et al. 2017; Bi et al. 2020; Rampazzo et al. 2013; Ricci et al. 2023), despite the non-detection of a broad component in H /i.pc lines. From the WHANdiagram (Fig. 13), we see that the central regions in both the narrow and broad components require the energetic input of an AGN which is consistent with the LLAGN picture, also coinciding with AGN models in BPT diagram of 𝛼 = -1 . 2 , -1 . 4 and log 𝑈 = -3 , -2 (Fig. 12). \nLooking at the narrow-line BPT diagrams we see that [O/i.pc/i.pc/i.pc] 𝜆 5007/H 𝛽 always present a lower value in the centre compared to the disc, also the clear gradient in the temperature profile where temperature increases outwards from the centre are all good evidence that shocks have some importance in the ionization balance for this object. \n[O/i.pc] 𝜆 6300 is a famous tracer for shocks and as can be seen in Fig. 12 shock models can explain the emission line ratios found. Ho et al. (2014) and Allen et al. (2008) argument that for 𝜎 > 150 km s -1 and [O /i.pc] 𝜆 6300/H 𝛼 > -1 . 0 would be enough to say that shocks are the dominant mechanisms. In this reasoning, we show in the bottom panel of Fig. 12 that the line ratios are consistent with shock models with magnetic field values between 2 and 4 𝜇𝐺 and an average velocity of 350 km s -1 . This coupled with the fact that we see the outward increasing temperature profile, surpassing the 20000 K in some points provides even more evidence that the blue-shifted component is an actual outflow that is shock-heating the surrounding medium. Additionally, the [S /i.pc/i.pc] 𝜆 6716 , 6731/H 𝛼 diagram also shows that the line ratio increases towards the redshifted and the blueshifted components ( e.g. comparing the positions of regions B, D and A in the diagrams it becomes clear). By comparing the narrow and broad component results, the gradient inverts, again providing more evidence that another source of excitation must be present which we interpret as shocks. It is worth mentioning that we cannot say if shocks are the dominant source of ionization, but surely they contribute to the ionization balance of this object. \nDue to the ubiquitous old stellar population, it is expected that HOLMES create a dispersed ionizing radiation field capable of ionizing the surrounding gas. More evidence of the presence of this ionization source is the WHAN diagram and the 𝑊 𝐻 𝛼 maps. Looking at the regions where 𝑊 𝐻 𝛼 < 1 Å the ionization is likely to be provided by HOLMES. In the region 𝑊 𝐻 𝛼 > 3 Å the most likely scenario is AGN ionization which is compatible with all the other diagrams. Looking at the spatial distribution of 𝑊 𝐻 𝛼 it is clear that it is between 1 and 3 Å where all the regions with distinct kinematical features coincide, contributing to the picture that, despite playing a secondary role, when combined with the kinematical and temperature findings, we conclude that shocks also contribute to the gas excitation. \nFinally, recent studies (e.g. Lagos et al. 2022; Ricci et al. 2023; Hermosa Muñoz et al. 2024; Hsieh et al. 2017) have shown that the physical processes dominating the gas ionization, in the central region of sources classified as LINERs, are due to LLAGNs, while the of nuclear component is the one with divergent results. For instance, Lagos et al. (2022) prefers the scenario of pAGB stars ionizing the extended component, whereas Ricci et al. (2023) states the importance of shocks for some of their sources. In this context, NGC 6868 proves to be a galaxy that shares characteristics with both studies, showing that most likely both mechanisms are dominating in different regions. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 16. Maps of important emission line ratios. From left to right, [N /i.pc/i.pc]/H 𝛼 , [O /i.pc/i.pc/i.pc]/H 𝛽 , [S /i.pc/i.pc]/H 𝛼 , [O /i.pc]/H 𝛼 . They were all corrected by extinction. \n<!-- image -->', '6 CONCLUSIONS': 'We analysed the central region of NGC 6868 using GMOS-IFU and mapped the properties of the ionized gas as well as the excitation mechanism behind the gas ionization. The complexity in the gas kinematics and ionization makes NGC 6868 an exciting laboratory to study the different mechanisms involved in powering LINER sources. Our main findings are summarised as flows: \n- · Channel maps and line profiles reveal complex kinematics and morphology, hinting at different processes acting on NGC 6868.\n- · Emission line fitting has revealed two kinematic components: a narrow and a broad.\n- · The narrow component traces an ionized gas disc and the broad component traces inflowing gas that is being driven to the vicinity of the LLAGN present in NGC 6868, settling in a dispersion-dominated component. Flux distributions follow these findings.\n- · The spatial distribution of the reddening obtained from Balmer decrement is consistent with the derived from the stellar continuum fitting albeit with a larger amplitude. This is likely due to the obscuration of the ionizing source, mainly in the central regions.\n- · We report the first measurement of electron temperature in NGC6868. We find a temperature of ∼ 14000 K for the central region with the outer parts surpassing the ∼ 20000 K. The density profile shows an inverse behaviour as the central regions show an enhancement ( ∼ 800 cm -3 ) when compared to the outer regions( ∼ 100 cm -3 ), hinting at a compression of the ionized gas.\n- · All the points from our observations are found within the LINER region in the BPT diagram. Coupled with the WHAN diagram, we conclude that the central region is ionized mainly by an LLAGN, while HOLMES create an ionizing radiation field that is responsible for the ionization at larger scales.\n- · The derived temperature profile, coupled with the kinematics and emission line ratios point towards the presence of shock-heated gas. \nOur work reinforces the necessity of a detailed analysis via IFU of LINERs to disentangle the different mechanisms that drive the observed emission. For instance, the presence of an outflow component in LINERs arises now as an important ingredient to explain the observed spectrum of these objects.', 'ACKNOWLEDGEMENTS': "The authors thank Jose Andres Hernandez Jimenez for helpful discussions. We thank the anonymous referee for insightful comments and discussion. This work was supported by Brazilian funding \nagencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and by the Programa de Pós-Graduação em Física (PPGFis) at UFRGS. JPVB acknowledges financial support from CNPq and CAPES (Proj. 0001) and the support of a fellowship from the 'la Caixa' Foundation (ID 100010434). The fellowship code is LCF/BQ/DI23/11990084. RR thanks CNPq (Proj. 311223/2020-6, 304927/2017-1 and 400352/2016-8), Fundação de amparo à pesquisa do Rio Grande do Sul (FAPERGS, Proj. 16/25510000251-7 and 19/1750-2) and CAPES (Proj. 0001). TVR thanks CNPq for support under grant 306790/2019-0 and 304584/2022-3. RAR acknowledges the support from CNPq (Proj. 303450/20223, 403398/2023-1, & 441722/2023-7), FAPERGS (Proj. 21/25510002018-0), and CAPES (Proj. 88887.894973/2023-00). MT thanks the support of CNPq (process 312541/2021-0) and the program L'Oréal UNESCO ABC Para Mulheres na Ciência . LGDH acknowledges support by National Key R&D Program of China No.2022YFF0503402 and National Natural Science Foundation of China (NSFC) project number E345251001. DRD acknowledges financial support from CNPq (Proj. 313040/2022-2). AFM has received support from PID2021-123313NA-I00 and 4RYC2021031099-I of the MICIN/AEI/10.13039/501100011033/UE. \nBased on observations obtained at the international Gemini Observatory and processed using the Gemini /i.pc/r.pc/a.pc/f.pc package, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).", 'DATA AVAILABILITY': 'The data are publicly available on /g.pc/e.pc/m.pc/i.pc/n.pc/i.pc archive under the project GS-2013A-Q-52. The reduced and processed data can be made available under reasonable request.', 'REFERENCES': 'Allen M. G., Groves B. A., Dopita M. A., Sutherland R. S., Kewley L. J., 2008, The Astrophysical Journal Supplement Series, 178, 20 \n- active galactic nuclei, 2nd ed edn. University Science Books, Sausalito, Calif\n- Papaderos P., et al., 2013, Astronomy & Astrophysics, 555, L1\n- Pizzella A., et al., 1997, Astronomy and Astrophysics, v.323, p.349-356, 323, 349\n- Ramella M., Focardi P., Geller M. J., 1996, Astronomy and Astrophysics, 312, 745\n- Rampazzo R., Panuzzo P., Vega O., Marino A., Bressan A., Clemens M. S., 2013, Monthly Notices of the Royal Astronomical Society, 432, 374\n- Revalski M., Crenshaw D. M., Kraemer S. B., Fischer T. C., Schmitt H. R., Machuca C., 2018, The Astrophysical Journal, 856, 46\n- Ricci T. V., Steiner J. E., Menezes R. B., 2014a, Monthly Notices of the Royal Astronomical Society, 440, 2419\n- Ricci T. V., Steiner J. E., Menezes R. 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2017ARA&A..55..389T	The gas surrounding galaxies outside their disks or interstellar medium and inside their virial radii is known as the circumgalactic medium CGM. In recent years this component of galaxies has assumed an important role in our understanding of galaxy evolution owing to rapid advances in observational access to this diffuse nearly invisible material. Observations and simulations of this component of galaxies suggest that it is a multiphase medium characterized by rich dynamics and complex ionization states. The CGM is a source for a galaxys starforming fuel the venue for galactic feedback and recycling and perhaps the key regulator of the galactic gas supply. We review our evolving knowledge of the CGM with emphasis on its mass dynamical state and coevolution with galaxies. Observations from all redshifts and from across the electromagnetic spectrum indicate that CGM gas has a key role in galaxy evolution. We summarize the state of this field and pose unanswered questions for future research.	2017-08-01T00:00:00Z	['2017arXiv170909180T', 'arXiv:1709.09180', '10.1146/annurev-astro-091916-055240', '2017ARA&A..55..389T', '10.48550/arXiv.1709.09180']	['Astrophysics - Astrophysics of Galaxies']	The Circumgalactic Medium	2,017	200	0.75	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	866	https://arxiv.org/pdf/1709.09180.pdf	{'No Header': "Annual Review of Astronomy and Astrophysics 2017. AA:1-46 \nThis article's doi: /10.1146/annurev-astro-091916-055240 \nCopyright c © 2017 by Annual Reviews. All rights reserved", '1': 'Jason Tumlinson , Molly S. Peeples 1 , & Jessica K. Werk 2 \n1 \nSpace Telescope Science Institute and Johns Hopkins University, Baltimore, Maryland; email: [email protected], [email protected] 2 University of Washington, Seattle, Washington, email: [email protected]', 'Keywords': 'gas, galaxies, galaxy evolution, cosmology', 'Abstract': "The gas surrounding galaxies outside their disks or ISM and inside their virial radii is known as the 'circumgalactic medium' (CGM). In recent years this component of galaxies has assumed an important role in our understanding of galaxy evolution owing to rapid advances in observational access to this diffuse, nearly invisible material. Observations and simulations of this component of galaxies suggest that it is a multiphase medium characterized by rich dynamics and complex ionization states. The CGM is a source for a galaxy's star-forming fuel, the venue for galactic feedback and recycling, and perhaps the key regulator of the galactic gas supply. We review our evolving knowledge of the CGM with emphasis on its mass, dynamical state, and co-evolution with galaxies. Observations from all redshifts and from across the electromagnetic spectrum indicate that CGM gas has a key role in galaxy evolution. We summarize the state of this field and pose unanswered questions for future research.", '1. A Very Brief History': "In the mid-1950s, Guido Munch observed neutral sodium (Na i ) and singly-ionized calcium absorption (Ca ii ) in the spectra of hot stars at high Galactic latitudes. Before these data were published as Munch & Zirin (1961), Munch showed them to Lyman Spitzer, who interpreted the lines as evidence for diffuse, extraplanar hot gas ( T ∼ 10 6 K), which keeps the colder clouds traced by Na i and Ca ii in pressure confinement (Spitzer 1956). And so was born the idea of the 'Galactic corona' and its exploration by absorption lines in the spectra of background objects. Following Schmidt's 1963 discovery of quasars, studies of 'extragalactic' gas rapidly progressed with spectroscopy of the intervening absorption \nlines by J. Bahcall, M. Burbidge, J. Greenstein, W. Sargent, and others. Bahcall & Spitzer (1969) then proposed that 'most of the absorption lines observed in quasi-stellar sources with multiple absorption redshifts are caused by gas in extended halos of normal galaxies'. In the 1980s, subsets of the QSO absorption lines were associated with galaxies (Bergeron 1986; Bergeron & Boiss'e 1991) while the 'Lyman alpha forest' emerged as their IGM counterpart (Sargent et al. 1980). Spurred by these developments, Hubble and Keck made great leaps in the 1990s towards a broader characterization of the number density and column density distribution of the IGM and CGM back to z ∼ 3. Pioneering studies from Hubble's Key Project on QSO absorption lines demonstrated that galaxy halos give rise to strong Ly α , C iv , and other metal lines (e.g. Lanzetta et al. 1995; Chen et al. 1998) in a gaseous medium that is richly structured in density, temperature, and ionization (Figure 1). \nIn the 2000s, large galaxy surveys such as SDSS uncovered the galactic baryon deficit, the mass metallicity relation, and quenching problems ( § 2). Meanwhile theorists implemented new physical prescriptions for gas accretion and feedback with new numerical methods and faster computers. It soon became impossible even to address these big mysteries of galaxies without appealing to gas flows between the ISM, the IGM, and by implication, the CGM. Yet most such models of gas flows were, and are still, tested against observations of starlight-the same observations that first posed the problems. By the mid-2000s, models and observations of gas flows in and out of galaxies had reached the point that the former were in urgent need of direct observations of the gas flows themselves. CGM studies leaped forward in the late 2000s with the installation of Hubble's Cosmic Origins Spectrograph , which was designed for reaching diffuse gas with 30 × the sensitivity of its predecessors, and with new techniques for stacking and combining X-ray and optical spectra. This, then, is the context in which our review occurs. We aim to survey recent progress in observing and modeling the gas flows that drive galaxy evolution and thus to tell the story of galaxy evolution writ large, from the perspective of the CGM. \nFor additional perspective on the issues raised here from a more Galactic point of view, we recommend the recent Annual Review on halo gas by Putman, Peek & Joung (2012b). For an up-to-date survey of accretion, see the forthcoming volume 'Gas Accretion onto Galaxies' (Fox & Dav'e 2017).", '2.1. The Major Problems of Galaxy Evolution': 'We will motivate and organize our review with four major galaxy evolution problems in which the CGM is implicated (Figure 2). Why do dark matter halos of different masses give rise to galaxies with drastically different star formation and chemical histories ( § 2.1.1, 2.1.2)? Why do such a small fraction of cosmic baryons and metals reside in the galaxies ( § 2.1.3, 2.1.4)? The prevailing answers to these questions all feature the regulation of gas flows into and out of galaxies-which necessarily pass into and through the CGM. We initially pose these problems at low redshift, but they all have highz counterparts, and their solutions require understanding the CGM and the flows that feed it at all cosmic epochs.', '2.1.1. How do galaxies sustain their star formation?. Star-forming galaxies pose a conun-': 'drum: their ISM gas can last for only a small fraction of the time they have been forming stars (Figure 2a), implying an external supply of gas that keeps the ISM in a quasiequilibrium state. The depletion time, τ dep ∼ M gas / ˙ M sfr changes by only ∼ 2 × over the', 'CGM:': 'Circumgalactic Medium \nIGM: Intergalactic Medium \nISM: Interstellar \nMedium \nSDSS: Sloan Digital Sky Survey', 'CMD:': "Color-Magnitude Diagram \nFigure 1 \n<!-- image --> \nA cartoon view of the CGM. The galaxy's red central bulge and blue gaseous disk are fed by filamentary accretion from the IGM (blue). Outflows emerge from the disk in pink and orange, while gas that was previously ejected is recycling. The 'diffuse gas' halo in varying tones of purple includes gas that is likely contributed by all these sources and mixed together over time. \nfactor of 30 between subL ∗ and superL ∗ galaxies. More generally, subL ∗ galaxies generally have extended bursty star formation histories, as opposed to the more continuous star formation found in more massive galaxies, suggesting differences in how and when these galaxies acquire their star forming fuel. As this fuel is from the CGM, we must explain how subL ∗ and L ∗ galaxies fuel star formation for longer than their τ dep . \n2.1.2. What quenches galaxies and what keeps them that way?. How galaxies become and remain passive is one of the largest unsolved problems in galaxy evolution (Figure 2b). Proposed solutions to this problem involve controlling the gas supply, either by shutting off IGM accretion or keeping the CGM hot enough that it cannot cool and enter the ISM. \nFigure 2 \n<!-- image --> \nFour important problems in galaxy evolution viewed with respect to M glyph[star] . (a) the gas depletion timescale τ dep ∼ M gas / ˙ M sfr for star-forming galaxies at z ∼ 0, with M gas from Peeples et al. (2014) and ˙ M sfr from Whitaker et al. (2012); the shading denotes ± 0 . 15 dex scatter in ˙ M sfr . (b) the galaxy bimodality in terms of M glyph[star] and specific star formation rate (Schiminovich et al. 2010). (c) the galactic baryon fraction, M glyph[star] / ((Ω b / Ω m ) M halo ) from Behroozi, Conroy & Wechsler (2010), with stars in red and interstellar gas in blue (from Peeples et al. (2014). (d) the 'retained metals fraction', metals for several galactic components relative to all the metals a galaxy has produced Peeples et al. (2014), with stars in red, interstellar gas in blue, and interstellar dust in orange. Vertical bars mark the properties of subL ∗ , L ∗ , and superL ∗ galaxies at log M glyph[star] / M glyph[circledot] = 9 . 5 (blue), 10.5 (green), and 11.0 (red), respectively. \nLow-mass galaxies tend to continue forming stars unless they are a satellite of or near a more massive galaxy (Geha et al. 2012). This finding suggests that the central galaxy's gaseous halo strips the satellite with ram pressure or 'starves' the satellite of fresh fuel.", 'Λ CDM:': "Cold-Dark-Matter Cosmology with a Cosmological \nConstant \nThese ideas have specific testable implications for the physical state of the CGM. \n2.1.3. Why do galaxies lack their fair share of baryons?. The ΛCDM model predicts that baryons follow gravitationally-dominant dark matter into halos, where the gas dissipates energy as radiation and cools into the center of the halo. Observed galaxies, however, harbor only small share of the halo's expected baryons in their stars and ISM, with M b glyph[lessmuch] (Ω b / Ω m ) M h (Figure 2c). Even at their most 'efficient', L ∗ galaxies have converted only ∼ 20% of their halos' baryons into stars (Figure 2c), with values of only about 5-10% in subL ∗ and superL ∗ galaxies (Behroozi, Conroy & Wechsler 2010; McGaugh et al. 2010). There are three basic possibilities: the baryons are in the halo but not yet detected, such as hot or diffuse gas; the baryons have been accreted and then ejected from the halo altogether; or the baryons have been prevented from accreting into the halo in the first place. While reality probably combines aspects of all three, in any combination they strongly suggest that the CGM is an excellent place to look for missing halo baryons in cold or hot gas, or for direct evidence of past ejection. \n2.1.4. Where are the metals?. While baryons come from outside the halo, metals are sourced locally by stars and the deaths of stars. Star-forming galaxies over ∼ 3 decades in stellar mass retain a surprisingly flat ∼ 20-25% of the metals they have ever produced (Peeples et al. 2014) in their stars, ISM gas, and dust. Metals have clearly been lost to outflows (Tremonti et al. 2004), but how these outflows scale with galaxy mass is unclear. Models that already struggle to reproduce the observed steep mass-metallicity relation (Somerville & Dav'e 2015) fail to retain the low, flat fraction of metals produced (e.g., Muratov et al. 2015; Zahid et al. 2012; Oppenheimer et al. 2016b). While 'missing baryons' concern accretion and feedback through the outer boundary of the CGM, metals address the disk/halo interface: do they leave the halo altogether, or recycle back into the galaxy's ISM on long timescales as a 'halo fountain' (Oppenheimer & Dav'e 2008) On what timescales are ejected metals recycled? How metal-enriched is outflowing material relative to the ambient ISM, i.e., what are the entrainment fractions and metal-loading factors? How does dust survive the journey out of galaxies, and what chemical clues does it hide? As we will show, following the metals as 'Nature's tracer particles' is a fruitful and revealing route to understanding of the CGM.", '2.2. Our Point of View': "How galaxies acquire, eject, and recycle their gas are core issues in galaxy evolution, on par with how they evolve in their shapes and how star formation works. To a large extent these gas flows are galaxy evolution. The CGM is a main venue for these flows: it is potentially the galactic fuel tank, waste dump, and recycling center all at the same time. This review will approach the growing body of empirical results and theoretical insights from the direction of these four major questions. Rather than asking, for example, 'what are the Mg ii absorbers?', we will ask 'what do the Mg ii absorbers tell us about the mass and kinematics of galactic outflows?'. We will thus favor physical insights and synthesis of discoveries over detailed discussions of methods, compilations of data, or exhaustive cataloging of the literature. We hope that this approach will improve understanding between those who study gas and galaxies (still disparate groups) and more effectively highlight open issues to be pursued in the future. \nFigure 3 \n<!-- image --> \nThese simulated views (from EAGLE, Schaye et al. 2015; Oppenheimer et al. 2016b) of the CGM are more sophisticated but possibly just as uncertain as Figure 1. The four columns render a single galaxy with M glyph[star] = 2 . 5 × 10 10 M glyph[circledot] at z = 0 in density (left), temperature (middle) and metallicity (right). The galaxy is shown at redshifts z = 3, 2, 1, and 0 from top to bottom. The dotted white circle encloses the virial radius at each epoch. \nPhysics: underlying physical properties and processes", 'Phenomenology:': "emergent properties and scaling relations \nLLS: Lyman Limit Systems, N HI > 10 16 . 2 cm -2 , the 'dense' CGM \nDLA: Damped Lymanα Systems, N HI > 2 × 10 20 cm -2 , generally ISM \nFor the purposes of our discussion, we define the CGM to be bounded at the outside by the virial radius R vir of a galaxy's dark matter halo, and on the inside by the disk or ISM. Neither boundary is well-defined, and precisely defining when gas passes through one of these boundaries can be either a valuable research contribution or a fruitless semantic exercise depending on circumstances. We focus on the physics of gas that fills out halos without too much attention to these exact definitions.", '3.1. Transverse Absorption-Line Studies': "Viewing the CGM in absorption against a bright background source like a quasar offers three major advantages over other methods: (1) sensitivity to extremely low column density, N glyph[similarequal] 10 12 cm -2 , (2) access to a wide range of densities, unlike emission-line measures that scale as density squared, and (3) invariance of detection limits to redshift and the luminosity of the host galaxy. These advantages come at a cost, however: absorption provides only projected, pencil-beam measures of gas surface density, usually limited to one sightline per galaxy by the rarity of background quasars. Within the local Universe (a few Mpc) it is possible to use multiple sightlines (e.g., Lehner, Howk & Wakker 2015; Bowen et al. 2016), and at higher redshift, multiply-lensed images from background quasars (e.g., Rauch & Haehnelt 2011; Rubin et al. 2015) to constrain the sizes of absorbers. In general, however, CGM maps made from absorption-line measurements are a statistical sampling of gas aggregated from many galaxies. With massive optical spectroscopic surveys, samples have grown to hundreds or thousands in low ions like Mg ii and Ca ii (e.g., Zhu & M'enard 2013a). Quasar/galaxy pairings have now been extended out to z ∼ 4 and beyond (Turner et al. 2014; Matejek & Simcoe 2012). \nThere are three basic ways of building absorber samples. First, 'blind' surveys select background quasars on brightness and/or redshift and so are optimal for samples that are unbiased with respect to foreground structure. Ground-based redshift surveys around previously observed quasar sightlines are now a time-honored method for constructing samples of quasar/galaxy pairs (e.g., Chen et al. 1998; Stocke et al. 2006; Rudie et al. 2012). The second, 'targeted', approach chooses background sources because they probe particular foreground structures, such as L ∗ galaxies (Tumlinson et al. 2013), subL ∗ galaxies (Bordoloi et al. 2014b), galaxies with known ISM content (Borthakur et al. 2015), or groups and filaments (Wakker et al. 2015; Tejos et al. 2016), by cross-matching the observable quasar with catalogs of these structures. Finally, maps of absorption in the Milky Way's CGM use essentially any quasar (or UV-bright halo stars), sometimes chosen to pass through known halo gas structures and sometimes not. Though most absorption-line work has been in the UV and optical, Chandra and XMM have been used to search for X-ray gas in individual absorbers, constraining the extent of CGM and IGM hot gas (Nicastro et al. 2005). \nIt is useful to distinguish between H i column density regimes that must be, or can be, treated differently in analysis. Lines up to log N glyph[similarequal] 15 can usually be analyzed with equivalent widths or Voigt profile fitting. The value log N glyph[similarequal] 15 is high for the Ly α forest but low for the CGM (there are of course a few exceptions, Tumlinson et al. (2013); Johnson et al. (2014), where H i is not seen at < 100 kpc even to low limits). At log N glyph[similarequal] 16, saturation becomes a major factor and robust column densities (as opposed to lower limits) must come from profile fitting or from the higher Lyman series lines, if the system is redshifted enough. Systems with log N glyph[similarequal] 16, are partial or complete LLSs. If the Lyman \nlimit is covered ( z > 0 . 24 for Hubble), the flux decrement at λ = 912(1 + z ) ˚ A allows a precise measurement of log N HI and improved ionization and metallicity diagnostics. Above log N HI glyph[similarequal] 18 (where N HI is the HI column density in cm -2 ), the Lyman limit is totally opaque, the highest Lyman series lines are saturated, and genuine column densities must come from fitting the Ly α profile for LLS and DLAs.", '3.2. Stacking Analyses': "Massive spectroscopic surveys have enabled another novel method for examining halo gas. 'Stacking' of hundreds or thousands of spectra is a powerful way to extract faint signals from absorption-line datasets. This technique requires catalogs of redshifts, for either foreground galaxies or absorbers, so that the spectra of background objects can be shifted to their rest frames and continuum-normalized and then co-added together. The co-addition beats down statistical noise, enabling measurements of weak absorption at the cost of averaging over individual absorber profiles. When the catalogs of foreground galaxies include properties such as mass, radius, star formation rate, color, environment, or orientation, the stacks can be performed with subsets of the data to examine the variation of mean profiles with these properties (York et al. 2006; Zhu & M'enard 2013b; Bordoloi et al. 2011). Stacking experiments that correlate the reddening of quasars due to foreground galaxy halos in the SDSS survey have revealed large quantities of dust in the CGM of galaxies (M'enard et al. 2010; Peek, M'enard & Corrales 2015). Stacking techniques can also exploit more numerous, but fainter, sources; for example, Steidel et al. (2010) characterized the CGM of z ∼ 3 galaxies by stacking the spectra of background galaxies. Stacking can detect weak signals in the mean properties of gas absorbers, but at the cost of averaging out kinematic and ionization structure that may contain significant physical meaning.", '3.3. Down the Barrel': "'Down-the-barrel' spectroscopy uses a galaxy's own starlight as a background source for detecting absorption. This method has been a fruitful one for studying galactic inflows and outflows from spectroscopy of star-forming galaxies. This method is commonly used in optical and near-UV lines such as Ca ii , Na i , Mg ii , and Fe ii (Martin 2005; Kornei et al. 2012; Bordoloi et al. 2011; Rubin et al. 2014) to study outflows from galaxies out to z ∼ 1, in UV lines for low-redshift star-forming galaxies (Henry et al. 2015; Heckman et al. 2015), or even to examine accretion (Rubin et al. 2012). Down-the-barrel measurements are critical pieces of the CGM puzzle because they directly trace current outflows at galactocentric radii that are inefficiently covered by background sources (because of the R 2 scaling of foreground cross-section). While down-the-barrel spectra are key for tracing the accretion and outflows that dominate CGM kinematics, they have the key limitation that the galactocentric radius of any detected absorption is unconstrained-it could be anywhere along the line of sightcomplicating mass and covering fraction estimates inferred from these spectra.", '3.4. Emission-line maps': "Emission-line observations search for photons emitted directly from CGM gas. As the emission measure scales as n 2 , and the CGM has n H ∼ 10 -2 or less, this is a stiff challenge. The MW halo has been extensively mapped for HVCs and other halo structure using radio emission at 21 cm. This technique has been applied to external galaxies (Putman, Peek & \nFigure 4 \n<!-- image --> \nA range of ion equivalent width (rest-frame) measurements for a compilation of published surveys. We progress from H i though seven metallic ions of increasing ionization potential. The surveys are COS-Halos Tumlinson et al. (2013); Werk et al. (2013), COS-Dwarfs (Bordoloi et al. 2014b), COS-GASS (Borthakur et al. 2015), MAGIICAT Nielsen et al. (2013), Liang & Chen (2014), the Keck Baryonic Structure Survey (Rudie et al. 2012; Turner et al. 2015), CASBaH (Tripp et al. 2011), Prochaska et al. (2011a), and the X-ray study of Yao et al. (2012) that imposes as stacked upper limit on O vii . \nJoung 2012b) but detections are limited to within ∼ 10 -20 kpc of the targeted galaxies. The soft X-ray band is optimal for gas at glyph[greaterorsimilar] 1 million K. The extremely low surface brightness of the gas makes these observations challenging and expensive, but a few individual halos have been detected and their hot gas budgets measured by Chandra and/or ROSAT (e.g., Humphrey et al. 2011; Anderson, Churazov & Bregman 2016). Stacking of individual galaxies techniques has also yielded mass density profiles for hot gas around nearby galaxies Anderson, Bregman & Dai (2013). When combined with halo size, density, and metallicity constraints from soft X-ray absorption line techniques, these maps have aided in the assessment of the total mass and baryon fraction of the hot CGM. \nEmission line maps are also possible at UV/optical wavelengths, though no less challeng- \ning than in the X-ray. Recent reports claim a detection of an extended O vi halo ( R ∼ 20 kpc) around a low-redshift starburst galaxy (Hayes et al. 2016). Extended Ly α emission has been seen out to ∼ 100 kpc away from z ∼ 2 . 5 galaxies and QSOs (Cantalupo et al. 2014; Prescott, Martin & Dey 2015). In another case, an extended filamentary structure connected to a galactic disk was detected using diffuse emission in the optical (Martin et al. 2015). Emission maps can constrain the density profile, morphology, and physical extent of the gas more directly than aggregated pencil-beam sightlines (Corlies & Schiminovich 2016). For X-ray emission from fully ionized gas, masses can be inferred more directly, avoiding the uncertain ionization corrections that plague absorption-line measurements ( § 4); indeed, the CGM's more massive cousin, galaxy clusters' intracluster medium, has been studied in detail via X-ray emission for decades (Vikhlinin et al. 2006). On the downside, emission line maps are still challenging technically; the surface brightnesses are extremely small compared to sky and detector backgrounds, and surface brightness dimming has a steep increase with redshift. In a recent study using stacks of fiber spectra from SDSS, Zhang et al. (2016) achieved detections of H α at 50 -100 kpc around low-redshift galaxies, demonstrating that very sensitive limits can be reached on galaxies in the aggregate. These observations remain challenging, but as 'taking a picture' of an astrophysical object remains the ideal, efforts to improve instrument technology and enable emission line mapping to reach samples of hundreds of galaxies across cosmic time is an important goal.", '3.5. Hydrodynamic Simulations': "Physical models and simulations are essential tools for understanding the CGM. In contrast to observations, they provide for controlled environments where physical properties, histories, and futures of gas are all known and can be manipulated to tease insights out of the otherwise unmanageable complexity of a multiphase gaseous medium. As reviewed by Somerville, Popping & Trager (2015), there are many schemes for simulating the development of the cosmic web and galaxies under the influence of dark matter, gravity, and hydrodynamics. The major methods at present are smoothed particle hydrodynamics (SPH, such as Gadget, Ford et al. 2013; Oppenheimer et al. 2016b, Gasoline, Christensen et al. 2016; Gutcke et al. 2017, and GIZMO Muratov et al. 2016), adaptive mesh refinement (AMR, such as Enzo, Hummels et al. 2013; Corlies & Schiminovich 2016), and moving mesh (Arepo and the Illustris simulation, Suresh et al. 2015). Large-scale cosmological simulations in Mpc-scale boxes can simulate hundreds of galaxies in their proper ΛCDM context (e.g., Oppenheimer & Dav'e 2006; Vogelsberger et al. 2014; Ford et al. 2014). At the opposite end of the scale, very high resolution simulations focused on the interaction between dense clouds and diffuse halos (e.g., Heitsch & Putman 2009; Armillotta et al. 2016) that can reach scales at glyph[lessmuch] parsec. Spanning these two regimes are the so-called 'zoom' simulations, which resolve enough of the large scale structure to accurately trace a single galaxy or a subset of galaxies selected out of larger boxes (Figure 2, Schaye et al. 2015). Even zooms must make assumptions about physics that they do not resolve, using 'sub-grid' prescriptions to stand in for such complex phenomena as star formation, metal mixing and transport, supernova and AGN feedback, and others. Sub-grid models are parameterized and tuned to yield specific metrics-like the stellar mass function at z = 0-and then the properties that emerge-such as SFRs, morphology, quenching, and the CGM-are analyzed and compared to data to constrain the physical prescriptions that went in. We will use simulations from a broad range of techniques and groups to look for insights into how \nthe CGM participates in galaxy evolution, and to help interpret data.", '4. The Physical State of the CGM': 'We now turn to the density profile, phase structure, and kinematics of the CGM. We first present the data that show the various ionization states and velocity distributions of the CGM absorption ( § 4.1). Next, we describe how the absorption line measurements may be translated into physical parameters such as density, temperature, and size in ( § 4.2). We then draw lessons from kinematics ( § 4.3) before considering the physical complexities and challenges inherent in the interpretation of these data ( § 4.4 and § 4.5).', '4.1. The Complex, Multiphase CGM': "As a matter of empirical inference, the CGM is 'multiphase' in its ionization structure and complex in its dynamics. The ionization structure is seen in Figure 4, which compiles measurements for six diagnostic ions as a function of impact parameter (a proxy for radius). These data indicate a wide range of density and ionization conditions up to a few 10 5 K with very little interpretation required. Observationally, 'multiphase' means many of these metal ions spanning an order of magnitude in ionization potential energy are commonly found within the same 'absorber system' occupying a galaxy's halo. An open question in the physics of circumgalactic gas is what this observed mulitphase ionization structure reveals about the small-scale multiphase density, temperature, and metallicity structure of the CGM. \nOver the last 20 years, the practice of using such empirical inputs in analytic arguments to infer the physical state and structure of the diffuse plasma has matured greatly (Mo & Miralda-Escude 1996; Maller & Bullock 2004). To produce an extended, multiphase CGM, authors have proposed several scenarios which we categorize as follows: (1) massive inward cooling flows driven by local thermal instabilities (e.g. McCourt et al. 2012); (2) boundary layers between moving cool clouds in a hot atmosphere (e.g. Begelman & Fabian 1990); and (3) the continual shocking and mixing of diffuse halo gas by galactic outflows (e.g. Fielding et al. 2016; Thompson et al. 2016). We discuss the applicability of some of these analytic models in § 4.4 and § 4.5. \nDirect evidence for a hot component (log T glyph[greaterorsimilar] 6) in the multiphase CGM comes from diffuse soft X-ray emission (Anderson & Bregman 2010; Anderson, Bregman & Dai 2013), and in absorption along QSO sightines (Williams et al. 2005; Gupta et al. 2012) for the Milky Way and external galaxies. Indirect evidence for a hot phase comes from highly ionized metals that correlate with the low-ionization HVCs (Sembach et al. 2003; Fox, Savage & Wakker 2006; Lehner et al. 2009; Wakker et al. 2012), suggesting boundary layers between a hot medium and the colder HVCs. Milky Way HVCs also show head-tail morphologies indicative of cool clouds moving through a hot medium (e.g., Bruns et al. 2000). Finally, the multiphase CGM is clearly manifested in hydrodynamic simulations, which exhibit a mixture of cool (10 4 K) and warm-hot (10 5 . 5 -10 6 K) gas within a galaxy virial radius with a density profile that drops with increased distance from the central host galaxy (e.g., Shen et al. 2013; Stinson et al. 2012; Ford et al. 2013; Suresh et al. 2017, Figure 3). For practical purposes we can regard the outer boundary of the CGM to correspond to R vir , but there is no empirical reason to believe that any special behavior occurs at that radius; current observations favor trends in column densities that scale with R vir but do not change in \nFigure 6 shows the basic schema for constraining CGM gas properties with these 'multiphase' ions. The grey-scale phase diagram renders the properties of all < R vir gas from a Milky Way mass EAGLE zoom simulation (Oppenheimer et al. 2016b). Accessible ions at \n<!-- image -->", 'Figure 5': "A selection of absorption-line data and Voigt profile fits from the COS-Halos survey (Werk et al. 2016), showing a range of metal ions and HI on a common velocity scale with the galaxy at v = 0 km/s on the x-axis. The black outlined beige curve traces H i , the purple Si ii , the blue Si iii , the green Si iv , and the orange shows O vi . \nform at that arbitrary boundary. \nEvidence for kinematic complexity is revealed as the detected ion species breaking into different 'components' with distinct velocities and linewidths. Shown in Figure 5, the various metal ions show significant but varied correspondence in their component structure. The combination of both aligned and misaligned components between ionization states may reflect clouds or streams with density structure or a population of clouds with different ionization states projected together along the line of sight to the same range of observed velocities. Cloud sizes are difficult to constrain in a model independent way, but multiplylensed images from background quasars (Rauch, Sargent & Barlow 2001; Rauch & Haehnelt 2011) prefer 1-10 kiloparsec scales. Fitting Voigt profiles to multi-component absorption yields column density N , Doppler b parameter, and velocity offset v for each component from the galaxy systemic redshift, as well as the total kinematic spread of gas in a halo (but this fitting is subject to issues caused by finite instrumetal resolution). Generally, the kinematic breadth of an absorber system is thought to reflect the influence of the galaxy's gravitational potential, bulk flows, and turbulence in the CGM.", '4.2. From Basic Observables to Physical Properties': 'We must characterize the ionization states, chemical composition, and density to properly describe the symbiotic relationship with the gas and stars in the central galaxy disk and the CGM. If it were feasible to obtain precise measurements for every ion of every abundant element, in all velocity components, then the gas flows, metallicity, and baryon budget of the multiphase CGM would be well-constrained. However, atomic physics dictates that only a subset of the ionization states of each element lie at accessible wavelengths. Taking oxygen as an example, O i and O vi place strong lines in the far-UV, while O ii -O v lines appear in the extreme-UV (400-800 ˚ A). O vii and O viii , arising in hot gas, have strong transitions in the soft X-ray ( ∼ 20 ˚ A). While it is therefore possible in principle to detect (or limit) every stage of oxygen, this potential has yet to be realized. \nNUV: Near UltraViolet, 2000 glyph[lessorsimilar] λ glyph[lessorsimilar] 3400 ˚ A \nFUV: Far UltraViolet, 900 glyph[lessorsimilar] λ glyph[lessorsimilar] 2000 ˚ A \nEUV: Extreme UltraViolet, 400 glyph[lessorsimilar] λ glyph[lessorsimilar] 900 ˚ A \nX-ray: λ glyph[lessorsimilar] 30 ˚ A', 'Low Ions:': 'IP < 40 eV, T = 10 4 -4 . 5 K', 'Intermediate Ions:': '40 glyph[greaterorsimilar] IP (eV) glyph[lessorsimilar] 100, T = 10 4 . 5 -5 . 5 K', 'High Ions:': "IP glyph[greaterorsimilar] 100 eV, T > 10 5 . 5 K \nCIE: Collisional Ionization Equilibrium \nPIE: PhotoIonization Equilibrium \nEUVB: Extragalactic UltraViolet Background \neach temperature and density are marked with colored squares and dashed lines. This plot is intended to be a useful guide for finding the most likely tracers of a given CGM gas phase. It cannot be used to extract precise temperatures and densities for any given ion since the metal ion positions on this phase diagram are model-dependent. The inset shows the most common strong lines from these species plotted as observed wavelength versus redshift; the rest frame wavelength is where each intercepts z = 0. Practically, FUV lines are available at z < 1 with Hubble and z > 2 from the ground, the EUV lines can be reached at z glyph[greaterorsimilar] 0 . 5 -1 with Hubble ( λ obs glyph[greaterorsimilar] 1100 ˚ A), and the X-ray lines can currently only be detected toward the small number of bright QSOs and blazars with reach of the sensitivity of Chandra and XMM . As a result, most CGM measurements rely on heterogeneous ion sets- several low ions from C, N, Si, and Mg, a few intermediate ions from C and Si, and a high ion or two from Ne and O. Therefore, the gas density and temperature can only be understood in the context of a model for its ionization state (and abundance patterns). \nMany assumptions are necessary to make progress toward physical models of the CGM. The two most generic classes of models are PIE and CIE. Generally, low and intermediate ions can be accommodated within PIE models, while high ions require CIE models. Species at intermediate ionization potentials, such as C iv and O vi , will sometimes show a preference for one or the other or have contributions from both. These two classes of model are not mutually exclusive: a gas that is collisionally ionized may have the ion ratios further affected by incident radiation, and there are numerous possible departures from equilibrium that further complicate modeling (e.g. Gnat & Sternberg 2007b). Generally, having access to more metal ion tracers means one is able to place more refined constraints on the models, while results from models with fewer ions are more model-dependent. \nRadiative transfer models like Cloudy (Ferland et al. 2013) are used to build PIE models (e.g. Bergeron & Stasi'nska 1986; Prochaska et al. 2004; Lehnert et al. 2013; Werk et al. 2014; Turner et al. 2015), which are parametrized by density n H , or equivalently the ionization parameter log U ≡ Φ /n H c , the observed neutral gas column density N HI , and a gas-phase metallicity, log [Z/H]. Here, Φ is the number of photons at the Lyman edge (i.e., the number ionizing photons), set by the assumed incident radiation field with a given flux of ionizing photons. Besides ionization and thermal equilibrium, another major underlying assumption of photoionization modeling is that the included metal ions arise from a single gas phase with the same origin (i.e., are co-spatial). The single cloud, single density approximation for PIE modeling of low-ions leads to uncertain 'cloud' sizes, determined by N H /n H ranging from 0.1-100 kpc (Stocke et al. 2013; Werk et al. 2014). In response, some models have begun to explore internal cloud density structure (Stern et al. 2016) or local sources of radiation (e.g. star-formation in the galaxy, the hot ISM, Fox et al. 2005; Werk et al. 2016). PIE models generally fail for highly-ionized metal species like O vi , sometimes C iv , and certainly for X-ray ions. For those we turn to CIE, where temperature controls the ionization fractions and a metallicity must be assumed or constrained to derive total hydrogen column N H . \nBeyond PIE and CIE, there are non-equilibrium ionization mechanisms that may reproduce the intermediate- and high-ion states that generally fail for PIE (e.g. C iv , N v , O vi ). These models include: (1) radiative cooling flows that introduce gas dynamics and self-photoionization to CI models (Edgar & Chevalier 1986; Benjamin 1994; Wakker et al. 2012), (2) turbulent mixing layers, in which cool clouds develop skins of warm gas in KelvinHelmholtz instabilities (Begelman & Fabian 1990; Slavin, Shull & Begelman 1993; Kwak & Shelton 2010), (3) conductive interfaces, in which cool clouds evaporate and hot gas condenses in the surface layer where electron collisions transport heat across the bound- \nary (Gnat, Sternberg & McKee 2010; Armillotta et al. 2016), and (4) ionized gas behind radiative shocks, perhaps produced by strong galactic winds (Dopita & Sutherland 1996; Heckman et al. 2002; Allen et al. 2008; Gnat & Sternberg 2009). These models all modify the column density ratios given by pure CIE, but do not change the basic conclusion that gas bearing these ionic species must be highly ionized, i.e. with a neutral fraction glyph[lessmuch] 1%. These large and unavoidable ionization corrections, when applied to H i column densities of log N HI ∼ 15-18, entail surface densities and total masses that are significant for the galactic budgets ( § 5). It is likely that combinations of PIE and CIE into these more complex models are more accurate descriptions of Nature than either basic process considered in isolation.", '4.3. Line Profiles and Gas Kinematics': "Linewidths, given by the Doppler b parameter, illuminate the CGM temperature structure and gas dynamics. The gas temperature, T , and any internal non-thermal motions are captured in the following parameterization: b 2 = (2 kT/m i ) + b 2 nt , for a species with atomic mass m i . When the low and high ions are assessed via Voigt profile fitting, the low ions are usually consistent with gas temperatures < 10 5 K, with a contribution from non-thermal broadening ( < 20 km s -1 , Tumlinson et al. 2013; Churchill et al. 2015; Werk et al. 2016). 'Broad Lyman alpha' (BLA; b glyph[greaterorsimilar] 100 km s -1 ) and Ne viii systems have been detected in QSO spectra at high S/N that directly probe gas at log T ∼ 5 . 7 (Narayanan et al. 2011; Savage, Lehner & Narayanan 2011; Tripp et al. 2011; Meiring et al. 2013). These UV absorption surveys indicate that the CGM contains a mixture of photoionized and/or collisionally ionized gas in a low-density medium at 10 4 - 10 5 . 5 K (e.g., Adelberger et al. 2003; Richter et al. 2004; Fox et al. 2005; Narayanan et al. 2010; Matejek & Simcoe 2012; Stocke et al. 2013; Werk et al. 2013; Savage et al. 2014; Lehner et al. 2014; Turner et al. 2015). \nThe velocity dispersion and number of components reveals the kinematic substructure of the CGM. Most significantly, gas near lowz galaxies across the full range of log M glyph[star] = 8 . 511 . 5 show the projected line-of-sight velocity spreads that are less than the inferred halo escape velocity, even accounting for velocity projection. Thus most of the detected CGM absorption is consistent with being bound to the host galaxy, with implications for outflows and recycling ( § 7). This is true for all the observed species from H i (Tumlinson et al. 2013) to Mg ii (Bergeron & Boiss'e 1991; Nielsen et al. 2015; Johnson, Chen & Mulchaey 2015b) to O vi (Tumlinson et al. 2011; Mathes et al. 2014). The strongest absorption seen in H i and low ions are heavily concentrated within ± 100 km s -1 . For low ionization gas, internal turbulent / non-thermal motions are b nt ∼ 20 km s -1 , while for high ionization gas the non-thermal/turbulent contributions to the line widths are 50-75 km s -1 (Werk et al. 2016; Faerman, Sternberg & McKee 2017). Similar total linewidths are seen in the z > 2 KODIAQ sample, possibly indicating similar physical origins at different epochs Lehner et al. (2014). \nMis-alignments of the high and low-ion absorption profiles in velocity space may indicate that the gas phases bearing high and low-ions are not co-spatial and thus that the gas is multiphase (e.g. Fox et al. 2013). Some systems, however, show close alignment between low and high ionization gas (Tripp et al. 2011) in a fashion that suggests each detected cloud is itself multiphase, perhaps in a low-ion cloud / high ion skin configuration. Heckman et al. (2002) and others (e.g. Grimes et al. 2009; Bordoloi, Heckman & Norman 2016) have argued \nFigure 6 \n<!-- image --> \nMetal absorption lines (ions) of the CGM from Mg i to O viii having 19 < λ rest < 6000 ˚ A shown on a phase ( T -n H ) diagram within R vir of the z = 0 EAGLE simulation shown in Figure 2. The points are colored according to ionization state, ranging from neutral (I; black) to highly ionized (X; magenta). The position of each point is set on each axis where its ionization fraction peaks in CIE (temperature axis) and a standard PIE model (density axis) (Gnat & Sternberg 2007b; Oppenheimer & Schaye 2013a); the range bars show the T and n range over which each species has an ionization fraction over half its maximum value (i.e., the FWHM). Complete line lists are available in Morton (2003). \nthat the relationship between O vi column density and absorption-line width for a wide range of physically diverse environments indicates a generic origin of O vi in collisionallyionized gas. However, the relationship exhibits considerable scatter, is impacted significantly by blending of multiple unresolved components (at least at the moderate R ∼ 20 , 000 resolution of COS), and may arise from other physical scenarios such as turbulent mixing (e.g. Tripp et al. 2008; Lehner et al. 2014). Generally, high-ions like O vi in the CGM exhibit systematically broader line widths than low and intermediate ions (e.g. Werk et al. 2016). Though complex and varied, absorber kinematics may provide important observational constraints on both ionization and hydrodynamic modeling, but new methods of analysis \nand new statistical tools will be required to realize their full potential.", '4.4. Challenges in Characterizing the Multiphase CGM': "Ionization modeling is limited by what might be considered 'sub-grid' processes that investigators must cope with to get from line measurements to useful constraints on models. The most basic of these arise in the data themselves. CGM absorption observations are generally not photon-noise limited, but line saturation is a major issue particularly for the most commonly detected species. Only lower limits can be derived from the equivalent widths of saturated lines; line profile fitting helps where the saturation is not too severe. Reliable columns of the crucial H i ion are often challenging except where the Lyman limit is available. Moreover, the blending of narrow components with small velocity offsets in data with finite spatial resolution make all line measurements somewhat ambiguous. It is often necessary to model an entire line profile as a single nominal cloud, though sometimes the ionization state can be constrained on a component-by-component basis. \nThere is often ambiguity about whether to adopt PIE, CIE, or combination nonequilibrium models. These issues are compounded by uncertainties in the additional model inputs. These include the relative elemental abundances, which need not be solar but are usually assumed to be. The EUVB is a particular problem as it may be uncertain especially at low redshift (Kollmeier et al. 2014), introducing up to an order of magnitude systematic error into some ionic abundances (Oppenheimer & Schaye 2013b). \nThough O vi is among the strongest and most frequently detected CGM metal absorption line, it amply demonstrates the problems encountered in precisely constraining the exact physical origins of ionized gas. For example, absorption-line studies in high-resolution and high-S/N QSO spectra and complementary studies of HVCs around the MW show that the ionization mechanisms of O vi are both varied and complex over a wide range of environments (e.g. Sembach et al. 2004; Tripp et al. 2008; Savage et al. 2014). Ionic column density ratios and line profiles sometimes support a common photoionized origin for O vi , N v , and low-ion gas (e.g. Muzahid et al. 2015), while other systems require O vi to be collisionally ionized in a ∼ 10 5 . 5 K plasma (e.g. Tumlinson et al. 2005; Fox et al. 2009; Tripp et al. 2011; Wakker et al. 2012; Narayanan et al. 2011; Meiring et al. 2013; Turner et al. 2016). Often, the multiple components for a single absorber show both narrow and broad absorption lines consistent with both scenarios. \nAll these thorny issues with ionization modeling highlight the difficultly of getting at the detailed 'sub-grid' physics of a complex, dynamic, ionized medium. We should maintain a cautious posture toward conclusions that depend sensitively on exact ionization states. Much of the detailed physics is still at scales that we cannot yet resolve. Nevertheless, in Section 5 we will see what we can learn by simplifying the situation to the most basic classes of models and proceeding from there.", '4.5. Gastrophysical Models': "The 'Galactic Corona' began with Spitzer's insight that cold clouds could be confined by a hot surrounding medium. This model has matured over the years into a strong line of theoretical research focused on the detailed physics of how the thermal, hydrodynamic, and ionization state of CGM gas evolves in dark matter halos. Placing multiphase gas into the context of the dark matter halo, Maller & Bullock (2004) suggested cold clouds cooled out of thermal instabilities in a hot medium, while maintaining rough pressure equilibrium \n(though see Binney, Nipoti & Fraternali (2009) for a counterpoint). Accretion may also be seeded by gas ejected from the disk, as in the 'galactic fountain' or 'precipitation' model (Fraternali & Binney 2008; Voit et al. 2015b, e.g.,). These scenarios start with very simple assumptions-such as hydrostatic hot halos, diffuse clouds in photoionization equilibrium, or particular radial entropy profiles. These simplifying assumptions are necessary because we do not know the large-scale pjhysical state of the CGM as a whole. Photoionization modeling of the low-ionization CGM using only the EUVB (Haardt & Madau 2001) strongly disfavors hydrostatic equilibrium with hot gas at T vir (Werk et al. 2014); the cool and hot phases appear to have similar densities, rather than similar pressures. Furthermore, if O vi -traced gas follows a hydrostatic profile at the temperature where its ionization fraction peaks, T ∼ 10 5 . 5 K, then its column density profile would be significantly steeper than observed (Tumlinson et al. 2011). There may be other means of supporting this gas, such as turbulence (Fielding et al. 2016), cosmic rays (Salem, Bryan & Corlies 2016), or magnetic fields. \nAdding to the uncertain physical conditions in the CGM is the fact that O vi likely represents a massive reservoir of warm gas ( § 5.2.3). Such a massive reservoir is apparently at odds with the short cooling times for O vi given by typical CI models; these timescales are often much shorter than the dynamical time, on the order of glyph[lessorsimilar] 10 8 yr. Yet, the short cooling times for O vi are in fact characteristic of many models for the multiphase CGM. In many formulations, the cooler low-ion traced gas precipitates out of the warmer O vi -traced phase, owing to thermal instabilities (Shapiro & Field 1976; McCourt et al. 2012; Voit et al. 2015b; Thompson et al. 2016; see also Wang 1995), while the O vi -traced gas may be continually replenished by a hot galactic outflow. In a similar vein, the O vi -traced warm gas could be cooling isochorically out of a hotter halo (e.g. Edgar & Chevalier 1986; Faerman, Sternberg & McKee 2017) but overcome its short expected lifetime by extra energy injection from star formation or AGN. \nFully understanding the broader context and origin of the multiphase CGM will require more than microphysical and phenomenological models alone can offer. Cosmological hydrodynamic simulations with self-consistent cosmic accretion and multiphase outflows are key to deciphering the panoply of observed absorption lines ( § 7). Moreover, much of the microphysics proposed as a natural source or maintainer of multiphase gas (e.g., thermal instabilities and turbulence) requires resolutions much higher than can be achieved by simulations that must simultaneously model a the enormous dynamic range required for galactic assembly. Yet essentially all cosmological hydrodynamic simulations do produce a multiphase CGM (see, e.g., Figure 3). In general, the combination of the simulated density and temperature profiles of the CGM results in different ions preferentially residing at different galactocentric radii, with low-ions preferring the denser, cooler inner CGM and higher ions filling the lower density, hotter outer CGM (Hummels et al. 2013; Ford et al. 2014; Suresh et al. 2015; see also Stern et al. (2016)). Yet inhomogeneous mixing of the different gas phases complicates predictions for gas cooling rates and the small-scale metal mixing which depend crucially on the unknown diffusion coefficient (Schaye, Carswell & Kim 2007). \nHydrodynamic simulations may be compared directly to observations via synthetic spectra, potentially helping to disentangle the degeneracy between physical space and observed velocity space. Constructing these synthetic spectra, however, faces many of the same challenges as modeling the ionization states of the observed gas: while the density, temperature, and metallicity of the simulated gas may be known, the EUVB and ionization mechanism must still be assumed in order to calculate ionization states (see, e.g., Hummels, Smith \n& Silvia 2016). Most simulations rely on the same radiative transfer codes (e.g., Cloudy , Ferland et al. 2013) that observational analyses do, though non-equilibrium chemistry and cooling are being included as computation power increases (Oppenheimer & Schaye 2013a; Silvia 2013). If these assumptions are incorrect, comparisons of derived results (such as masses) rather than observables (such as column densities) may lead to simulations getting the 'right answer' for the wrong reasons.", '5.1. The Missing Baryons Budget': 'Empirically constraining the total CGM mass as a function of stellar and/or halo mass is essential to quantifying models of galactic fueling and feedback. Under the condition Ω b /Ω m = 0 . 16 (Planck Collaboration et al. 2013), the total baryonic budget of subL ∗ to superL ∗ galaxies spans two orders of magnitude, ranging from 10 10 . 3 -10 12 . 3 M glyph[circledot] . Although the stars and ISM for superL ∗ galaxies are similar fractions of the total ( ∼ 5%), the absolute amount of mass that must be found is around 100 × larger for subL ∗ galaxies and 10 × larger for L ∗ galaxies. How much of this 80-90% missing mass is in the CGM? We organize this subsection by temperature, and review the observations, assumptions, and uncertainties in each calculation, using Figures 7 and 8 to synthesize current results. We note that a recent review by Bland-Hawthorn & Gerhard (2016) performed a similar radially-varying mass-budget compilation for the Milky Way and its halo and incorporates some of these same results. \nThe baryon census as presented here relies on the assumption that galaxies fall along well-defined scaling relations of ISM and CGM gas mass as a function of stellar mass, and that the scatter in these scaling relations is uncorrelated. We caution that there is tentative evidence that this is not necessarily the case: COS-GASS has shown galaxies with more cold gas in their ISM have more cold gas in their CGM (Borthakur et al. 2015). While the correlation between CGM and ISM exhibits a high degree of scatter, likely from patchiness in the CGM, it exists at > 99 . 5% confidence, and stacked Ly α profiles for low and high ISM masses clearly show the effect. The large-scale environment and gaseous interstellar content are difficult to explicitly account for in overall baryon budgets, and may account for some of the scatter in the various estimates. For example, Burchett et al. (2015) find that the detection of C iv around galaxies with M glyph[star] > 10 9 . 5 M glyph[circledot] drops significantly for galaxies in high-density regions (see also, Johnson, Chen & Mulchaey 2015a). Future work should control for these properties.', '5.2. CGM Masses by Phase': "5.2.1. Cold Gas, T < 10 4 K. Cold-gas tracers consist of neutral and low ions like H i , Na i , Ca ii , and dust. This is material that may have cooled from hotter phases that experienced thermal instability, or may arise in clouds entrained in multiphase outflows. Putman, Peek & Joung (2012b) estimated the total cold gas mass traced by HVCs in the Milky Way halo to be M = 2 . 6 × 10 7 M glyph[circledot] (including only HVCs detected via 21 cm emission, and excluding the Magellanic Stream system). The Magellanic Stream provides an additional contribution of ∼ 3 × 10 8 M glyph[circledot] , but it cannot be assumed to be a generic feature of galaxies. Thus, the total contribution from cold gas is M glyph[lessorsimilar] 10 9 M glyph[circledot] even if the ISM of the Clouds are included, making up less than 1% of the missing baryons for a Milky-Way like halo. We further note \nthat while dust masses have been estimated from stacks of reddened background QSOs (M'enard et al. 2010) and galaxies as 'standard crayons' (Peek, M'enard & Corrales 2015) indicating values comparable to the dust in the ISM of these galaxies (see § 6.3), both ISM and and CGM dust are at most only ∼ 1% of the missing halo baryons. Finally, using stacked optical spectra from SDSS, Zhu et al. (2013) derived a column density profile for gas bearing Ca ii H and K around ∼ L ∗ galaxies. For the purposes of Figure 5 we have converted this to a mass density profile, conservatively assuming that the calcium is entirely in Ca ii and Z = Z glyph[circledot] . The total mass for Ca ii itself is 5000 M glyph[circledot] , and when we scale to Z = Z glyph[circledot] we derive M = 2 × 10 8 M glyph[circledot] for the cold component, again ∼ 1% or less of the baryons budgets.", '4 -5': "5.2.2. UV Absorption Lines and the Cool 10 K CGM. The mass of the cool CGM ( ∼ 10 4 -5 K) is perhaps the best constrained of all the phases at low redshift, owing to the rich set of UV lines in this temperature range. Prior to COS, estimate for this phase were based on single ions with very simple ionization and metallicity corrections to arrive at rough estimates. Prochaska et al. (2011b) estimated M cool ≈ 3 × 10 10 M glyph[circledot] for all galaxies from 0.01 L ∗ to L ∗ , assuming a constant N H = 10 19 cm -2 out to 300 kpc. Using a 'blind' sample of Mg ii absorbers, Chen et al. (2010) estimated M cool ≈ 6 × 10 9 M glyph[circledot] for the Mg ii -bearing clouds alone. The former estimate simply took a characteristic ionization correction, while the latter counted velocity components as clouds and converted from a metal column density to N H using a metallicity, because neither study had the multiphase diagnostic line sets that could be used to self-consistently constrain gas density and metallicity. Both L ∗ and superL ∗ galaxies have provided the most reliable constraints, mainly due to their relative ease of detection in photometric and spectroscopic surveys at z < 0 . 5 (Chen & Mulchaey 2009; Prochaska et al. 2011b; Werk et al. 2012; Stocke et al. 2013). \nWith COS, it became practical to build statistically significant samples of absorbers that cover a broader range of ions. These estimates still rely on photoionization modeling, carried out under the standard assumption that the low-ions and H i trace cool ( T < 10 5 K) gas and the primary source of ionizing radiation is the extragalactic UV background (UVB). Using the COS-Halos survey, Werk et al. (2014) addressed the mass density profile and total mass for L ≈ L ∗ galaxies with PIE models that derive self-consistent n H and Z using a range of adjacent ionization states of low-ion absorption lines (primarily C ii , C iii , Si ii , Si iii , N ii , and N iii ). The resulting surface density profile appears in Figures 7, and yields M cool = 6 . 5 × 10 10 M glyph[circledot] for L ∗ galaxies out to R vir . Using the same COS-Halos sample with new COS spectra covering the Lyman limit, and taking a non-parametric approach with a robust treatment of uncertainties, Prochaska et al. (2017) recently refined the cool CGM mass estimate to be 9.2 ± 4.3 × 10 10 M glyph[circledot] out to 160 kpc. Stocke et al. (2013) used the complementary approach of estimating of individual cloud sizes and masses, along with their average volume filling factor, for galaxies in three luminosity bins ( < 0 . 1 L ∗ , 0 . 1 -1 L ∗ , and L > L ∗ ). They find volume filling factors that range from 3-5% for their modeled clouds, with length scales (N H / n H ) ranging from 0.1-30 kpc, totaling log M cool = 7 . 8 -8 . 3, 9 . 5 -9 . 9, and 10 -10 . 4, respectively. Finally, Stern et al. (2016) determine the total mass in the cool (and possibly warm CGM) of 1 . 3 ± 0 . 4 × 10 10 M glyph[circledot] for L ∗ galaxies given their 'universal' cloud density profile. In this phenomenological model each ion occupies a shell of a given n and T such that the fraction of gas in that particular ionization state is maximized. Thus, this calculation represents a conservative minimum of baryons that must be present. These ranges are shown in Figure 8. \nFor superL ∗ galaxies, Zhu et al. (2014) use stacking techniques to estimate the correlation function between luminous red galaxies with a mean stellar mass of 10 11 . 5 M glyph[circledot] and cool gas traced by Mg ii absorption in SDSS data for ∼ 850,000 galaxies with 0 . 4 < z < 0 . 75. The cool CGM around massive galaxies calculated in this way appears to completely close the CGM baryon budget for superL ∗ galaxies, at 17% of the total halo mass. The assumptions for metallicity and ionization corrections, however, make it uncertain. \n5.2.3. UV Absorption Lines and the Warm 10 5 -6 K CGM. In Figure 6, it appears as though ions like C iv , N v , O vi , and Ne vii trace the warm CGM at T ≈ 10 5 -6 K. However, this temperature range in particular is burdened by significant uncertainty in the precise ionization mechanism responsible for its purported ionic tracers (see § 4.4). If highions are partially photoionized, O vi for example, may trace a non-negligible fraction of T < 10 5 K gas that has already been counted toward the total baryon census in the previous section. For gas traced by O vi , Werk et al. (2016) point out that typical photoionization models like those used for the low-ions have difficulty accounting for the total column of O vi and column density ratios of N v / O vi without the need for path lengths in excess of 100 kpc. However, significant additional ionizing radiation at ∼ 100 eV may reduce this requirement. \nIn general, CIE models require a very narrow range of temperature to reproduce the O vi observations, T = 10 5 . 3 -5 . 6 K (Tumlinson et al. 2011; Werk et al. 2016). Furthermore, the kinematics of O vi relative to the low-ions, in particular large b values, seem to naturally support the idea that the O vi is in a hotter phase (Tripp et al. 2011; Muzahid et al. 2012; see also Tripp et al. 2001; Stern et al. 2016). Tumlinson et al. (2011) found that O vi traces a warm CGM component that contributes > 2 × 10 9 M glyph[circledot] of gas to the L ∗ baryon budget. This mass estimate is strictly a lower limit due to the conservative assumptions adopted: (1) solar metallicity; (2) the maximum fraction of oxygen in O vi allowed by CIE models, 0.2, and (3) the CGM sharply ends at 150 kpc. We adopt log M warm = 10 . 0 in Figure 8 for the COS-Halos galaxies (see also Faerman, Sternberg & McKee 2017). \nFor subL ∗ galaxies, Bordoloi et al. (2014b) estimate M warm using C iv . As these galaxies are at z < 0 . 1, the COS spectra do not cover the full range of Lyman series lines and ions available at z > 0 . 1, hindering detailed ionization modeling. COS only covers O vi at z > 0 . 2, where it is difficult to assemble statistically significant samples of confirmed subL ∗ galaxies, so an O vi -based mass estimate for low-mass galaxies is not currently possible. With these caveats in mind, assuming a limiting ionization fraction for C iv , Bordoloi et al. derive log M warm = 9 . 5, if the gas typically has solar metallicity. For gas with lower metallicity, e.g., 0.1 solar, the value is 10 times higher and rather closer to baryonic closure for subL ∗ galaxies (Figure 8). We caution that for C iv , detailed photoionization often places C iv with low-ionization state gas rather than with high-ionization state gas (e.g., Narayanan et al. 2011). Thus, the C iv -derived mass for subL ∗ galaxies is highly uncertain without detections of additional ionization states. \nOne of the most surprising results to emerge from Tumlinson et al. (2011) is that O vi appears to be absent around the non-star-forming, more massive galaxies in the COS-Halos sample. Thus, there is tentative evidence that ∼ 10 5 . 5 K gas is not a major component of the CGM of superL ∗ galaxies, which may be a result of massive galaxies having generally hotter halos or non-equilibrium cooling (Oppenheimer et al. 2016b). Thus, we do not have a good observational constraint for the warm CGM baryonic content for superL ∗ galaxies. The extreme-UV ion Ne viii redshifts into the COS band at z > 0 . 5, where a few detections \nFigure 7 \n<!-- image --> \nA synthesis of CGM mass density results for 'cold gas' (pink, Zhu & M'enard 2013b), 'cool gas' (purple, Werk et al. 2014), 'warm gas' traced by O vi (green, Tumlinson et al. 2011; Peeples et al. 2014), X-ray emitting gas (yellow, NGC1961, Anderson, Churazov & Bregman 2016), and dust (brown, M'enard et al. 2010). An NFW profile for M DM = 2 × 10 12 M glyph[circledot] is at the top in black. \n(Tripp et al. 2011; Meiring et al. 2013) hint that it may be present in halos out to 100200 kpc. However, the number of absorbers associated with particular galaxies is not yet sufficient to include it in mass estimates for the warm phase. \n5.2.4. The Hot T > 10 6 K Phase. Hot gas at the virial temperature ( T vir = G M halo m p / kR vir ) is a long-standing prediction. For M halo glyph[greaterorsimilar] 10 12 M glyph[circledot] , the temperature should be T glyph[greaterorsimilar] 10 6 K, and observable at X-ray wavelengths, although there are extreme-UV tracers such as Mg x and Si xii that have yet to yield positive detections (Figure 4). Only a few very luminous spirals and ellipticals have had their halos detected (Anderson & Bregman 2011; Dai et al. 2012; Bogd'an et al. 2013; Walker, Bagchi & Fabian 2015; Anderson, Churazov & Bregman 2016), and independent constraints the temperature, density, and metallicity profiles from soft X-ray spectroscopy is rarer still. Thus the fraction of baryons residing in the hot phase, and its dependence on stellar and or halo mass, are not yet determined. \nThree sets of constraints are relevant: the Milky Way, individual external galaxies, and \nstacked samples of external galaxies. Anderson & Bregman (2010) addressed directly the problem of whether hot gas could close the baryon budget for the Milky Way. From indirect constraints such as pulsar dispersion measures toward the LMC, cold gas cloud morphology, and the diffuse X-ray background, they limited the hot gas mass to M glyph[lessorsimilar] 0 . 5 -1 . 5 × 10 10 M glyph[circledot] , or only 2-5% of the missing mass. The choice of an NFW profile for the hot gas is a key assumption: if the density profile is assumed to be flatter ( β ∼ 0 . 5), the mass can be 3-5 times higher, but still only 6-13% of the missing baryons. The Gupta et al. (2012) claims that the baryon budget is closed for the Milky Way, based on the assumption of an isothermal, uniform density medium, have been questioned by evidence that the gas is neither isothermal nor of uniform density (Wang & Yao 2012). \nThe well-studied case of NGC 1961 (Anderson, Churazov & Bregman 2016) constrains the hot gas surface density out to R glyph[similarequal] 40 kpc, inside which M hot = 7 × 10 9 M glyph[circledot] compared with the stellar mass of 3 × 10 11 M glyph[circledot] and far from baryonic closure. Extrapolating to 400 kpc yields M hot = 4 × 10 11 M glyph[circledot] , but given the declining temperature profile it is likely that it declines to more intermediate temperatures, T glyph[lessorsimilar] 10 6 K, where EUV and FUV indicators provide the best diagnostics. Stacked emission maps of nearby galaxies provide the strongest evidence for extended hot halos. In a stack of 2165 isolated, K-selected galaxies from ROSAT, Anderson, Bregman & Dai (2013) found strong evidence for X-ray emission around early type galaxies and extremely luminous galaxies of both early and late type. The X-ray luminosity depends more on galaxy luminosity than on morphological type. Luminous galaxies show M = 4 × 10 9 M glyph[circledot] within 50 kpc, and M = 1 . 5 -3 . 3 × 10 10 M glyph[circledot] if extrapolated out to 200 kpc, comparable to the stellar masses. Yet high amounts of hot gas this far out would appear to be excluded by Yao et al. (2010), who stacked Chandra spectra at the redshifts of foreground galaxies and placed strict ( glyph[lessorsimilar] 1 m ˚ A) limits on O vii and O viii . The limits are also consistent with the limits on nearby galaxy emissivity earlier derived by Anderson & Bregman (2010). The key uncertainty is how far out the hot gas extends with the flat, β ∼ 0 . 5 density profile seen at R glyph[lessorsimilar] 50 kpc, but the Yao et al. (2010) limits imply that hot gas halos around nearby galaxies appear to host at most glyph[similarequal] 10 10 M glyph[circledot] . In their summary of the X-ray results, Werk et al. (2014) adopted M hot = 1-14 × 10 9 M glyph[circledot] from Anderson, Bregman & Dai (2013). \nThe thermal SZ effect-scattering of CMB photons by free electrons in a plasma-may constrain the hot gas content of galaxy clusters and halos down to the galactic scale. Planck Collaboration et al. (2013) and Greco et al. (2015) claim detections down to M glyph[star] = 2 × 10 11 M glyph[circledot] and a possible signal down to M glyph[star] = 6 × 10 10 M glyph[circledot] . These results create tension with the X-ray measurements, since the SZ detections imply a 'self-similar' relation between M halo and M hot down from the cluster scale ( M halo ∼ 10 14 M glyph[circledot] ), where we know hot baryons close the budgets, into the galactic range where this is much less clear. It may be that the hot gas extends well beyond the X-ray surface brightness limits at 50 kpc up to the Mpc scales where the SZ effect is measured. On the other hand, if every ≥ L ∗ halo was filled with T vir gas, it would violate constraints from the soft X-ray background (Wu, Fabian & Nulsen 2001). If halos with M halo glyph[lessorsimilar] 10 11 M glyph[circledot] depart from self-similarity, the cause could be the cooling and feedback that cause prevent halos from reaching their cosmic share of baryons. The kinematic SZ effect-in which photons receive a Doppler shift when scattering of a plasma with bulk motion-may be able to reach even lower masses for halo gas measurements (Hill et al. 2016). This work is in its early stages and we look forward to more progress that complements the UV and X-ray. \nCMB: \nCosmic \nMicrowave Background \nSZ: \nSunyaev-Zeldovich \nFigure 8 \n<!-- image --> \nUpper left: An accounting of CGM baryon budgets for all physical phases. The solid bars show the minimum values, while the hatched regions show the maximal values. The other three panels show simulated baryon budgets from Ford et al. (2014) in the upper-right, Illustris (Suresh et al. 2017) in the bottom left, and in the bottom right, the EAGLE halo shown in Figures 2 and 6 (Schaye et al. 2015; Oppenheimer et al. 2016b). \n5.2.5. Theoretical Considerations. From the discussion above and the synthesis in the top left panel of Figure 8, we see that CGM measurements have added significantly to the baryon budgets for galaxies, and may complete those budgets under some assumptions. There has been theoretical progress as well: hydrodynamical simulations generally agree that the CGM contains a budget of baryons at the same order of magnitude as the stellar masses. In the other three panels of Figure 8, we show there is less quantitative agreement for the temperature partitioning of the CGM as a function of stellar mass, despite these models having approximately the same predictions for the baryonic content of galaxies. \nA promising aspect of this quantitative disagreement is that different physical treatments of energetic and/or kinetic feedback do indeed lead to different total baryon fractions, and in particular to different trends in the fraction by phase. Thus, observations of how CGM gas masses are distributed by phase can favor or disfavor particular physical prescriptions, and thus already offer phenomenological tests of models. However, these comparisons additionally show how challenging it will be to perform stringent tests. Even where simulations with radically different physical prescriptions yield opposite trends, at any particular mass they only different by factors of glyph[lessorsimilar] 2 in the fraction of any phase. At present, this range is comparable to the systematic errors remaining in the observational characterization of the phases. Thus any claims that the data favors or disfavors any particular model should be made and interpreted carefully. As discussed in § 4.5, comparing the models to observations by using synthetic data and directly comparing observables such as column densities and line kinematics have the benefit of shifting the myriad assumptions discussed in § 4.2 onto the simulations.", '6.1. The Metals Census': "Total mass budgets by themselves do not fully reveal the flows that govern galaxy evolution. However, there is a ready means of distinguishing inflows from outflows: stars produce heavy elements sending passively-advecting 'tracer particles' out into the ISM, CGM, and IGM from stellar winds and supernovae. The metal content of galactic flows can help identify their origins and determine their fate, and break degeneracies between models matched to the four galaxy problems. The galactic metals census ( § 2) requires that we compute the total budget of 'available metals' produced by the galaxy by z = 0. This census was performed by Peeples et al. (2014) by compiling measurements on stars, ISM and CGM gas, and dust. As shown in Figure 9, the contributions bound in stars (red), interstellar gas (blue), and interstellar dust (orange)-the metals inside galaxies-add up to only consistently 20-30% over a factor of ∼ 1000 in stellar mass. 1 Ideally, this census would be done for each element individually, with the CGM divided into each ionization state of that element, e.g., oxygen (Oppenheimer et al. 2016b), but as that is observationally not yet generally feasible, the ionization corrections discussed in earlier sections must instead be done to account for unobservable ionization states. Qualitatively similar results are seen in simulations that have addressed this problem in particular (Muratov et al. 2016). This striking invariance must offer some important clues to the operation of galactic outflows \nFigure 9 \n<!-- image --> \nLeft: A metals census of the CGM around star-forming z ∼ 0 galaxies following Peeples et al. (2014), including a subL ∗ budget from Bordoloi et al. (2014b). As in Figure 7, stars are red, ISM gas is blue, ISM dust is orange the cool CGM is purple, the O vi -traced CGM is green, the X-ray traced CGM is yellow, and intergalactic dust is in brown. Right: A simulated budget from 55 relatively isolated log M glyph[star] ≥ 8 . 5 star-forming EAGLE halos, with a moving average smoothing (Oppenheimer et al. 2016b). In both panels, the denominator is the total mass of metals ever produced by the central galaxy; the CGM may have contributions from, e.g., satellites. \nand inflows, with potentially large implications for the processes of galaxy fueling, feedback, and recycling.", '6.2. Metals Observed as Gas': 'Even Lyman Spitzer might have recognized that the heavy elements observed in the CGM are in some sense the cause of, and solution to, all our problems. Apart from the (problematic) series of Lyman lines in the rest-frame FUV, virtually all our knowledge of the physical state, mass, kinematics, and evolution of the CGM gas come from lines of C, N, O, Si, Fe, Mg, Ca, and so on, whether they appear in the UV or X-ray. Yet, as described in Section 4, these critical diagnostics also present many problems of analysis and interpretation. To work through this, it helps to distinguish between measurements of metal content or metal mass on the one hand and metallicity on the other. This distinction hinges on whether or not the hydrogen content can be measured, which is notoriously difficult. Measurements of hydrogen suffer severe H i saturation effects, and juggling both metals and hydrogen compounds the difficulties of ionization corrections. When considering metal mass, we can often tolerate simpler ionization corrections or even direct sums of metal ion surface densities, sidestepping the large ionization corrections for H i ( § 4.4). \nThe COS-Halos survey (Tumlinson et al. 2011) used the O vi line observed with COS in a way that typifies measurements of metal content rather than metallicity. Their basic empirical finding is that O vi appears at column densities of log N OVI glyph[similarequal] 14-14 . 5 out to the \n150 kpc limits of the survey. Since O vi does not reach more than 20% of the total oxygen in most ionization conditions, they were able to place a robust lower limit of > 10 7 M glyph[circledot] of total oxygen for star forming galaxies. As it comes from direct integration of surface densities for a heavy element, does not refer to H, and uses a limiting ionization correction, this estimate avoids some of the trickiest aspects of metallicity measurements, and yet has significant implications for the budgets of galactic metals (Peeples et al. 2014). The O vi traces a high ionization component of the CGM gas; adding lower ionization gas to the budget requires the more complex ionization corrections and assumed relative abundances of oxygen and, e.g. Mg and Si, though it does not require the H i -dependent metallicity corrections that plague the baryon census. Altogether, 20-30% of available metals have been located in the R < 150 kpc CGM around ∼ L ∗ galaxies. \nBy contrast with the measurements of total metal mass, bona fide metallicities require robust measurements of the hydrogen surface density, which entails accurate measurements of N HI and reliable ionization corrections. For most strong CGM absorbers at z glyph[lessorsimilar] 0 . 2, the Lyman series lines are saturated and do not yield reliable H i column densities. However, beyond this redshift, and at log N HI > 16 . 2, Lyman limit systems enable adequately precise ( ± 0 . 2 -0 . 3) measurements of N HI and the ionization corrections are manageable. \nBy building a sample of LLSs from high-quality COS sightlines, Lehner et al. (2013) and Wotta et al. (2016) found that the distribution of metals in LLS clearly exhibits two peaks near 4% solar and 50% solar metallicity (Figure 10a). The metallicities are constrained by detections of low-intermediate ions such as C II-IV, Si II-IV, OII-IV, and Mg ii . This bimodal distribution qualitatively matches with expectations that accretion from the IGM into halos will have low metallicity, while accretion of gas previously ejected will have higher metallicity. The relative absence of intermediate values challenges our intuition that gas should naturally mix over time into a continuous distribution, and has posed a challenge to simulations (Hafen et al. 2016, etc.). But most of these systems have not yet been identified with galaxies. In contrast to the Lehner bimodality, Prochaska et al. (2017) find a unimodal distribution of metallicities within 160 kpc of L* galaxies with a median of ∼ 30% solar. These metallicities derive from tight constraints on N HI around L ∗ COS-Halos galaxies with well-defined masses and distance to the absorber. The contrast between the absorberselected Lehner et al. sample and the galaxy-selected COS-Halos sample may indicate that they arise in other selection effects, but it may also indicate variation in CGM metallicity in different subsets of the galaxy population. \nBy mining the Keck database of highz QSO absorbers, the KODIAQ survey studied a sample of LLSs at z > 2 (Lehner et al. 2016). This sample is shown in the left panel of Figure 10 compared to the expanded low-z sample of Wotta et al. (2016). The z > 2 distribution is unimodal and centered at [X/H] ∼ -2. A similar result was obtained for two samples of LLSs at still higher redshift, z = 3 . 5 -4, with unimodal distributions centered at [X/H] ∼ -2 . 5 (Glidden et al. 2016; Cooper et al. 2015). This is near the bottom edge of the low-metallicity peak at z < 1, indicating evolution in the average metallicity of high-column CGM over the few Gyr interval. Somehow, the bimodality emerges long after the initial buildup of metals, and is noticeable only in the z < 1 sample. Note that neither of these samples has specific galaxies attached-both are selected based on HI alone and the galaxies will have to be identified later. It is also possible that the column density range used for selection traces different galaxy masses, radii, and total column densities at the different redshifts, and so the apparent evolution does not occur in the same type of physical system (owing to a higher mean cosmic density). Nevertheless, it is now possible to compare the \nFigure 10 \n<!-- image --> \nTwo views of CGM metallicity: (a) Two LLS distributions from Lehner et al. (2013) and Wotta et al. (2016). This comparison clearly shows evolution in the LLS metallicities over time. (b) Trends in Mg ii and C iv line density per unit redshift: the low-ion Mg ii traces the cosmic star formation history, while C iv continually becomes more abundant. \ndistribution of CGM metallicities over ∼ 6-10 Gyr of cosmic time. \nIn particular, there are ever-increasing samples of z > 2 absorbers that do have associated galaxy information, allowing for a more direct comparison to the lowz COS studies (Figure 4). The Keck Baryonic Structure Survey (KBSS; Rudie et al. 2012) has engaged in a long campaign to characterize the CGM of star-forming galaxies at z ∼ 2 . 2, going back to pioneering studies of absorption associated with Lyman-break galaxies (Adelberger et al. 2003). These data show ion sets that overlap strongly with the lowz studies. Both H i and metals (O iv , N v , C iii , C iv , and O vi ) show strong statistical correlations with galaxies out to 100-300 kpc. Using stacking, Steidel et al. (2010) and Turner et al. (2015) examined the relative kinematics of metals and galaxies, finding essentially all outflow kinematics and little sign of inflow; there must be gas flowing in to mainain the observed star formation rates, but it may be occuring in thin filaments with low covering fraction. \nThese results across redshift can be viewed a different way, by examining the redshift evolution of strong lines that are likely to trace CGM gas. Figure 10b shows the comoving sightline density of Mg ii ( W rest ≥ 1 ˚ A) and C iv ( W rest ≥ 100 m ˚ A), which follow different trends at z < 2. The number density of strong Mg ii absorbers rises and then declines again toward z = 0. Absorbers above this limit occur within ∼ 100 kpc of galaxies (see Figure 4), so the resemblance of this curve to the cosmic SFR density (Hopkins et al. 2006) suggests that the strong Mg ii absorbers are linked to the fueling or feedback of star formation. Indeed, other evidence suggests that we are seeing the rise and decline of galactic superwinds (See 7.3). In contrast to the Mg ii , strong C iv absorbers continue their march upwards at low redshift. This trend in moderate-to-high ionization gas may indicate that ionized gas in occupying the bulk of the CGM volume becomes more common even as strong winds creating Mg ii absorbers decline with the cosmic SFR density.', '6.3. Metals Observed as Dust': "The interstellar medium is a mixture of gas and dust; this is no less true of the CGM. In a pioneering study, York et al. (2006) stacked a sample of 800 strong Mg II absorbers to find evidence of SMC-like dust reddening. M'enard et al. (2010) added the SDSS photometric galaxy catalogs to this style of analysis and found that the reddening extends over angular scales consistent with distances hundreds of kpc away from the luminous galaxies (7). To tie dust to specific galaxies and precise physical scales, Peek, M'enard & Corrales (2015) used passively evolving galaxies from SDSS as 'standard crayons' to examine the reddening imposed by foreground SDSS spectroscopic galaxies. They found a strong reddening effect out to 150 kpc in the bluest bands and a steeper drop past that radius than in the angular correlations of M'enard et al. (2010). The correlations with physical radius allow Peek, M'enard & Corrales (2015) to further estimate the typical total mass of dust for galaxies between 0 . 1-1 L ∗ of M dust glyph[similarequal] 6 ± 2 × 10 7 M glyph[circledot] . They found only a weak trends with stellar mass, M dust ∝ M 0 . 2 glyph[star] and no discernible trend with the galaxies' specific star formation rates. Thus the presence of dust in the CGM out to 100 kpc scales provides unambiguous evidence that the CGM is fed by galactic outflows, accounting for approximately 10% of the metals budget near L ∗ (Figure 9). This degree of reddening can be explained by outflows from normal star forming galaxies in simulations, provided the dust-to-gas ratio is similar to the Galactic value and the dust survives the trip (Zu et al. 2011). It is not yet clear why the dust properties show so little dependence on galaxy stellar mass, resembling the CGM H i and low ions more than the CGM high-ionization gas. It might be that the increasing reddening at low redshift indicate a steady buildup of metals in the CGM and a relative lack of recycling into future star formation. Dust observations could also be used to test the physical models of galactic outflows that employ radiation pressure on dust to drag gas out of galaxies (Murray, Quataert & Thompson 2005; Murray, M'enard & Thompson 2011). Further explorations of CGM dust promise to constrain galactic outflows and recycling in ways that complement studies of gas.", "7.1. The Problems: Galaxy Fueling and 'Missing' Metals": "Recent findings show that the CGM possesses a significant budget of baryons, but how are they feeding galaxies across the spectrum of galaxy masses (Figure 2)? Accreting gas passes through the CGM on its journey from the IGM to galaxies, where it presumably leaves some observable signatures that we can use to characterize the inflows. The rates of accretion onto galaxies and of outflow out of galaxies are crucial parameters in most models of galaxy evolution (Tinsley 1980). However, there is not agreement about where and how a galaxy's fuel source is regulated. It is often assumed gas inflow from the IGM is balanced by the sum of star formation, gas ejection as outflows, and any net buildup of gas in the ISM (Lilly et al. 2013; Dekel & Mandelker 2014; Somerville & Dav'e 2015). This formulation completely omits the role of the CGM, even at the phenomenological level, but this 'bathtub' model appears to nonetheless describe the many broad trends in galaxy scaling relations with redshift (Dekel & Mandelker 2014).Col These models, though they do not explicitly address the CGM's composition or physical state, nonetheless have specific implications for its content and evolution (e.g., Shattow, Croton & Bibiano 2015). Conversely, models that use physical principles to describe the regulation of flows between the CGM and ISM (Voit \net al. 2015b) can reproduce the same phenomenological galaxy scaling relations without detailed treatments of star formation inside galaxies. With observations of the CGM and its dynamics, we can potentially assess whether its role in regulating star formation is trivial, as the former models assume, or essential, as the latter models assume. Ideally, CGM observations would not only answer this question, but also reveal how it fuels star formation and manages outflows as a function of galaxy mass. \nThe observations we have discussed up to this point reveal the CGM (at low z ) as a massive gaseous medium with a rich internal kinematic structure that is, in bulk, consistent with being bound to the host galaxies. Yet the degeneracy between kinematics and the physical location of absorbing gas can easily get lost in transverse sightline observations. \nIn simulations the CGM can appear to have obvious and well-ordered large-scale structure, with accreting and outflowing gas occupying physically distinct regions such as filaments and biconical outflows (Shen et al. 2012; Corlies & Schiminovich 2016, see also Figure 3), but at low redshift, circumgalactic gas tends to be more well mixed, with instantaneous velocities having little bearing on the origin or fate of a particular pocket of gas (Ford et al. 2014; Muratov et al. 2015; Christensen et al. 2016), though this is also seen at z = 3 (van de Voort et al. 2012). In light of the observational projection effects, and theoretical cautions, we will now consider what can be learned from observing inflow and outflow directly in down-the-barrel observations, in which we interpret gas blueshifted relative to the galaxy as outflowing and redshifted gas as inflowing. These observations are better at probing gas in or near the disk-halo interface rather than the 'proper' CGM out in the halo. Considering them in conjunction with CGM finding from transverse sightlines promises insights into the dynamics of the CGM that are not otherwise available.", '7.2. Empirical Signs of Fueling and Inflows': "Gas accretion is perhaps the most fundamental process in their formation (Fox & Dav'e 2017), as they must acquire gas, but feedback is optional. In the prevailing theoretical paradigm, gas flowing into galaxies at glyph[lessorsimilar] 10 12 M glyph[circledot] should be dynamically and thermally cold, while more massive halos receive most of their baryons as hotter ( T > 10 5 ) gas (Dekel & Woo 2003; Dekel & Birnboim 2006; Kereˇs et al. 2005; Kereˇs & Hernquist 2009; Stewart et al. 2011, though see Nelson et al. 2013). Thus cold, dense, metal-poor CGM gas is often interpreted as direct evidence of accretion. First, cool, dense CGM gas is abundant in the form of LLSs. A large fraction of these are metal-poor at all redshifts Lehner et al. (2013); Glidden et al. (2016); Cooper et al. (2015). Metal-poor LLSs are evident as tracers of accretion in high resolution simulations (Fumagalli et al. 2011; Hafen et al. 2016). The cool, bound H i seen in the CGM of z ∼ 0 . 25 galaxies (Tumlinson et al. 2013) should have a short cooling time. Finally, the finding from COS-GASS that there is a correlation between interstellar and circumgalactic H i (Borthakur et al. 2015) implies a connection between circumgalactic fuel and star forming fuel. Though subL ∗ and dwarf galaxies have not yet had their 'cool' CGM masses measured directly, the widespread presence of Ly α at similar strength suggests they too possess significant budgets of cold halo gas. \nAll this evidence taken together strongly indicates that galaxies possess large reservoirs of CGM gas eligible for accretion. Yet evidence for fuel does not automatically constitute evidence for fueling : bound, cold gas has turned up in halos where its presence is surprising, such as the CGM of passive galaxies (Thom et al. 2012). The actual fate of this material is unclear: how can we claim the bound cold gas is fueling star forming galaxies but not \nthe passive galaxies? We therefore seek direct signatures of gas accretion onto galaxies. Yet these signatures are notoriously difficult to observe as incoming material may be metal poor, ionized, and obscured by outflowing material. Once gas is near the disk, proving empirically that it is accreting can be extremely difficult when it is seen in projection and its kinematics are easily confused with disk material. \nThe Milky Way itself provides direct and unambiguous evidence for inflow in the form of its blueshifted high-velocity clouds (HVCs) and the striking Magellanic Stream. The HVCs arise in many complexes of clouds lying within ∼ 10 kpc of the disk and have 100300 km s -1 blueshifted radial velocities that indicate they will reach the disk within 10 7 -8 yr. Their mass inflow rate falls between 0 . 1-0 . 5 M glyph[circledot] yr -1 , compared to the 1-2M glyph[circledot] yr -1 of star formation (Putman, Peek & Joung 2012b). These clouds are all detectable in 21 cm emission, meaning that they occupy the tip of the column density distribution of CGM gas seen around other galaxies. The inflow rate inferred for ionized gas is much larger than for the classical HVCs, ˙ M glyph[similarequal] 0 . 8 -1 . 4 M glyph[circledot] yr -1 (Lehner & Howk 2011), more comparable to the Milky Way's star formation rate. The Magellanic Stream is estimated to contain around 2 × 10 9 M glyph[circledot] of gas in neutral and ionized form Bland-Hawthorn et al. (2007); Tepper-Garc'ıa, Bland-Hawthorn & Sutherland (2015), and could provide ∼ 5 M glyph[circledot] yr -1 of gas to the Milky Way disk as it accretes (Fox et al. 2014). Unfortunately, HVCs both above and below the radio-detection threshold are difficult to detect in external galaxies, despite intensive searches (Putman, Peek & Joung 2012b), and satellites like the Magellanic Clouds and their Stream are not very common in L* galaxies. So we cannot generalize this result to mainstream galaxy populations. \nDown-the-barrel spectroscopy provides complementary information on inflows. Using this technique on z ∼ 0 . 5 galaxies with Keck spectroscopy and HST imaging, Rubin et al. (2012) detected clear signs of inflow at 80 -200 km s -1 in star forming galaxies of log M glyph[star] /M glyph[circledot] = 9 . 5 -10 . 5, inferring mass inflow rates of ˙ M glyph[greaterorsimilar] 0 . 2-3M glyph[circledot] yr -1 . It seems likely that these estimates significantly undercount inflow, since inflowing (redshifted) gas is often obscured by outflows (blueshifted) or by emission from the galaxy's ISM (this problem is esepcially noticeable at higher redshift, Steidel et al. 2010). Even if outflow is not present, the profiles are not sensitive to accretion from the lower half of the bimodal LLS metallicity distribution (Lehner et al. 2013), which could make up a large fraction of the available cold CGM gas. Recently, Zheng et al. (2017) reported the detection of enriched, accreting gas at the disk-halo interface of M33 via COS observations of SiIV absorption along several sightlines to bright O stars in the disk. Their kinematic modeling of the observed absorption features implies an accretion rate of 2.9 M glyph[circledot] yr -1 . While these results provide evidence for accretion of cold, metal-enriched gas directly into galaxy disks, evidence for more metal-poor 'cold-mode' accretion, and for gas entering further out in the disk ('on-ramp', Figure 1), is still lacking (though see Bouch'e et al. 2013), as is empirical characterization of how accretion rates vary with galaxy mass.", '7.3. The Preeminence of Outflows': "By consensus, outflows are an accomplice if not the perpetrator in each of the problems outlined in § 2. The existence of outflows is not in question: the large share of metals outside galaxies provides incontrovertible evidence for them ( § 6). COS-Halos found widespread O vi around star-forming galaxies-extended to ∼ 300 kpc by Johnson, Chen & Mulchaey (2015b)-but could not show that this ion becomes more prevalent with SFR. Even so, \nsimulations found that robust outflows were necessary to produce the observed reservoir of metals (e.g., Hummels et al. 2013; Ford et al. 2013; Suresh et al. 2015), such that the high metal ions provide a significant constraint on the time-integrated effects of outflows even if it does not show the effects of recent or ongoing outflows directly. After that, the important questions concern how they transport baryons, metals, momentum, energy, and angular momentum. There is empirical evidence and strong theoretical suggestions that the physical drivers and properties of galaxy winds-their velocity, mass loading, metal content, and likelihood of escape-depends on galaxy mass, circular velocity ( v circ ), star formation rate, and metallicity. Many investigators pursue CGM observations in the hope that they can help to constrain these outflows and how they scale with galaxy properties. \nDirect observational evidence for outflows is readily available at all redshifts (see Veilleux, Cecil & Bland-Hawthorn 2005, for a review). In the nearby universe, large-scale complex multiphase outflows are seen in starbursts (e.g., M82) and from the Milky Way's central regions (Fox et al. 2015). Down-the-barrel spectroscopy of the Na i D in local starbursts (Martin 2005) found that outflow velocities depend linearly on v circ . Rubin et al. (2012) and Bordoloi et al. (2014a) characterized similar flows using Mg ii at z ∼ 1. At z > 2, where the FUV-band ions used at z ∼ 0 appear at visible wavelengths, Steidel et al. (2010) used down-the-barrel spectroscopy to detect nearly 'ubiquitous' outflows in rapidly star-forming LBG galaxies, with no clear indications for redshifted inflow. While these results help constrain the mass loading and covering fraction of outflows, they do not show how far these winds propagate into the CGM. It may be that the bulk of the energy is transported out in the hot gas while the bulk of the mass leaves in the cold phase, but this is still an open question (Strickland & Heckman 2009). \nAbsorbers on transverse sightlines can directly constrain the impact of winds on the CGM. Cross-correlations of Mg ii absorbers with the orientation of galaxies on the sky at z glyph[lessorsimilar] 1, from both samples of individual galaxies (Kacprzak et al. 2012; Mathes et al. 2014) and stacked spectroscopy (Bordoloi et al. 2011; Zhu & M'enard 2013b) find that the strongest absorbers prefer the semi-minor axis of disk galaxies, as expected for biconical outflows emerging from the disk. The preference for the semi-minor axis disappears by ∼ 60 -80 kpc, indicating that winds propagate at least that far, or merge into the general medium near that radius (e.g., the z = 2 example in Figure 3). Studies of outflow covering fractions at z ∼ 1 reinforce a picture of outflows being roughly biconical, with little surface area ( ∼ 5%) solely dedicated to inflow (Martin et al. 2012; Rubin et al. 2014). Another strong clue about outflows comes from examining the CGM of starburst and post-starburst galaxies. Using an SDSS-selected sample, Heckman & Borthakur (2016) found unusually strong H i and multphase ions at 100 -200 kpc compared with the COS-Halos and COSGASS samples of galaxies at lower SFRs. These studies collectively show that SFR is a factor in determining the content of the CGM, perhaps as far out as R vir . \nDown-the-barrel measurements tell us that outflows are ubiquitous, and sightline measurements tell us that they reach 100 kpc scales. Together these findings suggest that a large part of the CGM is made of outflows, and to examine one is to illuminate the other. The open questions concern not only the basic scaling of velocity and mass loading with galaxy v circ -which has received much attention-but just as importantly the distribution of outflow temperatures, metallicities, and fate. These cannot (yet) be simulated from first principles but can be constrained by the combination of CGM and down-the-barrel observations. The former constrain the radial extent and the velocity fields of multiphase gas far from the disk, while the latter constrain the initial velocities, mass loading, and (possibly) \nmetallicities. \nA recent goal of models and simulations has been to discriminate between winds that are 'momentum-driven' (Murray, Quataert & Thompson 2005), which appear to improve the match of simulations to the galaxy mass-metallicity relation (Finlator & Dav'e 2008) and the metal content of the IGM (Oppenheimer & Dav'e 2006, 2008), and those that are 'energy-driven' (Murray, M'enard & Thompson 2011), which appear to better match the galaxy stellar mass function (Dav'e, Finlator & Oppenheimer 2012a) and new COS data (Ford et al. 2016). A momentum-driven outflow has a velocity v w ∝ v -1 circ , while an energy-driven flow has much faster outflows for low-mass galaxies with v w ∝ v -2 circ ; with a fiducial wind speed of ∼ 100kms -1 , an unimpeded flow reaches 100 kpc in only 1 Gyr, i.e., the scales on which metals are seen in the CGM ( § 6). Thus understanding the history of CGM metals and the velocities and mass flow rates of galactic flows go hand-in-hand. Real winds may depend less on the local potential well and more on the local star formation rate surface density (Kornei et al. 2012; Heckman et al. 2015). New hydrodynamic simulations of galaxies that resolve the multiphase ISM and explicitly include radiation pressure and thermal pressure (Hopkins, Quataert & Murray 2012) support this picture. Like essentially every other simulation suite on the market, however, models with this feedback scheme have too little O vi in the CGM while retaining too many metals in stars (Muratov et al. 2015).", '7.4. Following the Metals: The Role of Recycling': "Inflow and outflow are necessary processes in galaxy and CGM evolution; can one become the other by the recycling of outflows into fresh accretion of ejected gas? We have already established that, at least at low-redshift, galaxies require a long-term source of fuel, and that their CGM gas and metals are massive and bound. Recycling is a natural consequence; this gas 'should' reaccrete onto the galaxy if the cooling time is short. Indeed, the predominance of metal-enriched accretion is supported by essentially all cosmological simulations where the origins of gas joining the ISM has been tracked: significant fractions at gas accreting onto galaxies has previously been ISM gas-and often through multiple cycles (Ford et al. 2014; Christensen et al. 2016; Muratov et al. 2016), with the majority of star formation at late times fueled by recycled gas (Oppenheimer et al. 2010). Ford et al. (2014) found 60% of all star formation at z = 0 is powered by gas that was in the CGM a billion years before. This idea has the intriguing implication that a substantial fraction of all heavy elements on Earth once cycled through the Milky Way's halo at 100 kpc scales. The timescales are unclear: Christensen et al. (2016) find that half of outflow mass is recycled on timescale of 1 Gyr with a logarithmic tail, independent of halo mass, while Oppenheimer & Dav'e (2008) find that t rec ∝ M -1 / 2 halo ∼ 10 9 ± 0 . 5 yr, a timescale so short for massive galaxies that it is like not having an outflow at all, and so long for dwarfs that it essentially escapes forever. \nThus the idea of recycling is well-motivated, but the details are still murky. Is it a simple process in which gas launched at v < v esc encounters hydrodynamic resistance and eventually succumbs to gravity to fall back into the galaxy as part of a large-scale halo fountain? Or is the CGM well-mixed but multi-phase, with metal-rich gas precipitating out of the hot halo and raining onto the galaxy (Voit et al. 2015a; Fraternali et al. 2015; Thompson et al. 2016)? Here too can metals help disentangle the ins and outs. Intriguingly, dense CGM gas (Lehner et al. 2013; Wotta et al. 2016; § 6.2) is roughly equally divided between gas at a few percent solar (metal-poor IGM accretion) and 40% solar (recycling ejecta?). \nLRG: Luminous Red Galaxies \nWhile gas 'accreting' from the IGM generally has (or is assumed to have) very low metallicity (Lehnert et al. 2013; Cooper et al. 2015; Glidden et al. 2016), cases with metallicity well below the IGM (Ly α forest) at the same redshift are rare (Fumagalli, O'Meara & Prochaska 2011; Crighton, O'Meara & Murphy 2016). That is, either pristine cosmic accretion entrains metal-enriched circumgalactic gas on its way into the galaxy (e.g., Fraternali et al. 2015), or that even at the highest redshifts where accretion is potentially observable, it is at least partially comprised by material that has previously been in the ISM, i.e., that recycled mode accretion is critical to galaxy evolution even at early cosmic times. Yet most formulations of the 'bathtub model' assume that the accreting gas is pristine (e.g., Lu, Blanc & Benson 2015, though see Dav'e, Finlator & Oppenheimer 2012b). Entrainment is a commonly invoked phenomenon for galaxy outflows, where it refers to the wind fluid sweeping up ambient ISM and mixing it with the fresh supernova ejecta powering the outflow. (It is important to note that the metallicity of the outflowing material is necessarily higher than that of the ambient ISM, contrary to what is assumed in some popular simulation recipes, e.g., Vogelsberger et al. 2014.) Does 'recycled accretion' behave in a simular way but in the opposite direction, with pristine inflows sweeping up metal-polluted CGM material on its way from the IGM to the ISM? Or do galaxy winds preferentially re-accrete, sweeping up more pristine cosmic accretion? \nTaking all this evidence into account, we can see the outlines of an emerging picture of galaxy inflows, at least at low redshift. They arise in the massive reservoir of cold, metal enriched gas bound to a galaxy's potential well, and enter the disk in HVC-like clouds but also in smooth flows of ionized gas. There may be a metal-poor component that comes more directly from the IGM without spending much time in the CGM, or otherwise acquiring metals. All these aspects of the CGM-cold, bound, metal enriched, and accreting-align better with the phenomenon of 'recycled accretion' better than the bimodal 'hot / cold' accretion. Recycled accretion arises from the ejection of metal-enriched galactic winds that lack the energy to escape the halo entirely, or which encounter the CGM itself and lose energy to radiation from shocks and then eventually cool and re-enter the galaxy. It may be that 'recycling', rather than 'accretion and feedback' is the more accurate way of viewing how galaxies acquire their gas.", '8. The Paradox of Quenching': 'Passive and/or quenched galaxies possess little if any cold gas in their ISM, and blaming the CGM merely relocates the problem: how and why do these massive galaxies that once possessed a cold ISM lose and not regain it? Presumably their dark matter halos continue to add mass, but the accompanying gas does not enter the ISM and form stars like it once did. How galaxies achieve this transition is a deep and abiding problem in astrophysics, and the array of possible mechanisms for consuming, removing, and/or heating cold gas are beyond the scope of our review. We address the phenomenon of quenching by considering the CGM as a factor in, and indicator of, the quenching process.', '8.1. The Fate of Cold Accretion and The Problem with Recycling': "The accretion of gas into halos, its heating to around T vir , and eventual cooling and entry to the ISM was long the prevailing picture of galaxy fueling. In an important twist on this basic picture, Kereˇs et al. (2005) argued that star-forming galaxies are fed by 'cold \naccretion' never reaches T vir but entered a galaxy's disk via streams while remaining below T ∼ 10 5 K. Above log M glyph[star] / M glyph[circledot] ∼ 10 . 3 -10 . 5 (or M halo ∼ 10 12 M glyph[circledot] ), the dark matter halo has sufficient mass, and the CGM enough pressure, to support a virial shock and suppress the cold mode. The coincidence of this mass with the stellar mass that divides star-forming from passive galaxies (Figure 2) drew great attention to this scenario (e.g. Dekel & Birnboim 2006), leading to predictions that the halos of passive galaxies should possess little cold gas (Stewart et al. 2011). \nThe observational picture belies the clean transition seen in simulations and the stark division of observed star formation rates. While COS-Halos did find a dramatic difference in highly ionized O vi around star-forming and passive galaxies, the latter do not show as strong a deficit of CGM H i . As shown in Figure 11, the equivalent widths and covering fractions of H i do not drop as stellar mass increases across the range log M glyph[star] glyph[similarequal] 10 -11 (Thom et al. 2012). This is directly contrary to the expectation from, e.g. Stewart et al. (2011) that the covering fraction of strong H i should drop to nearly zero as galaxies transition to the hot mode of accretion. The inner CGM ( < 50 kpc), however, is not well covered by these observations (Figure 4); it is possible that high pressure hot gas close to the galaxy prevents this cold material from accreting, as some models predict (Schawinski et al. 2014). \nThe presence of cool gas in the halos of massive red galaxies is now well-established by Mg ii studies. Gauthier, Chen & Tinker (2010) and Bowen & Chelouche (2011) found covering fraction of f c = 10 -20% out to 100-200 kpc for > 1 ˚ A absorbers around LRGs. Using a sample of ∼ 4000 foreground galaxies at z = 0 . 5 -0 . 9 from the zCOSMOS survey, Bordoloi et al. (2011) found that the Mg ii equivalent width for blue galaxies is 8-10 times stronger at inner radii ( < 50 kpc) than for red galaxies, but even red galaxies possess evidence for cold gas. Using a new SDSS-based catalog of Mg ii QSO absorbers and LRGs, Zhu et al. (2014) mapped the mean profile out to glyph[greatermuch] 1 Mpc scales, and argue that the mean profile at this mass scale is even stronger than found by Bordoloi et al., extending at a detectable level out to 1 Mpc for LRGs. Johnson, Chen & Mulchaey (2015b) have pointed out that strong Mg ii absorbers are usually consistent with being bound to their host halos, meaning that the cold gas is contained with the dynamical influence of the galaxy. \nFrom a theoretical perspective, the quenching of galaxies is still a significant unsolved problem. Star formation must be curtailed, and later accretion and cooling of gas must be suppressed indefinitely to explain how galaxies remain passive for > 6 Gyr (Gallazzi et al. 2008). Theories vary in how they accomplish this: some models artificially truncate star formation based on halo mass (Somerville & Dav'e 2015), while others suppress the starforming fuel by heating the CGM itself (e.g., Gabor et al. 2010; Gabor & Dav'e 2012). Thus the CGM itself can be the proximate cause of quenching, even if the source of CGM heating is not yet identified. Unfortunately models that manipulate the CGM directly cannot be tested against CGM observations, or at least, they must be modified somehow to recover the cold gas seen in passive galaxy halos. \nBy contrast, models that include self-consistent subgrid treatments of feedback, whether 'thermal' (Schaye et al. 2015), 'mechanical' (Choi et al. 2015), or a combination of thermal, mechanical, and radiative (Vogelsberger et al. 2014) can be compared to CGM observations as tests of their success. As an example, the mechanical feedback model implemented by Choi et al. (2015) performed better than the 'standard' (Springel, Di Matteo & Hernquist 2005; Di Matteo et al. 2008) thermal feedback model in both suppressing galaxy formation and reducing the surface density of gas in the CGM by factors of 3-10 at 10-100 kpc. \nSuresh et al. (2015) addressed quenching using the Illustris simulations, which are tuned \nFigure 11 \n<!-- image --> \nThree views of the CGM and quenching. Top: a trend in Ly α equivalent width over three decades in stellar mass from COS-Halos (Tumlinson et al. 2013, purple) and COS-Dwarfs (Bordoloi et al. 2014a, orange). As shown by Thom et al. (2012), the presence of H i around red, passive galaxies indicates that their halos are not devoid of cold gas. Middle: Mg ii from COS-Halos and MAGIICAT (Nielsen et al. 2016, green). Bottom: the galaxy SFR bimodality from Figure 2. \nto the observed M glyph[star] /M halo and galaxy metallicities, but not the CGM. In Illustris, 'thermal' AGN feedback is deposited locally, inside the galaxy, when the SMBH is in its energetic 'quasar' mode. But in the ∼ 90% of the time when the SMBH is accreting quiescently, its 'radio mode' feedback is deposited non-locally as thermal energy over 100 kpc scales. This amounts to direct heating of the CGM, shifting cold gas to intermediate temperatures showing more O vi , and otherwise warm gas to high temperatures showing O vii and O viii . The net effect is that the Tumlinson et al. (2011) trend of strong O vi around star forming galaxies and weak O vi around passive galaxies is recovered. The 'cold' CGM is reduced, but not completely destroyed. To be consistent, any visible effects of feedback would need to persist even when the AGN is not active, as the COS-Halos galaxies in question are not AGN at the time we observe them. The EAGLE simulations presented by Oppenheimer et al. (2016b) show a similar conclusion with models of thermal feedback and non-equilibrium cooling: at higher mass, with more feedback, O vi is suppressed and the cold gas is depleted but not completely destroyed. These feedback effects force behaviors that generally resemble the data: they suppress star formation to create a red sequence, they force net gas loss from the inner CGM by heating gas that then bouyantly rises, and they shift the balance of gas ionization toward higher temperatures and higher ions. \nDespite these advances, the basic paradox of quenching remains: what happens to the halos of passive galaxies to quench their star formation, keep it quenched, and yet leave cold gas present in their halos? If passive galaxies possess cold gas and are not using it, can we be sure of the (naively obvious) conclusion that star-forming galaxies are using the diffuse gas they possess? Moreover, if the bulk of star formation at lowz comes from recycled accretion, then to understand both how galaxies get their gas and how galaxies quench, we must understand how both the internal and external fuel supplies are shut off.", '8.2. The CGM of AGN and Quasars': "If feedback from AGN is effective at quenching their star formation and their cold CGM in simulations, it naturally suggests that this effect will be visible in the gaseous halos of galaxies with ongoing AGN activity. While hard radiation fields of AGN may leave distinctive ionization signatures in halo gas even long after the AGN fades (Keel et al. 2012; Oppenheimer & Schaye 2013a), studies like COS-Halos with subsamples of passive galaxies have excluded active AGN for the most part, and even so have not seen any apparent signs of AGN effects on the CGM. No published study has systematically examined background QSO/foreground AGN pairs, though there is one such study underway with Hubble /COS 2 . \nAt z > 2, the 'Quasars Probing Quasars' (QPQ) program has seen clear evidence that galaxies hosting bright quasars show greatly enhanced gas budgets in H i and low ions (Prochaska, Lau & Hennawi 2014) though less excess in the high ions. This enhancement of neutral and low-ionization gas hints at a larger accretion rate for these robustly starforming galaxies. AGN may even yield a net gain of cold gas in the CGM Faucher-Giguere et al. (2016). The Ly α blobs observed at z > 2 may be gas accreting on to galaxies, with radiation powered by gravitational infall (Goerdt et al. 2010), though these data may be more consistent with illumination from buried AGN (Prescott, Martin & Dey 2015). The higher gas masses only exacerbate the problem of feedback and quenching-there is more gas to be removed, and it is still not clear how that gas is removed or heated and accretion suppressed thereafter. Future work should focus on following such galaxies down through cosmic time as their QSOs fade, star formation is quenched, and the galaxies later evolve passively. Post-AGN and post-starburst galaxies should be examined for CGM gas as much as is practical. Understanding this process is critical to properly understanding the role of the CGM in creating or reflecting the birth of the red sequence.", 'Data in Need of More Theory': '- 1. Are there any clean observational tests or theoretical discriminants between the various heuristic models of feedback?\n- 2. Are there self-consistent models of quenching that produce a red sequence of galaxies and yet leave a significant mass of cold CGM? How is the remaining cold gas kept from accreting?\n- 3. What do the detailed kinematic profiles of the multiphase suite of absorbing ions tell us about the physical and dynamic structure of the CGM?', 'Theory in Need of More Data': "- 1. What is the mass and composition of the CGM at high-redshift and in lowz M glyph[star] < 10 10 M glyph[circledot] galaxies, and how do these constrain galaxy evolution models?\n- 2. What is the small-scale density and kinematic structure of the CGM, and what does it tell us about the physics?\n- 3. What does the CGM do as galaxies quench? Does cool, neutral gas extend into the inner CGM of passive galaxies?\n- 4. Where are the metals that are still missing from the census? What are the elemental abundance ratios in CGM gas, and how do they depend on the galaxy's mass and star formation history?", '9.1. Progress and Problems': "New instruments and new thinking reveal the CGM as a complex, dynamic gaseous environment that may close galactic baryon budgets and regulate gas accretion, star formation, and chemical enrichment. The observational studies that underlie the mass density profiles in Figure 7 and mass budgets in § 4 and 5 have all been obtained since 2010. For years, questions about how and when gaseous halos influenced galaxy evolution consistently struggled with what was there. The bulk contents of the CGM are now better characterized than ever before. There remain missing pieces-the baryon and metals budget well below L ∗ remain to be filled in (Figure 8), and many of the metals remain missing-but we can already see signs that the most urgent questions motivating new studies take what and where as known, and go on to ask how and when . These sort of questions strike more directly at physics than at phenomenology. \n9.1.1. The Scale Problem. How a gaseous halo evolves is determined at any instant primarily by its density, temperature, metallicity, and radiation fields. But for an actual CGM (such as the simulated one in Figure 6) these physical quantities vary and evolve on many relevant scales, ranging from the sub-parsec sizes for single cold clouds to the > 100 kpc size of the whole CGM and even > Mpc scales in the IGM. If we are to answer the hows of accretion, feedback, recycling, and quenching, we must achieve a better understanding of the basic physical fields at higher spatial and kinematic resolution. This means finding ways to capture sub-parsec boundary layers and instabilities while also maintaining the glyph[greatermuch] kpc context. Yet this 5-6 order of magnitude range still cannot be captured simultaneously in numerical simulations. One approach would be to continue the development of physically rigorous analytic models (e.g., Voit et al. 2015b; Thompson et al. 2016; Fielding et al. 2016; Faerman, Sternberg & McKee 2017) that can isolate the key physical effects and then to incorporate these lessons into simulations at the subgrid level while their resolution improves with computing power. For instance, it might be possible to include subgrid models that account for unresolved interfaces between hot and cold gas, or to extract subgrid models for cosmological boxes from extremely high resolution idealized cloud simulations with carefully controlled physics. To complete the leap between phenomenology and physics, these intrinsically 'sub-grid' processes must come under control while the proper cosmological \nand galactic context is maintained. \nThe transport of metals and the information they provide would also benefit from addressing the scale problem. Metals trace feedback and drive cooling, so how they are distributed through CGM gas at small scales is a critical factor in a proper physical understanding of accretion and feedback. Dense CGM gas appears bimodal in metallicity, congruent with the idea of 'pristine accretion' and 'recycled winds'. What does this tell us about the small-scale structure of the CGM, the relationship between accretion and feedback, and the mixing of diffuse gas? These are among the thorniest of open questions, because of the huge dynamic range in metallicity that must be captured. This problem will be addressed by larger absorber and galaxy surveys, but perhaps poses its stiffest challenges to numerical simulations, because many of the relevant physical mechanisms for mixing gas at boundaries and interfaces are still well below the 'sub-grid' level of simulations. This is another case where coupling small-scale simulations of clouds to cosmological boxes could pay dividends. \nThe 'scale' problem exists also for data but might be better labeled a problem of resolution and confusion. In data, the rich multiphase and multiscale structures of CGM gas are seen through a complex rendering in absorption or emission lines from diagnostic ions. The line profiles of absorbers likely contain more information than we are currently able to extract and interpret. Systematic effects from line saturation, uncertain ionization and radiation fields, relative abundances, limited signal-to-noise, and finite spectral resolution all complicate the derivation of the true CGM density field, which in turn enters into mass estimates, energy balance, and timescales for the gas flows of interest. While we are learning to model and simulate the CGM at higher resolution with better physics, we should also aim to extract and use the full information available in the rich kinematic profiles of multiphase absorbers, which will likely require new analytic and statistical techniques. The importance and complexity of the CGM make it imperative to examine all of the information that Nature provides. \n9.1.2. Mass Flows and the Fate Problem. The CGM matters to galaxies as long as it provides them with fuel and recycles their feedback. Ultimately this is what we care about most - how does the CGM influence galaxy evolution? The most fundamental questions with which we began are still not completely answered: How does cold gas accrete and form stars over billions of years, and why does this cycle stop in massive galaxies? Does the CGM empty out or get consumed when galaxies quench? How much star formation is fueled by recycling and how much by new accretion? Can we ever hope to identify particular absorbers as accretion, feedback, or recycling, or are we destined never to separate them? These questions will drive the field as it advances from phenomenology toward more sophisticated physical understanding. Properly explaining these phenomena in terms of the hows of accretion, feedback, recycling, and quenching requires that we follow mass flows, not merely mass budgets. \nNow that we have a grip on the bulk contents of the CGM, it is time to develop and deploy the tools to probe these questions of how the gas flows operate. To follow flows, we will need to make at least three key advances. First, the mass budgets should be characterized more fully in all phases at stages of galaxy evolution, including those that are relatively short lived such as mergers and AGN. These analyses would additionally benefit from analyzing how outflows and inflows seen in down-the-barrel measurements relate to the kinematics viewed on transverse sightlines, an overdue synthesis deserving attention \nfrom both observations and theory. Second, we must attempt to directly constrain the timescales of CGM evolution using data alone-how do mass budgets and kinematics jointly constrain timescales? Third, we must look at simulations in a new way that focuses on the origins and evolution of the physical phases and how these appear in the data. A large measure of simulation work addressed to the CGM has focused on using column densities and kinematics to constrain uncertain mechanisms of feedback by matching real data to mocks from simulations. While these issues are being resolved, it is also valuable to look at simulations from a different phenomenological point of view. The study from Ford et al. (2014) provides an example; that paper identified particles as 'pristine accretion', 'recycled accretion', 'young outflows', and 'ancient outflows' and followed their evolution over time. These insightful categories turn out to be correlated with observable signatures. We believe there is great potential in viewing models and data from this angle, trying to identify the more distinctive or even unique manifestations of key physical processes defined by their 'fate' rather than their instantaneous properties or appearance.", '9.2. Future Prospects for Data': 'The next decade should bring a wide array of new instruments and numerical capabilities that will address these unsolved problems. \nWhile Hubble lasts (mid-2020s), UV absorber samples will grow, particularly those that focus on the z > 0 . 5 regime where a broader set of EUV ionization diagnostics is available (such as Ne viii ). This increase in coverage will in turn allow more careful treatments of ionization diagnostics component-by-component, hopefully with a better understanding of how CGM gas is spread across physical phases and across galaxy mass. COS remains the ideal instrument for this problem, and big advances are still possible in the metals budget, ionization and kinematic relationships of multiphase gas, and the relationships between CGM gas and special types of galaxies. Starting in 2018, the James Webb Space Telescope (JWST) will enable much deeper searches for faint galaxies near QSO sightlines, likely associating galaxies with samples of z > 4 absorbers that are already known (Becker, Bolton & Lidz 2015; Matejek & Simcoe 2012). Detections of H i emission (e.g., Martin et al. 2015; Arrigoni Battaia et al. 2015; Cantalupo et al. 2014) will provide useful tests of models for CGM mass and structure, but the problems of gas ionization state and metal transport will require much more challenging maps of emission from oxygen and carbon ions (see Hayes et al. 2016, for a pioneering effort). Such maps might emerge from IFU spectrographs such as MUSE and KCWI, and their successors on 30m class telescopes; limits can be further improved by stacking of multiple galaxies. The optimal galaxies would be those where absorption line probes are also available, so that emission-line and pencil-beam measurements can be compared. Emission maps of metal-bearing CGM gas (e.g. Bertone et al. 2010; Corlies & Schiminovich 2016) are a key goal of the Large Ultraviolet/Optical/Near Infrared Surveyor (LUVOIR 3 ), which will push to 50x the UV point source sensitivity of Hubble/COS and 100-fold multiplexing in UV spectroscopy. Planned for launch in the 2030s, LUVOIR would be able to directly image the CGM in metal-line emission, map the most diffuse gas with weak absorbers, and resolve the multiphase kinematics of CGM gas with R > 50 000 UV spectroscopy (Dalcanton et al. 2015). The hot gas phase would be addressed by the ESA-planned X-ray flagship known as the Advanced Telescope for High \nENergy Astrophysics ( ATHENA 4 ) in 2028, with a significant focus on understanding the cosmic evolution of hot gas in the IGM and CGM. \nThe size of our samples provide statistical power over the key galaxy variables: mass, redshift, shape, evolutionary state, and orientation to the sightline. Here, future UV absorber samples must be supplemented by optical absorber samples at z ∼ 3, and by deeper galaxy surveys at all redshifts. This is a problem for the next generation of giant groundbased telescopes, which will advance highz CGMstudies in rest-UV lines and support lowz studies by obtaining redshifts of subL ∗ galaxies near QSO sightlines at surveys at z < 1 to fill in the low-mass baryon and metals census, still a major missing piece. \nMassive fiber based surveys have proven effective at characterizing CGM gas and its flows with both intervening and down the-barrel measurements. This technique should only accelerate in the future, pushing to fainter sources, higher redshifts, and rarer foreground galaxies with future massively multiplexed spectrographs (e.g., eBOSS, PFS) on large telescopes. This technique excels at detecting weak signals in the CGM, and at examining more and more foreground galaxy properties with good statistics. With larger, deeper samples, we can look forward to addressing questions about the behavior of the cold/dense CGM in rarer galaxy types, such as quasars and AGN, mergers, and groups.', '9.3. Final Thoughts': "Galaxies were understood as island universes long before astronomers discovered the interstellar gas that forms their stars. The intergalactic medium was added to the big picture with the discovery of QSO absorption lines and the development of the dark-matter cosmology. Because it is much fainter than stars, and much smaller than the IGM, the CGM is arguably the last major component of galaxies to be added but it has nevertheless become a vital frontier. As to why , it is clear that much has been learned by viewing galaxy evolution from the perspective of the CGM. The circumgalactic medium can even provoke fascination: might the heavy elements on Earth cycled back and forth through the Milky Way's CGM multiple times before the formation of the Solar System? It appears that the solution to major problems in galaxy formation that are still unsolved will run through this elusive region of the cosmos.", 'DISCLOSURE STATEMENT': 'The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.', 'ACKNOWLEDGMENTS': "MSP and JT acknowledge support from NSF grant AST-1517908. We are grateful to Ann Feild of STScI for her expert artistic contributions, to Joop Schaye and Ben Oppenheimer for use of the EAGLE simulation shown in Figures 2, 6, and 8, to Josh Suresh for data from the Illustris simulation shown in Figure 8, to Sasha Muratov for data from the FIRE simulation (Figure 8), and to Ben Oppenheimer for the data from the specially-analyzed EAGLE halos shown in Figure 9. We also thank Lauren Corlies, Matt McQuinn, Andrew \nFox, Romeel Dav'e, and John O'Meara for insightful comments on a draft of this article. 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2024A&A...690A.334P	Context. Recent evidence from spectroscopic surveys points towards the presence of a metalpoor young stellar population in the low chemically thin disk. In this context the investigation of the spatial distribution and time evolution of precise unbiased abundances is fundamental to disentangle the scenarios of formation and evolution of the Galaxy. Aims. We study the evolution of abundance gradients in the Milky Way by taking advantage of a large sample of open star clusters which are among the best tracers for this purpose. In particular we used data from the last release of the GaiaESO survey. Methods. We performed a careful selection of open cluster member stars excluding those members that may be affected by biases in spectral analysis. We compared the cleaned open cluster sample with detailed chemical evolution models for the Milky Way using welltested stellar yields and prescription for radial migration. We tested different scenarios of Galaxy evolution to explain the data namely the twoinfall and the threeinfall frameworks which suggest the chemical thin disk is formed by one or two subsequent gas accretion episodes respectively. Results. With the performed selection in cluster member stars we still find a metallicity decrease between intermediateage 1 lt AgeGyr lt 3 and young Age lt 1 Gyr open clusters. This decrease cannot be explained in the context of the twoinfall scenario even by accounting for the effect of migration and yield prescriptions. The threeinfall framework with its late gas accretion in the last 3 Gyr is able to explain the low metallic content in young clusters. However we have invoked a milder metal dilution for this gas infall episode relative to previous findings. Conclusions. To explain the observed low metallic content in young clusters we propose that a late gas accretion episode triggering a metal dilution would have taken place extending the framework of the threeinfall model for the first time to the entire Galactic disk.	2024-10-01T00:00:00Z	['2024arXiv240817395P', '2024A&A...690A.334P', '10.1051/0004-6361/202451395', '10.48550/arXiv.2408.17395', 'arXiv:2408.17395']	['stars: abundances', 'Galaxy: abundances', 'Galaxy: disk', 'Galaxy: evolution', 'open clusters and associations: general', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics']	Mapping radial abundance gradients with GaiaESO open clusters Evidence of recent gas accretion in the Milky Way disk	2,024	200	0.63	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	2	https://arxiv.org/pdf/2408.17395.pdf	{'Evidence of recent gas accretion in the Milky Way disk': "M. Palla 1 , 2 , L. Magrini 3 , E. Spitoni 4 , F. Matteucci 4 , 5 , 6 , C. Viscasillas Vázquez 7 , M. Franchini 4 , M. Molero 4 , 8 , and S. Randich 3 \n- 1 Dipartimento di Fisica e Astronomia 'Augusto Righi', Alma Mater Studiorum, Università di Bologna, Via Gobetti 93 / 2, 40129 Bologna, Italy\n- 2 INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti 93 / 3, 40129 Bologna, Italy e-mail: [email protected]\n- 3 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy\n- 4 INAF - Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34131 Trieste, Italy\n- 5 Dipartimento di Fisica, Sezione di Astronomia, Università degli studi di Trieste, Via G.B. Tiepolo 11, I-34143 Trieste, Italy\n- 6 INFN - Sezione di Trieste, via A. Valerio 2, I-34100, Trieste, Italy\n- 7\n- Institute of Theoretical Physics and Astronomy, Vilnius University, Sauletekio av. 3, 10257 Vilnius, Lithuania\n- 8 Institut für Kernphysik, Technische Universität Darmstadt, Schlossgartenstr. 2, Darmstadt 64289, Germany \nReceived XXX; accepted XXX", 'ABSTRACT': 'Context. Recent evidences from spectroscopic surveys point towards the presence of a metal-poor, young stellar population in the lowα / chemical thin disk. In this context, the investigation of the spatial distribution and time evolution of precise, unbiased abundances is fundamental to disentangle the scenarios of formation and evolution of the Galaxy. \nAims. We study the evolution of abundance gradients in the Milky Way by taking advantage of a large sample of open star clusters, which are among the best tracers for this purpose. In particular, we use data from the last release of the Gaia -ESO survey. \nMethods. We perform careful selection of open cluster member stars excluding those members that may be a ff ected by biases in spectral analysis. The cleaned open clusters sample is compared with detailed chemical evolution models for the Milky Way, using well tested stellar yields and prescription for radial migration. Di ff erent scenarios of Galaxy evolution are tested to explain the data, i.e. the two-infall and the three-infall frameworks, suggesting that the chemical thin disk is formed by one or two subsequent gas accretion episodes, respectively. \nResults. With the performed selection in cluster member stars, we still find a metallicity decrease between intermediate age (1 < Age / Gyr < 3) and young (Age < 1 Gyr) open clusters. This decrease cannot be explained in the context of the two-infall scenario, even by accounting for the e ff ect of migration and yield prescriptions. The three-infall framework, with its late gas accretion in the last 3 Gyr, can explain the low metallic content in young clusters. However, we invoke a milder metal dilution for this gas infall episode relative to previous findings. \nConclusions. To explain the observed low metallic content in young clusters, we propose that a late gas accretion episode triggering a metal dilution should have taken place, extending the framework of the three-infall model for the first time to the entire Galactic disk. \nKey words. Galaxy: disk -- Galaxy: abundances - Galaxy: evolution - stars: abundances - open clusters and associations: general', '1. Introduction': 'A fundamental constraint to study the formation and chemical evolution of the Galaxy are abundance gradients along the Galactic disk. \nDi ff erent stellar and nebular Galactic tracers that correspond to di ff erent epochs in the evolution of our Galaxy have been used to probe radial abundance gradients. These are Open Clusters (OCs, e.g. Randich et al. 2003, 2022; Magrini et al. 2010; Yong et al. 2012), HII regions (e.g. Balser et al. 2011; Esteban et al. 2017; Méndez-Delgado et al. 2022), young massive O and B stars (e.g. Daflon & Cunha 2004; Bragança et al. 2019), Classical Cepheids (CCs, e.g. Lemasle et al. 2007, 2008; Luck & Lambert 2011; Genovali et al. 2015; Kovtyukh et al. 2022), planetary nebulae (PNe, e.g. Maciel et al. 2003; Henry et al. 2010; Stanghellini & Haywood 2010, 2018), and also field stars with \nprecise stellar ages (e.g. Anders et al. 2017; Santos-Peral et al. 2021). \nThe plethora of information from all these tracers has been carefully interpreted by means of models, allowing us to understand fundamental properties for the Milky Way (MW) disk formation. Among these, the inside-out mechanism (e.g. Matteucci & Francois 1989; Chiappini et al. 2001; Schönrich & McMillan 2017), variable star formation e ffi ciency (SFE), i.e. higher in the inner regions than in the outer ones (e.g. Colavitti et al. 2009; Grisoni et al. 2018; Palla et al. 2020b) and radial gas flows (e.g. Portinari & Chiosi 2000; Spitoni & Matteucci 2011; Bilitewski & Schönrich 2012; Cavichia et al. 2014). Moreover, studies on tracers of the "old" stellar gradients have given important information on the impact of the process of stellar radial migration (e.g. Minchev et al. 2018; Willett et al. 2023). \nOn the other side, the analysis of spectroscopic data from ground based surveys, such as the APOGEE (e.g. Hayden et al. 2015; Queiroz et al. 2020), the Gaia -ESO (GES, e.g. RecioBlanco et al. 2014; Kordopatis et al. 2015; Rojas-Arriagada et al. 2016), and the GALAH (Buder et al. 2019, 2021) ones, the accurate asteroseismic stellar ages (e.g. Pinsonneault et al. 2014, 2018; Miglio et al. 2021) and the kinematics and dynamical properties provided by the Gaia mission (Gaia Collaboration et al. 2016, 2018, 2021, 2023b) have pointed towards the existence of two sequences of stars in the [ α / Fe] 1 versus [Fe / H] abundance pattern in the MW disk: the so-called highα and lowα sequences. \nTo explain the wealth of these data, Palla et al. (2020b, 2022) (see also Spitoni et al. 2019, 2021) suggested the that presence of this feature (also known as α -bimodality) may be connected to to a delayed ( ≳ 3 Gyr) accretion of gas. The latter forges the lowα sequence of stars, while highα stars are formed promptly in a gas infall episode occurring in the first phases of Galactic formation. This scenario is confirmed by several models and simulations of the evolution of galactic disks (e.g. Noguchi 2018; Grand et al. 2018; Mackereth et al. 2018; Buck 2020) which suggested that the bimodality may be strictly connected to a delayed accretion of gas of primordial / metal-poor chemical composition. \nRecently, to reproduce the chemical abundances from Gaia DR3 Radial Velocity Spectrometer (RVS) spectra in the solar vicinity (Gaia Collaboration et al. 2023a; Recio-Blanco et al. 2023) and in particular a population of massive stars with evidences of a recent chemical impoverishment, Spitoni et al. (2023) suggested a novel scenario of chemical evolution in which the lowα population of stars is generated by two distinct gas accretion episodes, with the latter infall happening at very recent times ( < 3 Gyr of age). This scenario is constrained by the star formation histories for disk stars inferred from Gaia DR1 and DR2 color-magnitude diagrams (CMDs, Bernard 2017; Ruiz-Lara et al. 2020), which show evidence for short episodes of enhanced star formation (hereafter, SF) in recent times. In this model, the enhanced SF is triggered by the gas infall, which in turn causes the visible chemical impoverishment in the young stellar population as observed by Gaia RVS. \nIn the light of these recent development for the MW disk formation and evolution, we exploit the OCs from the last data release of the Gaia -ESO survey (Randich et al. 2022) to investigate the late evolution of radial chemical gradients in the Galaxy. OCs are in fact considered excellent tracers of the chemical properties of the disk stellar populations of our Galaxy, including the spatial distribution of elemental abundances, especially when observed with high-resolution spectroscopy (Spina et al. 2022). To this regard, the Gaia -ESO survey (Gilmore et al. 2012, 2022; Randich et al. 2013, 2022), is the only survey performed on a 8 m-class telescope, which put specific focus on the population of Galactic OCs. Gaia -ESO targeted OCs over a wide range of ages, distances, masses, and metallicities, observing large unbiased samples of cluster candidates, with a well-defined selection function (Bragaglia et al. 2022; Randich et al. 2022). \nOf this sample, Magrini et al. (2023) selected 62 OCs, with extended radial (up to R > 15 kpc) and age (up to 7 Gyr) ranges, analysing the shape of radial gradients in chemical elements spanning di ff erent nucleosynthetic origin (from Oxygen to Europium) and their time evolution. They found that the gradients of most of chemical elements, including the metallicity [Fe / H], \ncan be better approximated with a two-slope shape, steeper in the inner regions and rather flat in the outer ones. Shallower gradient slopes in outer regions were also observed in other studies using di ff erent tracers (OCs, e.g. Carbajo-Hijarrubia et al. 2024; CCs, e.g. da Silva et al. 2023), suggesting a flat SFE law for large Galactocentric distances at variance with previously theorised (e.g. Grisoni et al. 2018; Palla et al. 2020b). In addition, Magrini et al. (2023) found that the youngest clusters in the sample (age < 1 Gyr) have lower metallicity than their older counterpart, even though the e ff ect could be mitigated by avoid considering stars with low surface gravity. \nIn this work, we compare the above mentioned data sample with detailed chemical evolution models, which also account for the e ff ect of stellar radial migration. We start from well tested models under the revised two-infall scenario (Palla et al. 2020b; da Silva et al. 2023), which successfully reproduce data from high-resolution surveys in di ff erent Galactocentric regions (see also Spitoni et al. 2021) and were already adopted in the context of radial abundance gradients (Palla et al. 2020b; da Silva et al. 2023). To better investigate the observed behaviour in Gaia -ESO OCs, we then extend the comparison to the newly proposed scenario of the three-infall model (Spitoni et al. 2023), expanding this framework for the first time to the whole MW disk and discussing its feasibility in the context of the adopted dataset. \nThe paper is organised as follows. In Section 2 we describe the Gaia -ESO OCs sample and the additional dataset adopted in this work. In Section 3, we present the model framework used, i.e. from the model scenarios to the nucleosynthesis and radial migration prescriptions. In Section 4 we present the comparison between the di ff erent model predictions and the observations, also discussing the implications of the obtained results. Finally, in Section 5 we draw some conclusions.', '2.1. Gaia -ESO open clusters': 'Gaia -ESO is a large public spectroscopic survey that observed the major components of our Galaxy with FLAMES@VLT from 2011-2018 (see Randich et al. 2022; Gilmore et al. 2022, for a full description of the survey). The final release dr 5.1 is public and available at the ESO website from June 2023. Gaia -ESO made use of FLAMES with both the high-resolution spectrograph UVES (operating at a resolving power, R = 47 000) and the medium-resolution spectrograph GIRAFFE (R ∼ 20 000). It observed open star clusters for about 30% of its 340 nights. The observed clusters cover a wide range in age, distance, mass and metallicity (see Randich et al. 2022), with an unbiased selection of cluster candidates. Each cluster was observed with both UVES and GIRAFFE. In particular, UVES spectra cover a wide spectral range from 480.0 nm to 680.0 nm (U580) or from 420.0 nm to 620.0 nm (U520). The large spectral interval, combined with the high signal-to-noise ratios (S / N) and with the high-resolution, have enabled an unprecedented characterisation of a large sample of open clusters, observed and analysed in a homogeneous way (Bragaglia et al. 2022; Hourihane et al. 2023). For the about 80 observed clusters, it was possible to obtain precise stellar parameters and abundances of more than 30 di ff erent ions, including those of elements belonging to all the main nucleosynthesis channels, from the lightest ones, such as Li to the heaviest one, such as Eu. The detailed chemistry of the open cluster sample in Gaia -ESO, combined with uniform ages \nand precise distances from Gaia (e.g. Cantat-Gaudin et al. 2020) has been used, e.g., to calibrated age-sensitive abundance ratios, the so-called chemical clocks (Casali et al. 2019, 2020b; Viscasillas Vázquez et al. 2022), to investigate the nucleosynthesis of neutron-capture elements (Magrini et al. 2018, 2022; Van der Swaelmen et al. 2023; Molero et al. 2023), to study the evolution of Li abundance (Randich et al. 2020; Romano et al. 2021; Magrini et al. 2021b,a), and to study the shape and the evolution of the radial abundance gradients (Jacobson et al. 2016; Overbeek et al. 2017; Magrini et al. 2017; Spina et al. 2017, 2022; Magrini et al. 2023; da Silva et al. 2023). \nIn the present work, we make us of the sample of open clusters used in Magrini et al. (2023, hereafter Ma23), in which abundances for 25 chemical elements were provided for the 62 Gaia -ESO OCs older than 100 Myr. The adopted cluster ages are the same as in Viscasillas Vázquez et al. (2022) and are derived from the homogeneous analysis of Gaia photometric and astrometric data by Cantat-Gaudin et al. (2020) by means of an artificial neural network trained on a set of objects with well-determined parameters in the literature. Here, we consider the guiding radius (R guide ) as tracer of the OCs Galactocentic distance. These are computed as defined by Halle et al. (2015, 2018), i.e. as the average between the minimum and maximum radius of the orbits. For the latter, we calculate them by using the G alpy code with the axisymmetric potential MWP otential 2014 (Bovy 2015). For further information on the sample cluster parameters, as well as on their distribution in di ff erent quantities, we refer to Viscasillas Vázquez et al. (2022). For each cluster, the membership analysis was done using both radial velocities from Gaia -ESO and proper motions and parallaxes from Gaia edr 3 (Gaia Collaboration et al. 2021), as described in Magrini et al. (2021b), Viscasillas Vázquez et al. (2022) and Jackson et al. (2022).', '2.1.1. A restricted sample to avoid observational biases': 'From a purely observational point of view, the cluster sample in Ma23 showed an unexpected behaviour regarding the evolution with time of the metallicity gradient: the youngest clusters (age < 1 Gyr) in the inner disk have lower metallicity than their older counterparts and they outline a flatter gradient below that of the older population \nTo distinguish the real evolution of the gradient from possible spectral analysis e ff ects, Ma23 restricted the sample of member stars per cluster. In fact, an investigation of the internal abundances of each cluster showed that there are trends of [Fe / H] versus stellar parameters, in particular gravity and microturbulence. These trends are not specific to Gaia -ESO but are present in all the considered spectroscopic surveys (see Fig. 10 and Appendix in Ma23). This might be due to two e ff ects: the former is related to problems in modelling the atmospheres of giant stars, which a ff ect the spectral analysis of low-gravity giant stars, as noticed already in Casali et al. (2020a) and Spina et al. (2022), and the latter to the e ff ects of magnetic activity in young massive giant stars. Hence, the spectral analysis of giants (log g < 2.5) likely underestimates their [Fe / H] of about 0.1-0.2 dex due to the combination of these e ff ects. \nTherefore, in a conservative approach, in order to preserve the actual mean abundance of clusters, we consider a restricted sample of member stars, where only stars with log g > 2.5 and ξ < 1.8 km -1 are taken into account to compute the mean cluster abundances. The average abundances used in the present work, restricted in stellar parameters, are reported in Tab. C.1. In the Table we also report the di ff erent cluster parameters (i.e. age, \ndistances, orbital parameters), which are the same as the ones introduced earlier in 2.1 for the original Gaia -ESO OC sample.', '2.2.1. Gaia -ESO field stars': 'We select a sample of about 3800 field stars in Gaia -ESO dr 5.1 following the criteria of Viscasillas Vázquez et al. (2022), to which we refer for a complete description. Here are the basic steps of our selection and the computation of their ages. Our set of field stars is composed by both field stars (GES\\_FLD keywords GES\\_MW for general MW fields, GES\\_MW\\_BL for fields in the direction of the Galactic bulge, GES\\_K2 for stars observed in Kepler2 (K2) fields, GES\\_CR for stars observed in CoRoT fields) and non members of the 62 open clusters considered in this work (age > 100 Myr). We applied some further quality selections: SNR > 20; eT e ff < 150 K, e log g < 0 . 25, e [Fe / H] < 0 . 20 and e ξ < 0.20 km s -1 . We also apply a further cut in abundance errors, considering only those values that have an eA ( El ) < 0 . 1. \nThe selection function adopted in the Gaia -ESO survey for UVES observations (see Stonkut˙e et al. 2016) favours the main sequence turn-o ff (MSTO), which constitute the majority of the sample. By construction, the sample of field stars has a limited extent in Galactocentric distances, and is, therefore, used in our analysis for comparison with the model in the [X / Fe] vs [Fe / H] plane.', '2.2.2. Classical Cepheids in Da Silva et al. (2023)': "da Silva et al. (2023, hereafter DS23) provided the largest (1118 spectra, 356 objects) and most homogeneous spectroscopic sample for Galactic CCs with measured metallicity from optical high-resolution, high-S / N spectra. The sample is distributed across the four Galactic quadrants and it ranges from the inner (R ∼ 5 kpc) to the outer (R ∼ 25 kpc) disk. For the distances, measurements were based either on trigonometric parallaxes from Gaia DR3 or on near-infrared period-luminosity relations (Ripepi et al. 2022). Due to the steadily variation in target's physical properties due to their natural radial oscillations, special care was dedicated to the estimate of the di ff erent atmospheric parameters ( Tef f , log g , ξ ), which were verified using di ff erent approaches and / or diagnostics (see also da Silva et al. 2022). \nIn this work, we take advantage of the CCs sample from DS23 to have an additional observational probe for present-day metallicity gradients, in addition to the young OCs from Ma23.", '3. Chemical evolution of the Milky Way disk': 'In this Section, we present the main assumptions and features of the multi-zone chemical evolution models adopted in this work. In particular, in 3.1 we provide the details of the revised two-infall model proposed by Palla et al. (2020b) (see also DS23), whereas in 3.2 we describe the details of the threeinfall framework from Spitoni et al. (2023), which we expand throughout this work. \nFor both of the chemical evolution models listed above, the basic equations that describe the chemical evolution of a given element i are: \n˙ Gi (R , t ) = -ψ (R , t ) Xi (R , t ) + Ri (R , t ) + ˙ Gi , in f (R , t ) + ˙ Gi , Rf , (1) \nTable 1. Summary of the main parameters of the two-infall model (2INF) adopted in this study.Notes. All the above parameters are the same to the ones adopted in DS23. The negative sign on the radial flow speed indicates inward flows. \nwhere Gi (R , t ) = Xi (R , t ) G (R , t ) is the fraction of the gas mass in the form of an element i and G (R , t ) is the fractional gas mass. Xi (R , t ) represents the abundance fraction in mass of a given element i , with the summation over all elements in the gas mixture being equal to unity. \nThe first term on the right-hand side of Eq. (1) corresponds to the rate at which an element i is removed from the ISM due to star formation. The star formation rate (hereafter, SFR) is parametrised according to the Schmidt-Kennicutt law (Kennicutt 1998): \nψ (R , t ) = ν Σ gas (R , t ) k , (2) \nwhere Σ gas is the surface gas density, k = 1 . 5 is the law index and ν is the star formation e ffi ciency (SFE). \nRi (R , t) (see Palla et al. 2020a for the complete expression) takes into account the nucleosynthesis from low-intermediate mass stars (LIMS, m < 8M ⊙ ), core collapse (CC) SNe (Type II and Ib / c, m > 8M ⊙ ) and Type Ia SNe. For these latter, we assume the single-degenerate scenario and in particular the delay-timedistribution (DTD) by Matteucci & Recchi (2001). This choice can be considered a good compromise to describe the delayed pollution from the entire Type Ia SN population as it enables us to obtain abundance patterns that are similar to those obtained with other DTDs (see Palla 2021 for details). Ri (R , t ) output is also weighted by the initial mass function (IMF). Here, we adopt the IMF by Kroupa et al. (1993), which is preferred to reproduce the characteristics of the MW disk (Romano et al. 2005). \nThe last term of Eq. (1) refers to radial inflows of gas, which here are implemented following Portinari & Chiosi (2000) (see also Palla et al. 2020b for a detailed description of the implementation). In our models, we use a constant speed pattern, i.e. with vflow = 1 kms -1 across all radii, as suggested by Palla et al. (2020b). Low speeds, i.e. small radial inflow motion, are also suggested by previous chemical evolution studies (e.g. Bilitewski & Schönrich 2012; Mott et al. 2013; Vincenzo & Kobayashi 2020) as well as observations of external galaxies (e.g. Wong et al. 2004; Di Teodoro & Peek 2021). \nIn general, we ignore the e ff ect of Galactic winds on chemical evolution of the MW disk. In fact, by studying the Galactic fountains originated by the explosions of Type II SNe in OB associations, Melioli et al. (2008, 2009) and Spitoni et al. (2009) found that metals fall back to approximately the same Galactocentric region from where they were ejected. Moreover, Spitoni et al. (2009) computed the typical timescale of the fallback of this material, finding a value of 0.1 Gyr. These results were also later supported by cosmological simulation of galactic discs of virial mass > 10 11 M ⊙ , i.e. encompassing the MW, which showed that the majority of the mass ejected by the disc is reaccreted on short timescales and close to the ejection location (e.g. Hopkins et al. 2023 and references therein). Therefore, galactic winds are likely to produce a negligible e ff ect on the global chemical evolution of the Galaxy.', '3.1. The two-infall model': 'In the two-infall model formalism, the model assumes two consecutive gas accretion episodes feeding the MW disk, forming the so-called highα and lowα sequences observed in the Galactic disk. Therefore, the third term on the right-hand side of Eq. (1) can be expressed in this way: \n˙ Gi , in f (R , t ) = A (R) Xi , 1 in f (R) e -t τ 1 + + θ ( t -tmax , 1) B (R) Xi , 2 in f (R) e -t -t max , 1 τ 2 , (3) \nwhere Gi , in f (R , t ) is the infalling material in the form of element i and Xi , Jin f is the abundance of the same element for the J -th infall. τ 1 and τ 2 are the timescales of the two infall episodes, while tmax , 1 indicates the time of maximum infall, which is also the delay between the first and the second infall episodes. θ is the Heavyside step function, while A (R) and B (R) coe ffi cients are set to reproduce the present-day total surface mass density of the highα and lowα disks at di ff erent Galactocentric radii. These latter are assumed to have exponential profiles, in this way: \nΣ J (R) M ⊙ pc -2 = Σ 0 , J e -R / R d , J , (4) \nwhere the disk scale length R d , J is 3.5 kpc for the lowα disk ( J = 2) and 2.3 kpc for the highα one ( J = 1, see Palla et al. 2020b and references therein). The quantities Σ 0 , J are tuned to reproduce the total surface mass density in the solar neighbourhood as provided by McKee et al. (2015) of 47 . 1 ± 3 . 4 M ⊙ pc -2 . \nIn this work, we take advantage of the model prescriptions adopted in Palla et al. (2020b) and later revised in DS23. The model is thus a revised version of the two-infall paradigm (see also Spitoni et al. 2019) in which two consecutive gas accretion episodes are separated by a delay of tmax , 1 = 3 . 25 Gyr. The fairly large delay relative to the "classical" two-infall paradigm (of 1 Gyr, see Chiappini et al. 1997; Romano et al. 2010), allow us to reproduce [ α / Fe] vs. [Fe / H] abundance diagrams (e.g. Palla et al. 2020b; Spitoni et al. 2021) throughout the MW disk as well as stellar ages trends (e.g. Spitoni et al. 2019; Nissen et al. 2020) in the solar vicinity. \nGoing into more detail, in the first infall (forging the highα sequence) the timescale of gas accretion is fixed at τ 1 = 1 Gyr at all radii, with also a fixed SFE of ν = 2 Gyr -1 . For the second gas-infall episode (forming the lowα sequence), the timescale for gas accretion τ 2 increases linearly with radius according to the inside-out scenario (following Romano et al. 2000; Chiappini et al. 2001 law). In order to reproduce the slope observed in radial abundance gradients of CCs in DS23, as well as the gradients in other physical quantities such as SFR and gas density (e.g. Stahler & Palla 2005; Nakanishi & Sofue 2003, 2006), the SFE of the second infall episode is variable depending on the galactocentric radius, with values between ν = 5 Gyr -1 (at R = 4 \nkpc) and ν = 0 . 4 Gyr -1 (from R > 12 kpc). In addition, inward radial gas flows with a constant velocity of vflow = 1 km s -1 are also needed to reproduce the gradients, as discussed above. \nA summary of all the main parameters adopted in this work for the two-infall model are listed in Table 1.', '3.2. The three-infall model': 'In their recent work, Spitoni et al. (2023, hereafter Sp23) proposed a new chemical evolution framework for the Galactic disk components, constrained by the star formation histories inferred from CMD analyses in Gaia DR1 and DR2 (Bernard 2017; RuizLara et al. 2020). These works revealed enhanced SF activity within the last 2-3 Gyrs. This is mimicked in the chemical evolution model by including a recent (age < 3 Gyr) gas infall episode, which triggers this enhanced SF at late times. In this way, Sp23 were also able to reproduce the new abundance ratios provided by the General Stellar Parametriser-spectroscopy module for the Gaia DR3 (Gaia Collaboration et al. 2023a; RecioBlanco et al. 2023), which show a chemical impoverishment in the young population of stars in the solar neighbourhood. \nIn this work, we extend the approach presented in Sp23 to the whole MW disk, in order to investigate abundance gradients. \nIn this model, the functional form of the gas infall rate is: \n˙ Gi , in f (R , t ) = A (R) Xi , 1 in f (R) e -t τ 1 + + θ ( t -tmax , 1) B (R) Xi , 2 in f (R) e -t -t max , 1 τ 2 + + θ ( t -tmax , 2) C (R) Xi , 3 in f (R) e -t -t max , 2 τ 3 , \n(5) \nwhere τ 3 is the timescales of the third gas accretion episode, tmax , 2 is the Galactic time associated to the start of the third infall, and C (R) is the coe ffi cient to reproduce the present-day total surface density of the third accretion episode Σ 3. Here, the sum between the latter and the total surface density of the second infall ( Σ 2) is equal to the density profile as described in 3.1 for the lowα disk in the two-infall model. All the other variables are as in Eq. (3). In fact, the new model uses the framework presented in 3.1, but split the lowα sequence into two distinct gas accretion episodes in order to mimic the recent enhanced SF activity. Therefore, we will leave unchanged all the parameters adopted for the first two infall episodes, such as the infall timescales and the SFEs at di ff erent Galactocentric radii, as well as the IMF (from Kroupa et al. 1993). \nFor what concern the third additional gas accretion, instead, we set its starting time at tmax , 2 = 11 Gyr (as in Sp23) for all the Galactocentric radii. For the other parameters, i.e. the SFE ν 3, the timescale of infall τ 3 and the total surface mass density accreted by the third gas infall, we test di ff erent parametrisations, which are listed in Tab. 2: \n- 1. First, we adopt a setup very similar to the one proposed by Sp23 (3INF-1), which is shown in Tab. 2 upper row. Here, the parameters of the third infall, i.e. tmax , 2, τ 3, Σ 2 /Σ 3 are the same of the latter paper. Concerning the SFE ν 3, rather than fixing the value adopted in Sp23, we fix the proportion between the SFE in the second and third infall episode to be similar to the one used in that paper;\n- 2. to test further the viability of the three-infall scenario in the context of radial gradients, we allow to vary the third infall parameters relative to the values adopted in the model 3INF1. While leaving constant the starting time of the third infall (to be consistent with SF peaks as show by, e.g., Ruiz-Lara \nTable 2. 3rd infall parameters adopted in the models in this paper. \nNotes. For 3INF-1, τ 3 and Σ 2 /Σ 3 parameters are from Spitoni et al. (2023). For ν 3, we consider a similar proportion to the one used by Spitoni et al. (2023) for the solar vicinity, i.e. R = 8 kpc. \net al. 2020) we act on all the other physical parameters, i.e. the infall timescale τ 3, the SFE ν 3 and for the ratio between the baryonic mass accreted by the second and third infall Σ 2 /Σ 3. The parameter for this setup (3INF-2) are shown in Tab. 2 bottom row.', '3.3. Nucleosynthesis prescriptions': 'The nucleosynthesis prescriptions and the implementation of the stellar yields are fundamental ingredients for chemical evolution models. LIMS, massive stars and Type Ia SNe play a fundamental role in shaping the [X / Fe] vs. [Fe / H] abundance patterns as well as radial abundance gradients of the elements of the Periodic Table. \nIn this work, we mainly adopt the prescriptions listed below: \n- -for LIMS, we use the yield set from Ventura et al. (2013, 2018, 2020), also comprising the domains of super-AGB stars (6 < m / M ⊙ < 8 -9) and supersolar metallicities.\n- -for massive stars we use the stellar yields from Limongi & Chie ffi (2018). In particular, we adopt the "mixed vrot 2 set", i.e. the one used in the best model (MWG-12) in Romano et al. (2019) for the MW disk.\n- -for Type Ia SNe we adopt the stellar yields from Iwamoto et al. (1999) (W7 model) which are extensively used in chemical evolution literature (e.g. Romano et al. 2010; Prantzos et al. 2018; Palla et al. 2020b among others). \nHowever, underlying uncertainties in stellar evolution and nucleosynthesis theory may limit our chances of getting a firm grasp on the evolutionary scenario for the Galaxy. \nFor this reason, we also test other stellar yields for massive stars and Type Ia SNe. In particular, we run additional models taking advantage of the stellar yields from Kobayashi et al. (2006, 2011) for massive stars. Concerning Type Ia SNe, instead, we also adopt either yields from Leung & Nomoto (2018) (benchmark model) or Leung & Nomoto (2020) (bubble detonation pattern model), which represent some of the most recent Type Ia SN models for di ff erent progenitor classes, i.e. nearChandrasekhar mass white dwarfs (nearMCh , Leung & Nomoto 2018) and sub-Chandrasekhar mass white dwarfs (subMCh , Leung & Nomoto 2020, see Kobayashi et al. 2020; Palla 2021 for more details).', '3.4. Accounting for migration and observational uncertainties': 'The overall picture on MW shows also evident signatures of stellar migration, both on the theoretical and observational sides \n(e.g. Schönrich & Binney 2009; Minchev et al. 2011, 2018; Kordopatis et al. 2015), even when specifically focusing on OCs (e.g. Anders et al. 2017; Spina et al. 2021; Myers et al. 2022). Therefore, we include in the models stellar radial migration prescriptions from the literature to account for this phenomenon. In particular, we implement migration in the chemical evolution model by adopting the approach already tested in Palla et al. (2022) for MW disk stars. It is worth noting that OCs are more massive than single stars, and therefore the e ff ect of the interactions with perturbing structures should be in principle less pronounced than for field stars (e.g. Zhang et al. 2021; Ma23), especially for young-intermediate ages (see Viscasillas Vázquez et al. 2023). However, a model for radial migration valid for field stars is appropriate within our work, as it gives at least a robust upper limit on the e ff ect of radial migration on OCs, which is a necessary mechanism to properly explain their radii, abundances and ages trends in the Galactic disk. \nIn the following, we provide some details on the implementation, which is extensively described in Frankel et al. (2018). Here, migration is seen as a result of a di ff usion process, which is the e ff ect of repeated and transient torques on stars by features such as spiral arms or a bar, and it is treated in a parametrical way. Following Sanders & Binney (2015) and adapting their parameterization to a Galactocentric radius coordinate, the probability for a star to be currently at a Galactocentric radius R f , given that it was born at a radius R0 and at a certain age can be written as: \nln p (R f \| R0 , Age ) = ln( c 3) -(R f -R0) 2 2 σ RM ( Age / 10 Gyr) , (6) \nwhere σ RM is the radial migration strength and c 3 a normalization constant ensuring that stars do not migrate to negative radii. For σ RM we adopt a value of 3, as found by Frankel et al. (2020) as a result of their Bayesian fitting procedure of APOGEE red clump stars in the lowα disk. The value above mentioned refers to the churning 3 strength, as we already account for blurring effect by considering OCs guiding radius (R guide ) rather than their present-day Galactocentric radius R GC . \nIt is worth noting that the migration framework adopted is in 1D, i.e. it is not considering azimuthal variations in migration strength. In this way, it can be fully integrated within the chemical evolution models described previously in this Section. The 1D assumption is robust, as various works showed rather small azimuthal abundance variations in galaxies ISM (e.g. Kreckel et al. 2019). Another possible limitation of the adopted migration model is the assumption of no radial or temporal dependence on the migration strength. However, including these dependencies means adding further and uncertain assumptions on the inventory of speeds and strengths of spiral and bar patterns during Galactic evolution (see Frankel et al. 2018). In any case, the calibration of the model parameters on a sample of lowα disc stars extending up to radii ≃ 15 kpc (Frankel et al. 2020) allows for a radial and temporally averaged estimate of migration in a radial range that comprises all the OCs described in Section 2, except one (Br29). \nWe also account for the e ff ect of observational uncertainties on the chemical abundances of the predicted stellar populations in our model. \nFig. 1. Time evolution of the radial [Fe / H] gradient as predicted by the two-infall model. Filled circles with errorbars are the OC sample by Magrini et al. (2023), which are divided in three age bins: young ( Age < 1 Gyr, blue points), intermediate (1 < Age / Gyr < 3, green points) and old ( Age > 3 Gyr, red points). Solid lines are the results for the [Fe / H] gradient as predicted by the two-infall model at 0.5 Gyr (blue lines), 2 Gyr (green lines) and 5 Gyr (red lines). In this plot, and in the following Figures, we use the guiding radius of the orbit, computed as the average between the Apogalacticon and Perigalacticon radii (see 2.1), as an indication of the location of each cluster in the disk. \n<!-- image --> \nIn particular, we add at each Galactic time t a random error to the abundances of the stars formed at t (see also Spitoni et al. 2019; Palla et al. 2022). In this way, we have for each chemical element a "new abundance", defined as: \n[X / H] new ( t ) = [X / H]( t ) + N ([X / H] , σ [X / H]) , (7) \nwhere N is a random function with Normal distribution. In order to have fair comparison with the Ma23 data set adopted in this study, the standard deviation σ [X / H] corresponds to average spread observed within the OCs in the age intervals of interest, i.e. 0 . 1 < Age / Gyr < 1, 1 < Age / Gyr < 3, 3 < Age / Gyr < 7.', '4. Results': 'In this Section, we show the results of the comparison between the models presented in Section 3 and the Ma23 OC sample. In particular, in 4.1 we look at the prediction of the two-infall model. In 4.2 we instead show the gradients obtained by means of the three-infall scenario in the light of the data at disposal, discussing such a scenario in 4.3.', '4.1. Comparing observed gradients with the standard two-infall scenario': 'In Fig. 1, we show the time evolution of the [Fe / H] as predicted by the two-infall model and compared to the OC data presented in Ma23. From this sample, we remove the Blanco 1 cluster due to its extremely high internal spread observed in metallicity ( σ > 0 . 5 dex), which may hide problems in abundance derivation of their members. Coming back to Fig. 1, we see that the model captures the general trend of the data: the gradient slope clearly decreases going towards larger radii, in agreement with the trend shown by OCs. This result confirms the conclusion by DS23 of a flattening of the gradient at R ≳ 12 kpc, which requires a flat behavior of the SFE at large Galactocentric radii. \nHowever, if we focus on di ff erent age bins, we note that OCs with Age < 1 Gyr (blue points) are clearly below the prediction from the model. Moreover, we note that the metallicity of these OCs is lower than that of older clusters, i.e. the ones with 1 < Age / Gyr < 3 (green points). Therefore, such a decrease in metallicity with decreasing age cannot be reproduced by genuine chemical evolution model predictions in a scenario of continuous SF, as the one of the two-infall model in the age ranges investigated in this study: subsequent stellar generations are progressively enriching the ISM in metals, as demonstrated by model lines in Fig. 1. \nFor this reason, we try to explore if the inclusion of e ff ect of stellar migration and abundance uncertainties / spread within OCs (see 3.4 for details) may reconcile predictions and observations. The results are shown in Fig. 2. Here, the density plots represent the probability of finding a star with an abundance [X / H] at a Galactocentric radius R in a certain age bin, i.e. Age > 3 Gyr (left panel) 1 < Age / Gyr < 3 (central panel) and Age < 1 Gyr (right panel), according to the predictions by the two-infall model. In order to highlight the spread caused by the inclusion of the di ff erent e ff ects, the density plot in Fig. 2 (as well as subsequent Figures) is represented in log scale. By looking at the left and central panels of the Figure, we can see that the model predictions generally well capture the observed slope and spread within Ma23 OC sample, with just the exception of few objects with 10 ≲ R / kpc ≲ 12 in the intermediate age bin. On the other hand, the right panel clearly shows that young clusters are clearly overestimated by model predictions at di ff erent radii. This is also demonstrated by the plotted solid and dashed contour lines, showing the limits in which are contained the 68 and 95% of the predicted stars, respectively. Almost all the observed clusters are in fact outside the 2 σ of the distribution. \nHowever, as noted by Ma23, abundances in stars with lower surface gravity log g and higher mictroturbolence parameter ξ are susceptible to artifacts in stellar spectral analysis (see 2.1.1). In turn, this may a ff ect the reliability of the obtained OC abundances especially for young clusters, which are more prone to contain such stars. Therefore, we decide to remove stars with log g < 2 . 5 and ξ > 1 . 8 km s -1 , as suggested by Ma23, to minimise the possible bias introduced in stellar abundance determination within clusters. Despite such a choice costs a not negligible number of OCs for the data-model comparison, this is the only way we can assure robust estimations for di ff erent chemical abundances. \nIn Fig. 3, we show the comparison between the models already shown in Fig. 2 and the Ma23 OC sample with imposed cuts on individual stellar parameters (hereafter, restricted sample). As mentioned above, the old and intermediate age bins are barely a ff ected by the cut in stellar parameters and therefore the data-model comparison the left and central panel is very similar to what already shown in Fig. 2. For the right panel, we see instead that the restricted sample shows in general larger metallicities than those observed in the full OC sample. Nonetheless, the observed [Fe / H] are still clearly overestimated by the prediction of two-infall model, with most of the clusters still outside the 2 σ of the predicted distribution. \nThis is also highlighted in Tab. 3, where we show intercepts and slopes of gradient linear fits for the full and restricted OC samples and the two-infall model predictions in di ff erent age bins. The fits are obtained in the radial range below R < 12 kpc: this choice is done to avoid the influence of the gradient flattening at large radii, which is found to start around this Galactocen- \ntric distance (see, e.g. Carraro et al. 2007; Magrini et al. 2017; Donor et al. 2020; DS23). For the young age bin, the restricted sample shows a significant increase in the fit intercept relative to the full sample, but still this is around 0.2 dex lower than what predicted by the two-infall model. Therefore, even by considering stellar migration and abundance uncertainties e ff ects, these are not su ffi cient to explain the low [Fe / H] imprinted in the data sample. \nWe also checked whether uncertainties on cluster ages can be a source of bias for such a result. We consider either the cases of (i) "internal" uncertainties for the age derivation method adopted in this work and (ii) adoption of di ff erent isochrones grids and / or di ff erent photometric datasets to derive ages. For (i), we perform 1000 Monte Carlo resamplings of the age of the clusters according to their uncertainties. We consider an age uncertainty of log( Age ) = 0 . 2 dex, as found in Cantat-Gaudin et al. (2020) for clusters older than log( Age / yr) > 8 . 5 in their validation sample 4 . Even by randomly perturbing the cluster ages according to the uncertainties, we still find a decrease > 0 . 15 dex (precisely, from 0.70 to 0.52 dex) between the intercepts in the intermediate and young age bins relative to the ones shown in Table 3 for the restricted sample. For (ii), instead, we refer to Je ff ries et al. (2023), who compared di ff erent sets of age determinations for OCs in Gaia -ESO, including a large fraction of the OCs used in this work. We find that the di ff erence between the mean literature ages (from Je ff ries et al. 2023) and the ages adopted in this work is always lower than the uncertainty of log( Age ) = 0 . 2 dex adopted for our resampling test: therefore, the distribution of gradient in di ff erent age bins will remain similar even when changing the ingredients in the age derivation. \nTo further inspect the low [Fe / H] abundances by OCs at young ages, we also consider the metallicities derived for the CCs sample in DS23 (see Section 2). It is worth reminding that the latter sample provided homogeneous derivation of Fe abundances relative to the OC sample adopted here, preventing any additional observational bias. Both the datasets are shown in Fig. 4, together with the prediction for the young age bin by the two-infall model. It is worth noting that here and in the following Figures, we avoid showing the contours of the predicted stellar distribution to avoid overcrowding the plots. Despite the much larger abundance spread observed in DS23 sample, the Figure shows that the bulk of Cepheids observed in DS23 up to a radius about ∼ 10 kpc are also underabundant relative to the model predictions. For larger radii instead, the very large spread in [Fe / H] and the relevant abundance uncertainties for distant Cepheids (we do not show them here for sake of readability, but they arrive up to 0.25 dex), do not allow us to draw strong conclusions, despite the hint of a flattening of the radial metallicity gradient at large Galactocentric distances remains (see DS23 and references therein). In any case, the superposition between the two datasets, which we remind are obtained using di ff erent gradient tracers, strengthen the findings described in this Section. \nAs described in 3.3, we also run additional simulations using di ff erent yields for massive stars and Type Ia SNe. In this way, we test the dependence of the observed present-day gradient overestimation on nucleosynthetic calculations. \nFig. 2. Time evolution of the radial [Fe / H] gradient as predicted by the two-infall model at Age > 3 Gyr (left panel), 1 < Age / Gyr < 3 (central panel), and Age < 1 Gyr (right panel) for the two-infall model, including stellar migration and abundance uncertainties (see 3.4). Density plots show the normalised density (in log scale) of stars as predicted by the model in given Galactocentric bin of 0.2 kpc width. White contour lines show the limits within are contained the 68% (solid) and the 95% (dashed) of the predicted stellar distribution in a given radial bin. Solid lines show the results for the [Fe / H] gradient as predicted by the genuine chemical tracks, i.e. the ones shown in Fig. 1, at Ages = 0.5 Gyr (cyan line), 2 Gyr (green line) and 5 Gyr (magenta line). Data are the same as in Fig. 1 \n<!-- image --> \nFig. 3. Same as Fig. 2, but with data restricted to OCs members with log g > 2.5 and ξ < 1 . 8 km s -1 (restricted sample). \n<!-- image --> \nTable 3. Comparison between OC sample by Magrini et al. (2023) and two-infall model [Fe / H] gradient slopes and intercepts for R < 12 kpc at di ff erent ages. Data fits are shown for both the full OC sample and the one restricted to OCs members with log g > 2.5 and ξ < 1 . 8 km s -1 . \nResults for these runs are shown in Appendix A. They highlight that the theoretical predictions with di ff erent stellar yields are very similar, with an overestimation of the present-day gradient in the two-infall model and a similar evolution of the predicted gradient through cosmic time. This denotes that the conclusions of the evolutionary scenarios on the [Fe / H] gradient are marginally a ff ected by the nucleosynthesis prescriptions. Therefore, in the rest of paper, we will proceed by using the sets of yields adopted throughout this Section.', '4.2. A late time metal dilution: the three-infall model': 'To explain the unexpected decrease in the [Fe / H] gradient at late times, we explore the scenario proposed by Sp23 for the solar vicinity, where a late-time burst of SF (suggested by Gaia CMD analysis, e.g. Ruiz-Lara et al. 2020) is fueled by gas \naccretion, which in turn causes a metal dilution in the ISM gas. In particular, we extend such a scenario to the whole disk than the solar neighbourhood. \nWe start by adopting a setup for the third gas accretion very similar to the one by Spitoni et al. (2023), which is shown in Table 2 upper row. Here, the parameters of the third infall are the same of the latter paper, except for the SFE ν 3, for which we fix the proportion between the SFE in the second and third infall episode to be similar to the one used in Spitoni et al. (2023). In this way, we are able to preserve the gradient slope observed in the data at di ff erent ages: if we apply a flat SFE with radius for the third infall, this will significantly decrease the gradient slope, which is not really seen in the observations (see Table 3). Finally, at variance with Sp23, we adopt a primordial chemical composition for the third gas infall. However, we also perform an \nFig. 4. Same as Fig. 3, but showing only the results for the young age bin ( < 1 Gyr), with the addition of Classical Cepheids data sample from da Silva et al. (2023) (pink filled circles). \n<!-- image --> \nadditional run with an infall enrichment as in Sp23 (1 / 5 enriched with an abundance pattern as the one predicted for the highα phase at [Fe / H] = -0 . 75 dex), finding negligible di ff erences. \nIn Fig. 5, we show the predicted evolution of the [Fe / H] gradient by the three-infall model with parameters as for the 3INF1 setup. The results are shown for the last 3 Gyr, i.e. the ages at which this late gas accretion episode is actually acting and changing the gradient evolution. Fig. 5 shows that the e ff ect of metal dilution by the gas accretion is too strong. In particular, the upper panel highlights that we miss to reproduce the metal-rich stars in the intermediate age bin. This is due to the very strong gas dilution happening 2.7 Gyr ago, which leads the bulk of stellar production at subsolar metallicity, even at small Galactocentric radii. This is instead not seen in OC observations which, despite of the spread seen at individual radial distances, only marginally cover the region where the model expects most of the data. \nThis is also reflected by looking at Fig. 6, where we show the Age-[Fe / H] relation as predicted by the model 3INF-1 in di ff erent radial bins, i.e. inner (R < 7 kpc, left panel), solar (7 < R / kpc < 9, central panel) and outer (R > 9 kpc, right panel). Even accounting for migration and abundance uncertainties e ff ects (see density bins in the Figure), the prominent metal dilution prevents a good agreement with the data. This is especially evident at small Galactocentric radii (left panel), where the model fails to reproduce the age-metallicity trend within the last 3 Gyr. The situation is less dramatic at larger radii, where a large fraction of the OCs falls within the range of values allowed by the predictions. However, in the central and right panels we also observe that the genuine chemical evolution predictions by the model at 8 and 10 kpc (solid lines in central and left panels, respectively), slightly underestimate the metal content observed in young clusters. The slight underestimation of the present-day gradient by the model 3INF-1 is in fact observed in Fig. 5 lower panel. \nTo test further the viability of the three-infall scenario in the context of radial gradients, we allow to vary the third infall parameters relative to the values proposed in Sp23. While leaving constant the starting time of the third infall (to be consistent with SF peaks as show by, e.g., Ruiz-Lara et al. 2020) we act on all the other physical parameters, i.e. the infall timescale τ 3, the SFE \n<!-- image --> \nFig. 5. Time evolution of the radial [Fe / H] gradients at 1 < Age / Gyr < 3 (upper panel) and Age < 1 Gyr (lower panel) for the model 3INF-1, including stellar migration and average OC spread (see 3.4). Lines show the results for the [Fe / H] gradient as predicted by the model at Ages = 0.5 Gyr (cyan line), 1.8 Gyr (green dashed line) and 2.8 Gyr (green solid line). Data are the same as in Fig. 3. \n<!-- image --> \nν 3 and for the ratio between the baryonic mass accreted by the second and third infall Σ 2 /Σ 3, as already shown in Tab. 2 bottom row. \nIn addition, we allow a mild chemical enrichment for the infalling gas during the third gas accretion, with half of the gas enriched at a level of [Fe / H] = -0.75 dex with abundance pattern as the one predicted for the highα phase at that metallicity. This assumption is justified in the light of two possible invoked physical mechanisms behind the observed recent peak in SF (e.g. Isern 2019; Mor et al. 2019; Ruiz-Lara et al. 2020). On one hand, it is suggested a tight connection between the SF peak and the last pericentric passage of Sagittarius dSph (Ruiz-Lara et al. 2020; Roca-Fàbrega et al. 2021): the gas retained by Sagittarius after its first encounter with the Galaxy may have been definitely stripped in its second pericentric passage, contributing to the gas accretion together with steady cooling flow of gas from the hot corona. On the other side, Nepal et al. (2024) proposed that the peak in MW disk SF was triggered by intense MW bar activity, which was shown to trigger enhanced SF in galaxies both from a theoretical (e.g. Baba & Kawata 2020) and observational (with Integral Field Spectroscopy on local galaxies, e.g. Lin et al. 2020) points of view. Without discriminating between the two \nFig. 6. Age-[Fe / H] relation for the model 3INF-1 in di ff erent radial ranges, i.e. R < 7 kpc (left panel), 7 < R / kpc < 9 (central panel) and R > 9 kpc (right panel), including stellar migration and OC spread (see 3.4). The density plot show the normalised density of stars (in log scale) as predicted by the model in a given age bin of 0.25 Gyr width. Lines are genuine chemical tracks at 6, 8 and 10 kpc (solid) and 4 and 12 kpc (dashed). Filled circles represent the restricted sample within OCs in Magrini et al. (2023). \n<!-- image --> \nscenarios, it is therefore likely that a mild chemical enrichment have to be present in this late stage of Galactic evolution. \nIn Fig. 7 we show the predicted evolution of the [Fe / H] gradient by the three-infall model with parameters as for the 3INF-2 setup. In the upper panel, showing the metallicity gradients in the age bin 1-3 Gyr, we note that the dilution e ff ect is milder relative to what seen in Fig. 5. This is highlighted by the green solid and dashed lines, showing that the chemical evolution track before and almost 1 Gyr after the onset of the third infall, which have a metallicity di ff erence of around 0.2 dex. The smaller dilution allows us to capture the trend and spread observed in OCs in this age range: here, the location of data points broadly correspond to the regions with predicted higher probability density of stars, at variance with what happens in Fig. 5. The good agreement between OC data and the 3INF-2 model is also seen in Fig. 7 lower panel, with all the OCs in the youngest age bin falling within the range of values allowed by the model. \nAs done for model 3INF-1, we also compare the observed age-metallicity relations at di ff erent radii with the prediction of the model 3INF-2. These are shown in Fig. 8. As already noted in Fig. 7, here the dilution by the third gas accretion episode is much less prominent relative to the one seen in Fig. 6 for the model 3INF-1. This di ff erence can be explained by the larger infall timescale for model 3INF-2: here, the slower gas accumulation allows dying stars to progressively pollute with metals the ISM, leaving a more prolonged but less marked decrease in metallicty. This behaviour better reproduces the observed age-metallicity trends, as no sharp dilution e ff ects are seen in our restricted sample. Moreover, the slightly larger SFE ν 3 in model 3INF-2 allows a better agreement between the chemical evolution tracks at di ff erent radii (solid and dashed lines in Fig. 8) and the metal content in young clusters, which is instead underestimated by model 3INF-1. \nTo probe even further the three-infall scenario, we looked also at other chemical elements than Fe. However, in doing this we are more prone to the intrinsic uncertainties in models related to stellar nucleosynthesis, which are less significant when probing the Fe gradient (see Appendix A). To limit as much as possible this problem, we focus on two α -elements whose \n<!-- image --> \nFig. 7. Same as Fig. 5, but for the model 3INF-2. \n<!-- image --> \n[ α / Fe] vs. [Fe / H] abundance patterns are generally well reproduced throughout the whole MW metallicity range with the main sets of yield adopted in this work (see 3.3), i.e. O and Si (see \nFig. 8. Same as Fig. 6, but for the model 3INF-2. \n<!-- image --> \nPrantzos et al. 2018; Romano et al. 2019). Other α -elements, such as Mg, are severely underestimated when adopting massive stars Limongi & Chie ffi (2018) prescriptions (Prantzos et al. 2018; Palla et al. 2022), while most of Fe-peak elements also suffer of the additional uncertainty on the progenitor nature of Type Ia SNe, which severely alters the abundance trends (Kobayashi et al. 2020; Palla 2021). \nIn Fig. 9 we show the evolution of [O / H] (upper panels) and [Si / H] (lower panels) as predicted by the model 3INF-2, compared with our restricted sample. The Figure highlights a global agreement between the model scenario and the data. [Si / H] gradient resembles the trends already described in this Section for Fe, sharing similar gradients slopes with the [Fe / H] gradient in di ff erent age ranges. This reflects in a rather flat [Si / Fe] gradient throghout the last Gyr of galactic evolution. On the other hand, the predicted and observed [O / H] gradient is much shallower, resulting in a positive [O / Fe] gradient. For the O gradient, we also note that in the 1-3 Gyr age bin (Fig. 9 upper left panel) all OC abundances lay in the upper end of the range of values allowed by the model, not showing any sign of particular spread as opposed to Fe and Si. However, it is worth noting that for O in general we have to rely on a much lower number of abundance data, which limit us in understanding the real motivation of this discrepancy. Nonetheless, the [O / H] gradient in the youngest age bin (Fig. 9 upper right panel) is reproduced by the model predictions, well in agreement with our proposed scenario. \nIn Fig. 10 we also report the evolution of the OCs and the three-infall scenario in the [X / Fe] vs. [Fe / H] diagram in di ff erent radial regions at di ff erent ages. In particular we show the results for Si, as we can rely on a much larger sampling of OCs, for a more thorough comparison relative to O. In Fig. 10 upper panels we show the comparison for ages larger than 3 Gyr, i.e. before the onset of the third infall. Here, the model predictions including the e ff ects of migration and abundance uncertainties are shown in each radial region with colored areas. Moreover, to guide the eye, the genuine chemical tracks at 6, 8 and 10 kpc (in the left, central and right panel, respectively) are highlighted with color between 3 and 7 Gyr, i.e. the age range considered in these panels. Also, in the left and right panel tracks for 4 and 12 kpc in the age range considered are shown. Despite of the small data sampling in this age range, we can say that data are well within the predicted [Si / Fe] vs. [Fe / H] ranges considering \nmigration and abundance uncertainties e ff ects. In Fig. 10 lower panel we show instead the predictions for ages below 3 Gyr, the ones interested by the late gas accretion. As for the upper panel, we highlight the parts of the tracks with ages < 3 Gyr with color. Also in this case, the agreement between predictions and observations is remarkable, with all the clusters falling within the range of values allowed by model predictions.', '4.3. Discussion': 'Throughout this work we investigated the behaviour in radial abundance gradients as traced by Ma23 Gaia -ESO OCs, which show a decrease in their metal content towards the youngest population. Within the framework of our models of chemical evolution, that also include the e ff ects of stellar radial migration (following Frankel et al. 2018, 2020), we observe that the observed metal impoverishment of young OCs should be mainly caused by a chemical dilution from a recent gas infall episode, which in turn is triggering a late enhanced SF activity. These results confirm the conclusion by Sp23, who found the necessity of very late ( ≲ 3 Gyr of age) gas accretion episode to explain the abundance ratios in the solar neighbourhood observed in Gaia DR3 RVS spectra, and extend it to a much larger range of Galactocentric radii. \nHowever, our preferred scenario generally requires a milder metal dilution than in Sp23 in this late gas infall. In particular, our best scenario results to be more similar to the "weak dilution" scenario in Sp23, with smaller contribution of the late infall to the total mass budget and larger SFE relative to their best model. The di ff erence in the results can be explained by the di ff erent data set adopted in this work to calibrate our best model. In fact, in Sp23 the model was tuned to reproduce the abundances from young massive stars from Gaia RVS spectra (Gaia Collaboration et al. 2023a; Recio-Blanco et al. 2023) and cross matched targets in APOGEE DR17 (Abdurro\'uf et al. 2022, see their Fig. 14). Ma23 showed that such a class of objects may su ff er of biases in the determination of stellar abundances, with an evident decrease in their metallic content, and this in turn critically influence our view on [X / Fe] vs. [Fe / H] diagrams and abundance gradients. This is also shown in this work, where we see how massive giants with low log g can influence our view on the gradients of young stellar populations (see 4.1). As shown in Magrini et al. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 9. Radial and [O / H] (upper panels) and [Si / H] (lower panels) gradients at 1 < Age / Gyr < 3 (left panels) and Age < 1 Gyr (right panels) for the model 3INF-2, including stellar migration and average OC spread (see 3.4). Lines show the results for the [Fe / H] gradient as predicted by the model at Ages = 0.5 Gyr (cyan lines), 1.8 Gyr (green dashed lines) and 2.8 Gyr (green solid lines). Filled circles represent the restricted sample within OCs in Magrini et al. (2023). \n<!-- image --> \n(2023), this problem concerns spectral analysis in general and is present in various surveys (e.g. APOGEE, GALAH) regardless of the analysis method and spectral range. An e ff ort is needed to improve the model atmospheres of low gravity giant stars, and to include the e ff ect of the magnetic field to advance the analysis of these stars. Spectral analysis of these stars will have even more implications in the future because in the ELT era the brightest giant stars will give us detailed information about chemical abundances in distant galaxies (Roederer et al. 2024). \nIn the current situation, the choice of imposing motivated cuts in the stellar parameters of OC members allows us to be confident of being less prone to abundance systematics. Moreover, we compare the results of our model with all the di ff erent diagnostics available, i.e. abundance gradients in di ff erent chemical abundances as well as age-metallicity relations and [X / Fe] vs. [Fe / H] abundance patterns in di ff erent Galactocentric regions, all of them showing a good agreement between data and models. We also verify that the proposed scenario agrees with the observed values at solar radius and gradients of di ff erent physical quantities, such as SFR and gas surface densities (see Appendix B). In this way, we ensure that our proposed scenario of chemical evolution is robust in the context of the MW disk formation. \nNonetheless, the search for unbiased samples with abundances from high-resolution spectroscopy comes at expenses of \nthe sample size. In fact, our results are still limited by the moderate sample size of OCs within our restricted sample ( ≃ 50). More data are certainly needed to better probe the three-infall scenario of chemical evolution, imposing more stringent limits and maybe revising the values of the physical parameters that have been adopted in this work. In particular, further sampling in age and in Galactocentric radii will be fundamental to pursue this goal. As for example, it is worth noting that for moderate to young ages we clearly lack of data at R ≳ 12 kpc. This crucially limits our ability to draw firm conclusions on the evolution of Galactic outer regions, whose evolutionary trends are only suggested following a framework similar to that found for the innermost disk. However, reliable tracers of gradient evolution as OCs are clearly missing from the outer galaxy (see, e.g. Cantat-Gaudin et al. 2020) The CCs have started to be observed with high statistical significance also in the outermost regions of the Galaxy (e.g. da Silva et al. 2023; Trentin et al. 2024), but these objects only inform us on the present-day situation. Therefore, accurate abundances and ages of the stellar populations of the outermost Galaxy are needed, and planned (e.g. 4MOST, de Jong et al. 2012; WEAVE, Jin et al. 2024; PLATO, Rauer et al. 2022 ) and proposed (e.g. WST, Mainieri et al. 2024) facilities will undoubtedly help us in this search. \nFig. 10. [Si / Fe] vs. [Fe / H] evolution for the model 3INF-2 in di ff erent radial ranges, i.e. R < 7 kpc (left panels), 7 < R / kpc < 9 (central panels) and R > 9 kpc (right panels). The upper panels show results for ages > 3 Gyr, while the lower panels for ages < 3 Gyr. The shaded cyan, light green and magenta areas are the model prediction in a certain radial range taking into account the e ff ect of stellar migration and OC spread (see 3.4). The solid lines represent genuine chemical evolution tracks at 6, 8 and 10 kpc and are colored in the age range considered in the respective panel. Colored dashed lines are the same as colored solid lines but for the radii of 4 kpc (left panels) and 12 kpc (right panel). Colored filled circles represent the restricted sample within OCs in Magrini et al. (2023). Grey points are selected field stars from the Gaia -ESO survey. \n<!-- image -->', '5. Summary and conclusions': 'In this paper we studied the evolution of radial abundance gradients in the Milky Way (MW) by taking advantage of the sample of open clusters (OCs) from the last data release of the Gaia -ESO survey (Randich et al. 2022; Magrini et al. 2023). OCs, in fact, are among the best tracers of the shape and time evolution of the radial metallicity gradient due to their ages and distances, which can be properly measured by isochrone fitting of the complete sequence, and to high-resolution spectroscopic observations that provide precise abundances. Gaia -ESO dedicated about 30% of its observing time to provide the largest sample of precise and homogeneous stellar parameters and abundances of member stars in Galactic OCs. \nFrom a theoretical perspective, we started from the well tested revised two-infall model of chemical evolution (e.g. Palla et al. 2020b, see also Spitoni et al. 2019), which successfully reproduces data from high-resolution surveys, such as APOGEE (Ahumada et al. 2020), and extended the comparison to the newly proposed scenario of the three-infall model (Spitoni et al. 2023), proposed in the light of constraints given by Gaia star formation history (e.g. Ruiz-Lara et al. 2020) and abundance ratios (Gaia Collaboration et al. 2023a). For all the probed \nscenarios, our models take into account the e ff ects of stellar radial migration, by including well tested prescription from the literature (Frankel et al. 2018, 2020), allowing us to directly see its impact on the late evolution of radial abundance gradients. \nOur main considerations and conclusions are thus summarised as follows: \n- 1. conservatively, we excluded stars whose spectral analysis can generate metallicity bias. This choice reduces the trends between stellar parameters and metallicity within the same cluster. This solution does not solve the problem of analysing massive and / or low-gravity giants, but allows us to circumvent it until progress is made on atmosphere models for these stars and including magnetic activity;\n- 2. despite of restricting the OC sample to stars which should not hide biases in spectroscopic analysis, we find a metallicity decrease between intermediate age (1-3 Gyr) and young ( < 1 Gyr) OCs. We show that the radial metallicity gradient as traced by young OCs is overestimated by the predictions of the two-infall scenario, even by accounting for the e ff ect of stellar migration in the model. We also check whether the adoption of di ff erent nucleosynthetic yields for massive stars and Type Ia SNe, i.e. the main contributors to Fe enrichment, \nmay a ff ect this result. We find negligible di ff erences between the di ff erent runs, confirming the conclusion; \n- 3. to explain the observed low metallic content in young clusters, we propose that a late gas accretion episode triggering a metal dilution should have taken place. This is in agreement with the proposed scenario of the three-infall model by Spitoni et al. (2023) for the solar vicinity, which explains the recent star formation history and abundance ratios as derived by Gaia satellite, as a consequence of recent gas infall episode triggering enhanced SF at recent ages. It is worth noting that in this work, for the first time we extend the threeinfall scenario to the whole MW disk;\n- 4. at variance with the best model presented in Spitoni et al. (2023), we invoke a milder metal dilution for this late gas infall episode. In particular, our best scenario requires smaller contribution of the late infall to the mass budget forging the lowα disk (factor ∼ 1.5-4 lower), while larger infall timescale ( τ 3 ≃ 1 Gyr instead of τ 3 ≃ 0 . 1 Gyr) and star formation e ffi ciency ( ν 3 ≃ (2 / 3) ν 2 instead of ν 3 ≃ (1 / 2) ν 2). The di ff erence in this results can be explained by the di ff erent data sample adopted in this work. In fact, our model are thought to reproduce a sample of OCs cleaned of stars subject to biases in chemical abundance determination (Magrini et al. 2023, see also point 1 of this Section), whose class is instead considered in Spitoni et al. (2023) (see Gaia Collaboration et al. 2023a; Recio-Blanco et al. 2023). \nFurther data are definitely needed to probe the new threeinfall scenario of chemical evolution on the whole MW disk, imposing more stringent limits and maybe revising the values of the physical parameters that have been adopted in this work. \nHowever, the constraints coming from high-resolution, unbiased chemical abundances, precise ages and star formation histories from multiple tracers (e.g. Isern 2019; Mor et al. 2019; Ruiz-Lara et al. 2020) allow us to consider the proposed model as a robust and viable scenario for the MW disk formation. \nAcknowledgements. The authors thank the referee for the careful reading of the manuscript and the useful comments improving the paper content. MP acknowledges financial support from the project "LEGO - Reconstructing the building blocks of the Galaxy by chemical tagging" granted by the Italian MUR through contract PRIN2022LLP8TK\\_001. LM, ES, MF, and SR thank INAF for the support (Large Grant EPOCH) and MP, LM, and CVV for the MiniGrant Checs. LM, MF, and SR acknowledge financial support under the National Recovery and Resilience Plan (NRRP), Mission 4, Component 2, Investment 1.1, Call for tender No. 104 published on 2.2.2022 by the Italian Ministry of University and Research (MUR), funded by the European Union - NextGenerationEU- Project \'Cosmic POT\' Grant Assignment Decree No. 2022X4TM3H by the Italian Ministry of Ministry of University and Research (MUR). MM thanks the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 279384907 - SFB 1245, the State of Hessen within the Research Cluster ELEMENTS (Project ID 500 / 10.006) for financial support. This research was supported by the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany\'s Excellence Strategy EXC-2094 - 390783311. Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programmes 188.B3002, 193.B-0936, and 197.B-1074. These data products have been processed by the Cambridge Astronomy Survey Unit (CASU) at the Institute of Astronomy, University of Cambridge, and by the FLAMES / UVES reduction team at INAF / Osservatorio Astrofisico di Arcetri. These data have been obtained from the Gaia -ESO Survey Data Archive, prepared and hosted by the Wide Field Astronomy Unit, Institute for Astronomy, University of Edinburgh, which is funded by the UK Science and Technology Facilities Council. This work makes use of results from the European Space Agency (ESA) space mission Gaia . 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D. 2012, AJ, 144, 95\n- Zhang, M., Xiang, M., Zhang, H.-W., et al. 2021, ApJ, 922, 145", 'Appendix A: Probing present-day gradients overestimation with different yield sets': 'As mentioned in 3.3 and 4.1, we perform additional model runs testing di ff erent massive stars and Type Ia SN yields than the reference ones used in this paper (Limongi & Chie ffi 2018 and Iwamoto et al. 1999, respectively). \nIn the following Figures we show the results of this experiment, by reporting the gradient evolution (left panels) and the gradient in the young age bin considering the e ff ects of stellar migration and abundance uncertainties in the model (right panels) In particular, Fig. A.1 shows the predicted radial [Fe / H] gradient obtained by adopting the massive star yields from Kobayashi et al. (2006) instead of those from Limongi & Chie ffi (2018). Fig. A.2 instead reports the predicted radial [Fe / H] gradient obtained by adopting the Type Ia SN yields from an equal mixture of Leung & Nomoto (2018) (nearMCh progenitors) and Leung & Nomoto (2020) (subMCh progenitors) instead of those Iwamoto et al. (1999). Regarding the latter Figure, it is worth mentioning that no significant changes are obtained by employing di ff erent mixtures in Type Ia yields.', 'Appendix B: Gradients in physical quantities from the three-infall model': 'As mentioned in 4.3, we checked whether our three-infall scenario reproduces the constraints in other gradients than those in chemical abundances, i.e. the present-day gradients in the physical quantities. \nResults of the comparison between our best model 3INF-2 and literature constraints are shown in Fig. B.1 and B.2. In Fig. B.1, we show the prediction of the model for the SFR surface density. In the left panel, we show the time evolution of this quantity at di ff erent radii, with the present-day SFR observed in the solar vicinity (see Prantzos et al. 2018) shown as reference. In the right panel, we show instead the comparison between the predicted present-day gradient and measurements from the literature (Rana 1991; Stahler & Palla 2005; Green 2014). In Fig. B.2, we display the same scheme as Fig. B.1, but for the gas surface density, where the reference value for the solar vicinity and the observed gradients are taken from Dame (1993); Nakanishi & Sofue (2003, 2006). For a detailed discussion on the adopted data sets, we refer to Palla et al. (2020b).', 'Appendix C: The restricted OCs sample': 'As described in 2.1.1, to avoid observational biases in the computation of the mean abundance of OCs, we build a restricted sample of member stars, where only stars with log g > 2.5 and ξ < 1.8 km -1 are considered to compute the mean cluster abundances. \nIn Tab. C.1, we provide the average abundances obtained with this membership selection, as well as all the obtained cluster parameters (see also Viscasillas Vázquez et al. 2022). \n<!-- image --> \nFig. A.1. Predicted radial [Fe / H] gradient from the model 2INF using Kobayashi et al. (2006) instead of Limongi & Chie ffi (2018) yields for massive stars. Left panel: same as Fig. 1, but showing the data for the restricted OC sample. Right panel: same of Fig. 3, but showing only the results for the young age bin ( < 1 Gyr). \n<!-- image --> \n<!-- image --> \nFig. A.2. Predicted radial [Fe / H] gradient from the model 2INF using Leung & Nomoto (2018, 2020) instead of Iwamoto et al. (1999) yields for Type Ia SNe. Left panel: same as Fig. 1, but showing the data for the restricted OC sample. Right panel: same of Fig. 3, but showing only the results for the young age bin ( < 1 Gyr). \n<!-- image --> \n<!-- image --> \nFig. B.1. Left panel: SFR surface density time evolution at 4, 8, 12 and 16 kpc. The present-day observed value for the solar vicinity is taken from Prantzos et al. (2018). Right panel: present-day radial SFR surface density gradient. Data are from Rana (1991) (black points with errorbar), Stahler & Palla (2005) (black squares with errorbar), Green (2014) (black crosses). \n<!-- image --> \n<!-- image --> \nFig. B.2. Left panel: gas surface density time evolution at 4, 8, 12 and 16 kpc. The present-day observed value for the solar vicinity is taken from Dame (1993); Nakanishi & Sofue (2003, 2006). Right panel: present-day radial gas surface density gradient. The dashed curve is the average between the Dame (1993) and Nakanishi & Sofue (2003, 2006) data sets. The grey shaded region represents the typical uncertainty at each radius (see Palla et al. 2020b for more details). \n<!-- image --> \nTable C.1. Excerpt from the list mean cluster abundances and cluster parameters from our adopted restricted sample (see 2.1.1). \nNotes. For the orbital parameters, we use the G alpy code, with the axis-symmetric potential MWP otential 2014 (Bovy 2015). The complete Table will be available at the CDS.'}
2024JATIS..10c8001S	The advent of backilluminated complementary metaloxidesemiconductor CMOS sensors and their wellknown advantages over chargecoupled devices make them an attractive technology for future Xray missions. However numerous challenges remain including improving their depletion depth and identifying effective methods to calculate perpixel gain conversion. We have tested a commercial Sony IMX290LLR CMOS sensor under Xray light using an Femmlmprescripts mmlnone 55 radioactive source and collected Xray photons for 15 consecutive days under stable conditions at regulated temperatures of 21C and 26C. At each temperature the data set contained enough Xray photons to produce one spectrum per pixel consisting only of singlepixel events. We determined the gain dispersion of its 2.1 million pixels using the peak fitting and the energy calibration via correlation ECC methods. We measured a gain dispersion of 0.4 at both temperatures and demonstrated the advantage of the ECC method in the case of spectra with low statistics. The energy resolution at 5.9 keV after the perpixel gain correction is improved by 10 eV for singlepixel and all event spectra with singlepixel event energy resolution reaching 123.60.2 eV close to the Fano limit of silicon sensors at room temperature. Finally our long data acquisition demonstrated the excellent stability of the detector over more than 30 days under a flux of 10SUP4SUP photons per second.	2024-07-01T00:00:00Z	['2024JATIS..10c8001S', '10.48550/arXiv.2409.05954', 'arXiv:2409.05954', '10.1117/1.JATIS.10.3.038001', '2024arXiv240905954S']	['X-ray detectors', 'complementary metal-oxide-semiconductor sensors', 'energy calibration', 'X-ray spectral performance', 'Physics - Instrumentation and Detectors', 'Astrophysics - Instrumentation and Methods for Astrophysics']	Xray spectral performance of the Sony IMX290 CMOS sensor near Fano limit after a perpixel gain calibration	2,024	200	0.31	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	0	https://arxiv.org/pdf/2409.05954.pdf	{'X-ray spectral performance of the Sony IMX290 CMOS sensor near Fano limit after a per-pixel gain calibration': "Benjamin Schneider a,* , Gregory Prigozhin a , Richard F. Foster a , Marshall W. Bautz a , Hope Fu a , Catherine E. Grant a , Sarah Heine a , Jill Juneau a , Beverly LaMarr a , Olivier Limousin b , Nathan Lourie a , Andrew Malonis a , Eric D. Miller a \na Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA, USA b Universit'e Paris-Saclay, Universit'e Paris Cit'e, CEA, CNRS, AIM, 91191 Gif-sur-Yvette, France \nAbstract. The advent of back-illuminated complementary metal-oxide-semiconductor (CMOS) sensors and their well-known advantages over charge-coupled devices (CCDs) make them an attractive technology for future X-ray missions. However, numerous challenges remain, including improving their depletion depth and identifying effective methods to calculate per-pixel gain conversion. We have tested a commercial Sony IMX290LLR CMOS sensor under X-ray light using an 55 Fe radioactive source and collected X-ray photons for ∼ 15 consecutive days under stable conditions at regulated temperatures of 21°C and 26°C. At each temperature, the data set contained enough X-ray photons to produce one spectrum per pixel consisting only of single-pixel events. We determined the gain dispersion of its 2.1 million pixels using the peak fitting and the Energy Calibration by Correlation (ECC) methods. We measured a gain dispersion of 0.4% at both temperatures and demonstrated the advantage of the ECC method in the case of spectra with low statistics. The energy resolution at 5.9 keV after the per-pixel gain correction is improved by ≳ 10 eV for single-pixel and all event spectra, with single-pixel event energy resolution reaching 123 . 6 ± 0 . 2 eV, close to the Fano limit of silicon sensors at room temperature. Finally, our long data acquisition demonstrated the excellent stability of the detector over more than 30 days under a flux of 10 4 photons per second. \nKeywords: X-ray detectors, CMOS sensors, Energy calibration, X-ray spectral performance. \n* Benjamin Schneider, [email protected]", '1 Introduction': 'For X-ray astronomy, charge-coupled devices (CCDs) have been successfully used over the past decades, 1-5 and are still proposed for upcoming and future X-ray missions. 6-9 CCDs are a mature technology that has benefited from continuous development, achieving very low noise and excellent uniformity. The next generation of X-ray missions requires higher readout speed, a lower power consumption and a better radiation tolerance while maintaining high quality imaging and spectral performance. Recent improvements of complementary metal oxide semiconductor (CMOS) detectors and the use of back-side illuminated CMOS offer a promising alternative to CCDs for future X-ray instruments. In particular, CMOS detectors provide a high readout frame rate per second (fps), low readout noise, low power consumption, and can be operated at room temperature. In addition, the active pixel design of CMOS prevents the inherent and growing charge transfer inefficiency affecting CCD performance over time due to space radiation. Their relatively low cost compared to CCDs makes them an attractive technology for wide-field X-ray instruments. The wide-field X-ray telescope (WXT), on board the Einstein Probe (EP) 10 mission, launched in early 2024, and its pathfinder the Lobster Eye Imager for Astronomy (LEIA) flight experiment, 11 launched in 2022, are the first X-ray missions to use scientific CMOS sensors in space environment and have already demonstrated promising results. 12,13 However, many challenges remain before CMOS sensors can be considered a viable alternative for more X-ray applications. CMOS devices suffer from lower depletion depths and reduced X-ray quantum efficiency compared to CCDs, 14 \nthough manufacturers have been improving X-ray optimized designs. 15,16 The highly parallelized readout architecture of CMOS devices, which provides their high readout rate, also leads to higher variation in gain from pixel-to-pixel compared to that from the single readout node of a CCD array. This gain variation can degrade the spectral energy resolution. 15-19 Improved procedures to characterize the pixel-to-pixel gain variation inherent to CMOS devices can significantly improve their spectral energy resolution. One commercial CMOS sensor that has shown promise for detection of X-rays is the Sony IMX290LLR ( 1920 × 1080 2 . 9 µ m pixels), which is optimized for optical light and routinely used for astrophotography or security camera systems. However, its low noise ( ∼ 2 e- rms) and back-side illuminated design makes it suitable for detecting X-rays. Previous works have successfully tested the device under X-ray light and demonstrated its excellent spectral performance from 250 eV to 6.4 keV. 20,21 In this paper, we investigate an efficient technique to characterize per-pixel conversion gain calibration using a CMOS sensor. We measure and correct the per-pixel gains of the 2.1 million pixels composing the Sony IMX290 CMOS sensor. We then derive the 5.9 keV spectral performance of the device after correcting for gain dispersion. We finally report on the stability of the system over time, based on the long acquisition periods required to measure the gain of each pixel.', '2 Experimental setup': 'Our setup employs the IDS Imaging UI-3862LE-M camera offering a board-level, low-cost, compact and versatile system. The board is equipped with a Sony STARVIS I IMX290 CMOS sensor (see Table 1), which is a low-noise monochrome back-side illuminated device making it suitable to detect X-ray light. The board uses USB3.1 for command/control, power and image transfer enabling a readout speed up to 120 fps. As is standard for X-ray imaging, the cover glass over the top of the sensor package was removed to ensure the detection of X-ray photons below 10 keV. The camera was installed in a cryostat and operated under vacuum. A thermoelectric cooler (TEC) was used to cool down the camera and a proportional-integral-derivative (PID) controller was implemented to maintain a constant temperature ( ± 0.2°C) during measurements. The hot and cold sides of the TEC were clamped to copper interfaces to maximize the thermal conduction and were monitored by two resistance temperature detectors (RTDs). The hot side was connected to a power feedthrough to evacuate the heat produced by the camera and TEC in the cryostat. External to the power feedthrough, a cold plate was attached and liquid cooled by a chiller at 10°C, which maintained the hot side at a constant temperature and improved the TEC cooling capacity (see Fig. 1). \nPreliminary tests illuminated the sensor with a radioactive source of 55 Fe producing Mn-K α (5.90 keV) and Mn-K β (6.49 keV) emission lines. During that phase, the analog gain of the image sensor was tuned to obtain a conversion gain of 1 e-/ADU and maintain the electronic noise as low as possible. We also illuminated the system with a source composed of 210 Po and Teflon to produce fluorescence lines of C-K (0.53 keV) and F-K (0.68 keV). Both sources validate the ability of this sensor to detect soft X-ray photons down to 277 eV, as previously observed in 20. Only the results from the 55 Fe measurements are presented in this paper.', '3 Real time data processing': 'We used the uEye interface for Python (PyuEye), provided by IDS, to set the camera settings (e.g., fps, pixel clocks, gain) and access full raw frames. The hot pixel correction offered by the API \nFig 1 Picture of the experimental setup. The CMOS sensor is visible at the center of the image and surrounded by its readout electronics. The camera is connected to the cold side of the TEC (not visible) via a copper plate. The hot and cold sides are held together with PTFE plastic screws to minimize the heating load. \n<!-- image --> \nwas disabled to prevent single X-ray events from being considered as hot pixels. The camera was operated at 10 fps, with each frame producing 4.2 MB of data, thus generating ∼ 2.5 GB every minute. For experimental testing, the large amount of data generated raised multiple challenges, especially in terms of storage capacity and post-processing time. In addition, pixels hit by Xray photons represent only a small fraction of the entire image. To overcome these challenges, we developed an algorithm to process the frames in real time to only extract and save valid Xray events which drastically reduced the memory usage by only keep the relevant information deposited by X-ray photons in each frame. \nFor real-time X-ray event finding, the imaging sequence starts by taking a series of initial frames, of at least 200 frames, to compute the offset map. The map is then regularly updated to mitigate, for instance, the emergence of new flickering hot pixels over time. When a new frame is transferred, the offset map is subtracted in real time and the resulting pixel amplitudes are compared to an event threshold where all pixels above it are considered as valid X-ray events. The event threshold can be defined as a fixed value or pixel-dependent using the initial frames of the sequence. In practice, we use a fixed event threshold value for all pixels tuned based on the best resulting spectral performance observed. The island around each valid event is extracted and finally saved. Given the small pixel size of the sensor (2.9 µm) and the likely small thickness of the device ( < 10 µm), we consider islands from 3 × 3 up to 7 × 7 pixels to ensure the extraction of X-ray events extend over multiple pixels. Then, a list of events considered as valid X-ray photons is regularly saved. Possible remaining hot pixels are rejected during the offline post-processing based on their anomalous count rate. Although the frames can be partially reconstructed at a later \nTable 1 Main characteristics of the IDS UI-3862LE camera equipped with the Sony IMX290 CMOS sensor. \nCamera model \nUI-3862LE-M \nSensor model \nIMX290LLR-C \nImaging area \n5.610 mm × 3.175 mm \nNumber of pixels \n1936 × 1096 \nPixel size \n2.9 µm \nPixel clock range \n20 - 474 MHz \nFrame rate \nup to 120 fps \nExposure time \n0.011 ms - 120 s \nReadout noise \n1.9 e- rms \nADC \n12 bit \nShutter \nRolling shutter \nCamera power \n0.9 - 1.5 W \ntime, one of the main limitations of this approach is that event extraction cannot be run again with other parameters such as different event threshold values or larger event island sizes. The algorithm was successfully run on a Raspberry Pi 4B and a Intel Core i7-13700K processor. The number of frames per second that can be processed by the CPU depends on many factors, such as the counting regime or the event island size. For our study, running at 10 frames per second allowed us to extract ∼ 7,000 events per second.', '4.1 Calibration methods': "In CMOS architecture, charge-voltage conversion and the first amplification stage are implemented directly at the pixel level. This means that each pixel has its own amplification circuit, likely generating non-uniform conversion gain between pixels. In addition to the Fano and electronic noise, small variations in the pixel gains can broaden the energy line and degrade the energy resolution of the sensor. To overcome this degradation, correction of the pixel-to-pixel gain dispersion can potentially improve the spectral performance. The X-ray calibration process involves illuminating the detector with a source emitting X-ray photons at known energies, ideally with several lines to increase calibration accuracy. The correlation between the observed spectrum (in ADU) and the known spectrum (in keV) can be measured using different approaches. A common method is the so-called 'peak fitting'. It consists of fitting every line in the observed ADU spectrum with a Gaussian function to determine the line centroid and find the relation between these centers and the expected incident photon energy. A linear relation usually provides a reliable energy calibration, but a more complex relation (e.g., quadratic) can be employed to correct for non-linear readout electronic effects over a large energy range. Another possible approach is the 'Energy Calibration by Correlation' (ECC) 22,23 initially employed for CdTe semiconductor detectors. The method \nrelies on finding the maximum of correlation between a synthetic spectrum (template) of the incident source and the observed spectrum. Similarly to the peak fitting approach, the ECC method can be used to find linear or more complex relations. ECC offers multiple possible advantages over peak fitting. It allows a more robust and accurate calibration using lines and background as a whole, provides flexible ways of tracking the energy calibration evolution over time using a previous calibrated dataset, and has been shown to outperform peak fitting in the case of a low-statistics spectrum. 22 The latter is particularly interesting for energy calibration of CMOS sensors, where a large number of pixel gains needs to be calculated, causing significant challenges in generating sufficient statistics for each pixel. Both methods were used in our study, and their results have been compared to determine the strength of each approach (Sec. 4.4).", '4.2 Data collection for the energy calibration': 'The Sony IMX290 CMOS sensor is composed of 2.1 million pixels, and its precise energy calibration requires the measurement of a gain for each pixel. This requires generating a spectrum for each pixel, where each pixel spectrum is composed of single-pixel events to avoid mixing the gains of adjacent pixels. It also means collecting data under stable conditions to limit the contribution of external effects (e.g. temperature) on the gains measured. \nTo keep the acquisition time at a reasonable level, we illuminated the sensor with a bright 55 Fe radioactive source ( ∼ 38 MBq), which produces about 7,000 events/s on the detector. The camera was run at 10 fps to limit the pile-up and reduce the dark current to a negligible level for the operating conditions. To limit the data flow and save storage space, we performed real-time event extraction to capture only the relevant information, as described in Sec. 3. The camera temperature was kept within 0.2°C throughout the acquisition using a TEC and a PID loop. \nThe 55 Fe radioactive source produces two main energy lines at 5.90 keV and 6.49 keV. The fraction of single-pixel events obtained at 5.9 keV was measured to be ∼ 30%. Previous work on the IMX290 attempted to measure the gain dispersion by dividing the sensor area into 5 × 5 subareas and measured a dispersion of 0.14%. 20 Although this method can give a rough idea of the gain dispersion, it can still mix in small scale gain variations caused by mismatch variations of the pixel elements (e.g., transistors, photodiodes) during the manufacturing processes. \nWe performed 55 Fe simulations to determine the optimal number of single-pixel events per pixel to accurately recover a given gain dispersion. We estimated that for a gain dispersion of ∼ 0.1%, more than 1,000 events per pixel are needed with both calibration methods. A more precise estimate of the optimal number of individual events is discussed further in Sec. 4.3. Based on these estimates and the count rate produced by the radioactive source, we targeted 14 consecutive days of acquisition to achieve ∼ 1,200 events per pixel. \nOur acquisition sequence consists of two main steps. First, the offset map is calculated using 300 frames (30 s of data collection). The setup configuration prevents removal of the radioactive source during this acquisition time. We thus derived the offset map by applying a sigma clipping to mitigate the effect of X-ray events on the calculation. The second phase consists of collecting 6,000 consecutive frames (10 min of acquisition) and performing a real-time X-ray event extraction using the offset map as described in Sec. 3. We repeat this sequence every 10 min for 14 days to track the evolution of the offsets and to ensure efficient real-time event extraction. When the camera is turned on, the temperature may temporarily fluctuate faster due to the additional heat produced by the readout electronics until a new equilibrium is reached. In order to have only datasets with \nFig 2 Top panel: Count map of single-pixel events obtained after 14 days of acquisition with the 55 Fe source. The upper and right insets represent the sum of counts by column and row, respectively. The upper right inset shows a zoom of a small area of the sensor outlining the square patterns observed. Lower panel: Noise map derived from 300 frames at the beginning of one of the 10-min acquisition sequences. The upper and right insets show the average noise by column and row, respectively. The upper right inset shows a zoom in on a small region of the sensor, highlighting the pixels with higher noise than average producing the squared patterns observed on the count map. \n<!-- image --> \na stable camera temperature, the first 1.5 h of datasets were removed from our post-processing analysis.', '4.3 Post-processing and per-pixel spectrum': 'The data set collected over 14 days generated 1 TB of data. Processing such a large volume of data remains challenging with current laboratory computing resources. Thus, we performed the post-processing analysis on the MIT SuperCloud, 24 a supercomputing cluster at MIT that currently provides individual allocation of 384 CPUs. The post-processing analysis was designed and optimized to minimize computational time, resulting in approximately one day of computation on 384 CPUs. Single-pixel events were first extracted from each 10-minute event list file and then redistributed to their respective pixels along the 14 days of acquisition. The top panel of Fig. 2 shows the count map of the 2.1 million pixels from the measurement. We determined a median count of 1368 and observed a non-uniform distribution of counts across the sensor area. The lower left corner had a count of more than 2,000, while the upper right corner had a count of ∼ 500. This could be due to a misalignment of the 55 Fe source with the sensor creating an inhomogeneous illumination pattern. The potential impact of this inhomogeneity on our analysis is discussed further in Sec. 4.4. We also observed that a significant number of pixels exhibited a square pattern, where the central pixel has a higher count rate than the adjacent pixels (see the inset zoom region in Fig. 2). Similarly, when we extracted two-pixel events from the datasets, we noticed an opposite pattern trend in the count map, where the central pixel has a lower count rate than the adjacent pixels. These patterns appear to be associated with pixels with higher noise levels, as shown in the noise map in the bottom panel of Fig. 2. In this respect, pixels with higher noise levels cause the migration of single-pixel events into two-pixel events in adjacent pixels. This is a consequence of using the same split threshold for all pixels during the event extraction process. Implementing a pixel-dependent split threshold based on individual noise levels would mitigate this effect and result in a more homogeneous count map. \nWe ended up with one spectrum for each of the 2.1 million pixels, made up entirely of singlepixel events. Figure 3 shows an example of two spectra from two pixels in the same column (#662) but in different rows (#192 vs. #242). The two main emission lines produced by the radioactive source are clearly detected at 5.90 keV (Mn-K α ) and 6.49 keV (Mn-K β ). In addition, the Si-K escape line of Mn-K α (4.16 keV) is marginally detected. Both spectra were calibrated in energy with the same gain, and a small energy shift between them is noticeable, contributing to the broadening of the lines when combined. We measured a spectral resolution of 122 eV at 5.9 keV using a Gaussian fit, close to the intrinsic Fano limit at this temperature (119 eV) and consistent with a 2e- rms noise device.', '4.4 Gain measurement and optimal counts per spectrum': 'Two different calibration methods, peak fitting and ECC, were used to determine the gain for each pixel (see Sec. 4.1). We used the enhanced version of ECC with adaptive mesh refinement (AMR) 23 to discretize the parameter space and reduce the computational time. \nFor both methods, we assumed a linear calibration in energy, expressed as E = gain × PHA + offset , where PHA is the pulse height amplitude. Because the 55 Fe source produces two emission lines within a limited energy range (0.5 keV), it can be challenging to constrain a small residual offset sometimes required to improve the energy calibration of some detectors. We first performed \nFig 3 55 Fe spectra of single-pixel events from the same column (#662) but in different rows (#192 vs. #242) of the sensor area calibrated with the same gain value. The Mn-K α (5.90 keV) and Mn-K β (6.49 keV) lines produced by the 55 Fe radioactive source and the Si-K escape line of Mn-K α (4.16 keV) are detected in both spectra. A small horizontal shift is visible between the two spectra. The gray dashed line shows the best-fit Gaussian model used to measure the spectral energy resolution for the pixel ID (662, 242). \n<!-- image --> \nan energy calibration using the peak fitting method, considering two free parameters (gain and offset). The ratio of the offset to its error returned a median value of 1.2, suggesting that the residual offset, expected to be close to zero, cannot be well-constrained with the current data. This is supported by the real-time data processing, in which the offset map subtracted from the frames prior to extracting the X-ray events is updated every 10 minutes. This ensures that any systematic offset drift is minimized over time and that the residual offset should tend to zero. We therefore assumed an offset of 0 for the energy calibration in the rest of our analysis. \nThe gain map derived by the peak fitting method considering only one free parameter (gain) is shown in Fig. 4. The gain distribution shows a normal distribution with a mean of 294 ADU/keV and a 1-sigma dispersion of 0.4%. This observed variation exceeds that reported in the literature using 5 × 5 binning regions (0.14%), 20 suggesting potentially significant interdevice variability or discrepancies due to the method used. The upper inset of Fig. 4 represents the average gains for each column. A difference was observed between the average gains of odd and even columns. This difference is constant along the columns ( ∼ 1 ADU/keV) and larger than the standard deviation of the average column gains. This trend was not observed in the right inset of the 2D gain map (Fig. 4), where even and odd rows have consistent average gains along rows. The feature observed for the average gain columns is thought to be due to the parallelism of the readout specific to CMOS technology, where the signal from an entire row can be read out simultaneously. In this design, each column has its own second stage signal amplifiers, correlated double sampling (CDS) and analog-to-digital converters (ADC). In addition, these components can be distributed on either side of the sensor area to save space, with even-column components on one side and odd-column \nFig 4 Gain map derived from 55 Fe spectra of single-pixel events using the peak fitting method on the Mn-K α and Mn-K β lines. The upper and right insets represent the average gain by column and row, respectively. Average gains for even (odd) columns and rows are indicated by blue (black) lines. The upper right inset shows a zoom of a small area of the sensor outlining the small scale gain variation. \n<!-- image --> \ncomponents on the other. No significant difference was observed between the gains measured by ECC and peak fitting methods, with ˜ gain ECC = 293 . 66 ± 1 . 18 ADU/keV and ˜ gain peak fitting = 293 . 59 ± 1 . 18 ADU/keV. \nWe investigated the performance and advantages of each method with respect to the number of single-pixel events in the spectrum (sample size). Due to the inhomogeneity observed in the count map in Fig. 2, which could bias the results, we performed this analysis by randomly selecting 10,000 pixels within the 90th to 95th percentile of the count distribution. This criterion yields an initial pixel sample of 104,737 pixels with counts between 1664 and 1720 photons. For each pixel, we generated a set of spectra with different sample size (10, 50, 100, 200, 500, ..., events) by randomly selecting events from their initial distribution. For each sample size, we applied both ECC and peak fitting to the set of 10,000 spectra and measured their gains. The energy resolution of the Mn-K α line was then determined by combining the 10,000 spectra into a single spectrum after correcting for individual gain variations. The evolution of spectral resolution as a function of sample size is shown in Fig. 5. Both methods converge to the same energy resolution when spectra are composed of more than ∼ 500 single-pixel events. At lower counts, the ECC method outperforms peak fitting, showing an improvement in spectral performance for statistics greater than 10 photons. In contrast, peak fitting requires a minimum of 100 photons to achieve results similar to using a single gain for all pixels. Given a gain dispersion of 0.4%, this analysis provides guidance on the optimal statistics required to accurately correct for the per-pixel gain dispersion of the sensors. Since all pixels in our data set have more than 492 single-pixel events, it is unlikely that the inhomogeneous count pattern observed in the count map (Fig. 2) has affected our per-pixel gain correction. \nFig 5 Energy resolution of single-pixel events at 5.9 keV (Mn-K α ) as a function of sample size. The energy resolution is measured using a Gaussian fit on a spectrum combining 10,000 pixels from the high-count region of Fig. 2 and corrected for per-pixel gain variation using the peak fitting (red) and ECC (blue) methods. The dashed gray line represents the energy resolution assuming the same gain for all pixels. The Fano limit expected at room temperature is indicated by a dashed dotted line. \n<!-- image -->', '5.1 Per-pixel gain correction': 'Correcting the per-pixel variation of CMOS sensors is expected to improve the spectral performance of the device. To validate our gain measurement, we corrected the gain dispersion using the gain map visible in Fig. 4 on a 10-min data set, corresponding to about 4 million photons detected. The combined spectrum of single-pixel and all events before and after the gain correction is shown in Fig. 6. As expected, the energy resolution is improved after correcting the per-pixel gain variation from 136 . 1 ± 0 . 2 eV to 123 . 6 ± 0 . 2 eV ( ∼ 12 eV) for single-pixel events and from 144 . 5 ± 0 . 6 eV to 136 . 1 ± 0 . 6 eV ( ∼ 10 eV) for all events spectra. \nThe improvement observed after the per-pixel gain correction confirms our ability to measure and correct the gain variation. The spectral performance of single-pixel events is approaching the theoretical limit of silicon detectors of 119 eV at room temperature (300 K) for an electron-hole pair creation energy of 3.67 eV and a Fano factor of 0.118. 25 The difference observed for singlepixel events compared to the Fano limit could be mainly attributed to the electronic noise of ∼ 2erms measured for this device. In addition, the electronic noise might increase the observed difference between single-pixel and all events to some extent by the fraction of energy loss below the split threshold, known as the charge sharing effect, which depends on the detector geometry and multiplicities. This effect can be mitigated empirically or by Monte Carlo simulations, 5 but was not corrected for in this analysis. Additional broadening of the lines compared to the Fano limit \nFig 6 Spectrum of single-pixel events (left) and all events (right) before per-pixel gain correction (in black) and after per-pixel gain correction (in blue or red). The gray, dark blue and dark red dashed lines show the best Gaussian fit used to derive the energy resolution. \n<!-- image --> \ncould result from the use of the same split threshold for the entire sensor area, producing the square patterns observed in Fig. 2 and the overestimation of the energy of some events.', '5.2 Temperature dependence of gains': 'After a pause of 6 days, we acquired a second data set under the same configuration and operating conditions except that the camera temperature was increased from 21°C to 26°C. The objective was to investigate the temperature dependence of the gain, in particular the per-pixel dispersion. The left panel of Fig. 7 shows the comparison of the gain distribution at the two temperatures. The gain dispersion at both temperatures was consistent at 0.4%, but the distribution showed a median shift of 0.15 ADU/keV (0.05%) when the temperature was raised by 5°C. At first order, this suggests that all pixel gains are affected in a similar way as the temperature changes, and that the gain dispersion remains constant within the temperature variations observed during data acquisition ( ≲ 0.2°C). In a broader context, measuring the per-pixel gain dispersion at one temperature may offer the possibility of mitigating the gain dispersion at another temperature, at least within a certain temperature range. Additional data sets over a broader temperature range are needed to confirm the temperature dependence of the gains.', '5.3 Device stability over time': 'Our long acquisition data sets allow us to measure the stability of the device over several days at a high count rate ( ∼ 7,000 counts per second). In the right panel of the Fig. 7, the total number of counts per 10 min file from the Mn-K α ( 5 . 7 < E < 6 . 1 keV) is shown for both temperatures. Hot pixels have been removed from each file and the gain temperature dependence observed in the left panel of Fig. 7 has been corrected. We observed that at both temperatures the count rate in the Mn-K α line decreases progressively with time. The measured decay is consistent with the expected radioactive decay rate of 55 Fe (half-life decay of 2.737 years) over this period. However, we noticed an additional constant offset of ∼ 0.5% in the counts to the expected trend at 26°C. By \nFig 7 Left panel: Histogram of the 2.1 million gains at 21°C (blue) and 26°C (orange). Right panel: Total number of counts per 10-minute file from the Mn-K α line ( 5 . 7 < E < 6 . 1 keV) as a function of time at 21°C (blue) and 26°C (orange). The dashed black line represents the expected radioactive decay rate of 55 Fe during the observation period. The lighter orange circles indicate count rates at 26°C, reduced by a constant offset of 0.5% to obtain a 55 Fe decay rate consistent with the one measured on the 21°C data set. The exact cause for this offset remains to be explored in future work. \n<!-- image --> \nremoving this offset, we can obtain a decay consistent with the data set acquired at 21°C. The exact reasons for this offset will be the subject of future work. \nAll pixels with an abnormal count rate (more than 10 counts per 10 min) were considered as hot pixels. The number of hot pixels was very stable over time for both temperatures, with a median number of hot pixels of 39 ± 6 at 21°C and 44 ± 6 at 26°C. At 26°C, we also saved the offset and noise map every 10 minutes to track their evolution over time. We found that the noise was very stable, with an equivalent electronic noise of 2 . 10 +0 . 05 -0 . 01 e- rms. The offset level was configured to a value of 200 ADU, and we measured a value of 199 . 76 +0 . 60 -0 . 63 ADU, corresponding to a fluctuation of about 1 e- over 16 days. Our measurements demonstrated the excellent stability in terms of readout noise, hot pixels and dark current level of the camera over more than 30 days of data acquisition.', '6 Conclusion': 'We investigated the per-pixel gain calibration of the Sony IMX290 CMOS sensor composed of more than 2.1 million pixels. We collected X-ray photons from a 55 Fe radioactive source under stable conditions over a period of two weeks at room temperature (21°C), in order to obtain more than 500 single-pixel events per pixel and generate a single-event spectrum per pixel. We used two energy calibration methods (peak fitting and ECC) to measure the gain of all pixels, and investigated the possible advantages of each method. ECC outperformed peak fitting in the case of low statistics and offers a promising prospect for per-pixel gain correction of CMOS sensors. Our gain measurements showed good overall gain homogeneity and a gain dispersion of 0.4% for the entire sensor area. However, a constant difference along the columns of approximately 1 ADU/eV was noted between the average gains of the odd and even columns. Once corrected for per-pixel gain dispersion, we confirmed that the spectral performance was improved by ≳ 10 eV at 5.9 keV. The spectral performance of single-pixel events was found close to the Fano limit at room temperature as expected for a 2e- rms readout noise device and demonstrated the impressive \nspectral performance of the sensor relative to its low cost. We repeated the measurement at a temperature +5°C higher (26°C) and observed a small shift (0.05%) in the central value of the gain distribution, while the gain dispersion remained stable at 0.4%. Furthermore, both data sets confirmed the excellent stability of the device with regard to noise, hot pixels and offsets over 30 days of acquisition under a count rate of more than 7,000 counts per second. Further work on the Sony IMX290 CMOS sensor is needed to confirm its potential use in X-ray astronomy, such as measuring its X-ray quantum efficiency.', 'Disclosures': 'The authors declare no relevant financial interests or potential conflicts of interest related to this manuscript.', 'Code, Data, and Materials Availability': "The data and codes utilized in this study are not publicly available and are the property of the Massachusetts Institute of Technology. They can be requested from the authors at [email protected]. The Energy Calibration by Correlation (ECC) code is the property of the Commissariat 'a l'Energie Atomique et aux Energies Alternatives (CEA).", 'Acknowledgments': "The authors acknowledge the MIT SuperCloud and Lincoln Laboratory Supercomputing Center for providing (HPC, database, consultation) resources that have contributed to the research results reported within this paper. They also acknowledge support from the MIT Kavli Institute's Research Investment Fund.", 'References': "- 1 L. Struder, U. Briel, K. Dennerl, et al. , 'The European Photon Imaging Camera on XMMNewton: The pn-CCD camera,' A&A 365 , L18-L26 (2001).\n- 2 G. P. Garmire, M. W. Bautz, P. G. Ford, et al. , 'Advanced CCD imaging spectrometer (ACIS) instrument on the Chandra X-ray Observatory,' in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy. , J. E. Truemper and H. D. Tananbaum, Eds., Society of PhotoOptical Instrumentation Engineers (SPIE) Conference Series 4851 , 28-44 (2003).\n- 3 K. Koyama, H. 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2024A&A...691A..33R	Context. The Vera C. Rubin Observatory is set to discover 1 million supernovae SNe within its first operational year. Given the impracticality of spectroscopic classification at such scales it is mandatory to develop a reliable photometric classification framework. Aims. This paper introduces a novel method for creating spectral time series that can be used not only to generate synthetic light curves for photometric classification but also in applications such as Kcorrections and bolometric corrections. This approach is particularly valuable in the era of large astronomical surveys where it can significantly enhance the analysis and understanding of an increasing number of SNe even in the absence of extensive spectroscopic data. Methods. By employing interpolations based on optimal transport theory starting from a spectroscopic sequence we derive weighted average spectra with high cadence. The weights incorporate an uncertainty factor for penalizing interpolations between spectra that show significant epoch differences and lead to a poor match between the synthetic and observed photometry. Results. Our analysis reveals that even with a phase difference of up to 40 days between pairs of spectra optical transport can generate interpolated spectral time series that closely resemble the original ones. Synthetic photometry extracted from these spectral time series aligns well with observed photometry. The best results are achieved in the V band with relative residuals of less than 10 for 87 and 84 of the data for type Ia and II respectively. For the B g R and r bands the relative residuals are between 65 and 87 within the previously mentioned 10 threshold for both classes. The worse results correspond to the i and I bands where in the case of SN Ia the values drop to 53 and 42 respectively. Conclusions. We introduce a new method for constructing spectral time series for individual SNe starting from a sparse spectroscopic sequence and demonstrate its capability to produce reliable light curves that can be used for photometric classification.	2024-11-01T00:00:00Z	['arXiv:2409.10701', '2024A&A...691A..33R', '10.1051/0004-6361/202449170', '2024arXiv240910701R', '10.48550/arXiv.2409.10701']	['methods: data analysis', 'methods: statistical', 'supernovae: general', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Instrumentation and Methods for Astrophysics']	A novel optimal transportbased approach for interpolating spectral time series Paving the way for photometric classification of supernovae	2,024	200	0.51	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.10701.pdf	{'Paving the way for photometric classification of supernovae': "M. Ramirez 1 , 2 , G. Pignata 3 , 2 , Francisco Förster 2 , 4 , 5 , 6 , Santiago González-Gaitán 7 , Claudia P. Gutiérrez 10 , 11 , B. Ayala 1 , 2 , Guillermo Cabrera-Vives 2 , 12 , 13 , Márcio Catelan 2 , 8 , 9 , A. M. Muñoz Arancibia 2 , 5 , and \nJ. Pineda-García 1 , 2 \n- 1 Instituto de Astrofisica, Departamento de Fisica, Facultad de Ciencias Exactas, Universidad Andres Bello, Fernandez Concha 700, Las Condes, Santiago RM, Chile.\n- e-mail: [email protected]\n- 2 Millennium Institute of Astrophysics, Nuncio Monseñor Sotero Sanz 100, Of. 104, Providencia, Santiago, Chile.\n- 3 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile.\n- 4 Data and Artificial Intelligence Initiative (IDIA), Faculty of Physical and Mathematical Sciences, Universidad de Chile, Chile.\n- 5 Center for Mathematical Modeling, Universidad de Chile, Beauchef 851, Santiago 8370456, Chile\n- 6 Departamento de Astronomía, Universidad de Chile, Chile\n- 7 CENTRA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal.\n- 8 Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile.\n- 9 Centro de Astroingeniería, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile.\n- 10 Institut d'Estudis Espacials de Catalunya (IEEC), Gran Capità, 2-4, Edifici Nexus, Desp. 201, E-08034 Barcelona, Spain. \n11 \nInstitute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s \n/ \nn, E-08193 Barcelona, Spain. \n- 12 Department of Computer Science, Universidad de Concepción, Chile.\n- 13 Data Science Unit, Universidad de Concepción, Edmundo Larenas 310, Concepción, Chile. \nReceived January 8, 2024; accepted September 12, 2024", 'ABSTRACT': 'Context. The Vera C. Rubin Observatory is set to discover 1 million supernovae (SNe) within its first operational year. Given the impracticality of spectroscopic classification at such scales, it is mandatory to develop a reliable photometric classification framework. Aims. This paper introduces a novel method for creating spectral time series that can be used not only to generate synthetic light curves for photometric classification, but also in applications such as K-corrections and bolometric corrections. This approach is particularly valuable in the era of large astronomical surveys, where it can significantly enhance the analysis and understanding of an increasing number of SNe, even in the absence of extensive spectroscopic data. \nMethods. By employing interpolations based on optimal transport theory, starting from a spectroscopic sequence, we derive weighted average spectra with high cadence. The weights incorporate an uncertainty factor for penalizing interpolations between spectra that show significant epoch di ff erences and lead to a poor match between the synthetic and observed photometry. \nResults. Our analysis reveals that even with a phase di ff erence of up to 40 days between pairs of spectra, optical transport can generate interpolated spectral time series that closely resemble the original ones. Synthetic photometry extracted from these spectral time series aligns well with observed photometry. The best results are achieved in the V band, with relative residuals of less than 10% for 87% and 84% of the data for type Ia and II, respectively. For the B , g , R, and r bands, the relative residuals are between 65% and 87% within the previously mentioned 10% threshold for both classes. The worse results correspond to the i and I bands, where, in the case of SN Ia, the values drop to 53% and 42%, respectively. \nConclusions. We introduce a new method for constructing spectral time series for individual SNe starting from a sparse spectroscopic sequence, and demonstrate its capability to produce reliable light curves that can be used for photometric classification. \nKey words. methods: data analysis - methods: statistical - (Stars:) supernovae: general', '1. Introduction': 'Supernovae (SNe) are transient astronomical events that occur during the terminal phases of stellar evolution. SNe play an important role in galactic evolution, influencing both chemical evolution and energy dynamics across galaxies (e.g., Ceverino & Klypin 2009). Additionally, due to their luminous and homogeneous intensity, type Ia SNe serve as robust distance indicators, enhancing our understanding of cosmic scales (e.g., Riess et al. 1998; Perlmutter et al. 1999; Abbott et al. 2019). \nThe current classification scheme of SNe is mainly based on di ff erences in their spectra, but initially they were classified into Type I and Type II based on the presence or absence of hydrogen lines in their spectra (Minkowski 1941). Furthermore, a variety of subtypes emerge depending on the manifestation and / or strength of other chemical elements (Wheeler & Levreault 1985; Elias et al. 1985; Harkness et al. 1987; Wheeler & Harkness 1990). On one hand, Type I has been divided into Ia, Ib, and Ic, where SNe Ia show strong signatures of Si II near their peak, while SNe Ib are defined by the presence of strong He I features. \nSNe Ic do not show strong lines of either He or Si II. On the other hand, a subtype of type II SNe is SN IIb, where spectra at early phases are dominated by strong H I lines, but weaken with time while He I features strengthen. Finally, the spectra of SN IIn are dominated by prominent narrow emission lines of the Balmer series at all phases. From the explosion mechanism point of view, SNe can be broadly classified into thermonuclear and core collapse (CC) explosions. Thermonuclear SN coincides with the SN Ia observational class and is believed to be the result of the thermonuclear incineration of a carbon-oxygen white dwarf in a binary system (e.g, Hoyle & Fowler 1960). On the other hand, CC explosions include all the other SN observational types mentioned above (II, IIb, IIn, Ib and Ic) and are expected to be the result of the collapse of the iron-degenerate core of a massive star (M ⪆ 8 M ⊙ ) (for a review see Smartt 2009) \nThe Legacy Survey of Space and Time (LSST) that will be carried out by the Vera C. Rubin Observatory with the Simonyi Survey Telescope is projected to discover more than 10 7 SNe spanning a considerable redshift range during its ten years of operations (LSST Science Collaboration et al. 2009). Given the time investment that would be required to carry the spectroscopic classification described above on the LSST sources, the development of reliable photometric classification algorithms is fundamental. This will enable the complete realization of the enormous potential of the photometric dataset produced by current and future transient surveys. The performance of a photometric classification based on machine learning techniques (e.g., Ishida &de Souza 2013; Lochner et al. 2016; Charnock & Moss 2017; Boone 2019; Villar et al. 2019; Möller & de Boissière 2020), regardless of the specific algorithm used, strongly depends on the dataset employed for training (e.g., Richards et al. 2012; Karpenka et al. 2013; Millard & Richardson 2015). In this respect, spectro-photometric time series o ff er a valuable resource, as they enable the construction of synthetic light curves at various redshifts, which are ideal for use as training sets. \nA key aspect of training sets is that they must be representative of the diversity of the classes that are the targets for classification. The first spectral time series were constructed by combining spectra from multiple SNe belonging to the same class. These spectral "templates" are representative of the entire class of the objects that were employed in their construction. Synthetic light curves for individual objects are then generated by warping the spectra to match their observed color. Spectral templates have been generated for Type Ia SNe (e.g., Nugent et al. 2002; Hsiao et al. 2007; Lu et al. 2023) and CC SNe (e.g., Kessler et al. 2010, 2019). Nevertheless, this approach can potentially reduce the intrinsic diversity between the members of a given class, introducing biases if the spectral templates are used to generate training sets for photometric classification. This bias is particularly important for CC SNe, because of the large heterogeneity that they display within their classes. Building up spectral time series for individual SNe belonging to a given family preserves the diversity, making them particularly suitable for generating synthetic light curves for training sets. Vincenzi et al. (2019) compiled a set of 67 spectral time-series across various SN types (II, IIn, IIb, Ib, Ic, Ic-BL), integrating photometric and spectroscopic data from the literature with Gaussian processes (GPs). \nIn this context, we introduce a novel method for constructing spectral time series of SNe based on optimal transport (OT) and the Wasserstein barycenter. OT theory has found large applications in a variety of scientific fields, from economics to biology, physics, data science, and machine learning. OT has also been applied in the field of astronomy. Frisch et al. (2002), for example, demonstrated that the reconstruction of the early density \nfluctuations of the Universe is e ff ectively an optimization problem, leveraging optimal mass transportation techniques. Similarly, Levy et al. (2021) advanced this field by developing a fast semi-discrete OT algorithm, providing a unique and e ffi cient approach to modeling these early cosmic structures. Nikakhtar et al. (2022, 2023) apply OT for reconstructing biased tracers in redshift space and baryon acoustic oscillations, enhancing our understanding of the structure of the Universe. Rawson & Hultgren (2022) employ OT to interpolate high-resolution images from low-resolution data, selecting the best match through a process that optimizes for the smallest Wasserstein distance, e ff ectively refining the interpolation into a more precise reconstruction. Here, our method utilizes well-calibrated spectrophotometric data from individual events, and is therefore a versatile, data-driven approach applicable to various types of SNe. \nThis paper is organized as follows: In Sect. 2 we detail the data sample and discuss our selection criteria. Section 3 is dedicated to our methodology, where we elucidate the foundations of OT. Section 4 presents the tests conducted with models and the recipe we use for the production of the spectral time series. In addition, we validate our approach with observed photometry. Finally, we summarize our findings in Sect. 5.', '2. Data sample': 'The spectra and light curves used for this work were retrieved from the open supernova catalog (Guillochon et al. 2017), from The Weizmann Interactive Supernova Data Repository ( WISeREP 1 ) (Yaron & Gal-Yam 2012), and from the literature. The data cleaning process is described in Appendix A. \nFrom the whole sample, we selected SNe of type Ia and II for which photometry is available in at least two bands within the Johnson-Cousins (JC) or Sloan (SDSS) photometric systems and for which a minimum of three spectra have been obtained (at least one before maximum light in the case of type Ia SNe). These selection criteria yielded an initial sample of 458 type Ia and 138 type II SNe. We assess the calibration of the spectra by comparing the flux from synthetic photometry Fsyn with that from observed photometry Fobs across as many bands as possible through the following ratio: \nFsyn Fobs = Kx , (1) \nwhere the Kx corresponds to a given band x . Fsyn is calculated using the following equation: \nFsyn = F 0 R f ( λ ) R ( λ ) d λ R R ( λ ) d λ , (2) \nwhere f ( λ ) represents the flux density of the spectrum and R ( λ ) is the band response. Both of these variables are functions of the wavelength λ . F 0 is the zero point (see Appendix B). We only compute Fsyn when at least 95% of the response band is covered by the spectrum. \nFobs is obtained by interpolating the observed light curve using the Automatic Loess Regression (ALR) technique as outlined in Rodríguez et al. (2019) at the corresponding spectrum date. \nFor each spectrum, we compute the Kx values for as many bands as possible. This allows us to rescale the spectrum against \nthe photometry by dividing its flux by the median of the Kx values ( ˜ K ) and also to evaluate the quality of its calibration, computing the median absolute deviation (MAD; hereafter MAD ( K )) of these values. As the aim of this work is to assess the performance of OT, all the results presented in this paper are based on spectra with a relative error ( MAD ( K ) / ˜ K ) of less than 10%, which make up our "golden sample." The final golden sample consists of 110 SNe Ia and 31 SNe II, which are reported in Table C.1.', '3. Methodology': 'In this work, we employ OT to interpolate between SN spectra. Introduced by Monge (1784), OT is a mathematical framework designed to help find the most e ffi cient way to move "mass" between di ff erent distributions, understood in this context as an abstract representation of resources, probabilities, and distributed data, which in our case are spectra. For a more detailed explanation of the computational foundations, please refer to Villani (2009), Peyré & Cuturi (2020), and Zhang et al. (2021). \nMoving to the mathematical formalism, we first consider the simpler, 1D case, where we move something from position ( x 1 , x 2 , x 3 , . . . , xN ) to a new location ( y 1 , y 2 , y 3 , . . . , yN ). The problem is to find the optimal transport plan T ( xi ) = yi that minimizes the total transportation costs CT . The total cost is given by \nCT = N X i c ( xi , T ( xi )) , (3) \nwhere c is the cost function of moving from one point to another. Equation 3 establishes that, for each transport ( x ) -→ ( y ) , the unit cost of the transport depends on the quantity to be transported (the transportation plan), given by T , and the starting point. By adding over all possible origins and destinations, we get the total cost. With the OT plan found by minimizing Eq. 3, it is possible to construct the Wasserstein distance as in Kolouri et al. (2017). Wasserstein distance is a metric for quantifying the distance between two probability distributions, and this distance is a way of measuring how much work it takes to transform one distribution into another, also referred to as the cost of moving. The Wasserstein barycenter is then the distribution that results from minimizing the total sum of these distances (costs) to all other distributions. It is like finding a middle point, not in terms of physical distance, but in terms of how much you would have to change each distribution to reach this middle point. A parameter α , ranging from 0 to 1, is defined to control the interpolation between the two distributions. An α value of 0 interpolates entirely to the first distribution, while a value of 1 interpolates to the second one. We show a practical illustration of how OT works in Fig. 1. In the top panel, the starting and final distributions are shown in blue and red, while the Wasserstein barycenter and L 2 are shown in green and black, respectively. The barycenter L 2, also known as the Euclidean barycenter, is calculated as the average of each corresponding pair of points of the two distributions, that is, it provides a linear interpolation. As visible in the plot, the L 2 barycenter results in a bimodal distribution, while the Wasserstein barycenter produces a distribution that is transitional between the initial ones. The lower panel of Fig. 1 shows the interpolation path of the Wasserstein barycenter for di ff erent weights, that is, di ff erent values of α , illustrating the transition of the interpolations from one distribution to the other. \nMoving closer to the subject of this paper, in Fig. 2 we apply L 2 and OT methods to compute interpolations among black \nFig. 1. Comparison between the L 2 barycenter and the Wasserstein barycenter. The starting and final distributions are in blue and red, respectively. The black dashed distribution represents the barycenter calculated using the L 2 distance, whereas the green dashed distribution is computed using the Wasserstein distance. The bottom panel illustrates the interpolation path of the Wasserstein barycenter for di ff erent α values, highlighting its transition from the starting distribution to the final one. \n<!-- image --> \nbodies. As visible in the top panel, the OT method e ff ectively replicates the thermal evolution of black body radiation. On the contrary, the L 2 barycenter leads to interpolations that do not accurately reflect the physical changes expected in a real black body as its temperature varies (see bottom panel). \nFig. 2. Examples of black body interpolations using Wasserstein and L 2 barycenters. In both panels, black lines show black body radiation at different temperatures. In the top panel, OT interpolations are represented in blue, while in the bottom panel, linear interpolations are presented in red. \n<!-- image --> \nEncouraged by these results, in this work we use the Python Optimal Transport (POT) library from Flamary et al. (2021) to \nperform the interpolations in time between spectra of SNe. We specifically employ the barycenter computation function 2 for interpolating between two spectra, which takes a matrix A that contains the distributions, in our case the spectra, a loss matrix ( M ), the regularization term ( reg ), and the weights of each distribution. The normalized loss matrix M of size n × n was calculated with the utils function provided by the POT library 3 . \nIn this case, an α value of 0 interpolates entirely to the spectrum of the first date, while a value of 1 interpolates to the spectrum of the second date. Consequently, a value of 0.5 results in a spectrum interpolated exactly halfway between the two dates. Considering the scenario where the initial and final phases are given by pha = 2 and phb = 6, respectively, to interpolate to intermediate phases phc = [3 , 4 , 5], α values of [0.25, 0.5, 0.75] must be employed. These α values correspond to the proportional distances of phc within the interval defined by pha and phb . This parameter shapes the weight array input for the ot.bregman.barycenter() function using the form [1 -α , α ] for the two distributions.', '4. Analysis of the spectral interpolation performance': 'To assess the performance of OT for producing spectral time series, we conducted three sets of tests. To avoid spurious residuals introduced by the noise and miscalibration that naturally a ff ect the observed spectra, for the first two sets of tests, we use publicly available model spectra from Dessart et al. (2014) and Dessart et al. (2013), for Type Ia and II SNe, respectively. For consistency with the tests performed on observed data, we calculated synthetic photometry for the JC and Sloan bands from the spectral time series models; this photometry is used to compute ˜ K and MAD ( K ) for each interpolated spectrum. A phase to each spectrum is assigned, taking as a reference the epoch of the maximum light in the V band obtained by fitting a second-order polynomial around the peak for SNe Ia, while for SNe II we considered the midpoint of the transition phase ( tPT ) as defined in Olivares et al. (2010).', '4.1. Optimal transport on model spectra': 'For our purposes, the most basic form of interpolation involves pairing two spectra. Therefore, our initial set of tests focused on this procedure. The test is conducted in the following way: Let us assume we have spectra at phases ph 1, ph 2, and ph 3 with ph 1 < ph 2 < ph 3. We first interpolate a spectrum at ph 2 from the spectra at ph 1 and ph 3; this spectrum is then rescaled against the photometry, as detailed in Sect. 2, and compared to the actual ph 2 spectrum, computing the mean relative spectral residual, ϵ , which is defined as follows: \nϵ = 1 n n X λ = λ 0 \| fM λ -fI λ \| fM λ ! , (4) \nwhere fM and fI represent the flux of the model and the flux of the interpolated spectra, respectively. This procedure is applied across all spectral pair combinations, utilizing a moving grid and progressively increasing the time interval between them. \nAs mentioned in Sect. 3, OT employs an α value to define the position of the interpolation, which in this study corresponds \nto the phase of the spectrum we aim to compare with. It is worth mentioning that we only test interpolations with an α value ranging from 0.45 to 0.55. This is because an α of 0.5 is the most challenging case to interpolate, given that it represents the largest phase distance from the two spectra. For comparison, we also compute the same interpolations between the same pairs of spectra but employ standard linear interpolation. The results of this first set of tests can be seen in Fig. 3, where we indicate with ∆ ph the phase di ff erence between the pair of spectra. As expected, for small ∆ ph , the relative spectral residual is low for both interpolation methods. OT shows a slower increase in ϵ , which remains below 10% even for phase gaps of 40 days for type II and 25 days for type Ia SNe. In contrast, linear interpolation sees a faster increase in ϵ , exceeding 40% as phase gaps become larger. To account for the increment of ϵ introduced by an increasing ∆ ph between spectra, we fit a plane in the ∆ ph -ph -ϵ space for both SNe Ia and SNe II. This plane allows us to assign an uncertainty Σ ( ϵ ), which we use to penalize interpolation between pairs of spectra with large ∆ ph . \nFig. 3. Relative spectral residuals ϵ as a function of ∆ ph . Blue circles correspond to the linear interpolation and the red ones to the OT. The size and darkness of the circles increase with the phase of the interpolated spectrum. \n<!-- image --> \nIn the second set of tests, to include as much information as possible in generating a given interpolated spectrum, we consider not just one pair of spectra, but all possible combinations of pairs between four spectra. This approach is applied to both the linear and the OT method, ensuring that the same information is used in both cases. While a larger number of spectra could enhance the interpolation by providing more information, this scenario is not often realistic, given that not all SNe have an extensive number of available and well-calibrated spectra. The iterative procedure is illustrated in Fig. 4, where the black boxes represent the four spectra used in the interpolation, and the white boxes indicate the position of the spectrum being interpolated. Colored lines connect the pairs of spectra used to compute the interpolated one. As in the previous test, following interpolation, we rescaled the spectra against the photometry. If there are \nthree connections, this means that the weighted average spectrum is computed with these three interpolated spectra. As illustrated, once all possible interpolations among the existing spectrum pairs are completed, the grid moves to include an additional spectrum in the interpolation process, a concept exemplified in cycle 4. \nThe flux of the final spectrum f ( λ ) is computed as a weighted average over all the interpolated spectra, as follows: \nf ( λ ) = P n i = 1 fi ( λ ) ( MAD ( K ) 2 i +Σ ( ϵ ) 2 i ) P n i = 1 1 ( MAD ( K ) 2 i +Σ ( ϵ ) 2 i ) , (5) \nwhere fi ( λ ) is the flux for the di ff erent spectra for a given λ and MAD ( K ) has the same meaning as in Sect. 2. \nThe results of this second set of tests are displayed in Fig. 5, where the red circles represent the relative spectral residuals for OT interpolation and the blue circles represent those for the linear interpolation. As in Fig. 3, the size of the circles increases with phase. As expected, the residuals of the weighted spectra are larger than in Fig. 3 at a given ∆ ph . This is because the weighted average includes spectra with larger ∆ ph than in Fig. 3. However, the inclusion of more than one pair of spectra in computing the interpolated spectrum is crucial in the case of observed spectra because it reduces the e ff ect of noise and miscalibration.', '4.2. Optimal transport on observed spectra': 'To evaluate the performance of OT on observed data, we generated a set of time series from the golden sample of spectra outlined in Sect. 2. For each SN, we calculated interpolated spectra with a daily cadence from all the potential combinations of paired spectra within the sample. This implies that for a specific phase, we generate as many interpolated spectra as the number of combinations that include this phase. The process is the same as illustrated in Fig. 4; however, in this instance, we did not limit ourselves to using only four spectra. Our goal is to use as much data as possible, and so we included all spectra with ∆ ph shorter than 40 days. For a given epoch, for each interpolated spectrum, we compute MAD ( K ), Σ ( ϵ ), and then rescale the spectrum using the ˜ K value. As the edges of spectra are usually poorly calibrated, we incorporate a weight term, C ( λ ), which decreases linearly from 1 to 0 within a 50 Å window located at the boundaries of the interpolated spectrum. The flux of the combined spectrum is computed as follows: \nf ( λ ) = P n i = 1 fi ( λ ) Ci ( λ ) ( MAD ( K ) 2 i +Σ ( ϵ ) 2 i ) P n i = 1 Ci ( λ ) ( MAD ( K ) 2 i +Σ ( ϵ ) 2 i ) . \nϕ = Fsyn -Fobs Fobs , (7) \nΦ = Fsyn -Fobs σ obs , (8) \nwhere σ obs is the error associated with the observed flux. Figures 7 and C.1 present these results for ϕ and Φ , respectively. \nWe observe that the relative photometric residuals are generally below 10%, and we do not see significant di ff erences between SN Ia and SN II types, with the only exception being the I and i bands, where residuals are much lower in the latter than the former. We believe that these larger residuals are mostly due to the di ff erence in natural bands of the instruments with which the observed photometry was obtained. Both I and i bands include the Ca II NIR triplet feature and, especially for the I band, the red cuto ff can vary significantly between imagers (e.g., Pignata et al. 2008a), including a di ff erent fraction of the P-Cygni profile. In SN Ia, the Ca II NIR triplet feature is much stronger than in SN II, which can explain the larger residuals visible in the plots. In the case of Φ , we observe similar trends. In most bands, the di ff erence between synthetic and observed photometry falls within three times the error of the observed photometry. In the previous test, although the spectra are only scaled by a constant factor ˜ K , this factor still contains information from all the available photometric observations. To ensure that this information is not entering into the estimation of the relative residual for any given band, we conducted an additional test using a leave-one-out cross-validation technique. For this test, we selected only SNe with spectra covered by at least four bands. This approach allows us to compute ˜ K using at least three bands, even after leaving one out. For example, if we are computing the relative residual between the synthetic and observed photometry for the B band, and the available bands are BVRI , the scaling factor ˜ K is computed using only the VRI bands. The results of this test are shown in Figs. 8 and C.2. As can be seen, for some bands, the distribution of residuals became slightly wider ( BVgr ) or slightly narrower ( Ri ) with respect Figs. 7 and C.1, respectively. The only case where the distribution became significantly narrower is in the case of SN Ia for the I band, where the fraction of residuals below 0.1 magnitudes increased from 41.7% to 60.6%. The latter supports the hypothesis that a significant portion of the residuals in the I band are due to di ff erences in the natural bands of the instruments used to observe the SNe, rather than to a decrease in the performance of the interpolation method within the wavelength range covered by this band. These results demonstrate that the spectral time series we generate with OT can produce accurate and reliable synthetic light curves that closely resemble the observed photometry. The performance across these di ff erent SN types is consistent, showing the versatility of our method.', '(6) 5. Conclusions': "Having produced our spectral time series, examples of which are reported in Fig. 6, we proceed to compute the synthetic photometry using Eq. 2 and compare it with the observed one. This is done for those bands for which at least 95% of the total response is covered by the spectrum. Linear interpolation is applied to the resulting synthetic light curves, enabling us to evaluate Fsyn for the corresponding Fobs dates. We then compute the relative photometric residuals ϕ and Φ as follows: \nIn this study, we assessed the performance of the OT interpolation in producing spectral time series. Using SN models from Dessart et al. (2013) and Dessart et al. (2014), we first tested the OT interpolation between pairs of spectra, finding that even with phase di ff erences of 40 days, the relative spectral residuals ( ϵ ) stay below 20% and 10% for SNe Ia and SNe II, respectively (Fig. 3). To include more information in the generation of a given interpolation, we included all possible combinations between four synthetic spectra. Again the relative spectral residuals ( ϵ ) stay below 20% and 10% for SN Ia and SN II, respectively \nFig. 4. Averaging scheme for the second test: Black boxes represent the spectra used for interpolation, and white boxes indicate the target phase for interpolation. Colored lines link pairs of spectra, which are used to create the interpolated spectrum at the position marked by the white box. To compute the final spectrum at a given phase, a weighted average is calculated across all interpolated spectra. \n<!-- image --> \nFig. 5. Same as Fig. 3, but in this case Eq. 5 has been used to compute the flux of the spectra. We note that in this figure a given ∆ ph refers to the largest phase interval of the interpolated spectral pairs that enter in the weighted mean. \n<!-- image --> \n(Fig. 5). Our findings indicate that the error associated with the OT method increases at a slower rate with the phase di ff erence compared to the linear method. This means that OT demonstrates a superior capability in preserving the spectral shape as the phase gap increases. \nFinally, using the observed spectra of our golden sample of SNe described in Sect. 2, we computed spectral time series, from which we constructed synthetic light curves in the BVRIgri bands. We find that a significant portion of the relative photometric residuals ( ϕ ) for both SN types generally fall below 10% error (Fig. 7), indicating a good match between the synthetic and observed light curves. For Type Ia SNe, this is particularly evident in the B , V , r , and g bands, while the I and i bands show more variability. Nevertheless, we find evidence that, for the I band, di ff erences between the natural bands of the instruments used to observe the SNe contribute at least in part to the residuals. In the case of Type II SNe, the residuals also show a good match across all bands, with notably high percentages of residuals within acceptable error margins. When examining the residuals weighted by observational error ( Φ ), we observe that the majority of the di ff erences between synthetic and observed photometry for both types of SNe are within three times the error of the observed photometry, although some specific bands for Type Ia SNe show larger deviations (Fig. C.1). \nIn conclusion, the OT interpolation method emerges as a robust and innovative approach for creating spectral time series; it e ff ectively performs accurate interpolations even in scenarios with substantial phase gaps between spectra, demonstrating its capability to produce high-quality synthetic light curves. These spectral time series are highly suitable for generating training sets, which are essential for photometric classification algorithms. This aspect is particularly important in large astronomical surveys where extensive spectroscopic data may not be available. Additionally, the series may also be useful in performing K-corrections and bolometric corrections. \nAcknowledgements. The authors acknowledge support from National Agency for Research and Development (ANID) grants ANID-PFCHA / Doctorado Nacional / 2020-21202606 (MR), ANID-PFCHA / Doctorado Nacional / 202221221964 (BA). Support from the Chilean Ministry of Economy, Development, and Tourism's Millennium Science Initiative through grant ICN\\_12009, awarded to the Millennium Institute of Astrophysics (GP, MC, AMMA, FF); by \nFig. 6. Spectral time series for Type Ia SN2006cp and Type II SN2006be. The observed spectra are shown in blue, while our spectral time series, calculated with the weighted average OT, are displayed in orange. The black numbers represent the phase with respect to the maximum flux and the tpt , respectively. \n<!-- image --> \nFONDECYT Regular grant 1231637(MC), FONDECYT Regular 1200710 (FF) and by ANID's Basal project FB210003(MC,FF). BASAL project FB210005 (AMMA), BASAL Center of Mathematical Modeling Grant PAI AFB-170001 (FF). SGG acknowledges support by FCT under Project CRISP PTDC / FIS-AST31546 / 2017 and Project No. UIDB / 00099 / 2020. CPG acknowledges financial support from the Secretary of Universities and Research (Government of Catalonia) and by the Horizon 2020 Research and Innovation Programme of the European Union under the Marie Skłodowska-Curie and the Beatriu de Pinós 2021 BP 00168 programme, from the Spanish Ministerio de Ciencia e Innovación (MCIN) and the Agencia Estatal de Investigación (AEI) 10.13039 / 501100011033 under the PID2020-115253GA-I00 HOSTFLOWS project, and the program Unidad de Excelencia María de Maeztu CEX2020-001058-M.", 'References': "1997, NATO Advanced Study Institute (ASI) Series C, Vol. 486, Thermonuclear supernovae \n- Abbott, T. M. 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The z-scores tell us how far a value is from the average in units of standard deviation and is defined as follows: \nzi = xi -µ , (A.1) \nσ \nwhere µ is the mean and σ is the standard deviation of the population xi . In this work, we use a robust statistics approach using the median M and the median absolute deviation ( MAD ) rather than the µ and σ as proposed by Whitaker & Hayes (2018). The authors employed a modified z-scores outlier detection technique to identify spikes: \nzi = 0 . 6745 ∇ xi -M MAD . (A.2) \nHere MAD = median ( \| x -M \| ), ∇ xi = xi -xi -1, and the value 0.6745 corresponds to the 0.75th quartile of a normal distribution. We take the absolute value of the z-scores and select a threshold value of 3.5 as is recommended in Hoaglin (2013). \nSpikes are corrected by calculating the mean values within a 2m + 1 window surrounding them, where m represents an adjustable input value that determines the window size. By default, m = 3, but we may alter this value and the threshold to better accommodate the data. \nAdditionally, we have incorporated into the CRD the functionality to interactively remove cosmic rays. This allows users to input the specific wavelengths between which a spike is observed. This feature proves beneficial when dealing with particularly noisy spectra that contain numerous spikes, which may not necessarily be attributed to cosmic rays. Once the cosmic rays are detected it is removed connecting the two edges of the window with a straight line. \nWe visually inspected each spectrum and identified telluric lines using a reference spectrum. These lines were then removed through linear interpolation between the edges of each telluric feature.', 'Appendix B: Standard stars': "For the calibration of F 0 in Eq. 2 we compare the synthetic photometry of three stars reported in Stritzinger et al. (2005), specifically HR0718, HR4468, and HR4963 with the observed photometry reported in Cousins (1980, 1984) for the BVRI bands and Fukugita et al. (1996) for the g'r'i' bands. For computing the synthetic photometry we use the BVRI and g'r'i' bands reported in Bessell et al. (1998) and Fukugita et al. (1996), respectively. \nFor each band we also computed the root mean square, mean, and relative error σ of the values obtained for the three stars, which are reported in Table B.1. \nTable B.1. Mean F 0 value and relative error σ for each band.", 'Appendix C: Extra figures and tables': 'The following figures display the weighted relative photometric residuals Φ , as referenced in Sect. 4.2. Fig. C.1 corresponds to the standard test, while Fig. C.2 shows the results using the leave-one-out cross-validation test. Table C.1 presents the final golden sample as referenced in Sect. 2 \nFig. C.1. Relative photometric residuals Φ for each of the BVRIgri filters. This residuals are weighted by the error in the photometry. The red dotted line denotes the zones where \| Φ \| < 3 . 0. The legend also includes the fraction of relative residuals that fall outside the range shown in the histogram ( \| Φ \| > 10 . 0). \n<!-- image --> \nFig. C.2. Same as Fig. C.1, but with the relative residuals computed using the leave-one-out cross-validation test. \n<!-- image --> \nTable C.1. SNe included in this work. \nM. Ramirez et al.: A novel optimal transport-based approach for interpolating spectral time series \n( b ) References: (1) Silverman et al. (2012a); (2) Stahl et al. (2020); (3) Yaron & Gal-Yam (2012); (4) Brown et al. (2014); (5) Smitka et al. (2015); (6) Walker et al. (2015); (7) Smartt et al. (2015); (8) Cellier-Holzem et al. (2012); (9) Childress et al. (2016); (10) Je ff ery et al. (1992); (11) Lira et al. (1998); (12) Gómez & López (1998); (13) Mazzali et al. (1993); (14) Matheson et al. (2008); (15) Riess et al. 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2024arXiv240902804C	Primordial black holes PBHs are usually assumed to be described by the Schwarzschild or Kerr metrics which however feature unwelcome singularities. We study the possibility that PBHs are nonsingular objects considering three phenomenological regular tr timeradialsymmetric spacetimes including the wellknown Bardeen and Hayward ones featuring either de Sitter or Minkowski cores. We characterize the evaporation of these PBHs and constrain their abundance from gammaray observations. For all three metrics we find that constraints on ftextpbh the fraction of dark matter DM in the form of PBHs weaken with respect to the Schwarzschild limits because of modifications to the PBH temperature and greybody factors. This moves the lower edge of the asteroid mass window down by potentially an order of magnitude or more leading to a much larger region of parameter space where PBHs can make up all the DM. A companion paper is devoted to nontextittrsymmetric metrics including loop quantum gravityinspired ones. Our work provides a proofofprinciple for the interface between the DM and singularity problems being a promising arena with a rich phenomenology.	2024-09-01T00:00:00Z	['arXiv:2409.02804', '10.48550/arXiv.2409.02804', '2024arXiv240902804C']	['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory']	Primordial regular black holes as all the dark matter. I. Timeradialsymmetric metrics	2,024	200	0.33	['EPRINT_HTML', 'EPRINT_PDF']	9	https://arxiv.org/pdf/2409.02804.pdf	{'Primordial regular black holes as all the dark matter. I. Time-radial-symmetric metrics': "Marco Calz'a, 1, 2, ∗ Davide Pedrotti, 1, 2, † and Sunny Vagnozzi 1, 2, ‡ \n1 Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy § 2 Trento Institute for Fundamental Physics and Applications (TIFPA)-INFN, Via Sommarive 14, 38123 Povo (TN), Italy (Dated: December 9, 2024) \nPrimordial black holes (PBHs) are usually assumed to be described by the Schwarzschild or Kerr metrics, which however feature unwelcome singularities. We study the possibility that PBHs are non-singular objects, considering three phenomenological, regular tr (time-radial)-symmetric spacetimes (including the well-known Bardeen and Hayward ones), featuring either de Sitter or Minkowski cores. We characterize the evaporation of these PBHs and constrain their abundance from γ -ray observations. For all three metrics we find that constraints on f pbh , the fraction of dark matter (DM) in the form of PBHs, weaken with respect to the Schwarzschild limits, because of modifications to the PBH temperature and greybody factors. This moves the lower edge of the asteroid mass window down by potentially an order of magnitude or more, leading to a much larger region of parameter space where PBHs can make up all the DM. A companion paper is devoted to nontr -symmetric metrics, including loop quantum gravity-inspired ones. Our work provides a proof-ofprinciple for the interface between the DM and singularity problems being a promising arena with a rich phenomenology.", 'I. INTRODUCTION': "The Standard Model of particle physics (SM) and General Relativity (GR) have proven to be extremely successful at describing a huge range of terrestrial, astrophysical, and cosmological observations. However, their successes are limited by a number of shortcomings, potentially (especially in the case of SM) pointing towards the need for new physics which may better describe the matter and gravity sectors. On the more observational/phenomenological side, the SM lacks a candidate for the dark matter (DM) which accounts for ≃ 25% of the energy budget of the Universe [1, 2]. On the more theoretical side, continuous gravitational collapse in GR leads to the pathological appearance of curvature singularities [3, 4]. The nature of DM and the singularity problem are arguably two among the most important open questions in theoretical physics. \nThe solution to the DM problem could reside in the physics of some of the most peculiar objects in the Universe: black holes (BHs). It has long been realized that primordial BHs (PBHs), hypothetical relics from the primordial Universe formed from the collapse of large density perturbations upon horizon re-entry, are indeed excellent DM candidates [5-104] (see e.g. Refs. [105-114] for reviews): in fact, PBHs are the only viable DM candidate which does not invoke new particles surviving to the present day. Once believed to merely be objects of mathematical speculation, observational effects associated to BHs are now routinely detected [115], turning \nthese objects into extraordinary probes of fundamental physics [116-166]. As a result, the possibility of PBHs accounting for the entire DM budget is severely constrained by a wide range of considerations and constraints. The only (not entirely undebated) remaining open window of parameter space where PBHs could make up all the DM is the so-called ' asteroid mass window ', roughly for PBH masses 10 17 g ≲ M pbh ≲ 10 23 g [167-182]: lighter PBHs would have evaporated fast enough to either have disappeared by now or overproduced γ -rays in the MeV range, whereas heavier PBHs would have been detected through the microlensing of background stars. \nAlmost all works on PBHs assume that these are Schwarzschild or Kerr BHs [105-114]. All constraints and considerations on DM potentially being in the form of PBHs are therefore subject to this underlying assumption. The existence of the asteroid mass window, and the extension thereof, is of course no exception. The assumption in question is not at all unreasonable at least from the phenomenological point of view, given that there are at present no signs of tension between astrophysical observations and the Kerr-Newman family of metrics, and more generally the no-hair theorem. Nevertheless, from the theoretical point of view such an assumption might stir some unease, given the appearance of singularities in the Schwarzschild and Kerr metrics. The above considerations naturally lead to the following question: ' what if PBHs are non-singular '? The present work is a pilot study whose goal is to explore this question, which naturally merges the DM and singularity problems. \nWe entertain the possibility that PBHs are 'regular', i.e. free of curvature singularities [183-187], and therefore that DM may be in the form of primordial regular BHs (PRBHs). For concreteness, we consider so-called tr (time-radial)-symmetric metrics, for which the product of the coefficients for the dt 2 and dr 2 components of the line element in four-dimensional Boyer-Lindquist coordi- \nnates is equal to -1, and the function which multiplies the angular part of the line element is r 2 , i.e. r is the areal radius. More specifically, we focus on the following three regular, static spherically symmetric space-times, all of which are characterized by an additional regularizing parameter ℓ and recover the Schwarzschild space-time in the ℓ → 0 limit: Bardeen BHs [188], Hayward BHs [189], and Culetu-Ghosh-Simpson-Visser BHs [190-193]. These three space-times present a rich phenomenology, including either de Sitter or Minkowski cores. We focus our attention on observational constraints from PRBH evaporation, which set the lower edge of the asteroid mass window, discussing in detail how the evaporation process is modified with respect to that for Schwarzschild PBHs. We show that, as a result, the phenomenology of PRBHs can be quite different from that of Schwarzschild PBHs, with a larger range of masses where PRBHs could make up the entire DM component, opening up the asteroid mass window by up to an extra decade in mass. Keeping in mind that the metrics in question are phenomenological in nature, our results demonstrate that a common solution to the DM and singularity problems in the form of PRBHs is one which is worth taking seriously, and warrants further investigation, and more generally the interface of these two problems provides a promising arena. We stress that our work should not be intended as a comprehensive analysis of PRBHs, but rather as a pilot study, pointing towards a direction which has thus far received little attention and indicating promising directions for further work. \nThe rest of this paper is then organized as follows. In Sec. II we briefly introduce the regular space-times studied in the rest of the work. Various aspects of our methodology are discussed in Sec. III, with Sec. III A devoted to the calculation of the so-called greybody factors, Sec. III B to the computation of photon spectra resulting from Hawking evaporation, and Sec. III C to the comparison against observations. The resulting limits on the fraction of DM which may be in the form of primordial regular BHs are then critically discussed in Sec. IV. Finally, in Sec. V we draw concluding remarks. A number of more technical aspects concerning the greybody factors computation are discussed in Appendix A, whereas the time evolution of the primordial regular BHs we consider is studied in Appendix B. Unless otherwise specified, we adopt units where G = c = ℏ = 1. In closing, we note that a related study is being presented in a companion paper [194]: this focuses on nontr -symmetric metrics, which also include loop quantum gravity-inspired metrics, but at the same time complicate the study of the evaporation process. We recommend that the interested reader go through the present work prior to consulting our companion paper [194].", 'II. REGULAR BLACK HOLES': "It is well known, thanks to the Penrose-Hawking singularity theorems, that continuous gravitational collapse in GR sourced by matter contents satisfying reasonable energy conditions leads to the appearance of singularities [3, 4]. These are regions of space-time where curvature invariants, i.e. sets of independent scalars constructed from the Riemann tensor and the metric, diverge (with the archetypal example being the central singularity in the Kerr-Newman family of metrics). These singularities are arguably unsatisfactory as they lead to a potential breakdown in predictivity (see Refs. [195-197] for a different viewpoint). For this reason, they are oftentimes regarded as a manifestation of our lack of knowledge of (new) physics in the high-energy/high-curvature regime. A widespread belief is that quantum gravity effects on these scales would ultimately cure the singularity problem (and potentially lead to observable effects), although this is more of a hope supported only by a few first-principles studies [198-212]. \nEven in the absence of a widely agreed upon theory of quantum gravity, one can still hope to make progress in understanding and taming singularities, while also potentially gaining intuition about the possible features of such a theory, through a more phenomenological approach. Under the assumption that a metric description maintains its validity, one can introduce metrics which are free of singularities in the entire space-time, and describe so-called regular BHs (RBHs) [213]. It is often (albeit not necessarily always) the case that RBH metrics are controlled by an extra parameter, which in what follows we shall refer to as regularizing parameter (and denote by ℓ ), typically recovering the Schwarzschild metric (for non-rotating RBHs) in the limit ℓ → 0. Several RBH metrics have been studied over the past decades, see e.g. Refs. [214-264] for an inevitably incomplete selection of examples, as well as Refs. [183-187] for recent reviews on the subject. 1 While most of these metrics have been introduced on purely phenomenological grounds it is known that possible sources for several RBH metrics lie in theories of non-linear electrodynamics [268-272]. \nAs alluded to earlier, our interest in this work is to explore the possibility that primordial RBHs may play the role of DM. As a proof-of-principle in this sense we will establish constraints on f pbh , the fraction of DM in the form of PRBHs, focusing on the asteroid mass window, whose extent we will show can be further extended. To the best of our knowledge, we are aware of seven works in this little explored direction [273-279]. Ref. [274] studied primordial BHs with de Sitter interiors as DM candidates, but considering the case where the DM is actually constituted of remnants from the evaporation process. Ref. [275] studied the thermodynamics \nof primordial regular BHs, focusing however on the case where they do not evaporate, and therefore did not study constraints on f pbh . Ref. [276] studied the evaporation of a loop quantum gravity-inspired BH, and weakened constraints on f pbh were reported in a later proceeding [277] (which however does not appear to be widely known)., Ref. [278] studied signatures of primordial BHs with magnetic charge, which could be (as is often but necessarily the case) regular. Finally, Refs. [273, 279] entertained the case where the evaporation times of primordial regular BHs are significantly longer than the standard Schwarzschild case, and did not therefore explicitly compute the evaporation constraints we instead study here. The aim of this pilot study and our companion paper [194] is instead to provide a more comprehensive investigation of primordial regular BHs, considering a more diverse set of metrics and investigating constraints on f pbh in detail. \nIn our work, we shall consider three different nonrotating RBH metrics, as discussed in more detail in the following subsections. The static, spherically symmetric space-times we investigate are a subset of the Petrov type-D class of metrics. In four-dimensional BoyerLindquist coordinates, their line elements take the following general form: \nds 2 = -f ( r ) dt 2 + g ( r ) -1 dr 2 + h ( r ) d Ω 2 , (1) \nwhere d Ω 2 = dθ 2 + sin 2 ( θ ) dϕ 2 is the metric on the 2sphere. We also require our space-times to be asymptotically flat, which amounts to the following requirements: \nf ( r ) r →∞ ---→ 1 , g ( r ) r →∞ ---→ 1 , h ( r ) r →∞ ---→ r 2 . (2) \nIn addition, as stated earlier, we require our space-times to be tr -symmetric (the nontr -symmetric case is covered in a companion paper [194]), which imposes the following additional conditions: \nf ( r ) = g ( r ) , h ( r ) = r 2 , (3) \nimplying that the coordinate r is effectively the areal radius. With the conditions given by Eqs. (2,3) imposed upon Eq. (1), our most general line element therefore takes the following form: \nds 2 = -f ( r ) dt 2 + f ( r ) -1 dr 2 + r 2 [ dθ 2 +sin 2 ( θ ) dϕ 2 ] . (4) \nIn what follows, we refer to the function f ( r ) as being the 'metric function'. The three different RBH solutions we consider, which we will discuss very shortly in Sections II A- II C, are characterized by different functional forms of f ( r ). \nAn important parameter characterizing the behaviour of BHs is their temperature T . This is particularly crucial when evaluating evaporation constraints on PBHs, given that the temperature controls the strength of the emitted radiation, which in turn can be directly constrained by various observations. We treat the temperature of the RBHs as being the usual Gibbons-Hawking one, i.e. \nthe one evaluated by Wick rotating the metric in the standard way and imposing regularity in the Euclidean period [280]. The cyclic imaginary time → temperature identification is legitimate if one can formally identify the Euclidean action e -S with the Boltzmann factor e -βH in the partition function, as usually done in finite temperature quantum field theory: in turn, this can be done if one is assuming the standard Boltzmann-Gibbs distribution, but may not be the consistent if other entropies are assumed (see e.g. the recent discussion in Ref. [281]). Since, as we will reiterate later, the RBHs we study are introduced on phenomenological grounds and we remain agnostic as to their theoretical origin (which may in principle be rooted within alternative entropic frameworks), in what follows we assume the Boltzmann-Gibbs distribution, so that the temperature of the RBHs in question is the standard Gibbons-Hawking one, and is given by the following: \nT = κ 2 π = f ' ( r ) 4 π \| r H , (5) \nwhere the prime indicates a derivative with respect to r , and κ is the BH surface gravity, given by the following: \nκ = f ' ( r ) 2 \| r H . (6) \nIn Eqs. (5,6), r H is the horizon radius, which is the solution to the following equation: \ng -1 ( r H ) = f ( r H ) = 0 , (7) \nwith the first equality following from the choice of focusing on tr -symmetric space-times. In the case of Schwarzschild BHs, where the metric function is f ( r ) = 1 -2 M/r , one recovers the well-known expressions r H = 2 M and T Sch = 1 / 8 πM . However, in more general spacetimes the horizon radius in Eq. (7) is not guaranteed to have a closed form expression, and the same therefore holds for the temperature in Eq. (5). In Fig. 1 we show the evolution of the temperatures (normalized to the temperature of Schwarzschild BHs, T Sch = 1 / 8 πM ) of the three regular space-times we will introduce shortly, as a function of the regularizing parameter ℓ (itself normalized to the event horizon radius r H ). As the Figure clearly shows, for all three space-times the temperature is a monotonically decreasing function of ℓ . 2 We also see \nFIG. 1. Evolution of the temperatures (normalized to the temperature of Schwarzschild black holes, T Sch = 1 / 8 πM ) as a function of the regularizing parameter ℓ (normalized to the event horizon radius r H ) for the three regular black holes studied in this work: the Bardeen (blue solid curve, Sec. II A), Hayward (red dashed curve, Sec. II B), and Culetu-GhoshSimpson-Visser (green dotted curve, Sec. II C) regular spacetimes. In all cases the temperature is a monotonically decreasing function of ℓ . Note that the range of allowed values of ℓ/r H is different for the three regular black holes. \n<!-- image --> \nthat the temperatures of these BHs approach zero in the extremal limit. As has been observed earlier, in this limit the usual constraints on f pbh vanish, since these BHs do not evaporate [275]. \nA final caveat is in order before discussing the RBH metrics we consider. The latter are all regular in the sense of having finite curvature invariants R ≡ g µν R µν , R µν R µν , and K ≡ R µνρσ R µνρσ . However, a more stringent criterion for regularity is that of geodesic completeness, which does not necessarily imply finiteness of curvature invariants and viceversa. We recall that a spacetime is geodesically complete if any geodesic thereon can be extended to arbitrary values of their affine parameter. In other words, all geodesics extend (or can be extended) for all times. A number of 'popular' RBHs have indeed been shown to have finite curvature invariants but to be geodesically incomplete [293]. This includes the well-known Hayward RBH, which is among the ones we shall consider here. Nevertheless, given the significant interest in this metric, the fact that it is widely taken as prototype for RBHs, and our phenomenological goal of going beyond Schwarzschild PBHs, we will take this space-time into consideration, while cautioning the reader about the above issues, and therefore that the Hayward metric (but more generally all the metrics we will study) should be considered nothing more than phenomenological toy models at this stage. Note, in addition, that the stability of RBHs featuring inner horizons is currently a matter of debate in the literature [294-298].", 'A. Bardeen black hole': 'The Bardeen BH is easily one of the best known RBHs, and one of the first ones to ever have been proposed [188]. It is characterized by the following metric function: 3 \nf B ( r ) = 1 -2 Mr 2 ( r 2 + ℓ 2 ) 3 / 2 , (8) \nwhere, in terms of the BH mass M , the regularizing parameter satisfies ℓ ≤ √ 16 / 27 M ∼ 0 . 77 M in order for the metric to describe a BH and not a horizonless object. Note that the Schwarzschild metric function is recovered in the ℓ → 0 limit. A perhaps physically more motivated choice is to express quantities in units of the horizon radius r H , defined as the largest root of the equation f ( r H ) = 0, in which case the regularizing parameter is subject to the constraint ℓ ≲ 0 . 70 r H (which is the point at which T = 0 in Fig. 1). In order to obtain this limit we have computed the solution to f ( r H ) = 0 fixing M = 1, in order to extrapolate r H ( ℓ ), before analyzing for which real values of the parameter n the equation ℓ = nr H ( ℓ ) admits solutions. We note that the peculiar factor of √ 16 / 27 in the extremality condition appears when one demands that the cubic equation determining the horizon(s) location(s) of the Bardeen RBH, f B ( r ) = 0, admits one real non-zero root: with some algebraic manipulation, one sees that this condition amounts to the requirement that √ 27 ℓ 8 M 4 -16 ℓ 6 M 6 is real, from which the requirement ℓ ≤ √ 16 / 27 M follows. \nIt is worth noting that the Bardeen BH possesses a de Sitter (dS) core which replaces the central singularity of the Schwarzschild BH. This is evident by noting that, in the limit r → 0, the metric function goes as f B ( r ) ∝ r 2 , exactly as expected for an asymptotically dS space-time. Although originally introduced on phenomenological grounds, it is now known that the Bardeen RBH can emerge from a magnetic monopole source [299], potentially within the context of a specific non-linear electrodynamics theory [300]. Another possible origin for the Bardeen RBH are quantum corrections to the uncertainty principle [301]. Irrespective of its origin, and consistently with the approach pursued for the other spacetimes, we consider this solution as a model-agnostic phenomenological toy model.', 'B. Hayward black hole': 'Another widely known regular space-time is the Hayward RBH [189], characterized by the following metric \nfunction: \nf H ( r ) = 1 -2 Mr 2 r 3 +2 Mℓ 2 . (9) \nIf expressed in terms of BH mass M , the regularizing parameter for the Hayward BH is subject to the same limit as that of the Bardeen BH, i.e. ℓ ≤ √ 16 / 27 , M . On the other hand, if expressed in terms of the more physically motivated horizon radius, the limit is instead ℓ ≲ 0 . 57 r H (again this is the point where T = 0 in Fig. 1). We note that the Schwarzschild metric function is recovered in the ℓ → 0 limit. The factor of √ 16 / 27 in the extremality condition originates from considerations similar to those we made for the Bardeen RBH in Sec. II A: with some algebraic manipulation, one sees that this condition amounts to the requirement that √ 27 ℓ 4 M 2 -16 ℓ 2 M 4 is real, from which the requirement ℓ ≤ √ 16 / 27 M follows. \nJust as the Bardeen RBH possesses a dS core, so does the Hayward RBH. Indeed, introducing a dS core characterized by a (positive) cosmological constant Λ = 3 /ℓ 2 in order to prevent the central singularity was precisely the original justification for the Hayward BH which, just like its Bardeen counterpart, was proposed on purely phenomenological grounds. Nevertheless, potential theoretical origins for the Hayward BH have been investigated, and range from corrections to the equation of state of matter at high density [302, 303], finite density and finite curvature proposals [304-306], theories of non-linear electrodynamics [307, 308], and more generally as the result of corrections due to quantum gravity [309, 310]. Just as with the Bardeen RBH, we shall here treat the Hayward RBH as a model-agnostic phenomenological toy model for a singularity-free space-time.', 'C. Culetu-Ghosh-Simpson-Visser black hole': "The regular space-times considered so far featured dS cores, which in itself is a very common feature of several RBH metrics. Nevertheless, another interesting phenomenological possibility consists in considering 'hollow' RBHs wherein the central singularity is replaced by an asymptotically Minkowski core, where the associated energy density and pressure asymptote to zero. This is quite unlike the case of the dS core where the energy density asymptotes to a finite value associated to a positive cosmological constant, and the pressure asymptotes to an equal but opposite value. Possible theoretical/mathematical motivations for considering RBHs with Minkowski cores include the fact that the vanishing energy density can significantly simplify the physics in the deep core, whereas the otherwise messy solutions to polynomial equations (which often cannot be written down in closed form) can be traded for arguably more elegant special functions, resulting in the space-time being more tractable. Our physical motivation in considering this class of BHs is instead to broaden the range of physical \nproperties and phenomenological implications of PRBHs, going beyond the dS core RBHs studied thus far. \nWith this in mind, we consider a RBH featuring a Minkowski core, independently studied by Culetu [190, 191], Ghosh [192], as well as Simpson and Visser [193]. Although such a RBH does not have any particular name associated to it in the literature, here we refer to it as CGSV BH (from the initials of the four authors above). The space-time is characterized by the following metric function: \nf CGSV ( r ) = 1 -2 M r exp ( -ℓ r ) . (10) \nThe horizon radius r H , for which a closed form expression is not available in the Bardeen and Hayward cases, here is given by: \nr H = -ℓ W ( -ℓ 2 M ) , (11) \nwhere W denotes the Lambert function. Considering the principal branch W 0 , a real and positive horizon radius is present for: \nW 0 ( -ℓ 2 M ) ≤ 0 = ⇒ 0 ≤ ℓ < 2 M e , (12) \nor, alternatively, 0 ≤ ℓ < r H . While the CGSV BH was original introduced purely on phenomenological/mathematical grounds, it was shown in Refs. [311, 312] that such a space-time can emerge within the context of GR coupled to a specific non-linear electrodynamics source. In this case, denoting by g the non-linear electrodynamics coupling constant/charge, the regularizing parameter ℓ is given by ℓ = g 2 / 2 M , with M the BH mass. Nevertheless, as with all the other RBHs considered, here we shall treat the CGSV RBH as a toy model for a regular space-time possessing a Minkowski core.", 'A. Greybody factors': "A set of parameters playing a key role in describing the Hawking radiation spectra emitted from evaporating BHs are the so-called greybody factors (GBFs). These are functions of energy/frequency and angular momentum which govern the deviation of the emitted spectrum from that of a blackbody [313-315]. Although the emitted Hawking radiation at the horizon takes the blackbody form, the potential barrier due to space-time geometry will attenuate the radiation, so that an observer at spatial infinity will measure a spectrum which differs from that of a blackbody by a frequency-dependent function Γ( ω ). GBFs can be characterized by setting up a classical scattering problem around the BH potential barrier, with boundary conditions allowing for incoming wave packets \nfrom infinity or equivalently, due to the symmetries of the scattering problem, originating from the horizon. The scattering problem is governed by the so-called Teukolsky equation, which is a partial differential equation describing the propagation of perturbations of given spin in the BH background [316]. \nFor the static, spherically and tr -symmetric metrics given by Eq. (4) which we consider, the Teukolsky equation in spherical coordinates is separable. A key role in computing the GBFs is played by the radial Teukolsky equation, which we now report in full generality for the class of metrics in question. Using the Newman-Penrose (NP) formalism, the Teukolsky equation governing the evolution of (massless) perturbations of different spin can be condensed into a single master equation [316]: \n[ -r 2 f ∂ 2 t + s ( r 2 f ' f -2 r ) ∂ t ] Υ s + [ ( s +1)( r 2 f ' +2 rf ) ∂ r ] Υ s + [ 1 sin θ ∂ θ (sin θ∂ θ ) + 2 is cot θ sin θ ∂ ϕ + 1 sin 2 θ ∂ 2 ϕ -s -s 2 cot 2 θ ] Υ s + [ sr 2 f '' +4 srf ' +2 sf ] Υ s = 0 . (13) \nHere, Υ s represents a general perturbation of spin s , defined by the NP scalars relative to the respective perturbation. To not make the notation too heavy, we drop the l and m indices labelling the field mode, so Υ s is understood to really mean Υ lm s . We note that Eq. (13) is separable if one makes the following wave ansatz: \nΥ s = ∑ l,m e -iωt e imϕ S l s ( θ ) R s ( r ) , (14) \nwhere ω is the perturbation frequency, l is the angular node number, and m is the azimuthal node number. \nThe functions S l s ( θ ) contribute to defining the socalled spin-weighted spherical harmonics S s l,m ( θ, ϕ ) = ∑ S l s ( θ ) e imϕ , satisfying the following equation [317-320]: \n( 1 sin θ ∂ θ (sin θ ∂ θ ) + csc 2 θ ∂ 2 ϕ + 2 is cot θ sin θ ∂ ϕ + s -s 2 cot 2 θ + λ s l ) S s l,m = 0 , (15) \nwhere λ s l ≡ l ( l +1) -s ( s +1) is the separation constant. For the spin 0 case, these functions reduces to the usual spherical harmonics S 0 l,m = Y l,m . \nAnalogously to the Schwarzschild and Kerr BH cases [316], the decoupled radial Teukolsky equation takes the following form [321, 322]: \n1 ∆ s ( ∆ s +1 R ' s ) ' + ( ω 2 r 2 f +2 iωsr -isωr 2 f ' f + s (∆ '' -2) -λ s l ) R s = 0 , (16) \nwhere ∆( r ) ≡ r 2 f ( r ) and ' ≡ ∂ r . We set in purely ingoing boundary conditions, so the asymptotic solutions of Eq. (16) are given by: \nR s ∼ R in s e -iωr ⋆ r + R out s e iωr ⋆ r 2 s +1 ( r →∞ ) R s ∼ R hor s ∆ -s e -iωr ⋆ ( r → r H ) , (17) \nwhere r ⋆ is the tortoise coordinate defined by: \ndr ⋆ dr = 1 f ( r ) . (18) \nWe note that r ⋆ → r for large values of r , given that the metrics we consider are asymptotically flat. \nIn general, numerical integration methods are required to compute GBFs for general space-times, and this holds for our tr -symmetric RBHs as well. In our work, we make use of the so-called shooting method (see Appendix A for further details), which has already been successfully applied to these types of calculations in several earlier works [323-330]. \nTo begin with, we rewrite Eq. (16) in terms of the rescaled coordinate x , given by the following: \nx ≡ r -r H r H , (19) \nwhere r H is the largest real root of f ( r ) = 0. With this substitution Eq. (16) is rewritten as follows: \nx 2 ( x +1) 3 f R s +( s +1) x ( x +1) ( 2( x +1) f +( x +1) 2 ˙ f ) ˙ R s + V ( ω, x ) R s = 0 , (20) \nwhere ˙ ≡ ∂ x , and V ( ω, x ) is given by: \nV ( ω, x ) = ( ω 2 r 2 H ( x +1) 2 f +2 is ( x +1) ω -isr H ( x +1) 2 ˙ f f ω + s ( 2 f +4( x +1) ˙ f +( x +1) 2 f -2 ) -l ( l +1) + s ( s +1) ) x ( x +1) . (21) \nIn order to further simplify the problem, we work in units of horizon radius and therefore set r H = 1, so that r = x + 1. In these units, the metric functions of the three RBHs under consideration are given by the following: \nf B ( x ) = 1 -(1 + ℓ 2 ) 3 / 2 ( x +1) 2 ( ℓ 2 +( x +1) 2 ) 3 2 , f H ( x ) = 1 -( x +1) 2 (1 -ℓ 2 ) ( ( x +1) 3 -ℓ 2 ℓ 2 -1 ) , f CGSV ( x ) = 1 -e ℓ -ℓ x +1 x +1 , (22) \nfor the Bardeen, Hayward, and CGSV space-times respectively. \nSetting purely ingoing boundary in proximity of the horizon, the solutions to Eq. (20) can be expressed in the form of a Taylor expansion as follows [323, 324, 331-336]: \nR s ( x ) = x -s -iω τ ∞ ∑ n =0 a n x n , (23) \nwhere iω/τ is a function of the field spin and the regularizing parameter, and also depends on the space-time in question. We refer the reader to Appendix A for further details. The a n coefficients can be determined by substituting Eq. (23) in Eq. (20) and iteratively solving the resulting algebraic equations. The near-horizon solution is then used to set the boundary conditions and numerically integrate the radial equation up to large distances, where the general form of the solution is the following: \nR ( x ) r →∞ ---→ R in s e -iωx x + R out s e iωx x 2 s +1 . (24) \nThe GBFs can then be computed from the s R lm in ( ω ) coefficient. More specifically, the normalization of the scattering problem is set by requiring a 0 = 1. With this normalization, the GBFs then read: \nΓ s lm ( ω ) = δ s \| s R lm in ( ω ) \| -2 , (25) \nwhere the coefficient δ s is given by: \nδ s = τ ie iπs (2 ω ) 2 s -1 Γ ( 1 -s -2 iω τ ) Γ ( s -2 iω τ ) , (26) \nwhere Γ is the Γ function. Using the method discussed above, we compute the GBFs for perturbations of different spin on the backgrounds of the three regular metrics discussed earlier, for different values of the regularizing parameter ℓ . In the specific case s = 1, we have checked that calculating the GBFs up to l = 4 is sufficient for our purposes. The GBFs we calculate are then used to characterize the Hawking evaporation spectra. \nIn Fig. 2 we show the Γ s =1 l =1 GBFs for the Schwarzschild BH and the three regular space-times we study. For illustrative purposes we have focused on the Γ s =1 l =1 GBFs, since we are interested in photons ( s = 1) and the dominant emission mode is the l = 1 one. We also note that, since we are considering spherically symmetric space-times, the (2 l +1) different m modes are degenerate. We have fixed the regularizing parameter to ℓ = 0 . 3 r H for all three regular space-times, also for illustrative purposes. We see that, for all three regular space-times, the GBFs are slightly higher than their Schwarzschild counterparts (by ≲ 20% at most), and asymptote to the latter for both ω/M ≲ 0 . 3 and ω/M ≳ 0 . 7. While for definiteness we have focused on these specific values of s , l , and ℓ , we have explicitly checked that the features described above are present for other values of the parameters in question as well. \nFIG. 2. Greybody factors Γ s =1 l =1 as a function of ω/M for Schwarzschild BHs (black curve), as well as the Bardeen (blue solid curve), Hayward (red dashed curve), and Culetu-GhoshSimpson-Visser (green dotted curve) regular space-times. For illustrative purposes we only plot Γ s =1 l =1 , since we are interested in photons ( s = 1) and the dominant emission mode is the l = 1 one. We have fixed the regularizing parameter to ℓ = 0 . 3 r H for all three regular space-times. We see that in all three cases the GBFs are consistently (slightly) higher than their Schwarzschild counterparts. The features shown in this plot do not change sensibly for higher values of l and other values of ℓ . \n<!-- image -->", 'B. Evaporation spectra': 'We now discuss our computation of the photon spectra resulting from Hawking evaporation of the regular BHs discussed previously. In what follows, we only account for the primary photon spectrum. Nevertheless, we have checked that in the mass region of interest the impact of the secondary component of the spectrum, i.e. that resulting from the decay into photons of other unstable particles which are also produced during the evaporation process, is negligible. \nThe Hawking radiation rate (number of particles emitted per unit time per unit energy) of a given particle species i with spin s , as a result of Hawking evaporation, is given by the following [337-341]: 4 \nd 2 N i dtdE i = 1 2 π ∑ l,m n i Γ s l,m ( ω ) e ω/T ± 1 , (27) \nwhere n i is the number of degrees of freedom of the particle in question, ω = E i is the mode frequency (in natural units), Γ s l,m are the GBFs discussed previously, and we have implicitly set k B = 1. Note that \nFIG. 3. Primary photon spectra resulting from the evaporation of Bardeen black holes of mass 10 16 g for different values of the regularizing parameter ℓ (normalized by the horizon radius r H ): ℓ/r H = 0 . 15 (red dotted curve), 0 . 3 (green dashed curve), and 0 . 45 (magenta dash-dotted curve). The blue solid curve corresponds to the case ℓ/r H = 0, which recovers the Schwarzschild black hole. \n<!-- image --> \nFIG. 4. As in Fig. 3, but for Hayward black holes, with identical values of the regularizing parameter ℓ/r H and identical color coding. \n<!-- image --> \nthe plus (minus) sign in the denominator is associated to fermions (bosons). Following the methodology discussed in Sec. III A, we calculate the GBFs within all the BH space-times in question for photons ( s = 1), up to l = 4 (note that the angular node number l should not be confused with the regularizing parameter ℓ ). We have checked that adding higher l modes does not appreciably improve the resulting spectra. \nWe show examples of the resulting evaporation spectra in Figs. 3, 4, and 5. The spectra obviously depend on the mass of the evaporating PRBH, which we have set to M pbh = 10 16 g, as it sits roughly in the middle of \nFIG. 5. As in Fig. 3, but for Culetu-Ghosh-Simpson-Visser black holes, with identical values of the regularizing parameter ℓ/r H and identical color coding. \n<!-- image --> \nthe mass range of interest. Nevertheless, we stress that the features we discuss below do not depend on the chosen mass. The resulting spectra all peak approximately between 5 MeV and 10 MeV. \nFor the Bardeen, Hayward, and CGSV PRBHs we observe that an increase in the regularizing parameter ℓ leads to a decrease in the intensity of the spectra at all energies . This is na¨ıvely what one could expect by inspecting the temperature evolution shown in Fig. 1, since for all three these PRBHs the temperatures decrease relative to their Schwarzschild counterparts. However, as Eq. (27) shows, the temperature is not the only quantity at play, since the GBFs enter as well. As we have seen in Fig. 2, the GBFs for all three PRBHs are larger than their Schwarzschild counterparts, which would seem to counteract the effect of a lower temperature. A posteriori, however, the dominant effect turns out to be that of a lower temperature. We could actually have expected as much, since the resulting spectra given by Eq. (27) are linear in the GBFs, but depend exponentially on the temperature. Therefore, one might expect that a decrease in the temperature will have a much more dramatic effect than a comparable increase in the GBFs. This is precisely what we observe in Figs. 3, 4, and 5. \nAside from the amplitude of the spectrum, for the Bardeen and Hayward PRBHs we note that the position of the peak in the spectrum is only mildly affected by the regularizing parameter, an increase in which pushes the peak towards slightly lower energies. On the other hand, for the CGSV BH an increase in the regularizing parameter pushes the peak towards slightly higher energies. We do not exclude that this different behaviour may be related to the type of core being considered, and we defer a more detailed investigation of this point to future work. At any rate, we expect that the behaviour of the spectra discussed above should lead to constraints on f pbh which', 'C. Evaporation constraints': 'The spectra calculated in Sec. III B are then used to set evaporation constraints on f pbh ( M ) ≡ Ω pbh / Ω dm , the fraction of DM in the form of PBHs, where Ω pbh and Ω dm are the PBH and DM density parameters respectively. Specifically, the computed spectra are used to obtain predictions for the flux of photons resulting from Hawking evaporation, which are then directly compared against measurements of the extragalactic photon background across a wide range of energies (see e.g. Ref. [342] for a recent review). Evaporation constraints are the dominant ones in the 10 13 g ≲ M pbh ≲ 10 18 g mass range: the lower limit of the range is set by the requirement that PBHs have not yet evaporated at the time of recombination, whereas the upper limit is defined by measurements of the diffuse extragalactic γ -ray background (EGRB) in the energy range 100 keV ≲ E γ ≲ 5 GeV, given that the intensity of the Hawking radiation flux is inversely proportional to the mass of the evaporating BH. In what follows, we will direct our attention exclusively to PBHs for which M pbh ≳ 10 15 g: these have yet to fully evaporate today and, having formed deep during the radiation domination era, are therefore excellent non-baryonic DM candidates. There is another important reason for focusing on this mass range. As shown in Appendix B, PBHs (either Schwarzschild or the three PRBHs we consider) within this range have lifetimes much longer than the age of the Universe, are far from having fully evaporated at the present time, and have only lost a negligible fraction of their mass from formation until now. When using the symbol M pbh , we are therefore safe in denoting the values of the PBH mass both at formation and now. \nWe work under the commonly adopted assumption that PBHs are isotropically distributed on sufficiently large scales. Therefore, the flux resulting from their evaporation and reaching us today is given by the redshifted sum of the contributions from all evaporating PBHs in our Universe, and can be used to constrain the average extragalactic distribution of DM in the form of PBHs. Furthermore, we work within the (also commonly adopted) approximation of monochromatic mass distributions (which can be expected if the formation mechanism arises from an amplification of the power spectrum at a very specific scale), although the effect of extended mass distributions is the subject of active research [343357]. Finally, as discussed earlier, we only consider the primary photon contribution, as the secondary component resulting from the decay into photons of other unstable particles is verified a posteriori to be negligible given the mass range of interest. While all these are clearly approximations, albeit widely adopted ones, we are confident that they are appropriate given the aim of our work. Our main goal is in fact to examine how the limits on f pbh change when moving from the Schwarzschild \nPBH framework to that of the regular metrics presented in Sec. II, altering the asteroid mass window. It is more than reasonable to expect that the shift in constraints relative to the Schwarzschild case, δf pbh , is only weakly affected by the above approximations. In other words we expect these approximations to have similar impacts on the constraints on f pbh relative to Schwarzschild BHs for the different RBHs discussed in Sec. II, therefore leading to negligible effects on the relative shift δf pbh , which is the quantity we are ultimately interested in. At any rate, the adopted approximations also allow for a more direct comparison to several previous works, and we therefore consider them appropriate for our pilot study, while stressing that their impact should definitely be explored in future follow-up works. Finally, note that we are tacitly assuming that PBHs cluster in the galactic halo in the same way as other forms of DM (unless they are extremely large, which is not the case for the mass range of interest). \nIn what follows, we therefore assume that PBHs all have the same initial mass M pbh . Following Ref. [358] we approximate the number of emitted photons in the logarithmic energy bin ∆ E γ ≃ E γ as being given by ˙ N γ ( E γ ) ≃ E γ ( d ˙ N γ /dE γ ). The emission rate of photons from Hawking evaporation per volume at a cosmological time t is then given by [358]: 5 \ndn γ dt ( E γ , t ) ≃ n pbh ( t ) E γ d 2 N γ dtdE γ ( M pbh , E γ ) , (28) \nwhere n pbh ( t ) is the PBH number density at time t . By integrating and taking into account the redshift scaling of the photon energy and density we end up with: \nn γ 0 ( E γ 0 ) = n pbh ( t 0 ) E γ 0 ∫ t 0 t ⋆ dt (1 + z ) d 2 N γ dtdE γ ( M pbh , (1 + z ) E γ 0 ) = n pbh ( t 0 ) E γ 0 ∫ z ⋆ 0 dz H ( z ) d 2 N γ dtdE γ ( M pbh , (1 + z ) E γ 0 ) , (29) \nwhere t 0 denotes the present time, t ⋆ and z ⋆ are respectively the cosmic time and redshift at recombination, and H ( z ) is the expansion rate. Finally, n γ 0 ( E γ 0 ) is the present number density of photons with energy E γ 0 . The resulting photon flux (more properly, the rate of photons per unit time per unit area per unit solid angle) is then given by: \nI ( E γ 0 ) ≡ c 4 π n γ 0 ( E γ 0 ) . (30) \nIt is this quantity which can then be directly compared against observations. \nWe assume a spatially flat ΛCDM cosmological model in specifying the expansion rate entering into Eq. (29), with the same cosmological parameters as in Ref. [358]. This allows us to robustly cross-check our Schwarzschild constraints on f pbh against those reported in the seminal Ref. [358], although we stress that our constraints are very stable against reasonable changes in the values of the cosmological parameters. Once the cosmological model is fixed, all the relevant quantities in Eq. (29) are known except for the present-day PBH number density, n pbh ( t 0 ), which can be constrained from EGRB observations and is ultimately related to f pbh . More specifically, for any given value of the PBH mass M pbh , through Eqs. (27,29,30) we can compute the unnormalized photon flux I ( E γ 0 ) /n pbh ( t 0 ), and adjust the normalization n pbh ( t 0 ) by comparing against EGRB observations (as we will explain shortly). This procedures gives us an upper limit on n pbh ( t 0 ), which can be translated into an upper limit on f pbh as follows: \nf pbh ( M pbh ) ≡ Ω pbh Ω dm = n pbh ( t 0 ) M pbh ρ crit , 0 Ω dm , (31) \nwhere ρ crit , 0 = 3 H 2 0 / 8 πG is the present-day critical density, with H 0 the Hubble constant, and we recall that this procedure is done for various values of M pbh . \nWe compare our theoretical predictions against various measurements of the EGRB. Specifically, we use observations of the EGRB from the HEAO-1 X-ray telescope in the 3-500 keV range [359], the COMPTEL imaging Compton γ -ray telescope in the 0 . 8-30 MeV range [360], and the EGRET γ -ray telescope [361]. A few comments are in order concerning the adopted datasets. While these are by now a couple of decades old, they basically still represent the state-of-the-art in the energy range of interest. One could entertain other observations, including local galactic measurements of the galactic γ -ray background [362], positron flux [363], 0 . 511 MeV annihilation radiation [364-366], and various other sources. While these galactic observations could lead to potentially stronger limits, they depend strongly on the form of the PBH mass function (assumed to be monochromatic in our study), as well as the clustering properties of these PBHs. On the other hand, our limits on f pbh are effectively testing the average extragalactic distribution of DM. Finally, other measurements of the EGRB are available, e.g. from Fermi-LAT [367], but these are mostly important for energy ranges larger than the ones of interest, and therefore for PBHs lighter than the ones we are considering. Therefore, we believe the choice of datasets (which is the one adopted in several works estimating evaporation limits on PBHs) is appropriate given the objective of our study. \nTo set upper limits on n pbh ( t 0 ) - and therefore f pbh through Eq. (31) - we adopt the simple method first explained in the seminal Ref. [358], and later adopted in most of the works examining constraints on PBHs from \nFIG. 6. Photon fluxes resulting from the evaporation of primordial Bardeen black holes with regularizing parameter ℓ = 0 . 3 r H , and masses of M pbh = 10 15 g (brown curve), 2 . 3 × 10 15 g (purple curve), 5 × 10 15 g (red curve), 10 16 g (green curve), 3 × 10 16 g (green curve), and 5 × 10 16 g (blue curve). The triangles indicate 1 σ upper limits on the extragalactic γ -ray background flux as measured by HEAO-1 (maroon), COMPTEL (turquoise), and EGRET (teal). For each value of M pbh , the PBH fraction f pbh has been set to its upper limit, determined as soon as the first datapoint is overshot (the latter is different for different values of M pbh ). \n<!-- image --> \nthe EGRB. Specifically, for each value of M pbh , and for given values of the regularizing parameter ℓ , the maximum allowed value of f pbh is determined by requiring that the predicted photon flux does not overshoot any of the ERGB datapoints by more than 1 σ . An example is shown in Fig. 6 for Bardeen PRBHs with regularizing parameter ℓ = 0 . 3 r H : for each of the mass values M pbh represented, the upper limit on f pbh is set as soon as the first datapoint is overshot. As is clear from the Figure, different PBH masses result in different datapoints being overshot. For each of the PRBHs considered, we use this procedure to determine upper limits on f pbh for fixed, representative values of ℓ ( ℓ/r H = 0 . 15, 0 . 3, and 0 . 45), comparing the results to the Schwarzschild case which is recovered when ℓ = 0. 6 We note that the exact origin of the EGRB is currently a matter of debate [368]: \nFIG. 7. Upper limits on f pbh , the fraction of dark matter in the form of primordial regular Bardeen black holes, as a function of the black hole mass M pbh . The limits are derived for different values of the regularizing parameter ℓ (normalized by the horizon radius r H ), with the shaded regions excluded: ℓ/r H = 0 . 15 (red dotted curve), 0 . 3 (green dashed curve), and 0 . 45 (magenta dash-dotted curve). Note that the blue solid curve corresponds to the case ℓ/r H = 0, which recovers the Schwarzschild black hole, whereas the value of M pbh corresponding to the upper right edge of the f pbh constraints marks the lower edge of the asteroid mass window. \n<!-- image --> \nalthough it is commonly believed that distant astrophysical sources such as blazars give a major contribution to the EGRB, there is no complete consensus on the level of this contribution. In this light, our approach of simply requiring that the PRBH evaporation contribution to the EGRB does not exceed any observed datapoint is rather conservative (given that there could in principle be a PBH contribution to the EGRB, should it be conclusively determined that known astrophysical sources are unable to fully account for the latter).', 'IV. RESULTS': "For each of the PRBHs discussed in Sec. II, we now proceed to derive upper limits on f pbh as a function of the PRBH mass M pbh , for different values of ℓ , using the methodology presented in Sec. III C. The results are shown in Figs. 7, 8, and 9 for Bardeen, Hayward, and CGSV PRBHs respectively. For each case, we also plot the constraints on f pbh for the ℓ = 0 case (blue solid curve in all the Figures), which correspond to the standard Schwarzschild PBH scenario widely studied in the \nFIG. 8. As in Fig. 7, but for primordial regular Hayward black holes, with identical values of the regularizing parameter ℓ/r H and identical color coding. \n<!-- image --> \nFIG. 9. As in Fig. 7, but for primordial regular Culetu-GhoshSimpson-Visser black holes, with identical values of the regularizing parameter ℓ/r H and identical color coding. \n<!-- image --> \nliterature. As a sanity check, we have verified that our ℓ = 0 constraints exactly recover those of the seminal Ref. [358]. It is worth noting that, for any given value of ℓ , the value of M pbh corresponding to the upper right edge of the f pbh constraints (i.e. the value of M pbh for which the limit reads f pbh < 1) marks the lower edge of the asteroid mass window. \nFor all PRBHs, we saw earlier that the temperature and photon spectra decrease in intensity with increasing regularizing parameter ℓ (see the discussion in Sec. III B, and Figs. 3-5). As we could have expected, this behaviour leads to overall looser constraints on f pbh (for any given M pbh ) relative to the standard limits reported for Schwarzschild PBHs in the literature. In the case of near-extremal Hayward PRBHs ( ℓ = 0 . 45 r H ) this be- \nour is somewhat enhanced compared to the nearextremal Bardeen and CGSV PRBHs, with the upper limits on f pbh approximately three orders of magnitude looser than the corresponding Schwarzschild ones: again, this is somewhat unsurprising when comparing Fig. 4 to Figs. 3 and 5. This could also have been expected from Fig. 1, noting that the temperature of Hayward BHs decreases more rapidly with increasing ℓ/r H relative to the Bardeen and especially CGSV ones. Although the temperature is not the only factor at play in determining the resulting evaporation spectra, given that the GBFs also play a key role as per Eq. (27), it is reassuring to see that the temperature behaviour observed in Fig. 1 is qualitatively reflected in the limits on f pbh we derive. \nAs a result of the shifts discussed above, the lower edge of the asteroid mass window where PBHs could make up the entire DM component is modified for all three metrics considered. We recall that in the Schwarzschild case, the lower edge of the window lies at M pbh ≃ 10 17 g. For the PRBHs we consider, the looser constraints on f pbh result in the asteroid mass window further opening up by up to half a decade in mass or more. The maximum extension of the window is reached for the Hayward PRBH closer to extremality, in which case the lower edge decreases by about an order of magnitude to M pbh ≃ 10 16 g. Overall, we therefore observe that considering PRBHs in place of the standard Schwarzschild ones can relax the resulting constraints on f pbh , further opening up the asteroid mass window. The allowed region for the window lower edge spans over a decade in mass, at least for the PRBHs and range of ℓ considered. We note that, as ℓ moves towards the extremal limit, the temperatures of these PRBHs T approaches zero. This is indeed generally expected from thermodynamical arguments. In this case, which is the one studied in Ref. [275], the PRBHs do not evaporate, and therefore our evaporation constraints do not apply. See also Refs. [279] for a related study appearing after ours. \nThree comments are in order before concluding. Firstly we note that, for a given PRBH space-time, the curves describing the f pbh ( M ) limits are approximately, but not exactly parallel to the Schwarzschild ones (blue solid curves in Figs. 7-9). The reason is simply that, as ℓ is increased, the datapoint shown in Fig. 6 which is first being overshot and therefore responsible for determining the f pbh limit can potentially change (in part due to the spectrum slightly changing shape). \nNext, the constraints we have determined on f pbh at a fixed value of ℓ/r H implicitly assume that all PRBHs in the Universe carry the same value of 'hair' parameter ℓ . However, particularly given our agnostic stance with regards to the origin of these space-times, in principle the value of ℓ/r H can vary from PRBH to PRBH. To make an analogy, let us assume for a moment that ReissnerNordstrom BHs are astrophysically relevant. Then, since the electric charge Q is not tied to a universal parameter of the underlying Einstein-Maxwell Lagrangian, there is no reason to expect it to carry the same value across \nall BHs. In the language of Ref. [282], the regularizing parameter for all three PRBHs considered is a 'specific hair' rather than an 'universal hair' (see Ref. [282] for various examples of BH solutions carrying universal hair), unless one were able to tie ℓ to some fundamental parameter of the underlying theory, which however is not the case in the phenomenological approach we are following. In principle one should therefore account for the (non-monochromatic) ℓ distribution for PRBHs across the Universe to determine constraints on f pbh . We see no obvious way of doing this, while noting that such a procedure would most likely result in upper limits on f pbh lying between the Schwarzschild and extremal cases: this observation suffices for our pilot study, and we defer a more complete investigation to future work. \nOur final comment concerns the fact that evaporation limits on the PBH abundance are not the only ones at play. Indeed, as recently summarized in Ref. [110], there are essentially four classes of limits, each of which is relevant in a different mass range: evaporation, lensing, dynamical, and accretion constraints. Constraints from the accretion of background gas at early times are relevant in a completely different mass range (10 33 ≲ M pbh / g ≲ 10 40 - see Fig. 7 of Ref. [110] and Fig. 10 of Ref. [369]). Although these have been derived assuming Schwarzschild PBHs, moving to the PRBH picture we have considered will not shift the relevant mass range by the ≳ 18 orders of magnitude required for these constraints to compete with the evaporation ones, unless the physics of gas accreting around RBHs changes drastically with respect to the standard picture, which appears very unlikely. Dynamical constraints, most of which are associated to the destruction of different astronomical objects by the passage of nearby PBHs, are also relevant in a completely different mass range (10 34 ≲ M pbh / g ≲ 10 55 - see Fig. 7 of Ref. [110] and Fig. 10 of Ref. [369]), and considerations completely analogous to those we made for accretion constraints hold. 7 \nOf potentially more relevance to the present work are lensing constraints, which constrain the abundance of PBHs (and more generally massive compact halo objects) with masses M pbh ≳ 10 15 g. Indeed, it is lensing constraints which locate the upper edge of the asteroid mass window where PBHs can make up all the DM. Nevertheless, we expect that these constraints should not change when moving from the Schwarzschild PBH framework to the PRBHs considered in this work. Indeed, with all other quantities being fixed (mass of source, relative \ndistances, and so on), lensing constraints only depend on the lens mass M , and are unaffected by the metric structure of the lens. Therefore, at fixed mass M , we can assume that the lensing limits on Schwarzschild PBHs hold for our PRBHs as well. Note that, as already pointed out in footnote 2, the parameter M appearing in the RBH metrics can be unambiguously identified with the RBH mass, just as with the parameter M in the Schwarzschild metric. We can therefore conclude that for the PRBHs we are considering it should only be the lower edge of the asteroid mass window which is altered with respect to the Schwarzschild case, but not the upper edge. In other words, space-times for which the lower edge moves towards lower masses (as in the Bardeen, Hayward, and CGSV PRBH cases) genuinely correspond to an enlarged asteroid mass window. Therefore, the window where Bardeen, Hayward, and CGSV PRBHs could account for all the DM is much larger compared to that of Schwarzschild PBHs. \nOther potentially relevant constraints come from µ -distortions in the Cosmic Microwave Background, and gravitational waves (either a stochastic background due to a population of coalescing PBHs or produced via second-order tensor perturbations generated by the scalar perturbations producing the PBHs, or associated to resolved events). The latter are expected to be relevant in a much higher mass range (again, see Fig. 7 of Ref. [110] and Fig. 10 of Ref. [369]), whereas the former are somewhat dependent on the PBH formation scenario from highσ tails of density fluctuations, and in particular on the shape of the tail. At any rate, while the focus in the present pilot study has been solely on evaporation constraints from the ERGB, revisiting all these other important sources of constraints (including the ones we discussed earlier) is a worthwhile endeavour which we plan to explore in upcoming works.", 'V. CONCLUSIONS': "Over the past decade, primordial black holes have regained tremendous interest as viable dark matter candidates, with the so-called 'asteroid mass window' (10 17 g ≲ M pbh ≲ 10 23 g) where PBHs could potentially account for the entire DM currently still open. Nearly all works on PBHs assume that these are Schwarzschild or Kerr BHs. However, while phenomenologically perfectly valid, such an assumption may stir some unease on the theoretical side, due to the appearance of singularities in these metrics. In our work, we have conducted a pilot study aimed at addressing a question which naturally merges the DM and singularity problems, arguably two among the most important open problems in theoretical physics: ' What if PBHs are non-singular '? Our study of primordial regular BHs (PRBHs) has focused on three so-called tr -symmetric metrics (including the wellknown Bardeen and Hayward space-times), whereas our companion paper [194] considers nontr -symmetric met- \n, including various metrics inspired from loop quantum gravity. \nWe show that evaporation constraints on f pbh , the fraction of DM in the form of PRBHs, can be substantially loosened when moving away from the Schwarzschild picture, leading to the asteroid mass window further opening up. For the three PRBHs considered (the Bardeen, Hayward, and CGSV ones) the lower edge of the asteroid mass window is shifted by a decade in mass or more, leading to a larger region of parameter space where PRBHs could account for the entire DM component, which should be the target of the same probes proposed to test the standard window [171, 173, 175177, 179, 180, 377, 378]. The nature of the regular BH core (de Sitter or Minkowski) does not appear to play a significant role in this sense. Overall, we have shown that the phenomenology of primordial regular BHs can be particularly rich, making the associated simultaneous solution to the DM and singularity problems one worthy of further study. We note that the constraints we obtained would become even weaker had we moved closer to the extremal limit for the regularizing parameter ℓ : indeed, in this limit the temperatures of the RBHs we considered approaches zero, implying that there are no constraints from evaporation [275]. \nWe remark that the present work (alongside our companion paper [194]) should be intended as a pilot study, and there are a huge number of interesting follow-up directions. One interesting avenue for further work involves systematically revisiting other sources of constraints which have been studied in the Schwarzschild PBH case, including but not limited to lensing, accretion, and dynamical constraints: while we have argued that these should not alter our considerations on the asteroid mass window, a detailed study which would allow us to extend our constraints over a much larger region of M pbh -f pbh plane is nevertheless in order. In addition, the metrics we have considered are inherently phenomenological in nature, and it would therefore be worth extending our study to non-singular metrics which enjoy a strong theoretical motivation (our companion paper [194] goes partially in this direction), including potentially metrics which are coupled to the cosmological expansion [237, 242, 244, 261, 264]. Moreover, the formation mechanisms for these PRBHs is likely to be much more complex than the corresponding mechanisms for Schwarzschild PBHs: in fact, the space-times considered are not vacuum solutions of GR, which implies that Birkhoff's theorem does not (necessarily) hold, and correspondingly the endpoint of gravitational collapse may not be unique. At this stage it is impossible for us to make definitive statements on the matter without abandoning our agnostic viewpoint and assuming a specific underlying theoretical model, but the issue of PRBH formation (possibly within the context of inflationary models leading to an enhanced spectrum of curvature fluctuations over specific scales) is one which requires further study. Last but definitely not least, if PBHs are truly regular, \none would hope to ascertain this via signatures complementary to those we have studied: gravitational wave observations, VLBI imaging, motion around BHs, or energy extraction are potentially interesting observables in this sense. For instance, the shadows of Bardeen, Hayward, and CGSV BHs are smaller than their Schwarzschild counterparts by up to 20% [282], whereas the motion of stars around these BHs (take for instance the S2 star orbiting around Sgr A ⋆ ) would be altered compared to the motion around a Schwarzschild BH. These probes are being actively studied within the community as a means of distinguishing Schwarzschild BHs from alternatives thereto, and are likely to provide a promising route towards testing the regular nature of PRBHs, in the event that these are detected in the future and demonstrated to make up the DM, or even just a fraction thereof. We plan to address these and other related points in followup work.", 'ACKNOWLEDGMENTS': "We acknowledge support from the Istituto Nazionale di Fisica Nucleare (INFN) through the Commissione Scientifica Nazionale 4 (CSN4) Iniziativa Specifica 'Quantum Fields in Gravity, Cosmology and Black Holes' (FLAG). M.C. and S.V. acknowledge support from the University of Trento and the Provincia Autonoma di Trento (PAT, Autonomous Province of Trento) through the UniTrento Internal Call for Research 2023 grant 'Searching for Dark Energy off the beaten track' (DARKTRACK, grant agreement no. E63C22000500003). This publication is based upon work from the COST Action CA21136 'Addressing observational tensions in cosmology with systematics and fundamental physics' (CosmoVerse), supported by COST (European Cooperation in Science and Technology).", 'Appendix A: Details on the computation of greybody factors': "Here we provide a few more details on the computation of GBFs. We recall that we expressed the solutions to the radial Teukolsky equation, Eq. (20), in the form of a Taylor expansion as given by Eq. (23). This is also known as a Frobenius series, being a by-product of a method for solving second-order differential equations named after Frobenius. The method applies to equations which take the following form \nu '' + p ( x ) u ' + q ( x ) u = 0 , (A1) \nin proximity of its singular points, namely those where p ( x ) and/or q ( x ) diverge. One can notice that Eq. (20) can be rewritten in the form of Eq. (A1), with one of its singular point being at x = 0, i.e. at the event horizon. \nTo solve the radial Teukolsky equation we therefore proceed as follows: \n- · We work in units of the event horizon and rewrite Eq. (20) in order to remove the denominators \nA ( x ) R '' s + B ( x ) R ' s + C ( x ) R s = 0 , (A2) \nwhere the functions A ( x ), B ( x ), and C ( x ) are given by the following: \nA ( x ) = f 2 ( x +1) 2 , B ( x ) = ( s +1) f 2 ( x )(2( x +2) + ( x +1) 2 f ' /f ) , C ( x ) = +( x +1) 2 ω 2 +2 is ( x +1) ωf -is ( x +1) 2 ωf ' + sf ( ( x +1) 2 f '' +4( x +1) f ' +2 f -2 ) -l ( l +1) f + s ( s +1) f , \n- · The lowest power term around x = 0 of each coefficient can be written in the following form: \nA ( x ) ∼ x 2 τ 2 , B ( x ) ∼ x ( s +1) τ 2 , C ( x ) ∼ ω 2 -iωsτ , \nwhere τ = τ ( ℓ ) depends on the choice of RBH. \n- · We then build the following characteristic equation: \nm ( m -1) τ 2 + m ( s +1) τ 2 + ω ( ω -isτ ) = 0 , (A3) \nwhose solutions are the following: \nm 1 = -s -iω τ , m 2 = iω τ (A4) \n- · It is then possible to conclude that Eq. (20) admits solutions near the singular point x = 0 of the form given by Eq. (23). \nExplicitly, for the three RBHs in question, τ is given by the following: \nτ B = 1 -2 ℓ 2 ℓ 2 +1 , τ H = (1 -3 ℓ 2 ) , τ CGSV = 1 -ℓ . \nWe notice that in the Schwarzschild limit ℓ → 0, all of the above reduce to τ = 1 as one could expect.", 'Appendix B: PRBH time evolution': "Throughout the paper, in using M to denote the masses of PRBHs, we never specified whether we were referring to the initial mass, the present mass of the PRBH undergoing evaporation, or another quantity. This question is relevant since the mass of an evaporating BH is of course a monotonically decreasing quantity. As we shall see, this question is intertwined with the question of what \nis the lifetimes of these PRBHs, and whether they have already evaporated by now: the aim of this Appendix is to address these questions. \nIn Sec. III B we discussed the spectrum of photons emitted due to Hawking evaporation. In general, however, BHs will emit not only photons, but the whole spectrum of particles of the underlying theory. In fact, no specific coupling is required for this to occur. Enforcing energy conservation leads to the conclusion that the emission comes at the expense of the BH mass. Following the steps of Refs. [339-341, 379, 380], and considering a field of spin s , the associated depletion function is defined as follows: \nf s = M 2 ∫ ∞ 0 dω d 2 N dtdE ω = M 2 ∑ i,l,m 1 2 π ∫ ∞ 0 dω n i Γ s l,m ( ω ) e ω/T ± 1 ω = M 2 ∑ i,l (2 l +1) 2 π ∫ ∞ 0 dω n i Γ s l ( ω ) e ω/T ± 1 ω, (B1) \nwhere in the last equality we have exploited the fact that for spherically symmetric space-times the (2 l +1) different m modes are degenerate, and n i is the number of degrees of freedom of the particle in question. It is worth noting that there is no reference to µ , the mass of the field being evaporated by the BH. Given that the emission of particles with masses above the Hawking temperature T is Boltzmann-suppressed, in practice working within the usually adopted approximation where the particles are considered massless for µ < T , and absent at lower temperatures, is sufficiently accurate for our purposes (so we do not need to compute the GBFs for massive fields). \nAt a given particle temperature, summing over all the degrees of freedom accessible at a given BH temperature one can define the following function: \nF = N 0 f 0 + N 1 / 2 f 1 / 2 + N 1 f 1 + N 3 / 2 f 3 / 2 + N 2 f 2 , (B2) \nwhere N s count the degrees of freedom of spin s such that µ < T . This function is related to the BH mass loss rate as a function of time t through the following differential equation: \ndM dt = -F M 2 , (B3) \n- [1] A. Arbey and F. Mahmoudi, Prog. Part. Nucl. Phys. 119 , 103865 (2021), arXiv:2104.11488 [hep-ph].\n- [2] M. Cirelli, A. Strumia, and J. Zupan, (2024), arXiv:2406.01705 [hep-ph].\n- [3] R. 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In this light, and given that we have focused on PBHs with masses M > 10 15 g, which are far from having fully evaporated today, for our purposes we can safely approximate the BHs as being quasi-static throughout the lifetime of the Universe, while consequently denoting by M the values of the PBH mass both at formation and today. \nThe above is valid for Schwarzschild PBHs. However, as we have already seen earlier, the three PRBHs we considered are all colder compared to their Schwarzschild counterparts (at a given mass), and the slight increase in the GBFs is not sufficient to compensate for these decreased temperatures. In other words, the considerations we drew in Sec. III B lead us to conclude that these PRBHs have a longer lifetime compared to their Schwarzschild counterparts at a given mass, expectation that we have checked explicitly. 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2024MNRAS.533.4435R	We present radio observations of the longduration gammaray burst GRB 221009A that has become known to the community as the Brightest Of All Time or the BOAT. Our observations span the first 475 d postburst and three orders of magnitude in observing frequency from 0.15 to 230 GHz. By combining our new observations with those available in the literature we have the most detailed radio data set in terms of cadence and spectral coverage of any GRB to date which we use to explore the spectral and temporal evolution of the afterglow. By testing a series of phenomenological models we find that three separate synchrotron components best explain the afterglow. The high temporal and spectral resolution allows us to conclude that standard analytical afterglow models are unable to explain the observed evolution of GRB 221009A. We explore where the discrepancies between the observations and the models are most significant and place our findings in the context of the most wellstudied GRB radio afterglows to date. Our observations are best explained by three synchrotronemitting regions that we interpret as a forward shock a reverse shock and an additional shock potentially from a cocoon or wider outflow. Finally we find that our observations do not show any evidence of any latetime spectral or temporal changes that could result from a jet break but note that any lateral structure could significantly affect a jet break signature.	2024-10-01T00:00:00Z	['2024MNRAS.tmp.2064R', '10.48550/arXiv.2408.16637', '10.1093/mnras/stae2050', '2024arXiv240816637R', '2024MNRAS.533.4435R', 'arXiv:2408.16637']	['Astrophysics - High Energy Astrophysical Phenomena']	Rocking the BOAT the ups and downs of the longterm radio light curve for GRB 221009A	2,024	200	0.58	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	1	https://arxiv.org/pdf/2408.16637.pdf	{'Rocking the BOAT: the ups and downs of the long-term radio light curve for GRB 221009A': "- L. Rhodes, 1 ★ A. J. van der Horst, 2 , J. S. Bright 1 , J. K. Leung, 3 , 4 , 5 , G. E. Anderson 6 , R. Fender 1 , 7 ,\n- J. F. Agüí Fernandez 8 , M. Bremer, 9 P. Chandra 10 , D. Dobie 11 , 12 , W. Farah 13 , 14 , S. Giarratana, 15 , K. Gourdji 16 ,\n- D. A. Green 17 , E. Lenc 18 , M. J. Michałowski 19 ,T. Murphy 11 , 12 , A. J. Nayana 20 , A. W. Pollak 13 , 14 ,\n- A. Rowlinson 21 , 22 , F. Schussler 23 , A. Siemion 1 , 13 , 14 , 24 , 25 R. L. C. Starling 26 , P. Scott 17 , C. C. Thöne 27 ,\n- D. Titterington 17 A. de Ugarte Postigo 9 , 28\n- 1 Astrophysics, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK\n- 2 Department of Physics, George Washington University, 725 21st St NW, Washington, DC, 20052, USA\n- 3 David A. Dunlap Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada\n- 4 Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada\n- 5 Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel\n- 6 International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia\n- 7 Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa\n- 8 Centro Astronómico Hispano en Andalucía, Observatorio de Calar Alto, Sierra de los Filabres, Gérgal, Almería, 04550, Spain\n- 9 Observatoire de la Côte d'Azur, Université Côte d'Azur, Boulevard de l'Observatoire, 06304 Nice, France\n- 9 Institut de Radioastronomie Millimétrique, 300 Rue de la Piscine, 38400 Saint-Martin-d'Hères, France\n- 10 National Radio Astronomy Observatory, 520 Edgemont Rd, Charlottesville VA 22903, USA\n- 11 Sydney Institute for Astronomy, School of Physics, The University of Sydney, New South Wales 2006, Australia\n- 12 ARC Centre of Excellence for Gravitational Wave Discovery (OzGrav), Hawthorn, Victoria, Australia\n- 13 SETI Institute, 339 Bernardo Ave, Suite 200 Mountain View, CA 94043, USA\n- 14 Berkeley SETI Research Centre, University of California, Berkeley, CA 94720, USA\n- 15 Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807 Merate, (LC), Italy\n- 16 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia\n- 17 Astrophysics Group, Cavendish Laboratory, 19 J.J. Thomson Avenue, Cambridge, CB3 0HE, UK\n- 18 Australian Telescope National Facility, CSIRO Astronomy and Space Science, PO Box 76, Epping, NSW 1710, Australia \n19 \nAstronomical Observatory Institute, Faculty of Physics, Adam Mickiewicz University, ul. Słoneczna 36, 60-286 Poznań, Poland \n- 20 Department of Astronomy, University of California, Berkeley, USA\n- 21 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, the Netherlands\n- 22 ASTRON, the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedĳk 4, 7991 PD Dwingeloo, the Netherlands\n- 23 IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France\n- 24 Department of Physics and Astronomy, University of Manchester, UK\n- 25 University of Malta, Institute of Space Sciences and Astronomy, Msida, MSD2080, Malta\n- 26 University of Leicester, School of Physics and Astronomy, University Road, Leicester LE1 7RH\n- 27 Astronomical Institute of the Czech Academy of Sciences (ASU-CAS), Fricova 298, Ondřejov, 251 65, Czech Republic\n- 28 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': 'Wepresent radio observations of the long-duration gamma-ray burst (GRB) 221009A which has become known to the community as the Brightest Of All Time or the BOAT. Our observations span the first 475 days post-burst and three orders of magnitude in observing frequency, from 0.15 to 230 GHz. By combining our new observations with those available in the literature, we have the most detailed radio data set in terms of cadence and spectral coverage of any GRB to date, which we use to explore the spectral and temporal evolution of the afterglow. By testing a series of phenomenological models, we find that three separate synchrotron components best explain the afterglow. The high temporal and spectral resolution allows us to conclude that standard analytical afterglow models are unable to explain the observed evolution of GRB 221009A. We explore where the discrepancies between the observations and the models are most significant and place our findings in the context of the most well-studied GRB radio afterglows to date. Our observations are best explained by three synchrotron emitting regions which we interpret as a forward shock, a reverse shock and an additional shock potentially from a cocoon or wider outflow. Finally, we find that our observations do not show any evidence of any late-time spectral or temporal changes that could result from a jet break but note that any lateral structure could significantly affect a jet break signature. \nKey words: gamma-ray burst: individual: GRB 221009A - ISM: jets and outflows - radio continuum: transients', '1 INTRODUCTION': "Long-duration gamma-ray bursts (GRBs) are produced in highly relativistic jets, launched during the collapse of massive stars, and they are the most powerful explosions in the Universe. GRB 221009A has been dubbed the Brightest of all Time, or the BOAT (Burns et al. 2023). Lasting about 600 seconds, the variable, high energy, prompt emission was detected by the Neil Gehrels Swift Observatory - Burst Alert Telescope and X-ray telescope (BAT and XRT, respectively, Williams et al. 2023), Insight-HXMT and GECAM-C (An et al. 2023), Konus Wind and SRG/ART-X (Frederiks et al. 2023) and the Fermi Gamma-ray Space Telescope (Lesage et al. 2023). Placed at a redshift of 0.151 (de Ugarte Postigo et al. 2022; Malesani et al. 2023), the isotropic gamma-ray energy output has been measured as 1 × 10 55 erg, 1 . 5 × 10 55 erg and 1 . 2 × 10 55 erg by Lesage et al. (2023); Anetal. (2023); Frederiks et al. (2023, between 1-10,000 keV, 10 keV - 6 MeV and 20 keV - 10 MeV, respectively), nearly twice the value of the next most energetic, GRB 080916C (Greiner et al. 2009). Given its prompt emission properties, it has been established as a once in 10,000 year event (Burns et al. 2023). In fact, GRB 221009A was so bright that the prompt emission caused disturbances in the ionosphere (Hayes & Gallagher 2022). \nThe afterglow to GRB 221009A has been detected consistently between 0.4 GHz and 20 TeV (Laskar et al. 2023; LHAASO Collaboration et al. 2023). In terms of spectral coverage, it exceeds the all other TeV afterglows with radio detections like GRB 190114C or GRB 190829A (MAGIC Collaboration et al. 2019a; H. E. S. S. Collaboration et al. 2021). In terms of data quantity and quality, it exceeds the GHz-to-GeV afterglow of GRB 130427A (e.g. van der Horst et al. 2014; Levan et al. 2014; Ackermann et al. 2014; De Pasquale et al. 2016), although the latter had a much better sampling of optical light curves since it did not suffer from extinction in the way that GRB 221009A did (Fulton et al. 2023; Levan et al. 2023). Similar to GRB 130427A (Anderson et al. 2014), Bright & Rhodes et al. (2023) showed that GRB 221009A had a bright light curve peak at 15 GHz within the first day, followed by an overall decline at radio frequencies. This behaviour is quite different from the 'classical' well-sampled radio afterglows of, for instance, GRB 970508 and GRB 030329, which have peaks at timescales of weeks to months (Frail et al. 2000; van der Horst et al. 2005; Resmi et al. 2005). The origin of the early time radio peaks are thought to be from reverse shock, produced from a shock front propagating back through the jet. Details of the light curve behaviour, in particular over a wide frequency range, give important insights into the underlying physics at various scales, from the jetted explosion outflow to the accelerated particles generating the observed emission (e.g., Sari et al. 1998; Wĳers & Galama 1999). \nThe focus of this paper is the radio emission from GRB 221009A, covering three orders of magnitude in both observing frequency and days post-burst. While the TeV emission leads to various questions regarding possible emission processes at these high energies and the potential for detecting GRBs at TeV energies more frequently (MAGIC Collaboration et al. 2019a; H. E. S. S. Collaboration et al. 2021; Abe et al. 2024), the radio observations provide the necessary context for understanding the physics of the jetted GRB outflow, together with multi-wavelengths observations in the optical and X rays (e.g. Gill & Granot 2023; O'Connor et al. 2023). The light curve behaviour of GRB 221009A in the first days to weeks do not seem to follow expectations of the standard model that is typically used to describe radio afterglows (e.g. Wĳers & Galama 1999; Granot & Sari 2002). The extremely dense sampling of the light curves at various radio frequencies as presented in this paper is unprecedented \nand allows for detailed modelling that will lead to better descriptions of GRB jets and the relevant emission processes. \nThe dominant emission mechanism in GRB afterglows at radio frequencies is synchrotron radiation from extremely relativistic electrons accelerated by shocks at the front of a relativistic collimated outflow (Meszaros & Rees 1993; Sari et al. 1998). This is also the emission mechanism assumed to be at play in the GRB 221009A afterglow. While we are only considering one emission mechanism, i.e., synchrotron, there can be multiple emission sites. For instance, the jet sweeping up particles in the ambient medium leads to a forward shock, but will also lead to the formation of the aforementioned reverse shock which can be dominant at early times given the right conditions (Kobayashi & Sari 2000). Besides this shock structure in the radial direction, there can also be structure in the lateral direction. This structure could be smooth, for instance, a structured energy profile as a function of distance to the jet axis instead of a homogeneous energy profile (Rossi et al. 2002; Lamb et al. 2021; Salafia & Ghirlanda 2022); but there could also be multiple jet components (Starling et al. 2005), and potentially a cocoon around the jet (Ramirez-Ruiz et al. 2002; Nakar & Piran 2017; Izzo et al. 2019). This could lead to multiple synchrotron emission components, or emission components that evolve differently from the canonical tophat behaviour (van der Horst et al. 2014; Bright et al. 2019; Rhodes et al. 2022). \nBesides these macrophysical considerations, high-quality multiwavelength data as presented here reveals nuances in the microphysics of GRB afterglows. Afterglow modelling can lead to insights into the magnetic field strength and energetics, but also the total energetics, acceleration efficiency, and energy distribution of the accelerated electrons (Granot & Sari 2002; Eichler & Waxman 2005). To complicate this further, detailed simulations of particle acceleration and magnetic field amplification by relativistic shocks indicate that there is potentially a time dependence of the energies in magnetic fields and electrons (Rossi & Rees 2003), and this has also been adopted in multi-wavelength modelling of some GRB afterglows with peculiar behaviour (van der Horst et al. 2014; Bright et al. 2019; Misra et al. 2021; Salafia et al. 2022). \nGiven the extremely high quality of the radio data presented in this paper, and the dynamics of the synchrotron spectrum that is likely quite different from the standard behaviour, we take a fairly cautious approach in the modelling presented here. While a standard GRB synchrotron spectrum is still assumed, the temporal evolution of the spectrum is kept free of constraints where possible, to provide input on detailed modelling and theoretical efforts, and get a better handle on the interpretation of the wealth of these data from this unique source. We highlight here the use of the convention 𝐹 𝜈 ∝ 𝑡 𝛼 𝜈 𝛽 throughout this work to describe the temporal and spectral evolution. This paper is laid out in the following manner: in Section 2 we present the new radio observations and the data reduction methods used; in Section 3, we lay out the results of our observing campaigns and describe the model used to explain the data; in Section 4 we put our results in a broader context and interpret the data using various models; and we conclude in Section 5.", '2 OBSERVATIONS': 'Here we present the data reduction processes for the observations used in this work. The flux density measurements and upper limits for our new observations are summarised in Table 1. In addition to the datasets we present here, our work also incorporates the previously published radio data from Laskar et al. (2023); Giarratana et al. \n(2023) and Bright & Rhodes et al. (2023), and the X-ray data from Williams et al. (2023).', '2.1 AMI-LA': 'The Arcminute Microkelvin Imager - Large Array (AMI-LA) is an eight-dish interferometer based in Cambridge, UK (Zwart et al. 2008). It observes at a central frequency of 15.5 GHz with a bandwidth of 5 GHz, achieving an angular resolution of about 30 arcsec (Hickish et al. 2018). Bright & Rhodes et al. (2023) presented the first five days of observations from AMI-LA, and here we present the rest of the observing campaign. We continued to observe the position of GRB 221009A almost daily until 210 days post-burst when the first non-detection occurred. Between 210 and 320 days post-burst, we concatenated separate non-detections to obtain deeper limits. \nAMI-LA data is reduced using a custom software package: /r.pc/e.pc/d.pc/u.pc/c.pc/e.pc\\_/d.pc/c.pc (Perrott et al. 2013). The software performs bandpass and flux scaling using 3C286 and complex gain calibration using J1925+2106. Flagging and imaging is done in /c.pc/a.pc/s.pc/a.pc using the tasks rflag , tfcrop and clean (McMullin et al. 2007). The details of observing times and measured flux densities are provided in Table 1. We note that unlike in Bright & Rhodes et al. (2023), we do not split each observation up, because the duration of a given epoch is a negligible fraction of the total time since the burst was first detected, so no significant evolution is expected within an observation.', '2.2 ASKAP': 'We obtained target-of-opportunity observations of the GRB 221009AfieldwiththeAustralianSquare Kilometre Array Pathfinder (ASKAP, Johnston et al. 2007). Our observations were centred on 888 MHz, with a bandwidth of 288 MHz, taken using the square\\_6x6 beam footprint (see figure 20 of Hotan et al. 2021). The data products for these observations can be found under the project code AS113 with SBIDs: 44780, 44857, 44918, 45060, 45086, 45416, 46350, 46419, 46492, 46554 and 48611 in the CSIRO ASKAP Science Data Archive (CASDA 1 ). \nObservations of PKS B1934 -638 were used to calibrate the antenna gains, bandpass and the absolute flux-density scale. Flagging of radio frequency interference, calibration of raw visibilities, fullpolarisation imaging, and source finding on total intensity images were all performed through the standard ASKAPsoft pipeline (Guzman et al. 2019). The resulting image reached a typical rms of ∼ 50 𝜇 Jy beam -1 . We evaluated and corrected for the systematic fluxscale offset by comparing the flux density of field sources in each observation against those in the Rapid ASKAP Continuum Survey (RACS) catalogue (Hale et al. 2021).', '2.3 ATA': 'Located ∼ 200 miles north of San Fransisco, the Allen Telescope Array is a 42-element radio interferometer hosted at the Hat Creek Radio Observatory. Mounted on the focus of each element is a dualpolarization, log-periodic feed that is cryogenically cooled and sensitive to radiation in the range of 1 to 12 GHz. Analogue signals from the array are transmitted through fibre to a centralised signal processing room and are split into 4 independent chains that get multiplexed by 4 tunable local oscillators in a super-heterodyne system. The current correlator backend supports the digitisation of 2 out of \nthe 4 available tunings for 20 of the 42 antennas, where each tuning can be placed anywhere in the available RF range of the log periodic feed, with ∼ 700 MHz of usable bandwidth for each. \nThe radio counterpart of GRB 221009A was observed extensively with the ATA beginning just a few hours after the burst as reported in Bright & Rhodes et al. (2023). Here we build on that work and utilised the flexible frequency tunability of the ATA to monitor the 1-10 GHz spectral evolution over its entire outburst. Either 3C147, 3C48, or 3C286 was observed as flux calibrator at the beginning of each observing block, and a 10 minute observation of the phase calibrator J1925 + 2106 was interleaved for every 30 minutes of science target recording (regardless of observing frequency). We evolved our total integration time on source over the course of the follow-up campaign to account for the fading of GRB 221009A. Visibilities from each observation block were reduced using a custom pipeline using AOFLAGGER (Offringa 2010) and CASA (McMullin et al. 2007). Images for the flux, phase and science targets were formed using standard CASA tasks and by deconvolving with the CLEAN algorithm (Högbom 1974; Clark 1980; Sault & Wieringa 1994). We used two Taylor terms to account for the high fractional bandwidth (especially at low frequencies) and a Briggs robust value of 0.5 when imaging. Finally, flux densities for GRB 221009A were derived by fitting a point source (i.e., with a source size fixed to the dimensions of the main lobe of the dirty beam) to the science target.', '2.4 ATCA': "Wecarried out multiple observations of the radio counterpart to GRB 221009A using the Australia Telescope Compact Array (ATCA) under the project codes: CX515 (director's discretionary time), C3374 (PI: G. E. Anderson), C3542 (PI: G. E. Anderson). These observations were carried out using the 5.5/9, 16.7/21.2, 33/35, and 43/45 GHzreceiver configurations, with a bandwidth of 2048 MHz for each intermediate frequency. \nFor each observation, we reduced the visibility data using standard procedures in M/i.pc/r.pc/i.pc/a.pc/d.pc (Sault et al. 1995). We used a combination of manual and automatic radio-frequency interference flagging before calibration. For bandpass calibration, we used PKS B1934 -638 at 5.5/9 GHz, while at higher frequencies (16.7/21.2, 33/35 and 43/45 GHz) we used either B1921 -293 or B1253 -055; the spectral shape of B1921 -293 and B1253 -055 was accounted for by fitting to first order the measured flux densities of these calibrators at each intermediate frequency for each of the higher frequency observing bands. The flux-density scale was set using B1934 -638 for all observing frequency bands. For all observations, we used B1923 + 210 to calibrate for the time-variable complex gains. After calibration, where there was sufficient signal-to-noise, we split the 2048 MHzbandwidth into further sub-bands to obtain higher spectral resolution. We then inverted the visibilities and applied the multifrequency synthesis CLEAN algorithm (Högbom 1974; Clark 1980; Sault & Wieringa 1994) to the target source field using standard tasks in M/i.pc/r.pc/i.pc/a.pc/d.pc to obtain our final images. The flux densities of the radio afterglow candidate were extracted by fitting a point source to the radio source, in the case of a detection, while, in the case of a non-detection, the limits were obtained using the rms sensitivity in the residual image.", '2.5 e -MERLIN': "The enhanced Multi-Element Remotely Linked Interferometer Network ( e -MERLIN) is a radio interferometer made up of seven dishes \nspread across the UK. With a maximum baseline of 217 km, whilst observing at 5 GHz, it can resolve angular scales of 0.05'. We observed the position of GRB 221009A with e -MERLIN through a combination of rapid response time requests (PI: L. Rhodes, RR14001) and open time proposals (PI: L. Rhodes, CY13003, CY14001 and CY15206) at both L- and C-band. Our L- and C-band observations were centered at 1.51 and 5.08 GHz, respectively, both with a bandwidth of 512 MHz. We note that the first two epochs obtained at L-band have previously been published in Bright & Rhodes et al. (2023). \nAll observations were reduced using the e -MERLIN pipeline within /c.pc/a.pc/s.pc/a.pc (McMullin et al. 2007; Moldon 2021). The pipeline performs preliminary flagging for radio frequency interference and known observatory issues. It then performs two rounds of bandpass calibration and complex gain calibration, using OQ208 and J1905+1943, respectively, along with flux scaling using 3C286. Further flagging of the target field is conducted. We performed interactive cleaning and deconvolution using the casa task tclean .", '2.6 LOFAR': "Eight hours of Director's Discretionary Time with the Low Frequency Array (LOFAR; DDT20\\_003) were awarded to observe GRB 221009A. The allocated time was split into two observing runs of 4-hours, which took place on 18 and 20 July 2023 at matching local sidereal times. Each observing run was preceded by a 10-minute calibrator scan of 3C295. All observations were conducted in the HBA\\_dual\\_inner configuration where, in addition to the 22 core stations available, the inner tiles of 14 remote stations were also used. The single-beam observations were centred at 152.05 MHz with 380 subbands and data were recorded with an integration time of 1 second. Each subband consisted of 64 frequency channels of width 3.051 kHz. The data were subsequently averaged to 16 channels of 12.21 kHz per subband by the observatory during data preprocessing. Both target observations were calibrated for direction independent effects using LINC 2 with default settings, a pipeline developed by the observatory to correct for various instrumental and ionospheric effects present in interferometric LOFAR data (de Gasperin et al. 2019; van Weeren et al. 2016; Williams et al. 2016). Due to its relative proximity, Cygnus A was subtracted from the visibilities using the 'demixing' step in LINC. The data were further averaged to 4 channels of 48.82 kHz per subband and 4 seconds during calibration. The resulting calibrated data were concatenated into groups of 20 subbands and averaged in time to 8 seconds. These data products from both observations were subsequently jointly put through /d.pc/d.pc/f.pc/p.pc/i.pc/p.pc/e.pc/l.pc/i.pc/n.pc/e.pc 3 for direction-dependent calibration and imaging (Shimwell et al. 2019; Tasse et al. 2021). This resulted in a final image generated using a circular restoring beam of radius 3 arcseconds and 1.5 arcseconds pixel resolution.", '2.7 NOEMA': 'The NOrthern Extended Millimetre Array (NOEMA, situated in the southern French Alps) monitored GRB 221009A between October 10 th 2022 and April 25 th 2023 in the 3, 2 and 1mm bands. Interferometer configurations were medium-extended C and extended A \n2 \nhttps://linc.readthedocs.io/en/latest/index.html \nconfigurations with up to 12 antennas, primary flux calibrators were MWC349 and LKHA101. The data were reduced with the /c.pc/l.pc/i.pc/c.pc and /m.pc/a.pc/p.pc/p.pc/i.pc/n.pc/g.pc software packages that are part of the /g.pc/i.pc/l.pc/d.pc/a.pc/s.pc 4 package. Fluxes and their errors were derived from point-source UV-plane fits to the calibrated interferometric visibilities.', '2.8 uGMRT': 'We observed GRB 221009A with the upgraded Giant Metrewave Radio Telescope (uGMRT) in bands 5 (1000-1450 MHz) and 4 (550-900 MHz) under a DDT proposal (ddtC251, PI: P. Chandra). The observations were made at two epochs in both bands, once in January 2023 and then in March 2023. We recorded the data in 2048 frequency channels covering a bandwidth of 400 MHz with an integration time of ∼ 10 s. We used 3C286 and 3C48 as flux density and bandpass calibrators. J1924+3329 was used as a phase calibrator. \nThe data were analysed using the /c.pc/a.pc/s.pc/a.pc package (McMullin et al. 2007) following the procedure in Nayana et al. (2022). We also performed a few rounds of phase only and one round of amplitude and phase self-calibration to improve the image quality. The final flux densities were obtained by fitting a Gaussian at the GRB position.', '3 RESULTS & MODEL': "There have been several GRB 221009A afterglow modelling efforts whichhaveusedasubsetoftheradiodata published to date (including but not limited to Laskar et al. 2023; O'Connor et al. 2023; Levan et al. 2023; Gill & Granot 2023). Here, we present the results of our observing campaigns and described out modelling of the radio and X-ray afterglow.", '3.1 Light Curves and SEDs': "Theradio data presented in this paper spans three orders of magnitude in frequency space, from 0.15 MHz to 230 GHz, and lasts out to 475 days post-burst. Figure 1 shows the radio afterglow light curves split by observing frequency. Symbols with lower opacity denote all previously published data whereas the solid symbols mark data presented in this paper. We include all previous and newly published data to extract the clearest scenario of the afterglow. \nAbove 19 GHz, the afterglow is decaying at all times, with observations obtained between 1 and 200 days post-burst (the top two rows of Figure 1). The light curves between 90 and 105 GHz in Figure 1 show that the decay rate slowly steepens with time like a very smooth broken power law. Below 16 GHz, we observe the light curve peak in almost each observing band, except at 9-10 and 0.4 GHz since we were not observing early enough at those frequencies. Bright & Rhodes et al. (2023) interpreted this peak as emanating from the reverse shock, which we are tracking from 17.7 to below 1 GHz. The data between 1.3 and 3 GHz also show a second, distinct bump at around 50 days. In addition to the early peaks caught at 5 and 15.5 GHz, we also see evidence of further bumps during the decay phase. It is possible that the additional bumps originate from different spectral components. \nFigure 2 shows the broadband radio spectral energy distributions (SEDs) throughout our campaign. For the first 30 days, a lowfrequency turnover is visible and the below-turnover spectral index is consistent with 𝛽 ∼ 5 / 2 below the turnover. Above the turnover, we \nTable 1. A table of the new radio observations presented in this work. All non-detections are indicated by a '-' in the flux density column followed by the 3 𝜎 upper limit in the uncertainty column. The full list of radio observations are presented in supplementary material online. \nfindaflatspectrumextendingtothehighestfrequencies ( ∼ 200 GHz). A flat spectrum is inconsistent with optically thin synchrotron emission from a single component and so provides further evidence of multiple spectral components, similar to GRB 130427A (Perley et al. 2014). Only after 150 days post-burst does the spectrum steepen with typical optically thin spectral indices ( 𝛽 ∼ -0 . 5 to -1), more consistent with that from the late-time X-ray data (Williams et al. 2023). Williams et al. (2023) performed a joint fit to the UV, X-ray and gamma-ray data, which shows that the high-energy spectra can be described by either a single power law or a broken power law where the break, interpreted as the synchrotron cooling break 𝜈 c , sits in the XRT band. The broken power law is favoured but the fits are only performed on data up to one day post-burst whereas the X-ray light curve itself extends out to 200 days post-burst.", '3.2 Modelling': 'Here, we build on previous modelling efforts by combining our new observations from AMI-LA, ATA, ATCA,ASKAP, e -MERLIN,LOFAR,NOEMAanduGMRTwithradiodataavailableintheliterature. We also include the full Swift -XRT light curve (in flux densities at 10 keV; Williams et al. 2023). We do not include any optical or other high-energy data in our modelling work as there are too many contaminating components in these bands. At optical frequencies, there is significant extinction (Tiengo et al. 2023; Vasilopoulos et al. 2023) both from the Milky Way and the host galaxy, plus a contribution from the associated supernova. Above keV energies, there is an increasing contribution from the additional VHE component whose origin and emission mechanism is still debated (Aharonian et al. 2023; LHAASO Collaboration et al. 2023; Savchenko et al. 2024). \nWe consider models that use either two or three synchrotron spectral components that can evolve independently in time to explain the behaviour shown in the light curves (Figure 1) and SEDs (Figure 2). Each synchrotron spectrum is constructed of four power-law slopes divided by three frequency breaks: the synchrotron self-absorption break ( 𝜈 sa ), the characteristic or minimum electron energy break ( 𝜈 m ), and the cooling break ( 𝜈 c , above which radiative cooling is important). The peak of the spectrum, 𝐹 𝜈, max , is at whichever frequency break of 𝜈 sa or 𝜈 m is higher. The spectral index of each branch depends on the order of the frequency breaks. In the regime where 𝜈 sa < 𝜈 m < 𝜈 c , the spectral indices are 𝐹 𝜈<𝜈 sa ∝ 𝜈 2 , 𝐹 𝜈 sa <𝜈<𝜈 m ∝ 𝜈 1 / 3 , 𝐹 𝜈 m <𝜈<𝜈 c ∝ 𝜈 ( 1 -𝑝 )/ 2 and 𝐹 𝜈 c <𝜈 ∝ 𝜈 -𝑝 / 2 , where 𝑝 is the electron energy distribution index and is typically expected to be between 2 and 3 (although values slightly below 2 and above 3 have been reported, Kirk et al. 2000; Achterberg et al. 2001; Sironi et al. 2013). In the regime where 𝜈 m < 𝜈 sa < 𝜈 c , the spectral indices are 𝐹 𝜈<𝜈 m ∝ 𝜈 2 , 𝐹 𝜈 m <𝜈<𝜈 sa ∝ 𝜈 5 / 2 , 𝐹 𝜈 sa <𝜈<𝜈 c ∝ 𝜈 ( 1 -𝑝 )/ 2 and 𝐹 𝜈 c <𝜈 ∝ 𝜈 -𝑝 / 2 . As the jet expands and evolves, the spectral \nbreaks are expected to change as a power-law function of time, which depends on the jet dynamics and the density profile through which the jet is propagating, 𝜌 ∝ 𝑟 -𝑘 , where 𝑘 = 0 for a homogeneous medium and 𝑘 = 2 represents a stellar wind (Granot & Sari 2002; Granot & van der Horst 2014). \nWe use /e.pc/m.pc/c.pc/e.pc/e.pc to fit our respective models to the data (ForemanMackey et al. 2013). Each model uses 40 walkers and runs for at least 70000 steps or until convergence. All priors are uniform, and the only priors with fixed bounds were 𝑝 ∈ [ 1 . 5 , 3 . 5 ] to help rule out unphysical solutions. The best fit value for each parameter is the 50 th percentile post burn in of the posterior distribution, and the 84 th and 16 th percentiles are quoted as the upper and lower uncertainties, respectively.', '3.2.1 Two-Component Model': 'First, we fit the data with two separate synchrotron spectra. The first is the reverse shock identified in Bright & Rhodes et al. (2023), we find that the peak of the synchrotron spectrum is produced by 𝜈 sa and fit for the normalisation and evolution of the spectrum as well as 𝑝 . The second component is a forward shock that appears to dominate the optical and X-rays (e.g. Williams et al. 2023; Fulton et al. 2023; Shrestha et al. 2023), and also the late-time radio emission. Here we allow both 𝜈 sa and 𝜈 m to vary freely. We fit for the normalisation and evolution of 𝐹 𝜈, max , 𝜈 sa and 𝜈 m as well as 𝑝 . The resulting model parameters are provided in Table 2. \nWe find that the two-component model cannot reproduce the flat spectrum observed shown in Figure 2, the posterior distribution of 𝑝 for the reverse shock always ends up at the lower bound of the prior with values for 𝑝 below 1.5 or even below 1, and such a low value is unphysical and so we no longer consider this scenario.', '3.2.2 Three-Component Model': "Given the issues with a two-component model, we include a third component to alleviate the shallow value of p which was needed in the two-component model to explain the flat spectrum that is present during the first ∼ 150 days (Figure 2) and the additional bumps in the 5 and 15.5 GHz light curves around 5-10 days post-burst (see Figure 1). \nTo best explore the parameter space of the third component, first, we test both 𝜈 sa and 𝜈 m as the peak frequency of the third component and find that 𝜈 sa provides a better fit. Then we consider two different iterations of this extra shock with differing degrees of freedom, which are summarised in Table 3, in addition to the two shock components described in the previous section. In both iterations of our three-component model, the peak flux density of each of the three \nFigure 1. Radio afterglow light curves of GRB 221009A split by observing frequency (or frequency range). Any low-opacity data points are from previously published observations. All observations presented in this paper are shown with solid circles for detections and downwards-facing triangles for 3 𝜎 upper limits. \n<!-- image --> \nFigure 2. Broadband radio SEDs for GRB 221009A as a function of time. As in Figure 1, low-opacity data points denote previously published data, while solid points are observations presented in this paper. Because epochs have been chosen to demonstrate the spectral evolution, we note that not all data presented in Figure 1 are also shown here. \n<!-- image --> \nTable 2. The parameter values (50 th percentile) and their associated uncertainties (18 th and 64 th percentiles) derived for our best-fit two-component model. Any 𝛼 parameter refers to the temporal power law index of the parameter written in the subscript, as described in Section 3. For the reverse, forward and extra shock component, F 𝜈, max and 𝜈 sa are normalised to 1 day and 6.5 days, respectively. For each shock, p is the value of the electron energy spectral index. \ncomponents follows a smoothly broken power law (Rhodes et al. 2020): \n𝐹 𝜈 = 𝐹 𝜈, max GLYPH<18> 0 . 5 GLYPH<18> 𝑡 𝑡 𝑏 GLYPH<19> -𝛼 1 𝑠 + 0 . 5 GLYPH<18> 𝑡 𝑡 𝑏 GLYPH<19> -𝛼 2 𝑠 GLYPH<19> -1 𝑠 (1) \nwhere 𝐹 𝜈, max is the flux density at the break time 𝑡 𝑏 , 𝛼 1 and 𝛼 2 are the power-law indices, and 𝑠 is the smoothing parameter which we set to be 0.5. In Model 1, the synchrotron self-absorption break follows a single power law: 𝜈 sa = 𝜈 sa , 0 𝑡 𝛼 sa where 𝜈 sa , 0 is the location of the self-absorption break at 1 day post-burst. We set 𝛼 𝐹 𝜈, max , 1 = 3 and both 𝛼 𝐹 𝜈, max , 2 (defined in Table 3 as 𝛼 ) and 𝛼 sa can vary freely. We invoke a 𝛼 𝐹 𝜈, max , 1 = 3 as done in Peng et al. (2005) which is used in the regime where a blastwave that is initially off-axis has undergone significant deceleration and so the radiation begins to enter the observers' line of sight. In the paper, they do not consider the self-absorption break, but we find it fits well within the constraints of our work. Ryan et al. (2020) also consider off-axis afterglows from a numerical perspective and find steeper rise rates for 'far off-axis events'. We choose to be more conservative and use Peng et al. (2005) value. \nIn model 2, both the peak flux density and 𝜈 sa are both described with broken power laws where all the indices are fit for but the break time is the same. A full summary of our models to explain the extra forward shock is shown in Table 3. \nFigure 3 shows the results of the different iterations of our models. Unfortunately, not one of our models provides a perfect fit to the data, this may be due to a combination of unknown systematic uncertainties, and perhaps more importantly, this exquisite data set is showing evidence of more complicated physics and emission mechanisms that cannot be accounted for by the basic synchrotron models. As a result, we find quoting Bayesian evidence values inappropriate. However, we do find that model 1 provides the best fit. This is because our posterior distributions for all values of 𝑝 sit between 2 and 3 and do not require such uncomfortably large temporal index values. We present the parameters of this fit in Table 4. Figures 4 and 5 show our best- \nTable 3. Summary of the different iterations of the three-component model which explore the possible evolution of the third shock component. Each 𝛼 corresponds to a temporal index of the subscripted value, e.g., 𝛼 𝐹 𝜈, max , 1 corresponds to the first temporal index used to describe the behaviour of 𝐹 𝜈, max . We find that model 1, combined with a forward and reverse shock describes the data best. The model is shown compared to the data in Figures 4 and 5. The best fit parameters are shown in Table 4. \nTable 4. The parameter values (50 th percentile) and their associated uncertainties (18 th and 64 th percentiles) derived for our best-fit three-component model (model 1). Any 𝛼 parameter refers to the temporal power law index of the parameter written in the subscript, as described in Section 3 and Table 3. For the reverse, forward and extra shock component, F 𝜈, max and 𝜈 sa are normalised to 1 day, 6.5 days and t dec , respectively, where t dec is a parameter we fitted for. For each shock, p is the value of the electron energy spectral index. \nting model overlaid on the light curves and SEDs. Figure 4 shows that our model describes the long term evolution at all frequencies well. However, it cannot replicate the bumps and wiggles observed at 15.5, 5 and 0.4 GHz, despite that being one of the motivations for the three component model. Furthermore, it marginally over predicts the late time 0.8 GHz flux. Figure 5 demonstrates that the superposition of multiple components recreates the high frequency emission accurately and describes well the flat spectrum and broad turnover at earlier times post-burst. On the other hand, we find that it tends to place the 𝜈 sa much lower than the observed position. \nFigure 3. Evolution of the break frequencies and peak flux for the three-component model. Each panel corresponds to a different iteration of our model as described in Section 3 and Table 3. For each iteration, we show only the average value (50 th percentile value) of the posterior distribution for clarity. The lefthand vertical axis of each plot corresponds to the evolution of the frequency breaks (dotted and dashed lines for 𝜈 sa and 𝜈 m , respectively). The righthand vertical axis shows the evolution of the peak flux (solid lines) of each shock component. \n<!-- image -->", '4 DISCUSSION': 'Here we discuss the implications of our best-fitting three-component model and place them in context of other detailed radio studies of GRBs.', '4.1 Reverse shock': 'The dashed lines in Figures 4 and 5 show the contribution of the reverse shock from our model. Bright & Rhodes et al. (2023) used radio observations in the first five days post-burst to measure the evolution of F max and 𝜈 sa with time. They found that 𝐹 𝜈, max ∝ 𝑡 -0 . 70 ± 0 . 02 and 𝜈 sa ∝ 𝑡 -1 . 08 ± 0 . 04 , and concluded that the evolution of the spectral peak was too slow to match theoretical predictions and most likely a superposition of multiple emitting regions. When considering the full radio data set, we find a different, even slower reverse shock evolution: F max ∝ 𝑡 -0 . 59 ± 0 . 05 and 𝜈 sa ∝ 𝑡 -0 . 86 ± 0 . 03 , and that multiple shocks are contributing to the early 15.5 GHz observation. We find that the slow evolution of the reverse shock means that it contributes significantly to the low-frequency emission at all times. \nTo contextualise these findings, we compare our results to both thin and thick reverse shock models summarised in van der Horst et al. (2014). The distinction between thin and thick shell models refers to the depth and velocity spread of the shell that the shock is moving through. The reverse shock emission is produced as it propagates back through the shell at the front of the jet. In a thick shell scenario, the velocity spread of the ejected material is large enough such that the shock can accelerate to become relativistic, and the resulting light curves depend on the circumburst environment profile, as does the forward shock. In the thin shell scenario, the reverse shock remains Newtonian, and reverse shock light curves are dependent on the deceleration profile of the jet (Sari & Piran 1995; Mészáros & Rees 1999). With the results of our model, we cannot recreate our observations with physically realistic parameter values for either a thick or thin shell reverse shock model. We find that the reverse shock evolution that we measure is too slow compared to analytical models such as those in van der Horst et al. (2014). \nCompared to the number of detailed forward shock studies, there are very few GRBs where the reverse shock is observed in sufficient \ndetail to confidently examine certain reverse shock models. GRBs 130427A, 190114C and 190829A are the three most well-studied GRBs with bright reverse shock components (they also happen to all have - at least tentative - very high energy components like GRB221009A; Ackermann et al. 2014; MAGIC Collaboration et al. 2019b; H. E. S. S. Collaboration et al. 2021). The reverse shock component from GRB 190114C appears to match with theoretical models for reasonable physical parameters (Laskar et al. 2019). However, GRBs 130427A and 190829A could not be explained by analytical reverse shock models (van der Horst et al. 2014; Salafia et al. 2022). In the case of GRB 190829A, the best fit came from assuming a rapid decay in the magnetic field strength post-shock crossing (Salafia et al. 2022). It is possible that GRB 221009A requires a similarly complex model to explain the observed behaviour but that is beyond the scope of this work.', '4.2 Forward shock': "The dotted lines in Figures 4 and 5 denote the contribution from the forward shock. The forward shock component of our model dominates all of the high-frequency light curves (above 33 GHz) at all times. Moving to lower observing frequencies the forward shock contributes less, and below 10 GHz the forward shock component is always subdominant. At X-ray energies (Figure 6), the emission is always dominated by the forward shock component (the dotted line). Given how well our model fits the X-ray data, the cooling break 𝜈 c seems to be situated above the X-ray regime throughout the observations. Although we do not fit our model to the optical data, we have overlaid our model onto the optical data from Fulton et al. (2023) in Figure 7. The decay rate of our model matches that of the data except for the late time y-band data, which Fulton et al. (2023) suggested was due to a supernova component. Figure 7 reinforces that there is significant extinction affecting the optical emission from GRB 221009A (Fulton et al. 2023; Levan et al. 2023; Kann et al. 2023; Tiengo et al. 2023). We find that nearly 2 magnitudes of extinction are needed in the r-band, decreasing to ∼ 0 . 1 -0 . 2 magnitudes in the y-band. \nTraditional forward shock spectral models take the three frequency breaks and the peak flux density and calculate four afterglow param- \n2 \nFigure 4. The multi-frequency radio light curves for GRB 221009A overlaid with our best-fit three-component model (model 1). \n<!-- image --> \nFigure 5. Broadband radio SEDs for GRB 221009A as a function of time with our best-fit three-component model (model 1) overlaid. \n<!-- image --> \nFigure 6. The X-ray light curve for GRB 221009A at 10 keV, overlaid with our best-fit afterglow model. \n<!-- image --> \nFigure 7. Optical light curves of GRB 221009A from Fulton et al. (2023), overlaid with our best-fit afterglow model. While we do not fit our model to the optical data due to the large and mostly unconstrained extinction contribution as well as the supernova (Kann et al. 2023; Blanchard et al. 2024), our model reproduces the decay rate of the optical data well. It is clear that significant extinction, 2 magnitudes in the r-band, is needed to get the correct normalisation of our model with respect to the data. \n<!-- image --> \neters: the total kinetic energy, the circumburst density, the fraction of kinetic energy that goes into the electrons and magnetic fields (Sari et al. 1998; Chevalier & Li 1999). From there, if a jet break is detected (an achromatic break in the light curves), the opening angle of the jet can be calculated (Sari et al. 1999). For GRB 221009A, we cannot calculate these parameters for two main reasons. The first is that whilst we are able to track the evolution of 𝐹 𝜈, max , 𝜈 m and 𝜈 sa \nfor the forward shock, with the data we use in this work we are unable to localise 𝜈 c since it appears to be above the X-ray band (Williams et al. 2023), and 𝜈 c is needed to break the degeneracy between the different afterglow parameters. Secondly, to calculate the afterglow parameters, the observed evolution must match the model's prediction. Otherwise, the afterglow parameters derived at each time step will have different values. \nOur model finds that 𝜈 m ∝ 𝑡 -1 . 06 ± 0 . 06 , whereas theoretically it is expected that 𝜈 m ∝ 𝑡 -1 . 5 independent of circumburst environment density profile, strongly in disagreement with our findings. We also find that 𝐹 𝜈, peak and 𝜈 sa do not evolve in agreement with expectations from the standard afterglow model, instead we find that 𝐹 𝜈, peak ∝ 𝑡 -0 . 97 ± 0 . 03 and 𝜈 sa ∝ 𝑡 -1 . 4 + 0 . 2 -0 . 1 (we note that the temporal index for 𝜈 sa is pushing up on the bounds set for the priors in the /e.pc/m.pc/c.pc/e.pc/e.pc fit). Comparatively, for a stellar wind ( 𝑘 = 2) and homogeneous ( 𝑘 = 0) environment, 𝐹 𝜈, peak is expected to evolve as 𝑡 -0 . 5 and 𝑡 0 (Granot & Sari 2002), respectively, which is far slower than what we observe. The expected evolution of the synchrotron self-absorption break is also dependent on the circumburst environment's density profile: with 𝑡 0 and 𝑡 -0 . 6 for 𝑘 = 0 and 𝑘 = 2, respectively, again the temporal indices are too slow to match our model. \nUsing the relations from table 5 in van der Horst et al. (2014), we can derive individual circumburst density profiles from the evolution of both 𝜈 sa and F 𝜈, max . We find that 𝜈 sa ∝ 𝑡 -1 . 4 + 0 . 2 -0 . 1 and F 𝜈, max ∝ 𝑡 -0 . 97 ± 0 . 03 corresponds to 𝑘 = 2 . 8 + 0 . 10 -0 . 06 and 𝑘 = 2 . 64 ± 0 . 03, respectively. Both the evolution of 𝐹 𝜈, max and 𝜈 sa strongly favour a steeper circumburst density profile over a 𝑘 = 2 stellar wind profile. Such a density profile could arise from a changing mass loss rate of the progenitor star as it reaches the end stages of its life. Standard afterglow models predict that the evolution of 𝜈 m is independent of the circumburst environment, therefore we cannot assume that the slow evolution of 𝜈 m is due to environmental effects. In other GRBs, (e.g. Bright et al. 2019) the unexpected evolution of 𝜈 m is considered as a result of time-varying microphysical parameters or scintillation. In the case of GRB 221009A, we find no evidence for significant scintillation effects, and time-varying microphysical parameters would cause further changes in the evolution of 𝐹 𝜈, max and 𝜈 sa , which could potentially provide an alternative explanation, other than a steep 𝑘 value, for the observed behaviour.", '4.2.1 Late-time evolution': "Our latest observations were made with ATCA at 475 days post burst at 5.5 and 9 GHz. Our model finds that at such late times, the forward shock is the brightest emission component during the decay phase of the light curve at these radio frequencies. Many late-time radio and X-ray light curves extending out to hundreds of days show achromatic behaviour referred to as a jet break (e.g. Tanvir et al. 2010; Kangas & Fruchter 2021). As the jet decelerates, the beaming angle, dictating the fraction of jet that the observer can see, increases. Before the jet break, the light curve at a given frequency will decay at a shallower rate than the intrinsic evolution because a greater fraction of the jet is visible at every new time step. At the point where the opening angle is equal to the inverse of the bulk Lorentz factor, the jet break, the whole jet is within the beaming angle, so the light curve at all wavelengths will begin to decay at a steeper rate ( 𝑡 -3 𝑝 / 4 or 𝑡 -𝑝 , depending on whether lateral spreading is assumed or not, Sari et al. 1999; Gao et al. 2013) which matches the intrinsic evolution of the shock. By observing the jet break, it is possible to measure the opening angle of the jet. \nJet breaks have been observed at many different times post-burst, \nfrom a fraction of a day to tens of days or even later. For most GRBs, the afterglow quickly fades below detection limits before a jet break can be observed. In some long-lasting afterglows, no jet break is observed at all for a very long time, the best example being GRB 130427A where no jet break was observed out to at least 1.9 years post-burst (De Pasquale et al. 2016). Comparatively, we rule out the presence of a break in the light curve out to 1.3 years based on our latest ATCA observations. We note that the presence of lateral structure, as indicated by the need for a third shock component which is discussed in Section 4.3, could disguise the jet break signature which is predicted for top hat jets (Sari et al. 1999; Gao et al. 2013). \nWhilst the presence of the jet break is used to measure the jet opening angle, the measurement is also dependent on the jet's kinetic energy and the density of the circumburst environment. The fact that there has been no change in light curve behaviour out to over a year post-burst due to a jet break indicates that the kinetic energy of the jet could be higher than what is deemed 'normal' for a regular GRB jet, the circumburst density is very low, or a it has a wide jet opening angle. As already suggested by O'Connor et al. (2023), GRB 221009A may belong to a class of hyper-energetic GRBs (Chandra et al. 2008; Cenko et al. 2011; Martin-Carrillo et al. 2014), events whose kinetic energies are greater than 10 51 erg. Given the large isotropic-equivalent kinetic energies inferred from modelling so far, a large jet opening angle is unlikely as it would require the beaming-corrected kinetic energy to be physically challenging, approaching that of the isotopic equivalent kinetic energy. It has been suggested (e.g., Levan et al. 2023; O'Connor et al. 2023) that a jet break occurred within the first day post-burst. Our observations and modelling provide no evidence that such a jet break occurred. \nThere is also expected to be a change in the observed light curve behaviour as the jet leaves the stellar wind bubble produced by the progenitor star and enters the surrounding homogeneous interstellar medium. The stellar wind bubble is expected to be several tens of parsecs in size (Dwarkadas 2005; Eldridge et al. 2006). For GRB 130427A, a stellar wind to homogeneous transition is ruled out to 1.9 years post-burst. In that case, it was estimated that the jet had travelled between 50 and 105 parsecs, putting strong constraints on the presence/size of a termination shock, other nearby stars, etc. Our model for GRB 221009A disfavours any change in the structure of the circumburst environment out to 1.3 years, or that the stellar wind bubble produced by the stellar progenitor exists in a very low preexisting ISM density for the stellar wind to expand into. However, if the circumburst density profile is very steep, as our forward shock model suggests, it may be very difficult to observe such a transition. Cenko et al. (2011) suggested that the hyper-energetic events can occur in lower metallicity environments where the progenitor star maintains a higher angular momentum for longer and therefore evacuates a larger cavity with its stellar wind, therefore delaying any change in temporal behaviour. \nStudies of GRB progenitor systems predict termination shock radii to be less than 20 parsec (Fryer et al. 2006; Schulze et al. 2011). Using the radio source size growth rate from Giarratana et al. (2023), we estimate the distance travelled by the jet for three different assumed opening angles. For opening angles of 2, 5 and 10 · , the jet should have propagated ∼ 10, 4 and 2 parsecs, respectively. At the current epoch, our observations are still consistent with the sizes of termination shocks found in the literature (Fryer et al. 2006). Therefore, we can treat these values as lower limits on the termination shock size. Continued low-frequency radio observations will be vital in tracking the jet as it continues to expand into the surrounding medium.", '4.3 Extra shock': "As explained in Section 3, we ran two different iterations of the third shock component in our model to test different theoretical predictions (see Table 3 for a summary, and Figure 3 for the results). The dash-dotted lines in Figures 4 and 5 denote the contribution of this component. The most important aspect of the third spectral component is the delayed deceleration timescale over which the component comes into the observer's line of sight (Peng et al. 2005). We find that the deceleration time for the third component is 0.27 ± 0.02 days, the break time in our 𝐹 𝜈, max broken power law evolution. The delayed deceleration timescale is used to show that there is a possibility that the third component is either off-axis and therefore takes time to enter our line of sight, or that it is less relativistic than the main jet component and so needs longer to shock sufficient mass such that it undergoes significant deceleration. \nTo ensure that the data needs the 𝐹 𝜈, max ∝ 𝑡 3 rise, we also ran a separate model iteration which allows the rise index to vary (model 2 in Table 3). In this iteration, we find a broad posterior distribution, i.e., not a Gaussian posterior, extending from 𝐹 𝜈, max ∝ 𝑡 1 . 6 to the edge of the prior which is 𝐹 𝜈, max ∝ 𝑡 3 . Such a broad posterior could be indicative of some lateral structure in the outflow such that the whole shock front does not enter our line of sight at once (Mooley et al. 2018b; Ryan et al. 2020). \nAfter the peak, for a decelerating shock, afterglow models predict 𝐹 𝜈, max to decay between 𝑡 -1 . 7 and 𝑡 -1 . 8 , for 𝑝 = 3 . 1for a stellar wind and homogeneous medium, respectively (Granot & Sari 2002). Our observations find 𝐹 𝜈, max ∝ 𝑡 -0 . 71 ± 0 . 03 , significantly slower than the models predict. The break frequency 𝜈 sa is expected to decay as 𝑡 -1 . 1 and 𝑡 -1 . 3 , for 𝑘 = 0 and 2, respectively, whereas we find 𝑡 -0 . 46 + 0 . 03 -0 . 04 . Therefore, we find that the evolution of 𝜈 sa for this extra component is far slower than predicted by analytical blast wave models, contrary to the evolution in the forward shock case, which is too fast. \nWecanalsousetheobservedevolution to extract the density profile of the circumburst environment and 𝑝 , independently of the spectral fit (van der Horst et al. 2014). In this case, we take the observed 𝐹 𝜈, max and 𝜈 sa behaviour as a function of time and solve for 𝑝 and 𝑘 . However, solving for 𝑝 and 𝑘 does not provide physical solutions for either value, i.e., a negative value of 𝑝 . \nGiven the clear disagreement between our modelling results using three components and expectations from analytical shock models, it is possible that this third additional component is not produced by a relativistic shock but a slower outflow component such a circumstellar interaction from the supernova. The peak luminosity of the extra shock is around 10 30 erg s -1 Hz -1 which is still an order of magnitude higher than the most luminous radio-detected supernovae (e.g. Palliyaguru et al. 2019; Dong et al. 2021), and reaches such high luminosities within a day as opposed to 100-1000 s of days later. \nTherefore, we find that the origin of the additional spectral component is most likely a wider outflow or cocoon-like component, as opposed to circumstellar interaction from a supernova. Being slightly less relativistic than the jet, the cocoon will take less time to sweep up mass whose rest mass energy is equal to that of the outflow and therefore will experience delayed deceleration. It is also likely to be slightly off-axis compared to the forward and reverse shock-emitting jet. \nCocoons have been invoked in previous GRB systems (e.g. Mooley et al. 2018a; Izzo et al. 2019) where sufficiently high-quality data has been used to infer their presence. It is possible that cocoons are a more universal component of GRBs but our observations have been too sparse to find them.", '5 CONCLUSIONS': 'In this work, we have collated and presented the most detailed radio study of any long GRB to date. When combined with the published X-ray data, we find that the radio observations are best described with three synchrotron spectra, each evolving individually. A reverse shock component dominates the early-time low-frequency data below 20 GHz. The higher-frequency radio emission and X-ray data can be ascribed to a forward shock. Due to the high temporal and spectral coverage, we are also able to constrain the evolution and properties of a third component which we attribute to a potential cocoon-like outflow. Whilst it is possible to match the different spectra with different shock components, we find that in all cases the evolution of the self-absorbed regions of the afterglow does not match up with the models currently in the literature. Also the peak frequency and peak flux show temporal behavior that is inconsistent with theoretical afterglow models. Given the high signal-to-noise of our latest observations, we aim to continue observing the afterglow of GRB 221009A for years to come to detect a potential jet break, track the jet into the non-relativistic regime, and constrain the size of the wind bubble in which this GRB resides.', 'ACKNOWLEDGEMENTS': "The authors would like the thank the anonymous referee for their helpful comments. L.R. thanks Eric Burns, Courey Elliott, Geoffrey Ryan and Ashley Villar for useful conversations in the writing of this paper. K.G. acknowledges support through the Australian Research Council Discovery Project DP200102243. M.J.M. acknowledges the support of the National Science Centre, Poland through the SONATA BIS grant 2018/30/E/ST9/00208 and the Polish National Agency for Academic Exchange Bekker grant BPN/BEK/2022/1/00110. AR acknowledges funding from the NOW Aspasia grant (number: 015.016.033). RLCS acknowledges support from a Leverhulme Trust Research Project Grant. \nParts of this research were conducted by the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), project numbers CE170100004 and CE230100016. We thank the Mullard Radio Astronomy Observatory staff for scheduling and carrying out the AMI-LA observations. The AMI telescope is supported by the European Research Council under grant ERC2012StG-307215 LODESTONE, the UK Science and Technology Facilities Council, and the University of Cambridge. The Allen Telescope Array refurbishment program and its ongoing operations are being substantially funded through the Franklin Antonio Bequest. Additional contributions from Frank Levinson, Greg Papadopoulos, the Breakthrough Listen Initiative and other private donors have been instrumental in the renewal of the ATA. Breakthrough Listen is managed by the Breakthrough Initiatives, sponsored by the Breakthrough Prize Foundation. The Paul G. Allen Family Foundation provided major support for the design and construction of the ATA, alongside contributions from Nathan Myhrvold, Xilinx Corporation, Sun Microsystems, and other private donors. The ATA has also been supported by contributions from the US Naval Observatory and the US National Science Foundation. The Australia Telescope Compact Array is part of the Australia Telescope National Facility 5 which is funded by the Australian Government for operation as a National Facility managed by CSIRO. We acknowledge the Gomeroi \npeople as the Traditional Owners of the Observatory site. This scientific work uses data obtained from Inyarrimanha Ilgari Bundara / the Murchison Radio-astronomy Observatory. We acknowledge the Wajarri Yamaji People as the Traditional Owners and native title holders of the Observatory site. CSIRO's ASKAP radio telescope is part of the Australia Telescope National Facility 5 . Operation of ASKAP is funded by the Australian Government with support from the National Collaborative Research Infrastructure Strategy. ASKAP uses the resources of the Pawsey Supercomputing Research Centre. Establishment of ASKAP, Inyarrimanha Ilgari Bundara, the CSIRO MurchisonRadio-astronomyObservatoryandthePawseySupercomputing Research Centre are initiatives of the Australian Government, with support from the Government of Western Australia and the Science and Industry Endowment Fund. e-MERLIN is a National Facility operated by the University of Manchester at Jodrell Bank Observatory on behalf of STFC. This paper is based (in part) on data obtained with the International LOFAR Telescope (ILT) under project code DDT20\\_003. LOFAR (van Haarlem et al. 2013) is the Low Frequency Array designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientific policy. The ILT resources have benefited from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Université d'Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK; Ministry of Science and Higher Education, Poland. Based on observations carried out under project numbers S22BC, S22BF, W22BU and S22BE with the IRAM NOEMA Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). We thank the staff of the GMRT that made these observations possible. GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.", 'DATA AVAILABILITY': 'All the new data presented in this paper is given the supplementary material.', 'REFERENCES': '```\nAbe H., et al., 2024, MNRAS, 527, 5856 Achterberg A., Gallant Y. A., Kirk J. G., Guthmann A. W., 2001, MNRAS, 328, 393 Ackermann M., et al., 2014, Science, 343, 42 Aharonian F., et al., 2023, ApJ, 946, L27 An Z.-H., et al., 2023, arXiv e-prints, p. arXiv:2303.01203 Anderson G. E., et al., 2014, MNRAS, 440, 2059 Blanchard P. K., et al., 2024, Nature Astronomy, Bright J. S. & Rhodes L. et al., 2023, Nature Astronomy, 7, 986 Bright J. 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2024MNRAS.533.2113K	Timedomain astrophysics continues to grow rapidly with the inception of new surveys drastically increasing data volumes. Democratized distributed approaches to training sets for machine learning classifiers are crucial to make the most of this torrent of discovery with citizen science approaches proving effective at meeting these requirements. In this paper we describe the creation of and the initial results from the Kilonova Seekers citizen science project built to find transient phenomena from the GOTO telescopes in near realtime. Kilonova Seekers launched in 2023 July and received over 600 000 classifications from approximately 2000 volunteers over the course of the LIGOVirgoKAGRA O4a observing run. During this time the project has yielded 20 discoveries generated a goldstandard training set of 17 682 detections for augmenting deeplearned classifiers and measured the performance and biases of Zooniverse volunteers on realbogus classification. This project will continue throughout the lifetime of GOTO pushing candidates at evergreater cadence and directly facilitate the nextgeneration classification algorithms currently in development.	2024-09-01T00:00:00Z	['10.48550/arXiv.2406.02334', 'arXiv:2406.02334', '2024MNRAS.533.2113K', '10.1093/mnras/stae1817', '2024arXiv240602334K', '2024MNRAS.tmp.1933K']	['Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena']	Kilonova Seekers the GOTO project for realtime citizen science in timedomain astrophysics	2,024	200	0.56	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2406.02334.pdf	{'Kilonova Seekers : the GOTO project for real-time citizen science in time-domain astrophysics': "- T. L. Killestein, 1 , 2 ‡ ★ L. Kelsey, 3 ‡ † E. Wickens, 3 L. Nuttall, 3 J. Lyman, 2 C. Krawczyk, 3 K. Ackley, 2 M. J. Dyer, 4 F. Jiménez-Ibarra, 5 K. Ulaczyk, 2 D. O'Neill, 2 A. Kumar, 2 D. Steeghs, 2 D. K. Galloway, 5 V. S. Dhillon, 4 , 11 P. O'Brien, 6 G. Ramsay, 7 K. Noysena, 8 R. Kotak, 1 R. P. Breton, 9 E. Pallé, 11 , 12 D. Pollacco, 2 S. Awiphan, 8 S. Belkin, 5 P. Chote, 2 P. Clark, 3 D. Coppejans, 2 C. Duffy, 7 R. Eyles-Ferris, 6 B. Godson, 2 B. Gompertz, 14 O. Graur, 3 , 15 P. Irawati, 8 D. Jarvis, 4 Y. Julakanti, 6 M. R. Kennedy, 10 H. Kuncarayakti, 1 A. Levan, 13 S. Littlefair, 4 M. Magee, 2 S. Mandhai, 9 D. Mata Sánchez, 11 , 12 S. Mattila, 1 , 16 J. McCormac, 2 J. Mullaney, 4 J. Munday, 2 M. Patel, 6 M. Pursiainen, 2 J. Rana, 11 , 12 U. Sawangwit, 8 E. Stanway, 2 R. Starling, 6 17\n- B. Warwick, 2 K. Wiersema\n- 1 Department of Physics & Astronomy, University of Turku, Vesilinnantie 5, Turku, FI-20014, Finland.\n- 2 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK.\n- 3 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK.\n- 4 Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK.\n- 5 School of Physics & Astronomy, Monash University, Clayton VIC 3800, Australia.\n- 6 School of Physics & Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK.\n- 7 Armagh Observatory & Planetarium, College Hill, Armagh, BT61 9DG.\n- 8 National Astronomical Research Institute of Thailand, 260 Moo 4, T. Donkaew, A. Maerim, Chiangmai, 50180 Thailand.\n- 9 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK.\n- 10 School of Physics, Kane Building, University College Cork, Cork, Ireland. \n11 \nInstituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain. \n- 12 Departamento de Astrofísica, Univ. de La Laguna, E-38206 La Laguna, Tenerife, Spain.\n- 13 Radboud University, Postbus 9010, 6500 GL, Nĳmegen, Netherlands.\n- 14 School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK.\n- 15 Department of Astrophysics, American Museum of Natural History, Central Park West and 79th Street, New York, NY 10024-5192, USA.\n- 16 School of Sciences, European University Cyprus, Diogenes street, Engomi, 1516, Nicosia, Cyprus.\n- 17 Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK. \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': "Time-domain astrophysics continues to grow rapidly, with the inception of new surveys drastically increasing data volumes. Democratised, distributed approaches to training sets for machine learning classifiers are crucial to make the most of this torrent of discovery - with citizen science approaches proving effective at meeting these requirements. In this paper, we describe the creation of and the initial results from the Kilonova Seekers citizen science project, built to find transient phenomena from the GOTO telescopes in near real-time. Kilonova Seekers launched in July 2023 and received over 600,000 classifications from approximately 2,000 volunteers over the course of the LIGO-Virgo-KAGRA O4a observing run. During this time, the project has yielded 20 discoveries, generated a 'gold-standard' training set of 17,682 detections for augmenting deep-learned classifiers, and measured the performance and biases of Zooniverse volunteers on real-bogus classification. This project will continue throughout the lifetime of GOTO, pushing candidates at ever-greater cadence, and directly facilitate the next-generation classification algorithms currently in development. \nKey words: techniques: miscellaneous - (stars:) supernovae: general - surveys", '1 INTRODUCTION': "- ★ E-mail: [email protected] (TLK) ‡ Joint first authorship\n- † E-mail: [email protected] (LK) \nIn the current era of time-domain astronomy, we are operating close to the limit of human validation of transient phenomena due to the vast numbers of observations being taken on a daily basis. The expan- \nsive data volumes (TB per night) of current all-sky surveys such as the Gravitational-wave Optical Transient Observer (GOTO; Steeghs et al. 2022), Zwicky Transient Facility (ZTF; Bellm et al. 2019), Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al. 2018), and All-Sky Automated Survey for Supernovae (ASAS-SN; Kochanek et al. 2017), and the impending era of the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST; Ivezić et al. 2019) highlight the continuing need for novel, automated, machinelearned approaches of source classification in order to triage and follow-up candidates in a timely manner. \nModern transient discovery is predominantly based on difference imaging (e.g. Alard & Lupton 1998; Zackay et al. 2016). In this technique, 'template', 'reference' or 'background' images are subtracted from new 'science' images in order to remove non-varying sources from the image. These reference images are of the same part of the sky as the science image, but were taken at a prior time during the optimal sky conditions (dark moon phases, good seeing). Typically they are also of longer exposure than the science images, meaning that fainter sources can be detected. Subtracting the reference image from the new science image, after correcting for differential background and PSF mis-matches, results in a 'difference' image. This difference image may contain residual flux indicating that something has changed between the reference and science images - a potential transient or variable source has appeared. The photometry can then be extracted from the difference image, to measure positions and fluxes free of contamination from surrounding sources (e.g. Wozniak et al. 2002) or host galaxy light. \nThe majority of detections (referred to as candidates herein) in difference images detected via source extraction are artefacts, known as 'bogus' sources following the real-bogus paradigm introduced in Bloom et al. (2012). These artefacts broadly arise from bright star residuals, point-spread-function (PSF) mis-match, and/or misalignment. A vast literature has emerged to tackle this challenge - transitioning from traditional machine learning (ML) approaches (Bailey et al. 2007; Goldstein et al. 2015; Wright et al. 2015; Mong et al. 2020), through to deep learned classifiers (Cabrera-Vives et al. 2017; Duev et al. 2019; Killestein et al. 2021; Mong et al. 2023; Corbett et al. 2023) - with ever increasing performance. Naturally however, as surveys grow larger, more performant source classification algorithms are required to ensure that the number of (inevitable) false positives do not overwhelm human vetters. To achieve this goal, larger and larger data volumes are required to effectively train these algorithms, and fully sample the diversity of detections seen in survey data. As surveys get bigger, the method for dealing with these data volumes needs to improve. Such surveys quickly outstrip the capacity of individuals or small teams of scientists to effectively label. A complementary approach, which can be used to create a humanlabelled data-set for training machine-learning based classifiers, is to use citizen science. \nCitizen science enables collaboration between researchers and members of the public, by engaging the public to participate in research tasks and help make scientific discoveries. For tasks such as vetting of candidate transients, the person-power increase of opening this task up to the public is highly significant. Transient astronomy projects on the Zooniverse citizen science platform 1 such as Galaxy Zoo Supernovae (Smith et al. 2011) and Supernova Hunters (Wright et al. 2017), using data from the Palomar Transient Factory (PTF; Law et al. 2009; Rau et al. 2009) and Pan-STARRS1 (Chambers et al. 2016) respectively, have had great success involving the public \nin this way. In both cases, volunteers were provided with a set of target, reference and difference images for a candidate transient that had been flagged as interesting by a computer algorithm, and were asked a simple question to determine if the observation was real or bogus. This facilitates discovery of transient events, and creates a binary-labelled training set for ML algorithms to augment their performance in future iterations. \nAlongside the direct benefits for scientific analysis, citizen science provides an excellent opportunity for public engagement and outreach by enabling members of the public to help in key scientific discovery, and to achieve experiential learning (Bruner 1961; Kolb 1984). The Zooniverse platform was originally created for the flagship Galaxy Zoo project (Lintott et al. 2008), and has since become the predominant online platform for facilitating citizen science (Marshall et al. 2015). At the time of writing, the Zooniverse platform has 91 active projects on offer, with topics ranging from history, language, and literature to climate, nature, physics, and space; meaning that there is something of interest for everyone. Citizen science approaches have led to tangible scientific discoveries: In astronomy, the Galaxy Zoo project led to the discovery of 'green pea' galaxies, a new class of compact, star-forming galaxies (Cardamone et al. 2009). Similarly, the Planet Hunters project enabled the discovery of PH1b, the first known planet in a quadruple star system (Schwamb et al. 2013). \nWe have developed the Kilonova Seekers citizen science project 2 on the Zooniverse platform, providing an opportunity for members of the public to help the GOTO collaboration in the discovery of transient events that may have been otherwise missed or overlooked, and enabling them to participate in cutting-edge science in near realtime. \nIn this paper, we report findings from the launch of Kilonova Seekers on 2023 July 11, over a ∼ 6 month period until the end of the O4a observing run of the LIGO-Virgo-KAGRA (LVK) gravitationalwave detectors, on 2024 January 16. As the primary aim of GOTO is to follow up gravitational-wave alerts from LVK, the timeframes for Kilonova Seekers are strongly driven by the schedules of these observing windows. In Section 2 we begin by introducing GOTO and the need for a citizen science project. In Section 3 we discuss the Kilonova Seekers project in terms of the data used, the workflow and interface the volunteers interact with, the behind-the-scenes machinery, and the alerting and reporting mechanisms. We present in Section 4 statistics about volunteer classifications, demographics and engagement, with a particular focus on the valuable contribution of our 'power users'. In Section 5 we highlight the key scientific results and discoveries from the project, the overall performance of volunteers, and measure the selection function of the volunteers compared to the GOTO real-bogus classifier. Finally in Section 6 we summarise the project so far and our key findings, noting our future plans for the project throughout the lifetime of the GOTO survey. A full list of the citizen scientists who were involved with Kilonova Seekers can be found in Appendix A.", '2 THE GRAVITATIONAL-WAVE OPTICAL TRANSIENT OBSERVER (GOTO)': 'GOTO (Steeghs et al. 2022; Dyer et al. 2022) is a multi-site, widefield telescope array designed to observe electromagnetic counterparts to gravitational wave events - specifically the afterglow of \ncompact binary mergers involving a neutron star, known as kilonovae. GOTO operates in two distinct observing modes: "triggered follow-up" and "all-sky survey" (see Dyer et al. 2020), to rapidly target and tile over the regions associated with incoming alerts, such as gravitational-wave alerts from the LIGO-Virgo-KAGRA (LVK) detectors. While other transients, such as supernovae, take a few weeks on average to reach their optical peak brightness (Anderson et al. 2014; Taubenberger 2017; Perley et al. 2020), kilonovae peak around 1 day after merger (e.g. Li & Paczyński 1998; Kasen et al. 2013; Arcavi et al. 2017). Surveys optimised to find kilonovae must have quick responses to alert triggers, fast survey cadence, and efficient transient identification methods. GOTO\'s overall field of view is larger than the localisation skymap of GW 170817 (Abbott et al. 2017), the only gravitational wave (GW) event with a detected electromagnetic (EM) counterpart, and can cover the whole sky in 2-3 d - so is ideally suited for these types of searches. \nDue to a combination of the large sky coverage and fast cadence in all-sky survey mode, GOTO collects and generates large volumes of data (500 GB/24h raw, 2-5 TB/24h dataproducts) that make unfiltered human vetting challenging. To address these data volumes, GOTO uses a real-bogus classifier ( gotorb ) based on a convolutional neural network (CNN) to classify candidate transients in difference imaging (for more information, see Killestein et al. 2021). Each classification is given a probability of being real, and an associated confidence level between 0 and 1. This classifier is effective at filtering out bogus detections, with a 97 per cent recovery rate of real transients for a fixed false positive rate of 1 per cent. As seen with other citizen science projects such as Supernova Hunters (Wright et al. 2017), CNNs and human classifiers have different strengths, which when combined can make a more efficient process than only using one. CNNs are very good at processing large volumes of data, and human classifiers perform better than CNNs when the image is more ambiguous, and when there are not many examples to compare it to.', '3 THE CITIZEN SCIENCE PLATFORM': "Given the significant volumes of detections generated, only the highest-scoring candidates from a gravitational-wave follow-up can be prioritised for eyeballing by the GOTO collaboration. By the imperfect nature of classification algorithms, a number of false negatives will always exist below the chosen score threshold, potentially being astrophysically interesting. By lowering the score threshold, we can improve recovery rates, although naturally with increased false positives. \nBeyond the real-time necessity for fast transient searches, increasing the possible size of human-labelled datasets is important for training improved classification algorithms. The presence of label noise (inaccurate labelling, see e.g. Frénay & Verleysen 2013) is a strong limiting factor in pushing accuracies from 99% to 99 . 9% (and beyond) and can likely only be mitigated via grouping of labels, weighting by quality of data item, and clipping of bad or unrepresentative examples. \nCitizen science provides a methodology to scale data labelling tasks from small teams of expert scientists, up to thousands of individuals. Calibrated uncertainty quantification is also a crucial missing link in many current astronomical classifiers (e.g. Abdar et al. 2020). Although strides with Bayesian neural networks (e.g. Valentin Jospin et al. 2020) have neatly quantified uncertainties associated with choice of model, this often does not represent the uncertainty (or confidence) a human would assign to their prediction. The true nature of uncertainties in ML is a complex issue, however, nominal \nestimates are useful in active learning (where models may suggest which data is most informative to be labelled by a human, e.g. Ren et al. 2020), anomaly detection, and decision making rules under uncertainty. \nGiven these challenges, a citizen science approach is well-suited to generating the scale (and quality) of labelled datasets required to train improved classifiers, and drive searches for candidates that may otherwise be missed in real-time. Kilonova Seekers launched in July 2023, after a short beta-testing period with live volunteers. At its core, Kilonova Seekers streams uncurated difference image detections (referred to as 'candidates' herein) meeting certain cuts from the GOTO real-time pipeline (see Lyman et al., in prep.) to the Zooniverse platform, populating a workflow with pre-baked data visualisations (known as subjects) to receive annotations and classification from citizen scientist volunteers. Through custom infrastructure (see Section 3.2), we listen to the classification stream from Zooniverse in real-time, and use this to trigger alerts according to set rules on consensus. We elaborate further on the specifics of this process in the following sections.", '3.1 Data extraction, pre-processing, and presentation': "Kilonova Seekers ingests candidates as part of a scheduled task executed on a daily cadence during project launch, and increased to every three hours during the O4a observing run. Given the multi-site nature of GOTO, this leads to 8 uploads of data per day (weatherpermitting). A candidate corresponds to a single difference image detection - analogous to the concept of alerts in other transient surveys. For logistical reasons, Kilonova Seekers does not take into account multiple candidates at the same location being associated (i.e. operating at a source level) - which would require more complex logic to de-duplicate candidates, adding additional overhead. This is intentionally decoupled from how candidates are handled internally, to provide an independent dataflow. \nThenumbersofrealtransients and artefacts are heavily imbalanced (Bloom et al. 2012), thus we sample difference image detections uniformly in their real-bogus score (with values between 0 and 1 inclusive, see Killestein et al. 2021) through a process of histogram equalisation - selecting a uniform number of candidates per realbogus bin, with typical equal bin-size of 0.1. Although these choices necessarily bias the dataset generated, there still exists sufficient diversity to re-balance (and thus train classifiers on) the final dataset. \nCandidates are queried from the main difference photometry table generated by GOTO's kadmilos data processing pipeline (see Lyman et al., in prep.), up to a user-specified maximum to avoid flooding volunteers with candidates in the case of rich fields. A number of operational considerations drive the exact query used to ingest candidates - with our selection cuts being: \n- · Signal-to-noise greater than 10: to minimise the number of false alarm detections due to correlated noise in the initial stages.\n- · Avoidance of the Galactic plane ( \| 𝑏 \| < 10 · ): to minimise the number of variable sources being uploaded to Kilonova Seekers -both for practical rate-limiting purposes, as well as dataset imbalance considerations.\n- · Exclusion of specific GOTO unit telescopes (UTs): owing to ongoing hardware issues, one specific UT was disabled in the Kilonova Seekers live workflow to minimise impact on volunteers.\n- · Cuts on images with extremely high numbers of difference image detections: after excluding the plane, these are likely to be poor subtractions which affect class balance. We impose that number of \nFigure 1. Example subjects from Kilonova Seekers . The science, reference, and difference images are plotted, along with subframes and event information. The top layout shows SN2024gy, a Type Ia supernova in the nearby galaxy (13.5 Mpc) NGC 4216 flagged by volunteers. The bottom layout shows a cosmic ray artifact that was flagged by volunteers, visible in only one of the four sub-frames, and unfortuitously projected on top of a galaxy. \n<!-- image --> \ndetections in each difference image must be less than the 90 th percentile number of detections across all difference images. \n- · Real-bogus score: for the purposes of fast discovery, we adopt a real-bogus score of 0.7 or greater. This is slightly below the normal score threshold of 0.8 used internally, and corresponds approximately to the equality point betwen false positive rate and false negative rate, a common choice in ML contexts. \nWe extract a set of stamps, sized approximately 3 × 3 arcminutes, from the science, reference, and difference images, small cutouts of the main images centred on each candidate detection. The science and reference images are derived from stacked data products, a sigmaclipped combination of a number of individual sub-frames, to reject single-image outliers such as cosmic rays. Stamps are extracted at native GOTO pixel scale (1.4 arcsec/pixel). Pixel thresholds are set using the IRAF zscale algorithm (Tody 1986, 1993), per-channel to span their full range. In a break from the norm of other transient discovery projects on Zooniverse, we use colourised images: specifically the matplotlib bone colourmap. The tasteful blue shading is intended to minimise visual stress. To generate and upload a subject to Zooniverse, we construct a pre-baked layout that we populate with stamps and metadata for a given candidate. We prominently display the detection time into each stamp, to reinforce the real-time nature of uploads to the volunteers, and write which survey each image comes from: to alert volunteers to any images from gravitationalwave (GW), gamma-ray burst (GRB), or neutrino follow-up. The overplotted cross-hairs draw attention to the centre of the frame, and the box shows the field-of-view that the GOTO real-bogus classifier sees, providing important context. We illustrate a subject in Fig. 1. \nEarly in Gen. 1 Kilonova Seekers , we noticed volunteers overwhelmingly classifying cosmic rays (CRs) as real detections, in spite of their often non-PSF-like appearance and documentation on the field guide for these objects. This motivated the addition of the sub- \nframes panel for Gen. 2 - in which we display the individual images that compose the stack - to identify single-frame artifacts such as CRs that propagate into the stack. These are visible in Fig. 1, and are 32 × 32 pixels each, with a faint circle added to aid the user in identifying potential moving targets. \nBased on feedback from the volunteers, we added labels to show the volunteers which GOTO site the data originates from, and an event tag to explain which mode GOTO was in when the image was taken. As GOTO is focused on transient follow-up, driven by triggers from external facilities - the types of images that the volunteers are presented with may change on a daily basis. For example, in survey mode many galaxies may be present in the images, whereas if GOTO is following a specific alert, the telescopes may be pointed towards regions of greater source density, with images being dominated by nearby variable stars in our galaxy. To explain this clearly to our volunteers, we use the following event labels and provide links to the individual instruments listed here so that they can find more information if they are interested in learning more: \n- · All-sky survey - GOTO is scanning the sky systematically to find new sources.\n- · LVK alert [alert number] - GOTO is following a specific gravitational-wave alert from the LIGO-Virgo-KAGRA (LVK) detectors, searching for the potential optical counterpart. 3\n- · Fermi alert - GOTO is following a GRB alert from the Fermi Space Telescope. 4\n- · Swift alert - GOTO is following a GRB alert from the Swift Space Telescope. 5\n- · IceCube alert - GOTO is following a neutrino alert from the IceCube detector. 6\n- · Supplemental survey - GOTO is doing something else that isn't covered by the other event tags. \nSome metadata is deliberately censored from the volunteers, such as the sky location of each candidate, and exact discovery time. This is predominantly to prevent volunteers from seeking additional contextual information outside of the image, that would e.g. confirm a given detection is a minor planet and thus real, as well as for operational reasons to prevent any discoveries being correlated with GW event skymaps, or reported without scrutiny on TNS or social media channels. This policy will naturally evolve with workflow requirements, with in-development workflows (see Section 6) providing additional (albeit carefully chosen) contextual information for classifications.", '3.2 Workflow and ingestion': "Kilonova Seekers presents one unified workflow to the user, tailored to the real-bogus paradigm for source classification. Subjects are shown to volunteers randomly, from the pool of data that has not reached retirement (when voted upon by 15 volunteers). Volunteers are asked if a real source exists at the centre of the crosshairs in the science and difference images. Initial beta tests including a fuzzy maybe option showed volunteers overwhelmingly ( ≳ 50%) selected this option, hindering consensus estimates and making uncertainty estimation impossible. \nThe web workflow is depicted in Fig. 2. Kilonova Seekers also has a companion mobile workflow, delivered via the Zooniverse app. \nFigure 2. Screenshot of the live Kilonova Seekers main workflow. \n<!-- image --> \nThis has the same layout as the web workflow, but with the addition of an intuitive 'swipe left and right' interface familiar from other popular mobile apps. We defer a full discussion of the workflows and their utilisation to Section 4.2.", '3.3 Alerting and reporting': 'Alerts are intended to flag an object for further follow-up once a given candidate (subject) reaches a configurable consensus threshold. For Kilonova Seekers this is set at a threshold of 80% agreement, and a minimum of 8 votes for the majority option - set through empirical testing during beta. The high minimum vote threshold is crucial to avoid false consensus, where the wrong answer may be locked in by an early run of votes. This was determined empirically, but is further motivated statistically by ensuring an error of ∼ 10% in the derived agreement fraction. \nAlerting to the collaboration is delivered via Slack 7 (the communication platform used by the GOTO collaboration), using the Incoming Webhook API to post an alert card to a dedicated #knseekers-alerts channel for rapid triaging of candidates. One such alert card is displayed in Fig. 3 - with action links to direct the vetter to the internal GOTO Marshall (see Lyman et al,. in prep.), a web interface for further analysis of transients and reporting, or to the Kilonova Seekers Talk pages to check discussion on the object. Collecting key information via a collaborative platform provides a way to centralise discussion about candidates in a maintainable, open way. Real extragalactic transients are reported to the Transient Name Server (TNS 8 ) through the existing GOTO Marshall architecture. To \nFigure 3. Alert card for a Kilonova Seekers candidate that has reached consensus, published via Slack. Visible on the alert card are the consensus level for the candidate, links to both internal GOTO webpages and the Kilonova Seekers discussion forum, and the candidate itself. \n<!-- image --> \ncredit volunteers for their work, we append the names of 5 randomlyselected classifiers of a given transient to the TNS remarks section, subject to integrity checks (see Section 4). This randomisation occurs at point of consensus, and is done in this way to more fairly assign credit, rather than just the first (who may be in a more favourable \ntime-zone, for example). Regardless of this prompt report, all volunteers who correctly identify a given transient are credited on the project results page.', '3.4 Implementation details': "To power the real-time nature of Kilonova Seekers , we developed a web service to receive classifications from Zooniverse in low-latency (typically in ∼ s), combine them with contextual information from the GOTO Marshall, and generate alerts for promising transients. \nWe use Zooniverse's Caesar 9 tool to generate a stream of classifications, pushed into a PostgreSQL database hosted locally via a HTTP POST endpoint, exposed on the database machine. The web endpoints for Kilonova Seekers are write-only by design, delivered via Apache2 backed by the Python django framework. Schema validation via pydantic ensures only POST requests containing valid classifications are ingested, and enforces strong type safety by checking and enforcing that ingested data are of the right type, enhancing reliability. As Zooniverse predominantly use NoSQL databases internally and make heavy use of free-form JSON data throughout their APIs, we make no attempt to normalise these at ingest and instead use PostgreSQL's excellent native support for JSON(B) datatypes, despite it being a relational database at heart. This was largely driven by the requirement for the database to host ingests from multiple projects, including the internal GOTOzoo project used for GOTOtemplatevetting. Given that different projects may have different metadata (provided as JSON strings), we create project-specific database views for each project, to ensure queries can be written in simpler, more user-friendly ways, without having to parse the JSON strings each time. The full Kilonova Seekers database and real-time stack is hosted on low-power commodity hardware, specifically a cloud-hosted Raspberry Pi Model 4B. Although comparatively tiny, we found this hardware performed ably throughout the first 6 months of the project with over a 99.9% uptime - proving highly capable and handling peak throughputs of ∼ 100 classifications per second during the initial launch rush phase. We are currently in the process of migrating Kilonova Seekers to more powerful hardware, as we introduce active learning and online ML estimators to our workflows, though this is predominantly for operational stability and could easily remain in-situ. To provide monitoring of the health of the project, Grafana 10 and Prometheus 11 are used to construct real-time dashboards to visualise the rates, ratios of real-bogus, and bulk properties of incoming classifications. Metrics such as the daily number of active users and classification rate are crucial for informing ongoing engagement strategies and thus are prominent in the design. \nWeanticipate open-sourcing various aspects of the real-time flows of Kilonova Seekers in the near future, to enable the community to make use of pre-built utilities for real-time citizen science projects especially in light of new transient surveys coming online in the near future that aim to deliver citizen science components, for example the Vera C. Rubin Observatory (e.g. Higgs 2023).", '4 VOLUNTEERS': 'As a citizen science project, our Zooniverse volunteers are the key to the success of Kilonova Seekers . For us, it is not only important that \n10 \nhttps://grafana.com/ \nFigure 4. Cumulative classifications per day on Kilonova Seekers from launch until the end of O4a (2024 January 16). The blue shaded region corresponds to the dates of press releases, and active media coverage of the project during the launch period. The red shaded region towards the end of September shows the maintenance period after three months of operations, when we temporarily paused the scheduled uploads and implemented the Gen. 2 subjects based on feedback from the volunteers. The green shaded region highlights the increase in rate of classifications over the winter holiday period and the subsequent return to work. The solid red line corresponds to the date of an email newsletter sent out to registered volunteers, leading to a clear increase in classifications. The dashed line is the date we increased the data upload cadence from twice per day to every three hours. \n<!-- image --> \nthe project provide useful classifications for improving the GOTO real-bogus classifier, but that the volunteers contribute to meaningful scientific discovery, engage with our collaboration and the other volunteers, learn from the project, and most crucially, enjoy participating in the science of GOTO. \nIn this section we discuss the volunteer classifications, highlighting the valuable contribution of our most prolific users (in the top 25, herein power users); before exploring the volunteer demographics, engagement, and the speed and efficiency of their classifications.', '4.1 Volunteer Classifications': "Kilonova Seekers launched publicly on Zooniverse on 2023 July 11 at 14:30 UTC, achieving 1000 classifications within the first 30 minutes. Coinciding with the project launch, Kilonova Seekers was featured in press releases from the GOTO partner institutions and social media, and the Kilonova Seekers leads (T.L.K and L.K) were interviewed about the project on the radio for BBC Radio Solent 12 and on the 'Missing Links' show on Dublin City FM. 13 This period of active publicity is highlighted in blue on Fig. 4, where the impact of this can be seen by a steep gradient in the rate of classifications. \nAfter the initial launch rush, classifications settled down to an average of ∼ 4000 classifications per day over the course of the first 3 months of operations. We consider this time to be 'Gen. 1' of Kilonova Seekers . During this time, only GOTO-North was included, and we were operating the Kilonova Seekers project with a once-per-day upload cadence, along with the Gen. 1 image style that did not contain the subframes for easier detection of cosmic rays (as discussed in Section 3.1). As illustrated in Fig. 4 by the red shaded region, \nFigure 5. Classifications per day on Kilonova Seekers for the first 100 days after launch. This distinct classification curve shows that volunteers regularly classify on the project with the release of new data. \n<!-- image --> \nwe paused the scheduled uploads for a week to rapidly implement the Gen. 2 subjects based on feedback from the volunteers, and to upgrade the behind-the-scenes infrastructure ready for ingesting subjects from GOTO-South and the planned increase in upload cadence. We announced our new Gen. 2 subjects in an email newsletter once the maintenance was complete, as indicated in Fig. 4 by a solid red line. Classifications quickly increased again to an average of ∼ 3100 classifications per day after this maintenance period. \nGOTO-South at Siding Spring Observatory was integrated successfully into our upload pipeline, and we moved to a three-hour upload cadence on 2023 October 11, as indicated by the dashed line in Fig. 4. Classification rates did slow after this period to an average of ∼ 1700 per day, however this was largely due to poor weather at both sites due to the changing seasons, meaning there were fewer data to upload to the project. \nA particularly interesting feature of Fig. 4 is highlighted by the green shaded region. This indicates the Christmas holiday period (December 24 - Jan 1), when many people are off work for around a week. We found a significant increase in classifications during this time, suggesting that our users may have had more free time to engage with Kilonova Seekers - as evidenced by an increase in Talk posts from many of our users during this period. \nIn total, over the course of this initial run of Kilonova Seekers , between launch and the end of O4a, our volunteers achieved 643,124 classifications of 42,936 subjects. \nByfocusinginonthefirst100dayspostlaunch,wecancomparethe classification curve of Kilonova Seekers (Fig. 5) with other projects on the Zooniverse. As discussed in Spiers et al. (2019), the majority of projects on Zooniverse show high classifications on project launch that rapidly declines after the initial launch rush. Occasional peaks in activity may be seen after periods of project promotion, press coverage, or further data release. Other projects such as Supernova Hunters show a dramatically different classification curve (see Fig. 4 in Spiers et al. 2019), with more regular spikes in classification indicative of recurring activity. For Supernova Hunters , these spikes were on a weekly cadence, resulting from the weekly data upload and newsletter cadence of the project. Kilonova Seekers falls somewhere in-between these two trends. The project shows a clear initial launch peak and rapid decline, with smaller regular spikes in activity, likely corresponding to our regular daily upload cadence (barring any weather restrictions). \nFigure 6. The distribution in classifications among users from launch until the end of O4a. The median number of classifications is 11; however, we have a strong core user-base, with a number of users completing more than 10,000 classifications each. \n<!-- image -->", '4.1.1 Power Users': "As shown in Fig. 6, which shows the distribution in classifications among users, many Kilonova Seekers volunteers only undertake a few classifications. Similarly to those for Galaxy Zoo (Lintott et al. 2008) and Bursts from Space: MeerKAT (Andersson et al. 2023), the distribution follows a power law, where the majority of volunteers complete between 1 and 10 classifications on the project, with the number of volunteers declining for larger numbers of classifications. Additionally, this plot clearly shows the significant impact of our 'power users' who have each contributed thousands of classifications to the project. An alternative framework to look at this is via the Pareto-like (e.g. Lorenz 1905; Cowell 2011) plot in Fig. 7, where the cumulative fraction of classifiers, and their cumulative share of the classification effort is depicted. Around 90% of the classifications are performed by 10% of the volunteers, with a Gini index (Gini 1912) of 0.9, in line with other Zooniverse projects of a similar nature (e.g. Table 3 of Spiers et al. 2019). \nThe majority of these power users are the most active participants on the Talk pages, regularly asking questions about the project, sharing their experiences, and providing their thoughts and insights to help others. For the next generation of Kilonova Seekers we anticipate appointing and training some of these individuals as moderators to aid in the day-to-day running of the project. \nTo better understand the classification patterns of the volunteers, we present in Fig. 8 the average daily classifications for the power user group (the 25 users with the greatest number of classifications between launch and the end of O4a), displayed in 15 minute windows to see trends in volunteer classifications throughout an average day, calculated by dividing the total number of classifications per user per window by the window length in days. We split this into two based on initial daily upload schedule in Fig. 8a and based on the later change to upload new data every three hours in Fig. 8b. For the 92 days when we were uploading data every day at 12:00 UT, our most active users were predominantly doing their classifications immediately after the daily data upload. Whilst it is encouraging that volunteers were keen to classify the data immediately, and to be included on the discovery reports, these reports were quickly becoming dominated by the same few volunteers, and others were missing out. This gave further motivation to move to a more frequent data upload - alongside a more real-time data stream being beneficial for \nFigure 7. Pareto plot of the cumulative fraction of Kilonova Seekers participants from launch until the end of O4a, plotted against cumulative fraction of classifications. The dashed diagonal line represents perfect parity/equality in classification effort per participant. The Gini index is annotated, providing a quantitative measure of the inequality in contribution. \n<!-- image --> \nclassification speed and distributing the work more fairly. Uploading data more frequently enables volunteers across different timezones to see the data first: allowing them to participate in discovery, and be acknowledged on discovery reports. As illustrated in Fig. 8b, during the period where the data were uploaded every three hours, whilst the times that specific volunteers made no classifications remained consistent, there were no longer clear times when the most prolific volunteers did the majority of their classifications. In spite of these changes, some volunteers still seem to consistently work non-stop on the project, with gaps in Fig. 8b likely arising from binning/finite sampling.", '4.2 Volunteer Demographics': 'To date, Kilonova Seekers has attracted roughly 2000 volunteers, in over 20 distinct time zones, across 105 different countries. Fig. 9 displays the geographical distribution of volunteers on Kilonova Seekers , shaded according to classifiers per capita. Based on data obtained from Google Analytics, we have participants from every continent (except Antarctica). The wide accessibility of Zooniverse projects enables us to reach countries that may be traditionally underrepresented in astronomical communities. \nBased on the number of users per country, the United States is by far the largest contributor to Kilonova Seekers , with a total of 1284 users. At approximately half this value with a total of 615 users is the United Kingdom. However, considering average page views per user for individual countries in the time between launch and the end of O4a, we find that Portugal contains the most prolific Kilonova Seekers , with over 2750 views per user on average. \nKilonova Seekers is available to all users who can access the Zooniverse platform on the internet, which is available to computer, tablet and mobile users. Alongside the classic in-browser mode, Kilonova Seekers is available via the Zooniverse mobile app, available on both iOS and Android devices. The majority of classifications are done via a computer, indicated by Fig. 10, but roughly a third of classifications are done via mobile phones (inferred via user agent strings). As displayed in Fig. 11, the fraction of mobile classifications per user is bimodal, with the vast majority of volunteers either \nnot using a mobile phone at all or solely using their mobile phone to engage with Kilonova Seekers . Owing to this clear split in our user-base, it is important that future iterations of Kilonova Seekers (and other Zooniverse projects) do not contain too many images per page, to ensure continued readability on smaller mobile phone screens. Although the number of classifications specifically done via the mobile app is relatively small compared to those who use an internet browser (as indicated by the smaller pie chart in Fig. 10), it represents a non-negligible proportion of participants, necessitating that Kilonova Seekers remains compatible with the app, regardless of future updates, so that it remains accessible to all users. \nAs GOTO is a global collaboration with members from all across the world, it was important to offer Kilonova Seekers in the variety of languages that are spoken by the collaboration. At time of writing, Kilonova Seekers is available in English, Dutch, Spanish, and Indonesian. We were the first project on the Zooniverse platform to offer Indonesian, and are currently working on the Finnish, Japanese, Polish and Swedish translations, to be released in the near future. However, discussions on the Talk boards predominantly occur in English. These localisations are a volunteer effort driven by GOTO collaboration members, and thus we aim to scale up to support more languages as capacity/enthusiasm allows.', '4.3 Volunteer Engagement': 'The Kilonova Seekers team and the wider GOTO collaboration interact with the volunteers via the project \'Talk\' boards, a series of forum pages separated into categories and threads for different discussions. We encourage the volunteers to discuss subjects that they may be unsure of on their individual talk pages, and to ask the GOTO scientists questions by creating their own discussion threads. We use this platform as a key page for announcements to the volunteers from the Kilonova Seekers team, including details about new discoveries that they have made and updates about the project or status of the GOTO telescopes. Volunteers can "@" members of the Kilonova Seekers team on the Talk pages in the same way as popular social media platforms to alert them if they have a question or need help, and can also send private messages to the team and other volunteers. Through this, volunteers have told us how they have shared Kilonova Seekers with their families, friends, amateur astronomy groups, and have discussed the project in blogs and at conferences, widening the overall participation of the project. \nOn the project Talk pages, volunteers are able to tag their comments. Without any prompting from the team, volunteers started using very similar or the same hashtags as each other. Most of these indicate potential transients with tags such as #real or #transient , or highlight other astronomically interesting objects that are not part of the aims of the project e.g., #comet . The volunteers also use these tags to indicate common artefacts from the field guide, e.g., #badsubtraction and #satellite , along with artefacts they have encountered from prior similar citizen science projects, amateur astronomy, and even new ones of their own naming, which we have been able to use not only in our regular field guide updates, but also to update the GOTO hardware team on potential issues. For the next generation of Kilonova Seekers , we plan to implement a new multiclass workflow, and these tags will form the basis for the different labels we will include. \nAlongside the Talk pages, we engage our volunteers using newsletters. These provide an opportunity to update the volunteers on the status of the project, announce key findings, inform volunteers of changes to the project, and generally share our enthusiasm with the citizen scientists. We have found these to be particularly useful for \n(a) Average daily classification times for our top 25 users for the time between launch (11 th July 2023) and the 11 th September 2023 (a duration of 92 days), separated into 15 minute bins. During this period, new data were uploaded to Kilonova Seekers once per day at 12:00 UT. \n<!-- image --> \n(b) Average daily classification times for our top 25 users for the time between the 11 th September 2023 and the end of O4a (6 th January 2024; a duration of 97 days), separated into 15 minute bins. During this period, new data were uploaded to Kilonova Seekers every 3 hours. \n<!-- image --> \nFigure 8. Average classifications over the course of a day for our top 25 users (as defined by the 25 users with the highest number of classifications between launch and the end of O4a), divided into 15 minute windows. Each row corresponds to a unique user, in descending order to the total classifications over the initial phase of this project, i.e., the top row is the volunteer with the most classifications.Figure 9. Geographical distribution of volunteers on the Kilonova Seekers project. The intensity of a given country corresponds to the classifiers per capita, using information from Natural Earth 15 , log-normalised for visualisation purposes. \n<!-- image --> \nre-engaging volunteers who may have lost interest in the project over time, as can be seen in the upturn in classifications after a newsletter in Fig. 4. \nTo ensure that volunteers are credited appropriately for their contributions, discoveries are reported via a dedicated Kilonova Seekers results page, including the names or usernames of all of the volunteers who marked a candidate as \'real\'. Furthermore, we randomly select a subset of 5 names from the \'real\' list to add in a dedicated acknowledgement in the remarks section of the Transient Name Server (TNS) page for the object. In order to receive credit, volunteers must \nbe logged into their Zooniverse account when they make the discovery, so that they can be identified. When volunteers sign up to the Zooniverse platform, they have the option to give their real name. If they have chosen to provide this, their real name will be used for credits, otherwise we use their public username. We automatically filter out email addresses and web links from these text strings. \nFigure 10. Pie charts illustrating the different ways classifications are made on Kilonova Seekers . The larger pie chart indicates the percentages of classifications during O4a that were completed on computers, mobiles and tablets. The smaller, nested pie chart indicates the percentage of mobile classifications done via a mobile browser or the Zooniverse mobile app. \n<!-- image --> \nFigure 11. Distribution of the fraction of the total classifications per user performed on a mobile phone. This takes into account both mobile browser and mobile app classifications. \n<!-- image -->', '5 SCIENTIFIC HIGHLIGHTS': "In the six months between launch and the end of O4a, the Kilonova Seekers project reported a total of 29 objects to the Transient Name Server, which are listed in Table 1, where 20 of these were official discoveries, first made by Kilonova Seekers . \nAt present, the candidates that are flagged as interesting by the volunteers require cross-checking by the GOTO collaboration via the Slack alert cards (see Section 3.3). Real discoveries are then reported through the TNS via the GOTO Marshall. Anything that is a new discovery and has not appeared yet on the TNS with another group is immediately reported, but Kilonova Seekers candidates first identified by other groups are not yet routinely reported owing to limited person-power - something planned to improve via automation in future updates. \nTo date, 6 of the 20 transients first discovered by Kilonova Seekers during O4a have been classified spectroscopically. The first, AT 2023rob, was classified as a cataclysmic variable star (CV) by the Spectroscopic Classification of Astronomical Transients (SCAT; Tucker et al. 2022) survey (Hinkle 2023). The remaining were all classified as Type Ia supernovae (Kopsacheili et al. 2023; Do 2023; Davis et al. 2023; Fremling et al. 2024) by SCAT, the extended Public ESOSpectroscopic Survey of Transient Objects (ePESSTO+; Smartt \net al. 2015), and the Young Supernova Experiment (YSE; Jones et al. 2021). \nIn total over the period discussed in this paper, 1037 spectroscopically confirmed supernovae were reported to the TNS, of which 354 subjects associated with these known SNe were generated for Kilonova Seekers , assuming the subjects are associated with SNe using a narrow 1 '' cross-match radius. Of these, 259 reached the consensus threshold of 80% agreement and 8 or more positive votes. This implies a recovery fraction of 72% across this sample, broadly in line with more in-depth estimates presented in in Section 5.2. A large number of these transients are detected at low SNR, driving the lower recovery than perhaps anticipated - this figure increases rapidly with SNR, moving to 82% at SNR=20, 95% at SNR=50, and 100% at SNR=70. In the following subsections, we discuss in depth some of these early results from the Kilonova Seekers project.", '5.1 Rapid reporting': "One of the key accomplishments to highlight from Kilonova Seekers is the speed of classification and consensus from the volunteers. As we have volunteers from around the world, there is almost always someone online looking at the data in real-time, whether uploaded to Kilonova Seekers (e.g. Fig. 8), or internally within the collaboration. Between 2023 September 11 and the end of O4a, we changed the data upload cadence to the Zooniverse platform to every three hours, and found that the majority of new subjects uploaded were classified before the next data upload just three hours later. \nWe display in Fig. 12 the average classification speeds of the Kilonova Seekers volunteers per subject. We clip the maximum time per classification to 2 minutes to measure the actual attention paid to the classification - there were cases where classifications took on the order of 18 hours, which we interpret as situations where a volunteer stepped away from their device and submitted the classification at a later time. As shown in Fig. 12a, our power users typically take less time to classify a subject than the remainder of users, who have a wider range of classification times. However, the median classification time for both groups is roughly 5 s, meaning that if we take our total classifications for the period (see Section 4.1), our volunteers have dedicated at least 893 hours of classification time to the project during O4a. \nIn Fig. 12b, we break down the power-user classification times per user, and explore the distributions. There are clear differences here, with some users routinely taking under 10 s for every single classification they do, whilst others take substantially longer. This behaviour is unclear, and no conclusive explanation exists. Some power users may be reading and investigating the metadata for the subjects to find more insights that may help them make a classification - since these attributes are mentioned on the Talk boards by a small subset of volunteers. The final user on the plot is an extreme outlier - upon detailed inspection this user's classification times show a remarkable bimodality, with a similar 'early' peak to the other participants, but with a strong peak around 20 s, skewing their quartiles on this plot. \nA particularly significant scientific highlight for Kilonova Seekers was the discovery of AT 2023xqy (the Zooniverse subject for this discovery is displayed in Fig. 13). This object was observed by GOTO-South on 2023 November 13 at 11:06:02.592, and was reported to the TNS at 14:27:36 on the same day. It was observed, the data were reduced and uploaded to Zooniverse, the candidate was flagged as interesting, cross-checked and confirmed as real, and reported to the TNS within approximately 3 hours and 20 minutes of data being taken. This transient had a rapid rise in brightness. The last GOTO non-detection was 24 hours prior at a L-band magnitude \nDisco \nKilonova Seekers \npresent \name GOTO ies 2023-08-05 2023-08-08 ja 2023-09-05 2023-10-28 2023-11-11 2023-11-10 2023-11-12 2023-11-13 2023-11-18 2023-11-18 2023-11-28 2023-12-02 2023-11-30 2023-12-03 2023-12-03 2023-12-15 w l 2023-12-17 2023-12-17 2023-12-24 2023-12-26 2023-08-04 2023-10-26 2023-10-27 y 2023-11-10 2023-12-24 2024-01-06 2024-01-06 2024-01-06 2024-01-06 v er ies repor ted name, GOTO tak en directl y \n11 \nN \nGOTO \name \nN \nTNS \ner \nv \nDisco \ns \nSeeker \na \nv \nKilono \nGOTO23yt \nT2023pmm \nA \nGOTO23vt \nT2023pof \nA \nGOTO23a \nT2023rob \nA \nGOTO23bbl \nT2023wbu \nA \nGOTO23bia \nT2023xnj \nA \nGOTO23biq \nqf \nT2023x \nA \nGOTO23bip \nqg \nT2023x \nA \nGOTO23bjh \nqy \nT2023x \nA \nGOTO23blc \nT2023ydt \nA \nGOTO23blj \ner \nSN2023y \nGOTO23bms \nx \no \nT2023y \nA \nGOTO23bno \nT2023yqr \nA \nGOTO23bnn \nT2023yqs \nA \nGOTO23bnt \nSN2023yrs \nGOTO23bnz \nsp \nSN2023y \nGOTO23bus \nT2023aagc \nA \nGOTO23b \njf \nSN2023aa \nGOTO23bzu \nT2023abdm \nA \nGOTO23bzs \nT2023abdn \nA \nGOTO24P \nSN2023acla \nGOTO23uh \nc \nx \nSN2023o \nGOTO23bbc \ner \nSN2023v \nGOTO23bcc \nSN2023vqn \nGOTO23bh \nT2023xig \nA \nGOTO23caa \nT2023acdo \nA \nGOTO24J \nSN2024gy \nGOTO24Q \nSN2024hm \nGOTO24X \nT2024kh \nA \nGOTO24fq \nT2024agm \nA \ndisco \ns \nSeeker \na \nv \nKilono \n1: \nable \nT \nGOTO \nnal \ninter \nname, \nTNS \nthe \nfrom \nare \nedshifts \nR \nredshift. \nand \ntype \nMNRAS \nted \nepor \nR \n000 \n, 1-20 (2024) \nof 20.8. The transient was discovered 1 day later at a magnitude of 19.2 - suggesting this object rose in brightness by 1.6 mag/24 h, and implying the transient was caught early post-explosion. This finding was later confirmed by ATLAS on 2023 November 17. This speed of human vetting is simply not sustainable without the dedication of our citizen scientists.", '5.2 Validation dataset, detection efficiency, and volunteer benchmarking': 'Outside of the real-time transient discovery workflow, Kilonova Seekers provides a framework for generating a number of human benchmarks, and gold-standard datasets for training machine learning solutions, as a natural byproduct of the transient search workflow. We elaborate on a few ongoing analyses that provide substantial insights into the abilities of our volunteers, and map out the \'human factor\' present in transient follow-up, that few time-domain projects have previously explored in detail (e.g. Goldstein et al. 2015; Hayden et al. 2021). To measure the intrinsic performance of volunteers, and determine sensible classification baselines, we inject a number of validation datasets (both intentionally, and intrinsically via known objects) with known answers into the live project: \n- · Hand-labelled validation dataset: 300 examples, sampled uniformly in real-bogus score from detections prior to project launch, and hand-labelled by the Authors to ensure high accuracy.\n- · Minor planets: given the ingest pipeline is agnostic to contextual information, these detections with high real-bogus score naturally enter into Kilonova Seekers as part of the transient search workflow. We know a priori that these are real detections, and the spatial association enables us to retrieve high confidence low-signal-to-noise detections for testing. \nThe hand-labelled validation dataset is given an arbitrarily high retirement limit to ensure as many volunteers as possible see them for comparative analyses. For the analyses that follow, we neglect the possibility of label noise (inaccurate labelling by the team) in the validation datasets. For the hand-labelled set, these data were vetted by the Authors with both knowledge of the co-ordinates, and additional contextual information (historical variability, source crossmatches) to guide the labelling. For the minor planet dataset, we select only detections with high-confidence ( ≤ 4") matches to catalogued objects from the Minor Planet Centre, following Killestein et al. (2021). \nThrough analysis of the validation dataset, and binary classification labels from volunteers, we can assess both the cohort and individual performance of volunteers in a real-world setting. To ensure low sampling noise in our estimations of precision, we only consider volunteers who have completed 100 validation subjects or more, yielding noise of O(1%). We suspect the validation set size is sufficient to mitigate data-driven scatter in metrics. \nAs shown in Fig. 14, we plot the precision ( 𝑃𝑅 ) and recall ( 𝑅𝐶 ) for each volunteer evaluated on the hand-labelled validation dataset. \n𝑃𝑅 = 𝑇𝑃 𝑇𝑃 + 𝐹𝑃 (1) \n𝑅𝐶 = 𝑇𝑃 𝑇𝑃 + 𝐹𝑁 (2) \nwhere 𝑇𝑃 is the number of real transients correctly labelled as such by the volunteer, 𝐹𝑃 is the number of bogus transients incorrectly labelled as real, and 𝐹𝑁 is the number of real transients labelled as \nbogus. The 𝐹 1 score is a convenient metric derived as the harmonic mean of these quantities, given as \n𝐹 1 = 2 · 𝑃𝑅 · 𝑅𝐶 𝑃𝑅 + 𝑅𝐶 (3) \nwhere the precision and recall are defined as above. The volunteers broadly perform well on the validation dataset, achieving a median (class-weighted, 1 𝜎 uncertainty) F1 score of 78 + 13 -35 % and lie in a cluster in the upper right quadrant (precision and recall above 50%). and represents a class-balanced accuracy, weighting precision and recall equally. \nThere are a notable minority (20%) of volunteers who lie in the lower right quadrant (high precision, but low recall) - whom we interpret as \'underconfident\' volunteers. When they mark objects as real transients, they are likely to be correct, but they mark very few objects as real transients - perhaps owing to not fully trusting their own predictions. Reassuringly, very few volunteers lie in the low precision region of the plot, characterised by poor discriminative performance - we associate the upper left quadrant with \'overconfident\' volunteers, who recover the majority of real transients but mark many artifacts as real. We hope that, over time, volunteers PrecisionRecall scores will flow towards the upper right corner as they gain performance and familiarity with the workflow and project. \nIn Fig. 15, we compare the recovery of minor planets by the volunteers compared to the GOTO real-bogus classifier (see Killestein et al. 2021) as a function of the signal-to-noise of the detection. We cross-match all uploaded Kilonova Seekers subjects with Minor Planet Centre 16 ephemerides, and in total retrieve 92,640 classifications - which we know a priori are good transient detections. We compute the fraction of positive votes per signal-to-noise bin, chosen approximately to linearly span the range 3 to 20, where the majority of detections typically lie. Uncertainties are estimated from the normal approximation (Wald 1943) to the one-sided binomial proportion confidence interval: \n𝜎 ˆ 𝑝 = √︂ ˆ 𝑝 ( 1 -ˆ 𝑝 ) 𝑁 (4) \nwhich is an adequate and asymptotically-correct estimator, given the typically large 𝑁 per bin, and lack of bins with ˆ 𝑝 close to zero or one. \nFor comparison, we overplot the harmonic mean of real-bogus classifier scores - the closest analogy to the fraction of votes positive approach we use for volunteer labels. This is given as \n𝑃 = 1 𝑁 𝑁 ∑︁ 𝑖 = 1 1 𝑝 𝑖 (5) \nwhere 𝑝 𝑖 is the 𝑖 th classifier score in each bin, and 𝑁 is the total number of subjects per signal-to-noise bin. This plot highlights facets of the performance of both human vetters and the real-bogus classifier. The classifier score remains high across the SNR distribution, as expected. The marked bump at low ( ∼ 7) signal-to-noise in the classifier score is likely a result of the steep power-law slope in the magnitude distribution of minor planets - with many times more small bodies than larger in the training set (see Killestein et al. 2021). The human classifier scores show a smooth sigmoid curve, passing 50% recovery around a SNR of 6. Uncertainties (given by the error bar) are largely driven by sample size per bin, rather than human-derived \nFigure 12. Boxplots showing the classification times of the Kilonova Seekers volunteers. Maximum time per classification has been clipped to 2 minutes to remove those classifications where someone paused mid-classification and submitted at a much later time. Orange lines represent the median classification time, the boxes show the upper ( 𝑄 3) and lower ( 𝑄 1) quartile values, with width corresponding to the interquartile range (IQR) and the whiskers represent 𝑄 1 -1 . 5 × 𝐼𝑄𝑅 and 𝑄 3 + 1 . 5 × 𝐼𝑄𝑅 respectively. \n<!-- image --> \n(a) Boxplots showing the distributions of classification times of power users, selected as the top 25 most prolific classifiers on Kilonova Seekers , compared to the remainder of the user base (regular users). (b) Boxplots showing the distribution of classification times of our 25 power-users, sorted by median classification time.', '2023-11-13 11:06': "Figure 13. Kilonova Seekers subject for AT2023xqy. This transient was flagged by the volunteers as real and reported to the TNS within 3 hours and 20 minutes of data being taken by GOTO. \n<!-- image --> \nuncertainty. The real-bogus classifier score comfortably exceeds the human true positive rate, markedly so at lower signal-to-noise. It is perhaps not surprising that a classifier explicitly trained on minor planets outperforms a naive ensembling of human predictors - yet to our knowledge this is among the first validations of deep-learned classifiers outperforming human annotators in time-domain astronomy. Wecaution that the human-derived fraction of positive votes may not be well-calibrated probabilistically, taking into account discussions on variable precision and recall of volunteers above - nevertheless via thresholding and consensus these issues may be mitigated. \nOptimal schemes for thresholding or weighting (e.g. Marshall et al. 2016; O'Brien et al. 2024) are left to future publications, though we note that the uncertainty is a crucial component of our science aims, and so fraction of positive votes is diagnostic here. With priors on the true/false positive rates per volunteer from the validation set, Bayesian models of annotation (e.g. Paun et al. 2018) are a promising avenue for deriving well-calibrated and optimal inferences on how likely an object is to be real from volunteer labelling. \nNevertheless, this result underscores that classifier scores alone \nFigure 14. Precision-recall plot for the validation set, computed per volunteer with over 100 classifications. The dashed lines partition the precision-recall space into quadrants, corresponding to the 50% precision/recall boundary. The size of the plot markers is proportional to the number of classifications performed by that user. \n<!-- image --> \nare not sufficient to fully capture the uncertainty associated with a classification. Subjects that are genuinely challenging in a statistical sense, such as those at low signal-to-noise, should be treated with nuance to avoid over-interpretation. This underscores the necessity of uncertainty quantification in classification \nAlthough early in the project's lifetime, these validation datasets \nFigure 15. Fraction of positive votes per subject, binned by the SNR of the detection, derived from all live Kilonova Seekers minor planet detections. Uncertainties are estimated by the one-sided binomial score interval approximation, with error bars representing 2 sigma. The 50% recovery threshold sits around signal-to-noise 6. The harmonic mean of the real-bogus classifier score (Killestein et al. 2021) per bin is overplotted in orange, for illustration. \n<!-- image --> \nhave enabled a number of interesting scientific (and sociological) insights into the way volunteers approach classification tasks, their intrinsic efficiency at recovering transient objects, and the different dispositions of the volunteers to classification. More advanced validation experiments are currently underway - including injecting augmented variants of the validation set to track the evolving performance of the volunteers between Kilonova Seekers generations. One remaining, potentially insightful task is to re-run our validation workflowwithGOTOteammemberstocompareandcontrastFigs.14 and 15, and measure the selection function of project scientists (similar to the investigation of Wardlaw et al. 2018, for Martian surface feature detection and classification) - which could feed into downstream analyses to derive more informed recovery estimates/drive second-looks on more challenging data. \nBased on cuts inferred from the validation dataset, we define our gold-standard dataset as subjects with > 80% agreement, and more than 8 positive/negative votes from volunteers. Based on these cuts, we find a gold-standard dataset of 17,682 detections across O4a. This gold-standard dataset is informing the development of the next real-bogus classifier within GOTO, with a more detailed discussion of nuances associated with crowd-sourced training of transient classification models deferred to a future publication.", '6 CONCLUSIONS': 'In this paper, we have presented the first stage of Kilonova Seekers , a citizen science project designed specifically for real-time transient discovery, complementing the unique capabilities of the GOTO survey for gravitational-wave follow-up. \nIn the period from July 2023 to January 2024, Kilonova Seekers : \n- · Achieved 643,124 classifications of 42,936 subjects.\n- · Attracted roughly 2000 volunteers, in over 20 distinct time zones, across 105 different countries.\n- · Reported 29 objects to the TNS, where 20 of these are discoveries first reported by the project. 6 of these discoveries have been classified spectroscopically by other teams.\n- · Achieved turn-around times of as quick as 3 hours and 20 min- \nes between observation and TNS report, for candidates flagged as interesting by the volunteers. \n- · Created a gold-standard training set of 17,682 subjects for machine learning, with over 80% agreement among volunteers.\n- · Measuredthedetection efficiency of the volunteers at recovering transient sources, and compared this with the existing GOTO realbogs classifier. \nWith this initial phase of Kilonova Seekers , we have demonstrated concretely that citizen science can work both in real time and low latency - driving decision-making and discovery on large data-streams.', '6.1 Recent updates and future work': 'For the O4b observing run which is now underway, Kilonova Seekers has continued to grow rapidly and transitioned to an augmented hourly cadence upload, to further reduce the latency between discovery, upload, and consensus. This has led to a number of citizen science discoveries within 2 hours of images being taken. We intend to keep shortening this cadence towards zero-delay (uploads simultaneous with pipeline completion), as survey and platform capacity allow. A new injection of unbiased (spanning the full real-bogus range) candidates, which aggressively sample real-bogus scores across the whole range are proving an excellent seed dataset for novel deep-learned classifiers in development. In the time taken to prepare this publication, Kilonova Seekers has now reached 31 discoveries and achieved over 1 million classifications from volunteers. A full discussion of this second phase and ongoing discovery is deferred to future works. \nDevelopment of the Kilonova Seekers workflows continue, with multi-class, context-augmented workflows planned to be released later in 2024. This will enable volunteers to not only classify if a source is real or bogus, but to subdivide each of these classes into morphological types (e.g. supernova, nuclear transient, variable star). This workflow will further support the training of next-generation machine learning classifiers, and enable uncertainty-aware contextual classification. The introduction of this Kilonova Seekers Multiclass will mark Gen. 3 of the project, and be accompanied with a re-launch. This development is, of course, in addition to the original fast discovery workflow, to ensure continuity for volunteers and maintain compatibility with mobile app users. \nBased on the keen engagement with Kilonova Seekers , a number of parallel companion outreach and public engagement projects are under active development: empowering volunteers to do their own transient follow-up efforts with professional telescopes, learn about time-domain astrophysics through observing objects themselves, and generate meaningful scientific outcomes and publications on the objects they have discovered. \nThe time-domain community are eagerly following up alerts during the LIGO-Virgo-KAGRA O4b observing run, hoping these GW triggers will facilitate discovery of new electromagnetic counterparts. With the growth of the Kilonova Seekers project, this community is now markedly larger.', 'ACKNOWLEDGEMENTS': "Wethank the anonymous referee for their insightful comments which helped improve the quality of this manuscript. TLK acknowledges support via an Research Council of Finland grant (340613; P.I. R. Kotak), and from the UK Science and Technology Facilities Council (STFC, grant number ST/T506503/1). LK and LN thank the UKRI Future Leaders Fellowship for support through the grant \nMR/T01881X/1. EW thanks STFC for support through the grant ST/Y509486/1. JDL acknowledges support from a UK Research and Innovation Fellowship (MR/T020784/1). DMS acknowledges support by the Spanish Ministry of Science via the Plan de Generacion deconocimientoPID2020-120323GB-I00andPID2021-124879NBI00. SM acknowledges support from the Research Council of Finland project 350458. The Gravitational-wave Optical Transient Observer (GOTO) project acknowledges the support of the Monash-Warwick Alliance; University of Warwick; Monash University; University of Sheffield; University of Leicester; Armagh Observatory & Planetarium; the National Astronomical Research Institute of Thailand (NARIT); Instituto de Astrofísica de Canarias (IAC); University of Portsmouth; University of Turku. We acknowledge support from the Science and Technology Facilities Council (STFC, grant numbers ST/T007184/1, ST/T003103/1, ST/T000406/1, ST/X001121/1 and ST/Z000165/1). \nThis publication uses data generated via the Zooniverse.org platform, development of which is funded by generous support, including a Global Impact Award from Google, and by a grant from the Alfred P. Sloan Foundation. This research has made use of data and/or services provided by the International Astronomical Union's Minor Planet Center.", 'Software': 'This research has made use of /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (Astropy Collaboration et al. 2013, 2018, 2022), /g.pc/e.pc/o.pc/p.pc/a.pc/n.pc/d.pc/a.pc/s.pc (Jordahl et al. 2020), /i.pc/r.pc/a.pc/f.pc (Tody 1986, 1993), /m.pc/a.pc/t.pc/p.pc/l.pc/o.pc/t.pc/l.pc/i.pc/b.pc (Hunter 2007), /n.pc/u.pc/m.pc/p.pc/y.pc (Harris et al. 2020), /p.pc/a.pc/n.pc/d.pc/a.pc/s.pc (McKinney et al. 2010) and /s.pc/c.pc/i.pc/p.pc/y.pc (Virtanen et al. 2020).', 'DATA AVAILABILITY': 'GOTO images and source catalogs will be made available in a GOTO data release at a later date. Anonymised and/or aggregated classification data are made available upon reasonable request to the authors, but are anticipated to be released publicly at a later date. 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A full list of names of contributors (who gave permission for their name to be shared) since our launch is given below in alphabetical order, correct as of time of manuscript preparation: \n5, 958bacsal, A, A Piras, A Taylor, A\\_lot\\_ofimagination, Aaboli Samant, Aarush Naskar, Abby, Abditory, Abdulla G. Asanar, Abdurahman Mohamed, Abel, Abhĳeeth Veeranki, AbrilPerezH, abrosio, achmadsujana, Ada Ji, Adam, Adam Cash, Adam Gibson, Adam Martinez, Adam Schufeldt, Adam Straub, adamzwawy, Adekunle Adejokun, Aditi Brĳ, Adrian Morales, Adrian Smith, Adrianna Jones, Adrien Droguet, Afjal\\_khh, Agnid Nandi, Aguirre, Ahmad azizisani, Ahmed Estiak, Aiden Chadwick, Aimee Gonzalez Ferreira Sirvani Valentim, AJinSA, Aki Suvanto, Akiko Inamoto, aknepeter, Al Lamperti, Alaa Salah Afifi, Alan Teague, Albert, Albiona Leka, Alejandro, Alejandro Arróliga Vanegas, Alejandro Lopez, Aleksandr, Aleksandr Ketov, Aleksandr Timofeev, Aleksandra Pogorzelska, Alex, Alex Al-Sammarraie, Alex Andersson, Alex Gabriel, Alex Lammers, Alex Mitchell, Alex Zuniga, Alexander Becker, Alexander Blagrave, Alexander Davidson, Alexander Doens, Alexander G. Plasser, Alexandra Hercilia Pereira Silva, Alexandre Celier, Alexia Fotini Panagopoulos, Alexis Carrilllo, Alexis Casey, Alexis Daniel Gómez Alatorre, Alexis MANET, Alexis Tombrello, Alfredo Gimeno, Ali Kiwan, Ali Reza fani, Ali Tejani, Alice, Alice Bull, Alice Hu, Alima, alimamo, Alina Borissenko, Aliona Philippova, Alison Edwards, Allison Myers, Allison Umberson, Alma, almalthea, Alvin Echeverria, Alyssa Chandler, Amadeus Gabriel dos Santos Siqueira Silva, Amanda, AMAR PAL SINGH, Amaury Vincent, Amber Alvidrez, Amelia Chaber, Amirali Shahriarymanesh, Ammar Vora, Amoli Kakkar, Amy, Ana, Ana Haag, Ana Karen Tapia, ANA LUIZA MAXIMO AGUIAR DE ALMEIDA, Ana M. Pizarro Galán, Ana Paula Waaĳenberg, Ana Sofia de Oliveira Caldeira, analemma.sky, Anamaria Liana Axinte, Anargha Bose, Anastasia Eriksen, Anastasia Prybytko, anastasia scoggins, Anay Mishra, Andrea Bortoluzzi, Andrea Espinoza, Andrea Nava, Andrea Serio, Andrea Williams, Andrej Coleman, Andres Eloy Martinez Rojas, Andrew, Andrew Bickley, Andrew Boyer, Andrew Conan, Andrew Cooper, Andrew Del Santo, Andrew Obara, Andrew Shaw BSc(Hons) MCPara MRi, Andrew Waldie, Andrew Winkelman, Andrey Korobkov, Andrii Dzygunenko, Andry Nasief, Andrzej Bobinski, Andrzej Wojtowicz, Andy Tonthat, AndyTheAstronomer, Anel Madrigal Gonzalez, Angad Chadha, Angel Elbaz Sanz, Angela Brito, Angela Volpe, Angelika Reithmayer, Angelique Reder, Angelo De Lemos, Anil Vasudev, anita martins da cruz, Anita Springer, Anna, Anna Andriyanov, Anna Batueva, Anna Brisa Micheff Soares, Anna Clara de Souza Fraga, Anna Kruchinina, Anna Mackiewicz, Anna Plum, Anna Scott, Anna Vorobeva, Anna Zanone, AnnaJewel Pace, annparker, Anond Disyatat, anthony, Anthony R. Wells, Anthony Rainone, Anthony TREMBLIN, Antonio, ANTONIO JEFFERSON MONTE ALVERNE PAULINO, Antonio M. Puertas, Antonio Pasqua, Antony Davi Costa de Sena, anwilk, Anylem Gonzalez, Anđela Mogin, Aoife Boyd, Aoiffe Boyle, Aparna Joshi, Archana, Ariana montes, Arianne Ambion, arianny caetano, Arkanar, Arkaprova Dutta, Arkhipova Daria, Arla Heikkinen, Arlind.S, Arman Svoboda, armandina gutierrez, Armando I Zamora, armydragon637, Arnaud Dufourcq Lagelouse, arsama, Artemii Krykun, Arthur Almeida, Arthur Meunier, Arthur P. Pereira, arthur pereira martins, Arttu Sainio, arturovasquez, Artyom Yakubov, Aryan Vinod, Ash Washburn, Ashlee Kephart, Ashleigh Goh, Ashley Abrego, AShley Wilkinson, Ashley Willis, Ashton, Ashtyn Gibbs, Ashutosh, Ashwin Shenoy, Asim, asterisk\\_man, Athanasia Vlachou, Atlas, Aubrey Tyson, Aurelĳus A. 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ABD RAHIM, muhammadmustafa, Mustafa, Myar Misellati, Myranda Keightley, Márcia Saori Câmara Kishi, Márjorie Mourão, N. J. Leyland Elsom, Nadine Trimpin, Naomi K. Lee, naslund, Nastya Kulikova, Natalia, Natalia López Guzmán, Natalia Pirogova, Natalie Jones, Nataliia, Natasha A Frakes, Natasha Van Bemmel, NateA, Nathalie Amador, Nathalie Smits, Nathaly Hernandez, Nathan Coetzee, Nathan Fontanez, Nathan Scalzone, Nathan Yusuf, nathancbell, Navahra Lindsay, Navaneethakrishnan, Naveen kumar S, nchazarra, nebogipfel, Neil Cottrell, Neil Kenneth Burton, Neil Sharp, Neil Wickens, Nelvish Naiker, Nerine Storm Thompson, Nevaeh, NeZur, Ng Wen Xuan, Nica Roseme, Nicholas D. 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Hyzer, steve, Steve Davies, Steve Lake, steve\\_meadows, SteveF48, stevekrz, Steven Rosander, Stickywarp, Stitch1981, Stukalin Danil, Subhas Mazumdar, Sudeeksha Bhattacharyya, surhudm, Susanna Dargan, Susannah Bispham, Svetoslav Alexandrov, svr303, swaas, SY, Sydney Letlow, Syed Hozaifa Shabur, Syed Muhammad Abdullah Bin Hasan, Systrie, Tahseen Huda, Taira Agosto, tairving, Tan Poh Ching, Tania Ramirez, tania.ann.007, Tanishk Mohan, Tara, Tarant0ga, Tarpon Thompson, Tasfia Tarannum, Tatyana Reid, Taylor Benware, tdenton, tej Rajesh, tenfyr, Teodor Michalski, Terdoo Osu, Terrance P Felegie, Sr, Terry Georgas, Thaliyadath K Ravindranath, thecrazedlog, Theresa, Theresa Carl, Thomas BERTRAND, Thomas E Collett, Thomas Hoffmann, Thomas Jenkins, Thomas Kläger, Thomas Koceja, Thomas Laube, Thomas massey, Thomas R. Stewart, Thomas S Grady, Thomas Thomopoulos, thorsteinn, threepwood, Tia Rice, Tiep Tran, Tiffany Shaw-Diaz, Tim Dyson, Tim Jaramillo, Tim Loris Kunze, Tim Plank, Tim Pointing, Tim-Christopher Aust, tiorriot, tireniolu Oladipo, tkuhnle, Tobias Géron, tobidd, Tobrux, Toby Leeper, Tom, Tom Bickle, Tom Davison, Tom Osman, Tom Rich, Tomas Koval, Tomassci, tomjon, TomThaler, tony, Tony, Tony Grant, Tony Hoffman, Torsha Kundu, Torsha Majumder, tpetera, Trandafir Mircea, Trapeznikov Egor Valerevich, trash0, Travis Olah, Travis Rector, Trent Pierce, Tricia, trippyskippy, Trish Martinache, Tuesday Muse, TULSI GOYAL, Tyler Andreww, Tyler Bestram, Tyler Hirshberg, Tyronius McBasketball, Tyson Dabney, udayatejwani, Udunie, Uliana Shiliaeva, Uriel, Utsav Khan, Utsho, Vahavy, Vahid Kermani, Vaideeshwar Sivasubramanian, Vajrapani, Valentina, Valeria Tokareva, Valerie Blanco, valerie flores, Valerie Pegg, Valerie R seymour, Valmir Martins de Morais, Vandytim, Vanessa Eliseo, vanrock70, Varooni Manoj Sawant, Varsha, Vasileios Vlachakis, Vasiliy, Vasylenko A., Vaughan, venatrix, Venkatesh Deshpande, Vern Sowers, Viacheslav Zelenev, Victor, Victor Celli, Victor Cunha Da Silva, Victor Edwin, victor juan garcia porcel, Victoria Jackiw, Vikrant Kurmude, Viktor, Viktor Dobrenov, Viktor Wase, Vincent Hobeïka, Virgilio Gonano, Vishwanand Doobay, Vitor Luis Gonçales Dias, vivian, Vlad, Vlad Mihai, Vladimir, volare09, Walker Wells, Walter E Moody Jr, Walter MacDonald, wangqintao, warrenchen, Weatherly-Battaglia, WEBs\\_in\\_space, WELTON VAZ DE SOUZA, Wenceslao Santiago Germán, Wendy, Wendy Smith, wendy27, Wenjie Zhou, Wentao Huang, Weronika, Wesley Gabriel de Oliveira Melo, Wesley Teh Ee Wen, wesley webb, wgoltz, Wilfried Domainko, Wilfried Esken, Wilhelm A. Weidmann, Will, Will Haresch, William B Hernandez, william birney, William Midgley, William Paul Rhodes, William Russell, Willow Colson, Windsor S Smyser, WOJCIECH MIKINA, WolverineWazza, wrackard, Xander sprangers, Xavier Dartevel, Xenon Chase, xiaoguangliu, xKingx, YaBoy, Yanilsa H Marte Rodriguez, yanyam, Yavin\\_4, YGBS, Yogurtz, Yohan Terpend, Yolly Reyes, Younes Oubkis, Yufan Fane Zhou, Yun Chen, Yuri Peruzzi, Yuri Sushkov, Yvonne Harrison, Zac Thomas, Zaccaria Vidali, Zach Ortega, Zach Schroeder, Zachary Fleisig, Zachary Thede, Zachary Vaillancourt, zacho, zak bennett, Zakck Goolsbee, zan, Zander Polk, Zarriah Fisher, Zdeněk Flanderka, Zebraorpanda, zedcat, zena barabandi, Zenith Diehl., Zhiyuan Cheng, Zĳun He, Ziyun Tang, Zoe Kateri, Zoe Vickhouse, Zoryana, Zovacor, Ágata Moretti Zaneti, Íris Eduarda Forcel Roncada, Александр, Владимир Асташин, Даниил, Дмитрий Дуда, Дмитрий Сергеевич Фёдоров, Евгений Юрченко, Матвей, Николь Вельковская , 杨皓添 , 林于顺 , 蔡頌恩 , 马若瑜 , ᄀ ᅵ ᆷᄀ ᅧ ᆼᄒ ᅧ ᆫ , ᄋ ᅵ ᄉ - ᆼ ᄋ ᅮ , ᄒ ᅡ리 ᄀ ᅧ ᆼ \nThis paper has been typeset from a T E X/L A T E X file prepared by the author."}
2024ApJ...967..158C	Young stellar objects are thought to commonly undergo sudden accretion events that result in a rise in bolometric luminosity. These outbursts likely coincide with the onset of planet formation and could impact the formation of planets. The reason behind this dramatic enhancement of accretion is an active area of research and the mass of the system is a critical parameter. Using the Northern Extended Millimeter Array we survey five outbursting sources three FU Ori one EX Or and one peculiar source with the primary goal of determining the systems mass using an optically thin line of CO. We estimate the mass of a central region for each object that using both continuum emission and CSUP17SUPO J 21. The CSUP17SUPO emission likely includes both disk and inner envelope material thus acts as an upper limit on the disk mass ranging from 0.33 to 3.4 MSUBSUB for our sources. These derived masses suggest that the inner 1000 au contains enough mass along the line of sight for these sources to be gravitationally unstable.	2024-06-01T00:00:00Z	['10.48550/arXiv.2409.04527', '2024arXiv240904527C', '10.3847/1538-4357/ad4a5a', 'arXiv:2409.04527', '2024ApJ...967..158C']	['Protoplanetary disks', 'Young stellar objects', 'FU Orionis stars', 'Submillimeter astronomy', 'Interferometry', '1300', '1834', '553', '1647', '808', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics']	High Mass Inner Regions Found in Five Outbursting Sources	2,024	200	0.51	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	1	https://arxiv.org/pdf/2409.04527.pdf	{'High Mass Inner Regions Found in Five Outbursting Sources': "Jenny K. Calahan, 1, 2 Edwin A. Bergin, 1 Merel van 't Hoff, 1 Ke Zhang, 3 Nuria Calvet, 1 and Lee Hartmann 1 \n1 University of Michigan, 323 West Hall, 1085 South University Avenue, Ann Arbor, MI 48109, USA 2 Center for Astrophysics \| Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA 3 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison., WI 53706", 'ABSTRACT': "Young stellar objects are thought to commonly undergo sudden accretion events that result in a rise in bolometric luminosity. These outbursts likely coincide with the onset of planet formation, and could impact the formation of planets. The reason behind this dramatic enhancement of accretion is an active area of research, and the mass of the system is a critical parameter. Using Northern Extended Millimeter Array, we survey five outbursting sources (three FU Ori, one EX Or, one 'peculiar' source) with the primary goal of determining the system's mass using an optically thin line of CO. We estimate the mass of a central region for each object that using both continuum emission and C 17 O J=2-1. The C 17 O emission likely includes both disk and inner envelope material, thus acts as an upper limit on the disk mass, ranging from 0.33-3.4 M ⊙ for our sources. These derived masses suggest that the inner ∼ 1000 au contains enough mass along the line of sight for these sources to be gravitationally unstable. \nKeywords: protoplanetary disk, astrochemistry", '1. INTRODUCTION': "FU Orionis-type objects (FU Ori) and EX Ori-type objects (EXOr) have been theorized to be a common evolutionary phase between the Class I and Class II stages for low-mass pre-main sequence stars (Hartmann & Kenyon 1996; Quanz et al. 2007; Vorobyov & Basu 2015). An FU Ori object is classified as such after undergoing a sudden and extreme (several magnitudes) increase in brightness in optical and near-infrared wavelengths (Herbig 1977). This outburst has been attributed to short-lived and massive accretion events, predicted to accrete up to 10 -4 - 10 -5 M ⊙ /yr (Hartmann & Kenyon 1996; Audard et al. 2014; Fischer et al. 2023) onto the star. While ExOrs also undergo sudden increases in observed magnitude, they are less extreme and outbursts happen more frequently than FU Oris. There are a number of mechanisms that could cause this sudden and fast accretion, including thermal or gravitational instabilities (Bell & Lin 1994; Armitage et al. 2001; Boley et al. 2006; Zhu et al. 2010), perturbations from a close-by massive companion (Bonnell & Bastien 1992), or sporadic infall from the circumstellar envelope (Vorobyov & Basu 2006; Vorobyov et al. 2013). \nRobust mass constraints on the disk and inner envelope are essential in determining what mechanism(s) is/are responsible for FU Ori outbursts. Gravitational instabilities are associated with triggering a magnetorotational instability (MRI) which would cause the sudden burst of accretion, and this would require the disk to be relatively massive (M disk /M ∗ ≳ 0.1) (Hartmann & Kenyon 1996; Liu et al. 2016; Cieza et al. 2018) and thermal instabilities require regions of the disk that have high enough opacity to trap heat (Zhu et al. 2009). Regardless of the mechanism, FU Ori systems need to be massive enough to sustain the high accretion rate for many years. \nCO is commonly used as a mass tracer for molecular clouds, protostellar envelopes, and protoplanetary disks, as it is a highly abundant molecule, is chemically stable, and easily detectable. The measured ratio of CO/H 2 = 10 -4 in the ISM (i.e. Frerking et al. 1982; Kramer et al. 1999; Parvathi et al. 2012), and is consistent with the ratio found in young Class 0/I disks (van 't Hoff et al. 2020; Zhang et al. 2020). FU Ori objects present a unique laboratory for the purpose of mass determination of young stellar systems due to their recent outbursts. The outburst heats up the surrounding disk and envelope, pushing the snowlines beyond their regular radial \nextent (Cieza et al. 2016; van 't Hoff et al. 2018), exposing more of the disk's mass to be probed with CO. The radial location of the CO sublimation front (temperature ≈ 20 K) during an FU Ori outburst can almost quadruple or even increase tenfold (at the intermediate stage of luminosity rise, and the outburst maximum, respectively) (Molyarova et al. 2018). \n12 CO emission is found to be optically thick towards typical molecular clouds and protoplanetary systems, thus can only provide an lower limit to the mass. Given this, we use lesser abundant isotopologues of 12 CO to probe denser regions. 13 CO and C 18 O have often been used to probe higher density regions. C 18 O emission (both the J = 1-0, and 2-1 transitions) is generally assumed to be optically thin, and has been used as the mass tracer for FU Ori objects and other young stellar objects. Recent work shows, however, that C 18 O emission may not be completely optically thin. Available observations of FU Ori objects find that C 18 O J =1-0 can have measured τ > 1 (Fehér et al. 2017). Towards those same disks, emission from the brighter and more abundantly populated C 18 O J =2-1 line is expected to have an even higher τ . Recent observations with Northern Extended Millimeter Array (NOEMA) towards three non-FU Ori Class I objects showed that C 18 O and C 17 O J =2-1 were both optically thick in the disk region, and 13 C 18 O was required to reach optically thin emission levels (Zhang et al. 2020). \nTo determine the total mass of the inner regions of these outbursting source, an optically thin tracer (C 17 O or 13 C 18 O) is used, along with the measured temperature (derived from C 18 O) and an assumed abundance of the optically thin tracer to H 2 . We present NOEMA observations of five outbursting sources with the primary science targets of 13 CO, C 18 O, C 17 O, and 13 C 18 O. We will use the optically thin tracer to estimate the mass for each source. We focus on the central region, as defined by the peak of the continuum, covering areas with equivalent radii of hundreds of au.", '2.1. Source Selection': "For our sample, we isolate sources that have been published and observed with NOEMA or ALMA in C 18 O J =1-0 or 2-1, and are located within the observational range of NOEMA (Fehér et al. 2017; Principe et al. 2018; Ábrahám et al. 2018). There are nine total sources that fit this criteria; we chose the brightest five sources, each with C 18 O emission peaking directly on source. Within the northern hemisphere, these objects have the highest likelihood of a C 17 Oand 13 C 18 Odetection. Source properties can be found in Table 1. Our sources range from \nclassic FU Ori objects (V1057 Cyg, V1735 Cyg, V1515 Cyg), to a peculiar source with unique spectral features but otherwise FU Ori characteristics (V1647 Ori), to an EXor/UXor type star with a higher period of luminosity variability (V2492 Cyg). \nV1057 Cyg and V1515 Cyg, along with the namesake source FU Ori were the first three discovered and defined the class of FUOr-type objects (Herbig 1977). V1735 Cyg was discovered later, and exhibited the characteristic sharp rise in luminosity followed by a slow decay with short-period modulations (Szabó et al. 2022). V1647 Ori was at first categorized as an FU Ori type object due to its extreme rise in luminosity, however has exhibited multiple significant outburst events since the first one captured, thus it has been classified as either borderline FU Ori/EX Or, or 'peculiar' source (Audard et al. 2014; Connelley & Reipurth 2018; Fischer et al. 2023). V2492 Cyg was seen to go into an outburst stage in 2010, and since then its variability in brightness has been attributed to a combination of episodic changes in extinction and accretion (Hillenbrand et al. 2013; Giannini et al. 2018; Ibryamov & Semkov 2021).", '2.2. NOEMA Data Reduction': "These results are derived from NOEMA observations that took place July - October 2021 as a part of project S21AL. Each source was observed for ≈ 3.4 hrs with beam sizes ≈ 1. '' 58 × 1. '' 50 (V1735 Cyg, V1057 Cyg) 2. '' 45 x 1. '' 69 (V1647 Ori) 1. '' 04 × 0. '' 92 (V2492 Cyg, V1515 Cyg). We utilized Band 3, specifically in the range of 202-211 GHz (lower sideband) and 218-226 GHz (upper sideband). This setup covers our key science lines, simultaneously covering the 13 CO, C 18 O, C 17 O, and 13 C 18 O J=2-1 transitions. The initial round of data quality assurance and reduction was done with a staff member at IRAM (Institut de Radioastronomie Millimetrique). The raw data was self-calibrated and CLEAN-ed using the GILDAS package mapping 1 . The first step was to find the systematic velocity and correct the UV-tables to the accurate values, which aids in line detection. Next we isolate the continuum by zeroing out every line and other feature such as dead pixels or end-of-band wiggles. With the channels containing only continuum emission left over, we average over every ∼ 200 channels before self-calibration and imaging. That solution is then imposed on the original spectra that contains the emission lines. The self-calibrated data is then \nTable 1. Source Properties \nNote -Distances taken from parallaxes in Gaia Release 3 (Gaia Collaboration et al. 2020), however it is worth noting that outburst soruces are particularly difficult to determine distances to. Outburst date and luminosity taken from review Audard et al. (2014) and references therein. Stated ranges indicate the timespan where the onset of the outburst could have started. \ncontinuum-subtracted and imaged. For the goal of determining the mass of each source, we utilized the highvelocity resolution (0.5 km/s or 62.5 kHz) spectra of 13 CO, C 18 O, C 17 O and 13 C 18 O. V1735 Cyg likely contains significant extended emission in the three brightest CO isotopologues, as its solution contains artifacts that proved to be less strong if the smallest baselines were excluded. However, to maintain consistency in the analysis of all sources, the full NOEMA configuration including all baseline was used for the final results.", '3. RESULTS': "In our NOEMA sample we have observed 13 CO, C 18 O, and C 17 O towards all five sources, and the emission is partially to fully resolved. Towards one of them, V1057 Cyg, we also have an unresolved detection of 13 C 18 O. In the moment zero maps, all sources have 13 CO and C 18 O peaked on source, coincident with the continuum. C 17 O peaks on source for V1057 Cyg and V1515 Cyg, however the other three have C 17 O surrounding or nearby the central star. It is likely that the continuum may be blocking some central C 17 O emission for these three sources. In all sources, there is extended gas emission as compared to the continuum (with V1515 Cyg being being the most compact, see Figure 1). \nWe seek to determine the mass of the central region from both the continuum observations and an optically thin gas line. For the continuum mass derivation, we use the following equation: \nM cont = S ν d 2 κ ν B ν ( T ) . (1) \nHere, S ν is the flux density, d is the distance, B ν ( T ) is the Plank function for a given dust temperature T, and κ ν is the dust opacity coefficient which we assume to be 2.2 cm 2 g -1 ( κ ν = 10( ν/ 1000 GHz) ; Beckwith et al. 1990; Cieza et al. 2018). For the dust temperature, we uniformly use T=25 K for both the dust and gas mass \ndeterminations, as that is approximately the brightness temperature for the continuum of each object. We use a gas-to-dust mass ratio of 100, and thus find inner-region masses of 0.39-0.85 M ⊙ (see Table 3). \nOur ideal gas-mass tracer will be an optically thin transition of CO, and so we first calculate the optical depths of each of these isotopologues. The brightness temperature of a molecule is defined by: \nT B = T ex (1 -e -τ ) (2) \nwhere T ex is the excitation temperature and τ is the optical depth. Taking the ratio between the brightness temperatures of two isotopologues will provide a relationship between the optical depths of the two lines: \nT B ( 13 CO ) T B ( C 18 O ) = T ex ( 13 CO )(1 -e -τ 13 ) T ex ( C 18 O )(1 -e -τ 18 ) . (3) \nThe excitation temperatures of each isotopologue of the same rotational transition can be assumed to be equal. And if you can assume that the 13 CO is optically thick, you can simplify this relation to \nT B ( 13 CO ) T B ( C 18 O ) ≈ 1 1 -e -τ 18 . (4) \nOne can further state that τ 18 = τ 13 /X where X is the abundance ratio between the two isotopologues. Thus, the ratio of the optical depths is equal to the abundance ratio. In our case we use ratio values from Qi et al. (2011) which were found from models that reproduce multiple observations of CO isotopologues in a protoplanetary disk, and are consistent with the ISM (Wilson 1999): 12 C/ 13 C=67, 16 O/ 18 O=444, and 18 O/ 17 O=3.8. If R = T B ( 13 CO ) T B ( C 18 O ) then you can express the optical depth of each isotopologue line as follows: \nτ 13 = -444 67 ln(1 -1 R ) (5) \nFigure 1. Moment zero maps of 13 CO, C 18 O, C 17 O J=2-1 for each source observed in this survey. \n<!-- image --> \nTable 2. Observational Properties \nτ 18 = -3 . 8 ln(1 -1 R ) (6) \nτ 17 = τ 18 / 3 . 8 (7) \nUsing the observed flux ( S ν ∝ T B ) of each isotopologue, we calculated the optical depths for each observed line. We started with moment zero maps derived from our cleaned spectral cubes using bettermoments (Teague & Foreman-Mackey 2018). All five sources show optically thin emission in the C 17 O transition, while C 18 O is marginally optically thick ( τ ≈ 1 in most sources), and all 13 CO is optically thick. \nThe mass of each source is therefore calculated using C 17 O emission. In an environment that is optically thin and in LTE, the column density of the upper energy state of a certain molecular transition is defined as: \nN thin u = 4 πS ν ∆ ν A ul Ωhc , (8) \nwhere S ν ∆ ν is the integrated flux density. The frequency range, ∆ ν , is defined by the width of the C 17 O emission profile in frequency space, and we use the same frequency range for each source's mass estimate. A ul is the Einstein-A coefficient that represents the rate with which an energy level is depopulated through spontaneous emission, and this is a constant for the C 17 O J=21 transition and equal to 10 -6 . 2 s -1 (Müller et al. 2005). Ω is the solid angle of the emitting area. \nAfter the column density of the J=2 level is calculated, an LTE assumption can easily relate it to the total C 17 O column density, N T : \nN T = N u g u Q ( T rot )[ e -E u /kT ] -1 , (9) \nwhere g u is the statistical weight of the J=2 level, Q ( T rot ) is the partition function, and E u is the energy \nFigure 2. The 13 C 18 O J=2-1 detection towards V1057 Cyg, summed over the central few pixels with an equivalent area to the beam. \n<!-- image --> \nof the J=2 level. All of these constants are found in the JPL line catalogue (Pickett et al. 1998). \nOnce the total C 17 O column density is found, we can use the isotope ratios to back out the total CO column and then H 2 column density. We use C 16 O/C 18 O=1687 and CO/H 2 = 10 -4 (Wilson 1999; Frerking et al. 1982; Bergin & Williams 2017). Once the H 2 column density is derived per pixel, we sum over the surface area of a set of given pixels to produce a total mass. Using the continuum data, we define a central area of the image that contains at least all of the 15 σ continuum emission (see Figure 3). Three sources (V1057 Cyg, V1515 Cyg, and V1647 Ori) only included 20 σ flux so that the central areas were comparable sizes between sources. The final calculated masses from C 17 O range between 0.253.4 M ⊙ . These regions likely contains both disk and inner envelope emission in the corresponding C 17 O image (See Figure 4), in particular, emission from the envelope along our line of sight. All moment zero maps were used by implementing the bettermoments packages (Teague & Foreman-Mackey 2018). \nThe average C 17 Ocolumn densities, mass column densities, derived mass from C 17 O, continuum flux, the corresponding equvalent emitting radius, and the ratio between the final gas and dust masses are shown in Table 3. These derived masses from the C 17 O are likely upper estimate on the mass of the disk, while also being lower-end estimates on the total mass within this emitting area as not all of our sources have their C 17 O peak overlap with the continuum peak. \nFor the source V1057 Cyg, we also have a detection of 13 C 18 O which can be used as another mass tracer. The 13 C 18 O flux is not located in the central pixel, but is detected after summing over the area of the beam (see Fig. 2). We find an inner-region mass of 2.6 M ⊙ using \nTable 3. Derived Column Densities and Mass \n13 C 18 O. This used the sum of the flux over the same area used for the mass determination from both the continuum and C 17 O, and a 12 C 16 O/ 13 C 18 O = 29,748. Compared to the mass determined from C 17 O, it is slightly lower however well within a factor of two, thus we interpret this to be consistent. Both optically thin tracers suggest V1057 Cyg to be the most massive source within our sample.", '4. DISCUSSION': 'Mass is a fundamental property when it comes to understanding the protoplanetary disk environment and FU Ori outburst events. The quickest and most often used technique to extract disk masses is using dust thermal continuum observations. Assuming optically thin dust emission and a gas-to-dust ratio of 100, a gas mass can be calculated from the dust observations. We use this technique to estimate a lower-end mass of the central region, due to the fact that this continuum emission is likely optically thick (i.e. Tobin et al. 2020; Kóspál et al. 2021). Within similar objects, the dust that is probed at ∼ 220 GHz has been shown to be optically thick, and corresponding C 17 O emission often shows a dip corresponding with the center of the continuum emission (see V1647 Ori and V1057 Cyg in Figure 4). Using optically thin observations of C 17 O, we also calculate a mass of the central region, and treat this as an upper-end estimate of the disk mass. In each of our sources, C 17 O emission is optically thin, however it comes with some caveats. The emission likely probes the mass of the in-falling envelope and disk while the continuum probes large dust grains confined to the disk. Thus, the C 17 O emission likely probes more mass over a larger area, especially along the line of sight. However, the dust emission is expected to be optically thick and blocking C 17 O emission from the backside of the disk/envelope. C 17 O will be an upper limit if a source has an envelope, however if there is no envelope the C 17 O mass could be closer to a lower limit given an optically thick dust disk. Three of our disks have total inner-region masses that have similar values when derived from the continuum \n(with gas/dust =100) and from an optically thin gas tracer (C 17 O). V1735 Cyg, V1647 Ori, and V1515 Cyg have masses that agree within a factor of less than two, suggesting that there may be little envelope contribution. V2492 Cyg, our EXOr source, has a gas mass that is just over 2 times more massive than what would be assumed from continuum observations alone. The final gas mass along the line of sight with a radial extent of ∼ 1000 au around V1057 Cyg is 3.4 M ⊙ , over four times more massive than what would be assumed from continuum alone. V2492 Cyg and V1057 Cyg likely have a significant contribution from the surrounding envelope, perhaps feeding the disk and playing a role in the mechanism that caused an episodic accretion event for these sources. \nWithin our small sample, there appears to be no trend in mass versus outburst classification, peak luminosity, nor most recently observed brightness (most up-to-date photometic observations for each source: Kopatskaya et al. 2013; Peneva et al. 2009; Ninan et al. 2013; Szabó et al. 2022; Ibryamov & Semkov 2021). Our two most massive sources are an FU Ori (V1057 Cyg) and EX Ori object (V2492 Cyg) which exhibit markedly different peak luminosities and time since outburst. The fact that these two sources are the two most massive in our sample agrees with previous work that measured the mass of these two sources (Fehér et al. 2017) but does not agree with a trend seen when observing the continuum probed with ALMA that EX Ori sources tend to be less massive than FU Ori sources (Cieza et al. 2016). The most thorough samples of FU Ori masses come from continuum emission alone, with ≈ 18 determined masses (Cieza et al. 2018; Kóspál et al. 2021, and references therein). Compared to previously observed sources, our five sources reside alongside the most massive that have been seen with ALMA: V883 Ori, Haro 5a IRS, V900 Mon, and L1551 IRS 5 N. Our sample was biased to target the brightest outbursting sources in the Northern Hemisphere, thus it is not particularly surprising that we find higher masses than the average source in Cieza \nFigure 3. Continuum maps corresponding to ∼ 221 GHz, extracted from the upper sideband of our NOEMA observations. The bulk of the continuum emission resides within the black contours, which is used to calculate a continuum mass. All contours correspond to at least where 15 σ emission is found, and the corresponding areas for each source are similar spatial scales. These areas are used to calculate a gas mass from C 17 O, see Figure 4. \n<!-- image --> \net al. (2018); Kóspál et al. (2021). Direct comparisons of the continuum mass are not straightforward between papers, as different emitting areas are used, temperatures, and distances. However, it is insightful to determine if our sample hold true with a trend seen thus far, that the majority of FUOr sources are gravitationally unstable. \nThe Toomre Q parameter is often used to determine if a disk is gravitationally stable or not (Toomre 1964). A Qparameter greater than one suggests the disk is stable, while less than one is unstable. We use the following expression to calculate the Toomre Q parameter: \nQ ≡ c s Ω πG Σ . (10) \nThe angular speed of the gas disk ( Ω ) is defined by √ GM ⋆ /R 3 disk and the surface density ( Σ ) is calculated from the final total mass and emitting radius using the formula √ M disk /πR 2 . The sound speed of the gas (c s ) is calculated using √ k B T/µm H where k B is the boltzmann constant, T is taken to be the same temperature as used for the mass calculations (25 K), µ is the mean molecular weight which we use 2.3 (i.e. Armitage 2019). Using the values from Table 3, we find Toomre Q parameters from 0.06 - 0.79, with V1057 Cyg corresponding to the most gravitationally unstable. All sources in this \nsample have a Toomre Q parameter below one, suggesting that the inner regions of all of these objects contain enough mass to trigger a sudden outburst event. Previous mass determinations of FU Ori sorces also find that the majority of these objects tend to be gravitationally unstable (Cieza et al. 2018; Kóspál et al. 2021). \nDue to our beam-size, we are unable to resolve disk structure, and our continuum observations are unresolved on scales of a typical disk. Additionally, moment 1 maps of C 17 O showing the intensity weighted average velocity do not show typical signatures of disk rotation. Thus we are unable to make strong constraints on the protoplanetary disk size. Given our beam sizes of ∼ 1000 au, we are also unable to see clear Keplarian rotation from the disk in a position-velocity diagram. However, with such a relatively high mass within < 1000 au diameter region around each source, our results are consistent with previous work in determine FU Ori-type object central masses. What is clear about these systems is that they are full of complex structure. The moment 0 maps of 13 CO and 18 CO for each source show extended emission, and for three sources seem to suggest asymmetric features that could be part of infalling structures or streamers (V1057 Cyg, V2492 Cyg, and \nFigure 4. Moment zero maps of C 17 O with the white contour indicating the central area from which a mass is determined, as defined by the continuum flux. \n<!-- image --> \nV1735 Cyg). There is very likely larger scale emission that is resolved out due to our beam size, and follow up work using single dish data will be able to focus on the large scale environment of each source. The triggering of these outbursting sources may not only rely on the disk mass and dynamics, but also the surrounding environment and infalling material.', '5. CONCLUSION': 'Using the NOEMA interferometer, we observed five outbursting sources, three FU Ori objects and two EX Ori/Peculiar sources. C 17 O was detected in each source and was found to be optically thin, thus a mass tracer. We found masses of the centrally peaked clumps from 0.31-3.4 M ⊙ using both dust and C 17 O emission to determine a range of masses for the central region of each source (along the line of sight with a radial extent of \n< 1000 au around the central star). While these masses seem to represent upper limits on the total disk mass, it falls in line with a general trend seen in FU Ori objects suggesting massive disks that may be gravitationally unstable.', '6. ACKNOWLEDGMENTS': 'Matplotlib (Hunter 2007), Astropy (Astropy Collaboration et al. 2013, 2018), NumPy (Harris et al. 2020) \nJ.K.C. acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1256260 and the National Aeronautics and Space Administration FINESST grant, under Grant no. 80NSSC19K1534. \nE.A.B. acknowledge support from NSF Grant#1907653 and NASA grant XRP 80NSSC20K0259', 'REFERENCES': 'Ábrahám, P., Kóspál, Á., Kun, M., et al. 2018, ApJ, 853, \n28, doi: 10.3847/1538-4357/aaa242 Armitage, P. J. 2019, in Saas-Fee Advanced Course, Vol. 45, Saas-Fee Advanced Course, ed. M. Audard, M. R. Meyer, & Y. Alibert, 1, doi: 10.1007/978-3-662-58687-7\\_1 \nArmitage, P. J., Livio, M., & Pringle, J. E. 2001, MNRAS, 324, 705, doi: 10.1046/j.1365-8711.2001.04356.x \nAstropy Collaboration, Robitaille, T. P., Tollerud, E. J., \net al. 2013, A&A, 558, A33, \ndoi: 10.1051/0004-6361/201322068 \nAstropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., \net al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f'}
2024MNRAS.534.1339M	Many compact objects black holes and neutron stars exist in binaries. These binaries are normally discovered through their interactions either from accretion as an Xray binary or collisions as a gravitational wave source. However the majority of compact objects in binaries should be noninteracting. Recently proposed discoveries have used radial velocities of a bright star main sequence or evolved that are indicative of a massive but dark companion which is inferred to be a compact object. Unfortunately this burgeoning new field has been hindered by false positives including the Unicorn V723 Mon which was initially believed to be a red giantblack hole binary before being refuted. In this work we investigate the evolution of stellar binary populations over time using the binary evolution code COSMIC to simulate binary populations and determine the probability of a candidate object being either a true Unicorn actual compact objects in binaries or a false positive. We find that mainsequence MS stars have a higher true Unicorn probability than red giants or naked helium stars an exposed core of an evolved star particularly if the companion is more massive and is inlineformulatexmath idTM0001 notationLaTeXgetexmathinlineformula3 times less luminous than the MS star. We also find that a topheavy initial mass function raises the true Unicorn probability further that supersolar metallicity reduces the probability and that most true Unicorns are found at periods inlineformulatexmath idTM0002 notationLaTeXletexmathinlineformula100 d. Finally we find that a significant fraction of true Unicorns do not evolve into Xray binaries during the age of the Universe.	2024-10-01T00:00:00Z	['10.48550/arXiv.2409.05190', '2024MNRAS.534.1339M', '2024arXiv240905190M', '2024MNRAS.tmp.2104M', 'arXiv:2409.05190', '10.1093/mnras/stae2146']	['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena']	True unicorns and false positives simulated probabilities of dark massive companions to bright stars	2,024	200	0.54	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	0	https://arxiv.org/pdf/2409.05190.pdf	{'True Unicorns and False Positives: Simulated Probabilities of Dark Massive Companions to Bright Stars': 'Andrew M. Miller 1 , 2 , Alexander P. Stephan 3 , 2 , 4 &David V. Martin 5 , 2 , 6 \[email protected] \n- 1 Department of Physics and Astronomy, The University of Toledo, Toledo, OH 43606, USA\n- 2 Department of Astronomy, The Ohio State University, 4006 McPherson Laboratory, Columbus, OH 43210, USA\n- 3 Department of Physics & Astronomy, Vanderbilt University, Nashville, TN 37235, USA\n- 4 Center for Cosmology and AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA\n- 5 Department of Physics & Astronomy, Tufts University, Medford, MA 02155, USA\n- 6 NASA Sagan Fellow \nFirst submitted to MNRAS on March 3, 2023', 'ABSTRACT': "Many compact objects (black holes and neutron stars) exist in binaries. These binaries are normally discovered through their interactions, either from accretion as an X-ray binary or collisions as a gravitational wave source. However, the majority of compact objects in binaries should be non-interacting. Recently proposed discoveries have used radial velocities of a bright star (main sequence or evolved) that are indicative of a massive but dark companion, which is inferred to be a compact object. Unfortunately, this burgeoning new field has been hindered by false positives, including the 'Unicorn' (V723 Mon) which was initially believed to be a red giant/black hole binary before being refuted. In this work, we investigate the evolution of stellar binary populations over time, using the binary evolution code COSMIC to simulate binary populations and determine the probability of a candidate object being either a 'true Unicorn' (actual compact objects in binaries) or a false positive. We find that main sequence stars have a higher true Unicorn probability than red giants or naked helium stars (an exposed core of an evolved star), particularly if the companion is more massive and is ≥ 3 times less luminous than the MS star. We also find that a top-heavy initial mass function raises the true Unicorn probability further, that super-solar metallicity reduces the probability, and that most true Unicorns are found at periods ≤ 100 days. Finally, we find that a significant fraction of true Unicorns do not evolve into x-ray binaries during the age of the universe. \nKey words: binaries: visual, X-rays: binaries, stars: evolution", '1 INTRODUCTION': "compact object population and hindering our understanding of the processes which produced them. \nHistorically, neutron stars and stellar mass black holes have been discovered in interacting binaries. These interacting objects include X-ray binaries, where material from a donor star is accreted onto a compact object, emitting X-rays, (Giacconi et al. 1964; Overbeck & Tananbaum 1968; Shakura & Sunyaev 1973), and more recently, the merger of two compact objects and the production of gravitational waves (Abbott et al. 2016; LIGO et al. 2021). Compact objects can also exist in non-interacting binaries (Guseinov & Zel'dovich 1966; Trimble & Thorne 1969). There are two primary motivations to search for non-interacting compact objects: 1) it is expected that the majority of compact objects live in non-interacting binaries (Mashian & Loeb 2017), and 2) solely observing interacting binaries provides a biased sample of compact objects, distorting the observed \nCompact objects in non-interacting binaries can be found in several ways, including using pulsars which are eclipsed (Yan et al. 2021) or have timing variations (Lorimer et al. 2021). Astrometry is another method, and the final Gaia data release should contain hundreds or thousands of compact objects in non-interacting binaries (Mashian & Loeb 2017; Yamaguchi et al. 2018; Wiktorowicz et al. 2020; Chawla et al. 2022). Gaia mayhavethepotential to characterize every black hole with a luminous companion located within 1 kpc (Andrews et al. 2019). However, if Gaia data on the luminous star is incomplete, then only the total mass of the system is known, and additional methods are needed to break the binary's mass degeneracy and determine the companion's true mass. \nAtechnique which has found popularity recently is using radial \nvelocities (RVs). In this technique, RVs indicate a binary pair with two stellar mass objects, however both the spectroscopy and spectral energy distribution (SED) are consistent with only a single 'bright' star. Based on the masses of the objects, it is argued that if the companion were a regular star, then it would be visible. Therefore, the companion must be massive but 'dark', and hence a compact object (neutron star or black hole). \nThompson et al. (2019) studied the red giant 2MASS J05215658+435922. ASAS-SN photometry showed a roughly sinusoidal 83.2-day periodicity, which was attributed to star spots on a rotating star. Initial APOGEE radial velocities suggested a massive companion. Follow-up with the TRES spectrograph confirmed a dark companion with a minimum mass of 2 . 9 M ⊙ , comparable to the mass of the bright giant (3 . 2 M ⊙ ). The fact that an only slightly less massive companion was not visible was evidence for a compact object. There was also an absence of X-ray emission, and hence this was deduced to be a black hole in a non-interacting binary. The observations do not directly distinguish between a black hole and a neutron star for the dark companion, however a black hole was favoured since the conventional upper mass limit for a neutron star is 2 . 9 -3 . 0 M ⊙ (Finn 1994; Kalogera & Baym 1996). At ≥ 2 . 9 M ⊙ , this would be the lowest mass black hole discovered, placing it in the 'mass-gap', an interesting region of parameter space in between neutron stars and black holes. Currently, this discovery has not been overturned, and is still believed to be a black hole. \nMore recently, Gaia astrometric measurements were combined with radial velocity and spectroscopic follow-up from multiple instruments to discover Gaia BH1, a main sequence G star orbiting a ∼ 10 M ⊙ black hole (El-Badry et al. 2023a), as well as Gaia BH2, a ∼ 1 M ⊙ red giant orbiting a ∼ 9 M ⊙ black hole (El-Badry et al. 2023b). At 480 pc, Gaia BH1 represents the nearest black hole discovered to date. These were followed by Gaia NS1, a main sequence G star orbiting a ∼ 1.9 M ⊙ neutron star (El-Badry et al. 2024b), and Gaia BH3, a giant-branch G star on an 11.6 year orbit with a ∼ 32.7 M ⊙ black hole (Gaia Collaboration et al. 2024), which at 590 pc is the second-nearest black hole discovered. These methods were also used to identify a population of 21 main sequence stars orbiting neutron star candidates with masses between 1 . 261 and 1 . 898 M ⊙ (El-Badry et al. 2024a). For some of these candidates, a white dwarf has not been ruled out due to the mass degeneracy between high-mass white dwarfs and low-mass neutron stars. \nWhile these methods have been successful, they have also resulted in false positives. The Thompson et al. (2019) discovery was followed by similar discoveries in V723 Mon (aka 'the Unicorn') by Jayasinghe et al. (2021) and 2M04123153+6738486 (aka 'the Giraffe') by Jayasinghe et al. (2022), using comparable methods and datasets. As in Thompson et al. (2019), both targets have no visible X-ray emission, although Jayasinghe et al. (2021) did invoke an accretion disc around V723 Mon's black hole to explain the observed spectral energy distribution (SED). These were also proposed as mass-gap black holes in non-interacting binaries. Shortly after, El-Badry (2022) refuted the presence of a black hole in both binaries. A re-analysis of the SED and spectroscopy through spectral disentangling showed that the companions, whilst faint, are not completely dark, and hence are not compact objects. For V723 Mon, the companion is particularly nefarious, since it is a rapidly-rotating subgiant, where the spectral lines are so heavily broadened that they become effectively invisible. \nOur Galaxy should contain as many as 10,000 stellar/compact object binaries (Chawla et al. 2022). However, the number of objects which could impersonate these binaries in terms of mass and luminosity is unknown. Since this burgeoning field of non-interacting \nbinary discovery has faced early challenges, we seek to investigate the probability that a compact object candidate, once it has been identified as part of a sky survey, is either a genuine compact object or an imposter star. We define non-interacting binaries containing an actual compact object as 'true Unicorns'. While in Jayasinghe et al. (2021), the 'Unicorn' name was applied specifically to V723 Mon, in this study we apply it broadly to all non-interacting binaries containing one star and one compact object, a class of objects which V723Monwasbelieved to be before it was refuted. We define a 'false positive' as a stellar binary whose SED appears to indicate only a single star, however spectral disentangling would reveal the presence of a second, less luminous star. If this companion is extremely faint, its contribution to the SED can be virtually undetectable (El-Badry et al. 2018), requiring significant effort to classify the binary. We seek to analyze the types of stars and stellar populations for which true Unicorns are likely to outnumber false positives, and to provide criteria that will give observers a higher probability of identifying binary systems containing a non-interacting compact object. \nFor this purpose, we simulate stellar binary populations and track their evolution for 13 Gyrs. We first explore a generic, highmass Galactic population. We then explore a high-mass population similar to what would be found in the Arches Cluster, a dense stellar cluster toward the central region of our galaxy with a young age of ∼ 3 . 7 ± 0 . 2 Myrs (Hosek 2019; Gallego-Calvente et al. 2021). This cluster is believed to have a top-heavy initial mass function (Hosek 2019) as well as super-solar metallicity (Martins et al. 2008; Sabhahit et al. 2022), which we predict will produce a higher probability of compact objects compared to false positives. In Section 2 we outline the physical parameters of our binary populations and the code used to evolve them, and present our general results in Section 3 before concluding in Section 4.", '2.1 Binary Population Synthesis': "Weconstruct a population of binaries based on literature distributions for four key parameters: initial mass function (IMF), mass ratio 𝑞 = 𝑚 2 / 𝑚 1 , orbital period 𝑃 bin , and orbital eccentricity 𝑒 bin . \nThe IMF is generally described by the power law \n𝜉 ( 𝑚 ) ∝ 𝑚 -𝛼 , \nwhere we use two values for 𝛼 in order to compare two separate varieties of stellar populations in our galaxy. We use 𝛼 𝐾 = 2 . 7 to model the IMF of the solar neighborhood and overall galactic star population, based on Kroupa et al. (1993) (see also Salpeter 1955). We use 𝛼 𝐻 = 1 . 8 based on Hosek (2019) (see also Kim et al. 2006), which describes the IMF of the Arches cluster. This second IMF is top-heavy compared to the general Kroupa IMF, producing a higher number of massive stars. Through this, we examine whether young, dense clusters near the Galactic center might be home to more binaries containing non-interacting compact objects. We also probe the general effect caused by a change in IMF. The 𝛼 values we reference here are valid for initial primary stellar masses above 1 M ⊙ . \nWe model the values of 𝑞 , log 𝑃 bin , and 𝑒 bin as Gaussian distributions based on the results of Duquennoy et al. (1991). The distribution parameters, such as the average expected value 𝜇 and the standard deviation 𝜎 , are listed in Table 1. The eccentricity distribution is valid for binary periods less than about 1000 days, which covers most of the period range focused on in our study (see Section 2.4). \nTable 1. Distribution parameters for the variables used in constructing our binary populations, namely primary star mass 𝑚 1, mass ratio 𝑞 , orbital period 𝑃 bin , and orbital eccentricity 𝑒 bin , based on Duquennoy et al. (1991). In particular, we give the 𝛼 values for the IMFs based on Kroupa et al. (1993) and Hosek (2019), as well as expected value 𝜇 , and standard deviation 𝜎 for the Gaussian distributions. We also give the minimum and maximum variable values. \nUsing the distributions described in Table 1, we construct three separate populations of 5 × 10 5 binary pairs each, one employing the IMF of Kroupa et al. (1993), the second based on the IMF of Hosek (2019), and the third based on the Hosek IMF with super-solar metallicity as per Martins et al. (2008) and Sabhahit et al. (2022). We choose 2 M ⊙ as the minimum mass at formation for the primary star of each binary, as we are interested in relatively massive binary pairs that could either produce or mimic a hidden compact object. We treat each population as a cluster with all stars roughly equidistant from Earth, allowing us to focus on objects' luminosity instead of flux.", '2.2 Measuring Radial Velocities': 'To ensure that the binaries we examine could truly be detected through radial velocity measurement, we calculate the radial velocity semi-amplitude of the more luminous star in each binary pair. \nTo calculate the RV semi-amplitude, we use: \n𝐾 Bright = GLYPH<18> 2 𝜋𝐺 𝑃 orb GLYPH<19> 1 / 3 𝑚 Bright sin 𝑖 GLYPH<16> 𝑚 Bright + 𝑚 Dark GLYPH<17> 2 / 3 1 √ 1 -𝑒 2 \nwhere 𝑚 Bright is the mass of the bright star in each binary, and 𝑚 Dark is the mass of the less luminous companion. √ \nFor sin 𝑖 , we assign the value 1 / 2 to all binary pairs. This represents an average inclination within a stellar population, and is consistent with the expectation that lower inclinations (<30 · ) are geometrically disfavored in binaries (El-Badry et al. 2022).', '2.3 Identifying X-Ray Binaries': "As the focus of our study is to identify the number of binary systems where object classification could be open to misinterpretation, we seek to remove x-ray binaries from our population sample. X-ray binaries provide direct evidence of compact objects through the presence of an accretion disk and detectable x-ray radiation. This differentiates them from most stellar binaries that could act as false positives. To remove them, we must first define which objects within our populations could be detected through x-ray emission at different times. \nHigh-mass x-ray binaries (HMXBs) are predominantly formed when O and A type stars fill at least 80 to 90 percent of their Roche lobe, creating a wind-driven accretion stream onto the surface of a compact object companion (Hirai & Mandel 2021). The stars typically have mass ≥ 10 M ⊙ (Corral-Santana et al. 2015). During overflow they lose mass at a rate approaching 10 -6 M ⊙ yr -1 (King \n1995). These objects are rare in a galaxy the size of the Milky Way, where we should see only 1-2 observable black hole HMXBs at a time (Romero-Shaw et al. 2023). \nLow-mass x-ray binaries (LMXBs) are formed when stars with mass ≤ 1 M ⊙ transfer material via Roche lobe overflow through their inner (L1) Lagrangian point onto the surface of a compact object companion (Charles & Coe 2006). Objects detected as persistent LMXBs lose mass at a rate near 10 -8 M ⊙ yr -1 , while transient LMXBs lose mass at a rate near 10 -9 M ⊙ yr -1 (Tanaka & Shibazaki 1996). \nWhile intermediate-mass x-ray binaries (IMXBs) have been discovered (e.g., Tananbaum et al. 1972) and should be forming at a rate ≥ 5 times that of LMXBs (Pfahl et al. 2003), relatively few have been identified within the Galaxy to date (Hunt et al. 2021). IMXBs tend to quickly evolve into LMXBs through mass loss, making them difficult to characterize (Podsiadlowski et al. 2002; Pfahl et al. 2003). Their mass loss rates are also not well understood, given that stars of 1-10 M ⊙ are typically not large enough to generate the high wind-driven loss needed to produce an observable x-ray source (Tauris & van den Heuvel 2006). \nTo identify x-ray binaries, we adopt the methodologies of Podsiadlowski et al. (2003), Liotine et al. (2023), and Misra et al. (2023) to calculate the x-ray luminosity of each BH and NS undergoing accretion within our populations. The x-ray luminosity of the compact object is defined by \n𝐿 x = 𝜂 / 𝑀 acc 𝑐 2 \nwhere / 𝑀 acc is the mass-accretion rate and 𝑐 is the speed of light. Since not all accreting material will contribute to the compact object's growth, / 𝑀 acc is related to the actual change in the compact object's mass ( / 𝑀 CO ) by \n/ 𝑀 acc = / 𝑀 CO 1 -𝜂 \nThe radiative efficiency, 𝜂 , is the fraction of accreted rest mass which is converted into energy and radiated away, and is defined as \n𝜂 = 𝐺𝑀 CO 𝑅 acc 𝑐 2 \nwhere 𝐺 is the gravitational constant, and 𝑀 CO is the mass of the compact object. For neutron stars, 𝑅 acc is simply the object's radius, while for black holes, 𝑅 acc is the spin-dependent innermost stable circular orbit around the object (Podsiadlowski et al. 2003), defined as \n𝑅 ISCO = 6 𝐺𝑀 BH 𝑐 2 \nWe define x-ray binaries (XRBs) as any binary containing a nondegenerate star and an accreting BH or NS with a calculated x-ray luminosity > 10 35 erg s -1 (Misra et al. 2023). At this luminosity, we should be detecting both transient and persistent sources. One of the least luminous transient XRBs detected, XTE J1118+480, peaks at ∼ 3.6 × 10 35 erg s -1 (Dunn et al. 2010; Corral-Santana et al. 2015). Our sample may not include low luminosity Be XRBs (Pfahl et al. 2002), which show persistent x-ray luminosity in the range of 10 34 -10 35 erg s -1 with bursts ∼ 10 times higher (Sguera et al. 2023). \nWe further classify our XRB population based on the bright star's mass: HMXBs, with masses ≥ 10 M ⊙ ; LMXBs, with masses ≤ 1 M ⊙ ; and IMXBs, with masses from 1-10 M ⊙ . \nTable 2. List of evolutionary stellar states produced by COSMIC , and the acronyms used in this study to refer to each group of states. COSMIC also produces white dwarfs, which appear within our populations but are not listed here since they are outside the focus of this study.", '2.4 Modelling': "The binary populations we create with the parameters outlined in Section 2.1 are evolved using COSMIC (Breivik et al. 2020). COSMIC is a binary population synthesis and evolution code based on the older BSE code (Hurley et al. 2002), however it greatly expands upon the functionalities of BSE by including more detailed stellar and binary interaction processes and allowing for the evolution of more massive stars. COSMIC has been a popular tool of late, including the recent Weller & Johnson (2023) predictions for Milky Way Mapper in SDSS-V, which searches for progenitors of compact object collisions that will produce gravitational waves, as well as Liotine et al. (2023), which probes the lack of observed HMXBs that are predicted to evolve into binary black hole mergers. COSMIC has also been used to probe Gaia's ability to detect stellar/compact object binaries in the Galaxy (Andrews et al. 2019; Breivik et al. 2019; Chawla et al. 2022). \nFor our work we use COSMIC version 3 . 4 . 0 to evolve each binary for 13 Gyrs. We use solar metallicity (0 . 0134) for each star (Asplund et al. 2009) within the Kroupa and Hosek IMF populations. We also perform additional modelling for the Hosek IMF at higher metallicity (see Section 3.3). \nThe general input physics parameters for COSMIC are outlined in Breivik et al. (2020). For this study we have used COSMIC 's default values, representing reasonable parameters based on current understanding of stellar evolution. These values are listed in Appendix A. Of note is the 'remnantflag' parameter which determines the mass prescriptions for neutron stars and black holes, providing an option between rapid and delayed supernova explosion models. The mass distribution of black holes has been found to depend strongly on the choice of this parameter (Chawla et al. 2022). The rapid model assumes a mass gap between neutron stars and black holes. Breivik et al. (2019) notes a decrease in the overall number of black holes when building populations under this model, and notes inconsistencies with observation. Therefore, we have employed the delayed model, which assumes no mass gap and is the default in COSMIC . \nThe output of our COSMIC simulations consists of the evolutionary states of the binary pairs at a given time, as well as additional characteristics such as mass, orbital period, and effective temperature. Table 2 shows the COSMIC evolutionary states which we focus on in this study, as well as the acronyms used to refer to these states. \nCOSMIC provides the bolometric luminosity of each star as an output at each time interval. When we refer to an object as 'bright' \nor 'dark,' we are referring to its total bolometric luminosity and not its appearance in visible wavelengths. In categorizing stars by their bolometric (as opposed to visible) luminosities, we are consistent with the methods used to further analyze candidate non-interacting compact objects once they are detected through visible-light surveys. The combined total luminosity of both stars in a binary determines the shape of the spectral energy distribution (SED), and in the case of faint companions, further analysis of the spectrum and SED are needed to confirm the presence of either two stars, or of one star and a compact object. \nCOSMIC tracks binary evolution in two ways: 1) by sampling the binary characteristics each time a key evolutionary change occurs, including the exact time of the change; and 2) by sampling the binary parameters at a series of user-defined time steps, for which we used 37 logarithmic steps. We then combine these two different outputs and sort our binaries' evolution into a series of logarithmic time interval bins. Logarithmic bins were chosen as stellar populations on the whole tend to exhibit more change at early ages due to the rapid evolution of massive stars. At later ages, the population as a whole changes more slowly as lower-mass stars continue their gradual evolution. Combining the two COSMIC outputs ensures that no significant evolutionary change is missed in our data. \nWithin each time interval bin, we track the number of true Unicorns , which is the name we assign to non-interacting binaries containing one compact object, consistent with the class of binary postulated by Jayasinghe et al. (2021). We also track potential false positives , which are stellar binaries that mimic true Unicorns by exhibiting similar mass and luminosity characteristics, such as the objects identified in El-Badry (2022), El-Badry et al. (2022), and El-Badry & Burdge (2022). Since false positives contain one star less luminous than the other, spectral disentangling should be needed to determine the presence of the 'dark' star. These binaries would require additional effort and spectral analysis to categorize. \nWe seek to ensure that our count of true Unicorns and false positives reflects what an observer would find when viewing our stellar population at an instantaneous moment within a given time bin. Therefore, the count of objects within each time bin is weighted by the amount of time that object spends as a true Unicorn or false positive. For example, if the time bin's length is 1 Gyr, and the binary forms a false positive for only 10 Myrs within that bin, we count the binary as 0.01 false positives for that time bin. If the time bin's length is less than the amount of time the binary spends as a false positive (3 Myrs, for example), we count the object as 1 false positive for that time bin. \nWe employ a number of criteria to define which objects fall within the true Unicorn and false positive object classes during each time interval. Within each interval, we remove any binary where the less luminous object has mass <1 . 4 M ⊙ , since we are primarily concerned with binaries that either contain a BH or NS (true Unicorns), or which contain a less luminous star that mimics the mass of a BH or NS (false positives). This means that some neutron stars are removed from our population, since there is a range of masses for which neutron stars overlap with white dwarfs. White dwarfs are the remnants of low-mass stars, with observed white dwarf masses ranging from 0 . 136 -0 . 162 M ⊙ (Kawka & Vennes 2009; Kaplan et al. 2014) to 1 . 327 -1 . 365 M ⊙ (Caiazzo et al. 2021). Neutron stars form following supernovae when the compact object has a minimum post-supernova mass ≥ 0 . 95 -1 . 29 M ⊙ (Strobel et al. 1999; Lattimer & Prakash 2004). While the minimum mass for a NS built of cold catalyzed matter can be much lower at ∼ 0.09 M ⊙ (Haensel et al. 2002), it is only possible to reach this size through the interaction of short-period NS-NS binaries (Yudin et al. 2020). \nBy excluding any binaries where the less luminous object has mass <1 . 4 M ⊙ , we avoid objects for which the classification between neutron star and white dwarf would be uncertain, as it is with some of the NS candidates reported in El-Badry et al. (2024a). Within our populations, ∼ 65-70% of low-luminosity NS fall below the mass cut and are removed. However, some low-mass neutron stars are still counted. ∼ 14-21% of NS with mass <1.4 M ⊙ are hot, recently-formed objects (<100 Myrs in age) with higher bolometric luminosity than their stellar companions. In these binaries, the NS is the 'bright' object in the SED while the stellar companion is the 'dark' object, allowing the binary to be counted as a true Unicorn in spite of the low NS mass. Within our simulations, we find no white dwarfs with masses higher than 1 . 39 M ⊙ . Throughout this study, we use 'compact objects' to refer to BH and NS only, since white dwarfs are not a focus of this work. \nWithin each time interval, we also remove any binary pair where the bright star has a radial velocity semi-amplitude K < 1 km/s. While many instruments are capable of detecting smaller values, 1 km/s represents a conservative lower limit of Gaia RV detection (Jordan 2008), with typical Gaia RV semi-amplitudes expected to range from several to hundreds of km/s (Chawla et al. 2022). By excluding objects with K < 1 km/s, we ensure that binaries used in our model would be detectable through large-scale surveys. \nWe also remove binaries with orbital periods >1826 days (5 years) within each time interval. Our COSMIC output generates stars with orbital periods far too long to be observed in the timescale of human civilization which nonetheless have an RV semiamplitude of 𝐾 ≥ 1 km/s. We seek to ensure that binaries in our sample have a short enough period that their RV semiamplitude can be calculated using a reasonable observing timeframe of a few years. Many of the star/compact object binaries discussed in Section 1 have periods below this limit, lending credence to our selection. 2MASS J05215658+435922 has an orbital period of 83.2 days (Thompson et al. 2019), Gaia BH1 has a period of 185.6 days (El-Badry et al. 2023a), Gaia NS1 has a period of 731 days (El-Badry et al. 2024b), and Gaia BH2 has a period of 1277 days (El-Badry et al. 2023b). The 21 neutron star candidates proposed in El-Badry et al. (2024a) have periods ranging from 189-1046 days. The high-mass Gaia BH3 is an outlier with an 11.6-year orbit (Gaia Collaboration et al. 2024). Most objects of interest are expected to have periods on order of a few hundred days, as there is an observed lower incidence of binary BH companions in Gaia data with periods of 400-1000 days (El-Badry et al. 2023a). \nWithin each time interval, we also exclude any binary pairs which have formed an active x-ray binary (LMXB, IMXB, or HMXB), since we are only interested in tracking the number of compact objects whose presence is not clearly shown through x-ray radiation at a given time. To identify x-ray binaries within COSMIC , we measure the change in mass of each BH and NS between each time interval ( / 𝑀 CO ), and use this to calculate the mass-accretion rate and x-ray luminosity using the equations in Section 2.3. \nFinally, we exclude all binaries that have either merged or been disrupted within each time interval, as noted by COSMIC .", '2.5 True and False Positive Definitions': 'We define true Unicorns as a NS or BH orbited by a MS, RGB, or NHS (in COSMIC , a naked helium star is defined as an evolved star severely stripped by mass loss during core helium burning, resulting in exposure of the nuclear processed material in the core (Hurley et al. 2000)). False positives are defined as a pair of stars (MS/MS, RGB/MS, RGB/RGB, NHS/MS, or NHS/RGB) which mimic true \nTable 3. Binary pairings which constitute false positives and true Unicorns. Acronyms refer to: BH=Black Hole, MS=Main Sequence Star, NHS=Naked Helium Star, NS=Neutron Star, RGB=Red Giant. \nUnicorns in mass and luminosity, such as those objects characterized in El-Badry (2022), El-Badry et al. (2022), and El-Badry & Burdge (2022). \nWe examine the numbers of true Unicorns and false positives within our population under two different definitions: \n- · Criteria 1: The object with lower luminosity has a higher mass than the object with greater luminosity. The \'dark\' object must have a mass of at least 1 . 4 M ⊙ , ensuring that it can only be a BH, NS, or a star mimicking the mass of a BH or NS (see Section 2.4.)\n- · Criteria 2: The dark object can be either more or less massive than the bright object, but must still have a mass of at least 1 . 4 M ⊙ . \nFor binaries that meet the false positive definition under Criteria 1, we can expect that a significant degree of stellar evolution has taken place. Unless the more massive companion is a main sequence star experiencing high mass loss or mass transfer, it is likely to be in its post-main sequence stage since it is less luminous. We also analyze our population under the Criteria 2 definition to ensure that we are not under-counting the numbers of true Unicorns and false positives, as binaries may contain a dim star or compact object which is less massive than its bright companion. An example of how these criteria are used to define an object at different times is discussed in Section 3.5. Together, these two criteria set a lower and higher limit on the probability of potential false positives within a population. Table 3 summarizes the binary types which we define as false positives and true Unicorns. \nWe also examine the numbers of true Unicorns and false positives under five different luminosity ratios: \n- · L Dark <L Bright : The dark object is less luminous than the bright object by any amount.\n- · L Dark <1.5 L Bright : Dark object is at least 1.5 times less luminous.\n- · L Dark <3 L Bright : Dark object is at least 3 times less luminous.\n- · L Dark <5 L Bright : Dark object is at least 5 times less luminous.\n- · L Dark <10 L Bright : Dark object is at least 10 times less luminous. \nSED analysis of the false positive V723 Mon shows that the subgiant companion has ∼ 2/3 the luminosity of the giant primary star (El-Badry 2022). We can reasonably assume that a binary with this luminosity ratio or greater would require spectral disentangling to identify the \'dark\' companion. By considering different luminosity ratios, we place higher and lower limits on the probability of false positives based on how much less luminous we define the \'dark\' companion to be. \nOur population of interest is the number of true Unicorns and false positives found within each time interval, with the count weighted to reflect what an observer would instantaneously find at that time (see Section 2.4). We define a "false positive probability" as the percentage of binaries which might appear to contain a compact object candidate but are in fact made up of two stars. We define a \nTable 4. Binaries within our synthetic population similar to V723 Mon, as evolved under a Kroupa IMF (Kroupa et al. 1993), and based on the mass and temperature estimates in (El-Badry 2022). This shows that our simulations produce objects similar to a known false positive, although they do not reproduce V723 Mon exactly. \n"true Unicorn probability" as the percentage of binaries which might appear to be contain a candidate and do actually contain a BH or NS.', '3.1 Kroupa Initial Mass Function': "Figure 1 shows the results of our stellar evolution models for the population formed under a Kroupa IMF (Kroupa et al. 1993; Salpeter 1955), giving the probability that objects of interest are either a true Unicorn or false positive during different time intervals. Under Criteria 1, in which we assume the dark object to be greater in mass than the bright object (Figure 1, bottom left), we observe that binaries begin evolving into true Unicorns as early as 5 Myrs after cluster formation. The total false positive probability is high at early ages, remaining >90% until 2.4 Gyrs. By 3.8 Gyrs, the false positive probability falls to near zero, which could in part be explained by a deficit of massive stars at late ages. Of the 5 × 10 5 binaries in our population, 385 (0 . 08 ± 0 . 01%) form a true Unicorn at some point in their evolution, while 34,019 (6 . 80 ± 0 . 04%) form a false positive. \nUnder Criteria 1, we observe that ∼ 77% of the time a false positive is identified within various time intervals, the object is a NHS/MS binary with the NHS being the more luminous star. Such binaries have been proposed as explanations for several BH candidates, including LB-1 (Liu et al. 2019; Irrgang et al. 2020) and NGC1850 (El-Badry & Burdge 2022). The next most common false positives are RGB/RGB binaries ( ∼ 13%), and RGB/MS binaries with the RGB the more luminous star ( ∼ 9%). \nBinaries similar to V723 Mon, with an RGB undergoing severe stripping (mass <0.5 M ⊙ ) and a less luminous subgiant companion (T eff between 5000-6000 K), make up <1% of false positive detections. While our simulations do not reproduce an exact replica of V723 Mon, they produce binaries with masses and temperatures similar to those calculated in El-Badry (2022), confirming COSMIC 's ability to reasonably reproduce known false positives. Table 4 compares two of our synthetic binaries to V723 Mon. \nFigure 2 shows the true Unicorn probabilities as defined under Criteria 1 for MS, RGB, and NHS, using each of the different IMFs and luminosity ratios in this study. The colored region above and below each line indicates the calculated level of uncertainty. Under the most basic luminosity ratio of L Dark <L Bright (Figure 2, upper left), the likelihood of observing an RGB true Unicorn remains low for most ages, rising briefly to 94.6% at 3.0 Gyrs (uncertainty is ∼ 100% due to small number statistics). NHS true Unicorns are also unlikely, with probability <2% for all time intervals. For MS stars, the true Unicorn probability rises to 85 . 7 ± 11 . 1% at 9 Myrs, and remains between 80-100% ( ± 4 -13%) from 11-49 Myrs. Probability then falls gradually from 67 . 0 ± 3 . 5%at 62 Myrs to 6 . 6 ± 3 . 1%at 1.1 Gyrs, then rises to 78 . 7 ± 3 . 4% at 1.5 Gyrs and 100 ± 16 . 1% at 1.9 Gyrs. After this age, probability remains 100%, but uncertainty is \nalso ∼ 100% due to small number statistics. At 13 Gyrs, uncertainties decrease slightly for RGB ( ± 52.0%) and MS stars ( ± 73.2%), due in part to the larger size of this final logarithmic time bin. \nAs we consider progressively stricter luminosity ratios (Figure 2, left column), changing our definition of how the 'dark' object is defined in each binary, the number of true Unicorns decreases only slightly with each change. This is due to hot, recently-formed neutron stars (<100 Myrs in age) close to the luminosity of their small stellar companions being removed as the luminosity ratio changes. The number of binaries forming a true Unicorn at some point in their evolution dips to 382 for L Dark <3 L Bright and 380 for L Dark <10 L Bright . The number of false positives decreases more rapidly, driving the overall change in true Unicorn probability with each different luminosity ratio. The total number of binaries forming a false positive at some point in their evolution falls to 27,516 (5 . 50 ± 0 . 03%) for L Dark <3 L Bright , and 24,103 (4 . 82 ± 0 . 03%) for L Dark <10 L Bright . \nUnder different luminosity ratios, MS true Unicorns become more likely. Defining the 'dark' companion under L Dark <1.5 L Bright , MStrue Unicorn probability rises to 100 . 0 ± 24 . 2%at 5 Myrs, dips to 75 . 0 ± 20 . 1% at 7 Myrs, then rises back to ∼ 100% ( ± 3 -17%) from 9-79 Myrs, 129-705 Myrs, and 1.5-2.4 Gyrs. Under L Dark <3 L Bright , the MS true Unicorn probability is 100 . 0 ± 31 . 7% at 5 Myrs, and remains at this level ( ± 4-34%) until 3.0 Gyrs, where small-number statistics drive uncertainties upward. For binaries with a bright RGB, the true Unicorn probability changes little with different luminosity ratios. For binaries with a bright NHS, we see only one area where probability increases with changing luminosity ratios, from 434-553 Myrs. \nUnder Criteria 2, in which we assume the dark object is more massive than 1 . 4 M ⊙ but may be more or less massive than the bright star, we observe a higher number of true Unicorns and a much higher number of false positives compared to Criteria 1. 700 binaries (0 . 14 ± 0 . 01%) form true Unicorns at some point in their evolution, while 65,644 (13 . 1 ± 0 . 1%) form false positives. Both categories are roughly doubled from Criteria 1. \nIn Figure 1 (bottom right), we see that the total false positive probability under Criteria 2 remains above 99% until 2.4 Gyrs, then decreases to near-zero at 3.0 Gyrs. ∼ 89% of the time a false positive is identified, the object is a MS/MS binary. This is followed by RGB/MS binaries, with the RGB the more luminous star ( ∼ 7%), and NHS/MS binaries, with the NHS the more luminous star ( ∼ 3%.) \nFigure 3 shows the true Unicorn probabilities as defined under Criteria 2 for MS, RGB, and NHS, using each of the different IMFs and two of the luminosity ratios in this study (the others are qualitatively similar, and are not shown). As before, the colored region above and below each line indicates the calculated uncertainty. The RGB true Unicorn probability (Figure 3, upper left) remains low for most ages, climbing to ∼ 95% at 3.0 Gyrs, where uncertainty is again ∼ 100% due to small number statistics. At 13 Gyrs, uncertainty dips as it did under Criteria 1. NHS true Unicorn probability also remains low, with a peak of ∼ 5 . 0 ± 0 . 5% from 4-7 Myrs. MS true Unicorn probability is much lower than under Criteria 1, remaining below 1% until 1.5 Gyrs. It then climbs to 100 . 0 ± 8 . 2% at 1.9 Gyrs, remaining high through 2.3 Gyrs (100 . 0 ± 63 . 0%). At 3.0 Gyrs uncertainty again reaches ∼ 100%. We observe that 'dark' objects are considerably more likely to be a BH or NS if they meet the Criteria 1 definition of being more massive than the bright MS star. While this makes intuitive sense, the level of difference between probabilities under Criteria 1 and Criteria 2 is striking. \nAs we compare different luminosity ratios under Criteria 2 (Figure 3, left column), we observe very small increases in RGB and \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 1. Results of stellar binary evolution modelling under a Kroupa IMF, focusing on the probabilities of false positives (red shades) and true Unicorns (blue shades) within the population. Top left: shows the rate at which these objects appear at different time intervals as a percentage of the overall population of 5 × 10 5 binaries. This plot utilizes Criteria 1, wherein we require the dark object to be more massive than the bright object, and assumes L Dark <L Bright by any amount. Bottom left: provides the false positive and true Unicorn probabilities within our population of interest under Criteria 1. This gives the likelihood that any given binary with a bright star and a more massive dark object greater than 1 . 4 M ⊙ forms a true Unicorn. Top right: shows the rate at which these objects appear as a percentage of overall population under Criteria 2, which is a looser criteria wherein we do not require the dark object to be more massive than the bright object (note the change in y-axis scale from Top left plot). Bottom right: shows the true Unicorn and false positive rates under Criteria 2. Acronyms refer to: BH=Black Hole, MS=Main Sequence Star, NHS=Naked Helium Star, NS=Neutron Star, RGB=Red Giant. \n<!-- image --> \nNHS true Unicorn probabilities. MS true Unicorn probability under L Dark <10 L Bright increases by <1% compared to L Dark <L Bright until 899 Myrs, when it reaches 11 . 0 ± 1 . 8%, then 70 . 0 ± 1 . 8% at 1.1 Gyrs and 100% from 1.5 to 3.0 Gyrs ( ± 2-10%). Although the MS true Unicorn probability is low under Criteria 2, there is a window of time from ∼ 1-3 Gyrs where true Unicorns are more likely if the 'dark' object is confirmed to be significantly less luminous than the bright star.", '3.2 Hosek Initial Mass Function': 'Within our population produced under the IMF described by Hosek (2019), the total number of both true Unicorns and false positives is higher than under a Kroupa IMF due to the greater fraction of massive stars. Under Criteria 1, we find that 776 objects (0 . 16 ± 0 . 01%) form true Unicorns and 47,781 (9 . 56 ± 0 . 04%) form false positives at some \npoint in their evolution. Under Criteria 2, we find that 4102 objects (0 . 82 ± 0 . 01%) form true Unicorns and 90,476 (17 . 1 ± 0 . 1%) form false positives. A plot of the total true Unicorn and false positive probabilities within each time bin can be found in Appendix B. Under Criteria 1 (Figure B1, bottom left), the total false positive probability remains ≥ 89% until 1.9 Gyrs, where it declines to 66%. By 3.0 Gyrs the false positive rate drops to near-zero (small-number statistics remain an issue at this age, although less so than under a Kroupa IMF). Under Criteria 2 (Figure B1, bottom right), the total false positive probability remains ≥ 98% until 1.9 Gyrs, where it declines to 83%. By 3.0 Gyrs it drops to near-zero. \nUnder Criteria 1 and the basic luminosity ratio of L Dark < L Bright , the RGB true Unicorn probability (Figure 2, upper middle) is only ∼ 1%higher than under a Kroupa IMF until 1.9 Gyrs, where it rises to 14 . 3 ± 23 . 2%, then to 100 . 0 ± 52 . 2% at 3.9 Gyrs. NHS true Unicorn probability is also ∼ 1% higher than under a Kroupa IMF,', '8 Miller et al.': "Figure 2. Results of stellar binary evolution modelling, providing the likelihood that an object of interest will be a true Unicorn at different times. True Unicorns are defined under Criteria 1, wherein we require the dark object to be more massive than the bright object. The MS true Unicorn Probability is in orange, RGB true Unicorn probability is in blue, and NHS true Unicorn probability is in gray. The colored regions above and below each line indicate the uncertainty. Column 1 : binaries are formed under a Kroupa IMF. Column 2 : binaries formed under a Hosek IMF. Column 3 : binaries formed under a Hosek IMF with metallicity of 1.4 Z ⊙ . Row 1 : the dark companion in the binary can be less luminous than the bright star by any amount (L Dark <L Bright ). Rows 2, 3, 4, and 5 : L Dark <1.5 L Bright , L Dark <3 L Bright , L Dark <5 L Bright , L Dark <10 L Bright , respectively. \n<!-- image --> \nFigure 3. Results of stellar binary evolution modelling, providing the likelihood that an object of interest will be a true Unicorn at different times. The MS true Unicorn Probability is in orange, RGB true Unicorn probability is in blue, and NHS true Unicorn probability is in gray. The colored regions above and below each line indicate the uncertainty. True Unicorns are defined under Criteria 2, wherein we allow the dark object to be either more or less massive than the bright star. Column 1: binaries are formed under a Kroupa IMF. Column 2: binaries formed under a Hosek IMF. Column 3: binaries formed under a Hosek IMF with metallicity of 1.4 Z ⊙ . Row 1: the dark companion in the binary can be less luminous than the bright star by any amount (L Dark <L Bright ). Row 2: the dark companion has < 10 times the luminosity of the bright star (L Dark <10 L Bright ). Additional luminosity ratios are not shown here, as they are qualitatively similar to those shown. \n<!-- image --> \nrising to 88 . 6 ± 38 . 4% at 2.4 Gyrs, and declining to near-zero at 3.0 Gyrs. At early ages, MS true Unicorn probability does not follow an obvious trend compared to the Kroupa IMF, becoming 50% more likely at 4 Myrs, 30% less likely at 5 Myrs, 35% more likely at 7 Myrs, and so on. A trend arises in the 62 Myr to 1.5 Gyr timeframe, where MS true Unicorn probability is 30-60% higher than a Kroupa IMF, reaching 97 . 4 ± 5 . 6% at 1.5 Gyrs and 100 . 0 ± 27 . 6% at 1.9 Gyrs. While uncertainties beyond this age are still high ( ∼ 50-80%), they are lower than under a Kroupa IMF. \nUnder Criteria 2, the RGB true Unicorn probability (Figure 3, upper middle) is a few percent higher than under a Kroupa IMF from 5-24 Myrs. At 3.0 Gyrs, it rises to 97 . 2 ± 47 . 5%, then averages ∼ 100 ± 56% afterward. NHS true Unicorn probability is also a few percent higher from 4-9 Myrs, peaking at 2.4 Gyrs (88 . 6 ± 37 . 1%) before falling to near-zero. MS true Unicorn probability increases by ∼ 1%from 5-9 Myrs, then peaks at 1.5 Gyrs (27 . 2 ± 2 . 8%). As before, probability is ∼ 100% from 1.9 Gyrs on, with lower uncertainties than under a Kroupa IMF (averaging ± 70%). \nComparing different luminosity ratios, as we advance from L Dark <L Bright to L Dark <10 L Bright under Criteria 1 (Figure 2, middle column), we observe increases of a few percent in the RGB true Unicorn probability at early times, with significant decreases at middle ages. Under Criteria 2 (Figure 3, middle column), we observe a 1-8% increase from 5-30 Myrs and greater decreases from 1.9-3.0 Gyrs. The NHS true Unicorn probability under Criteria 1 rises by a few percent as we advance from L Dark <L Bright to L Dark <10 L Bright (5-25% from 209 Myrs-1.5 Gyrs). Under Criteria 2, NHS true Unicorn probability increases by 4-14% during the 4-9 Myr \ntimeframe, but decreases significantly at 2.4 Gyrs. For MS stars under L Dark <10 L Bright , the MS true Unicorn probabililty is ∼ 100% for early ages under Criteria 1, with uncertainties of ≤ 20%. After 3.0 Gyrs, the average uncertainty is ± 77%. Under Criteria 2, we again find a window where probabililty increases but uncertainties are still low. At 899 Myrs, probability reaches 55 . 8 ± 2 . 6%, then ∼ 100% from 1.1 to 3.0 Gyrs ( ± 8% on average) . During this time period, we observe a high MS true Unicorn probability when the 'dark' object is much less luminous than the bright star. \nWe observe that in general, populations formed under a Hosek IMF are more likely to produce MS stars orbiting a compact object than a Kroupa IMF, particularly when the 'dark' object is more massive than the bright star. This is a reasonable expectation, as a topheavy IMF produces a greater number of massive stars, and therefore a greater number of compact object companions. At its current age, 3 . 7 ± 0 . 2 Myrs (Hosek 2019; Gallego-Calvente et al. 2021), the Arches Cluster should already be forming a small population of these binaries, with a MS true Unicorn probability of 52 . 3 ± 4 . 0% compared to ∼ 0% under a Kroupa IMF. Top-heavy-IMF star clusters represent a potentially productive environment to search for noninteracting compact object binaries. However, this is only taking into account the IMF and ignoring other differences between stellar populations.", '3.3 Hosek Initial Mass Function with Super-Solar Metallicity': "For the stellar populations discussed in Sections 3.1 and 3.2, we have used solar metallicity for all stars (Asplund et al. 2009). However, \nobservation show that the Arches Cluster, upon which the Hosek IMF is based, may exhibit a super-solar metallicity of 1.3-1.4 Z ⊙ (Martins et al. 2008; Sabhahit et al. 2022). To more realistically model the Arches Cluster, which we use to represent young star clusters in general, we evolve an additional population of 5 × 10 5 binaries in COSMIC using the Hosek (2019) IMF and a higher metallicity value of 𝑍 = 0 . 01876. \nUnder Criteria 1 (Figure 2, upper right), we observe decreases of ∼ 1% in the RGB true Unicorn probability compared to the 1 Z ⊙ Hosek population. At 1.9 Gyrs, probability drops to zero. At 3.9 Gyrs, probability rises to 100 . 0 ± 63 . 6%. The NHS true Unicorn probability also sees a small decrease, then drops to zero at 2.4 Gyrs. We observe a decrease in the MS true Unicorn probability of 40% at 4 Myrs, then a smaller decrease of ∼ 3-10% from 5-705 Myrs and 15-25% from 899 Myrs-1.5 Gyrs. At 1.9 Gyrs, probability reaches 100 . 0 ± 11 . 7%, after which it remains ∼ 100% ( ± 93% on average). \nUnder Criteria 2 (Figure 3, upper right), we observe decreases of a few percent in the RGB true Unicorn probability compared to the 1 Z ⊙ population, with a drop to zero at 2.4-3.0 Gyrs. Probability remains ∼ 100% from 3.9 Gyrs onward ( ± 77%on average). The NHS true Unicorn probability is low for most ages ( ≤ 1%). We observe small decreases in MS true Unicorn probability compared to the 1 Z ⊙ population, with larger decreases of ∼ 15% at 1.5 Gyrs and ∼ 35% at 1.9 Gyrs. From 2.4 Gyrs onward, probability remains ∼ 100% ( ± 93% on average). \nThese decreases in true Unicorn probability agree with expectation. Under a metallicity slightly higher than solar value, BH formation will slow as massive stars experience high mass loss, causing them to collapse into neutron stars (Heger et al. 2003). For stellar populations formed under a super-solar metallicity, any true Unicorn is more likely to contain a NS than a BH. We observe an ∼ 11% increase in the number of NS under 1.4 Z ⊙ . \nTheeffectsofachangeinluminosity ratio have been discussed in Sections 3.1 and 3.2, and are qualitatively similar for this population (Figures 2 and 3, right columns). For binaries defined under L Dark <10 L Bright , we still observe a significant rise in MS true Unicorn probability from 1.1-3.9 Gyrs, with relatively low uncertainties ( ± 14% on average). For MS stars of this age, false positives are unlikely to be found among 'dark' objects significantly less luminous than their bright companions. \nWhile a super-solar metallicity does decrease the likelihood of the Arches Cluster forming true Unicorns at its current age, 3 . 7 ± 0 . 2 Myrs (Hosek 2019; Gallego-Calvente et al. 2021), we still find a 9 . 6 ± 3 . 8% MS true Unicorn probability at 4 Myrs under Criteria 1 (requiring that the companion is more massive than the MS star). This increases to 100 . 0 ± 6 . 5% if the companion is confirmed to be at least 1.5 times less luminous than the bright star. The Arches Cluster and any similar young clusters with top-heavy IMFs should be relatively productive environments for locating true Unicorns, even under higher metallicity.", '3.4 Orbital Characteristics': "Weseek to examine whether true Unicorns might have unique orbital parameters distinct from false positives. Figure 4 (top plot) shows the orbital period vs. semi-major axis at various times (provided by COSMIC ) of the binaries evolved under a Kroupa IMF which we identify as true Unicorns and false positives in our model. Figure 4 (bottom plot) shows the orbital period provided by COSMIC vs. the radial velocity semi-amplitude (as defined in Section 2.2) of the more luminous object in each pair. These plots use the definition of Criteria 2, where the dark object can be either more or less massive \n<!-- image --> \nFigure 4. Orbital characteristics of stars in the synthetic binary population formed under a Kroupa IMF. Top plot: Orbital period in days vs. semi-major axis in astronomical units (AU) for all objects which we identify as true Unicorns or false positives under Criteria 2. Colored points indicate known objects with compact object companions, described in Section 1. Bottom plot: Orbital period in days vs. radial velocity semi-amplitude in km/s for false positives and true Unicorns. \n<!-- image --> \nthan the bright star, and L Dark <L Bright , where the dark star can be less luminous than the bright star by any amount. At more restrictive luminosity ratios, the number of false positives decreases overall, but the distributions remain similar. Also included are the orbital characteristics of known objects (see Section 1), including 2MASS J05215658+435922 (Thompson et al. 2019), Gaia BH1 (El-Badry et al. 2023a), Gaia BH2 (El-Badry et al. 2023b), Gaia NS1 (El-Badry et al. 2024b), and the NS candidates discovered in El-Badry et al. (2024a). \nComparing orbital period vs. semimajor axis (Figure 4, top plot), we observe that most true Unicorns are clustered in a lowperiod regime. True Unicorns form an orbital period >400 days only 11 . 8 ± 1 . 0%of the time, and >100 days only 26 . 4 ± 1 . 4%of the time. Areview of Gaia data shows that compact object companions should be found more often at shorter periods (El-Badry et al. 2023a), and \nour models agree with this. Under Criteria 1, only 13 . 2 ± 1 . 8% of true Unicorns have a period >100 days, indicating that periods are even shorter when the dark object is more massive than the bright star. \nWe also observe that for most true Unicorns with a period equal to that of a false positive, the true Unicorn has a wider orbital separation. This again matches expectation. NS and BH masses tend to be higher than the average star, and Kepler's third law stipulates that more massive objects in a binary will produce a wider orbit at a given period (El-Badry et al. 2023a). Age-related rotational slowing (Barnes 2003) can also play a part in widening orbits, as can supernova kicks. In Figure 4 (top plot), several of the discovered BH and NS binaries lie outside the true Unicorn parameter space of the plot, however this is simply due to the high starting mass of our synthetic population (each of our binaries contains a primary star with initial mass ≥ 2 M ⊙ ). If we were to plot the stars within each object's local population, we would see their orbits as being wider than the stars around them. \nComparing orbital period vs. RV semi-amplitude (Figure 4, bottom plot), we find that true Unicorns do not appear to follow a particular trend compared to false positives. However, true Unicorns do become less common at orbits <10 -1 days (since small periods are likely dominated by x-ray binaries), and at radial velocities >400 km/s. Many of the actual objects discovered fit within the true Unicorn parameter space. \nUnder a Hosek IMF (both 1 Z ⊙ and 1.4 Z ⊙ metallicities), we observe qualitatively similar period vs. semi-major axis and period vs. semi-amplitude distributions. The only significant difference is a higher overall number of true Unicorns and false positives.", '3.5 Example Evolution': "Figure 5 provides an example of the stellar evolution of a single binary under our model. The purpose of this plot is to illustrate how the categories we define for each object of interest (true Unicorn, false positive, x-ray binary) are defined at a given time, and do not apply to the entire lifetime of the binary. The evolution of a binary pair may carry it through many stages over its lifetime. Masses, radii, luminosities, and Roche lobe filling fractions are given for both stars, as well as the orbital period. This binary is somewhat atypical within our populations given its high initial mass, however it illustrates the various states of our model well. \nCOSMIC tracks binary evolution in two ways: 1) by sampling the binary characteristics each time a key evolutionary change occurs; and 2) by sampling the binary parameters at a series of user-defined time steps, which we have set to 1 Myr for this particular binary. As before, we combine the time step output with the output from evolutionary changes to ensure that no significant developments were missed. \nThe binary begins as a MS/MS pair, then evolves to a RGB/MS pair at 4.145 Myrs, which we define as a false positive under Criteria 2 (a bright RGB with a darker, less massive MS companion). The pair then evolves into a NHS/MS binary at 4.151 Myrs, which meets the definition of a false positive under Criteria 1 (a bright MS star and a darker, more massive NHS companion). At 4.719 Myrs, the NHS has lost mass, and the object returns to a false positive definition under Criteria 2 (a bright MS star with a darker, less massive NHS.) At 4.748 Myrs, the pair evolves into a true Unicorn as defined under Criteria 2 (a BH/MS pair with the compact object less massive than the bright star). \nAt 14 Myrs, the MS star (with a current mass of 11.8 M ⊙ ) fills >90% of its Roche lobe, and the BH forms an accretion disk with \nFigure 5. Representative example of the evolution of a stellar binary pair from our synthetic population. The object is sampled at 1 Myr timesteps and at any key changes in stellar evolution. Mass (M ⊙ ), radius (R ⊙ ), visible luminosity (L ⊙ ), period (days), and Roche lobe filling fraction (R/R Roche ) are given for both stars. Initial mass of the primary star is 56.5 M ⊙ , and initial mass of the secondary is 11.8 M ⊙ . Initial orbital period is 3516 days. This plot demonstrates how a binary can change definition over its lifetime. \n<!-- image --> \nan x-ray luminosity >10 35 erg s -1 . This creates a high-mass x-ray binary (HMXB). The BH is now detectable through x-ray emission, and no longer meets the conditions of a non-interacting true Unicorn. As the BH accretes mass, the MS star shrinks to 4.2 M ⊙ at 18 Myrs, changing from a HMXB to an IMXB (intermediate-mass x-ray binary). At 31.9 Myrs, the bright star evolves onto the red giant branch, and accretion briefly slows. The binary becomes a true Unicorn under Criteria 1 (a bright RGB star and a more massive non-interacting BH). \nAt 32.1 Myrs, the RGB overflows its Roche lobe, and the binary becomes an IMXB once again. The orbital period also begins to expand. At 36 Myrs, the RGB's mass falls below 1 M ⊙ , and classification changes to a low mass x-ray binary (LMXB). At 56.3 Myrs, the bright RGB star has lost sufficient mass to form a NHS. The mass loss rate slows, and at 57 Myrs, the x-ray luminosity of the BH accretion disk drops to 3.8 × 10 31 erg s -1 . The binary is no longer detectable through x-ray emission. It is again considered a true Unicorn under Criteria 1 (a NHS with a more massive, non-interacting BH companion). \nAt 165.9 Myrs, the NHS condenses into a white dwarf, and is no longer an object of interest for our model.", '3.6 X-Ray Binary Evolution': "The focus of our study is to identify the number of binary systems where object classification is uncertain and it is difficult to differentiate between a compact object and a massive, low-luminosity star. Since x-ray binaries provide direct evidence of compact objects through the presence of an accretion disk and detectable x-ray radiation, we have removed interacting x-ray binaries from our population within each time interval (see Section 2.3). However, many true Unicorns do form x-ray binaries at some point during their evolution. In relatively short orbits, periods of mass transfer are likely to occur. In a typical star-forming galaxy, we expect that ≥ 10% of compact objects form high-mass x-ray sources at least once during their lifetimes (Mineo et al. 2012; Gilfanov et al. 2022). Within our more limited population of true Unicorns (with their defined orbital and mass constraints), we seek to determine whether compact objects form x-ray binaries at a similar rate. \nWithin our population of 5 × 10 5 binaries formed under a Kroupa IMF, 1265 objects form either LMXBs, IMXBs, and/or HMXBs at some point in their evolution. Of these, 191 also evolve into true Unicorns under Criteria 1 (424 under Criteria 2's definition). Under a Hosek IMF, 8924 objects form x-ray binaries at some point in their evolution, 425 of which also evolve into true Unicorns under Criteria 1 (3238 under Criteria 2). \nComparing to the totals of true Unicorns in Sections 3.1 and 3.2, we observe that 49 . 6 ± 3 . 6% of true Unicorns formed under a Kroupa IMF and defined under Criteria 1 evolve into x-ray binaries (60 . 6 ± 2 . 9% under Criteria 2). Of these, ∼ 2% under Criteria 1 form HMXBs ( ∼ 33% under Criteria 2). Under a Hosek IMF, 54 . 8 ± 2 . 7% of true Unicorns defined under Criteria 1 evolve into x-ray binaries (78 . 9 ± 1 . 4% under Criteria 2). Of these, ∼ 4% under Criteria 1 form HMXBs ( ∼ 69% under Criteria 2). \nAs a subset of the overall compact object population, around half of true Unicorns under Criteria 1 (the compact object is more massive than the bright star) form XRBs at some point in their evolution, though the rate at which they form HMXBs is below average. Given that more massive objects in a binary will produce wider orbits (El-Badry et al. 2024a), it is reasonable to expect a lower level of accretion for these pairs. True Unicorns defined under Criteria 2 (the compact object is either more or less massive than the \nbright star) account for a higher number of x-ray emitting pairs, and are also much more likely to form HMXBs than typical black holes. \nIn Sections 3.1, 3.2, and 3.3, we have shown that binaries with a more massive compact object and a less massive stellar companion are more likely to be observed than binaries with a less massive compact object and a more massive stellar companion. Since these more massive compact objects are also less likely to emit detectable xrays, the importance of radial velocity measurements and astrometry in combination with spectroscopy is well demonstrated. Even under a top-heavy IMF, which favors the formation of massive stars and compact objects, ∼ 45% of BH and NS binaries in relatively close orbits will never be detectable through x-ray emission during a 13 Gyr lifetime.", '3.7 Assumptions and Caveats': 'We model all stars as having formed from a single starburst event. Star clusters such as the Arches cluster likely experience only one epoch of formation, after which the remaining gas is cleared by stellar winds (Krause et al. 2020). Within a broader galactic population, however, additional stars do form at later times following supernova events, which will likely add to the number of both false positives and true Unicorns over time. An exploration of this effect is beyond the scope of this work. Nevertheless, as long as the age of a bright star in a binary can be independently determined, our models provide a reasonable probability that the star is orbiting a non-interacting compact object. \nWe have also modelled our population as 5 × 10 5 binary pairs, disregarding the evolutionary impact of higher-order multiples like triples and quadruples. Generally, stellar multiplicity fractions increase with stellar primary mass, driving an increase in the number of triple stars, quadruple stars, and beyond (Raghavan et al. 2010). Additional multiplicity may introduce complex dynamics on a physical star population, such as Kozai-Lidov mechanisms (Lidov 1962; Kozai 1962; Naoz 2016), tidal interactions (e.g., Naoz & Fabrycky 2014), gravitational wave events (e.g., Hoang et al. 2018), and dynamically driven stellar mergers (e.g., Rose et al. 2023). \nOur sample of true Unicorns includes a significant population of binaries experiencing ongoing mass transfer. Our observational characterization of an "interacting binary" is that it produces x-rays. While x-ray binaries are the product of ongoing mass accretion, not all accreting compact objects will emit detectable x-rays. Indeed, for the Unicorn, Jayasinghe et al. (2021) proposed an accretion disc around the black hole, despite no detected x-rays. The disc was used to explain a relative dilution of blue wavelengths in the SED. While we have removed x-ray binaries from our data within each time interval, some binary pairs in our data may be interacting in other ways, some of which may produce radiation in other wavelengths. \nMassive stars are also capable of producing x-ray emission through means other than accreting onto a compact object (Rauw et al. 2015), some of which can produce a high enough x-ray luminosity to rival x-ray binaries. These processes include hydro-dynamic shocks produced by instability in the stellar wind (e.g., Feldmeier et al. 1997), the collision of wind-driven magnetically channeled gas (e.g., Babel & Montmerle 1997; ud-Doula & Owocki 2002), and large-scale shocks from wind-wind interactions between large binary stars (Stevens et al. 1992). These processes are not modeled within our simulations, however they would provide another method through which massive false positive stars could be identified and differentiated from compact objects through their x-ray emission.', '4 CONCLUSIONS': 'To investigate the formation and evolution of compact objects, it is essential to locate them in non-interacting binaries, however this can be difficult to do observationally. In our models, we used COSMIC to generate synthetic populations of binary stars formed from both a Kroupa IMF (Kroupa et al. 1993) and a Hosek IMF (Hosek 2019). To effectively study compact object formation, we set the initial mass of the primary star in each binary to ≥ 2 M ⊙ . We counted the numbers of true Unicorns, defined as tightly-orbiting binaries containing a compact object more massive than 1 . 4 M ⊙ , and false positives, defined as tightly-orbiting binaries with one bright star and one dark star whose mass and luminosity mimic a compact object and would likely require spectral disentangling to characterize the binary. We define our "false positive probability" as the percentage of binaries which might appear to contain a compact object candidate but are in fact made up of two stars. We define our "true Unicorn probability" as the percentage of binaries which might appear to be contain a candidate and do actually contain a BH or NS. While other studies have examined the number of compact objects that should exist within the Galaxy, this work examines how often stellar binaries can mimic these compact objects. With this study, we make a number of predictions: \n- · Within a population of massive stars, main sequence stars have a significantly higher true Unicorn probability compared to red giants or naked helium stars, particularly at young ages. Under a Kroupa et al. (1993) IMF, if we require the dark companion to be more massive than the bright star (which we call Criteria 1), the true Unicorn probability is between 80-100% for stars aged 9-49 Myrs, falls from 67 . 0 ± 3 . 5% at 62 Myrs to 6 . 6 ± 3 . 1% at 1.1 Gyrs, then rises again to 78 . 7 ± 3 . 4% at 1.5 Gyrs and 100 ± 16 . 1% at 1.9 Gyrs. After this age, small number statistics make prediction less certain. In clusters formed under a top-heavy Hosek (2019) IMF, the MS true Unicorn probability is generally higher, and in particular for stars aged 62 Myr to 1.5 Gyr (probability increases by 30-60%). The false positive probability decreases to near-zero if we require the \'dark\' companion in each binary to be ≥ 3 times less luminous than the bright star. Therefore we can predict conditions under which a main sequence star with a non-interacting compact object companion is substantially more likely than a false positive: for a bright, young MS star with a less luminous companion (provided the companion has mass >1.4 M ⊙ and is more massive than the MS star, and the orbital period is ≤ 5 years) if the companion is confirmed to be at least 3 times less luminous than the MS star, there is an extremely high probability of the companion being a non-interacting compact object. No false positives were found orbiting young MS stars below this luminosity cutoff. This applies to the populations formed under a Hosek (2019) IMF as well.\n- · Under Criteria 2, in which we do not require the dark companion to be more massive than the bright star, the probability of locating a non-interacting compact object compared to a false positive is generally low. For MS stars, we see that only stars aged 1-2 Gyrs have a high true Unicorn probability (which expands to 900 Myrs-3 Gyrs if the companion is significantly less luminous). While compact objects can be less massive than their stellar companions, the probability of false positives is high for those objects.\n- · If we require the less luminous object in a binary to be more massive (Criteria 1), the most common type of false positives formed are NHS/MS binaries ( ∼ 77%), with the NHS (an evolved star severely stripped by mass loss, leaving the core exposed) being the more luminous object. If we do not require the less luminous object to \nbe more massive (Criteria 2), the most common false positives are MS/MS binaries ( ∼ 89%). \n- · A super-solar metallicity will reduce the number of black holes and true Unicorns in a binary population (Heger et al. 2003), reducing the MS true Unicorn probability by as much as 10-25% on average (for companions more massive than the MS star). This decrease becomes negligible when the \'dark\' companion is ≥ 3 times less luminous than the bright star.\n- · We observe that a young star cluster with a top-heavy IMF such as the Arches Cluster (modelled in this study) represents a more productive environment in searching for non-interacting compact objects. We find a 9 . 6 ± 3 . 8%MStrue Unicorn probability at 4 Myrs for an Arches Cluster analogue modelled under a top-heavy IMF and 1.4 Z ⊙ metallicity (provided the companion is more massive than the MS star), which increases to 100 . 0 ± 6 . 5% if the companion is confirmed to be at least 1.5 times less luminous than the bright star.\n- · In line with physical expectation, we find that a true Unicorn will tend to have a wider orbital separation than a false positive with the same period. We also find that for binaries with a non-interacting compact object companion more massive than the bright star, the orbital period will be <100 days ∼ 87% of the time, confirming that a significant fraction of these objects should be found in short periods. Binaries with long orbital periods are more likely to be false positives.\n- · A single binary pair can change definition between a true Unicorn and a false positive at different points in its lifetime. We provide an example of the evolutionary path of such a binary in Section 3.5.\n- · Asignificant fraction of true Unicorns, although non-interacting for most of their lives, will interact at some point and become detectable through x-rays (50-60% for a Kroupa IMF, 55-80% for a Hosek IMF). This stresses the importance of using a combination of radial velocity measurement, astrometry, and spectroscopy to locate these objects, given that roughly half of binaries with a bright star and a more massive BH/NS companion in relatively close orbits will never be detectable through x-ray emission during a 13 Gyr lifetime. \nIn the search for non-interacting binaries containing a black hole or neutron star, radial velocity measurements and astrometry combined with spectroscopy and SED analysis constitutes a promising detection method. Only a small number of such objects have been discovered to date, and false positives have complicated the process. However, our models show that while binaries with non-interacting compact objects are indeed rare, there are conditions under which the probability of false positives decreases significantly. If the age of the bright star can be calculated through other means, our models show that binaries of certain types, ages, and orbital parameters are much more likely to contain a true a non-interacting compact object than a star acting as an imposter.', '5 ACKNOWLEDGEMENTS': "We thank our anonymous reviewers for carefully reviewing the manuscript and providing valuable comments. A.M.M. thanks Mark Reynolds and Dominick Rowan for helpful advice and suggestions. A.P.S. thanks Katelyn Breivik for helpful discussions regarding the COSMIC code. A.M.M. acknowledges partial support from the Ohio State University Summer Undergraduate Research Program (SURP) in Astrophysics. A.P.S. acknowledges partial support from the President's Postdoctoral Scholarship by the Ohio State University and the Ohio Eminent Scholar Endowment. This research made use of COSMIC (Breivik et al. 2020), Matplotlib (Hunter 2007), Numpy \n(van der Walt et al. 2011), Pandas (McKinney 2010), and Scipy (Virtanen et al. 2020).", '6 DATA AVAILABILITY': 'The output from all of the COSMIC simulations can be provided upon request to the corresponding author.', 'APPENDIX A: ADDITIONAL COSMIC INPUT PARAMETERS': "COSMIC is a binary population synthesis and evolution code based \nTable A1. General input physics parameters for COSMIC version 3 . 4 . 0, as detailed in Breivik et al. (2020). For our study we have used COSMIC 's default values, which appear in this table. \non the older BSE code (Hurley et al. 2002). It greatly expands upon the functionalities of BSE by including more detailed stellar and binary interaction processes, and allowing for the evolution of more massive stars. \nFor this work we used COSMIC version 3 . 4 . 0 to evolve the synthetic binary pairs within our stellar populations. The general input physics parameters for COSMIC are outlined in Breivik et al. (2020). For our study we have used COSMIC 's default values, which we have listed in Table A1.", 'APPENDIX B: EVOLUTIONARY RESULTS FOR HOSEK IMF POPULATION': 'Figure B1 shows the results of our stellar evolution models for the population evolved under a Hosek IMF (Hosek 2019). This gives the likelihood that different objects of interest are either a true Unicorn or false positive at different time intervals. We define true Unicorns as a NS or BH orbited by a MS, RGB, or NHS. False positives are defined as a pair of stars (MS/MS, RGB/MS, RGB/RGB, NHS/MS, or NHS/RGB) which mimic true Unicorns in mass and luminosity as did the binaries observed in Jayasinghe et al. (2021) and Jayasinghe et al. (2022). \nFigure B1. Results of stellar binary evolution modelling under a Hosek IMF, focusing on the probabilities of false positives (red shades) and true Unicorns (blue shades) within the population. Top left: shows the rate at which these objects appear at different time intervals as a percentage of the overall population of 5 × 10 5 binaries. This plot utilizes Criteria 1, wherein we require the dark object to be more massive than the bright object, and assumes L Dark <L Bright by any amount. Bottom left: provides the false positive and true Unicorn probabilities within our population of interest under Criteria 1. This gives the likelihood that any given binary with a bright star and a more massive dark object greater than 1 . 242 M ⊙ forms a true Unicorn. Top right: shows the rate at which these objects appear as a percentage of overall population under Criteria 2, which is a looser criteria wherein we do not require the dark object to be more massive than the bright object. Bottom right: shows the true Unicorn and false positive rates under Criteria 2. Acronyms refer to: BH=Black Hole, MS=Main Sequence Star, NHS=Naked Helium Star, NS=Neutron Star, RGB=Red Giant. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nTime (Myrs) \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \n838'}
2024A&A...682A..34M	The eROSITA telescope array aboard the Spektrum Roentgen Gamma SRG satellite began surveying the sky in December 2019 with the aim of producing allsky Xray source lists and sky maps of an unprecedented depth. Here we present catalogues of both pointlike and extended sources using the data acquired in the first six months of survey operations eRASS1 completed June 2020 over the half sky whose proprietary data rights lie with the German eROSITA Consortium. We describe the observation process the data analysis pipelines and the characteristics of the Xray sources. With nearly 930 000 entries detected in the most sensitive 0.22.3 keV energy range the eRASS1 main catalogue presented here increases the number of known Xray sources in the published literature by more than 60 and provides a comprehensive inventory of all classes of Xray celestial objects covering a wide range of physical processes. A smaller catalogue of 5466 sources detected in the less sensitive but harder 2.35 keV band is the result of the first true imaging survey of the entire sky above 2 keV. We present methods to identify and flag potential spurious sources in the catalogues which we applied for this work and we tested and validated the astrometric accuracy via crosscomparison with other Xray and multiwavelength catalogues. We show that the number counts of Xray sources in eRASSl are consistent with those derived over narrower fields by past Xray surveys of a similar depth and we explore the number counts variation as a function of the location in the sky. Adopting a uniform allsky flux limit at 50 completeness of FSUB052 keVSUB gt 5 10SUP14SUP erg sSUP1SUP cmSUP2SUP we estimate that the eROSITA allsky survey resolves into individual sources about 20 of the cosmic Xray background in the 12 keV range. The catalogues presented here form part of the first data release DR1 of the SRGeROSITA allsky survey. Beyond the Xray catalogues DR1 contains all detected and calibrated event files source products light curves and spectra and allsky maps. Illustrative examples of these are provided. P The catalogue is available at the CDS via anonymous ftp to A hrefhttpscdsarc.cds.unistra.frcdsarc.cds.unistra.frA ftp130.79.128.5 or via A hrefhttpscdsarc.cds.unistra.frvizbincatJAA682A34httpscdsarc.cds.unistra.frvizbincatJAA682A34A	2024-02-01T00:00:00Z	['10.48550/arXiv.2401.17274', '2024A&A...682A..34M', '10.1051/0004-6361/202347165', '2024arXiv240117274M', 'arXiv:2401.17274']	['catalogs', 'surveys', 'X-rays: general', 'Astrophysics - High Energy Astrophysical Phenomena']	The SRGeROSITA allsky survey. First Xray catalogues and data release of the western Galactic hemisphere	2,024	201	0.77	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	132	https://arxiv.org/pdf/2401.17274.pdf	{'First X-ray catalogues and data release of the western Galactic hemisphere ⋆': 'A. Merloni 1 ⋆⋆ , G. Lamer 2 , T. Liu 1 , 25 , 26 , M. E. Ramos-Ceja 1 , H. Brunner 1 , E. Bulbul 1 , K. Dennerl 1 , V. Doroshenko 3 , M. J. Freyberg 1 , S. Friedrich 1 , E. Gatuzz 1 , A. Georgakakis 4 , F. Haberl 1 , Z. Igo 1 , 5 , I. Kreykenbohm 6 , A. Liu 1 , C. Maitra 1 , A. Malyali 1 , M. G. F. Mayer 1 , K. Nandra 1 , P. Predehl 1 , J. Robrade 7 , M. Salvato 1 , 5 , J. S. Sanders 1 , I. Stewart 1 , D. Tubín-Arenas 2 , P. Weber 6 , J. Wilms 6 , R. Arcodia 1 , 8 , E. Artis 1 , J. Aschersleben 9 , A. Avakyan 3 , C. Aydar 1 , 5 , Y. E. Bahar 1 , F. Balzer 1 , W. Becker 1 , 10 , K. Berger 6 , T. Boller 1 , W. Bornemann 1 , M. Brüggen 7 , M. Brusa 11 , 12 , J. Buchner 1 , V. Burwitz 1 , F. Camilloni 1 , N. Clerc 13 , J. Comparat 1 , D. Coutinho 1 , S. Czesla 14 , 7 , S. M. Dannhauer 9 , L. Dauner 6 , T. Dauser 6 , J. Dietl 9 , K. Dolag 15 , T. Dwelly 1 , K. Egg 6 , E. Ehl 1 , S. Freund 1 , 7 , P. Friedrich 1 , R. Gaida 1 , C. Garrel 1 , V. Ghirardini 1 , A. Gokus 6 , 16 , G. Grünwald 1 , S. Grandis 17 , I. Grotova 1 , D. Gruen 15 , A. Gueguen 1 , S. Hämmerich 6 , N. Hamaus 15 , G. Hasinger 18 , 19 , K. Haubner 6 , D. Homan 2 , J. Ider Chitham 1 , W. M. Joseph 3 , A. Joyce 6 , O. König 6 , D. M. Kaltenbrunner 1 , A. Khokhriakova 1 , W. Kink 1 , C. Kirsch 6 , M. Kluge 1 , J. Knies 6 , S. Krippendorf 15 , M. Krumpe 2 , J. Kurpas 2 , P. Li 2 , Z. Liu 1 , N. Locatelli 1 , M. Lorenz 6 , S. Müller 1 , E. Magaudda 3 , C. Mannes 9 , H. McCall 9 , 20 , N. Meidinger 1 , M. Michailidis 3 , K. Migkas 9 , 21 , D. Muñoz-Giraldo 3 , B. Musiimenta 11 , 12 , N. T. Nguyen-Dang 3 , Q. Ni 1 , A. Olechowska 1 , N. Ota 9 , F. Pacaud 9 , T. Pasini 10 , 22 , E. Perinati 3 , A. M. Pires 2 , C. Pommranz 3 , G. Ponti 23 , 1 , K. Poppenhaeger 2 , G. Pühlhofer 3 , A. Rau 1 , M. Reh 6 , T. H. Reiprich 9 , W. Roster 1 , S. Saeedi 6 , A. Santangelo 3 , M. Sasaki 6 , J. Schmitt 7 , P. C. Schneider 7 , T. Schrabback 17 , N. Schuster 15 , A. Schwope 2 , R. Seppi 1 , M. M. Serim 3 , S. Shreeram 1 , E. Sokolova-Lapa 6 , H. Starck 1 , B. Stelzer 3 , J. Stierhof 6 , V. Suleimanov 3 , C. Tenzer 3 , I. Traulsen 2 , J. Trümper 1 , K. Tsuge 6 , T. Urrutia 2 , A. Veronica 9 , S. G. H. Waddell 1 , R. Willer 1 , J. Wolf 1 , 5 , 24 , M. C. H. Yeung 1 , A. Zainab 6 , F. Zangrandi 6 , X. Zhang 1 , Y. Zhang 1 , X. Zheng 1 \n(A ffi liations can be found after the references) \nReceived 12 June 2023; accepted 26 November 2023', 'ABSTRACT': 'The eROSITA telescope array aboard the Spektrum Roentgen Gamma (SRG) satellite began surveying the sky in December 2019, with the aim of producing all-sky X-ray source lists and sky maps of an unprecedented depth. Here we present catalogues of both point-like and extended sources using the data acquired in the first six months of survey operations (eRASS1; completed June 2020) over the half sky whose proprietary data rights lie with the German eROSITA Consortium. We describe the observation process, the data analysis pipelines, and the characteristics of the X-ray sources. With nearly 930 000 entries detected in the most sensitive 0.2-2.3 keV energy range, the eRASS1 main catalogue presented here increases the number of known X-ray sources in the published literature by more than 60%, and provides a comprehensive inventory of all classes of X-ray celestial objects, covering a wide range of physical processes. A smaller catalogue of 5466 sources detected in the less sensitive but harder 2.3-5 keV band is the result of the first true imaging survey of the entire sky above 2 keV. We present methods to identify and flag potential spurious sources in the catalogues, which we applied for this work, and we tested and validated the astrometric accuracy via cross-comparison with other X-ray and multi-wavelength catalogues. We show that the number counts of X-ray sources in eRASS1 are consistent with those derived over narrower fields by past X-ray surveys of a similar depth, and we explore the number counts variation as a function of the location in the sky. Adopting a uniform all-sky flux limit (at 50% completeness) of F 0 . 5 -2 keV > 5 × 10 -14 erg s -1 cm -2 , we estimate that the eROSITA all-sky survey resolves into individual sources about 20% of the cosmic X-ray background in the 1-2 keV range. The catalogues presented here form part of the first data release (DR1) of the SRG / eROSITA all-sky survey. Beyond the X-ray catalogues, DR1 contains all detected and calibrated event files, source products (light curves and spectra), and all-sky maps. Illustrative examples of these are provided. \nKey words. catalogues - surveys - X-ray: general', '1. Introduction': "Wide-area multi-wavelength (and multi-messenger) sky surveys play a key role in the development of astrophysics. The design of these surveys is driven both by the desire to open up new discov- \nery spaces, and by the realisation that charting the structure of the Universe in detail may help solve long-standing open questions of fundamental physics (see e.g. Peebles 1980; Weinberg et al. 2013). \nSky surveys also occupy a central position in the relatively short history of X-ray astronomy (see e.g. Elvis 2020, for a succinct recap). These surveys were initiated by the SASA Uhuru satellite (1970-1973, Giacconi et al. 1971), a mission designed to conduct a survey of the X-ray sky in the 2- \n20 keV energy range, resulting in the identification of 339 Xray sources (Forman et al. 1978). Likewise, an X-ray survey was carried out by the cosmic X-ray experiment aboard the HEAO-1 observatory (1977-1979, Rothschild et al. 1979) covering the 0.25-25 keV energy range, enabling the discovery of 842 X-ray sources (Wood et al. 1984). The Einstein (HEAO-2) X-ray Observatory (1978-1981, Giacconi et al. 1979) performed a medium-sensitive survey in the 0.3-3.5 keV energy range, covering roughly one-third of the sky and providing the detection of about 4000 X-ray sources (Harris 1990). However, the first comprehensive all-sky X-ray survey was performed by the ROSAT X-ray observatory in the 0.1-2.4 keV energy range (1990-1999, Truemper 1982). \nThe ROSAT All-Sky Survey (RASS) and its associated catalogues of X-ray sources, The Bright Source Catalogue (BSC) (Voges et al. 1999), Faint Source Catalogue (FSC) (Voges et al. 2000), and the more comprehensive second catalogue release 2RXS (Boller et al. 2016), marked a new milestone in the quantity and quality of X-ray detections. With more than 10 5 unique sources detected in only six months of survey observations, RASS outnumbered all previous all-sky catalogues by more than two orders of magnitude. \nSince the turn of the century, XMM-Newton and Chandra , with their large collecting area and high spatial resolution, respectively, have increased the number of known X-ray sources significantly. Chandra and XMM-Newton serendipitous source catalogues, however, only cover a small fraction of the sky, due to their relatively small fields of view and their pointed observation strategy. \nThese and other X-ray surveys provide a unique view of many celestial phenomena. X-ray emission is a universal signature of accretion of matter onto compact objects. In binary systems, these mark the graveyards of stellar evolution (see e.g. Shapiro & Teukolsky 1983; Warner 1995; Becker & Truemper 1997; Fender et al. 2004; Remillard & McClintock 2006; Haberl 2007; Özel & Freire 2016), where black holes, neutron stars, and white dwarfs accrete matter from a companion. The brightest Xray sources in the sky, including Sco X-1, the first extrasolar one discovered (Giacconi et al. 1962), fall into this category. At cosmological distances, X-rays signpost the growth of the supermassive black holes (SMBHs) that sit at the centres of galaxies and which may strongly influence their formation and subsequent evolution (Brandt & Hasinger 2005; Hopkins et al. 2008; Hickox et al. 2009; Fabian 2012; Alexander & Hickox 2012; Kormendy & Ho 2013; Brandt & Alexander 2015). \nX-ray emission is seen from gas heated by the shocks generated by rapidly expanding supernova remnants, which are in addition likely responsible for a large fraction of the accelerated particles that di ff use through interstellar space (see e.g. Vink 2012; Blasi 2013, and references therein). Strong X-ray emission is also seen from the coronae of stars (Schmitt 1997; Pizzolato et al. 2003; Wright et al. 2011), which plays an important role in the determination of the physical characteristics of the orbiting planets' atmospheres, with implications for the potential habitability of such planets (Lammer et al. 2003; Sanz-Forcada et al. 2011; Poppenhaeger et al. 2021; Foster et al. 2022). \nAfurther source of X-ray emission of great wider importance is from hot gas associated with large-scale structures. Most of the baryons in the Universe are indeed predicted to be locked into a warm-hot (X-ray emitting) tenuous phase (Cen & Ostriker 1999; Davé et al. 2001; Nicastro et al. 2018; Tanimura et al. 2020). Direct or indirect detections of these baryons often require sensitive mapping of large volumes in X-rays. Moreover, in the hierarchical distribution of matter characteristic of \nour observed Universe, the densest knots of the large-scale structure are signposted by the hottest and most massive concentration of di ff use baryons. The clusters of galaxies that identify them are thus extremely sensitive tracers of the underlying geometry and growth history of the cosmos, and therefore prime cosmological tools (Bahcall 1977; Cavaliere & Fusco-Femiano 1978; Sarazin 1986; Rosati et al. 2002; Voit 2005; Arnaud 2005; Norman 2005; Borgani 2008; Vikhlinin et al. 2009; Borgani & Kravtsov 2011; Allen et al. 2011; Reiprich et al. 2013). It is the potential for sensitive cosmological measurements with galaxy clusters that provided the main motivation for the development of eROSITA in the early 2000s, when it became apparent that a significantly larger number of clusters of galaxies compared to those provided by narrow-field instruments would be required to e ff ectively constrain the fundamental parameters of cosmological models (Haiman et al. 2005; Merloni et al. 2012; Pillepich et al. 2012). In particular, these authors argued that sample sizes of order 10 5 clusters were required to provide constraints competitive with other prime cosmological measurement tools. \neROSITA (extended ROentgen Survey with an Imaging Telescope Array; Predehl et al. 2021), on board the Spektrum Roentgen Gamma (SRG) orbital observatory (Sunyaev et al. 2021), was developed in the period 2007-2019. It is a sensitive, widefield focusing X-ray telescope array, optimised to deliver large e ff ective area and field-of view (hence also grasp and survey speed) in the soft X-ray band. The angular resolution is su ffi -cient to distinguish between the two largest extragalactic source populations, that is, AGN and clusters of galaxies, via measurement of their X-ray extent. \nThe observing strategy was designed to achieve the needed sensitivity in the soft X-ray band (below 2 keV) to detect at least the requisite 10 5 clusters of galaxies by scanning the entire sky eight times over a period of four years (the eROSITA All-Sky Surveys, hereafter: eRASS). \nSRG was launched on July 13, 2019 from Baikonur, Kazakhstan, using a Proton-M rocket and a BLOK DM-03 upper stage. On its three months cruise to the second Lagrangian point (L2) of the Earth-Sun system the spacecraft and instruments underwent commissioning and checkout. Since mid-October 2019, SRG was placed in a six-month-periodic halo orbit around L2, with a semi-major axis of about 750 000 km within the ecliptic plane and semi-minor axis of about 400 000 km perpendicular to it (Sunyaev et al. 2021); periodic orbit correction manoeuvres over the intervening years have slightly reduced the size of the orbit in order to satisfy ground segment visibility constraints. \nFollowing First Light (Maitra et al. 2022; Haberl et al. 2022), a two-months long Calibration and Performance Verification (CalPV) programme was executed between October and December 2019. The large body of publications based on CalPV observations (see Campana et al. 2021, and the associated articles of the A&A special issue) have demonstrated the capabilities of eROSITA, and confirmed the main design characteristics. In particular, the eROSITA Final Equatorial Depth Survey (eFEDS; Brunner et al. 2022), designed to provide uniform exposure over a su ffi ciently large field (140 deg 2 ) about 50% deeper than what is expected for eRASS at the end of the 4 year all-sky survey programme, has shown that large samples of X-ray sources of di ff erent classes can be detected, identified, and characterised making use of the synergy with existing multi-wavelength surveys (see, e.g. Ghirardini et al. 2021; Liu et al. 2022a,b; Salvato et al. 2022; Klein et al. 2022; Schneider et al. 2022; Bulbul et al. 2022; Pasini et al. 2022; Ramos-Ceja et al. 2022, Nandra et al., submitted). \nThe eROSITA data are shared equally between German and Russian scientists, following an inter-agency agreement signed in 2009. Two hemispheres of the sky have been defined, over which each team has unique scientific data exploitation rights. These data rights are split by Galactic longitude ( l ) and latitude ( b ), with a division marked by the great circle passing through the Galactic poles ( l , b ) = (0 · , + 90 · ); (0 · , -90 · ) and the Galactic Center Sgr A* ( l , b ) = (359 . 9442 · , -0 . 04616 · ): data with -0 . 05576 · < l < 179 . 9442 · (eastern Galactic hemisphere) belong to the Russian consortium, while data with 359 . 9442 · > l > 179 . 9442 · (western Galactic hemisphere) belong to the German eROSITA consortium (eROSITA-DE). Here we only describe and release the data collected in the western Galactic hemisphere. \nAfter a brief introduction to the salient technical aspects of eROSITA and its calibration (Sect. 2), in Sect. 3 we describe in detail the eROSITA observing procedures in all-sky survey mode. Section 4 is then devoted to a summary of the main data processing stages. Most of the details of the software system used to process eROSITA data have already been presented in Brunner et al. (2022), and we refer the interested reader to that work for more information. The catalogues generated by our standard processing pipeline analysing the first eROSITA allsky survey (eRASS1) are presented in Sect. 5. There we present the preliminary astrometric correction applied to the detected sources as well as our general attempt to isolate and flag potential spurious sources and other artefacts. Section 6 describes further consistency checks on the X-ray catalogue performed by comparing the eRASS1 sources with those of the XMMNewton serendipitous catalogue (photometry) and other multiwavelength Quasi-Stellar Object (QSO) catalogues (astrometry). Section 7 then gives an overview of all the available products, including all-sky maps, and source-specific light curves and X-ray spectra. To conclude, we summarise our work in Sect. 8.", '2. eROSITA technical facts': 'In this section, we provide a compact summary of the main technical characteristics of eROSITA and its calibration; more details, including a summary of the on-ground calibration, can be found in Predehl et al. (2021). Specific technical descriptions of the instrument subsystems can be found in Meidinger et al. (2021) (camera system) and Friedrich et al. (2012) (mirror system), while a thorough description of the on-ground calibration campaign and its results can be found in Dennerl et al. (2020). Indeed, most of the data analysis and pipeline settings adopted for the reduction of the eRASS1 data presented here rely on this extensive on-ground calibration of the instrument. As for all other X-ray space observatories, in-flight calibration is a longterm endeavour; here we present some preliminary results that demonstrate the fidelity of the calibration and the reliability of the data released. An in-depth analysis of the in-flight calibration will be presented elsewhere, and on the website of the first data release (DR1).', '2.1. Instrument characteristics': "eROSITA consists of seven identical and co-aligned X-ray telescopes arranged in a common optical bench. A system of carbon fibre honeycomb panels connects the seven mirror assemblies on the front side with the associated seven camera assemblies on the focal plane side. \nEach of the mirrors comprises 54 mirror shells in a Wolter-I geometry, with an outer diameter of 360 mm and a common focal \nlength of 1 600 mm (Friedrich et al. 2008; Arcangeli et al. 2017). The average on-axis resolution of the seven mirrors, as measured during the on-ground calibration, is 16 . 1 '' Half-Energy Width (HEW) at 1.5 keV. The unavoidable o ff -axis blurring typical of Wolter-I optics is compensated by a 0.4 mm shift of the cameras towards the mirrors. This puts each telescope slightly out of focus, leading to a slight degradation of the on-axis performance (18 '' HEW), but improved angular resolution averaged over the field of view. Indeed, in the scanning observational mode adopted for the all-sky survey (see section 3.1), it is the field-of-view average HEW that matters. This is discussed in Section 2.2.1 and Appendix A. \nEach Mirror Assembly has a CCD camera in its focus (Meidinger et al. 2014). The eROSITA CCDs are advanced versions of the EPIC-pn CCDs on XMM-Newton (Strüder et al. 2001). They have 384 × 384 pixels in an image area of 28 . 8 mm × 28 . 8 mm, yielding a square field of view of 1 · . 03 × 1 · . 03. Each pixel corresponds to a sky area of 9 '' . 6 × 9 '' . 6. The nominal integration time for all eROSITA CCDs is 50 msec. The additional presence of a frame-store area in the CCD reduces substantially the amount of so-called out-of-time events, which are recorded during the CCD read-out, a significant improvement with respect to the EPIC-pn camera on XMM-Newton . For optimal performance during operations, the CCDs are cooled down to about -85 · by means of passive elements (Fürmetz et al. 2008). During survey operation (i.e. in scanning mode of observations), the angle between the scanning direction projected onto the sky and the CCD read-out direction in the focal plane is not the same for the seven cameras 1 . This contributes to averaging out any possible non-circular symmetry of the PSF as well as camera-induced defects. For calibration purposes, each camera has its own filter wheel with a radioactive 55 Fe source and an aluminium and titanium target providing three spectral lines at 5.9 keV (Mn K α ), 4.5 keV (Ti K α ) and 1.5 keV (Al K α ). \nThe electronics for onboard-processing of the camera data is provided by seven sets of Camera Electronics (CE), each one mounted and interfacing to the Cameras. Each of the CEs provide the proper voltage control and readout timing of the associated camera, and performs the on-board data processing within the time constraints of the camera integration time.", '2.2. eROSITA calibration': "The quantities derived from the eROSITA calibration are stored in a calibration database (CalDB), which is accessed by various processing tasks. Information about the content of the CalDB and how important entries were derived can be found in Brunner et al. (2022), Appendix B. A 'live' online repository can be found on the DR1 website 2 . Here we present a brief assessment of the current status of the in-flight calibration program.", '2.2.1. PSF calibration': "Given the scientific objectives of eROSITA, i.e. imaging the whole sky at soft-to-hard X-ray energies with good sensitivity to low surface brightness di ff use and extended emission, clusters of galaxies in particular, an accurate calibration of the tele- \nscopes' Point-Spread Function (PSF) is critical. As we describe in greater detail below (§ 4), the catalogues presented here have been constructed using a single photon mode, in which the PSF of each telescope module at the location of each detected event on the CCD is accounted for, using a shapelet-based model calibrated on the extensive dataset accumulated on ground before the launch (Dennerl et al. 2020). A description of the shapeletbased PSF modelling, and its usage in scanning mode can be found in Appendix B.1 of Brunner et al. (2022). \nWhile work is ongoing to accurately characterize eROSITA's PSF based on the survey data themselves, here we report on a preliminary analysis that confirms the reliability of the onground calibration adopted for the DR1 datasets. Appendix A shows a direct comparison of the average PSF shape (obtained by combining all seven Telescope Modules) from stacking point sources detected in the all-sky survey with the PSF model from the PANTER on-ground calibration and its shapelet representation. The di ff erences between the PSF models are within the ∼ 20% level in the inner 1 ' , although the shapelet PSF drops below the measured PSFs beyond this radius, where it is not used in the source detection process, except for normalisation. The measured HEWs from the source stacking method applied to survey data are 30.0 '' in the 0.2-2.3 keV band and 34.4 '' in the 2.3-5.0 keV band, very close to the pre-flights estimates of 28.3 '' and 36.2 '' , respectively, for the shapelet representation, and 32.0 '' and 38.0 '' for the PANTER ground-based values. The PSF does not appear to be varying across the sky, at least within the limited statistics of our preliminary analysis (see Tab. A.1). In Appendix A we also show an estimate of the PSF azimutal symmetry, and compare positional o ff sets of eRASS1 point sources against Gaia QSOs along equatorial and ecliptic coordinates, demonstrating the goodness of our circular symmetric approximation for the positional uncertainty of the detected sources.", '2.2.2. Energy calibration': 'After launch, the energy calibration obtained on ground was checked by using extensive measurements with the internal 55 Fe calibration sources. These measurements showed that the inflight energy resolution of the detectors was essentially the same as on ground (Tables 1 and 2 in Dennerl et al. 2020). The 55 Fe calibration measurements were also used to derive updated values of the Charge Transfer Ine ffi ciency (CTI) and Gain, in order to minimise their impact on the absolute energy scale. Additional tests of the energy calibration were made with dedicated observations of astrophysical calibration targets, especially of the isolated neutron star RX J1856-3754 and of the supernova remnant 1E0102.2-7219. These demonstrated that the energy calibration is su ffi ciently accurate for the analysis of survey data with their limited photon statistics 3 . Appendix B provides more details on these energy calibration experiments. The currently available energy calibration has already been used successfully for spectroscopic studies (e.g. Camilloni et al. 2023; Mayer et al. 2023; Ponti et al. 2023b; Yeung et al. 2023).', '2.2.3. Flux calibration: Effective area and vignetting': 'The challenge of accurately flux-calibrating space-based X-ray telescopes is as old as X-ray astronomy itself (see e.g. Marshall \net al. 2021; Madsen et al. 2021, and references therein for recent discussions). While the Quantum E ffi ciency (QE) of the CCD detectors can be accurately measured on ground, but is subject to degradation in the harsh space environment, the main di ffi culty rests with the paucity of suitable stable standard candles and the impossibility to accurately reproduce in the laboratory the observing conditions in space needed to calibrate a telescope e ff ective area and vignetting function (i.e. variation of e ff ective area across the focal plane). Clusters of galaxies, possibly the brightest intrinsically non-varying X-ray sources in the sky, are usually adopted as cross-calibrators among di ff erent Xray observatories, but even after decades, uncertainties remain (Nevalainen & Molendi 2023). For eROSITA, we can take advantage of the all-sky survey nature of the observations to build large samples that can be used to validate the flux calibration in a statistical sense. For point sources, we report here (Section 6.1) a comparison with XMM-Newton in the 0.5-2 keV band, resulting in a possible residual flux systematic uncertainty of just about 6%, similarly to what was found by Maitra et al. (2022) in the early phases of the mission. We have also tested the e ff ect of possible PSF calibration uncertainties on the reconstructed flux, by running an alternative source detection pipeline which makes use of the PSF image preliminary stacks, and found only a 3% source counts systematic o ff set compared to the catalogue described here. \nThe situation for clusters of galaxies in the survey is still under examination (but see e.g. Whelan et al. 2022; Sanders et al. 2022, for pointed Calibration observations), with Bulbul et al. (2024) reporting a systematic flux deficit with respect to Chandra of about 15%, while Liu et al. (2023a) and Migkas et al. (submitted) found a temperature o ff set with respect to both XMM-Newton and Chandra (with eROSITA measuring lower tempertures than both) that increases with the cluster temperature itself. Work is ongoing to determine to what extent this is induced by calibration uncertainties or by cluster-related astrophysical e ff ects (such as multi-temperature ICM distributions).', '3.1. Scanning strategy': "In order to complete an all-sky survey, the SRG observatory rotates continuously with a period of four hours around an axis pointed near the direction of the Sun. This gives a scan rate of 0.025 deg s -1 . The rotation axis slowly shifts by approximately one degree per day following the motion of the Earth (and of the L2 point) around the Sun. Following this scanning pattern, eROSITA observes the entire sky in about six months, and observes each point in the sky typically six times ('visits') for up to 40 seconds over a day at the ecliptic equator; towards the ecliptic poles, the sources remain observable for more than 24 hours, and are therefore scanned more than six times. Indeed, all great circles (individual scans) intersect in the north and south ecliptic poles in the sky, creating regions of deep exposure and longer visibility periods. In addition, a slight inhomogeneity in the sky coverage is introduced by the elongated halo orbit around L2 and the nonuniform angular movement of the spacecraft rotation axis, which compensate the separations between Sun and Earth, as seen from the spacecraft, in order to maintain the Earth in the cone of the downlink antenna (further details in Predehl et al. 2021; Sunyaev et al. 2021). \nTable 1. Timeline of the main eROSITA operations and major events during eRASS1.", '3.2. All-sky survey operations': "After the commissioning of the instrument and the CalPV phase, the first all-sky survey started on December 12, 2019, and was completed on June 11, 2020, for a total survey duration of 184 days. During this period, ground contacts with SRG and eROSITA took place every day, without interruption. A timeline of the most significant operations milestones during eRASS1 is presented in Table 1. \nTable 2. Timeline of extended diagnostic and engineering exposures during eRASS1. \nNotes. 'Raw Frames' are read-out cycles where each pixel is transmitted (by-passing the event processor), 10 such frames were commanded for each TM consecutively (due to the large amount of telemetry there is some time gap in between). After the seven TMs a second round of 'raw frames' was commanded, giving in total 10 × 7 × 2 = 140 such frames per epoch. \nAs described in Coutinho et al. (2022) and Predehl et al. (2021), the Mission Control Center, located in Moscow at NPO Lavochkin (NPOL), has mainly used two deep-space antennas for the science downlink (Ussuriysk and Bear Lakes) 4 . In total, approximately 75 GB of telemetry data from eROSITA were dumped during eRASS1, with an average of ∼ 407 MB per day. \nThe overall observing e ffi ciency of eROSITA during eRASS1 was ∼ 96.5%. This e ffi ciency has been calculated by taking the average of the Good Time Intervals (GTI) with respect to the total observing time for each camera, as they are operating independently. The main observation disruptions responsible for the loss of e ffi ciency come from Camera Electronics (CE) and Interface and Thermal Controller (ITC) anomalies: the CEs and ITC are susceptible to Single Event Upsets (SEU), which can interrupt the functioning of the instrument. Moreover, Telescope Modules (TM) 5 and 7 su ff er from a time-varying light leak, which can lead to loss of data and consequently to gaps in GTI \n(see Sect. 4.6 and Predehl et al. 2021; Coutinho et al. 2022, for further details). Figure 1 shows the cumulative observing e ffi -ciency as a function of time for each TM during eRASS1. \nRegular 'Filter Wheel Closed' (FWC) observations were carried out during the survey to monitor the instrumental background, starting in February 2020. The filter wheel of one camera per day was set to the closed position for 30 minutes. To reduce the impact on the exposure of the survey observations, the same camera had an FWC observation every seven days. In total, around 18 short FWC observations for each camera were performed in eRASS1 (exposure fraction 0.3%), providing a precious data set to model and monitor the background (see Yeung et al. 2023, for more details about the instrumental background model). In addition, more extended FWC observations were performed during periods of orbit corrections (see Table 2), along with other diagnostic and engineering exposures. The viewing direction of the telescope was reset after the orbit corrections (and monitoring pointed observations) to ensure survey coverage without gaps. \nWith the exception of the short-lived SEU-induced malfunctions of the camera electronics and of the ITC (on Feb. 10, 2020), all eROSITA sub-systems were fully functional during eRASS1, and they have not su ff ered any permanent damage, apart from the expected degradation caused by external environmental e ff ects along SRG's L2 orbit. In Coutinho et al. (2022) the interested reader can find more details about the technical performance of eROSITA during the first two years of operations (from eRASS1 to eRASS4).", '3.3. Pre-processing and archiving': "The eROSITA data received at a ground station are forwarded in real time via a socket connection to the SRG operations centre and to IKI (the Space Research Institute of the Russian Academy of Sciences), where they are stored in binary files with a size of approximately 7.5 MB each. These files are transferred to MPE via a data exchange server. The files are then picked up by the pre-processing pipeline, which converts the telemetry into FITS files and forward them to two separate pipelines: the preprocarchiver and the Near Real Time Analysis (NRTA). \nThe archive is organised into 'erodays', i.e. fixed intervals of 4 hours (also corresponding to the duration of one revolution of SRG in all-sky survey mode). Once an eroday is completed, the data are moved to the regular archive and the post-ingest pipeline is triggered. \nThe NRTA pipeline has been developed in order to (i) monitor interactively the instrument parameters and check the health of all sub-systems on the shortest possible timescales, and (ii) alert the team and the community of possible interesting timedomain phenomena observed by eROSITA. The NRTA and its functionality is described in Appendix C. At this point, the \nFig. 1. Cumulative observing e ffi ciency (as a percentage of the elapsed survey time) as a function of time for each eROSITA camera (labeled according to the associated Telescope Module, TM) during eRASS1. With the exception of TM5, badly a ff ected by light leak, all cameras achieved an e ffi ciency of more than 95% by the end of eRASS1, with TM1 reaching close to 99%. The largest loss of net observing time was caused by an ITC malfunction on February 10, 2020. \n<!-- image --> \narchived data are ready to be processed by the standard analysis pipeline.", '4. Standard data processing': 'The eRASS1 data were processed with the eROSITA standard data processing pipeline, operated at MPE. The pipeline consists of modules for event processing (TEL processing chain), event file and image creation (EXP processing chain), exposure and background map creation and source detection (DET processing chain), as well as for the creation of source-specific products such as spectra and light curves (SOU processing chain). Each processing chain executes a number of software tasks, which are part of the eROSITA Science Analysis Software System (eSASS). \nIn comparison with the data processing version (001) from the Early Data Release (EDR) 5 , the event calibration in the processing version (010) of eRASS1 has a stronger telescope module (TM) specific noise suppression of doubles, triples and quadruples 6 , a better computation of the subpixel position, a corrected flagging of pixels next to bad pixels, and improved accuracy of projection. The list of eSASS task and calibration database versions used for this work is provided in Appendix E. A list of standard data products included in the eROSITA DR1 data release is provided in Sect. 7. A detailed description of the main eSASS tasks, associated eROSITA calibration database, and calibrated data products is available in Brunner et al. (2022) (Appendix A-C). In the rest of this section we briefly summarize the organisation of the pipeline and the function of each pipeline chain. \nFig. 2. Spherical projection of the eRASS1 1B exposure map centred on the western Galactic hemisphere (i.e. the eROSITA-DE area) with overimposed in green the tiling of the sky into equal-area overlapping 3 . 6 · × 3 . 6 · tiles into which data are organised and analysed by the pipeline. \n<!-- image -->', '4.1. TEL chain': 'The TEL chains perform the functions of event file preparation (tasks evprep , ftfindhotpix ), pattern recombination (task pattern ), energy calibration (task energy ), and attitude calculation (tasks attprep , telatt , and evatt ). The TEL chains are executed separately for each telescope module for data intervals of one eroday. \nThe sky is divided into 4700 non-overlapping, unique areas ("sky tiles"), organized into 61 equatorial declination zones (see Fig. 2 for a visual representation of the adopted sky tiling). The size of these sky tiles is exactly three degrees in declination and approximately three degrees in right ascension (on the side facing the equator), resulting in an average area of about 8.78 square degrees per tile. Each unique sky tile is embedded in an overlapping, square sky field of size 3 . 6 · × 3 . 6 · centred on the sky tile. The minimum overlap between neighbouring sky fields, introduced in order to avoid edge e ff ects in the source detection, ranges between 15 \' for polar fields and 18 \' for equatorial fields 7 . After the completion of the event calibration, the event data of each TEL chain are sorted into these overlapping sky fields, based on the right ascension and declination of each photon (tasks telgti , telselect , telstage ). Source detection and further source-level analysis are then performed in each sky field.', '4.2. EXP chain': "The event data collected in each sky tile are merged on a 'per TM' basis (task expmerge ) and sky pixel event coordinates with pixel size 0 '' . 05 centred on the sky tile centre are computed (task \nradec2xy ). Combining these TM-specific files, filtered event files and images that include all TMs (pixel size 4 '' , 3240 × 3240 pixels) are created in a variety of energy bands suitable for source detection (task evtool ). The event filtering excludes invalid patterns (i.e. pixel patterns with a low probability of having been caused by a single X-ray photon), events on or close to bad pixels, as well as events outside a circular detection mask of radius 0.516 · . In addition, good time intervals free of background flares are created, if necessary (task flaregti ) 8 .", '4.3.1. Exposure maps': 'Exposure maps are created on a tile-by-tile basis by the task expmap . For each energy band and for each TM, a map of the active pixels of the CCD (that is, excluding flagged bad pixels, their orthogonal neighbours, as well as out-of-FOV pixels) is first divided by a vignetting map. The latter is generated as a weighted mean of the energy-dependent vignetting function across the respective energy band, the weighting being a power law of spectral index 9 Γ = 1 . 7. The vignetted CCD image is then projected onto the sky (using the same projection as the template image supplied to the task) at a series of time samples (nominally one per second) of the TM attitude, using only samples during GTIs for that TM. The resulting TM-specific sky exposure maps are then combined in a weighted mean for each band, the weights (corresponding to the fraction of area supplied by each TM with respect to the nominal all-TM area) being read from the calibration database. The results are N maps in the N specified energy bands, each giving the spatially varying mean (vignetted) exposure in that given band, in seconds. \nAs an example, Fig. 3 shows the combined all-sky exposure map (where darker regions mark longer exposures) for the 0.6-2.3 keV band. The main qualitative features of this map are determined by the scanning law and the associated event milestones discussed above (i.e. scan interruptions, overlaps, etc.); these features are common to all exposure maps. Quantitative measures, however, such as the overall normalisation of the exposure, may di ff er for di ff erent energy bands, mainly due to the variations of the vignetting function of the telescope at di ff erent energies.', '4.3.2. Source detection': "The eSASS source detection pipeline was applied to each of the 3 . 6 · × 3 . 6 · tiles with the following steps: \n- 1. Creation of masks to define image regions with valid data (task ermask );\n- 2. Initial source finding (task erbox );\n- 3. Background determination (task erbackmap );\n- 4. Search for candidate sources (task erbox );\n- 5. Creation of PSF fitting catalogue (task ermldet );\n- 6. Forced PSF photometry in additional energy bands (task ermldet );\n- 7. Aperture photometry in all energy bands and the corresponding sensitivity maps (task apetool ); the aperture is chosen as a fixed one at 75% encircled energy fraction 10 ; \n8. Creation of sensitivity maps (task ersensmap ). \nSteps 3. and 4. were iterated twice for an improved separation of sources and background. The source detection setup is very similar to the configuration used for the creation of the eFEDS catalogues (Brunner et al. 2022), where the individual detection tasks are described in more detail. \nReliable detection and characterisation of X-ray extended sources is crucial for any subsequent application and analysis of galaxy cluster samples. It should be noted that for both point sources and extended sources the rate, count and flux values reported in the eRASS1 catalogues are based on the scaling of the best fit PSF or PSF-folded extent model. By definition, these quantities correspond to values integrated to infinite radii while the model fits are performed on circular sub-images of 1 ' radius. Therefore, the flux related source parameters may be subject to significant systematic uncertainties, in particular for large extended sources. In Bulbul et al. (2024) the interested reader will find a detailed description of the procedure used to define clean clusters samples starting from the extended sources catalogues, and also to derive robust physical quantities from the X-ray data alone. \nIn each sky tile two catalogues were created: the single-band catalogue with sources detected in the broad 0.2-2.3 keV band (hereafter '1B') for maximal sensitivity, and a 3-bands catalogue (0.2-0.6 keV, 0.6-2.3 keV, 2.3-5.0 keV, hereafter '3B'), for which detection is carried simultaneously and single-band and combined likelihoods for each source are computed.", '4.3.3. Sensitivity maps: ersensmap': 'For both the 1B catalogue and the 3B catalogue sensitivity maps were calculated with the task ersensmap . As described in Brunner et al. (2022), the task ersensmap uses the eSASS exposure maps and background maps to estimate the detection limits in eROSITA observations. For eRASS1, the maps for the master catalogue contain the 0.2-2.3 keV source flux required to reach a typical detection likelihood of 5.0 at the respective pixel position for consistency with the threshold of the PSF fitting catalog created by ermldet . The map values for the 3-band catalogue correspond to the respective 0.2-5 keV flux. For the conversion between the map fluxes and count rates in the detection images the energy conversion factors (ECFs) listed in Table 3 were used. Based on spectral analysis of eFEDS AGN (Liu et al. 2022c), typical eROSITA-detected AGN have a median power-law slope of 2.0. So we adopted this slope and an absorbing column density N H = 3 × 10 20 cm -2 (typical value of the Galactic absorption) to calculate the ECFs (Brunner et al. 2022). The impact of the assumed power-law slope is negligible here.', '4.3.4. Sensitivity maps: apetool': 'The apetool sensitivity maps use aperture photometry to determine the selection function of a sample of X-ray point sources detected in a given spectral band. The selection function is defined as the probability of detecting an X-ray point source with a given count rate or flux in the band of interest across the eROSITA field of view, across all observations of the source throughout the duration of eRASS1. In generating these sensitivity maps, the stacked PSF at a given sky position is the exposuretime weighted superposition of the individual PSFs from all the \nFig. 3. E ff ective (vignetted) eRASS1 exposure map (Galactic coordinates, Aito ff projection). The values in the map show the exposure time multiplied by the average of the ratio of the (vignetted) e ff ective area to the on-axis e ff ective area in the energy band 0.6-2.3 keV. E ff ective exposure values range from ∼ 100 s at the ecliptic equator to more than 10 000 s close to the ecliptic poles (not visible on the colour scale of this image). The eROSITA-DE (western) Galactic hemisphere is on the right of the central meridian in this map. \n<!-- image --> \nTable 3. ECFs used to calculate the 1-band and 3-band sensitivity maps.Notes. For the 3B maps the ECFs refer to the conversion between the individual bands count rates and the total 0.2-5 keV flux. These ECFs are for conversion between an observed count rate and an observed flux, not corrected for absorption, under the spectral model assumption. \npixels (detector coordinates) that contribute to that sky position as eROSITA slews across the sky. The apetool sensitivity maps can only be used in combination with the aperture photometry of individual X-ray sources provided in the eROSITA catalogues. We refer to the appendices of Brunner et al. (2022) for a full description of the apetool functionality and Georgakakis et al. (2008) for details on the calculation of X-ray sensitivity maps based on aperture photometry. \nThere are two key parameters that are relevant for the apetool sensitivity maps: the radius of the circular aperture adopted for photometry and the Poisson probability that the observed counts within an aperture are produced by random fluctuations of the background (Poisson false detection probability, or False Alarm Probability, FAP). The lower the latter, the less likely it is that the counts within an aperture are produced by the background, thereby suggesting the presence of an astrophysical source. For a given background level (as specified in the background maps), we use Poisson statistics to estimate the minimum number of photons within the aperture, N min so that the corresponding FAP is below an adopted threshold, thus yielding X- \nray source detections to a given confidence level. The apetool sensitivity map is an image of N min across the field of view of the eROSITA observations. The sensitivity map can be further combined with the eROSITA exposure and background maps to determine the probability of detecting a source with FAP ≤ P thresh integrated over the eROSITA field of view. This probability as a function of count rate or flux (once an ECF is adopted) is referred to as the sensitivity or area curve and is also provided by apetool . This allows us to study the flux limit at each position, based on the local background level and exposure depth. Two applications of the apetool sensitivity maps (computation of flux limits and number counts) are described in Sects. 5.4 and 5.5.', '4.4. SOU chain': 'Source products (spectra, background spectra, response matrices, ancillary response files and light curves) were created using the srctool task, for the subset of bright sources with a detection likelihood greater than 20 in the single-band 1B catalogue. In terms of net counts, this corresponds to a sample with a median number of ∼ 21, and 90 th (10 th ) percentile of ∼ 79 ( ∼ 12) counts. Further details about the source products are given in Sect. 7.2.', '4.5. Bright sources: pile-up and other losses': "Source parameters derived from the detection pipeline, such as total flux or temporal or spectral variability, carry systematic uncertainties, in particular for very bright X-ray sources. This is mainly due to the pile-up e ff ect (Ballet 1999; Davis 2001; Tamba et al. 2022), which occurs if two or more photons hit the same CCDpixel in the same (50 ms) read-out cycle and the sum of the charges created will enter the event analyser ('energy pile-up'). Pile up also occurs if these photons are recorded in two adjacent pixels where, after recombination of the individual charges, a higher energy value is reconstructed ('pattern pile-up'). It is im- \nportant to note that for pile-up the total energy band is relevant and also events below the lower event trigger threshold, including optical photons. The probability for these e ff ects mainly depends on the source brightness in photons / cycle / pixel and on the actual shape of the PSF, for example, it is reduced for a deteriorated PSF far o ff -axis. This can lead to apparent spectral and temporal variability within a scan through the FOV, but also between one (e.g., more central) scan and another (more o ff -axis). All this complexity in general requires detailed simulations for an accurate estimate of pile-up e ff ects (see e.g. König et al. 2022). Preliminary studies using SIXTE (Dauser et al. 2019), indicate that for eRASS1 pile-up starts to become important (more than a few per-cent e ff ect) for point sources brighter than ≈ 10 -11 erg s -1 cm -2 in the 0.2-2.3 keV band. \nPhoton counts can also be lost in eROSITA due to e ff ects in the event analyser and telemetry limitations between camera electronics and ITC. The 'event quota' mechanism for each camera ensures a reasonable telemetry rate, even in the case of, e.g., a CCD column becoming bright between two ground contacts. In the current implementation, the event quota is triggered if in one TM there are more than 50 events for four consecutive readout cycles after onboard rejection of minimum ionising particles (MIP) and bad pixels. In that case, for one minute only frames containing less than 50 events are telemetered. After one minute this is reset and all frames are transmitted until the trigger criteria are fulfilled again. Due to this event quota, complete read-out frames are lost, and all sources within the FOV are a ff ected, differently from pile-up, which only a ff ects the piled-up source itself. This mechanism was designed for instrumental reasons, but very bright point sources, such as Sco X-1 (or even bright optical stars) and extended sources (for example Puppis A) are also (celestial) triggers. Missing read-out cycles are properly handled in the exposure computation, but there remains a bias towards fainter read-out cycles during active event quota triggers. \nFinally, the source parameters of some of the brightest sources in the catalogues ( ML\\_CTS\\_1 > 1000) su ff er from poor convergence of the PSF fits due to the large number of individual events to be included in the modelling. This may result in larger than expected deviations in both photometry and astrometry.", '4.6. Other known data processing issues': "During the commissioning phase of eROSITA, it was noticed that TM5 and TM7 (the CCDs not equipped with an on-chip Al optical blocking filter) were contaminated by optical light: a small fraction of sunlight reaches the CCD, by-passing the filter wheel. The intensity of this optical contamination depends on the orientation of the telescope with respect to the Sun. This 'light leak', mostly restricted to very low photon energies (typically < 0 . 3 keV), generates a non-negligible amount of telemetry data and decreases the low energy coverage and spectroscopic capabilities for these two cameras. In order to reduce the amount of transmitted data from TM5 and TM7, their primary thresholds are higher (about 125-145 eV) than the ones in the other TMs (about 65-95 eV). The fact that the amount of contamination by optical light is spatially and temporally variable makes it very di ffi cult to derive an accurate energy calibration for these TMs. More details about the light leak in eRASS1 can be found online on the DR1 web portal 11 . \nThe e ff ect of optical loading, i.e. the appearance of fake Xray sources, or the distortion of the measured X-ray properties \nof sources associated to (very) bright optical stars, is discussed in § 5.2.4. \nIn some cases, the source detection algorithm failed to converge during its error estimate procedure, leaving some sources without reliable uncertainty estimates for the position, counts, or extent. Many of the a ff ected sources have low detection likelihood or are related to spurious detections in areas of extended emission. In some cases, however, also significant detections can be a ff ected by this issue. We flag these sources with the labels FLAG\\_NO\\_RADEC\\_ERR , FLAG\\_NO\\_CTS\\_ERR and FLAG\\_NO\\_EXT\\_ERR , respectively (see Tables 5 and 6). \nFinally, in calculating the number of seconds elapsed between the time reference datum of the mission (00:00 hrs Moscow time on January 1 st , 2000) and a given later date, leap seconds should be added. Five leap seconds occurred between the reference datum and the start of the mission. However, the pipeline software used to create the DR1 data omitted to add these seconds. Therefore, when converting UTC times in the badcamt (bad time intervals) and timeoff (instrumental one second time shifts) calibration components to spacecraft clock a five second shift is introduced, leading to the exclusion of five seconds of good data. In both cases, changes in status (i.e. date entries in the component table) tend to occur when the camera is in an anomalous state, not receiving data (see Brunner et al. 2022, Appendix B). As the cameras are not yet registering photons at these times, not considering leap seconds does not have any other negative consequences (such as inclusion of bad time interval in the data) in this case. These five leap seconds will be properly corrected in the next data release.", '5. The eRASS1 X-ray catalogues': 'Following the approach devised for the Performance Verification eFEDS survey (Brunner et al. 2022), we present here two distinct X-ray catalogues: a catalogue of sources detected in the 0.2-2.3 keV band (selected from the 1B detection process; Main catalogue) and a catalogue of sources detected in the 2.3-5 keV band (selected from the 3B detection process; Hard catalogue). \nWe generate the single-band 1B catalogue including sources down to a low detection likelihood ( DET\\_LIKE\\_0 ⩾ 5), to maximise completeness. We then make use of the eRASS1 digital twin simulations (Comparat et al. 2019, 2020; Seppi et al. 2022) to estimate the amount of spurious detections as a function of the detection likelihood threshold (see also Liu et al. 2022c). Based on these all-sky survey simulations, we define our Main sample as the one comprising all extended sources and all point sources with DET\\_LIKE\\_0 ⩾ 6. Table 3 of Seppi et al. (2022) indicates that the Main catalogue should contain ≈ 14% spurious detections. This reduces to about 1% for DET\\_LIKE\\_0 > 10. Pointlike sources with 5 ⩽ DET\\_LIKE\\_0 < 6 are released as a (highly contaminated) Supplementary catalogue. \nTo extract the hard sample from the 3B catalogue, we apply a threshold for the detection likelihood in the 2.3-5 keV band of DET\\_LIKE\\_3 ⩾ 12. The threshold is higher than the one applied for the Main catalogue as the lower sensitivity and higher background in that energy range significantly increases the number of spurious detection at a given detection likelihood (Liu et al. 2022c). Based on the same simulations described in Seppi et al. (2022), we estimate for this threshold a spurious sources fraction of about 8-10% in the hard-band selected sample. We note here that, as shown in Seppi et al. (2022), the expected fraction of spurious sources depends on the exposure, with lower fraction of spurious detections predicted for higher exposures. Here, for simplicity, we have adopted all-sky average estimates. \nTable 4. Basic eRASS1 catalogue properties. \n3B detection [2.3-5 keV only] \nCatalogue \nDET\\_LIKE\\_3 \nEXT\\_LIKE', 'of Sources': "Hard \nHard, PS \nHard, Ext. \n⩾ \n12 \n⩾ \n12 \n= \n0 \n5087 \n⩾ \n12 \n> \n0 \n379 \nNotes. 'PS' indicates point sources (i.e. those with EXT\\_LIKE = 0), and 'Ext.' indicates extent-selected sources with EXT\\_LIKE > 0. We note here that there is a slight di ff erence in EXT\\_LIKE parameter in the 1B and 3B detections, as for the latter all photons in the 0.2-5 keV band are used to evaluate the source extent. \nTable 4 presents a summary of the catalogues selection criteria and properties. In Fig. 4 we show the distributions of net counts for the Main and the Hard samples, respectively. \nA summary of the content of each catalogue is presented in the Appendix D (column descriptions, units). Below we describe in greater detail the catalogue creation procedure, the astrometric verification steps and our attempt to flag potential spurious sources.", '5.1. Catalogue creation and preliminary astrometric correction': "The catalogues resulting from the PSF fitting with task ermldet were re-formatted using the eSASS task catprep and then merged into hemisphere catalogues. Since the survey sky tiles overlap each other, the sources outside the nominal, nonoverlapping area were removed from each sky tile catalogue. In order to avoid the loss of significant sources detected by chance just outside the nominal areas in two adjacent tiles, detections in a ± 30 '' strip near the nominal borders were matched, using a matching radius of r = 15 '' for point sources and r = 30 '' for extended sources, and only one detection for each match was kept in the merged catalogue. These matching radii approximately correspond to half of the HEW and one HEW of the survey averaged PSF (see Fig. A.1). \nAs we discussed above, the eROSITA field of view and survey scanning strategy imply that a source near the ecliptic equator is visited about six times within a time span of one day in each of the eROSITA six-months surveys. The time span and the number of visits increase with the ecliptic latitude of the source. In the catalogues, the epoch of survey coverage was estimated for each source by using the attitude time series for camera TM1 to calculate the times of the observation closest to the optical axis as well as the first and last appearance of the source in the camera field of view. These epochs are listed in the columns MJD , MJD\\_MIN , and MJD\\_MAX . \nThe positional uncertainty of X-ray sources is an important parameter for their association with multi-wavelength counterparts, especially given the relatively large PSF of eROSITA. The ermldet task provides a statistical estimate of this quantity for individual X-ray sources based on the spatial distribution of their X-ray photons and a PSF model (see Brunner et al. 2022, Appendix A.5, for more details). These measurements may under- \n⩾ \n0 \n5466 \nestimate the true positional errors because of, e.g., calibration uncertainties and other systematic e ff ects (e.g. Webb et al. 2020). In particular, the astrometric accuracy of the eROSITA all-sky catalogues can be a ff ected by the following systematic factors: \n- -Errors in the timing between attitude measurements and event arrival;\n- -Boresight calibration errors;\n- -Systematic errors introduced by the PSF fitting. \nAs for the latter point, eRASS1 data, just like those of the CalPV phase, have been analysed using the PSF model derived from on-ground calibration. The analysis presented in Appendix A demonstrate that this e ff ect is negligible, so we discuss only the former two below. \nDue to the SRG scanning geometry, a timing mismatch between the attitude measurements and event timing would result in an o ff set along the scanning direction, i.e., ecliptic latitude β . An o ff set in the boresight between the SRG attitude solution and the eROSITA cameras would result in a constant astrometric o ff set in ecliptic coordinates during a 180 · scan between the two ecliptic poles. Assuming that any timing or boresight o ff sets vary only slowly with time, in order to correct for this e ff ect we divided the merged catalogue into stripes of 1 · width in ecliptic longitude λ , corresponding to ∼ 1 day of survey scanning. For each stripe the X-ray positions of point-like detections at ecliptic latitudes ( -60 · to + 60 · ) were matched with mid-infrared counterparts from the AllWISE catalogue (Cutri et al. 2021). After applying a colour cut (0 . 3 mag < W 1 -W 2 < 1 . 7 mag and 2 . 2 mag < W 2 -W 3 < 3 . 6 mag) to select for likely QSO matches, median values of the o ff sets β X -β IR and ( λ X -λ IR) × cos( β ) were calculated. All X-ray positions within each latitude stripe were then corrected using the median o ff sets in ecliptic longitude and latitude ( ∆ λ × cos( β ), ∆ β ). The applied o ff sets range between -4 . 0 '' and + 3 . 9 '' and -1 . 5 '' and + 2 . 5 '' , respectively. \nThe resulting statistical errors are given in each coordinate as upper and lower bounds (columns RA\\_LOWERR , RA\\_UPERR , DEC\\_LOWERR , DEC\\_UPERR ). An error estimate averaged over both dimensions and directions is given as \nσ RA = (RA\\_LOWERR + RA\\_UPERR) / 2 \nσ DEC = (DEC\\_LOWERR + DEC\\_UPERR) / 2 \nRADEC\\_ERR = p ( σ RA) 2 + ( σ DEC) 2 , (1) \nin line with other X-ray catalogues (e.g. Webb et al. 2020). \nThe upper panels of Figure 5 displays the distribution of RADEC\\_ERR as a function of DET\\_LIKE\\_0 for the point sources in the Main and Hard catalogues, respectively. For sources where the calculation of RADEC\\_ERR failed (see Sect. 4.6), we calculate RADEC\\_ERR using the empirical correlation extracted from Fig. 5. It should be noted here that RADEC\\_ERR does not represent the 68% error radius for two parameters. Under the assumption of a circular error region, the averaged 1-dimensional 68% error as required e.g. for the comparison with a Rayleigh distribution can be derived with σ = RADEC\\_ERR / √ 2. We further elaborate on the astrometric accuracy of the eRASS1 catalogue in Sect. 6.2, where we present a validation method based on a comparison with external catalogues, which reveals the extent of the systematic uncertainty beyond the statistical one described here. \nThe lower panels of Fig. 5 display the distribution of flux significance as a function of detection likelihood for the point sources in the main and hard catalogues. For the large majority of the sources, the flux measurements have large uncertainties; in order to have at least 3σ flux measurements in the Main catalog, one could adopt an approximate threshold of DET\\_LIKE\\_0 > 20. \nFig. 4. Distributions of net counts for the Main (0.2-2.3 keV; Left panel ) and Hard (2.3-5 keV; Right panel ) catalogues. Point sources ( EXT = 0) and extended sources ( EXT > 0) are plotted in blue and orange, respectively. Point sources with any SP flag (see Table 6) are displayed as blue shaded histograms. The sources that appear in both the Main and the Hard catalogues are plotted in red in the left panel. \n<!-- image -->", '5.2. Flagging of problematic sources': 'The imperfect nature of the source detection process inevitably leads to contamination of the eRASS1 catalogue by spurious sources and / or inaccuracies in the derived source properties. The most clearly identifiable examples of spurious detections can be found within the vicinity of extremely bright X-ray point sources, such as Sco X-1, or bright, large extended sources, like supernova remnants, nearby galaxies, or galaxy clusters (see Fig. 6), whereas less clear-cut cases can be found at the lowest detection likelihoods, and their contribution to the eRASS1 catalogue quantified via simulations. Optical loading of the CCDs could also introduce fake X-ray sources in the catalogue. \nTo warn users of a potential spurious origin for a detection, despite a possibly high-detection likelihood, we have flagged sources that are located within overdensities in the eRASS1 source catalogue associated with systems that could create problems for the automatic background estimation in the detection pipeline (supernova remnants, extremely bright X-ray point sources, Galactic star clusters, local galaxies, and galaxy clusters). We also flag catalogue entries matched to very bright optical stars, as we describe below.', '5.2.1. Identification of overdense regions': "In order to identify those regions on the sky where many potentially spurious sources are clustered, we performed an empirical search for regions with a suspiciously large number of detected sources compared to their surroundings: after performing a uniform cut on the detected flux at F 0 . 2 -2 . 3 keV > 5 × 10 -14 erg s -1 cm -2 , to reduce dependence on the spatially varying exposure, we computed a density map of point-like and extended sources in the single-band catalogue, using a pixelisation of 0 . 25 deg 2 . A 'background' source density map was then created by applying a median filter with a radius of 10 · . By comparing the two maps, we identified all regions with a local source density more than twice the background. The exact shape of the overdensities was extracted by 'zooming in' on each identified region, creating a smoothed histogram of the local source distribution, and selecting all regions with a density larger than three \ntimes the local background which contribute more than 20 excess sources. \nWhile this procedure yields many overdensities caused by truly spurious source excesses, some resulting regions are expected to correspond to accumulations of real astrophysical sources. We thus manually identified the correspondence of each of the ∼ 80 localised overdensities to astrophysical sources, using the SIMBAD database (Wenger et al. 2000). Our overdense regions were then classified, and the enclosed sources flagged, according to their correspondence to i) di ff use emission associated with known supernova remnants ( FLAG\\_SP\\_SNR ), ii) excess emission in the vicinity of extremely bright point sources ( FLAG\\_SP\\_BPS ), iii) Galactic star clusters ( FLAG\\_SP\\_SCL ), iv) nearby galaxies ( FLAG\\_SP\\_LGA ). \nThis classification might be useful, for instance, if one were interested in studying the population of Milky Way point sources, as one would likely want to mask spurious sources caused by mis-classified di ff use emission from supernova remnants, but might not want to mask Galactic stellar clusters.", '5.2.2. Galaxy cluster catalogues': "Another flag is applied for possible spurious sources which are located close to known galaxy clusters. To do that, we make use of published X-ray cluster catalogues, including MCXC (Piffaretti et al. 2011), XXL365 (Adami et al. 2018), XCS (Mehrtens et al. 2012), eFEDS (Liu et al. 2022a), and X-CLASS (Clerc et al. 2012). Sources lying within R = 0 . 5 × R 500 from the cluster center are flagged as FLAG\\_SP\\_GC\\_CONS . When R 500 is not provided in the published catalogue, we adopt a radius of R = 500 kpc. We note that, to avoid over-counting, optical cluster catalogues are not used in this step, because large o ff sets between clusters' X-ray centers and optical centers are frequently observed (see, e.g., Seppi et al. 2022, 2023). Cluster catalogues selected on the basis of the Sunyaev-Zeldovich (SZ) e ff ect are not included either, due to the relatively large positional uncertainties in SZ surveys. Therefore, the catalogue of known galaxy clusters we used in the above approach is a rather conservative and incomplete compilation, and the identified spurious sources should be considered as a supplement for the identifi- \nFig. 5. Distributions of RADEC\\_ERR (upper panels) and flux measurement significance (in terms of ML\\_RATE\\_1 / ML\\_RATE\\_ERR\\_1 , lower panels) as a function of detection likelihood for point sources in the eRASS1 single-band detected catalogue (left panels) and in the Hard catalogue (right panels). In the left panels, the 40%, 68%, and 95% contours are plotted in orange, while the cyan line indicates an empirical correlation that describes the mode of the distribution, as reported in the label. Sources from the Main catalogue are shown in blue, those from the Supplementary catalogue in grey. The right panels ( Hard catalogue) have only the 68% contour plotted. Note that the lower right panel displays the likelihood and flux significance in the 2.3-5 keV band. \n<!-- image --> \nTable 5. Spurious and problematic source flag description. \nation of overdensities. Further cleaning is performed only for galaxy cluster candidates in the extended source catalogue. An extended source is flagged as a possible spurious detection when it is too close to its neighbour. Visual inspections are also performed on the extended source catalogue to remove any remaining obvious spurious sources and correct for mis-classified cases. We refer the readers to Bulbul et al. (2024) for more details of the cleaning procedure we performed in the extended source catalogue. \n5.2.3. Summary of spurious sources flagging procedure \nThe flagging procedure described above identifies cases where high local background levels render the automatic pipeline detection process unreliable. This is illustrated in Fig. 7, which shows the background rate distributions for sources with and without flags: the flagged sources are mostly found in regions with enhanced background rate. \nFig. 6. Top panel : Aito ff projection of the eRASS1 1B catalogue in Equatorial coordinates (J2000), with each grey point representing a detected source within the catalogue, and the coloured points denoting sources that have been flagged as potentially spurious according to the scheme presented in Table 5. Darker stripes are due to larger sources density due to a higher exposure in those parts of the sky. Bottom left : Zoom in plot of sources detected within the vicinity of Sco X-1, with pink sources flagged as potentially spurious using the FLAG\\_SP\\_BPS column in the catalogue. Bottom right : Similar for sources within the vicinity of the Vela SNR. \n<!-- image --> \nA summary of the di ff erent spurious sources flags added is presented in Table 5, while Table 6 reports the number of sources for each of the flag categories. After removing all those sources that are flagged by at least one of the potential spurious categories, the Main catalogue contains 890 036 point sources, with a median sky density of approximately 37 deg -2 . \nA note of caution is in place here: although steps are taken here to greatly reduce the number of high-detection-likelihood contaminants, it is still recommended that users double-check the relevant science images for their sources of interest before publication (e.g., if their point sources lies within a likely galaxy cluster but is not flagged as spurious here).", '5.2.4. Optical loading': "The eROSITA detectors are prone to optical loading, i.e. the accumulation of low energy (optical / UV) photons within a CCD pixel over the frame integration time of 50 ms, whereas the relevance of the e ff ect is governed by the optical brightness and detector position of the respective source. If the summed energy exceeds the X-ray detection threshold, any optically bright source will start to generate false X-ray events. Due to the na- \nTable 6. Number of sources flagged as potentially spurious. \nNotes. 'PS' stands for Point Sources, 'Ext.' for extent-selected, and we separate the Main (' M ') and the Hard (' H ') catalogues. The 'Any SP Flag' row indicates the number of sources that are flagged by any of the five identifiers of potential spurious sources in overdense regions, SP\\_SNR , SP\\_BPS , SP\\_SCL , SP\\_LGA , and SP\\_GC\\_CONS . The last three lines give the number of sources for which statistical error estimate from the pipeline failed (see Sect. 4.6 for more details). \nFig. 7. Top panel: distribution of local background rate level ( ML\\_BKG\\_1 / ML\\_EXP\\_1 , in units of counts arcmin -2 s -1 ) for sources with and without flags and DET\\_LIKE\\_0 > 10. Bottom panel: the fraction of each corresponding sub-sample among all the sources above any background level. \n<!-- image --> \nture of the e ff ect, these events appear predominantly at lower Xray energies. The strength of the optical loading signal strongly increases with increasing source brightness; however, for very bright sources 'saturation-like' e ff ects may occur, when events are removed by the pattern filtering procedure. The optical loading signal further depends on the o ff -axis angle of the source, as the optical PSF is sharper close to FOV center and photons are focussed on a smaller detector area. i.e. fewer pixels. During eRASS data taking, any optically bright source passes several times on di ff erent scan paths through the FOVs and thereby a time dependent optical loading signal may be generated. If a su ffi cient number of X-ray events is created, source detection triggers and the object makes it into the eRASS catalog. Due to the required intrinsic brightness, stars are by far the main contributors to optical loading sources in our data. \nAmong the practical consequences for astrophysical studies are the presence of fake X-ray sources, pseudo X-ray variability and the distortion of real X-ray sources, as the source may of course be optically bright and also an intrinsic X-ray emitter. If the optically bright source is X-ray dark, or at least faint enough to fall well below the eRASS sensitivity limits, all the registered events would come from optical loading. These fake X-ray sources always have very soft spectra and show pseudo variability. If the optically bright sources are also X-ray bright, these characteristics are likewise present, but the detected signal is a mixture of true X-ray photons plus optical loading events and contamination e ff ects. Any potential disentangling between these e ff ects depends on the individual source properties and on the detailed science objectives, but in general the X-ray properties of sources contaminated by optical loading are uncertain. \nTo best characterise the optical loading e ff ect, X-ray dark sources are required and two suitable stellar populations are used here: main-sequence and mildly evolved stars with spectral types late-B to mid-A and red giants, i.e. K / M stars with luminosity class III-I. Cross-matching these with the eRASS1 catalogs shows that optical loading is expected to a ff ect DR1 sources if they exceed certain brightness thresholds, and likely a ff ected X-ray sources are flagged in the catalog. The adopted brightness limits are B, V, G ≤ 4 . 5 mag and J ≤ 3 mag; if one or more criteria are fulfilled, then the source is tagged with FLAG\\_OPT . In total, 750 (17, 14) sources are flagged as potentially contaminated by optical loading in the Main ( Supplementary , Hard ) catalogues, respectively. Input for cross-matching are the Tycho-2 (Høg et al. 2000), 2MASS (Cutri et al. 2003), Gaia DR3 (Gaia Collaboration et al. 2023) catalogues plus Simbad database. Where proper motion information is available, optical catalog entries are updated to epoch 2020 positions, and a uniform matching radius of 15 '' is used. A full treatment depends on the individual source properties and is beyond the scope of this work, but users should be aware that specific attention is required when dealing with eROSITA data of brighter stars.", '5.3. Association of soft and hard band selected sources': 'As discussed above, we performed source detection with two di ff erent settings: a single X-ray band (1B, 0.2-2.3 keV) and a three-band detection (3B, 0.2-0.6, 0.6-2.3, and 2.3-5 keV), subsequently down-selected based on the 2.3-5 keV band significance. These produce, respectively, soft and hard X-ray selected source samples. The data used in the 1B and 3B detection are nonetheless largely overlapping. The only photons that are included in the 3B detection but not in the 1B are those in the 2.35 keV band, where the instrumental e ff ective area is relatively low and the background relatively high. For this reason, most of \nthe Hard catalogue sources are expected to have a matching entry in the 1B catalogues. This was the case also in the eFEDS survey, where 90% of hard band sources were found to have a counterpart in the main, soft X-ray selected catalogue (Nandra et al., in prep.). It is nonetheless of particular interest to identify those sources that are only detected in the hard band, as they signpost objects with extremely hard spectra, most obviously due to heavy obscuration. The association between sources in the two catalogues is not straightforward, however. The complexity arises due to various factors, such as the positional uncertainties, the morphological classification (extent measurement) and blending with nearby objects. In this section, through a specific matching procedure between the 1B and 3B sources, we provide a \'weak\' association that can be used to select sources only detected in the hard band and a \'strong\' association that can be used to select entries in the X-ray catalogues with a high degree of confidence that the X-rays originate from the same astrophysical source. \nThe weak association is based only on positional information. Since for extended sources the value of the extent parameter EXT (i.e. the best-fit core size of the beta model fitted to an extended source) is very broadly distributed, ranging from ∼ 10 \'\' to the maximum allowed value of 60 \'\' , to search for a counterpart to those sources we adopt a large matching radius of four times the EXT value. For point sources ( EXT = 0), we simply adopt a matching radius of 10 \'\' , which is 99 percentile of the point source RADEC\\_ERR in the main catalog. For each 1B source, we search for 3B sources within its matching radius, and for each 3B source, we search for 1B sources within its matching radius. Then we merge the results, which is equivalent to adopting the larger radius among the two. As a result of such loose association criteria, one source could be associated with multiple sources, some of which are false matches and some of which have di ff erent morphological classifications. To classify these matches, we define P2P, E2E, P2E, and E2P matches to denote pairs of point source (P) and extended sources (E); the former and the latter letters (P or E) indicate the classifications in the 1B and 3B catalogues, respectively. Hard band sources which do not have any counterpart in the soft band catalogues within the match radius are designated as hard band only sources. \nThe \'strong\' associations require more strict criteria. First, we require that each pair of sources have the same morphological classification, i.e., we adopt only E2E or P2P pairs. With E2E associations, one source can be matched to multiple ones. We adopt only the nearest match in such cases, so that the involved sources are unique. In the cases of P2P matches, one source is never matched to multiple counterparts, but in addition to a maximum separation of 10 \'\' , we further require i) separation / error < 3, where error is the larger of the two RADEC\\_ERR , and ii) the 1B and 3B measured 0.2-2.3 keV (combining band 1 and 2 in the 3B case) source counts cannot di ff er by > 50%. In this way, we enforce not only that the matched sources are su ffi ciently close in sky position, but that also have a consistent broadband brightness, such that sources with the same position but di ff erent extents (caused by de-blending uncertainties in crowded regions) can be excluded. We store the counterpart unique source ID ( UID ) in the UID\\_Hard column of the Main and Supplementary catalogues and the UID\\_1B column of the Hard catalogue as indicators of strongly associated sources. \nFor the weak associations, we also store the counterpart UID in the UID\\_Hard or UID\\_1B columns but multiplying it by -1. In this way, catalogue columns UID\\_Hard > 0 or UID\\_1B > 0 indicate that a source has a strong association in the other catalogue, while catalogue columns UID\\_Hard < 0 or UID\\_1B < 0 \nFig. 8. Distributions of 0.6-2.3 keV (x-axis) and 2.3-5 keV (y-axis) count rates of the hard-band selected sources. The "Hard-only" sources are marked in red colour. The solid line indicates the ratio (0.084) that corresponds to an unabsorbed power-law spectrum with a slope of 1 . 7. The dashed line indicates a 1:1 ratio. \n<!-- image --> \nindicate that a source may have a (weak-association) counterpart in the other catalogue. Finally, UID\\_1B = 0 marks the Hard sources that have no counterpart within their matching radius in the 1B catalogues. Following this procedure, we find 780 \'Hardonly\' sources; of these, 764 are point-like ( EXT = 0) and 16 extended ( EXT > 0). Figure 8 displays the count rate distribution of the hard-band selected sources in the 0.6-2.3 and 2.3-5 keV bands. The Hard-only sources identified above (red points) have significantly higher hardness than the others, by construction. \nThe matrix of possible identifications of Hard sources in the 1B catalogues, following our association criteria, is given in Table 7. For point sources, about 84% have a strong association, 15% are Hard-only and just 1% have weak associations in the 1B catalogues. These fractions do not change if one considers only sources without any spurious flag. Among extended sources in the Hard catalogue, on the other hand, a larger fraction is flagged as potentially spurious. Of the remaining ones (136 in total), about 80% have a strong association with a 1B catalogue extended source, about 9% have a weak association in the 1B catalogues, and 11% (15 in total) are Hard-only. Such Hard-only extended sources are most probably mis-classified point sources; indeed, only 1 / 15 of these has extent likelihood EXT\\_LIKE > 6 and extent EXT > 30 \'\' .', '5.4. The flux limit and completeness of eRASS1 in the 0.5-2 keV band': "To quantify the flux limit of the eRASS1 survey in the commonly adopted energy band 0.5-2 keV, we use the aperturephotometry-based method based on the apetool task (see Sect. 4.3.4). A local flux limit can be defined based on the local sensitivity curve as the flux where the probability of collecting a su ffi cient number of source photons to constitute a detection at the given threshold reaches a particular value, for example 50%. To calculate a sensitivity curve as a function of flux, we need an ECF to convert count rate to flux. Considering the nonuniform Galactic absorption, we assume a power-law spectral model with Γ = 2 and the total Galactic absorption column density (Willingale et al. 2013) at each source position, and calculate two ver- \nTable 7. Summary of identification of Hard (3B) sources in the 1B (Main and Supplementary) catalogues.Notes. FLAG = 0 selects sources with all the five spurious flags in the Hard catalogue being 0. Here, P2P, E2P, P2E and E2E refer to possible combinations of matched pairs between the 1B and 3B catalogs, based on the morphology of the sources: point-like (P) or extended (E); the former and the latter letters (P or E) indicate the classifications in the 1B and 3B catalogues, respectively. See text for more details. \nFig. 9. Bottom panels: Hammer-Aito ff projection maps, in Galactic coordinates, of the logarithm of the 0.5-2 keV flux limit calculated as the flux at 50% sky covering fraction (see text for details). The left and right panels correspond to values uncorrected and corrected for Galactic absorption, respectively. Six particular regions where the flux limits are increased by bright X-ray sources are marked with red circles; from the north (top) to the south (bottom) they are: the Virgo cluster, Sco X-1, the Vela SNR, the Crab pulsar, the LMC, and the SMC. Top panels: histogram of the logarithm of the flux limit in erg s -1 cm -2 . The colour bar on the X-axis illustrates the intensity scale of the corresponding map in the bottom panels. \n<!-- image --> \nions of ECF for each source, one correcting the 0.5-2 keV flux for absorption and the other not. \nForced photometry results at source positions in the 0.5-1 (band P2) and 1-2 keV (band P3) bands are available in the catalogue and can be combined into the 0.5-2 keV values. We sum the total counts ( APE\\_CTS\\_P2 + APE\\_CTS\\_P3 ) and background counts ( APE\\_BKG\\_P2 + APE\\_BKG\\_P3 ) to calculate APE\\_CTS\\_S and APE\\_BKG\\_S , where the su ffi x 'S' (Soft) indicates the 0.52 keV band. To combine the vignetted exposure times in P2 and P3, we measure the weighted average as \nAPE\\_EXP\\_S = (1 + w ) / 1 APE\\_EXP\\_P2 + w APE\\_EXP\\_P3 ! , \nwhere w is the relative weight factor, which depends on the ECF adopted to create the exposure maps in the P2, P3 and S bands. By comparing these three maps, we find a proper factor of w = 0 . 6268. The distribution of sources in the space of background counts and exposure time shows a strong concentration along a linear correlation, which indicates the typical \nbackground ( APE\\_BKG\\_S / APE\\_EXP\\_S ∼ 0 . 003 counts / s). There is a small fraction ( ∼ 1%) of obvious outliers below 0.002 counts / s, which indicates an underestimated background in low S / N regions. We adopt a minimum value of 0.002 counts / s to correct them. Having computed APE\\_CTS\\_S , APE\\_BKG\\_S , APE\\_EXP\\_S , and the ECF, we can calculate the Poissonian probability ( APE\\_POIS\\_S ) of each source and the sensitivity curve determined by the local background using the Python package 'scipy.special' 12 , which provides the same function as in the apetool task. The values of APE\\_POIS\\_S are reported in the Main and Supplementary catalogues, so that a sample with a fixed FAP threshold can be defined by the user as needed. \nWe here adopt an aperture size for photometry corresponding to a radius that includes 75% of the PSF photons (i.e. encircled energy fraction of 0.75), while the Poisson false detection probability threshold is set to P thresh = 4 × 10 -6 (following Georgakakis et al. 2008). To project the flux limits to a map, \nwe subdivide the sky into HEALPix 13 pixels (Zonca et al. 2019; Górski et al. 2005), adopting a HEALPix resolution order of 6. Through Voronoi tessellation, we also pixelise the sky into cells, each of which contains only one source. In each HEALPix pixel, we average the sensitivity curves of all the sources and weigh each source by its Voronoi cell area. Adopting the 50% flux of the averaged sensitivity curve, we obtain a flux limit for each HEALPix pixel and thus a flux limit map of the hemisphere. \nWe create two distinct maps, as displayed in Fig. 9. The left panel shows the flux limit of a (galactic) X-ray point source without consideration of the galactic absorption. The flux limit map follows the exposure map and the di ff use background map well. The right panel shows the case of an (extragalactic) Xray source obscured by the total Galactic N H. In the absorption corrected case, the high N H in the Galactic plane boosts the flux limit. The hemisphere median flux limits are approximately 5 × 10 -14 erg s -1 cm -2 (before Galactic absorption correction) and 6 × 10 -14 erg s -1 cm -2 (after Galactic absorption correction), respectively. If we were to increase the probability threshold to 80%, the corresponding flux limit would increase by about 50%. \nWe provide here some basic characteristics for two examples of sub-sample selection with well-defined statistical properties: a flux-limited one, and one above a fixed Poisson false-probability threshold. A selection of point sources obeying F 0 . 5 -2keV > 5 × 10 -14 erg s -1 cm -2 (observed, i.e. not corrected for absorption) returns a sample with 207 439 entries (after removing all those that are flagged by at least one of the potential spurious source categories), with a median sky density of approximately 9.8 deg -2 . On the other hand, a selection of point sources obeying APE\\_POIS\\_S < 4 × 10 -6 returns a sample with 229 266 entries (after removing all those that are flagged by at least one of the potential spurious source categories), with a median sky density of approximately 8.6 deg -2 . \nFinally, an estimate of the completeness of the eRASS1 catalogue is provided by the analysis of the detailed 'digital twin' eRASS1 simulation presented in Seppi et al. (2022). There it is shown how the point sources' completeness as a function of 0.5-2 keV flux changes with increasing exposure. Based on that work, Figure10 shows the estimated completeness for point sources, expressed as the ratio of detected to simulated AGN from the analysis of the simulations as a function of the net exposure time (for the Main catalog with DET\\_LIKE\\_0 > 6).", '5.5. Number counts of point sources': 'Before eROSITA, deep extra-galactic X-ray surveys 14 were generally confined to particular regions with areas of at most tens of square degrees. The largest contiguous X-ray survey before eRASS1 is the eFEDS (Brunner et al. 2022), which covers an area of 140 deg 2 . In such (relatively) small extra-galactic regions, the point source number density can be considered uniform. Serendipitous surveys, built by combining narrow field exposures (with Chandra , XMM-Newton , Swift ) can provide even larger sky coverage, albeit very patchy. However, in eRASS1 it becomes apparent that the X-ray point source number density is nonuniform across the sky. The first-order reasons for this are obviously the inhomogeneous Galactic absorption (see e.g. Ponti et al. 2023a) and the non-uniform distribution of the Galactic X-ray population itself. Even after excluding these Galactic fea- \nFig. 10. Estimated completeness of eRASS1 from the simulations of Seppi et al. (2022) as a function of input 0.5-2 keV flux for di ff erent exposure times. The completeness is estimated considering only point sources ( EXT\\_LIKE = 0) with DET\\_LIKE\\_0 > 6. \n<!-- image --> \ntures, the number density of distant AGN may not necessarily be uniform either, due to the inhomogeneous large-scale structure and the potential anisotropy (e.g., a dipole structure; see Secrest et al. 2021) of the Universe. More details about the eROSITA point-source number density maps are presented in Liu et al. (in preparation). In this section, we present the point-source number count distributions averaged in a few wide Galactic latitude ranges. \nWe divide the hemisphere into four Galactic latitude levels, 0-10 · , 10-20 · , 20-40 · , and 40-90 · . As introduced in the previous Sect. 5.4, we calculate the point source number counts using the method described by Georgakakis et al. (2008) based on the products of apetool for the 3B catalogue in the 0.6-2.3 keV band. Through the N H-dependent ECF (see Table 3), we convert the count rate of each source into absorption-corrected flux. The uncertainties of the cumulative X-ray number counts are estimated using a bootstrap re-sampling approach. For a given eROSITA sub-sample selected within a given Galactic latitude interval, the X-ray sources are randomly drawn with replacement to generate 100 new samples with the same size as the original one. The X-ray number counts are then generated for each of the 100 samples following the same approach as with the real data and the uncertainty at fixed flux is then estimated as the 1σ rms scatter of the 100 number count realisations. \nThe resulting cumulative number count distributions as a function of the absorption-corrected flux and the corresponding sensitivity curves are displayed in Fig. 11. The number counts obtained in sky regions with \| b \| > 20 · (high Galactic latitudes) are consistent with each other and with the results of Georgakakis et al. (2008), while a ∼ 30% excess is seen in the counts at high fluxes reported in Hasinger et al. (2005). These latter are based on a total of about 200 type-1 AGN selected from the ROSAT Bright Survey (RBS; Schwope et al. 2000). The systematic o ff set with the eRASS1 results may be related to e.g. uncertainties in the sensitivity calculations of the ROSAT sample, or \n<!-- image --> \nFig. 11. Left : Cumulative number counts as a function of flux for eRASS1 X-ray point sources selected in the 0.6-2.3 keV band. The apetool sensitivity maps and the eRASS1 aperture photometry for individual sources are used to construct the number counts in di ff erent Galactic latitude intervals, \| b \| = 0-10 · (brown dashed curves), 10-20 · (yellow solid curves), 20-40 · (purple dashed curves) and 40-90 · (blue solid curves) following methods described in Georgakakis et al. (2008). The shaded region associated with each of these curves corresponds to the 68% uncertainty in the determination of number counts estimated using the bootstrap resampling method described in the text. The curves are plotted up to the flux of 2 × 10 -14 erg s -1 cm -2 where the apetool sensitivity curve corresponds to about 1% of its maximum value. For comparison also shown are the 0.5-2 keV number counts determined by (Hasinger et al. 2005, black solid line) using ROSAT, XMM-Newton and Chandra surveys. The red shaded region and red dashed curve are the 0.5-2 keV number counts estimated from Chandra extra-galactic survey fields (Georgakakis et al. 2008). The extent of the red shaded region at fixed flux corresponds to the 1 σ uncertainty. For fluxes brighter than fX (0.5-2 keV) ≈ 10 -12 erg s -1 cm -2 the red dashed curve is an extrapolation of the best-fit double power determined by Georgakakis et al. (2008). Right : The corresponding sensitivity curve, i.e. area coverage vs. flux limit in the 0.6-2.3 keV band, for di ff erent galactic latitude intervals. \n<!-- image --> \nthe inclusion of more extended sources in RBS because of the larger ROSAT PSF. \nThere is a significant excess for the source population at low Galactic latitude. The absorbed spectral model adopted in the flux calculation is only valid for un-obscured AGN, and the cosmic variance of AGN is of small amplitude and unrelated to Galactic latitude. So the excess must be due to Galactic sources, for which the invalid absorption correction biases high the source fluxes and thus the number counts. Therefore, the low-latitude samples trace the distribution of Galactic X-ray sources (see Freund et al., submitted).', '5.6. The resolved fraction of the Cosmic X-ray Background in eRASS1': 'The cosmic X-ray background (CXB) radiation, discovered by Giacconi and collaborators at the dawn of X-ray astronomy in 1962, can be considered as the ultimate inventory of the energy released by high-energy processes throughout the history of the Universe. As the CXB radiation is dominated by accretion onto black holes, detailed modelling of the CXB over the years has accompanied our deeper understanding of the physical properties of AGN, and of their cosmological evolution. Indeed, deep extra-galactic X-ray surveys have resolved about 80-90% of the CXB using synthesis models of the obscured and un-obscured AGNpopulation (e.g Setti & Woltjer 1989; Comastri et al. 1995; Treister & Urry 2005; Hickox & Markevitch 2006; Gilli et al. 2007; Ueda et al. 2014; Aird et al. 2015; Ananna et al. 2019). \nHere we provide an estimate of the fraction of the CXB radiation resolved into individual sources by eRASS1 in the soft \nband (1-2 keV), where the eROSITA sensitivity is around its maximum, and the CXB contribution to the measured background emission dominates over both the foreground Galactic components and over the instrumental background (see e.g. Ponti et al. 2023b). We first adopt a mean value of the CXB intensity in the 1-2 keV band of 12.7 keV cm -2 s -1 sr -1 by fitting the data compilation of Gilli et al. (2007). We then compute the fraction of this intensity in the same energy range contributed by highly reliable eRASS1 sources with DET\\_LIKE\\_0 > 10 (excluding all flagged ones 15 ; see Table 5) above two flux limit thresholds: F 0 . 5 -2 keV > 5 × 10 -14 erg s -1 cm -2 or F 0 . 5 -2 keV > 1 × 10 -14 erg s -1 cm -2 . As discussed in Sect. 5.4 above, the former represents an approximately uniform flux limit achieved over the entire sky, while the latter includes fainter sources detected in the areas of increased exposure near the ecliptic poles. \nThe two bottom panels of Fig. 12 show a HEALPix map (order 4) of the 1-2 keV CXB intensity resolved fraction in the eROSITA-DE hemisphere. On the right hand side, the fraction computed using the eRASS1 flux limit appears more uniform, with the exception of the low-Galactic latitude regions, where Xray stars in nearby ( < 500 pc) star-forming regions (Sco-Cen association and Gould belt; Schmitt et al. 2022) increase the cumulative average emission. On the left hand side, instead, a significantly higher fraction of the CXB is resolved into fainter sources close to the south ecliptic pole (SEP). Finally, in the top panels of Fig. 12 we show the histogram of the number of HEALPix order 4 pixels at a given resolved fraction in the extra-galactic sky ( \| b \| > 30 · ) for the two cases. The solid lines are the median \nvalues: 0.19 and 0.24, respectively. Including only point sources (i.e. removing all those with EXT\\_LIKE > 0) the median value of the resolved fraction is reduced by about 2%, to 0.17 and 0.22, respectively. On the other hand, if we include all (non-flagged) sources with DET\\_LIKE\\_0 > 6 the median value for the CXB resolved fraction remain almost unchanged (0.20) for the uniform flux limit case ( F 0 . 5 -2 keV > 5 × 10 -14 erg s -1 cm -2 ), but increases to 0.33 for F 0 . 5 -2 keV > 1 × 10 -14 erg s -1 cm -2 , due to the large number of fainter sources detected with lower significance in the deeply exposed area near the SEP.', '5.7. Particular regions': 'In the soft X-ray band, a few nearby sources of X-ray emission are very prominent, including the eROSITA bubbles (Predehl et al. 2020), the two Magellanic clouds, the Vela SNR, Virgo cluster, Crab Pulsar, and Sco X-1. They enhance the e ff ective background for the detection of distant sources and thus appear clearly in the flux limit maps displayed in Fig. 9. As discussed in Sect. 5.2.1, our source detection algorithm is not optimized for such regions with bright or extended source contamination. This results in an increased number of spurious sources caused by background fluctuations. There are large uncertainties in the measurement of background and source properties in such regions. \nIn addition to these, the SEP region also poses particular issues for source detection, because the exposure is deeper than the typical depth across the sky by more than two orders of magnitude. Our detection algorithm is not optimized for such deep exposures either. As discussed by Liu et al. (2022c), one di ffi culty of source detection is de-blending nearby sources (or confusion). This problem is more severe where the fields are crowded, like in the regions near the ecliptic poles. Although the eRASS1 catalogues presented here are complete (i.e. includes also sources in the SEP region), an optimized SEP catalogue will be presented in Liu et al. (in preparation), where a more sophisticated data processing and source detection optimized for crowded regions with deep (and spatially variable) exposure will be introduced.', '6.1. Comparison with XMM source fluxes': "Before eRASS1, the previous best all-sky X-ray catalogue was provided by the ROSAT all-sky survey (2RXS: Boller et al. 2016). In terms of pure source numbers, the largest X-ray catalogue previous to eRASS1 is the XMM-Newton serendipitous catalogue (4XMM: Rosen et al. 2016; Webb et al. 2020). Covering only a small fraction of the sky, 4XMM has a similar resolution to eROSITA and generally deeper exposure than eRASS1. Therefore, we use the 4XMM-DR12 catalogue for a consistency check of our eRASS1 sources. \nWe selected 1250 . 8k point sources from the eRASS1 1B catalogue and 432 . 9k point sources ( SC\\_SUM\\_FLAG = 0 and SC\\_EXTENT = 0) from 4XMM-DR12 over the entire sky, and performed a simple positional cross-correlation between them, adopting a maximum separation of 10 '' . We found 16 . 5k detection pairs. The majority of the X-ray sources are in the low S / N regime, where their fluxes su ff er large uncertainties and Eddington bias. Taking the ratio of 0.2-2 keV flux to flux uncertainty as a measurement of flux measurement reliability, we selected sources with at least 1 σ reliability in both catalogues and compared their fluxes in the upper panel of Fig. 13. For faint sources, the Eddington bias is still visible in terms of the overestimated \neROSITA fluxes at the low-flux end. For the brightest sources with 5 σ reliability, the XMM-Newton and eROSITA measured fluxes are consistent, with the mode of the distribution of the flux ratio consistent with unity within 6%. Assuming the XMMNewton fluxes in this band are correct, this can in turn be interpreted as an estimate of the flux calibration uncertainty of eROSITA in the 0.2-2 keV band. \nWe also compared the hard-band fluxes of the Hard sources with that measured in the 4XMM catalogue, as displayed in the lower panel of Fig. 13. Here we are using slightly di ff erent bands (2.3-5 keV for eROSITA and 2-4.5 keV for XMM-Newton ), but we note here that assuming a power-law spectrum with a slope of Γ = 1 . 7 the fluxes in these two bands are expected to be almost identical, with a di ff erence < 1%. Because of the small sample size (only 131 matches) and the small eROSITA source photon counts in the hard band, this comparison is dominated by selection biases, even though residual e ff ective area calibration issues cannot be ruled out. \nA small fraction of the sources show significant variability, almost all of which turn brighter in eRASS1. This is because, typically, XMM-Newton observations are much deeper than eRASS1, and thus the sources that become fainter are filtered out by selection bias. The analysis of these variable sources will be presented elsewhere.", '6.2. Astrometric validation': "This section presents a methodology that provides independent constraints on the X-ray positional uncertainty using external multi-wavelength source catalogues with known and accurate positions. \nWe assume that each source i has a circular, Gaussian positional uncertainty with a standard deviation of σ i . It is then possible to cross-correlate two catalogues and count the number of matches as a function of angular distance. An example of such an angular separation distribution is shown in Fig. 14 and consists of a linear part at large angular distances and a pronounced peak at small separations. The former represents random matches between X-ray sources and the external catalogue, the latter is related to true associations. The number of random matches at a given angular separation depends only on the sky density of the external catalogue and the total number of X-ray sources: \nN rand( θ ) = NX · 2 π · θ · ρ · d θ, (2) \nwhere N rand( θ ) is the number of associations for angular separations between θ and θ + d θ and ρ is the sky density of the external catalogue. The peak in Fig. 14 can be expressed as the superposition of NX Rayleigh distributions, each of which depends on the positional uncertainty of the corresponding X-ray source, σ i , so \nN assoc( θ ) = F NX X i = 1 Pi ( θ \| σ i ) , (3) \nwhere Pi ( θ \| σ i ) is the Rayleigh distribution that describes the probability of true associations separated by angular distance θ given the positional uncertainty of the X-ray sources σ i . This parametrisation assumes that the positional errors of the external catalogue are much small than σ i and therefore can be ignored. The factor F , which takes values between zero and one, represents the fraction of X-ray sources expected to have counterparts in the external catalogue. In Fig. 14 the total number of sources at a given angular separation θ can be expressed as the sum of the \nFig. 12. Resolved CXB fraction in eRASS1. Bottom panels: HEALPix order 4 map of the 1-2 keV CXB intensity resolved fraction (eROSITA-DE hemisphere, Hammer-Aito ff projection, in Galactic coordinates). On the bottom left we include all sources with DET\\_LIKE\\_0 > 10, no flags and X-ray flux F 0 . 5 -2 keV > 1 × 10 -14 erg s -1 cm -2 . On the bottom right we include all sources with DET\\_LIKE\\_0 > 10, no flags and and X-ray flux F 0 . 5 -2 keV > 5 × 10 -14 erg s -1 cm -2 (i.e. the hemisphere median flux limit at 50% completeness). Top panels: distribution of the resolved fraction in the extragalacitc sky ( \| b \| > 30 · ) in 13.4287 square-degree pixels for the two cases. The solid lines are the median values: 0.24 and 0.19, respectively. \n<!-- image --> \nterms given by Equations 2 and 3. Put di ff erently, it is possible to model the observationally determined angular separation distribution of Fig. 14 using the equations above and hence to infer parameters such as F , ρ and the positional uncertainty. The number of matches at a given angular separation bin θ is a Poisson variate with expectation value λ ( θ ) = N rand( θ ) + N assoc( θ ). The likelihood of the observations in Fig. 14 can then be expressed as the product of the Poisson probabilities at each angular separation bin: \nL = N θ Y j = 1 Pois GLYPH<16> Nj \| λ ( θ j ) GLYPH<17> , (4) \nwhere the index j is for all angular separation bins N θ and Pois( Nj \| λ ( θ j )) is the Poisson probability of Nj matches given the expectation value λ ( θ j ). We use the U ltra N est nested sampling algorithm (Buchner 2016, 2019, 2021) to perform Bayesian inference on the likelihood of Equation (4) and determine posteriors for the various model parameters. For this exercise we parameterise the positional uncertainties of individual sources as \nσ i = POS\\_ERR = q A · σ 2 + σ 2 0 , (5) \n√ \nwhere σ = RADEC\\_ERR / 2 and RADEC\\_ERR is the catalogued positional uncertainty produced by ermldet (see Eq. 1). In equation 5 above, σ 0 represents systematic uncertainties and A is a multiplicative factor that scales the ermldet uncertainties. Under these assumptions, the four model parameters that are inferred by modelling the distribution of Fig. 14 are ρ , F , A , σ 0. We caution that this parametrisation assumes that the external \ncatalogue is assumed to have the same sky density, ρ , across the sky. Although the sample selection (see below) minimises such variations, it is inevitable that ρ has an intrinsic scatter that is not accounted for in the current version of the methodology. \nWe use the catalogue of AGN from Gaia and unWISE Data (Gaia / unWISE; Shu et al. 2019) to cross-match against the eRASS1 X-ray source catalogue. We only consider Gaia / unWISE sources with probability being a QSO PROB\\_RF > 0 . 8 and G -band magnitude < 20 . 5 mag. The latter criterion is adopted to minimise variations in the sky density of QSO candidates because of the variable depth of the GAIA survey as a result of the scanning law of the mission. For this magnitude cut it is empirically found that the sky density of QSO candidates in the extra-galactic sky (Galactic latitudes \| b \| > 20 · ) is nearly homogeneous. We further limit the eRASS1 catalogue to sources with ermldet detection likelihood DET\\_LIKE\\_0 > 7 (to increase the purity of the sample) that are not spatially extended (parameter EXT = 0) and are not identified as potentially spurious by the algorithms described in Sect. 5.2. We analyse separately eRASS1 sources with low ( < 30) and high ( > 30) number of net counts to accommodate a dependence of the positional uncertainty corrections on source brightness. Figure 14 shows the angular separation distribution from the cross-correlation of the high count eRASS1 sample with the Gaia / unWISE catalogue. The model with parameters set to the median of the corresponding posteriors is shown with the dashed line in that figure. The model represents the observations reasonably well. For the high-counts eRASS1 sub-sample, the parameters that are relevant to the positional error are estimated as A = 1 . 3 ± 0 . 1 and σ 0 = 0 . 9 '' ± 0 . 1 '' . \nFig. 14. Distribution of the angular separation between eRASS1 X-ray source positions and the Gaia / unWISE QSOs. The red dots correspond to the observed number of associations at a given angular separation bin, δ RX -Opt . We use only eRASS1 X-ray sources with Galactic latitudes in the range \| b \| = 30 -70 · (i.e. extragalactic sky), DET\\_LKE\\_0 > 7, with more than 30 counts, are not spatially extended (parameter EXT = 0) and are not identified as potentially spurious by the algorithms described in Sect. 5.2. The dashed line is the model described in the text for parameters fixed to the median of the corresponding posteriors. \n<!-- image --> \n<!-- image --> \nFig. 13. Flux comparison between eRASS1 and 4XMM catalogues. The upper panel displays the 0.2-2 keV fluxes of point sources. The brightest sources with at least 5 σ flux measurements are plotted in red points with green 1-sigma uncertainties. The other sources are plotted in blue points without error bars. The black lines indicate 1:1 and deviation from 1:1 by a factor of 2. The lower panel shows a hard-band comparison for the Hard sources. \n<!-- image --> \nFor the low-count sub-sample, the 3 σ upper limit of the σ 0 term is about 0.2 '' and consistent with zero within uncertainties. This indicates that the positional error budget for low count sources is dominated by the multiplicative term of Eq. (5), as the typical positional uncertainties of sources at the low count regime are already quite large. For simplicity, we then adopt a single scaling function to determine the positional uncertainty from Eq. 5, which we report as POS\\_ERR in the catalogue \nPOS\\_ERR = p 1 . 3 · σ 2 + 0 . 9 2 . (6) \nIndeed, for sources with < 30 counts the di ff erence in positional error between the above more general expression in Eq.(6) and a modified version of this equation where the additive term is assumed to be zero is only ∼ 0.1-0.2 '' on average. \nAs a further test, we have also tried to calibrate the positional uncertainty of the eRASS1 sources by comparing the catalogue entries with both Chandra and XMM-Newton serendipitous catalogues. The resulting values of the scaling terms A and σ 0 in Eq. 5 are consistent within the uncertainties with those derived above.", '6.3. On the effect of source blending': 'With the relatively large PSF of eROSITA (see Section 2.2.1), source blending could become a issue in crowded regions of the sky. Empirical tests of the e ff ect of blending (or lack thereof) on the catalog would require cross-match against X-ray catalogs where blending is expected to be negligible. Such tests, however, are potentially degenerate with source variability and nonuniformity (of background, exposure, etc.) of the matched catalogues. An alternative method is through simulations, where the full observation and detection process is simulated starting from a realistic sky source population. These have been performed for the eFEDS field (Liu et al. 2022c) and for eRASS1 (Seppi et al. 2022). The biggest impact of source blending is on the detection of extended objects, that is, blending of point sources leads to spurious extended sources in the catalog. Blending also causes incompleteness, since multiple individual sources are considered as only one. At the depth of eFEDS, the incompleteness caused by blending is between 1% and 3%, dependent on source brightness and extent. For eRASS1, at the DET\\_LIKE\\_0 threshold of the main catalog, the fraction of blended sources is less than 1% (see Seppi et al. 2022, Table 3), which is definitely much smaller than the expected level of contamination from spurious detections. Only in the south ecliptic pole region, where the exposure is deeper than average by a factor of more than hundred, this e ff ect becomes more significant. This will be discussed in a separate paper (Liu et al. in prep.).', '7. eROSITA-DE Data Release 1 (DR1)': 'The primary data products of the eROSITA-DE DR1 16 consist of eRASS1 calibrated event files, which contain the information generated by the operating cameras during eRASS1 observations. The data taken during the CalPV phase, which were released as part of the Early Data Release, are not part of DR1. \nFig. 15. Four examples of half-sky maps released as part of DR1. The maps are colour-coded by the logarithm of the count rate intensity (in cts s -1 arcmin -2 ), displayed in Zenith Equal Area (ZEA) projection in Galactic coordinates, with pixel size of 0.09 deg 2 and centered on ( l , b ) = (270, 0). We show the 0.2-0.6 keV ( top left ), 0.6-2.3 keV ( top right ), 2.3-5 keV ( bottom left ) and 0.2-2.3 ( bottom right ) bands. Please note that each map has a di ff erent (logarithmic) colour scale. \n<!-- image --> \nThe eSASS software package to help processing and analysing eROSITA data is also made public, with the name eSASS4DR1. \nAs discussed above, the eSASS pipeline divides the all-sky observations into 4700 sky tiles for practical purposes of computational tractability (see Fig. 2). eROSITA-DE have proprietary rights on 2248 of these tiles, while 199 of them have shared rights between the German and Russian consortia. eROSITA-DE DR1 finally comprises 2447 sky tiles, of which 199 are only partially filled with eROSITA-DE data. This public eRASS1 data was processed with version 010 of the eSASS pipeline. \nThe released calibrated event lists of each sky tile contain photons: \n- -with energies between 0.2-10 keV;\n- -flagged as flag=0xE000F000 , i.e., good events from the nominal field of view, excluding bad pixels; \n- -with patterns pattern = 15, i.e., including single, double, triple, and quadruple events. \nBesides calibrated event lists, the DR1 team also releases the following products per sky tile: \n- -Counts and count rate maps,\n- -Exposure maps,\n- -Background maps,\n- -Tables containing cumulative survey area as a function of limiting flux (for sources with likemin > 5 ),\n- -Sensitivity maps based on aperture photometry. Includes aperture-averaged exposure maps, aperture-integrated background maps, and area curves (survey area sensitive to a given count rate),\n- -Sensitivity maps (for sources with likemin > 5 ), \n<!-- image --> \n<!-- image --> \nFig. 16. Two examples of light curves (top panels) and spectra (bottom panels) for two sources contained in the eRASS1 Main catalogue, generated from the released source products. On the left hand side, we show the light curve and spectrum of the first eROSITA Quasi Periodic Eruptor (eRO-QPE1; Arcodia et al. 2021) (1eRASS J023147.1-102011; DETUID : eb01\\_038099\\_020\\_ML00005\\_002\\_c010). The light curve ( top left ) is binned in individual eroday visits, separated by 4-hours intervals. The spectrum ( bottom left ) is taken from all data, and fitted with an absorbed black body ( zbbody ). On the right hand side, we show one of the brightest AGN in the SEP region (1eRASS J061148.2-662434; DETUID : em01\\_093156\\_020\\_ML00001\\_012\\_c010; Bogensberger et al., in prep.). Given the high ecliptic latitude, the source is visited over two separate long periods of about 14 and 24 days, respectively. The light curve ( top right ) shown has the original binning in 10 seconds intervals. The spectrum ( bottom right ) is fitted with an absorbed power law. Note that both best-fit lines (light blue) are simple physical models aiming to guide the eye and do not intend to encompass all the potential detailed features present in the spectra. The background model is shown as black dotted lines in both figures. For more details on eROSITA spectral modelling and appropriate handling of background spectra, refer to Sect. 3 of Liu et al. (2022b). \n<!-- image -->', '-Source data products.': 'The data are made available to users using a web-based tool called eRODat. This interface allows one to interactively view the eROSITA sky, using Aladin Lite (Boch & Fernique 2014) to show the eROSITA HiPS (Hierarchical Progressive Surveys) maps and source positions. It also allows the user to identify the sky tiles associated with a given source position or a list of positions, and download the data products for those tiles. In addition, the user can search for sources around a position from the eROSITA X-ray catalogues and view those sources in various surveys, view the details of the catalogue entries, or download the individual source products. Data can be either downloaded by navigating the archive structure to obtain the required data files, following direct links as appropriate, or by adding the products to a virtual basket. The contents of the basket can be download immediately as a single tar file or eRODat can generate a shell script to later download those products. \nDR1 also includes the publication of the eROSITA upper flux limit server (Tubín-Arenas et al. 2023). The upper limits are calculated using X-ray photometry on the eROSITA standard calibration data products (counts image, background image, and exposure time), following the Bayesian approach described by Kraft et al. (1991). Pre-computed upper limits are \navailable for every pixel position in the eROSITA-DE sky at a confidence interval of 99.87% (this corresponds to a one-sided 3 σ level). These values are stored using the hierarchical indices (HEALPix) framework to enable a fast search. The products are delivered for the 1B energy band (0.2-2.3 keV), as well as in all sub-bands of the 3B DET run (soft: 0.2-0.6 keV, medium: 0.6-2.3 keV, hard: 2.3-5.0 keV, and the combined total band: 0.2-5.0 keV). The upper flux limit data will be available in two ways: either by downloading the pre-computed data products or through a web tool.', '7.1. Half-sky maps': 'Half-sky counts, count rate and exposure maps are o ff ered (in HiPS format) in seven di ff erent energy bands: \n- - 0.2-2.3 keV (Main)\n- - 0.2-0.6 keV\n- - 0.6-2.3 keV\n- - 2.3-5.0 keV (Hard)\n- - 0.2-0.5 keV\n- - 0.5-1.0 keV\n- - 1.0-2.0 keV, \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 17. Two more examples of light curves (top panels) and spectra (bottom panels) for two sources contained in the eRASS1 Main catalogue, generated from the released source products. On the left hand side, we show the light curve and spectrum of the bright star AD Leo (1eRASS J101935.6 + 195211; DETUID : em01\\_155069\\_020\\_ML00001\\_011\\_c010). The light curve ( top left ) is shown with the original 10s binning. The spectrum ( bottom left ) is fitted with a multi-temperature thermal collisionally-ionised plasma emission model ( apec+apec+apec , with tied abundances). On the right hand side, we show the light curve and spectrum of SMSS J114447.770-430859.3, the most luminous QSO of the last 9 Gyr (Onken et al. 2022; Kammoun et al. 2023) (1eRASS J114447.6-430858; DETUID : em01\\_176132\\_020\\_ML00001\\_002\\_c010). The light curve ( top right ) is binned in individual eroday visits, while the spectrum ( bottom right ) is fitted with an absorbed power law. Note that both best-fit lines (light blue) are simple physical models aiming to guide the eye and do not intend to encompass all the potential detailed features present in the spectra. The background model is shown as black dotted lines in both figures. For more details on eROSITA spectral modelling and appropriate handling of background spectra, refer to Sect. 3 of Liu et al. (2022b). \n<!-- image --> \nwhile the background maps and the sensitivity maps based on aperture photometry are released only for the first four energy bands above. As an example, Fig. 15 shows the half-sky images in Zenith Equal Area projection colour coded by count rate intensity in four bands: 0.2-0.6 keV, 0.6-2.3 keV, 2.3-5 keV and 0.2-2.3 keV, respectively. These images have been created using all TMs (also those a ff ected by light leak), so some artifacts are still present in the lowest-energy map. On the other hand, at the highest energy (2.3-5 keV) the count rate maps are dominated by the (unvignetted) particle background; small temporal fluctuations of such background (including rare Coronal Mass Ejections hitting SRG) are imprinted on those maps as stripes at fixed ecliptic longitudes 17 . \nZheng et al. 2024 have created custom-made half-sky eRASS1 maps in the standard ROSAT broad bands (R1-R7), with the goal of comparing the eRASS1 and RASS large scale emission in great detail. By only restricting to non-light-leak TMs (1, 2, 3, 4 and 6), and by carefully subtracting the nonvignetted particle background using FWC data, they produce clean maps almost free from artifacts. For more details on top- \nics such as flux estimation of large-scale di ff use emission for the non-light-leaked TMs, we refer the reader to that work.', '7.2. Source products': "Together with the catalogues of detected sources, eROSITA DR1 also contains a number of source-specific products, which are generated by the eSASS task srctool . In particular, we release source-specific products for 200 217 Main catalogue entries with DET\\_LIKE\\_0 > 20. These products include source and background spectra, their respective ARFs and RMFs, light curves, and source and background event lists. All these source products are o ff ered per telescope module and combined. In this section we describe how these products are generated. \nsrctool creates spectra, background spectra, response matrices, ancillary response files, and light curves for an input catalogue of sources. The tool chooses a circular source extraction region to optimise the signal to noise ratio of the source spectrum, given the local background surface brightness and the shape of the PSF, clipping the radius to a minimum of 15 '' and maximum of 99% of the PSF encircled energy fraction, assuming a circular PSF, and taking into account any excluded neighbouring contaminating sources. A detailed explanation of the al- \nthm, including the background definition procedures, can be found online 18 . The final products are produced for each telescope module separately and combined for the telescope module groups TM0 (all seven TMs), TM8 (only those TMs with onchip filter) and TM9 (only those TMs without on-chip filter). The products for each source are placed into separate archive files in order to save disk space usage and reduce the number of files in the archive. At odds with the rest of the pipeline, srctool was configured to use input GTIs calculated from the flaregti task, rather than the standard GTIs. Light curves were created in 10s time bins, discarding those bins when the source is not visible. The light curves were created in energy bands 0.2-0.6, 0.6-2.3 and 2.3-5.0 keV. \nAs eROSITA scans across the sky during the survey, a source samples di ff erent parts of each telescope and detector over time. The role of srctool is to account for the varying vignetting of the telescope, bad pixels and the e ff ect of photons lost outside the extraction region due to the shape of the PSF. These effects are calculated by stepping through time a set sample points spatially sampling the source. The average corrections are accounted for in the ancillary response matrix for spectral fitting, while the time-variable e ff ects are included in the outputted light curve rates. To avoid double counting these corrections, they are not included in the exposure time of the output files, which instead contain the time eROSITA is looking at the source. Those sources with an extent of zero from the source detection pipeline are assumed to be point sources, while extended sources are assumed to follow a beta model with the core radius given by the extent and an outer power-law slope fixed by the exponent β c = 2 / 3 (see Brunner et al. 2022, Appendix A.5). Table E.1 in the Appendix summarises the changes to the srctool package for DR1. \nIn Figs. 16 and 17 we present four examples of light curves and spectra of four point sources, generated from the released products.", '8. Summary and outlook': "In this paper we give an overview of the operations, observations, data reduction and analysis of the first eROSITA allsky survey for the hemisphere data (western Galactic) whose proprietary rights lie with the eROSITA-DE Consortium. We present the catalogues of X-ray sources extracted with the standard eSASS pipeline and describe quality control tests we have performed on the catalogued sources (astrometry, photometry, fidelity, etc.). Finally, we summarise the content of the data release associated with the eRASS1 survey so to facilitate interested users from the scientific community at large to interact with the rich data sets generated by SRG / eROSITA. \nThe scientific and information content of the eRASS1 survey is large, varied and di ffi cult to summarize in a few words. While our focus here is mainly on the eRASS1 X-ray catalogues and on the properties of the X-ray sources that are included there, other works focus on other relevant aspects of the all-sky survey data content. For example, already soon after the completion of eRASS1, Predehl et al. (2020) reported the discovery in the eROSITA all-sky image of large-scale X-ray bubbles in the Milky Way halo (the so-called 'eROSITA bubbles'). Following up on that work, Zheng et al. (2024) have studied maps of the di ff use emission as a function of energy and compared them with the ROSAT ones. Locatelli et al. (2023) have then used \nTable 8. eRASS1 by (rough) numbers. \nNotes. We list simplified estimates of the approximate number of calibrated photon counts (in millions) registered by eROSITA in eRASS1 in the 0.2-2.0 keV band in one hemisphere, split into separate physical components: Cosmic X-ray Background (CXB), Milky Way hot Circum-Galactic Medium (MW hot CGM), Instrumental background (FWC), Local Hot Bubble (LHB), Solar Wind Change Exchange (SWCX), Point Sources (PS), Extended Sources (Ext.). For the latter component, we quote in parenthesis the number of net counts from confirmed clusters of galaxies (Bulbul et al., 2024). See text for more details. \nthose maps to derive structural parameters of the Milky Way Circum-Galactic Medium (CGM) emission. Using dense molecular clouds of known distance as shadowing tools Yeung et al. (2023) have derived constraints on the properties of the Local Hot Bubble from its soft X-ray emission measured by eROSITA. \nBased on these analysis, here we try to give a simplified answer to the question of what the survey contains by providing a photon (i.e. calibrated events) budget of eRASS1 split into its major physical components. In order to do so, we separate source from background counts on the basis of the catalogues source net counts in di ff erent bands. As for the background and foreground emission, we adopt here the simplifying assumption that the total spectrum in the eFEDS extra-galactic field is representative of the all-sky, and we adopt the best fit model of the di ff use emission spectrum presented in Ponti et al. (2023b) to allocate photons to physical components. The result of this exercise is shown in Table 8. A more detailed inventory, as a function of position in the sky, can be found in Zheng et al. (2024). \nTime-domain variability analysis of the X-ray sources in eRASS1 will be presented in Boller et al. (in prep.), while Grotova et al. (in prep.) focus on the population of extra-galactic nuclear transients potentially associated to the tidal disruption of stars by SMBHs, which eROSITA is extremely sensitive to (see e.g. Malyali et al. 2021; Sazonov et al. 2021; Malyali et al. 2023; Homan et al. 2023; Liu et al. 2023b). \nThe details of the cross-identification methodology and of the classification schemes for the various classes of X-ray emitters contained in the eRASS1 catalogues will be described elsewhere. Specifically, the following papers will release the counterpart identification of: clusters of galaxies (Bulbul et al., 2024; Kluge et al., submitted); AGN (Salvato et al., in prep., Waddel et al., submitted); Blazars (Hämmerich et al., in prep.); Cataclysmic Variables (Schwope et al., in prep.); coronal emitting stars (Freund et al., 2024); X-Ray Binaries (Avakyan et al., in prep.); X-ray source population in the Magellanic Clouds system (Maitra et al., in prep., Kaltenbrunner et al., in prep.). \nWith almost one million entries, most of them never detected before, the eRASS1 half-sky catalogues represent a major advance for our knowledge of the X-ray Universe. Table 9 and Fig. 18 present a comparison of eRASS1 with other catalogues \nTable 9. Comparison among main catalogs from previous X-ray missions operating, at least partly, in the 'classical' X-ray energy range ( ∼ 0.2-10 keV). \nNotes. The column N objects lists the approximate number of sources in each catalogue and f Area is the fraction of the sky observed. ∗ The 4XMM-DR12 Hard catalogue (not shown in the figure) is derived from the 4XMM-DR12 by taking all sources for which the 2-5 keV flux is larger than the quoted 2-5 keV flux error. \nReferences: (1): Forman et al. (1978); (2): Warwick et al. (1981); (3) Wood et al. (1984); (4) heasarc.gsfc.nasa.gov/W3Browse/einstein/ ipc.html , Harris (1990); (5): Boller et al. (2016); (6): heasarc.gsfc.nasa.gov/W3Browse/rosat/wgacat.html ; (7) cxc.cfa.harvard. edu/csc/about2.1.html ; (8) Webb et al. (2020); (9) www.cosmos.esa.int/web/xmm-newton/xmmsl2-ug ; (10) Evans et al. (2020); (11) Brunner et al. (2022). \nfrom X-ray missions operating in the 'classical' X-ray energy range 0.1-10 keV, and highlight the steady progress in X-ray survey capabilities culminating with the eROSITA catalogues we discuss here. Simply considering the union of all the unique objects catalogued by any previous X-ray mission (without removing possible overlaps), the eRASS1 main catalogue presented here increases the number of known X-ray sources in the published literature by more than 60%. \nThe sensitive all-sky survey nature of the project implies that data are accumulated for a large variety of astronomical source classes, and for a plethora of possible science applications, well beyond the main mission-design-driving objectives; in other words, eROSITA data are endowed with tremendous legacy value. Indeed, existing all-sky and / or wide-area optical / IR surveys such as Gaia , SDSS, PanSTARRS, DES and Legacy Survey, HSC, VISTA / VHS, WISE already demonstrate the power of combining data sets for a deeper understanding of the high-energy Universe. Such a synergy extends also to longer wavelengths, at a time when the SKA precursors (LOFAR, MWA, ASKAP, MeerKAT) alongside the major observatories (APERTIF, JVLA) are surveying large swaths of the sky at unprecedented depth and speed. Beyond imaging, also large spectroscopic surveys have included among their main programs systematic follow-up of eRASS X-ray sources. Indeed, it is expected that SDSS-V (Kollmeier et al. 2017) and 4MOST (de Jong et al. 2019; Merloni et al. 2019; Finoguenov et al. 2019), by the end of their first survey period, will have accumulated hundreds of thousands of optical spectra of eROSITA X-ray sources. \nLooking forward, future eROSITA data releases will comprise data from multiple all-sky surveys. Besides providing a deeper view of the X-ray sky, this will open up the systematic study of variability on months / year timescales. Extrapolating from the current data quality, and with the foreseen improvements of the calibration in sight, the next data releases will include high-fidelity catalogues with several million X-ray sources, closely following the expectations laid down in the early phases of the project (Merloni et al. 2012). \nAcknowledgements. We thank the referee, Prof. F. Bauer, for the careful review and for the many constructive suggestions that significantly improved the final version of the paper. This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint Russian-German science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL with support from the Max-Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg, the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The eROSITA data shown here were processed using the eSASS / NRTA software system developed by the German eROSITA consortium. Part of this work was supported by the German Deutsche Forschungsgemeinschaft, DFG project number Ts 17 / 2-1. E. Bulbul, A. Liu, V. Ghirardini, C. Garrel and X. Zhang acknowledge financial support from the European Research Council (ERC) Consolidator Grant under the European Union's Horizon 2020 research and innovation programme (grant agreement CoG DarkQuest No 101002585). M. Brusa, A. Georgakakis and B. Musiimenta acknowledge funding from the European Union's Horizon 2020 research and innovation program under the Marie SkłodowskaCurie grant agreement No 860744. Z. Igo, C. Aydar, J. Wolf acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2094 - 390783311. G. Ponti acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 865637). P. Li is supported by the Alexander von Humboldt Foundation. D. Tubin-Arenas acknowledges support by DLR grant FKZ 50 OR 2203. 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J., Ponti, G., et al. 2023, A&A, 676, A3 \nZheng, X. & Ponti, G. e. a. 2024, A&A, in press \n- Zonca, A., Singer, L., Lenz, D., et al. 2019, Journal of Open Source Software, 4, 1298\n- 1 Max-Planck-Institut für Extraterrestrische Physik, Gießenbachstraße, D-85748 Garching, Germany\n- 2 Leibniz Institut für Astrophysik Potsdam, An der Sternwarte 16, D14482 Potsdam, Germany\n- 3 Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, D-72076 Tübingen, Germany\n- 4 Institute for Astronomy and Astrophysics, National Observatory of Athens, V. Paulou and I. Metaxa, 11532, Greece\n- 5 Exzellenzcluster ORIGINS, Boltzmannstr. 2, 85748, Garching, Germany\n- 6 Dr. Karl Remeis-Sternwarte and Erlangen Centre for Astroparticle Physics, Friedrich-Alexander Universität Erlangen-Nürnberg, Sternwartstraße 7, 96049 Bamberg, Germany\n- 7 Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany\n- 8 MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA \n- 9 Argelander Institute for Astronomy, University of Bonn, Auf dem Hügel 71, 53121 Bonn, Germany\n- 10 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,53121 Bonn, Germany\n- 11 Dipartimento di Fisica e Astronomia "Augusto Righi", Università di Bologna, via Gobetti 93 / 2, 40129 Bologna, Italy\n- 12 INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93 / 3, 40129 Bologna, Italy\n- 13 IRAP, Université de Toulouse, CNRS, UPS, CNES, Toulouse, France\n- 14 Tautenburg Landessternwarte, Sternwarte 5, 07778 Tautenburg, Germany\n- 15 University Observatory Munich, Faculty of Physics, LudwigMaximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany\n- 16 Department of Physics & McDonnell Center for the Space Sciences, Washington University in St. Louis, One Brookings Drive, St. Louis, MO63130, USA\n- 17 Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstraße 25 / 8, 6020 Innsbruck, Austria\n- 18 TU Dresden, Institute of Nuclear and Particle Physics, 01062 Dresden, Germany\n- 19 DESY, Notkestraße 85, 22607 Hamburg, Germany\n- 20 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA\n- 21 Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands\n- 22 Istituto di Radioastronomia IRA-INAF, via Gobetti 101, 40129 Bologna, Italy\n- 23 INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate, (LC), Italy\n- 24 Max-Planck Institut für Astronomie, Königstuhl 17, 69117 Heidelberg\n- 25 CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei 230026, China\n- 26 School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China', 'Appendix A: The eROSITA Survey PSF': "The PSF of the eROSITA telescopes can be obtained from the survey itself to verify the PANTER measurements and compare to its shapelet representation used in the source detection pipeline. We are undertaking a project to obtain the PSF from source stacking and present here preliminary results. 938 sources were taken from the full eROSITA sky, selected from a catalogue of eROSITA point sources with the largest number of counts. To optimise the signal and to avoid bad pixels we used eRASS:4 for TMs 1, 2, 3 and 6, eRASS:3 for TMs 5 and 7, and eRASS:2 for TM4. In this analysis we examined the PSF range out to 4 arcmin radius, to be consistent with the ground based measurements, although there is additional flux beyond this radius, estimated to be about 2.7% from the best fit core PSF (excluding stray light) of Churazov et al. (2023) (their equation A.1). We excluded the very brightest piled-up sources and those in crowded fields or near large extended sources. The position of each source was measured by simultaneously fitting 2D models to two X-ray images, one where the source was inside 15 arcmin of the optical axis, and another when it was outside. We matched against the Gaia G band catalogue to identify those sources potentially a ff ected by optical loading and used the Galactic latitude to flag those potentially a ff ected by scattering halos. We manually inspected the images and profiles of the sources to look for anomalous profiles, such as evidence of extension, or identify neighbouring contaminating sources. For sources which could be contaminated in the outskirts by other sources or by non-flat background, we excluded the source, or manually reduced the outer radius used from 4 arcmin to 1, 2 or 3 arcmin as appropriate. Lightcurves created with srctool were used to quantify pileup, where the inner stacking radius used was chosen as a function of time for a source according to pileup rate thresholds (rates of 15, 44, 100, 155, 250 and 600 s -1 produced inner radii of 20, 30, 40, 60, 90 and 240 arcsec, respectively, based on simulations to estimate the e ff ect of pileup). Source profiles were stacked in each energy band, and for each combination of inner and outer radius. A PSF model plus flat background model was simultaneously fitted to the stacked profiles in all the combinations of radial range, allowing the PSF normalisation and background level to vary in each combination of radial range, to produce a model of the PSF as a function of radius. This process was repeated for each energy band and combinations of TMs. \nFigure A.1 shows a comparison between the result of our source stacking analysis, ground-based PANTER measurements (Dennerl et al. 2020) and survey-averaged shapelet models of the ground-based data (Brunner et al. 2022). Also shown are comparisons of the PSF for the di ff erent eROSITA TMs. We note here that the survey-averaged shapelet PSF is never fitted to the data by ermldet in photon imaging mode, but rather, a PSF appropriate for each photon is created. Within 1 arcmin radius the deviations of the PSF are within 20%. Much of the deviation in the shapelet PSF is due to 'steps' occurring at intervals of 16 arcsec radius. At larger radius the shapelet PSF in the soft band is lower than the stacked or ground-based PSF. This contributes to the shapelet cumulative fraction being higher in the centre, as the profiles are normalised at 4 arcmin radius. The stacked profiles typically have smaller full-width-halfmaximums (FWHMs) than the PANTER and shapelet PSFs, but comparable HEW. The in-flight eROSITA camera sub-pixel positioning allows a better determination of the PSF core than the 9.6 arcsec pixel would otherwise suggest. \nFinally, we tested the uniformity of the PSF across the sky. For this, we divided the sample of 938 point sources described \nabove into four quadrants in ecliptic coordinates (with dividing lines at ecliptic longitude of 180 · and ecliptic latitude of 0 · ) and recompute the HEW of the stacked PSF for TM0 in the 0.22.3 keV band. The values are consistent with the all-sky average HEW(30.0 '' ) within 0.3 '' . \nTable A.1. Point Spread Function Half-Energy Width (HEW) in the 0.22.3 keV band, computed by stacking point sources in eRASS1 in four di ff erent quadrant across the sky. \nIn the future we plan to improve our stacking procedure. These improvements could include automated detection of contaminating sources and removal of sources with inconsistent profiles. Future processing versions will improve the accuracy of the boresight calibration as a function of time. By choosing which sources to include in the stacking as a function of radius to optimise the signal to noise, we would also reduce the e ff ect of the sky background. Finally, we will also verify the obtained PSFs by studying di ff erent subsamples of sources.", 'Appendix A.1: On the azimutal symmetry of PSF and positional uncertainties': "A key assumption for the astrometric calibration and positional uncertainty determination of our source detection algorithm (see Section 5.1), and of the srctool extraction tool is that the eROSITA PSF can be approximated with a azimutally symmetric function and that the associated positional errors are also symmetric. \nHere we validate this assumption in two ways. First of all, we show in Fig. A.2 the profile of the average stacked PSF (using the same 938 sources from the all-sky survey as described in the previous section) along ecliptic longitude and latitude. No significant di ff erence in the two profiles can be seen. Secondly, we show in Fig. A.3 the measured positional o ff set between eRASS1 X-ray source positions and Gaia / unWISE QSOs (see Section 6.2) in both equatorial and ecliptic coordinates directions. We fit these distributions with a Gaussian function, and obtain for the o ff set the following values of mean and Standard Deviation for the two cases: ( ∆ α, ∆ δ )mean = ( -0 . 11 '' , 0 . 08 '' ); ( ∆ α, ∆ δ )SD = (3 . 5 '' , 3 . 5 '' ); ( ∆ λ, ∆ β )mean = ( -0 . 11 '' , 0 . 16 '' ) and ( ∆ λ, ∆ β )SD = (3 . 5 '' , 3 . 5 '' ). From this, we conclude that the symmetry approximation is justified, and that the residual o ff set (of size < 0 . 2 '' ) is small enough when compared to the positional uncertainty to be safely ignored. \nFig. A.1. Comparison of di ff erent survey-averaged eROSITA PSFs. Shown are the PSFs in the bands 0.2-2.3 keV (left panels), 2.3-5.0 keV (central panels), or for the individual TMs from stacking in 0.2-2.3 keV (right panels). The PSF models shown are the survey-averaged shapelet PSFs, those obtained by stacking sources (see text in Appendix A for more details) and those from ground-based measurements using PANTER. The top row of panels show the PSF surface brightness profiles, normalised within 4 arcmin radius, where the vertical lines plot the FWHM values. The second panels down show the fractional di ff erence of the surface brightness of each PSF from the average stacked profile. The third panels down show the cumulative signal as a function of radius, plotting the HEWs as vertical lines. The lowest panels show the fractional di ff erence of the cumulative profiles to the average stacked profile. The shapelet PSFs are those used for fitting the energy band given, the stacked PSF is weighted by the spectra of the input sources, and the PANTER PSF is obtained at the monochromatic energy specified, which is chosen to be representative of the source photons in the band. The shapelet PSF is plotted as a dashed line outside a radius of 1 arcmin, the maximum used for fitting in the source detection pipeline. In the two leftmost columns, the PSF images were rebinned with 4 arcsec pixels where necessary for a fair comparison. In the rightmost column, we show the best fitting model and uncertainties, rather than stacked, rescaled and rebinned count profiles. The FWHMs quoted are sensitive to the pixelisation used and the inner value, and so should be used with care. HEWs were computed from minimally-binned images. \n<!-- image --> \nFig. A.2. Stacked PSF profile from the eRASS1 point sources in the 0.22.3 keV band along ecliptic longitude (red) and latitude (blue), demonstrating the symmetry of the PSF. The measured Half-Energy Widths (HEW) are consistent to within 0.6 '' . \n<!-- image --> \n<!-- image --> \nFig. A.3. Distribution of measured positional o ff set between astrometrically corrected eRASS1 sources with DET\\_LIKE\\_0 > 7 and Gaia / unWISE QSOs (see Section 6.2) along equatorial (left) and ecliptic (right) coordinates in arcseconds. The dashed lines represent the best fit gaussian function to these distributions, the parameters of which are reported in the text. \n<!-- image -->", 'Appendix B: Testing the energy calibration': "An excellent target for testing the energy calibration is the oxygen-rich SNR 1E 0102-7219 (in the following abbreviated as 1E0102), the brightest SNR in the SMC. It is characterized by strong emission lines of O, Ne, and Mg, exhibits only little 'contaminating' emission from Fe, it is su ffi ciently compact to utilise the high spectral resolution provided by slitless Xray gratings ( XMM-Newton RGS and Chandra HETG), yet extended enough to minimise any problems with pile-up. This object has been adopted as a standard calibration source by the International Astronomical Consortium for High-Energy Calibration (IACHEC), which has developed a standard (purely empirical) model specifically designed for calibration (Plucinsky et al. 2017). \nIn Fig. B.1 we show spectra of 1E0102 taken in dedicated calibration observations on 2019 Nov 7 - 8 (60 - 61 ks, 16 ks for TM6), and on 2021 Nov 26 - 27 (47 - 49 ks, 21 ks for TM4), thus covering a time span of more than 2 years. The data were taken with the same onboard processing mode which was used in eRASS. A major di ff erence, however, were the CCD temperatures, which ranged during the first observation between -85.5 and -84.6 C, and during the last observation between -77.9 and -77.0 C. When comparing both observations it should also be considered that, although 1E0102 was observed on-axis, its precise location on the CCDs was di ff erent between both observations. \nSpectra were extracted with the eSASS tasks evtool and srctool for each observation and each TM (except for TM5 and TM7, which are a ff ected by the light leak), separately for a source and a background region. A circular source extraction region was used, with a radius of 1 arcmin centered on the FK5 coordinates α = 16 . 006258 deg , δ = -72 . 032394 deg, and the background was extracted from a circle with a radius of 5 arcmin centered on the FK5 coordinates α = 16 . 169575 deg , δ = -71 . 828828 deg. \nThe fits are based on the standard IACHEC model for 1E0102, which consists of 52 narrow Gaussian emission lines, superimposed on an absorbed continuum. The emission lines are organised into 4 groups, corresponding to emission from O VII, OVIII, Ne IX, and Ne X 19 . \nFor an assessment of the quality of the energy reconstruction we performed a combined fit of TMs 12346 from both observations (10 spectra in total), using only single pixel events. We applied the standard IACHEC model and treated the normalisations of the O VII, O VIII, Ne IX, and Ne X line complexes as free, but TMindependent, parameters (4 free parameters). Only the overall normalisation was adjusted individually for each TM (10 free parameters). We allowed for TM specific shifts of the energy scales by XSPEC 'gain fits', with all slopes fixed to 1.0 and the 10 individual o ff sets as additional free parameters. This resulted in a common fit of 10 spectra with 24 free parameters. The fit yields χ 2 = 4055 . 2 for 2290 degrees of freedom, or χ 2 r = 1 . 77 (Fig. B.1), and the mean energy shift is -1 . 3 eV for the first and + 2 . 2 eV for the last observation, with a scatter of ± 1 . 0 eV and ± 3 . 0 eV. \nThese long pointed observations of a line-rich SNR represent a benchmark test of the energy calibration. Considering that the calibration requirements for eRASS spectra are more relaxed due to the much shorter exposure times, we conclude that the energy calibration is su ffi ciently accurate for the sources detected in the eROSITA all-sky survey. \nFig. B.1. eROSITA spectra of the SNR 1E 0102-7219, taken with TM12346 in Nov 2019 and 2021. \n<!-- image -->", 'Appendix C: The NRTA pipeline': "As outlined in Sect. 3.1, the field of view of eROSITA, which has a diameter of roughly one degree, scans the sky continuously with a rotational period of four hours and visits each position in the sky typically for six consecutive scans. Each position is then revisited roughly six months later. This cadence provides a unique opportunity to identify and study transient phenomena in the X-ray sky and o ff ers a compromise between time resolution and sensitivity. The survey schedule of eROSITA does not allow for interruptions in order to perform pointed observations, therefore fast identification of transient events and the communication to other facilities, also in other wavelength bands, is vital. To this end the NRTA pipeline was developed. Its purpose is to analyse the science data on ground as soon as it is ready at MPE and alert the appropriate team of scientists for a given event who can then decide the correct course of action. Beyond this, the NRTA also aids in some technical aspects of the maintenance of the eROSITA instrument. \nOne of the highest priorities is the identification of bright Xray transients, especially if they are not known. To this end, the NRTA runs an implementation of the Bayesian Blocks algorithm developed by Scargle et al. (2013) on the raw detector count rates of TMs 1, 2, 3, 4, and 6. The algorithm is disabled for TMs 5 and 7 because their optical light leak causes strong fluctuations in the count rates. When a bright source passes the field of view a significant excess of the count rate is expected which can be recognised by the algorithm. This excess is expected to have a duration of roughly 40 s. After additional filtering to exclude such periods of increased count rate caused by artefacts from the CE, each of these time windows is marked for source detection. An exemplary detector light curve of TM 6 for a bright source passing through the field of view and the resulting Bayesian Blocks is shown in Fig. C.1. \nFor all sources, either detected by source detection or ingested as point sources externally into the pipeline, a variety of products are extracted using the srctool task from the eSASS, most notably the spectrum and the light curve for the pass of the source through the field of view. Based on these data and possibly the information from the source detection, additional custom parameters tailored for specific science cases are calculated, like count rates in di ff erent energy bands, the signal-to-noise ratio, \nFig. C.1. Example detector light curve (blue) of TM 6 of one telemetry file with bins 5 s in length. The segmentation created by the Bayesian Blocks algorithm is shown in red and the identified intervals of bright sources in the field of view are shaded green. Note that the very short block at t ≈ 1400 s was caused by a remaining corrupt frame. The block of elevated count rate towards the end of the light curve was caused by an unknown transient X-ray source detected on New Year's Eve 2019 (Wilms et al. 2020). \n<!-- image --> \nhardness ratios, and the minimal distance of the source to the centre of the field of view. To compare the sources with entries in other external catalogues, the Nway Bayesian algorithm (Salvato et al. 2018) is used to match the positions of the sources against 54 catalogues covering all wavelengths bands. \nThe notification to the science teams is based on alerts. An alert for a source is generated if a given set of criteria, generically called 'triggers', which can be combined using logical and , or , and not operators, holds true. A variety of these criteria are implemented in a flexible manner and can be used, for example, to test whether a numerical parameter for a source falls within a given range, or equals a specific value, or has a match in a specific catalogue with a specific value. A blacklist is used to suppress alerts from areas which are known to produce large amounts of spurious detections, for example from around Sco X-1 (compare Fig. 6). If a trigger is evaluated as positive for a given source, an e-mail notification with basic information about the source is sent to the science team responsible for the trigger. The alert can be inspected in a web-based front-end which can display more advanced information about the source (images, light curves, and external surveys). A team of two scientists is assigned in weekly rotations to inspect the generated alerts and communicate with the science teams. Some triggers are automatically closed by the system and stored for later bulk analysis. All results of the NRTA are stored on a separate archive and available for later inspection. Typically, it takes less than six hours after the data have arrived for processing until the generation of an alert, which is su ffi cient given that data can be stored on-board for up to 24 hours before being transmitted to the ground station. \nDuring eRASS1 the NRTA analysed the occurrences of 21.4 million possible sources. Roughly one million of these were found during source detection in regions given externally or identified as containing a bright source, and the remaining ones were ingested as known point sources to monitor. In total, roughly 150 000 alerts were generated. Out of these, we identified several time critical events, which were published mostly as Astronomer's Telegrams (Rutledge 1998). The first such event \noccurred shortly after the begin of the survey on December 31, 2019 and was caused by a bright, and yet otherwise unidentified, X-ray transient designated SRGt J123822.3-253206 with a flux of 2 × 10 -10 erg s -1 cm -2 in the 0.2-10 keV band (see Fig. C.1, Wilms et al. 2020). Another bright transient, SRGt J071522.1191609, with a flux of 1 . 3 × 10 -11 erg s -1 cm -2 , was detected on April 14, 2020 (Gokus et al. 2020). On May 22, 2020, the NRTA detected a flare from the millisecond pulsar PSR J1023 + 0038 with a flux of 3 . 6 × 10 -11 erg s -1 cm -2 in the 0.3-10 keV band (Koenig et al. 2020). On June 6, 2020, the Be / X-ray binary RX J0529.8-6556 in the LMC was found in outburst by the NRTA for multiple consecutive scans with an average flux of 2 . 5 × 10 -11 erg s -1 cm -2 in the 0.2-8 keV band (Haberl et al. 2020; Treiber et al. 2021).", 'Appendix D: Column descriptions': 'Table D.1 contains a description of the catalogue data model. Our single-band and three-band detections result in the 1B and 3B catalogues, which have di ff erent columns. Each 1B and 3B source has a unique source ID, namely, UID , or DETUID , which are equivalent. The UID is only unique in the 1B or 3B catalogue, and most of the 1B and 3B sources are identical. The 1B3B association results (§ 5.3) are saved in additional columns of UID\\_1B or UID\\_Hard . The 1B catalog is divided into the Main and the Supp catalogues, which have identical columns. The Hard catalogue is a subsample of the 3B catalogue. \nMost of the source information is calculated in multiple bands and thus a band su ffi x is printed in the column names. Tables D.2 and D.3 provide a dictionary for the energy band definitions in the 1B and 3B catalogues, respectively. A few columns are from the PSF-fitting of source detection using the ermldet task, e.g., DET\\_LIKE , ML\\_RATE , ML\\_CTS , ML\\_FLUX , EXT , and EXT\\_LIKE . In the 3B detection, the band index 1, 2, and 3 indicate 0.2-0.6, 0.6-2.3, and 2.3-5 keV, and band 0 indicates allband summary values. In the 1B detection, only band 1 (0.22.3 keV) is involved, and the all-band summary value (band 0) is identical to that of band 1. For the 1B catalogues, we performed forced PSF-fitting ( ermldet ) and forced aperture photometry ( apetool ) at fixed source positions. They are reported in the bands P1-P9 (Table D.2). Combining band P1 (0.5-1 keV) and P2 (1-2 keV), we calculated the values in band "S" (0.5-2 keV, see § 5.4). \nTable D.1. eRASS1 catalogs column desciption. All errors are provided as 68% confidence intervals (1σ ). \nTable D.2. Dictionary of energy band su ffi xes in the eRASS1 1B (Main and Supplementary) catalogsTable D.3. Dictionary of energy band su ffi xes in the eRASS1 Hard catalog', 'Appendix E: Software and calibration versions used in this work': 'The eROSITA data presented in this work were processed in the time period Nov. 2021 to Jan. 2022. The software and calibration versions used (pipeline configuration 010) are therefore di ff erent from those of the earlier eROSITA early data release (EDR, pipeline configuration 001), as well as from later work based on proprietary eROSITA data (pipeline configuration 020). Here we provide a summary of the main di ff erences between the EDR, DR1 and later software (Tab. E.1) and calibration (Tab. E.2) versions. Further details on eSASS software versions are available on the eROSITA DR1 website 20 . \nTable E.1. EDR / DR1 eSASS task version changes \nNotes. Only tasks with functional improvements are listed. Task versions for pipeline configuration 020 are as of 2023-01-15. The EDR dataset only provided calibrated event lists; task versions available in the EDR user eSASS package are listed in square brackets. Task srctool has di ff erent but functionally equivalent version numbers in the DR1 pipeline and in the corresponding DR1 user eSASS (user eSASS version marked in square brackets). \nTable E.2. EDR / DR1 calibration version changes \nNotes. Listed are the versions numbers which are used in the respective processings. That does not necessarily mean, that they are used for eRASS1 processing, because their application is only necessary at a later date. \nDi ff erent versions are separated by " / ". In case a calibration file does not exist for a TM it is marked by "-".', 'Appendix F: Essential Dictionary and List of Acronyms': 'Table F.1 presents a list of useful terms and acronyms found in this paper. \nTable F.1. Essential dictionary and list of acronyms'}
2024arXiv240908451C	Because of the previously observed stability of the 171day period the superorbital modulation of the lowmass Xray binary 4U 182030 was considered a consequence of a third star orbiting around the binary. This study aims to further verify this triple model by testing the stability of superorbital period using the light curves collected by Xray sky monitoringscanning telescopes from 1987 to 2023. Both power spectral and phase analysis results indicate a significant change in the superorbital period from 171 days to 167 days over this 36year span. The evolution of the superorbital phase suggests that the superorbital period may have experienced an abrupt change between late 2000 and early 2023 or changed gradually with a period derivative of dot Psup3.58 pm 0.72 times 104 dayday. We conclude that the superorbital period of 4U 182030 was not as stable as anticipated by the triple model which strongly challenges this hypothesis. Instead we propose an irradiationinduced mass transfer instability scenario to explain the superorbital modulation of 4U 182030.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.08451', 'arXiv:2409.08451', '2024arXiv240908451C']	['Astrophysics - High Energy Astrophysical Phenomena']	The Puzzling Superorbital Period Variation of the Lowmass Xray Binary 4U 182030	2,024	201	0.49	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.08451.pdf	{'The Puzzling Superorbital Period Variation of the Low-mass X-ray Binary 4U 1820-30': 'Yi Chou ( /uni5468 /uni7FCA ), 1 Jun-Lei Wu ( /uni5433/uni541B /uni78CA ), 1 Bo-Chun Chen ( /uni9673 /uni6CCA /uni931E ), 1 and Wei-Yun Chang ( /uni5F35 /uni744B /uni82B8 ) 1 \n1 Graduate Institute of Astronomy, National Central University 300 Jhongda Rd. Jhongli Dist. Tauyuan, 32001, Taiwan', 'ABSTRACT': 'Because of the previously observed stability of the 171-day period, the superorbital modulation of the low-mass X-ray binary 4U 1820-30 was considered a consequence of a third star orbiting around the binary. This study aims to further verify this triple model by testing the stability of superorbital period using the light curves collected by X-ray sky monitoring/scanning telescopes from 1987 to 2023. Both power spectral and phase analysis results indicate a significant change in the superorbital period from 171 days to 167 days over this 36-year span. The evolution of the superorbital phase suggests that the superorbital period may have experienced an abrupt change between late 2000 and early 2023 or changed gradually with a period derivative of ˙ P sup = ( -3 . 58 ± 0 . 72) × 10 -4 day/day. We conclude that the superorbital period of 4U 1820-30 was not as stable as anticipated by the triple model, which strongly challenges this hypothesis. Instead, we propose an irradiation-induced mass transfer instability scenario to explain the superorbital modulation of 4U 1820-30.', '1. INTRODUCTION': '4U 1820-30, discovered by Giacconi et al. (1974), is an ultra-compact low mass X-ray binary (LMXB) located near the center of globular cluster NGC 6624. It was the first X-ray source known to exhibit TypeI X-ray burst Grindlay et al. (1976), indicating that the accretor in this binary system is a neutron star. Its 685 s orbital period, discovered by Stella et al. (1987) from its sinusoidal-like orbital modulation in the X-ray light curve, makes 4U 1820-30 being the most compact LMXB. The mass-losing companion is a Roche-lobe filling helium white dwarf with a mass of 0.06-0.08 M /circledot (Rappaport et al. 1987). Mass transfer in the system is induced by the orbital angular momentum loss through gravitational radiation which should result in a positive orbital period derivative with a lower limit of ˙ P orb /P orb > 8 . 8 × 10 -8 yr -1 (Rappaport et al. 1987). However, observed orbital period derivatives reported by Tan et al. (1991); van der Klis et al. (1993a,b), Chou & Grindlay (2001) (hereafter CG01), and Peuten et al. (2014) were negative with the latest value of ˙ P orb /P orb = ( -5 . 21 ± 0 . 13) × 10 -8 yr -1 updated by Chou & Jhang (2023), evaluated from ∼ 46 years of orbital phase evolu- \nCorresponding author: Yi Chou \[email protected] \ntion. This contradiction is believed due to the binary system accelerating by the gravitational potential in NGC 6624 (Tan et al. 1991; Chou & Grindlay 2001; Peuten et al. 2014; Chou & Jhang 2023). Additionally, superhump modulation with a period of 691 . 6 ± 0 . 7 s, ∼ 1% significantly longer than the orbital period, was observed in the both FUV (Wang & Chakrabarty 2010) and X-ray (Chou & Jhang 2023) bands. From the superhump period, the mass of companion of 4U 182030 is estimated as 0.07 M /circledot (Wang & Chakrabarty 2010; Chou & Jhang 2023). \nIn addition to orbital and superhump variations, 4U 1820-30 exhibits superorbital modulation with a period much longer than the orbital period. Priedhorsky & Terrell (1984) discovered the Xray flux modulation by a factor of 2 with a period of 176 . 4 ± 1 . 3 days using the light curve detected by Vela 5B from 1969 to 1976. This periodicity was further confirmed by Smale & Lochner (1992). However, by analyzing the light curve collected between 1996 and 2000 by All Sky Monitor on-board Rossi X-ray Timing Explorer (RXTE ASM), CG01 revised the superorbital period to 171 . 39 ± 1 . 93 days. Combining the times of flux minima of the superorbital modulation (hereafter superorbital minima) detected by Vela 5B and All Sky Monitor onboard Ginga (Ginga ASM), CG01 further constrained the period to be 171 . 033 ± 0 . 326 days and claimed that the superorbital period was stable over ∼ 30 years with \| ˙ P sup /P sup \| < 2 . 2 × 10 -4 yr -1 . Based on the stabil- \nthe superorbital period, GC01 proposed that this long-term variability is due to a hierarchical third star orbiting around the binary system (Grindlay 1986, 1988) (hereafter the triple model). The hierarchical third component can cause the eccentricity of inner binary system to vary with a period ( P ecc ) as \nP ecc = K P 3 2 P orb (1) \nwhere P 3 is the orbital period of third star, P orb is the binary orbital period and K is a constant of unity (Mazeh & Shaham 1979). Because the mass transfer rate is highly sensitive to the Roche lobe radius, which is proportional to the binary separation, the variation of binary eccentricity can cause the mass loss rate and the accretion rate to change with a period of P ecc and thus P sup = P ecc . For the 4U 1820-30 system, the orbital period of the third companion is estimated to be ∼ 1.1 days for K ∼ 1, and beat sidebands resulting from coupling binary modulation and ∼ 1.1 day periodicity may be observable in the power spectrum. Although these beat sidebands were not detected in RXTE observations (CG01), Chou & Jhang (2023) suggested that the 691 . 6 ± 0 . 7 s periodicity observed in the X-ray band might be caused by a hierarchical triple orbiting around the binary system with an orbital period of 0.8 days. Moreover, CG01 found that the active times of Type-I X-ray bursts were clustered within ± 23 days of expected superorbital minima, which aligns with the observation that the bursts can be seen only in low state (Clark et al. 1977; Stella et al. 1984). This fact implies that the superorbital modulation of 4U 1820-30 is due to changes in the accretion rate rather than external occultation or absorption effects, which is consistent with the triple model. \nThe periodicity of 171 days was further confirmed by ˇ Simon (2003); Wen et al. (2006); Zdziarski et al. (2007a); Kotze & Charles (2012) using additional RXTE ASM data and by Farrell et al. (2009) using the data collected by Burst Alert Telescope onboard the Neil Gehrels Swift Observatory (Swift BAT). Applying the triple model, Zdziarski et al. (2007a) demonstrated that the factor of 2 superorbital modulation in X-ray light curve can be explained by the eccentricity of inner binary oscillating between 0 and 0.004. The discovery of the dependence of orbital modulation profile on the accretion rate Zdziarski et al. (2007b) also supports the triple model. The hard X-ray light curve collected from Swift BAT showed that the superorbital modulation can be observable only for the energy bands less than 24 keV (Farrell et al. 2009). Conversely, by comparing the peak widths of the power spectra made from light \ncurves detected by RXTE ASM and Swift BAT with the corresponding simulated light curves, Farrell et al. (2009) found that the peak widths from real data are marginally wider than the ones from simulated data, concluding that this may be caused by the superorbital period change. Kotze & Charles (2012) adopted the dynamic power spectrum technique to analyze the superorbital variability of several X-ray binaries, and found no significant superorbital period change for 4U 1820-30 except for a weakening of power during MJD ∼ 5120052200. \nOwing to the monitoring/scanning X-ray telescopes, 4U 1820-30 has been observed for decades and is still being monitored by the Swift BAT and the Monitor of All-sky X-ray Image (MAXI). In this work, we aim to further verify the stability of the superorbital period, which is the crucial evidence for the triple model of 4U 1820-30 system, and to establish an updated ephemeris for superorbital modulation. In this paper, we introduce the instruments used to obtain the light curves for this research, including Ginga ASM, RXTE ASM, Swift BAT and MAXI, as well as the light curve collected by RXTE Proportional Counter Array (RXTE PCA) while it processed the monitoring observations of the galactic center and plane (Markwardt 2006), in Section 2. A preliminary test of superorbital period stability test was performed using the power spectrum made by the entire light curve of each instrument (Section 3.1). A more detail measurement of superorbital period variation was obtained by analyzing the superorbital phase evolution and updating the ephemerides (Section 3.2). The new ephemerides allows us to verify whether the Type-I Xray bursts occur clustered around the expected superorbital minima. In Section 4, we discuss the instability of superorbital period, which poses a serious challenge of the triple model, and explore the possible interpretations for the superorbital period variation of 4U 1820-30.', '2. OBSERVATIONS': "The Ginga ASM consists of two identical gas proportional counters with six fan-beam collimators to restrict field of view (FOV) of 1 · × 45 · . It is sensitive to the X-ray photons with energies between 1 and 20 keV, and has a total effective area of 420 cm 2 . It provided realtime alerts of X-ray transient phenomena and long-term historical records of X-ray sources. The Ginga ASM monitored the sky from 1987 February to 1991 October. Further details of the Ginga ASM are described by Tsunemi et al. (1989). The Ginga ASM light curve of 4U 1820-30 collected from MJD 46861 to 48532 was analyzed in this study. \nThe RXTE ASM (Levine et al. 1996) is an instrument mounted on RXTE to monitor the variable and the transient X-ray sources. It consists of three scanning shadow cameras, each containing a position-sensitive proportional counter, to observe the sky through a onedimensional coded mask with an FOV of 6 · × 90 · . It is designed to detect the cosmic X-rays in the photon energy range of 1.5-12 keV, which can be further divided into 1.5-3, 3-5 and 5-12 keV energy bands. In addition to these energy bands, the light curves with two different time resolutions, dwell (a 90 sec exposure) and one-day binned, were also archived. During its mission, from the beginning of 1996 to early 2012, the RXTE ASM scanned the entire sky every 90 minutes. In this research, the 1.5-12 keV RXTE ASM light curve collected between MJD 50088 and 55831 was selected to analyze the superorbital modulation of 4U 1820-30. \nIn addition to the regular pointing observations, the RXTE PCA also conducted the monitoring observations of the galactic center and plane starting from 1999 (Markwardt 2006). The PCA is an instrument with an effective area of 6500 cm 2 designed to detected the X-ray photons in the energy range of 2-60 keV (Jahoda et al. 1996). Despite being a non-imaging instrument, its 1 · FOV, constrained by collimators, allowed for identification and detection of X-ray sources. It scanned over galactic bulge and plane approximately twice per week (Markwardt 2006), providing sufficient time resolution to resolve the superorbital modulation of 4U 1820-30. The light curve of 4U 1820-30 collected by PCA in this program from February 5, 1999 (MJD 51214) to October 30 2011 (MJD 55846) is available on the program website 1 . \nThe BAT, an instrument on Swift, is a coded-mask telescope with a large FOV (1.4 steradian) to monitor the hard X-ray sky in the energy range 15-150 keV since 2004 (Barthelmy et al. 2005). Apart from triggering alerts for gamma-ray bursts, its angular resolution ( ∼ 20') and large photon collecting area (5200 cm 2 ) enable monitoring of the known cosmic X-ray sources as the Swift satellite orbits around the Earth every ∼ 96 minutes. This capability allows for the study of longterm variability these sources. In this work, we analyzed the daily binned light curve of 4U 1820-30 observed from February 14, 2005 (MJD 53415) through August 1, 2023 (MJD 60157). \nThe MAXI, installed on the Japanese Experiment Module of International Space Station (ISS), is designed to alert the transient X-ray sources \nFigure 1. Light curves collected by five instruments for analysis in this work. The bin size of these light curves is 10 days. \n<!-- image --> \nand monitor the long-term variations of the X-ray sources (Matsuoka et al. 2009). It contains two types of slit cameras with two different detectors: a gas proportional counter with an effective area of 5250 cm 2 for detecting the X-ray photons in the energy range of 230 keV, and a solid state camera of an effective area of 200 cm 2 sensitive to the X-ray photons in the energy range of 0.5-12 keV. MAXI can scan almost the entire sky twice during each ISS orbit ( ∼ 90 minutes). In this study, we analyzed the daily binned light curve of the energy range of 2-20 keV collected between 2009 August 12 (MJD 55055) and 2023 August 1 (MJD 60157), available on the MAXI website 2 , to study the superorbital modulation of 4U 1820-30. \nThe light curves of 4U 1820-30 collected by these five instruments are shown in Figure 1.", '3.1. Power Spectral Analysis': 'In the power spectral analysis, all the light curves were rebinned into daily averages for consistency. All the data points with a signal-to-noise ratio less than 2 σ were filtered out for further analysis. To probe the superorbital periods of various observations, the Lomb-Scargle (LS) periodogram (Scargle 1982) was applied to generate the power spectra. The errors of signal frequencies were estimated by the method proposed by Horne & Baliunas (1986): \nδf = 3 σ N 4 N 1 / 2 0 TA (2) \nFigure 2. Power spectra derived from the light curves collected five instruments in this work. The vertical dashed line indicates the superorbital frequency of the CG01 ephemeris (f=2.136 cycle/year). \n<!-- image --> \nwhere A is the amplitude of the signal, σ 2 N is the variance of the noise after the signal being removed, T is the time span of the light curve and N 0 is the number of data points. A was evaluated by fitting a single sinusoidal function to the light curve with the frequency fixed at the signal frequency obtained from the power spectrum and σ 2 N was estimated by the root-mean-square (rms) of the noise after the best fitted sinusoidal function was removed from the light curve. \nThe power spectra are depicted in Figure 2, and the detected superorbital periods are outlined in Table 1. It is apparent that the superorbital period deviates from the expected stability suggested by triple model, showingt a tendency to decrease over time. By incorporating the superorbital period reported by (Priedhorsky & Terrell 1984) from the Vela5B light curve, we estimated the timescale of the superorbital change by fitting a linear function to the detected superorbital periods over time (see Figure 3). This result in a period derivative of ˙ P sup /P sup = ( -7 . 37 ± 0 . 33) × 10 -4 yr -1 , corresponding to an evolution timescale of 1,357 years. This period derivative exceeds the upper limit proposed by CG10 ( \| ˙ P sup /P sup \| < 2 . 2 × 10 -4 yr -1 ). However, the linear fitting yielded a reduced χ 2 of 10.2, suggesting that the superorbital evolution of 4U 1820-30 is likely more complex than the constant period derivative model suggests. Further variations in the superorbital period from phase analysis will be demonstrated in Section 3.2. \nAdditionally, to compare the amplitudes of superorbital modulation, we folded these five light curves using two kinds of linear ephemerides. The first one is the \nFigure 3. Superorbital periods measured from 6 different instruments, including the one from Vela5B observation reported by Priedhorsky & Terrell (1984). The horizontal lines indicate the durations of the corresponding light curves, and the dashed line represents the best fit of a linear model to estimate the period change rate, which yields a period derivative of ˙ P/P = ( -7 . 37 ± 0 . 33) × 10 -4 yr -1 \n<!-- image --> \noptimal ephemeris proposed by CG01 (hereafter CG01 ephemeris), \nT N = JD 2 , 450 , 909 . 9 + 171 . 033 × N = MJD 50909 . 4 + 171 . 033 × N (3) \nThe other one is the local ephemeris, with a folding period corresponding to the best period obtained by the power spectrum (see Table 1),along with an arbitrary phase zero epoch for each light curve. The rms amplitudes folded by both types of ephemerides are listed in Table 1. The rms amplitudes of the profiles folded by the corresponding local ephemerides are larger than the those folded by CG01 ephemeris, indicating that the CG01 ephemeris is no longer suitable. This shows that the superorbital period of 4U 1820-30 has undergone significant changes during 1987 to 2023.', '3.2. Superorbital Phase Evolution': 'In this research, we aimed to trace the long-term evolution of the superorbital phase of 4U 1820-30, necessitating the analysis of superorbital phases measured from different instruments. However, time lags between different energy bands are often observed in astronomical time series. For instance, soft phase lags of pulsed emissions are commonly noted in accreting millisecond X-ray pulsars (Cui et al. 1998; Patruno & Watts 2021). Hence,a coherence test was conducted to verify if there was a significant time lag between any of two light curves from different instruments. However, this test could be \nTable 1. The Superorbital Period and RMS Amplitude Measured from the Light Curves Collected by Different InstrumentsTable 2. Coherence Test for RXTE ASM, RXTE PCA, Swift BAT and MAXI Light Curves \nonly performed on the light curves with overlapping observation times. Among the five light curves we analyzed in this study, except for Ginga ASM, other 4 light curves had overlapping observation times for each other, resulting in 6 pairs of light curves for the coherence test. For each pair of light curves, only overlapping parts were selected for coherence test. The power spectra were obtained the superorbital periods for the corresponding light curves. The superorbital modulation profiles of both light curves were conducted by folding the mean period measured from the power spectra with an arbitrary but fixed phase zero epoch. We discovered that all the profiles could be well fitted with a four-component sinusoidal function, that is, r ( φ ) = a 0 + ∑ 4 k =1 [ a k cos (2 πkφ ) + b k sin (2 πkφ )]. To measure the possible time delay between the two instruments, we applied the cross-correlation between the best-fitted modulation profiles. The test results are shown in Table 2. The phase difference is generally no more than 0.026 cycle, which is much smaller the phase jitters ( ∼ 0.1 cycle, see CG01). We conclude that there is no significant time delay is observable among these 4 instruments for the superorbital modulation of 4U 182030. \nTo trace the evolution of superorbital phase, we segmented the light curves and folded them to derive the modulation profiles. For instruments highly sensitive to superorbital modulation, like RXTE ASM and MAXI light curves, two cycles (2 × 171 days) per seg- \nent sufficed to yield clear profiles. In the case of Swift BAT observations, where no superorbital modulation was detected for the photon energy larger than 24 keV (Farrell et al. 2009), we selected four cycles as a data segment to ensure significant profile detection. As for the RXTE PCA light curve, due to the observation gaps, we adopted four cycles per segment to create the profiles. However, only three data segments provided sufficient phase coverage for further analysis. Given the very low sensitivity of Ginga ASM, a clear profile could only be obtained by folding the entire light curve. \nFollowing the approach of CG01, we selected the superorbital minimum as the fiducial point of the superorbital phase. Ideally , we would fold a light curve segment using a fix ephemeris, such as CG01 ephemeris (Eq. 3) to determine the phase (i.e. φ CG 01 ). However, as indicated in Section 3.1, the CG01 ephemeris is unlikely to be an optimal ephemeris for the all observations, particularly for recent ones (e.g. Swift BAT and MAXI observations), which could lead to profile deformation. To precisely determine the φ CG 01 , we folded the light curve segments using the optimal linear ephemeris specific to each instrument (local ephemeris). This involved folding the data by the period obtained from power spectral analysis (see Table 1) and an arbitrary but fixed phase zero epoch. A typical modulation profile is depicted in Figure 4. The phase ( φ local ) of a data segment was determined by fitting a four-component sinusoidal function and identifying the phase corresponding to the intensity minimum (fiducial point). This phase value, along with the local ephemeris, facilitated the evaluation the superorbital minimum time ( t m ) closest to the mid of observation time of the data segment. Subsequently, t m was then folded by CG01 ephemeris (Eq. 3) to obtain the phase φ CG 01 . \nThe superorbital orbital phases ( φ CG 01 ) are listed in Table 3, and their evolution is illustrated in Figure 5. It is noteworthy that while the superorbital modulation displays strong periodicity in the power spectra, the modulation profile varies from cycle to cycle, exhibiting the quasi-periodic nature as described in Zdziarski et al. (2007a). This variability induces phase jitters of ∼ 0.1 cycle, evident in Figure 5 and CG01. \nFigure 4. A typical superorbital modulation profile of a data segment created by folding the light curve collected by RXTE ASM from MJD 54158.64 to 54497.86 ( ∼ 2 cycles) with a folding period of 169.09 days from power spectral analysis (Section 3.1) and an arbitrary phase zero epoch. The solid line represents the optimal fit of a 4-component sinusoidal function to locate the superorbital minimum phase φ local . A BLS feature found by ˇ Simon (2003) can be also observed. \n<!-- image --> \nThese phase jitters are considerably larger than the error estimated from photon statistics ( ∼ 0.005 cycle). Despite the presence of phase jitters, a discernible phase evolution trend can be discerned in Figure 5. However, independently evaluating errors from phase jitters is difficult, which depend on the evolution model. In the subsequent analysis, we utilized the unweighted fitting method outlined by Press et al. (2002) to update the ephemeris for the superorbital modulation of 4U 1820-30.', '3.2.1. Linear Model': 'According to the triple model, The period should be remain stable from long-term perspective because the superorbital modulation is induced by a hierarchical third component orbiting around the binary system. Therefore, our initial approach involved fitting a linear function to the phase evolution as depicted in Figure 5. The parameters of the optimal linear function are listed in Table 4 yielding a period of 168 . 21 ± 0 . 15 days with a phase zero epoch of MJD 50920 ± 4 . 56. We assessed the root-mean deviation (RMSD), defined as: \nRMSD ≡ √ √ √ √ 1 ν N ∑ i =1 [ φ i -φ ( t i ) ] 2 (4) \nwhere φ i is the detected phase, φ ( t i ) is the expected phase value at t i evaluated from best fit model, and ν is the degree of freedom. The RMSD is 0.1 for the linear model. However best-fitted period in this model \nTable 3. Superorbital Phase ( φ CG 01 ) of 4U 1820-30 \nFigure 5. Evolution of superorbital phases folded by the CG01 ephemeris from 1987 to 2023. The dotted, solid and dashed lines represent the best fits for linear, glitch and quadratic models, respectively. The shaded area indicates the low power state between MJD 50773 and 52627, and the vertical dash-dot line represents the glitch time MJD 52264 evaluated by the glitch model, with the horizontal error bar indicating the 1 σ uncertainty of glitch time. \n<!-- image --> \nsignificantly differs from the reported superorbital periods that detected in early RXTE ASM observations, as listed in Table 5, as well as the superorbital period of 176 . 4 ± 1 . 3 days reported by Priedhorsky & Terrell (1984) from Vela5B observation. Furthermore, the expected phase at the midpoint of Ginga ASM observation time (MJD 47677.56) is 0 . 382 ± 0 . 033, about 7.3 σ deviated from the detected value of 0.14 (see Figure 5). Therefore, the linear model is unlikely to describe the superorbital phase evolution of 4U 1820-30.', '3.2.2. Glitch Model': "Table 5 presents the reported superorbital periods detected by early RXTE ASM observations, which are roughly consistent with the period in the CG01 ephemeris (171 days). However, for later observations, particularly those from Swift BAT and MAXI, the period is approximately 167.4 days. One the possibility is that the superorbital period underwent an abrupt change (glitch), likely between years 2000 and 2005 (see Figure 5). \nOnthe other hand, Kotze & Charles (2012) conducted a dynamic power spectrum analysis and observed weaker superorbital modulation power during the period MJD 51200-52200, hereafter, referred to as the low power state. Additionally, signals with shorter periods, ∼ 85 days (first harmonic) and ∼ 65 days emerged in the dynamic power spectrum, indicating a change in the modulation profile during that time interval. However, considering the window size used to generate the dynamic \nLinear model \nφ = a 0 + a 1 ( t - T 0 ) \na 0 = ( T 0 - T 0 ,CG 01 ) /P CG 01 a \na \n1 \n= ( \nP \n0 \n- \nP \nCG \n01 \n) \n/ \n( \nPP \nCG \n01 \n) \nParameter \nValue \na 0 \n0 . 065 ± 0 . 027 \na 1 (cycle/day) \n( - 9 . 83 ± 0 . 54) × 10 - 5 \ncov ( a 0 , a 1 ) (cycle/day) \n- 1 . 21 × 10 - 7 \nT 0 (MJD) \n50920 . 48 ± 4 . 65 \nP (days) \n168 . 21 ± 0 . 15 \nGlitch model \nφ \n= \n{ \na \na \n0 \n' \n0 \n+ \na \n+ \na \n1 \n' \n1 \n( \nt \n- \nT \n( \nt \n- \nT \n0 \n0 \n) \nif \nt \n≤ \nT \n) \nif \nt > T \na 0 = ( T 0 -T 0 ,CG 01 ) /P CG 01 \na 1 = ( P 1 -P CG 01 ) / ( P 1 P CG 01 ) \na ' 0 = ( T 0 -T 0 ,CG 01 ) /P CG 01 + n g ( P 1 -P 2 ) /P 2 \na ' 1 = ( P 2 -P CG 01 ) / ( P 2 P CG 01 ) \nT g = T 0 + n g P 1 \nTable 4. Parameters of Superorbital Modulation of 4U 1820-30 \nQuadratic model \nφ = a 0 + a 1 ( t - T 0 ) + a 2 ( t - T 0 ) 2 \na 0 = ( T 0 - T 0 ,CG 01 ) /P CG 01 \na 1 = ( P 0 - P CG 01 ) / ( P 0 P CG 01 ) \na 2 = 1 / 2 ˙ P/ ( P 0 P CG 01 ) \na T 0 ,CG 01 =JD 2,450,909.9=MJD50909.4 and P CG 01 =171.033 days from Eq.9 of GC01. \ng \ng \n; \n. \nTable 5. Superorbital periods of 4U 1820-30 from Early RXTE ASM Observations a \npower spectrum of 4U 1820-30 in Kotze & Charles (2012) (5 cycles), this low power state time interval should be extended to approximately MJD 50773 to 52627. \nCompared with the the phase evolution (i.e. Figure 5), it appears likely that the glitch occurred around MJD 52500, near end of low power state. Marginal evidence supports this assumption. As listed in Table 5, the reported superorbital period by Zdziarski et al. (2007a) was 170 . 6 ± 0 . 3 days, slightly smaller than those reported by CG01, ˇ Simon (2003), and Wen et al. (2006). This discrepancy may be due to that significant portion of data ( ∼ 37%) analyzed by Zdziarski et al. (2007a) were collected after MJD 52500. Similarly, the power spectral analysis of the entire RXTE ASM light curve yielded a superorbital period of 169.09 days (see Table 1), falling between 171 and 167.4 days, because about half ( ∼ 56%) of the data were collected after MJD 52500. Conversely, the periods detected form the power spectra of Swift BAT and MAXI were nearly identical at approximately 167.4 days, because both observations were made after MJD 52500 (see Table 1). \nWe therefore fitted the phase evolution with the glitch model using the ephemeris described by Eq.5 in (Wolff et al. 2009) \nT N = { T 0 + P 1 N if N ≤ n g ; T 0 + P 1 n g + P 2 ( N -n g ) if N > n g . (5) \nwhere T 0 is the phase zero epoch, P 1 and P 2 are the periods before and after glitch, respectively, n g is the glitch cycle count, and the glitch time T g ≡ T 0 + P 1 n g . The fitting results are shown in Figure 5, and the parameters are listed in Table 4. We obtained significantly different superorbital periods of 170 . 67 ± 0 . 64 days and 167 . 66 ± 0 . 18 days before and after the glitch time MJD 52264, respectively, and ∆ P sup /P sup = -0 . 018 ± 0 . 007. \nFigure 6. Superorbital modulation profiles folded by glitch ephemeris listed in Table 4 for the light curves collected by (a) Ginga ASM, (b) RXTE ASM, before glitch, (c) RXTE PCA, before glitch, (d) RXTE ASM, after glitch, (e) RXTE PCA, after glitch, (f) Swift BAT, and (g) MAXI. \n<!-- image --> \nThe RMSD for this glitch model is 0.078. Comparing this with the RMSD of 0.1 from the linear model, the Ftest yielded a p-value of 0.04, indicating that the glitch model is better than the linear model. Figure 6 shows the modulation profiles folded by the glitch ephemeris. All the superorbital minima (fiducial points) are close phase zero, implying that this ephemeris effectively describes the superorbital phase evolution of 4U 1820-30 from 1987 to 2023.", '3.2.3. Quadratic Model': 'While the glitch model effectively describes the superorbital phase evolution, we cannot rule out the possibility that period difference between early RXTE ASM observations and recent ones stems from a smooth change in the superorbital period. Farrell et al. (2009) observed that the peak width of the superorbital signal in the power spectrum made from Swift BAT light curve was marginally wider than that from simulation, suggesting that a change in superorbital period. Hence, we apply a simple model assuming a constant period derivative ( ˙ P sup ) to fit a quadratic function to the superorbital phase evolution.The fitting results are depicted in Figure 5, and the parameters are listed in Table 4. A period derivative of ˙ P sup = ( -3 . 58 ± 0 . 72) × 10 -4 day/day, or ˙ P sup /P sup = ( -7 . 71 ± 1 . 54) × 10 -4 yr -1 was obtained from the fitting, and a quadratic ephemeris \nT N = ( MJD 50914 . 41 ± 3 . 89) + (169 . 53 ± 0 . 29) × N +( -3 . 03 ± 0 . 61) × 10 -2 × N 2 \n(6) \nFigure 7. Superorbital modulation profiles folded by quadratic ephemeris listed in Table 4 for the light curves collected by (a) Ginga ASM, (b) RXTE ASM, (c) RXTE PCA, (d), Swift BAT, and (e) MAXI. \n<!-- image --> \nwas established. This period derivative value is consistent with the one evaluated from power spectra in Section 3.1. The RMSD for the quadratic model is 0.083. Compared to this with the RMSD of 0.078 from glitch model, the F-test yielded a p-value of 0.34, which indicating that these two models are about equally adept at describing the superorbital phase evolution. Figure 7 illustrates the modulation profiles folded by the quadratic model. Similar to the glitch model, all the superorbital minima (fiducial points) are located around phase zero. It provides an evidence that the quadratic model is suitable for describing the superorbital phase evolution of 4U 1820-30.', '3.3. X-ray Burst Active Times': "As previously mentioned in Section 1, Type-I X-ray bursts of 4U 1820-30 are exclusively observable during the low state (Clark et al. 1977; Stella et al. 1984). Further confirmed by CG01, indicated the Type-I X-ray bursts were detected only within ± 23 days around superorbital minima for bursts reported before 1985. This supports the notation that the superorbital modulation stems from changes in the accretion rate changes rather than occultation effects. However, CG01's statistics only included four burst active dates. Subsequently, more X-ray bursts of 4U 1820-30 were detected. With the updated superorbital ephemerides, this evidence can be further substantiated. Although the possibility exists that the X-ray bursts occur in another low state, which may deviate significantly from the superorbital minima (e.g.the brief low state found by ˇ Simon (2003)), it is \nlikely that most of X-ray burst active times would cluster around the superorbital minima. \nIn this study, we collected the reported burst active dates of 4U 1820-30 from Grindlay et al. (1976); Vacca et al. (1986); Haberl et al. (1987); Zdziarski et al. (2007b); Garc'ıa et al. (2013), and (Yu et al. 2024), as well as four superbursts discovered by Strohmayer & Brown (2002); in't Zand et al. (2011); Serino et al. (2021a), and (Serino et al. 2021b). The deviations in burst active dates relative to the nearest superorbital minima predicted by CG01 and three ephemerides derived in this study are shown in Figure 8. It is evident that large deviations can be observed after the low power state (MJD 52630) when using the superorbital minima predicted by the CG01 ephemeris. The root-mean-square deviations are 41.2, 32.7 27.4 and 27.0 days for CG01, linear, glitch and quadratic ephemerides, respectively. This provides supportive evidence that glitch and quadratic ephemerides are better than CG01 and linear ephemerides in describing the superorbital phase evolution of 4U 1820-30. Notably, X-ray bursts occurring between MJD 50399 and MJD 50343 exhibited large deviations ( ∼ 60 days) for all ephemerides. Upon examining the RXTE ASM light curve, we observed that these X-ray bursts occurred around a BLS with a count rate of only 12.0 cts/s compared to the mean count rate of 21.0 cts/s. This observation reaffirms that superorbital modulation is primarily caused by accretion variations rather than occultation effects.", '3.4. X-ray Burst Active Times': "As previously mentioned in Section 1, Type-I X-ray bursts of 4U 1820-30 are exclusively observable during the low state (Clark et al. 1977; Stella et al. 1984). Further confirmed by CG01, indicated the Type-I X-ray bursts were detected only within ± 23 days around superorbital minima for bursts reported before 1985. This supports the notation that the superorbital modulation stems from changes in the accretion rate changes rather than occultation effects. However, CG01's statistics only included four burst active dates. Subsequently, more X-ray bursts of 4U 1820-30 were detected. With the updated superorbital ephemerides, this evidence can be further substantiated. Although the possibility exists that the X-ray bursts occur in another low state, which may deviate significantly from the superorbital minima (e.g.the brief low state found by ˇ Simon (2003)), it is likely that most of X-ray burst active times would cluster around the superorbital minima. \nIn this study, we collected the reported burst active dates of 4U 1820-30 from Grindlay et al. (1976); \nFigure 8. The deviations of Type-I X-ray burst active dates relative to the nearest expected superorbital minima evaluated by CG01, linear, glitch and quadratic ephemerides. The circles are the superbursts reported by Strohmayer & Brown (2002); in't Zand et al. (2011); Serino et al. (2021a,b). The region between two horizontal dashed lines is the ± 23 days burst active interval suggested by CG01, and the vertical dash-dot line in the plot of glitch ephemeris is the glitch time evaluated by the glitch model (Section 3.2.2). \n<!-- image --> \nVacca et al. (1986); Haberl et al. (1987); Zdziarski et al. (2007b); Garc'ıa et al. (2013), and (Yu et al. 2024), as well as four superbursts discovered by Strohmayer & Brown (2002); in't Zand et al. (2011); Serino et al. (2021a), and (Serino et al. 2021b). The deviations in burst active dates relative to the nearest superorbital minima predicted by CG01 and three ephemerides derived in this study are shown in Figure 8. It is evident that large deviations can be observed after the low power state (MJD 52630) when using the superorbital minima predicted by the CG01 ephemeris. The root-mean-square deviations are 41.2, 32.7 27.4 and 27.0 days for CG01, linear, glitch and quadratic ephemerides, respectively. This provides supportive evidence that glitch and quadratic ephemerides are better than CG01 and linear ephemerides in describing the superorbital phase evolution of 4U 1820-30. Notably, X-ray bursts occurring between MJD 50399 and MJD 50343 exhibited large deviations ( ∼ 60 days) for all ephemerides. Upon examining the RXTE ASM light curve, we observed that these X-ray bursts occurred around a BLS with a count rate of only 12.0 cts/s compared to the mean count rate of 21.0 cts/s. This observation reaffirms that superorbital modulation is primarily caused by accretion variations rather than occultation effects. \nOn the contrary, out of the four superbursts detected, only the one on MJD 51430 (Strohmayer & Brown \n2002) fell within ± 23 days region, whereas the other three occurred outside of this timeframe for both glitch and quadratic ephemerdes (see Figure 8). Upon examining the RXTE ASM and MAXI light curves, we observed that the count rates on the dates of superbursts were 57%, 150%, 121% and 138% of the corresponding mean count rates of the light curves for the superbursts detected on MJD 51430 (Strohmayer & Brown 2002), MJD 55272 (in't Zand et al. 2011), MJD 59449 (Serino et al. 2021a) and MJD 59543 (Serino et al. 2021b), respectively. It is likely that the low state constraint for the regular X-ray bursts of 4U 1820-30 (Clark et al. 1977; Stella et al. 1984) does not apply to the superbursts. More observations are required for further confirmation.", '4.1. Challenge of the Triple Model': "The initial aim of this study was to further validate the stability of superorbital period of 4U 1820-30, a crucial piece of evidence for the triple model as described in Section 1, which explains its superorbital modulation with a period of ∼ 170 days. Given that 4U 1820-30 resides in NGC 6624, a star-crowded region, previous studies suggested a high likelihood of the binary capturing a third star and forming a stable hierarchical triple system (Grindlay 1988). Black (1982) proposed a stability criteria for such a triple system as \nµ ≤ µ crit = 0 . 175 ∆ 3 (2 -∆) 3 / 2 (7) \nwhere µ = ( m 2 + m 3 ) / 2 m 1 , ∆ = 2( R -1) / ( R + 1), R = R 3 /R 1 , m 1 , m 2 and m 3 are the masses of binary primary, secondary and tertiary companion, respectively, R 1 is the binary separation and R 3 is the maximum separation of the binary primary and the tertiary companion. Applying Eq. 7 to 4U 1820-30 system with the assumption that m 1 = 1 . 4 M /circledot , m 2 = 0 . 07 M /circledot (Chou & Jhang 2023), m 3 = 0 . 5 M /circledot (CG01), binary period of 685 s and third star orbital period of 1.1 days 1, according to the Kepler's third law, we found that µ = 0 . 204 and µ crit = 24 . 18, which satisfies the stability criteria proposed by Black (1982). Additionally, CG01, based on their analysis of the RXTE ASM light curve collected between 1996 and early 2000 and in conjunction with fidicial points detected by Vela5B and Ginga, found no significant superorbital period derivative, setting an upper limit of \| ˙ P/P \| < 2 . 2 × 10 -4 yr -1 , thereby confirming its stability and lending support to the triple model. Therefore, for the 4U 1820-30 system, with additional subsequent observations after 2000, one would expect that the observed superorbital pe- \nuld closely match the value found by CG01 (171 days) and that the period derivative could be further constrained. \nHowever, upon analyzing X-ray light curves collected by the sky monitoring/scanning instruments from 1987 to 2023, we discovered a significant change in the superorbital period from 171 days to 167 days, identified through both power spectral analysis (Section 3.1) and phase analysis (Section 3.2) over a time span of ∼ 36 years. This suggests that the ephemeris proposed by CG01 is no longer suitable for describing the superorbital modulation of 4U 1820-30, and the period is not as stable as anticipated by triple model. By analyzing the superorbital phase evolution, we suggested that the superorbital period may have experienced an abrupt change during late 2000 to early 2003 ( T g =MJD 52264 ± 466) or may be constantly changing with a period derivative of ˙ P = ( -3 . 58 ± 0 . 72) × 10 -4 day/day. \nThe significant difference between the period detected from Vela5B observation, 176 . 4 ± 1 . 3 days (Priedhorsky & Terrell 1984), and Ginga observation, 171 . 12 ± 1 . 99 days (see Table 1) suggests that the superorbital period may have experienced another glitch between 1976 and 1987. If 4U 1820-30 is a hierarchical triple system, from Eq. 1, a glitch in superorbital period may be induced by changes in either the binary orbital period or the third star orbital period as \n∆ P sup P sup = 2 ∆ P 3 P 3 -∆ P orb P orb (8) \nGlitches in orbital periods have been observed in some of total eclipsing LMXBs, like EXO 0748-676 (Wolff et al. 2009), XTE J1710-281 (Jain & Paul 2011; Jain et al. 2022) and AX J1745.6-2901 (Ponti et al. 2017). These glitches likely result from by magnetic, solar-type cycles of the companion star, affecting the mass distribution of companion and leading to variations in its quadruple moment (Wolff et al. 2009). However, the magnitudes of these glitches are typically in the order of milliseconds, with ∆ P orb /P orb ∼ 10 -7 -10 -6 . For 4U 182030 system, the superorbital period glitch was measured as ∆ P sup /P sup = 1 . 8 × 10 -2 . No glitch has ever been observed in the binary orbital phase evolution (see Figure 4 in Chou & Jhang 2023), implying that the orbital period glitch of the third companion was as high as ∆ P 3 /P 3 = 9 × 10 -3 , about 4 orders of magnitude larger than those from eclipsing LMXBs. However, the superorbital period change may not occur abruptly but within a finite short time interval. Suppose the timescale to be 900 days, estimated from the uncertainty of T g , the mean orbital period derivative of the third star would be as high as ˙ P 3 ≈ 2 × 10 -3 day/day. Thus, the triple \nmodel is unlikely to explain this large superorbital period change in such a short time. \nThe low power state is a particular phase during the superorbital modulation evolution of 4U 1820-30. The dynamic power spectrum demonstrated in Figure 24 of Kotze & Charles (2012) indicates that in addition to the weaker power detected in the superorbital period, the powers of its first harmonics and a signal of period ∼ 65 days became significant. This suggests that the modulation were more complicate than usual. Our phase analysis results also show that the superorbital phases had larger fluctuation during the low power state (see Figure 5) with an RMSD of 0.11, evaluated by the best glitch model fitting, compared to 0.075 for the phases outside low power state. It is probable that the superorbital period was 171 days before the low power state, became unstable during the low power state, and stabilized at 167 days later. However, it is unclear what the cause of this phenomenon is. \nConversely, the superorbital phase evolution is also well-fitted by a quadratic model, although there is a significant difference (2.2 σ ) between the period evaluated as ephemeris extrapolated to the midpoint of Vela5B observation time (172 . 84 ± 0 . 94 days) and the observed period (176 . 4 ± 1 . 3 days) as reported by Priedhorsky & Terrell (1984). If 4U 1820-30 is a hierarchical triple system, from Eq. 1, the relation of period derivatives can be written as \n˙ P sup P sup = 2 ˙ P 3 P 3 -˙ P orb P orb (9) \nAlthough the exact binary orbital period derivative being unknown, it is believed it is ˙ P orb /P orb ∼ +10 -7 yr -1 (Chou & Jhang 2023). Therefore, the observed superorbital period derivative ˙ P sup /P sup = ( -7 . 71 ± 1 . 54) × 10 -4 yr -1 is contributed by the tertiary companion, with a value of ˙ P 3 /P 3 = -3 . 9 × 10 -4 yr -1 , corresponding to a variation timescale of only ∼ 2,600 years. From Eq. 7, we infer that the triple system should be stable; therefore, such a fast period change is unlikely to occur. Furthermore, the acceleration from the gravitational potential in NGC 6624, estimated as a c /c ∼ -10 -7 yr -1 (Peuten et al. 2014), is insufficient for the orbital period derivative of the third companion derived from Eq. 9. Thus, the triple model can hardly explain the observed superorbital period derivative. \nIf 4U 1820-30 is not a triple system, the constraints regarding the observed value of the binary orbital period derivative may be relaxed. To explain the discrepancy between the positive theoretical value ( ˙ P orb /P orb > 8 . 8 × 10 -8 yr -1 , Rappaport et al. 1987) and the negative observed value ( ˙ P orb /P orb = ( -5 . 21 ± 0 . 13) × 10 -8 yr -1 , \nChou & Jhang 2023) of the binary orbital period derivative, it has been proposed that 4U 1820-30 is being accelerated by the gravitational potential within the globular cluster NGC 6624 (Tan et al. 1991; Peuten et al. 2014). However, Peuten et al. (2014) suggested that the maximum radial acceleration from the gravitational potential from NGC 6624 itself ( \| a c,max /c \| = 1 . 3 × 10 -9 yr -1 is an order of magnitude smaller than the value required to explain the observed binary period period derivative. Therefore, Peuten et al. (2014) proposed three possible scenarios to provide additional acceleration for 4U 182030, a flyby stellar mass dark remnant, a intermediatemass black hole at the center of NGC 6624, and a central concentration of dark remnants. Only the last scenario was preferred because the first two scenarios tend to destroy the triple system (Peuten et al. 2014). However, if 4U 1820-30 is a pure binary system, the first two scenarios become viable explanations for the observed binary orbital period derivative.", '4.2. Thermal Disk Instability': 'The fact that the type-I X-ray bursts of 4U 182030 can only be observed in the low state (Clark et al. 1977; Stella et al. 1984) has been reconfirmed in this work (see Section 3.4). This implies that the superorbital modulation of 4U 1820-30 is caused by variations in accretion rate rather than by external absorption or precession of accretion disk. Kotze & Charles (2012) listed eight possible mechanisms to account for the superorbital modulations observed in X-ray binaries, but only the third body (i.e., triple model) and the X-ray state changes can possibly be responsible for the superorbital modulation of 4U 1820-30. If the triple model is ruled out due to the instability of superorbital period, the only remaining mechanism is the X-ray state changes. X-ray state changes refer to variations in mass accretion rate between high and low states due to thermal disk instability, as observed in dwarf novae and soft X-ray transients. Priedhorsky & Terrell (1984) proposed that this mechanism could explain for the superorbital modulation of 4U 1820-30. However, Menou et al. (2002) pointed out that if thermal disk instability could occur in 4U 1820-30 system, the mass transfer rate ˙ m ≤ ˙ m crit = 4 . 4 × 10 16 g s -1 . From the mean flux of < F bol > = 8 . 7 × 10 -9 erg cm -2 s -1 for 4U 182030 (Zdziarski et al. 2007b) and the distance of 8.019 kpc for NGC 6624 (Baumgardt & Vasiliev 2021), we obtained a mean luminosity of < L > = 6 . 7 × 10 37 erg s -1 and a mass accretion rate of ˙ m 1 = 3 . 6 × 10 17 g s -1 for a neutron star with a mass of 1.4 M /circledot and a radius of 10 6 cm. It is approximately an order of magnitude larger \nthan the ˙ m crit . Therefore, this mechanism is unlikely to explain the superorbital modulation of 4U 1820-30.', '4.3. Irradiation-induced Mass Transfer Instability': "Zdziarski et al. (2007b) discovered that the binary orbital modulation amplitude and the offset phase in 4U 1820-30 depend significantly on the accretion rate, which is highly related to the superorbital modulation phase. The orbital modulation in the X-ray band is believed to be caused by absorption from structures in the disk rim where the accretion flow from the companion impacts the outer edge of the disk (Stella et al. 1987). As the mass loss rate changes, variations of the accretion stream induce changes in the absorption of outer edge structures and the position of impact point. This makes the amplitude and phase of orbital modulation dependent on the mass loss rate, and subsequently, on the accretion rate after a viscous time of ∼ 10 5 s (Zdziarski et al. 2007b). The variation in mass loss rate could be explained by the triple model as described in Section 1, where the eccentricity variation of the binary system induced by the third companion result in changes to the mass loss rate. Although the superorbital modulation is probably not a consequence of a third companion, the the discovery by Zdziarski et al. (2007b). implies that the superorbital modulation of 4U 1820-30 is due to changes in mass loss rate. \nOne possible cause of variations in the accretion flow, aside from the presence of a third companion, is irradiation-induced mass transfer instability. This model has ever proposed to explain the flux variation of soft X-ray transients (Hameury et al. 1986) and was included into the hybrid model proposed by ˇ Simon (2003). Due to small binary separation of 4U 1820-30, the irradiation on the companion by the X-ray emission from the neutron star and the inner part of accretion disk is strong. Because the companion of 4U 1820-30 is only partially degenerate (Rappaport et al. 1987), irradiation on the non-degenerate envelope enhances the mass loss of the companion. Chou & Jhang (2023) estimated that at least 40% of the mass lost from the companion is ejected from the binary system. Such a strong outflow is probably caused by the irradiation on the companion, as proposed by Tavani (1991). However, a part of X-ray irradiation on the companion is blocked by the accretion disk, with the area that depending on the scale height of disk rim. When the scale height of disk rim is small, a larger irradiation area enhances the mass loss rate and the accretion flow, which increases the scale height of the accretion disk rim. Conversely, when the scale height of accretion disk rim is large, a larger portion of the companion's surface is shielded by the disk. This results in \na reduction in the mass loss rate, as well as in the accretion flow and the scale height of the accretion disk rim. This a cyclical process may explain the quasi-periodic superorbital modulation of 4U 1820-30. \nSuppose the accretion disk in 4U 1820-30 is geometrically thin and optically thick (Pringle 1981). The shielded region on the companion can be estimated. For 4U 1820-30 with a neutron star mass of m 1 = 1 . 4 M /circledot , a companion mass of m 2 = 0 . 07 M /circledot (Chou & Jhang 2023), and an orbital period of P orb = 685 s, we derived the binary separation of a = 1 . 33 × 10 10 cm from Kepler's third law. The discovery of the superhump in 4U 1820-30 system (Wang & Chakrabarty 2010) indicates that the rim of the accretion disk reaches a 3:1 resonance radius, giving a disk radius of r d = 6 . 4 × 10 9 cm. The scale height of the accretion disk can be evaluated as √ kT/µm H / Ω k (Spruit 2010) where T and Ω k are the temperature and the Keplerian angular velocity at the disk radius r , m H is the mass of a hydrogen atom, µ = 4 for a helium-dominated disk, and k is the Boltzmann constant. The temperature at the disk rim is estimated as \nT = ( 3 G ˙ m 1 m 1 8 πσr 3 d ) 1 4 (10) \n(Pringle 1981) where σ is Stefan-Boltzmann constant and G is gravitational constant. The accretion rate of 4U 1820-30 ˙ m 1 = 3 . 6 × 10 17 g s -1 . Thus, the temperature at the disk rim is T = 2 . 7 × 10 4 K and the scale height is H = 2 . 8 × 10 7 cm. For a Roche lobe filled companion, the radius of companion is R 2 = R L = 2 / 3 4 / 3 [ q/ (1 + q )] 1 / 3 a (Paczy'nski 1971) where q = m 2 /m 1 . For 4U 1820-30 system, q=0.05, so R 2 = 1 . 68 × 10 9 cm. The scale height of the irradiation shielded by the accretion disk on the companion around L 1 point is h = H ( a -R 2 ) /r d = 5 . 0 × 10 7 cm, which equivalent to a latitudes of sin -1 ( h/R 2 ) = 1 . 7 · on the companion surface. Although the shielded latitude is small, it covers the L 1 point if the orbital plane and the accretion disk are coplanar. The accretion stream flows from a small region around the L 1 point on the surface of companion. If the accretion disk rim partially obscures this region, even a marginal change in irradiation on this region due to variations in the scale height of accretion disk rim could induce a significant change in mass loss because of weak effective gravitational field around L 1 point. Such a large variation in mass loss rate could result in quasi-periodic superorbital modulation, causing a 2-3 fold change in X-ray flux of 4U 1820-30. However, more observations and theoretical studies are required to verify this irradiation-induced mass transfer instability \nscenario, including the evolution of superorbital period discovered in this study.", '5. SUMMARY': 'The triple model was once considered a plausible explanation for the superorbital modulation observed in 4U 1820-30. The stability of the superorbital period is the crucial evidence for verifying this model. CG01 suggested that the superorbital period was stable at 171 days and early RXTE ASM data support this 171-day periodicity, indicating stability of superorbital period. \nIn this study, we analyzed the data collected by Ginga ASM, RXTE ASM, RXTE PCA, Swift BAT, and MAXI over a time span of 36 years to verify the triple model for the 4U 1820-30 system. The superorbital periods derived from the power spectra of these five instruments show a significant change from 171 days to 167 days between 1987 and 2023, suggesting the instability of the superorbital period. Phase analysis revealed that the superorbital period may have experienced a period glitch between late 2000 and early 2003, or may have changed smoothly with a period derivative of ˙ P sup = ( -3 . 58 ± 0 . 72) × 10 -4 day/day. Two ephemerides, glitch and quadratic, were established to describe the expected superorbital minimum times of 4U 1820-30. These updated ephemerides accurately describe the superorbital minimum times with a mean phase jitters of ∼ 0 . 08 cycles. The fact that the Type-I X-ray bursts can be observed only in the low state implies that a high probability of detecting the bursts around the superorbital minimum. By examining previously reported burst detection dates with different ephemerides, we found that the burst dates are more clustered around the superorbital phase zero when folded with the glitch and quadratic ephemerides, rather than with linear and CG01 ephemerides. This is not only reconfirms the low state constraint for regular X-ray bursts as suggested by Clark et al. (1977) and Stella et al. (1984), but also provides supportive evidence that the glitch and quadratic ephemerides better at describe the superorbital minimum times. \nThe instability of the superorbital periodicity in 4U 1820-30 discovered in this work seriously challenges the triple model. According to Eq. 1, the superorbital period change could be due to either the binary period variation or the orbital period change of the third companion. However, the binary orbital modulation has been monitored for over 46 years, and neither period glitch nor period derivative of an order of ˙ P orb /P orb ∼ 10 -4 yr -1 has ever been observed (see Chou & Jhang 2023). Therefore, the superorbital period changes likely reflect the orbital period variation of the third compan- \n. While orbital period glitches have been observed in some eclipsing LMXBs, the magnitude of these change is much smaller than that the superorbital period glitch observed 4U 1820-30. The period derivative derived from the quadratic model indicates that the timescale of the orbital period evolution of the third companion is ∼ 2600 years, which is inconsistent with the stability expected in a hierarchical triple system. If the triple model does not apply to the 4U 1820-30 system, two previously unfavorable scenarios proposed by Peuten et al. (2014) - a stellar mass dark remnant and an intermediate mass black hole - can be reconsidered to explain the discrepancy between the theoretical and observed binary orbital period derivatives. \nThe absence of regular Type-I bursts in the high state suggests that the superorbital modulation of 4U 1820-30 results from variations in the accretion rate rather than from the occultation effect caused by the precession of a tilting or warping accretion disk. Thermal disk instability is unlikely to be the cause of the superorbital modulation due to the high accretion rate of 4U 1820-30. On the other hand, because the amplitude of orbital modulation highly depends on accretion rate (Zdziarski et al. 2007b), the superorbital modulation could result from variation in mass transfer from companion. Given the instability of superorbital period of 4U 1820-30, such variation is unlikely to be induced by a third companion. We proposed that irradiation-induced mass transfer instability may be responsible for the superorbital modulation of 4U 1820-30. The accretion stream is expected to flow from a small region around the L 1 point on companion, where the effective gravitational field is weak. Therefore, the accretion stream is highly sensitive to the X-ray irradiation onto this region. The irradiation onto this region may be partly blocked by the accretion disk rim, whose scale height also depends on the accretion stream. Small variations in the scale height can lead to significant changes in accretion stream. A cyclical process could result in quasi-periodic superorbital modulation in 4U 1820-30. \nUsing the data collected by X-ray monitoring/scanning X-ray telescopes, we discovered the instability of the superorbital period of 4U 1820-30. Our study, we found that both the glitch model and the quadratic model describe the superorbital phase evolution well. However, additional observations are necessary to validate these models or to provide a better ephemeris for the superorbital modulation of 4U 182030. This period instability suggests that the triple model is unlikely suitable to explain the superorbital modulation of 4U 1820-30. Although we proposed that the irradiation-induced mass transfer instability may be responsible for the superorbital modulation, further observations and theoretical works are required to verify this model, including the periodicity, modulation amplitude and profile, and the puzzling phase evolution, which identified in this study. Fortunately, Swift BAT and MAXI are continuously monitoring the X-ray sky. Additionally, the newly operational Wide-field X-ray Telescope on-board the Einstein Probe (Yuan & Osborne 2015), which is sensitive to 0.5 to 4.0 keV X-ray photons and scans the entire night sky in three satellite orbits, can provide further data to better understand the nature of the superorbital modulation of 4U 1820-30. \nThis research has made use of data and software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC. We also express our gratitude to the RXTE team for archiving the RXTE PCA monitoring observations of the galactic center and plane data, and to MAXI team for archiving the MAXI data. \nFacilities: ADS, HEASRAC, Ginga (ASM), RXTE (ASM), RXTE (PCA),Swift (BAT), MAXI \nSoftware: \nheasoft(v6.30)', 'REFERENCES': "Barthelmy, S. D., Barbier, L. M., Cummings, J. 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2024Univ...10..356T	This study considers the specific case of a flat minimally coupled to gravity quintessence cosmology with a dark energy quartic polynomial potential that has the same mathematical form as the Higgs potential. Previous work on this case determined that the scalar field is given by a simple expression of the Lambert W function in terms of the easily observable scale factor. This expression provides analytic equations for the evolution of cosmological dark energy parameters as a function of the scale factor for all points on the Lambert W function principal branch. The Lambert W function is zero at a scale factor of zero that marks the big bang. The evolutionary equations beyond the big bang describe a canonical universe that is similar to inlineformulammlmath idmm1mmlsemanticsmmlmommlmommlsemanticsmmlmathinlineformulaCDM making it an excellent dynamical template to compare with observational data. The portion of the W function principal before the big bang extends to the infinite prebang past. It describes a noncanonical universe with an initially very low mass density that contracts by rolling down the dark energy potential to a singularity big bang at the scale factor zero point. This provides a natural origin for the big bang. It also raises the possibility that the universe existed before the big bang and is far older and that it was once far larger than its current size. The recent increasing interest in the possibility of a dynamical universe instead of inlineformulammlmath idmm2mmlsemanticsmmlmommlmommlsemanticsmmlmathinlineformulaCDM makes the exploration of the nature of such universes particularly relevant.	2024-09-01T00:00:00Z	['10.3390/universe10090356', '2024arXiv240906792T', '2024Univ...10..356T', 'arXiv:2409.06792', '10.48550/arXiv.2409.06792']	['cosmological constraint', 'dark energy', 'theoretical model', 'General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']	NonCanonical Dark Energy Parameter Evolution in a Canonical Quintessence Cosmology	2,024	201	0.33	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']	0	https://arxiv.org/pdf/2409.06792.pdf	{'No Header': 'Citation: Thompson,R.I.Special Issue for the Journal Universe "[Dark Energy andDarkMatter]". Journal Not Specified 2024 , 1 , 0. https://doi.org/ \nReceived: Aug. 12 2024 \nAccepted: Sept. 2, 2024 \nPublished: Sept. 5. 2024 \n<!-- image --> \nArticle', 'Non-Canonical Dark Energy Parameter Evolution in a Canonical Quintessence Cosmology': "Rodger I. Thompson 1,†,‡ * \nPublisher's Note: MDPIstaysneutral with regard to jurisdictional claims in published maps and institutional affiliations. \nCopyright: © 2024 by the author. Submitted to Journal Not Specified for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). \n- 1 Department of Astronomy and Steward Observatory University of Arizona; Tucson, AZ. 85721,USA; [email protected] \nAbstract: This study considers the specific case of a flat, minimally coupled to gravity, quintessence cosmology with a dark energy quartic polynomial potential that has the same mathematical form as the Higgs potential. Previous work on this case determined that the scalar field is given by a simple expression of the Lambert W function in terms of the easily observable scale factor. This expression provides analytic equations for the evolution of cosmological dark energy parameters as a function of the scale factor for all points on the Lambert W function principal branch. The Lambert W function is zero at a scale factor of zero that marks the big bang. The evolutionary equations beyond the big bang describe a canonical universe that is similar to Λ CDMmaking it an excellent dynamical template to compare with observational data. The portion of the W function principal before the big bang extends to the infinite pre bang past. It describes a noncanonical universe with an initially very low mass density that is contracting by rolling down the dark energy potential to a singularity, big bang, at the scale factor zero point. This provides a natural origin of the big bang. It also raises the possibility that the universe existed before the big bang and is far older and was once far larger than its current size. The recent increasing interest in the possibility of a dynamical universe instead of Λ CDMmakes the exploration of the nature of such universes particularly relevant. \nKeywords: cosmological constraint; dark energy; theoretical model", '1. Introduction': 'Flat, minimally coupled to gravity quintessence was one of the first and is probably one of the simplest rolling scalar field cosmologies. Even so quintessence has a complex range of properties that depend on the mathematical form of the dark energy potential and its arguments, as well as the boundary conditions on the cosmological parameters such as the current dark energy equation of state w 0 . The dark energy potential in this study is a quartic polynomial with the same mathematical form as the Higgs potential that is hence referred to as the Higgs Inspired, HI potential. In a Ratra Peebles format [1,2] the HI potential is \nV ( κθ ) = M 4 (( κθ ) 2 -( κδ ) 2 ) 2 = M 4 (( κθ ) 4 -2 ( κθ ) 2 ( κδ ) 2 +( κδ ) 4 ) (1) \nwhere M is a constant with units of mass in terms of the reduced Planck mass Mp , and κ is the inverse reduced mass 1 Mp The scalar field is θ and δ is a constant, both with units of mass, which makes κθ and κδ dimensionless. All of the HI potential dimensionality is in the M 4 term. Previous work [3] on this combination of cosmology and potential determined that the scalar field θ ( a ) has a simple solution in terms of the scale factor a and the Lambert Wfunction [4]. \nκθ ( a ) = κδ √ -W ( χ ( a )) . (2) \nThe argument χ ( a ) of the W function is a power law function of the scale factor a . \nχ ( a ) = qa p . (3) \nFigure 1. The upper panel shows portions of the principal, solid line, and minus, dashed line, branches of the Lambert W function W ( χ ) . The big bang lies at the 0,0 origin. Evolution is from right to left as indicated by the arrow in the figure. The dotted vertical line in the pre bang region marks the pre bang W ( χ ( a )) = 1 point. The slightly thicker portion of the χ = 0 axis indicates the width of the post bang universe. The lower panel shows the post bang universe. The extent of the solid line shows the present universe for w 0 = -0.999 and the extent of the dashed line for w 0 = -0.99. The terminator of each line is at a scale factor of one. \n<!-- image --> \nBoth q and p are constants determined by the boundary conditions and the HI dark energy potential. The expression ( κθ ) 2 -( κδ ) 2 appears in the HI potential and several other places in the evolutionary functions. In terms of the W function it is. \n( κθ ) 2 -( κδ ) 2 = -( κδ ) 2 ( W ( χ ( a )) + 1 ) . (4) \nUsing Eqn. 4 the HI potential in the W function format is \nV ( a ) = ( M κδ ) 4 ( W ( χ ( a )) + 1 ) 2 = ( M κδ ) 4 ( W ( χ ( a )) 2 + 2 W ( χ ( a )) + 1 ) (5) \nThe combination of the Lambert W function scalar, HI potential and the Quintessence cosmology is hereinafter referred to as the LHQ cosmology and universe. \nFigure 1 shows small portions of the W function principal and minus branch on both sides of the big bang. Although the Lambert W function minus branch is not considered in this work it is included to show that the maximum negative value of χ ( a ) lies at the end of the principal branch and beginning of the minus branch. At the terminal point of the principal branch χ = -1 e and W ( χ ) = -1. \nFigure 1 displays many of the unique aspects of the LHQ cosmology and its evolution. In keeping with its definition as the ratio of the size of the universe relative to the current size set as one, the scale factor a is positive on either side of the big bang. As shown later in section 3.2.1 q is negative in the post bang region and positive in the pre bang region, making χ negative in the post bang and positive in the pre bang regions. Figure 1 \nshows that The Lambert W function has the same sign as χ . This, in turn, indicates that the scalar field is real and positive in the post bang region and imaginary in the pre bang region. All of the pre bang cosmological parameters are, however, real. This divides the LHQ cosmology universe into a canonical post bang era containing the canonical universe and the noncanonical pre bang era. Although only plotted out to a pre bang χ of 5 the W function principal branch extends to infinity and the equation for the scalar is mathematically valid at all points on principal branch. \nArecent publication by the DESI collaboration [5] has claimed evidence for a preference of a dynamical cosmology over Λ CDM, drawing more interest to dynamical universe models. Several publications, both positive and negative, have appeared relative to the DESI results, eg.[6-12], employing parametric analysis, except for [12]. This highlights the need for accurate, analytic, solutions based on physics, rather than parameterizations, for the evolution of dark energy cosmological parameters. \nThe investigation centers on two regions of the principal branch. The first is the far past in the pre bang epoch that extends to the infinite past. The study adopts a more practical starting point at a pre bang scale factor of 1000. The region between a pre bang scale factor of one and 1000 is the far past region. The region between a post bang scale factor of one and a pre bang scale factor of one, straddling the big bang is the second region, designated the transition zone. In comparison to the far past the transition zone is miniscule, however, it includes all of the known post bang universe, the big bang, and the pre bang approach to the big bang. \nThe existence of analytic simple equations for the dark energy scalar and potential provides the primary motivation for this investigation. These equations along with the quintessence cosmology, reviewed in Sect. 2 and the Friedmann constraints provide the tools to calculate mathematically deterministic solutions for cosmological parameters as functions of the scale factor at all points on the W function principal branch. The evolution of the dark energy parameters is determined by the principal branch track of the Lambert Wfunction illustrated in Fig. 1. The matter and radiation average densities are strictly functions of their boundary condition densities and the scale factor. \nThe remainder of the manuscript is arranged as follows. After this introduction the basic quintessence equations are reviewed in Sect. 2. The mathematical properties of the cosmology are then established in terms of the Lambert W function in Sect. 3. These properties are the equations for the evolution of cosmological parameters as analytic functions of the scale factor. The boundary conditions are set in Sect. 4. The physical properties for the far past and transition zone are established in Sec.5. These contain both mathematical and graphical representations of the evolution of cosmological parameters. A set of conclusions are presented in Sect. 6. The study uses natural units with c , ¯ h and 8 π G set to one. The unit of mass is the reduced Planck mass Mp . In these units κ = 1 but is retained in equations to show the proper units.', '2. Flat Minimally Coupled Quintessence': 'Even though they are well known, some of the more useful quintessence equations are presented here for easy reference. The equations are expressed in the familiar form with ϕ as the true scalar. The true scalar is ϕ = M κθ for the Ratra Peebles scalar κθ used in this work. Equations 2 and 1 plus the equations in this section, the Friedmann constraints in Sec. 3.1, along with the boundary conditions in Sec 4 form a complete set of tools to produce deterministic solutions for the evolution of the primary cosmological parameters. \nThe action is \nS = ∫ d 4 x √ -g [ R 2 -1 2 g µν ∂µ∂νϕ -V ( ϕ )] + Sm + Sr (6) \nwhere R is the Ricci scalar, g is the determinant of the metric g µν and V ( ϕ ) is the dark energy potential. Sm and Sr are the actions of the matter and radiation fluids. By definition the big bang has not occurred in the pre bang epoch so the radiation term is zero in the \npre bang action. The dark energy equation of state is P ϕ ρϕ with a dark energy density and pressure of \nρϕ ≡ ˙ ϕ 2 2 + V ( ϕ ) , P ϕ ≡ ˙ ϕ 2 2 -V ( ϕ ) . (7) \nThe kinetic term ˙ ϕ 2 2 is often referred to as X . It follows that \nP ϕ + ρϕ = ˙ ϕ 2 , P ϕ + ρϕ ρϕ = ( w + 1 ) (8) \nwhich provides the useful relation \n˙ ϕ 2 = ρϕ ( w + 1 ) . (9) \nThe average matter and radiation densities are simply \nρ m = ρ m 0 a 3 , ρ r = ρ r 0 a 4 (10) \nwhere ρ m 0 and ρ r 0 are the matter and radiation densities at a scale factor of one. Only these average densities are considered in this study.', '3. The Mathematical Properties of The LHQ Universe': 'This section develops the analytic functions for the evolution of cosmological parameters in both the pre and post bang epochs. Some duplication of derivations in [3] occurs but are included in order to have most derivations in this manuscript rather than having to refer back to the previous publication.', '3.1. Friedmann constraints': 'The first and second Friedmann constraints, given below, are important tools in deriving the cosmological parameter evolution equations. \n3 H 2 = ρ de + ρ m + ρ r . (11) \n3 ( ˙ H + H 2 ) = -ρ de + ρ m + ρ r + 3 P 2 . (12) \nThe radiation density term ρ r is absent in the pre bang constraints. The cosmological parameter evolutionary equations are expressed in terms of the Lambert W function when possible.', '3.2. Mathematical properties, equations and functions': 'The equations and functions needed to describe the physical properties of the LHQ universe are developed in the following sections.', '3.2.1. The dark energy scalar constants and current values': 'From [3] the constants p and q in χ are \np = 8 ( κδ ) 2 q = -e c ( κδ ) 2 ( κδ ) 2 c = 2 ( κδ ) 2 ln ( κθ 0 ) -( κθ 0 ) 2 . (13) \nand the present day scalar κθ 0 is \nκθ 0 = -4 -√ 16 + 12 Ω θ 0 ( w 0 + 1 )( κδ ) 2 2 √ 3 Ω θ 0 ( w 0 + 1 ) (14) \nFigure 2. The value of q is plotted as a function of w 0 for pre and post bang w 0 values between -0.99 and -1.01. The sign of q transitions from negative for w 0 less negative than minus one to positive for w 0 values more negative than minus one. \n<!-- image --> \nwhere Ω θ 0 is the ratio of the present dark energy density to the critical density. \nFigure 2 shows the solution for q in Eqn. 13 in terms of w 0 using Eqns. 13 and 14. The post bang values of q are negative which produce a negative value of χ and a real scalar. The pre bang values are positive making the scalar imaginary. Although both the pre bang scalar and its time derivative are imaginary all of the observable cosmological parameters are real. Positive values of the W function require being on the positive portion of the principal branch. Figure 2 shows this is achieved with w 0 values more negative than minus one. This requirement is addressed in the boundary conditions for the pre bang values of w 0 in Sect. 4.', '3.2.2. The time derivative of the scalar': 'The time derivative of the scalar is \n˙ θ = d θ da da dt = d θ da Ha (15) \nwhich requires an equation for d θ da . \nThe derivative of the W function W ( χ ) with respect to its argument χ is \ndW ( χ ) d χ = W ( χ ) χ ( W ( χ ) + 1 ) . (16) \nThe derivative of the scalar with respect to χ is then \nd θ d χ = δ 2 ( -W ( χ )) -1 2 ( -dW ( χ ) d χ ) (17) \nwhich results in \nd θ d χ = δ 2 √ -W ( χ ) χ ( W ( χ ) + 1 ) . (18) \nThe derivative with respect to the scale factor, using Eqn. 3, is \nd θ da = d θ d χ d χ da = p δ √ -W ( χ ) 2 a ( W ( χ ) + 1 ) (19) \nwhere p is the power of the scale factor in Eqn. 3. The time derivative of the scalar is then \n˙ θ = p δ √ -W ( χ ) 2 a ( W ( χ ) + 1 ) Ha = p δ √ -W ( χ ) 2 ( W ( χ ) + 1 ) H = 4 √ -W ( χ ) ( κδ )( W ( χ ) + 1 ) H (20) \nusing Eqn. 13 for the constant p . It shows that both the scalar and its time derivative are zero at the singularity where the W function is zero.', '3.2.3. The HI potential constants δ and M': 'The kinetic term is zero at the singularity making the dark energy EoS -V V = -1. This requires a thawing post bang w that starts at minus one and thaws to less negative than minus one at larger scale factors. The previous study showed values of δ near one put the zero point in the future and produced thawing evolutions. The constant δ is therefore set at 0.95 to satisfy the requirements. \nM is a constant therefore its value can be calculated at any point on the principal branch which is taken to be a post bang scale factor of one. The time derivative of the scalar at a = 1 is \n˙ θ 0 = p ( κδ ) √ -W ( χ ( 1 )) 2 ( W ( χ ( 1 )) + 1 ) H 0 = 4 √ -W ( χ ( 1 )) ( κδ )( W ( χ ( 1 )) + 1 ) H 0 . (21) \nThe first Friedmann constraint at the present time is. \n3 H 2 0 = -8 W ( χ ( 1 )) ( κδ ) 2 ( W ( χ ( 1 )) + 1 ) 2 H 2 0 +( M κδ ) 4 ( W ( χ ( 1 )) + 1 ) 2 + ρ m 0 + ρ r 0 . (22) \nRearranging Eqn. 22 gives \n3 H 2 0 ( 1 + 8 W ( χ ( 1 )) 3 ( κδ ) 2 ( W ( χ ( 1 )) + 1 ) 2 ) = ( M κδ ) 4 ( W ( χ ( 1 )) + 1 ) 2 + ρ m 0 + ρ r 0 . (23) \nSolving Eqn. 23 for M 4 by replacing ˙ θ 0 with Eqn. 21 and ( κθ ) 2 -( κδ ) 2 with Eqn. 4 yields \nM 4 = 3 H 2 0 ( 1 + 8 W ( χ ( 1 )) 3 ( κδ ) 2 ( W ( χ ( 1 ))+ 1 ) 2 ) -ρ m 0 -ρ r 0 ( κδ ) 4 ( W ( χ ( 1 )) + 1 ) 2 . (24) \nSince κθ 0 is a function of w 0 there is a different M value for each of the two post bang values of w 0 . The calculated values of M are listed in Table 1 along with the choice of δ .', '3.2.4. The Hubble parameter': 'Rearranging Eqn. 23 and replacing H 0 with H and ( 1 ) with ( a ) provides a solution for the post bang Hubble parameter. \nH ( a ) = √ √ √ √ √ ( M κδ ) 4 ( W ( χ ( a )) + 1 ) 2 + ρ m 0 a 3 + ρ r 0 a 4 3 + 8 W ( χ ( a )) ( κδ ) 2 ( W ( χ ( a ))+ 1 ) 2 (25) \nThe pre bang Hubble parameter is the same as the post bang except for the absence of the radiation density and a minus sign indicating contraction.', '3.2.5. The time derivative of the Hubble parameter': 'From [3] the time derivative of the Hubble parameter, needed for the second Friedmann constraint, is \n˙ H ( a ) = -1 2 (( M κ ) 4 ˙ θ 2 + ρ m 0 a 3 + ρ r 0 a 4 ) (26) \nwhere the radiation density term was added to the previous study result for the post bang epoch. Equation 20 provides the equation for ˙ θ in terms of the W function.', '3.2.6. The kinetic term X': 'Equation 25 for the Hubble parameter provides the complete equation for ˙ θ in terms of the W function. \nκ 2 ˙ θ = 4 √ -W ( χ ) κδ ( W ( χ ) + 1 ) √ √ √ √ √ ( M κδ ) 4 ( W ( χ ( a )) + 1 ) 2 + ρ m 0 a 3 + ρ r 0 a 4 3 + 8 W ( χ ( a )) ( κδ ) 2 ( W ( χ ( a ))+ 1 ) 2 . (27) \nMultiplying through by 1 κδ ( W ( χ ( a ))+ 1 ) gives \nκ 2 ˙ θ = 4 √ -W ( χ ) √ ( M κδ ) 4 ( W ( χ ( a )) + 1 ) 2 + ρ m 0 a 3 + ρ r 0 a 4 3 ( κδ ) 2 ( W ( χ ( a )) + 1 ) 2 + 8 W ( χ ( a )) . (28) \nUsing Eqn. 28 the post bang kinetic term is \nX ( a ) = -8 W ( χ ( a )) ( ( M κδ ) 4 ( W ( χ ( a )) + 1 ) 2 + ρ m 0 a 3 + ρ r 0 a 4 3 ( κδ ) 2 ( W ( χ ( a )) + 1 ) 2 + 8 W ( χ ( a )) ) . (29) \nThe pre bang kinetic term is identical except for the absence of the radiation term. The value of ˙ θ and the kinetic term are real and positive in the post bang epoch. In the pre bang era ˙ θ is imaginary making the kinetic term negative and real.', '3.2.7. The dark energy density and pressure': 'The quintessence dark energy density and pressure evolutionary functions are given in Eqns. 7. Using equation Eqn. 29 the post bang dark energy density is \nρ de ( a ) = -8 W ( χ ) ( ( M κδ ) 4 ( W ( χ ) + 1 ) 2 + ρ m 0 a 3 + ρ r 0 a 4 3 ( κδ ) 2 ( W ( χ ) + 1 ) 2 + 8 W ( χ ) ) +( M κδ ) 4 ( W ( χ ( a )) + 1 ) 2 . (30) \nThe post bang dark energy pressure is \nP de ( a ) = -8 W ( χ ) ( ( M κδ ) 4 ( W ( χ ) + 1 ) 2 + ρ m 0 a 3 + ρ r 0 a 4 3 ( κδ ) 2 ( W ( χ ) + 1 ) 2 + 8 W ( χ ) ) -( M κδ ) 4 ( W ( χ ) + 1 ) 2 (31) \nThe pre bang functions are identical except for the absence of the radiative term. \n3.2.8. The dark energy equation of state \nFrom the definition of the dark energy EoS w is the ratio of the dark energy pressure and density. The dark energy EoS is \nw ( a ) = X ( a ) -( M κδ ) 4 ( W ( χ ( a )) + 1 ) 2 X ( a ) + ( M κδ ) 4 ( W ( χ ( a )) + 1 ) 2 (32) \nwhere X ( a ) are the pre and post bang kinetic terms.', '4. Boundary Conditions': 'Table 1 shows the boundary conditions for the pre and post bang epochs. Most are post bang values where observational evidence exists. Post bang H 0 = 73 (km/sec)/Mpc. The pre bang scale factor one Hubble parameter values are near, but not exactly 73. They do not appear in Table 1 because they are not boundary conditions. The two post bang w 0 values are -0.99 and -0.999, purposely set close to Λ CDM. Pre bang are -1.01 and -1.001 symmetric to post bang. Ω θ 0 is 0.7 at a = 1 in both epochs. The matter and radiation ratios are 0.2999 and 0.0001 post bang respectively with the pre bang matter at 0.3. \nTable 1. The post and pre bang boundary conditions \nThe dimensionfull units are: H 0 , ( km / sec ) / Mpc , ρ θ 0 , ρ M 0 and ρ M 0 , M 4 p , M and δ , Mp . The other boundary conditions are dimensionless.', '5. The Physical Properties of the LHQ Universe': 'This section examines the evolution of cosmological parameters in the far past region and the transition zone. The physical properties of the pre bang universe calculated by the mathematically deterministic functions from Sec.-3.2 are presented in Sect. 5.1 followed by the transition zone physical properties in Sect. 5.2.', '5.1. The physical properties of the pre bang epoch': 'Although the equations and mathematical properties of the pre bang epoch are identical to the post bang the physical properties are significantly different. Some differences are due to the purely imaginary scalar and its derivatives. A major difference is a contracting universe as the scalar rolls down the HI potential with a negative Hubble parameter. At the beginning of the far past, a pre bang scale factor of 1000, the universe is cold and starless. It remains so for the entire pre bang region that terminates at pre bang scale factor of one. Non-dark matter is most likely elementary particles, quarks, leptons, gauge bosons and the Higgs boson. The density of matter is 10 -9 of the current post bang density matter density. Dark energy dominates with a density more than 2000 times greater than the current post bang density. In the equation for the dark energy density the HI potential dominates the much smaller kinetic term. This condition continues up to the present time. The far past start scale factor of 1000 is deep into the region where both the W function and the scalar are significantly greater than one. This means that the several ( W + 1 ) terms are equal to Wand the first term of the HI potential is dominant. In the following the abscissas of the plots are in Log 10 of the scale factor to properly display the evolutions at all scale factors between one and 1000. The ordinates are linear unless stated otherwise such as in the top panel of Fig. 3.', '5.1.1. Chi, the W function and the scalar': 'Figure 3 shows the evolution of χ ( a ) , the Lambert W function, and the scalar κθ from a scale factor of 1000 to 1. The top χ ( a ) panel is a log log plot but the bottom two panels are the standard linear log plots. Time evolution is from right to left with the scale factor contracting. The Lambert W function is positive and real while the scalar is positive and imaginary. The nature of the early time W function and scalar evolution persists until it nears the transition zone where it flattens out. The evolution of χ , W ( χ ) and κθ are very similar for the two pre bang w 0 boundary conditions of -1.01 and -1.001. All three of the functions are monotonically decreasing as the scalar rolls down the HI potential. The imaginary scalar makes the square of the scalar negative instead of the post bang positive values. The flattening of the evolution of the W function and the scalar near the \ntransition zone is an indicator that the transition zone evolution is atypical of the general LHQ evolution.', '5.1.2. The HI potential': 'Figure 4 shows the pre bang evolution of the HI potential in ( M P ) 4 . The imaginary scalar renders all terms of HI potential positive. The upper panel shows the evolution in the standard log linear format along with the evolution of the kinetic term X demonstrating the dominance of the HI potential in the pre bang region. The evolution is a smooth monotonic decline. The w 0 = -1.01 case has a slightly higher value than the w 0 = -1.001 case. The lower panel shows that the W 0 = -1.01 case crosses below the w 0 = -1.001 case as it enters the transition zone. Comparison of the two panels indicates that the HI potential decreased by roughly a factor of 2300 from a pre bang scale factor of 1000 to one.', '5.1.3. The dark energy density and pressure': 'Figure 5 shows the pre bang evolution of the dark energy density and pressure. As expected, the range of evolution is similar to that of the HI potential since both the density and pressure are potential dominated in the pre bang epoch. The magnitude of the pressure exceeds that of the density since both the kinetic and potential terms in the pressure are negative while in the density the potential term is positive but the kinetic term is negative. Both the dark energy density and pressure have almost flat evolutions as they approach the transition zone. The dark energy density and pressure drop by approximately a factor of 10 4 between a scale factor of 1000 and one.', '5.1.4. The Hubble parameter': 'Figure 6 displays the evolution of the Hubble parameter which is negative in all parts of the far past. The Hubble parameter is dark energy dominated and becomes less negative with time following the decrease in dark energy density shown in the upper panel of Fig. 5. The evolution is linear in the log linear plot from a scale factor of 1000 to a scale factor of 3. The Hubble parameter evolution is essentially the negative of the W function evolution in Fig. 3. This is due to the dominance of W ( χ ) over one in the ( W ( χ ) + 1 ) terms in Eqn. 25. The pre bang Hubble parameter evolution flattens at a scale factor of 3 as it approaches the transition zone.', '5.1.5. The time derivative of the scalar and the kinetic term': 'Figure 7 shows the kinetic term and the time derivative of the pre bang scalar. The pre bang time derivative of the scalar is negative and purely imaginary since the scalar is decreasing and purely imaginary. It has a similar evolution as previous parameters with a steep decrease in magnitude and a flattening of evolution as it approaches the transition zone. The kinetic term is negative but real since it is proportional to the square of an imaginary number.', '5.1.6. The dark energy equation of state': 'The upper panel of Fig. 8 shows the pre bang evolution of the dark energy EoS w . The most prominent feature is the negative dip at scale factors between two and three reaching w values near -2.5. The lower panel in Fig. 8 shows how the dip occurs. At scale factors near the transition zone the kinetic term plays a larger, but not dominant, role as the HI potential declines. The ratio of the pressure to the density reaches a maximum at a scale factor of three producing the dip. At smaller scale factors the kinetic term is approaching zero making w ≈ -V / V = -1 as it enters the transition zone. The overall evolution of w is from a value almost minus one at a scale factor of 1000, becoming more negative as it evolves to a minimum near -2.5 near a scale factor of 3 then returning toward minus one as it approaches the transition zone. \nFigure 3. The w 0 = -1.01 cases are plotted with dashed lines and the w 0 = -1.001 cases with solid lines. These line styles are used in the remainder of the pre bang plots unless stated otherwise. The top panel shows the evolution of χ ( a ) in a log, log plot. The other panels are log linear plots. Time evolution is from right to left. \n<!-- image --> \nFigure 4. The figures show the distant past evolution of the HI dark energy potential in the pre bang epoch. The upper panel shows the evolution of the HI potential in the standard log linear plot. The smaller kinetic term is at the bottom of the panel. The lower panel shows the detail of the cross over of the potentials near the pre bang scale factor of one. \n<!-- image --> \nFigure 5. The figures show the evolution of the HI dark energy density and pressure in the pre bang epoch. The upper panel shows the evolution of the dark energy density and the lower panel shows the pressure. \n<!-- image --> \nFigure 6. The distant past evolution of the Hubble parameter in the pre bang epoch. \n<!-- image --> \nFigure 7. The top panel shows the evolution of the time derivative of the scalar ˙ θ . The time derivative is purely imaginary. The bottom panel show the kinetic term X = ˙ θ 2 2 \n<!-- image --> \nFigure 8. The upper panel shows the distant past evolution of the dark energy equation of state w . The evolution of the dark energy density and pressure creating the w dip is shown in the lower panel. \n<!-- image -->', '5.1.7. Friedmann constraints in the pre bang era': 'This section examines the compliance of the derived parameters to the two Friedmann constraints, Eqns. 11 and 12 in the pre bang zone. The pre and post bang equations and deviations are identical except the pre bang lacks the radiation density in the first constraint. The equations for fractional deviations from the first, F1, and second, F2, Friedman constraints are \ndev = 3 H 2 -( ρ de + ρ m + ρ r ) 3 H 2 and 3 ( ˙ H + H 2 ) + ( ρ de + ρ m + ρ r + 3 P 2 ) 3 ( ˙ H + H 2 ) . (33) \nFigure 9 shows the constraint deviations for the first and second Friedmann constraints with w 0 = -1.001. Their deviations are all less than one part in 10 15 . The nature of the fractional errors suggest that they are the digital limitations of the Mathematica calculations which are on the order of a few times 10 -16 . The plots of the w 0 = -1.01 case are similar to the -1.001 case satisfying the Friedmann constraints at a high level.', '5.1.8. Pre bang chronology': 'This section examines the predicted pre bang chronology. The time derivative of the scale factor is simply H ( a ) a for the Hubble parameter H ( a ) and the scale factor a . The time as a function of a is \ndt = da H ( a ) a (34) \nwhere H ( a ) is the pre bang Hubble parameter given by the negative of Eqn. 3.2.4 without the radiation term. The time in gigayears from a scalar factor of 1000 to a scale factor of a is calculated by numerically integrating Eqn. 34. Figure 10 shows the time derivative of the scale factor as a function of the scale factor. The time since a scale factor of 1000 is shown at \nFigure 9. The panels show the error level of the first and second Friedmann constraints for w 0 = -1.001 case . The Friedmann constraints are satisfied to better than one part in 10 15 at all points in the pre bang zone. \n<!-- image --> \nthe bottom for scale factors of 1000, 100, 10 and 1. The time derivative of the scale factor is of course negative in the pre bang epoch and is quite large at the pre bang scale factor of 1000. The contraction from a scale factor of 1000 to 100 takes only 1.1 gigayears. The contraction to a pre bang scale factor of 1 takes 32.6 gigayears, less than three times the age of the post bang universe. The time to contract from a pre bang scale factor of one to the big bang is 14.9 gigayears, slightly longer than the age of the post bang universe. These numbers illustrate the significant difference between the rapid contraction in the very early pre bang universe relative to the contraction and expansion rates in the transition zone. It further emphasizes that the transition zone is a remarkable time in the evolution of the LHQ universe. \nFigure 10. The time derivative of the scale factor is plotted for pre bang scale factors between 1000 and 1. The time in gigayears for the scale factors of 1000, 100, 10, and 1 is given at the bottom of the figure. \n<!-- image -->', '5.1.9. Still a Quintessence cosmology': "The pre bang epoch has two properties associated with a phantom cosmology: a dark energy EoS more negative than minus one and a negative kinetic term. It is, however, still a quintessence cosmology rather than a phantom cosmology with well known instability issues [13-18]. In a phantom cosmology the kinetic term is negative because it has been changed to negative in Eqns. 7 to give the 'wrong sign' in the definitions of the dark energy density and pressure [19]. In this study the equations retain their quintessence form. The negative sign for the kinetic term is due to the imaginary value of ˙ θ . Since the equations retain their quintessence form the sound speed, c 2 s = P , X ρ , X , is one in both the pre and post bang epochs. From [20] this indicates that the LHQ universe is stable in both epochs.", '5.2. The physical properties of the transition zone': 'The transition zone contains the end of the pre bang epoch, the big bang and the entirety of the post bang observable universe between post bang scale factors from zero to one. It is a region of relatively little dark energy evolution but of a highly evolving matter density. The big bang is a natural consequence of the contraction of the pre bang universe to a singularity. The big bang is, therefore, no longer the origin of time and the universe. This is counter to the canonical view of the big bang being the origin of both time and the universe, eg. [21-25] and many others that invoke a universe that is the result of a quantum entanglement collapsing to a given quantum state. In LHQ the universe is in a given state far before the big bang. In this work our chosen scale factor of 1000 for the beginning of the evolutionary calculations is presumed to be after the state has been established. A quantum analysis of the establishment of the LHQ state is beyond the scope of this investigation. Two possibilities, however, are worth consideration. One is that the universe has always been in the LHQ state and does not involve a quantum entanglement. The other possibility invokes a multiverse option that there is a quantum entanglement that exists in the distant past before the big bang. New universes form from the collapse of part of these entanglements at regular intervals and we are in the one that collapsed to the LHQ state. Neither of these possibilities have been examined in this work beyond pure speculation. \nThe big bang is, however, an essential event required to produce favorable conditions for the formation of stars and galaxies that is not possible in the pre bang epoch. Contraction to a singularity and transition to a post bang universe is not a new concept [26-30]. Several studies of cyclic universes [31-34] proposed universes that cycled through numerous big bangs, however, with some entropy issues due to the multiple occurrences. The LHQ universe is not cyclic. It contains only one big bang.', '5.2.1. Expansion normalized variable examination of the transition zone': 'Asuccessful transition from the pre bang to post bang epoch requires the universe to evolve through the big bang rather than bouncing back into the pre bang epoch such as described in [35]. Expansion Normalized, EN, variables [36] are employed to determine whether a transition is possible. The calculation is made near either side of the singularity. The quintessence kinetic x and potential y variables are defined in [36] as \nx = κ ˙ θ √ 6 H y = κ √ V √ 3 H . (35) \nIn the pre bang epoch x is imaginary since ˙ θ is imaginary but x is real in the post bang epoch. The value of y is real in both epochs. \nFigure 11 shows the evolution of the EN variables in the transition zone. It indicates that a transition through the singularity from the pre to post bang epoch is possible. The figure also shows the dominance of the y potential variable over the kinetic x variable. Even though the dark energy potential V does not go to zero the y value is zero at the singularity due to the infinite Hubble parameter there. The kinetic evolution is near zero on both sides \nFigure 11. The evolution of the EN variables x and y at the pre and post bang boundary. The arrows indicated the direction of the evolution. The numbers at the ends of the track indicate the appropriate w 0 values for the track. \n<!-- image --> \nof the singularity for a significant portion of the plot, particularly for the w 0 value closest to minus one.', '5.2.2. χ ( a ) , the Lambert W function and the scalar': 'The flatness of the kinetic term evolution in the EN plot is an indicator of a region of suppressed evolution of dark energy on either side of the singularity. It is due to the slow evolution of χ ( a ) for scale factors less than one. Equation 13 for χ ( qa p ) shows that the power of the scale factor is larger than 8 which dampens the evolution for scale factors less than one. \nThe dampening of the evolution of χ ( a ) is apparent in the upper panel of Fig. 12. The middle panel of Fig. 12 shows the Lambert W function evolution and the bottom panel the evolution of the scalar κθ . The scalar has a flat evolution between scale factors of 0.5 on either side of the singularity. The w 0 = -0.99 case has significantly more evolution than the w 0 = -0.999 case showing the effect of w 0 on evolution. All three of the parameters in Fig. 12 are zero at the singularity. The change of the scalar from imaginary to real is an important aspect of the transition. The scalar is much smaller than one throughout the transition zone, particularly for the w 0 case closest to minus one. This is a significant factor in the evolution of the HI potential discussed in the next section.', '5.2.3. The HI dark energy potential': 'At the singularity the HI dark energy potential is equal to the constant term ( M κδ ) 4 since the scalar κθ = 0. Figure 13 shows that the evolution of the HI potential is essentially flat in the transition zone. The w 0 = -0.999 case deviation is only 0.1% at a scale factor of one. The slow evolution occurs because the scalar remains much less than one in the transition zone as shown in the lower panel of Fig. 12. This makes the first two terms of the HI potential in Eqn. 1 much smaller than the constant third term. Figure 13 shows that the HI potential continues to decrease as it exits the transition zone. The decrease is due to the second term of HI potential changing from positive in the pre bang epoch to negative in the post bang era as the scalar changes from imaginary to real. The flatness of the dark energy potential evolution in the transition zone is a combination of the scalar being almost \nFigure 12. The upper panel shows the evolution of χ ( a ) through the transition zone. As was the case in the pre bang plots the w 0 = -0.999 case is the solid line and the w 0 = -0.99 case is the dashed line. The evolution is from the pre to post bang epoch, right to left. The middle panel shows the evolution of the W function and the bottom panel the evolution of the scalar. \n<!-- image --> \nFigure 13. The figure shows the evolution of the HI potential, Eqn. 1, in the transition zone for the two different w 0 cases. Compared to their absolute values there is relatively insignificant evolution of the HI potential in the transition zone. \n<!-- image --> \nFigure 14. The upper panel shows the transition zone evolution of the Hubble parameter from scale factors between 0.01 and 1.0. in units of M 2 P on either side of the singularity. The lower panel is for scale factors between 0.4 and 1.0. The tracks for the two w 0 cases are the same to the width of the plot line but not identical. \n<!-- image --> \nFigure 15. The figure shows the fractional deviation of the two LHQ Hubble parameter evolutions from the Λ CDMevolution for the post bang era. It is plotted with time evolution from right to left to be consistent with the previous plots. \n<!-- image --> \nzero near the singularity due to the W function, the presence of the constant term in the HI potential, and the dampening described in Sect. 5.2.2. As discussed in Sect. 5.2.6 this leads to an evolution of cosmological parameters in the transition zone very close to Λ CDMwith no fine tuning.', '5.2.4. The Hubble parameter': 'The negative pre bang and positive post bang Hubble parameters create the expected discontinuity marking the transition from a contracting to an expanding universe at the singularity. Figure 14 shows the transition zone Hubble parameter evolution near the singularity. The upper panel tracks terminate at a scale factors of 0.01 to show the discontinuity. The lower panel tracks terminate at scale factors of 0.4 to better show the evolution at larger scale factors The tracks for the two different w 0 values overlap at the resolution of the plot. This demonstrates the insensitivity of the Hubble parameter to w 0 , consistent with previous studies [37,38]. \nFigure 15 shows the fractional deviation of the post bang Hubble parameter from Λ CDM, defined as ( H LHQ -H Λ ) H Λ , for the two w 0 values. The maximum fractional deviation \nFigure 16. The top panel shows the transition zone evolution of the time derivative of the scalar ˙ θ . It is negative and imaginary in the pre bang epoch and transitions to positive and real in the post bang epoch. The bottom panel shows the evolution of the kinetic term X which is negative in the pre bang epoch and positive in the post bang epoch. \n<!-- image --> \nis 0.006 for the w 0 = -0.99 case and only 0.00005 for the w 0 = -0.999 fiducial case. The Hubble parameter is not a sensitive discriminator between static and dynamical cosmologies for w 0 values close to minus one. This makes it hard to verify Λ CDMor falsify LHQ based solely on observations of the Hubble parameter.', '5.2.5. The time derivative of the scalar and the kinetic term.': 'The upper panel of Fig. 16 shows the evolution of the time derivative of the scalar ˙ θ in units of M 2 P and the lower panel shows the evolution of the kinetic term X = ˙ θ 2 2 in units of M 4 P . Like the scalar, ˙ θ is zero at the transition and stays near zero for scale factors of ≈ 0.5 on either side of the transition, particularly for the w 0 = -0.999 case. ˙ θ is negative and imaginary in the pre bang epoch. It becomes real and positive at the big bang transition point. \nThe kinetic term X evolution is quite flat in the transition zone. This is consistent with the dominance of the potential in the EN variables in Fig. 11. Its value is real and negative in the pre bang epoch since it is the square of an imaginary number.', '5.2.6. The dark energy density and pressure': 'Figure 17 shows the evolution of the dark energy density and pressure. The dark energy densities for both w 0 values are identical at a post bang scale factor of one since the density is a boundary condition. Both the dark energy density and dark energy pressure have smooth transitions from the pre bang to post bang era. Their evolutions are almost flat through the transition zone but are not zero due to the constant in the HI potential. The slight increase in the dark energy density in the post bang era is primarily due to an increase in the kinetic term X as shown in fig. 16. \nFigure 17. The top panel shows the evolution of the dark energy density in the transition zone. The bottom panel shows the evolution of the dark energy pressure. The post bang dark energy scale factor one densities are identical because it is a boundary condition. \n<!-- image --> \nFigure 18. The evolution of the dark energy equation of state w in the transition zone. \n<!-- image --> \nThe almost constant dark energy density in the transition zone, particularly for the w 0 = -0.999 case, acts like a cosmological constant. It means that LHQ has essentially the same success in matching the observations as Λ CDMin the post bang observable universe. Alarge part of this success is due to the w 0 values very close to minus one. Much larger deviations from minus one would not be as successful as shown by the higher deviation of the w 0 = -0.99 case. On the other hand if observational constraints become much more rigorous favoring, Λ CDM,the LHQ cosmology can match them by decreasing the deviation of w 0 from minus one. Even though the match to the observations may be the same the scalar field source of dark energy in LHQ is profoundly different than the cosmological constant source in Λ CDM. The very different properties of the two universes outside of the transition zone are also profound.', '5.2.7. The dark energy equation of state': 'Figure 18 shows the evolution of w in the transition zone. It shows that w is or is very close to minus one between the pre and post bang scale factors of 0.5 on either side of the singularity. This corresponds to the flat evolution of the dark energy density and pressure between the same scale factors in Fig. 17. In the post bang era w demonstrates a thawing evolution. The w 0 = -0.999 case w does not deviate from minus one until very recent times and even then it would be difficult to reliably distinguish it from minus one. Confirmation \nFigure 19. The panels show the error level of the first Friedmann constraint for w 0 = -0.99 case in the top panel and the second Friedmann constraint in the bottom panel. \n<!-- image --> \nof such a deviation of w from minus one would negate Λ CDMand be significant evidence for a dynamical cosmology such as LHQ. The LHQ predictions for the evolutions of the observables such as the Hubble Parameter and the dark energy EoS meet all of the current observational constraints.', '5.2.8. Friedmann constraints in the transition zone': 'Figure 19 shows the fractional deviation for the w 0 = -0.99 case of the first Friedmann constraint in the upper panel and the second Friedmann constraint in the lower panel. All of the fractional deviations are less than 10 -15 for the first Friedmann constraint indicating that it is satisfied. The spikes in the second Friedmann constraint plots are caused by the denominator in Eqn. 33 becoming zero when ˙ H and H 2 are equal but with opposite signs. The plots are discrete points therefore the height of the spikes depends on how close the discrete plot point is to the zero point. The plot away from the spikes is similar to the first Friedmann constraint plot. Even with the spikes the second Friedmann constraint is also well satisfied. The plots for the w 0 = -0.999 case are similar.', '6. Conclusions': 'LHQ is a self consistent, analytic set of deterministic cosmological parameter evolutions. The evolutions are mathematically valid at all points on the Lambert W function principal branch. The principal branch spans from the infinite past to a terminal point in the future. LHQ has two distinct epochs, the pre big bang epoch and the post big bang epoch. The post bang epoch, from the big bang to the present time is a canonical quintessence universe with a quartic polynomial dark energy potential. As shown in fig. 1 it is after the big bang at χ and W ( χ ) of zero where both χ and W ( χ ) are negative resulting in a real and positive scalar. The bottom panel of fig. 1 shows that the entire canonical universe only occupies a minute region compared to the total span of the W function principal branch. \nThe pre bang epoch spans from the infinite past to the big bang singularity, a vast space on the principal branch relative to the post bang space. It raises the possibility of the universe existing before the big bang. The big bang is, in fact, a natural consequence of the contraction of the pre bang universe to the big bang singularity. In order to explore a possible pre bang universe this work assumes that the mathematically valid evolutions \nin the pre bang epoch are physically valid as well. Under this assumption the calculated cosmological parameter evolutions reveal a noncanonical universe that is greatly, perhaps infinitely, older and at early times larger, than the canonical post bang universe. The evolution of the total LHQ universe starts with a very low matter density, high dark energy density and vastly larger universe that is contracting rather than expanding. Its basic equations are the same quintessence equations of the post bang universe but it is cold, starless, and has a scalar whose value is imaginary rather than real. It is rolling down the HI potential reaching a matter singularity at a scale factor of zero. The values of χ , W ( χ ) and the scalar κθ are also zero but the HI potential is not. Since the scalar is zero the HI potential is equal to its constant term M 4 ( κδ ) 4 producing a constant dark energy density similar to Λ CDMat the instant of the big bang. The HI potential is proportional to ( W ( χ ( a )) + 1 ) 2 , eqn. 5, therefore it takes a long time for W ( χ ( a ))) to increase enough to significantly alter the HI potential. This makes the LHQ universe similar to Λ CDMin most of the transition zone. \nThe Λ CDM like behavior in the transition zone is not achieved by fine tuning but occurs due to the selection of two important arguments, the HI potential constant term κδ and the dark energy EoS boundary values w 0 . The Λ CDMdark energy EoS is a constant w = -1. In [3] is was found that κδ values near one produced thawing dark energy EoS evolutions with w = -1 at a scale factor of zero, hence the choice of κδ = -0.95. A goal of the study was to provide post bang analytic evolutions close to Λ CDMfor use in likelihood comparisons between Λ CDMand LHQ therefore the w 0 values are set close to minus one, -0.99, -0.999, -1.001 and -1.01. This limits the evolution of w in the transition zone to be between minus one and the boundary condition w 0 values. \nThe canonical post bang LHQ universe has significant flexibility as a dynamical cosmology template for comparison to observations. The investigation in [3] determined that dark energy EoS evolution can be thawing, as in the case here, thawing but transition to freezing, or freezing only, by changing the value of the constant κδ in the HI potential to be near one, two or three respectively. This provides the means to determine the relative likelihoods over a wide range of EoS evolutions. Varying the present day boundary conditions on w 0 increases that range. The same is true for all of the boundary conditions listed in Table 1. This is one of the significant advantages of having analytic equations as a function of the scale factor for the evolutions of cosmological parameters as opposed to numerical solutions that do not provide the same insights as analytic equations. \nAlthough not a subject of the present investigation it is worthwhile to mention one aspect of the future evolution of the LHQ universe. The termination of the Lambert W function principal branch at χ ( a ) = -1 e is the minimum and zero point of the HI potential. Any evolution onto the minus branch requires a reduction of the scale factor and an increase in the magnitude of the HI potential. As such the end of the principal branch may be a benign stable equilibrium point for the LHQ universe. This would provide a future with no big rip or crunch. This possibility will be the subject of a subsequent investigation. \nThe LHQ universe is an important addition to a list of cosmologies for comparison with current and the expected future cosmological observational data. It provides a noncanonical possibility of a universe that exists before the big bang which is far older and once much larger than the canonical Λ CDMor post bang LHQ universes. Exciting new observations deserve investigations that are outside of the box if we are to discover new aspects of cosmological physics not present in our current canonical universe models. \nAcknowledgments: The author acknowledges the very useful and informative discussions with Sergey Cherkis on the mathematical properties of the Lambert W function. \nFunding: This research received no external funding \nData Availability Statement: No new data was produced in this study. \nConflicts of Interest: The author declares no conflicts of interest.', 'References': '- 1. 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2024arXiv240908888G	We present preliminary results of a detailed 3D study of supernova remnants in the nearby spiral M51 using data from the SIGNALS survey obtained with the imaging Fourier transform spectrometer SITELLE at the CanadaFranceHawaii telescope CFHT. Data cubes covering the entire galaxy were gathered in three spectral ranges SN3 647685 nm R 5000 SN2 482513 nm R 600 and SN1 363386 nm R 1000. The spectral resolution of the SN3 cube allows a precise spatially resolved measurement of the velocity dispersion of each object. While most of the SNRs were known from previous surveys based on imagery and longslit spectroscopy we now provide 2D line flux and kinematic maps for all of them and found 20 new candidates. Most of the SNRs show velocity dispersions sigma in the range 3080 kms which is typical for middleaged SNRs. Finally we compare the properties of SNRs with those of thousands of HII regions included in the same dataset.	2024-09-01T00:00:00Z	['10.48550/arXiv.2409.08888', '2024arXiv240908888G', 'arXiv:2409.08888']	['Astrophysics - Astrophysics of Galaxies']	Characterization of M51 supernova remnants with the imaging spectrometer SITELLE	2,024	202	0.51	['EPRINT_HTML', 'EPRINT_PDF']	0	https://arxiv.org/pdf/2409.08888.pdf	{"Characterization of M51's supernova remnants with the imaging spectrometer SITELLE": "Billy Gamache, Laurent Drissen, Carmelle Robert and Mykola Posternak \nD'epartement de physique, de g'enie physique et d'optique, Universit'e Laval, Qu'ebec (QC), G1V 0A6, Canada", 'Abstract': 'We present preliminary results of a detailed 3D study of supernova remnants in the nearby spiral M51 using data from the SIGNALS survey obtained with the imaging Fourier transform spectrometer SITELLE at the Canada-France-Hawaii telescope (CFHT). Data cubes covering the entire galaxy were gathered in three spectral ranges: SN3 (647 - 685 nm, R = 5000), SN2 (482 - 513 nm, R = 600) and SN1 (363 - 386 nm, R = 1000). The spectral resolution of the SN3 cube allows a precise, spatially resolved measurement of the velocity dispersion of each object. While most of the SNRs were known from previous surveys based on imagery and long-slit spectroscopy, we now provide 2D line flux and kinematic maps for all of them and found 20 new candidates. Most of the SNRs show velocity dispersions ( σ ) in the range 30 -80 km/s, which is typical for middle-aged SNRs. Finally, we compare the properties of SNRs with those of thousands of HII regions included in the same dataset.', '1 Introduction': 'Extragalactic supernova remnant (SNR) population studies are key elements for our understanding of the chemical evolution of galaxies. Each population has its own unique signature that reflects the star formation rate, metallicity and interstellar medium (ISM) properties of the host galaxy. The SNRs of M51, a well-studied, nearby ( ∼ 8.6 Mpc) interacting system, was the subject of a recent study by Winkler et al. (2021) (hereafter W21). Using Hubble Space Telescope images and Gemini Multi-Object Spectrograph (GMOS), they found 179 candidates and obtained the spectrum of 66 of them. Following their lead, we used data from the SIGNALS survey to characterize all of these candidates and search for new ones.', '2 Data and methodology': "M51's data are part of the SIGNALS survey(Rousseau-Nepton et al., 2019), wich targets nearby galaxies to study star formation through their emission lines with SITELLE (Drissen et al., 2019), an imaging Fourier transform spectrometer attached to the 3.6-m CFHT. This instrument produces data cubes with a 11 ' × 11 ' field of view sampled at 0 . 32 '' per pixel; the spectral resolution can be tailored to the need of the program up to R = 10000. For this \nFigure 1: Spectrum of the newly found SNR candidate GD24-252 in the SN3 bandpass. \n<!-- image --> \nstudy, we used an R = 5000 cube with the SN3 filter (647 - 685 nm) for the detection process and spectral analysis. Its spectral range includes H α , [NII] λλ 6548 , 83 and [SII] λλ 6716 , 31. Further analysis will be made with the SN2 (R = 600 for H β and [OIII] λλ 4959, 5007) and SN1 (R = 1000 for [OII] λ 3727) filters. \nAs a first step, we identified all of the W21 candidates in the emission line maps and searched for new objects having a [SII] λ 6716 + λ 6731/H α ratio larger than ∼ 0.4 (the 'traditional' criterion to detect extragalactic SNRs). This led to an initial list of 283 objects. For each of them, we selected a domain of integration using a flux treshold as well as a background region. Using the SITELLE dedicated software ORCS (Martin et al., 2021), we extracted the integrated, background-subtracted spectrum of each region and fitted each emission line with a sincgauss profile. 1 ORCS then provides the amplitude, the flux, the velocity, the velocity dispersion (from the σ of the gaussian) and the S/N ratio for each line.", '3 Preliminary results and future work': "As suggested by Long et al. (2018) and Points et al. (2019), our team took advantage of the moderately high spectral resolution provided by the SN3 cubes of the SIGNALS survey to include the velocity dispersion as a supplementary criterion to identify SNRs: see for instance Vicens-Mouret et al. (2023) and Duarte Puertas et al. (2024), who introduced the parameter ξ = [ SII ] Hα × σ . For our preliminary analysis of the M51 data, we thus chose to consider as excellent SNR candidates the objects for which the integrated spectrum simultaneously shows (a) an [SII]/H α ratio larger than 0.35, (b) a velocity dispersion of the [NII] lines, σ [ NII ] , \nFigure 2: ξ and H α flux for the candidate SNRs and the control sample in M51. \n<!-- image --> \nlarger than 30 km s -1 , as well as (c) a S/N ratio larger than 3 in all the lines. This led to a catalogue of 104 objects (84 from W21 and 20 new), which we considered for further analysis. Fig. 1 shows the spectrum of one of these new SNR candidates. \nFor comparaison, we used a control sample of over 1000 emission regions (most likely HII regions) in the same data cube for which we also extracted the sky-subtracted integrated spectrum. Figure 2 compares the SNR candidates population with the control sample in terms of the ξ [ NII ] parameter . The two populations are well separated, showing how SNRs are dominated by the shock heating process, but also by the kinematics of the gas. \nThe median value of [SII]/H α ratio is 1 for SNR candidates, while the velocity dispersion of their [NII] lines ranges between 30 and 160 km/s. Their [NII]/H α ratio is also unusually high, with a median value of 1 . 6. This property has already been noticed by Winkler et al. (2021) and is associated with the high metallicity in the galaxy gas. This is shown in figure 3 which compares M51 SNR line ratios to those of two other spirals. We can clearly see the odd nature of M51's SNRs. In addition to an strong average value, the [NII]/H α ratio of the SNR population also shows a clear negative galactocentric gradient, as expected. But interestingly, this is not the case of the HII regions, wich show a surprising positive gradient (excluding the central AGN and its surroundings). \nThe velocity dispersion measurements of the SNR population also show intriguing features. We sometimes found large differences between the velocity dispersion of the H α and the [NII] lines. For 33 cases, the difference is greater than 30 km/s, H α always beeing the narrower line. Visual inspection of the spectra motivated us to model the spectra with two components for H α and [NII] in a few cases. We were able to fit 20 spectra with a broad and a narrow component with a S/N above 3 (5 with a S/N above 5). The velocity dispersion of the narrow component is in the range 10 -25 km/s while the broad one varies between 50 and 150 km/s. The broad component is tought to originate from the shock-heated plasma, but the narrow component is more mysterious. Despite our care in subtracting a proper local background, we cannot exclude the presence of some residuals. Another possibility is that it \nFigure 3: Line ratios for the SNRs in M51, in comparaison with those of NGC6946 (Long et al., 2019) and M33 (Duarte Puertas et al., 2024). \n<!-- image --> \noriginates from a photoionised precursor (Medina et al., 2014). \nWe are pursuing these preliminary results along the following lines: we will first study the impact of lowering the threshold on the [SII]/H α ratio and σ on the number of SNR candidates, which might disfavor SNRs with slow shock velocities and/or older objects (Kopsacheili et al., 2020) . We will also include other emission lines in the analysis using the SN1 and SN2 data in hand: for instance, among the SNR candidates in our sample, 47 shows strong [OIII] λ 5007 emission and 27 strong [OII] λ 3727.", 'Acknowledgments': "Based on observations obtained with SITELLE, a joint project of Universit'e Laval, ABB, Universit'e de Montr'eal, and the CFHT. We wish to recognize and acknowledge the very significant cultural role that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most grateful to have the opportunity to conduct observations from this mountain.", 'References': 'Drissen, L., et al. 2019, MNRAS, 485, 3930 \nDuarte Puertas, S., Drissen, L., Robert, C., Rousseau-Nepton, L., Martin, R. P., Amram, P., & Martin, T. 2024, MNRAS, 2024MNRAS.tmp.1695D \nKopsacheili, M., Zezas, A., & Leonidaki, I. 2020, MNRAS, 491, 889 \nLong, K. S., Blair, W. P., Milisavljevic, D., Raymond, J. C., & Winkler, P. F. 2018, ApJ, 855, 140 \nLong, K. S., Winkler, P. F., & Blair, W. P. 2019, ApJ, 875, 85 \nMartin, T., Drissen, L., & Prunet, S. 2021, MNRAS, 505, 5514 \nMedina, A. A., Raymond, J. C., Edgar, R. J., Caldwell, N., Fesen, R. A., & Milisavljevic, D. 2014, ApJ, 791, 30 \nPoints, S. D., Long, K. S., Winkler, P. F., & Blair, W. P. 2019, ApJ, 887, 66 \nRousseau-Nepton, L., et al. 2019, MNRAS, 489, 5530 \nVicens-Mouret, S., Drissen, L., Robert, C., Rousseau-Nepton, L., Martin, R. P., & Amram, P. 2023, MNRAS, 524, 3623 \nWinkler, P. F., Coffin, S. C., Blair, W. P., Long, K. S., & Kuntz, K. D. 2021, ApJ, 908, 80'}
2024A&A...691A..19V	The study of gasphase metallicity and its spatial distribution at high redshift is crucial to understand the processes that shaped the growth and evolution of galaxies in the early Universe. Here we study the spatially resolved metallicity in three systems at z 6 8 namely A2744YD4 BDF3299 and COSMOS24108 with JWST NIRSpec IFU lowresolution R 100 spectroscopic observations. These are among the highestz sources in which metallicity gradients have been probed so far. Each of these systems hosts several spatial components in the process of merging within a few kiloparsecs identified from the restframe UV and optical stellar continuum and ionised gas emission line maps. The sources have heterogeneous properties with stellar masses logMSUBSUBMSUBSUB 7.69.3 star formation rates SFRs 115 MSUBSUB yrSUP1SUP and gasphase metallicities 12logOH 7.78.3 which exhibit a large scatter within each system. Their properties are generally consistent with those of the highestredshift samples to date z 3 10 though the sources in A2744YD4 and COSMOS24108 are at the high end of the massmetallicity relation MZR defined by the z 3 10 sources. Moreover the targets in this work follow the predicted slope of the MZR at z 6 8 from most cosmological simulations. The gasphase metallicity gradients are consistent with being flat in the main sources of each system. Flat metallicity gradients are thought to arise from gas mixing processes on galaxy scales such as mergers or galactic outflows and supernova winds driven by intense stellar feedback which wash out any gradient formed in the galaxy. The existence of flat gradients at z 6 8 sets also important constraints on future cosmological simulations and chemical evolution models whose predictions on the cosmic evolution of metallicity gradients often differ significantly especially at high redshift but are mostly limited to z 3 so far.	2024-11-01T00:00:00Z	['2024arXiv240303977V', '10.48550/arXiv.2403.03977', '10.1051/0004-6361/202449855', '2024A&A...691A..19V', 'arXiv:2403.03977']	['galaxies: high-redshift', 'galaxies: abundances', 'galaxies: ISM', 'galaxies: evolution', 'techniques: imaging spectroscopy', 'techniques: high angular resolution', 'Astrophysics - Astrophysics of Galaxies']	Gasphase metallicity gradients in galaxies at z 68	2,024	202	0.64	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	27	https://arxiv.org/pdf/2403.03977.pdf	{'Gas-phase metallicity gradients in galaxies at z ∼ 6 -8': 'G. Venturi 1,2, ⋆ , S. Carniani 1,2 , E. Parlanti 1 , M. Kohandel 1 , M. Curti 3 , A. Pallottini 1 , L. Vallini 4 , S. Arribas 5 , A. J. Bunker 6 , A. J. Cameron 6 , M. Castellano 12 , A. Ferrara 1 , A. Fontana 12 , S. Gallerani 1 , V. Gelli 7,8 , R. Maiolino 9,10,11 , E. Ntormousi 1 , C. Pacifici 13 , L. Pentericci 12 , S. Salvadori 14,2 , and E. Vanzella 4 \n- 1 Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy\n- 2 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy\n- 3 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748, Garching, Germany\n- 4 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Gobetti 93 / 3, I-40129, Bologna, Italy\n- 5 Centro de Astrobiología (CAB), CSIC-INTA, Cra. de Ajalvir Km. 4, 28850, Torrejón de Ardoz, Madrid, Spain\n- 6 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK\n- 7 Cosmic Dawn Center (DAWN), Jagtvej 128, 2200, Copenhagen N, Denmark\n- 8 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200, Copenhagen N, Denmark\n- 9 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\n- 10 Cavendish Laboratory - Astrophysics Group, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK\n- 11 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK\n- 12 INAF - Osservatorio Astronomico di Roma, via di Frascati 33, I-00078 Monte Porzio Catone, Italy\n- 13 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n- 14 Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, Largo E. Fermi 1, 50125, Firenze, Italy \nReceived 5 March 2024; accepted 12 August 2024', 'ABSTRACT': 'The study of gas-phase metallicity and its spatial distribution at high redshift is crucial to understand the processes that shaped the growth and evolution of galaxies in the early Universe. Here we study the spatially resolved metallicity in three systems at z ∼ 6-8, namely A2744-YD4, BDF-3299, and COSMOS24108, with JWST NIRSpec IFU low-resolution ( R ∼ 100) spectroscopic observations. These are among the highestz sources in which metallicity gradients have been probed so far. Each of these systems hosts several spatial components in the process of merging within a few kiloparsecs, identified from the rest-frame UV and optical stellar continuum and ionised gas emission line maps. The sources have heterogeneous properties, with stellar masses log( M ∗ / M ⊙ ) ∼ 7.6-9.3, star formation rates (SFRs) ∼ 1-15 M ⊙ yr -1 , and gas-phase metallicities 12 + log(O / H) ∼ 7.7-8.3, which exhibit a large scatter within each system. Their properties are generally consistent with those of the highest-redshift samples to date ( z ∼ 3-10), though the sources in A2744-YD4 and COSMOS24108 are at the high end of the mass-metallicity relation (MZR) defined by the z ∼ 3-10 sources. Moreover, the targets in this work follow the predicted slope of the MZR at z ∼ 6-8 from most cosmological simulations. The gas-phase metallicity gradients are consistent with being flat in the main sources of each system. Flat metallicity gradients are thought to arise from gas mixing processes on galaxy scales, such as mergers or galactic outflows and supernova winds driven by intense stellar feedback, which wash out any gradient formed in the galaxy. The existence of flat gradients at z ∼ 6-8 sets also important constraints on future cosmological simulations and chemical evolution models, whose predictions on the cosmic evolution of metallicity gradients often di ff er significantly, especially at high redshift, but are mostly limited to z ≲ 3 so far. \nKey words. Galaxies: high-redshift - Galaxies: abundances - Galaxies: ISM - Galaxies: evolution - Techniques: imaging spectroscopy - Techniques: high angular resolution', '1. Introduction': "Gas-phase metallicity, its scaling relations with other galactic properties, and its spatial distribution are fundamental tools to study and understand the evolution of galaxies (e.g. Maiolino & Mannucci 2019). The interplay between gas accretion, star formation, outflows, and mergers regulates the growth of galaxies and the buildup of their metal content, leading to the correlation between gas-phase metallicity and stellar mass in galaxies known as mass-metallicity relation (MZR; e.g. Davé et al. 2011; Pallottini et al. 2014; Somerville & Davé 2015). The spatial distribution of metals in galaxies, usually described through radial metallicity gradients, bears the imprint of these underlying processes. Therefore, spatially resolved studies of metallicity in the \nearly Universe, where the formation and primordial growth of galaxies were taking place, are key to better constrain the main mechanisms that have contributed to shape them. \nNegative (radially decreasing) gradients are usually interpreted as resulting from an inside-out galaxy formation scenario, in which stars start forming earlier in the inner parts of galaxies and thus have more time to chemically enrich the inner regions than the outer ones (e.g. Samland et al. 1997; Prantzos & Boissier 2000; Davé et al. 2011; Pilkington et al. 2012; Gibson et al. 2013; Hemler et al. 2021; Tissera et al. 2022). Flattened gradients could arise from radial mixing of gas and redistribution of metals on galaxy scales, induced by supernova (SN) winds within the galaxy or galactic outflows of metal-enriched material expelled from the galaxy and re-accreted in the outer regions (so-called galactic fountains), driven by intense stellar feedback (e.g. Gibson et al. 2013; Ma et al. 2017), as well as by galaxy \nFig. 1. Simple sketch summarising the three main classes of radial gasphase metallicity gradients and possible physical origin. Negative (decreasing) gradients can be interpreted as the result of the inside-out growth of a galaxy, in which star formation and the following chemical enrichment start earlier in the inner regions. Flat gradients can result from radial mixing processes, such as galaxy mergers as well as SN winds redistributing the metals in the interstellar medium (ISM) and galactic outflows of metal-enriched gas evacuated (or re-accreted) in the outer regions. Positive (increasing) inverted gradients may arise from accretion of pristine gas to the central regions of the galaxy; this can also lead to flattened gradients. \n<!-- image --> \nmerging and interactions (e.g. Rupke et al. 2010b,a; Rich et al. 2012; Torres-Flores et al. 2014). Finally, positive (radially increasing; also called inverted) gradients may be produced by accretion of external pristine gas towards the central regions of the galaxy (e.g. Ceverino et al. 2016); this could also contribute to flatten the metallicity gradient, if not invert it. Particularly strong metal-loaded galactic outflows could also contribute to produce a positive gradient, by moving the metal-rich gas from the central starburst regions to the outskirts (e.g. Tissera et al. 2022). A schematic cartoon that summarises the above framework is shown in Fig. 1. \nIn the local Universe, most spiral galaxies show negative metallicity gradients, with more metal-enriched gas in the inner than in the outer regions of the galaxy (e.g. Magrini et al. 2009, 2017; Stanghellini & Haywood 2010; Luck et al. 2011 for the Milky Way; Zaritsky et al. 1994; Magrini et al. 2010; Kewley et al. 2010; Bresolin 2011; Sánchez et al. 2014; Berg et al. 2015; Ho et al. 2015; Belfiore et al. 2017 for other local galaxies). By using metallicity diagnostics that trace past metal enrichment, primarily planetary nebulae (PNe; up to 5-10 Gyr ago, i.e. z ≲ 2) some studies have suggested that the Milky Way and nearby galaxies had flatter gradients in the past which \ngrew progressively steeper with cosmic time (Stanghellini et al. 2014; Stanghellini & Haywood 2018; Magrini et al. 2016). A dependence of gradient slope with mass has been observed in the local Universe, with more massive galaxies exhibiting progressively steeper gradients, while low-mass galaxies having almost flat gradients (Belfiore et al. 2017). This relation between mass and metallicity gradient may be the result of an evolutionary sequence in mass, with more massive, more evolved galaxies having steeper gradients and low-mass, less evolved ones, analogues of highz galaxies, having flatter gradients (e.g. Maiolino &Mannucci 2019). \nCosmological simulations of galaxy evolution make very different predictions on metallicity gradients at high redshift (up to z ∼ 3), which can strongly vary depending on prescriptions on star formation and stellar feedback models and on their relative contribution to gas mixing as compared to mergers (e.g. Gibson et al. 2013; Ma et al. 2017; Hemler et al. 2021; Tissera et al. 2022). For example, according to the FIRE simulations (Hopkins et al. 2014), mergers and rapid variations of metallicity gradients in highz galaxies, induced by starburst episodes which drive strong outflows and disrupt the gas disc, would flatten any negative gradients previously developed, especially in low-mass galaxies where feedback mechanisms are more e ffi -cient (Ma et al. 2017). Instead, the TNG50 simulations (Pillepich et al. 2019; Nelson et al. 2019) which feature less bursty feedback, predict negative gradients at all redshifts, steeper (more negative) with increasing z (Hemler et al. 2021). The metallicity gradients resulting from the MUGS ('normal' feedback; Stinson et al. 2010) and MaGICC ('enhanced' feedback; Brook et al. 2012) simulation suites are compared in Gibson et al. (2013). The enhanced feedback (including pre-SN early stellar feedback from massive stars) re-distributes energy and re-cycled ISM material over large scales, through re-accretion in the outer parts of the galaxy of gas expelled via outflows. In this case, relatively flat and temporally invariant abundance gradients are predicted, in contrast to the steeper negative gradients at increasing z emerging from the normal feedback scenario. Finally, the EAGLE simulations (Schaye et al. 2015) find that the median gradient is zero (flat) at all redshifts, but with the scatter around the median increasing with z due to individual galaxies transitioning between steep (either negative or positive) and flatter gradients (Tissera et al. 2022). The authors report that this behaviour results from the higher frequency of major mergers at higher z leading to episodes of enhanced accretion of low-metallicity gas which trigger intense star formation and ejection of metalenriched gas. \nNon-cosmological chemical evolution models have also been employed to study the evolution of metallicity gradients with cosmic time, mostly for the Milky Way. Mott et al. (2013), assuming an inside-out formation of the disc, a constant star formation e ffi ciency (SFE) along the disc, and the presence of radial flows with varying speed, predict an inversion of the gradients (from negative to positive) at z ≳ 1, due to a strong infall of primordial gas in the innermost disc regions at early times; only a variable SFE does not lead to an inversion of the gradients at high z in their models. Kubryk et al. (2015) study the role of the radial motions of gas and stars on the evolution of abundance profiles in the Milky Way disc, finding steep abundance profiles at high z which flatten with time, as a result of the inside-out formation of the disc. Mollá et al. (2019) investigate the role of the growth of the stellar disc, the e ff ect of infall rate and star formation prescriptions, as well as the pre-enrichment of the infall gas, and find a smooth evolution of the gradients with a slight flattening from z = 4 to 1. Finally, the first principles-based model of \nSharda et al. (2021) finds that the gradient in Milky Way-like galaxies has steepened over time and also predicts the evolution of metallicity gradients with redshift in galaxy samples matched in both stellar masses and abundances, finding that disc galaxies transition from the advection- to the accretion-dominated regime from high to low z ; in general little evolution of the gradients is predicted for z ≳ 1. \nObservations at the cosmic noon epoch (i.e., z ∼ 1-4) have found heterogeneous results, with both negative, positive, and flat metallicity gradients (e.g. Cresci et al. 2010; Yuan et al. 2011; Queyrel et al. 2012; Swinbank et al. 2012; Stott et al. 2014; Troncoso et al. 2014; Jones et al. 2013, 2015; Leethochawalit et al. 2016; Wuyts et al. 2016; Wang et al. 2017, 2022; Carton et al. 2018; Förster Schreiber et al. 2018; Curti et al. 2020a). Nevertheless, the majority of highz measurements is consistent with little or no cosmic evolution of metallicity gradients, which are found to be flat or only moderately negative or positive ( ≲ \| ± 0 . 1 \| dex kpc -1 ; see e.g. the compilation in Curti et al. 2020a). \nAt higher redshift ( z ∼ 7), Vallini et al. (2024) also found flat gradients by making use of a physically motivated Bayesian model to derive metallicities from rest-frame far-infrared lines ([O iii ] 88 µ m and [C ii ] 158 µ m) from Atacama Large Millimeter / submillimeter Array (ALMA) observations. However, as the authors point out as a caveat, there may be systematics in comparing their results with gradients of O / H abundance obtained with the more standard methods based on optical lines, since the adopted model does not take into account the likely enhancement of O / C at low metallicities (see e.g. Maiolino & Mannucci 2019), which would result in the model returning higher values of metallicity. Therefore, while their results consistent with flat gradients at z ∼ 7 are very significant, they require independent confirmation. \nThe James Webb Space Telescope (JWST) has recently opened up the possibility of measuring metallicity gradients in the first 1-2 Gyr of the Universe by using rest-frame optical emission lines (Wang et al. 2022; Rodríguez Del Pino et al. 2023; Arribas et al. 2023) which are extensively used and calibrated at lower redshifts (e.g. Curti et al. 2017, 2020b). In this work, we study the spatially resolved gas-phase metallicity and investigate the shape of metallicity gradients at high redshift ( z ≳ 6), by tracing the warm ( T ∼ 10 4 K) ionised gas emission. Specifically, we observed three highz systems with the Near-InfraRed Spectrograph (NIRSpec) on board JWST in its Integral Field Unit (IFU) mode with the low-spectral resolution PRISM / CLEAR( R ∼ 100) disperser-filter combination. These are A2744-YD4, part of the proto-cluster A2744-z7p9OD ( z ∼ 7.88), BDF-3299 ( z ∼ 7.11), and COSMOS24108 ( z ∼ 6.36). Basic information on the targets is given in Table 1. We trace gas-phase metallicity by making use of strong-line calibrators relying on rest-frame optical and nearUV emission lines. We also present the basic properties of each source, specifically stellar mass, obtained from spectral energy distribution (SED) fitting, and star formation rate (SFR), from H β or H α . The star formation history of these targets is studied in detail in a separate paper (Kohandel et al., in prep.). \nThroughout this work, the reported wavelengths are in vacuum and we adopt a flat Λ CDM cosmology with H 0 ≃ 67.7 km s -1 Mpc -1 , Ω M ≃ 0.31, and ΩΛ ≃ 0.69 (Planck Collaboration et al. 2020).", '2. Description of observations, data reduction, and analysis': "The observations employed in this work were carried out between October and December 2022 as part of the JWST GO \nprogram ID 1893 (PI: S. Carniani; Carniani et al. 2021). The three sources were observed for 15.102 ks ( ∼ 4 h) on-source each (22.059 ks with overheads; ∼ 6 h) with 1.678 ks of background exposure time each (2.830 ks with overheads). The observations were performed with the PRISM / CLEAR disperser-filter combination and an eight-point dither 'medium' cycling pattern (step ∼ 0.5 '' ). The JWST NIRSpec ∼ 3 '' × 3 '' IFU PRISM observations simultaneously span the spectral range 0.6-5.2 µ m with a spectral resolution ranging between R ∼ 30-330.", '2.1. Data reduction': "We retrieved the raw data from the MAST archive and we ran the three stages of the pipeline using version 1.11 with CRDS (calibration reference data system) context 'jwst\\_1094.pmap'. First, at stage 1 'calwebb\\_detector1', the pipeline applied the detector-level corrections (e.g. check for saturation, dark exposure subtraction, flagging of bad pixels and cosmic-ray persistences) and performed ramp fitting for individual exposures. We then calibrated the count rate images by executing stage 2 'cal-webb\\_spec2' of the pipeline, which corrects for flat field and performs the wavelength calibration. The background was subtracted from each exposure during this stage by using the observations of the dedicated background for each target. We processed the background targets up to stage 2 of the pipeline, then we applied the background step for the science target. In the background step, the pipeline subtracts the background exposure from each target exposure in the detector space. Finally, each calibrated exposure was combined in stage 3 'calwebb\\_spec3' by using 'drizzle' weighting and a spaxel size of 0.05 '' to obtain the final cubes. During stage 3, we applied the outlier rejection step built in the pipeline and then a sigma clipping to remove any residual outliers in the final cube.", '2.2. Data analysis': 'In this section, we describe the analysis of the NIRSpec R100 IFU data. In brief, we obtained the emission line fluxes from emission-line modelling of the spectra, which were used to infer the gas-phase metallicity. This was done on both an integrated basis (to get the integrated metallicity of each target) and a spatially resolved one, in this case both spaxel-by-spaxel (to produce maps) and in concentric radial annuli (to obtain radial gradients). From the integrated spectra, we also obtained the stellar mass ( M ∗ ) of each source from SED fitting and their SFR from the emission-line modelling. We provide more details in the following.', '2.2.1. Emission line fitting': "The main goal of this work is to measure radial gas-phase metallicity gradients. To do so, we infer the oxygen abundance relative to hydrogen (12 + log(O / H)), a proxy of gas-phase metallicity, by making use of the optical and near-UV strong-line diagnostic ratios reported in Table 2. We adopt the new diagnostic ratio ˆ R = 0.47 R2 + 0.88 R3, first introduced in Laseter et al. (2024), in place of the more traditional R23 = ([O ii ] λλ 3727,30 + [O iii ] λλ 4960,5008) / H β , since the former is more suited for highz galaxies, while the latter is a projection mostly driven by lowmetallicity local analogues. We adopt the best-fit polynomial calibrations from Curti et al. (2017, 2020b), slightly revisited in Curti et al. (2023a,b) to better probe the low-O / H regime, to infer the gas metallicity from the combination of the above ratios. \nTable 1. Basic information on the observed targets. \nNotes. ( a ) From this work. ( b ) The reported scale is corrected for the lensing magnification factor of ∼ 2 (Morishita et al. 2023; Bergamini et al. 2023). \nTable 2. Emission line diagnostic ratios used in this work together with their compact notation. \nWetested other gas-phase metallicity calibrations (e.g. Nakajima et al. 2022; Sanders et al. 2024). These gave 12 + log(O / H) values similar to the Curti et al. ones within 0.1 dex, consistent with the uncertainties, and virtually no di ff erence in the metallicity radial gradients. \nThe full width at half maximum (FWHM) of the point spread function (PSF) of JWST depends on wavelength. Specifically, the PSF FWHM of NIRSpec IFU ranges from around 0.09 '' at 1 µ m up to around 0.16 '' at 4.5 µ m, and the variation with wavelength is stronger in the direction perpendicular to the IFU slices than along them (D'Eugenio et al. 2023). Since the aim of this work is to obtain spatially resolved metallicities from emission line ratios sometimes far in wavelength (see Table 2), we applied a wavelength-dependent spatial smoothing to the data cube prior to the line fitting with the aim of achieving the same spatial resolution at all wavelengths. We adopted the PSF FWHM curves from D'Eugenio et al. (2023), empirically calibrated by matching NIRSpec IFU observations with NIRCam ones of the same target reported in their Eqs. 3 and 4, for the along- and across-slice cases, respectively. The smoothing was done by convolving the cube with a wavelength-dependent 2D Gaussian kernel in order to obtain a smoothed cube having the same spatial resolution at each wavelength, specifically the PSF FWHM reported in D'Eugenio et al. (2023) at the wavelength of the highest-wavelength line of interest for the along-slice case (which has the largest FWHM among the two cases). This corresponds to FWHMgoal ∼ 0.16 '' (at [O iii ]) for A2744-YD4, ∼ 0.14 '' (at [O iii ]) for BDF-3299, and ∼ 0.18 '' (at H α -[S ii ]) for COSMOS24108. Specifically, the 2D Gaussian kernel to be used for the convolution, at a given wavelength λ , was defined as follows: FWHM 2 kern;dir ( λ ) = FWHM 2 goal ( λ ) - FWHM 2 mod;dir ( λ ), where FWHMmod;dir is the model FWHM from D'Eugenio et al. (2023) and 'dir' indicates the direction, either along or across the slices. \nIn general, we employed the original unsmoothed data cube for the flux maps of continuum and emission lines and for integrated measurements from circular apertures, while we adopted the smoothed data cube for the maps and radial profiles involving line ratios and metallicity, which would have otherwise been a ff ected by the wavelength-dependence of the PSF. In case of \nA2744-YD4, the highestz target of our sample, we adopted the smoothed data cube for every map, to obtain visually clearer maps as compared to the more noisy ones from the unsmoothed cube. \nThe data analysis consisted in modelling the rest-frame optical and near-UV emission lines available in the observed spectral range. The emission lines are spectrally unresolved in the PRISM spectra (the spectral resolution is σ res ∼ 1250 km s -1 at ∼ 3 µ m and σ res ∼ 450 km s -1 at ∼ 5 µ m), therefore a single Gaussian function per line was used. We fitted the [O iii ]-H β , the [O ii ]-[Ne iii ]-H γ , and the H α -[N ii ]-[S ii ] line complexes separately. The [N ii ] 6550,85 was included to allow for a better modelling of the H α line profile, which was otherwise showing a marked residual in its redward wing. This was done only when fitting integrated spectra from circular apertures or concentric annuli for radial profiles, not in the lower-S / N case of spaxelby-spaxel fitting, for which no residual wing indicative of [N ii ] was detected above the noise. For each line complex, the velocity was tied to be the same for all the lines, while we allowed the line width to vary to match the PRISM spectral resolution, given that all lines are spectrally unresolved as mentioned. We fixed the flux ratios [O iii ] λ 5008 / λ 4960 and [N ii ] λ 6585 / λ 6550 to their theoretical value of 3 (Storey & Zeippen 2000). We included an underlying first-order polynomial to model the continuum. We accounted for the Balmer break of the continuum at around 3645 Å rest-frame, when needed, by employing three (first-order) polynomials, one modelling the jump and the other two on each side of it. \nThe line fluxes used to infer the metallicity or the SFR were first corrected for dust extinction, when possible (see below), using the H γ / H β ratio (for A2744-YD4 and BDF-3299) or H α / H β (for COSMOS24108), only when both lines had S / N > 3. We adopted a Calzetti et al. (2000) reddening curve, with an RV = 4.05, suitable for highz low-metallicity (12 + log(O / H) < 8.5) star-forming galaxies at the wavelengths of interest here of ≳ 3000 Å (see e.g. Shivaei et al. 2020). We assumed the theoretical extinction-free H γ / H β and H α / H β ratios of 0.466 and 2.87, respectively, valid for case-B recombination and an electron temperature of T e ∼ 10 4 K (Osterbrock & Ferland 2006), typical of the warm ionised gas emitting rest-frame optical lines. \nWe selected the line ratio diagnostics for the metallicity estimation in an adaptive way, based on the line S / N. Specifically, a certain ratio (e.g. [Ne iii ] / [O ii ]) was employed only when all the lines involved in it had S / N > 3. In this way, for each fitted spectrum we only selected the sub-sample of line ratio diagnostics whose line fluxes exceeded the S / N threshold, instead of using all the ratios (even those involving S / N < 3 lines) or none of them (when just one or a few of the lines were below the S / N threshold). Finally, given a set of diagnostic ratios, the best-fit metallicity was obtained by minimising the chi-squared defined simultaneously by the di ff erent observed ratios and their relative calibration curves, weighted by the observed uncertainties and \nthe intrinsic dispersion of the calibration (added in quadrature; see Curti et al. 2020b). \nWe first modelled the emission lines in integrated spectra extracted from circular apertures centred on each sub-source present in the three studied systems. From these, we obtained the integrated gas-phase metallicity (as described above) and SFR of each sub-source. The SFR was obtained from the extinctioncorrected flux of H α , using the relation from Kennicutt & Evans (2012). In A2744-YD4 and BDF-3299, for which H α was not available, the SFR was obtained from the extinction-corrected flux of H β , converted to H α by adopting the theoretical H α / H β ratio of 2.87. We employed circular apertures of 0.15 '' radius for A2744-YD4 and COSMOS24108, and of 0.1 '' for BDF-3299, due to the spatial vicinity of the sources in this latter system. \nThe emission line modelling was then performed both on a spaxel-by-spaxel basis (for both the unsmoothed and the smoothed data cubes) and in concentric radial annuli centred on each source (for the smoothed data cube). The latter was done to increase the S / N on the emission lines and get more robust estimates of metallicity with the goal of obtaining metallicity gradients. For this annular line modelling, we extracted integrated spectra at each radius by collapsing the spaxels within concentric circular annuli. The annuli, having radial width of 1 spaxel (0.05 '' ) each, were centred on each of the main spatial components detected in each targeted system, with a variable maximum aperture radius depending on the component extension (as described in the next sections separately for each target). \nThe extinction-correction of emission line fluxes used to infer metallicity was done only for the case of integrated spectra from circular apertures and from concentric radial annuli. This was not possible for the spaxel-by-spaxel case because, among the useful Balmer lines tracing extinction, H γ is detected in almost no spaxels in A2744-YD4 and BDF-3299. Therefore, the maps of metallicity were obtained without accounting for possible extinction. For COSMOS24108, also including H α in the spectral range, H β is detected in a large enough number of spaxels to attempt for an extinction correction of the line fluxes for the metallicity. However, the resulting map is very similar to the non-extinction-corrected one, only more noisy, therefore we report the metallicity map obtained without the extinction correction.", '2.2.2. SED fitting': 'With the goal of obtaining the stellar mass of each target, we fitted the spatially integrated spectra extracted from the data cube by using the SED fitting code bagpipes (Carnall et al. 2018). We adopted the stellar population models by Bruzual & Charlot (2003) and included the nebular emission with cloudy (Ferland et al. 2017) with the ionisation parameter ( -3 . 0 < log U < 0 . 0) as a free parameter and adopting a [C / O] = [C / O] ⊙ . We assumed a Kroupa (2001) initial mass function truncated at 0.01 and 100 M ⊙ and a Calzetti et al. (2000) attenuation curve. Finally, we used a non-parametric star-formation history model with continuity priors (see Leja et al. 2019) and with four time bins: 0 < t < 10 Myr, 10 < t < 50 Myr, 50 < t < 100 Myr, and 100 < t < 300 Myr.', '3. A2744-YD4': "A2744-YD4 is part of the proto-cluster A2744-z7p9OD at z ∼ 7.883 located behind the strong lensing cluster Abell 2744. The proto-cluster includes 22 sources at 7 < z < 9 identified \nthrough combined deep HST and Spitzer IRAC photometry (Laporte et al. 2014; Zheng et al. 2014; Atek et al. 2015; Ishigaki et al. 2016) as part of the Hubble Frontier Fields program (Lotz et al. 2017). Seven of them were recently spectroscopically confirmed to be at z = 7.88 through JWST NIRSpec MSA spectroscopy (Morishita et al. 2023). This makes A2744-z7p9OD the most distant proto-cluster known so far. The system is highly over-dense, with an excess of surface number density from the field average δ = ( n -n ) / n ∼ 130 + 66 -51 (Ishigaki et al. 2016). \nThe ∼ 3 '' × 3 '' IFU FOV of our observations includes five sources of the proto-cluster known as the 'quintet', namely YD1, YD4, YD6, YD7, and ZD1, as well as a new source identified by Hashimoto et al. (2023) and named s1 (Fig. 2, top-left). We only partially cover the East component of YD7 (named YD7E), while its emission toward to West lies outside the FOV. The lensing magnification factor of the source in the FOV is µ ≃ 2.0 (Morishita et al. 2023; Bergamini et al. 2023). YD4, at the centre of our IFU observation, was reported to be at z = 8.38 based on Ly α and ALMA [C ii ] 158 µ m and [O iii ] 88 µ m line emission (Laporte et al. 2017, 2019; Carniani et al. 2020). Recently, Morishita et al. (2023) measured z = 7.88 for YD4 based on the detection of high-S / N[O iii ] 4959,5007 Å and H β emission lines in high-spectral resolution ( R 2700) NIRSpec MSA observations, thus ruling out the previous redshift measurement. \nThe top-left panel of Fig. 2 shows the map of the continuum emission in the range 2-3 µ mobserved wavelength ( ∼ 0.230.34 µ m rest-frame). In addition to the sources named YD4, YD6, YD1, s1, YD7-E, and ZD1, the continuum map also shows the presence of a source at the northernmost part of the FOV, which is also present in the NIRCam image but is not labelled in Hashimoto et al. (2023); we name this source s2. However, given that this is at the edge of the FOV, where many artefacts are present in the NIRSpec IFU data, we consider this source as tentative and report its label in parenthesis in the figure. Moreover, in the map of the continuum obtained from the unsmoothed data cube integrated over the spectral range 1.2-2 µ m (Fig. A.3 in the Appendix 1 , top-left), where the spatial resolution is the highest, the eastern tail of YD1 appears as a separate peaked spatial component. We label this extra source as YD1-E. \nFig. 2, bottom panel, illustrates the integrated spectrum (aperture radius = 0.15 '' ) associated with the main component of the A2744-YD4 system, that is, YD4 itself. The rest-frame optical lines [O ii ], H β , and [O iii ] are detected with high S / N ( ≳ 6 on the peak). The spectroscopic redshift based on Lyman break and optical lines ( z ∼ 7.88) is consistent with that found by Morishita et al. (2023) and Hashimoto et al. (2023) from the NIRSpec R 2700 data. We find a stellar mass of log( M ∗ / M ⊙ ) ∼ 8.7 from the SED fitting and a SFR ∼ 2 M ⊙ yr -1 from H β for YD4 (Table 3). These were corrected for the magnification factor of 2 due to the lensing. The gas-phase metallicity, relying on line ratios, is instead not a ff ected by it. For the other spatial components in the system, we find log( M ∗ / M ⊙ ) ∼ 7.5-8.6 and SFR ∼ 1-3 M ⊙ yr -1 . \nThe [O iii ] flux map resulting from our spectral emissionline modelling is displayed in Fig. 2, central panel (the maps of H β and [O ii ] are shown in Fig. A.3 1 ). From the [O iii ] map, we identify two new emission regions which are weak or absent in continuum, which we label as YD6-[OIII]-E and YD4-[OIII]W. Overall, the [O iii ] maps obtained from the PRISM data presented here are deeper and reveal fainter and more extended features than those from high-resolution grating data presented in \nTable 3. Properties of the individual sources. \nNotes. (1) Source name. (2) Redshift from [O iii ] λ 5008. (3) Stellar mass from SED fitting with bagpipes . (4-9) [O ii ] λλ 3727,30, [Ne iii ] λ 3870, H β , [O iii ] λ 5008, H α , and [S ii ] λ 6718,33 measured fluxes (in units of 10 -20 erg s -1 cm -2 ), (10) SFR (in M ⊙ yr -1 ), and (10) gas-phase metallicity from emission line Gaussian fitting. For the sources in the A2744-YD4 system, the reported M ∗ and SFR are corrected for the lensing magnification factor of ∼ 2 (Morishita et al. 2023; Bergamini et al. 2023); the line fluxes are instead the observed ones. All the properties are obtained from integrated spectra extracted from circular apertures (having radius of 0.15 '' for A2744-YD4 and COSMOS24108 and of 0.1 '' for BDF-3299) centred on each source. For the line fluxes and SFR, we also report the 3 σ upper limits in case of no detection. \nFig. 2. Maps for A2744-YD4 from JWST NIRSpec IFU. All the maps are obtained from the (wavelength-dependent) spatially smoothed data cube (details in Sect. 2.2). No extinction correction is applied. A cut of S / N > 3 on the peak flux of each line is applied. Median stellar continuum emission in the observed spectral range 2-3 µ m (0.23-0.34 µ m rest-frame; left); [O iii ] integrated flux (centre); map of oxygen abundance, 12 + log(O / H) (right). Contours mark the continuum from first panel. NIRSpec PRISM / CLEAR spectrum extracted from a circular aperture with radius of 0.15 '' centred at the location of the target YD4 (bottom). \n<!-- image --> \nHashimoto et al. (2023) (who do not report the detection of any other emission line due to the lower S / N of their data). We can count nine sources in total in the ∼ 3 '' × 3 '' NIRSpec IFU FOV, among those detected in continuum and in line emission, namely YD4, YD6, YD6-[OIII]-E, YD4-[OIII]-W, YD7-E, ZD1, YD1, YD1-E, and s1, and possibly a tenth source, s2. All these sources lie at z ∼ 7.88, either confirmed spectroscopically (YD4, YD6, YD6-[OIII]-E, YD4-[OIII]-W, YD1, YD1-E, and s1; e.g. Morishita et al. 2023, Hashimoto et al. 2023, and this work) or from photometry (ZD1), except for YD7-E (and the tentative source s2) whose redshift is not assessed. The spectra of all targets, but YD4, are reported in Figs. A.1 and A.2 in the Appendix 2 . \nIn the top-right panel of Fig. 2, we show the map of oxygen abundance, 12 + log (O / H), a proxy for gas-phase metallicity. The metallicity is inferred by making use of the strongline diagnostics reported in Table 2 as described in Sect. 2.2 (the maps of the [O iii ] λ 5008 / H β and [O iii ] λ 5008 / [O ii ] λλ 3727,30 emission line ratios are shown in Fig. A.3 2 ). The ratios involving lines which are not close in wavelength (all but [O iii ] and H β ) may be a ff ected by extinction. Unfortunately, the only robustly detected Balmer line at the spaxel level is H β , and only in a handful of spaxels, therefore estimating the dust extinction spaxel-by-spaxel is not possible. Based on this, the abundance map should not be taken as a robust measurement; moreover, the map is very noisy. Nevertheless, the map seems to suggest that, in the northern system, YD1 is embedded between two sources at lower metallicity, s1 and YD1-E. No clear pattern of metallicity shows up in the southern system (the YD4 one) from the map.", '4. BDF-3299': "BDF-3299 is a spectroscopically confirmed star-forming galaxy (SFRUV ∼ 6 M ⊙ yr -1 , SFRdust ≲ 12 M ⊙ / yr) at z ∼ 7.109 (Castellano et al. 2010; Vanzella et al. 2011; Maiolino et al. 2015; Carniani et al. 2017), located in an over-density of galaxies (Castellano et al. 2016). We report the maps for BDF-3299 from the NIRSpec IFU observations in Fig. 3. \nThe median continuum in the observed spectral range 1.1-2 µ m ( ∼ 0.14-0.25 µ m rest-frame) shows that BDF-3299 is composed of three spatial components. The two brighter ones are located at the NW (the strongest of the two in the continuum) and at the centre of the system, respectively; a third, fainter one resides in the E part. We label these sources as BDF-3299-a, BDF-3299-b, and BDF-3299-c, respectively, in decreasing order of continuum brightness, as labelled in Fig. 3, left panel. The other continuum emitters in the FOV are lower-redshift sources. \nFig. 3, bottom panel, shows the integrated spectrum (aperture radius = 0.1 '' ) associated with the brightest component in continuum emission, BDF-3299-a. The rest-frame optical lines [O ii ], H β , and [O iii ] are detected with high S / N ( ≳ 6 on the peak). From the lines, we estimate a redshift of 7.114, slightly higher but roughly consistent with the previously reported one from Ly α (7.109; Vanzella et al. 2011) and [C ii ] (7.107; Carniani et al. 2017). We obtain a stellar mass of log( M ∗ / M ⊙ ) ∼ 7.9 from the SED fitting and a SFR ∼ 3.5 M ⊙ yr -1 from H β for BDF3299-a (Table 3). The spectra of the other spatial components in the system, namely BDF-3299-b and c, are reported in Fig. B.1 in the Appendix 2 ). From these, we find log( M ∗ / M ⊙ ) ∼ 7.6-8.2 and SFR ∼ 1.5-4 M ⊙ yr -1 . \nThe [O iii ] emission (Fig. 3, central panel) peaks on BDF3299-b rather than on BDF-3299-a (the strongest in continuum emission). The line ratio maps are quite noisy and only few spaxels are above the S / N threshold of 3 (Fig. B.2). Therefore, the same applies to the metallicity map, which generally shows low values, of 12 + log(O / H) ≲ 7.8, in the few spaxels where it can be estimated (Fig. 3, right).", '5. COSMOS-24108': "COSMOS-24108 ( z ∼ 6.36; SFRUV ∼ 29 M ⊙ / yr, SFRdust ≲ 6.2 M ⊙ / yr) shows two, or possibly three, spatial components in Hband HST rest-frame UV imaging (Pentericci et al. 2016). In Fig. 4, left panel, we show the 1-2 µ m observed continuum ( ∼ 0.14-0.27 µ m rest-frame) from the NIRSpec IFU data. This shows two main spatial components, and a third, fainter one in the SE part of the system, consistent with those seen with HST in a similar spectral band. We label these three spatial components as COSMOS24108-a, b, and c, in order of continuum brightness. The continuum at redder wavelengths, between 3-3.5 µ m ( ∼ 0.39-0.46 µ mrest-frame; Fig. C.2 in the Appendix 2 ), is dominated by the southernmost of the two main components, and only an extended emission towards the northernmost source is present at these wavelengths, instead of a more clearly separate spatial component as at lower wavelengths. Therefore, the northernmost source has bluer continuum than the southernmost one. In the SW corner of the FOV, another galaxy at a lower redshift is also present. \nFig. 4, bottom panel, shows the integrated spectrum (aperture radius = 0.15 '' ) associated with the brightest component in continuum emission, COSMOS24108-a. The rest-frame optical lines [O ii ], H β , [O iii ], H α , and [S ii ] are detected with high S / N ( ≳ 4 on the peak). We obtain a stellar mass of log( M ∗ / M ⊙ ) ∼ 9.3 from SED fitting and a SFR ∼ 15 M ⊙ yr -1 from H α for COSMOS24108-a (Table 3). The integrated spectra of the other spatial components in the system are shown in Fig. C.1 in the Appendix 2 . For these, we find log( M ∗ / M ⊙ ) ∼ 8.4-8.9 and SFR ∼ 2-15 M ⊙ yr -1 . \nThe [O iii ] ionised gas line emission (Fig. 4, central panel; H β , [O ii ], and H α maps are reported in Fig. C.2 2 ) is much more extended than the continuum, revealing two additional bright clumps to the SE of the system, one of the two also tentatively detected in continuum (see contours). We label these as COSMOS24108-[OIII]-Ea and [OIII]-Eb. The H α map (Fig. C.2, bottom-left) shows a bridge of gas (visible also in [O iii ], though weaker relative the rest of the emission) connecting the main system to the closer of these two clumps, COSMOS24108-[OIII]-Ea. By comparing the ionised gas line emission with the 1-2 µ m continuum, we see that the former peaks in between the two main continuum components, with a preferential extension towards the northernmost, weaker continuum component (COSMOS24108-b) rather than to the southernmost, brighter one (COSMOS24108-a). \nFig. 4, right panel, reports the map of gas metallicity (the [O iii ] λ 5008 / H β and [O iii ] λ 5008 / [O ii ] λλ 3727,30 line ratio maps are shown in Fig. C.2 2 ). The metallicity exhibits differences of up to ∼ 0.5-0.6 dex among the di ff erent sources in the system. The northernmost of the two main continuum sources, COSMOS24108-b, appears to have lower metallicity (12 + log (O / H) ∼ 8.0-8.1) than the southernmost one, COSMOS24108-a ( ∼ 8.2), while the SE minor continuum component, COSMOS24108-c, has larger metallicity ( ∼ 8.4). Towards the two ionised gas clumps to the SE, COSMOS24108- \nFig. 3. Maps for BDF-3299 from JWST NIRSpec IFU. The flux maps are obtained from the original data cube, while the line ratio and metallicity maps from the spatially smoothed data cube (details in Sect. 2.2). Median stellar continuum emission in the observed spectral range 1.1-2 µ m ( ∼ 0.14-0.25 µ m rest-frame; top-left). The rest is as in Fig. 2. The PRISM / CLEAR spectrum (bottom) is extracted from a circular aperture with radius of 0.1 '' centred at the location of the most luminous component in continuum, BDF-3299-a. \n<!-- image --> \nFig. 4. Maps for COSMOS24108 from JWST NIRSpec IFU. Same as in Fig. 3. The continuum map (left) is the median in the 1-2 µ m observed range ( ∼ 0.14-0.27 µ m rest-frame). The PRISM / CLEAR spectrum (bottom) is extracted from a circular aperture with radius of 0.15 '' centred at the location of the most luminous component in continuum, COSMOS24108-a. \n<!-- image --> \n[OIII]-Ea and COSMOS24108-[OIII]-Eb, the metallicity is the lowest, with values of ∼ 7.8-7.9.", '6. Integrated mass-metallicity relation': "We obtained the properties of each sub-source belonging to each target, by extracting integrated spectra from circular apertures of radius of 0.15 '' for A2744-YD4 and COSMOS24108 and of 0.1 '' \nfor BDF-3299 centred on each sub-source (Table 3). We find emission line ratios of log([O iii ] / H β ) ∼ 0.4-0.9 for A2744-YD4, log([O iii ] / H β ) ∼ 0.7-0.8 for BDF-3299, and log([O iii ] / H β ) ∼ 0.5-0.9 and log([S ii ] / H α ) ≲ -0.8 for COSMOS24108. These are consistent with the range of values found for z ∼ 6-8 starforming galaxies in the JADES survey (Cameron et al. 2023), of log([O iii ] / H β ) ∼ 0.5-0.8 and log([S ii ] / H α ) ≲ -0.8 (when only considering detections and not upper limits), in a range of stel- \n10.0 \n<!-- image --> \nFig. 5. Star formation rate versus stellar mass (left) and gas-phase metallicity versus stellar mass (right) for the single spatial components in the systems from this work (red, green, and blue circles). In grey circles we show for comparison the compilation of z ∼ 3-10 galaxies from JADES (Curti et al. 2023b; including GNz-11, Bunker et al. 2023), CEERS (Nakajima et al. 2023, as re-computed for consistency in Curti et al. 2023b), and EROs (Curti et al. 2023a; Laseter et al. 2024). Left: The main sequences of star formation (SFMS) at z ∼ 0 and z ∼ 8 from Popesso et al. (2023) are displayed for reference. Right: Best-fit mass-metallicity relations (MZRs) from Curti et al. (2023b) for these z ∼ 3-10 targets, Li et al. (2023) at z ∼ 3 (GLASS-JWST), Sanders et al. (2021) at z ∼ 3.3 and 2.3 (MOSDEF), and Curti et al. (2020b) at z ∼ 0.08 (SDSS). The highz MZRs predicted by EAGLE ( z ∼ 8; Schaye et al. 2015), FIRE ( z ∼ 6; Ma et al. 2016), IllustrisTNG ( z ∼ 8; Torrey et al. 2019), FirstLight ( z ∼ 6 and z ∼ 8, for log( M ∗ / M ⊙ ) ≲ 9 at z ∼ 6; Langan et al. 2020; and z ∼ 8, for log( M ∗ / M ⊙ ) ≳ 9 at z ∼ 6; Nakazato et al. 2023), and SERRA ( z ∼ 8; Pallottini et al. 2022) cosmological simulations are also displayed. \n<!-- image --> \nlar masses (log( M ∗ / M ⊙ ) ∼ 6.5-9.0) and SFRs ( ∼ 0.1-30) comprising those of the sources in this work. In some cases, like COSMOS24108-a with log([O iii ] / H β ) ∼ 0.8 and log([S ii ] / H α ) ∼ -0.8, the ratios are at the high end of the values reported in (Cameron et al. 2023). \nWe inferred the gas-phase metallicity and the SFR from emission-line fitting and the stellar mass from SED fitting, as described in Sect. 2.2. In Fig. 5, we display the metallicity versus SFR and stellar mass diagrams for the spatial components identified in the images (Table 3). We compare these with the values reported by Curti et al. (2023b) for the z ∼ 3-10 galaxies from the JADES (including GNz-11; Bunker et al. 2023), CEERS(Nakajima et al. 2023, re-computed by Curti et al. 2023b for consistency), and EROs (Curti et al. 2023a; Laseter et al. 2024) samples (grey circles). The best-fit relation for this z ∼ 310 compilation obtained by Curti et al. (2023b) is also shown. We also report for reference the best-fit mass-metallicity relations (MZRs) for galaxies at z ∼ 0.08 from SDSS (Curti et al. 2020b), z ∼ 2-3 from MOSDEF (Sanders et al. 2021), and z ∼ 3 from GLASS-JWST (Li et al. 2023). \nOur targets are generally compatible with the values found for the z ∼ 3-10 galaxies, especially the sources in BDF-3299 which sit on the MZR defined by this highz collection. We note that most of the sources in A2744-YD4 and COSMOS24108 ( z ∼ 8 and 6, respectively) are at the high end of the JADES, CEERS, and EROs points, where the z ∼ 2-3 MZRs from MOSDEF and GLASS-JWST lie. This suggests that the A2744-YD4 and COSMOS24108 systems may comprise more evolved sources as compared to the majority of sources at the same redshifts and be instead more similar to galaxies at cosmic noon. \nWe further display the predictions for the highz MZR from a number of cosmological simulations, specifically EAGLE ( z ∼ 8; Schaye et al. 2015), FIRE ( z ∼ 6; Ma et al. 2016), IllustrisTNG ( z ∼ 8; Torrey et al. 2019), FirstLight (from Langan et al. 2020 at z ∼ 6 and z ∼ 8 and Nakazato et al. 2023 at z ∼ 8, for log( M ∗ / M ⊙ ) \nTable 4. Metallicity gradients inferred for the main sources in this work. \n≲ 9 and ≳ 9, respectively, at z ∼ 6), and SERRA ( z ∼ 8; Pallottini et al. 2022) 3 . We note that the sources analysed in this work, both as a whole and within the individual A2744-YD4 and COSMOS24108 systems, are better aligned with simulations predicting steeper MZR slopes than the median of the observations at z ∼ 3-10.", '7. Gas-phase metallicity gradients': "To more robustly constrain the spatially resolved metallicity, we extracted the line fluxes from spatially integrated spectra from concentric radial annuli, as described in Sect. 2.2. This allowed us to increase the S / N of the emission lines at large radii and determine the dust extinction from the faint Balmer lines. We thus determined the gas-phase metallicity as a function of the distance from the centre of each source, as defined in the following. \nThe resulting metallicity gradients are reported in Fig. 6. In the system A2744-YD4, we determined the metallicity gradients only for the two brightest sources YD4 and YD1 (cen- \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 6. Metallicity gradients for the main sources in A2744-YD4 (top two rows), BDF-3299 (mid-bottom), and COSMOS24108 (bottom). Radial annuli have radial width of 1 spaxel (0.05 '' ). The wavelength-dependent spatially smoothed cube was used (details in Sect. 2.2). Metallicity gradients (right) and [O iii ] maps (left; also from smoothed cube), where the concentric annuli are reported on the source (the shaded region in the COSMOS24108 map was masked out from the gradient; see text). The vertical grey dashed line marks the spatial resolution ( σ res . ). The fluxes of emission lines far in wavelength are corrected for extinction based on H γ / H β in A2744-YD4 and BDF-3299, while in COSMOS24108 H α / H β is used for that and also [S ii ] and H α are used to infer the metallicity. A cut of S / N > 3 on the peak flux of each line is applied. \n<!-- image --> \nCOS24108-main-a \ntred around their continuum peak) since the metallicity measurements for the other sources su ff ered from large uncertainties due to low S / N of emission lines. For BDF-3299, since we could have only extracted two radial points for each spatial component (BDF-3299-a, b, and c) due to the compactness of this system, we obtained the metallicity gradient profile for the whole BDF-3299 system, centred on the brightest source in line emission, BDF-3299-b. For the COSMOS24108 system, we centred the gradient around the continuum emission peak, corresponding to the spatial component COSMOS24108-a (Fig. 4, left panel). We masked the region surrounding the low-metallicity [O iii ] clump COSMOS24108-Ea (shaded circle in Fig. 6), since it has a markedly low metallicity (see Fig. 4, right panel) and can be clearly spatially separated from the rest of the system. Given that the line emission peaks between the two main continuum components, COSMOS24108-a and COSMOS24108-b, and not on either of them (see Fig. 4, central panel), we also extracted the gradient centred around the [O iii ] peak, instead of around the continuum. This is reported in Fig. C.3 in the Appendix 4 . The two gradients are not statistically di ff erent within the uncertainties. In Fig. C.3 we also report the metallicity gradient from two additional smaller apertures centred on the emission line clumps to the SE of the main system, COSMOS24108-Ea and COSMOS24108-Eb. \nFor all the sources, A2744-YD4, YD1, BDF-3299, and COSMOS24108, the metallicity gradients span a range between ∇ r log( Z ) ∼ -0 . 05 dex kpc -1 and ∇ r log( Z ) ∼ 0 . 15 dex kpc -1 (Table 4). We note that, within the uncertainties, all metallicity gradients in our sample are flat. \nWe also estimate the gradients relative to the e ff ective (halflight) radius, R e, as sometimes done in observational (e.g. Belfiore et al. 2017) and theoretical studies (e.g. Mollá et al. 2019); in this case, the gradient should not depend on other properties of galaxies (as suggested by e.g. Garnett et al. 1997). We estimated the e ff ective radius with a 2D Gaussian modelling of the continuum emission for the whole BDF-3299 and COSMOS24108 systems (including all the spatial components in continuum), and for the single YD4 and YD1 components, for consistency with the way metallicity gradients were calculated; for the same reason, we employed the wavelength-dependent spatially smoothed data cube, as done for the gradients. We stress that, therefore, the R e thus obtained should not be considered as the intrinsic ones (i.e. de-convolved for the spatial resolution of the observations), and, in the case of BDF-3299 and COSMOS24108, they are not the R e of the single spatial clumps. The obtained R e are ∼ 0.74, 0.70, 0.75, and 1.2 kpc for YD4, YD1, BDF-3299, and COSMOS24108, respectively. The resulting gradients are reported in Table 4. These are flat, and in any case ≳ -0.1 dex R -1 e when considering the uncertainties; in the local Universe, such gradients are typical of low-mass galaxies (log( M ∗ / M ⊙ ) ≲ 9.5-10; Belfiore et al. 2017).", '8.1. Driving mechanisms of the observed gradients': "We find flat gradients within the uncertainties for the sources in A2744-YD4, BDF-3299, and COSMOS24108 analysed in this work. As introduced in Sect. 1 and sketched in Fig. 1, negative (radially decreasing) gradients are expected as a consequence of the inside-out galaxy formation scenario, while inverted (radially \nincreasing) gradients may arise as a consequence of pristine gas accretion towards the central regions. Possible explanations for flat radial gradients are radial gas mixing processes, occurring as a consequence of mergers, SN-driven winds and / or large-scale gas circulation (galactic fountains, i.e. metal-loaded outflows expelled from the galaxy and re-accreted in the external regions), which redistribute the gas in the galaxy and wash out any preexisting radial gradient of metallicity. \nAll the targets in this work show multiple sources within a few kpc and disturbed morphologies. This indicates that they are experiencing, or have experienced, interactions and galaxy merging processes, which are expected to be more frequent in the early Universe according to cosmological simulations (see e.g. Kohandel et al. 2020; Pallottini & Ferrara 2023). Specifically, galaxies at z ∼ 6 with stellar masses ≳ 5 × 10 8 M ⊙ living in dense environments, similar to those studied in this work (see Table 3), are expected to have already experienced multiple merger events (Gelli et al. 2020). On the other hand, while in the lowspectral resolution ( R ∼ 100) NIRSpec PRISM data analysed in this work the emission lines are spectrally unresolved, [O iii ] and H β do not show evidence for asymmetric wings or any complex line profile indicative of outflows in the higher resolution ( R ∼ 2700) NIRSpec IFU or MSA data from Hashimoto et al. (2023) and Morishita et al. (2023), respectively, for the case of YD4, YD1, and s1 in the A2744-YD4 system (YD4 having both IFU and MSA R ∼ 2700 data). Therefore, this indicates that strong outflows are not occurring at present in this system. However, in principle we cannot exclude that the redistribution of metals across the galaxy may also be the result of past intense SN winds mixing the ISM or re-accretion in the outer regions of metalenriched gas expelled by past galactic outflows, whose trace is absent in present-day spectra. All in all, mergers and / or possibly either past galactic outflows or SN wind mixing seem the most likely mechanism driving the observed flat metallicity gradients. \nWestress that, even if we did our best in extracting the metallicity gradient for each separate spatial component, complications arise because in some cases these sub-sources are very close to each other and may contaminate each other's metallicity (as appears to be the case of YD1 and YD1-E; Fig. 2) or there is confusion as to which source the detected gas belongs (as in the case of COSMOS24108-a and COSMOS24108-b, where gas is located between the two sources). Moreover, the sources are not settled in radially symmetric metallicity distributions. All these aspects can contribute to yielding a flat gradient. Nevertheless, the spatial vicinity of the sources and the displacement of gas and stars are the result of the ongoing merging processes. Therefore, ultimately, mergers appear to be the most likely cause for the flatness of the metallicity gradients.", '8.2. Metallicity gradients across cosmic time': "In Fig. 7, we report the metallicity gradients from a compilation of observational studies, from z = 0 up to z ∼ 8 (this study), namely the highest redshift probed so far. We report the gradients in dex kpc -1 , given that most values reported in literature are in these units rather then in dex R -1 e . The flat gradients within the uncertainties that we find at z ∼ 6-8 are compatible with the other currently available measured gradients at these redshifts, namely Vallini et al. (2024) from FIR lines with ALMA and Arribas et al. (2023) from rest-frame optical lines with JWST / NIRSpec. Specifically, the results from Vallini et al. (2024) showcase the potential for synergies between spatially resolved observations with JWST and ALMA, though the spatial resolutions may be \n<!-- image --> \nAge of the Universe [Gyr] \nFig. 7. Gas-phase metallicity gradient (in dex kpc -1 ) as a function of redshift (bottom x axis) and cosmic time (upper x axis). The targets analysed in this work are marked with red (A2744-YD4 and YD1), green (BDF-3299), and blue (COSMOS24108). The other symbols are a compilation of gas-phase metallicity gradients from literature. Specifically, at z ∼ 0, we report the measurements from Rupke et al. (2010c, full and empty squares for isolated and merging galaxies, respectively) and (Belfiore et al. 2017, MaNGA, median). The evolution of metallicity gradients with cosmic time for the Milky Way, M33, and M81 from H II regions ( z ∼ 0) and PNe (up to a few Gyr ago), from Magrini et al. (2009, 2010), Stanghellini & Haywood (2010, 2018), and Stanghellini et al. (2014) is shown. We report the gradients estimated at various z from Yuan et al. (2011), Swinbank et al. (2012, HiZELS), Queyrel et al. (2012, MASSIV, median), Stott et al. (2014, KMOS-HiZELS, median), Troncoso et al. (2014, AMAZE + LSD), Jones et al. (2013, 2015, GLASS), Leethochawalit et al. (2016, CASSOWARY), Wuyts et al. (2016, KMOS 3D ), Carton et al. (2018), Wang et al. (2017, 2019, 2020, 2022, GLASS + GLASS JWST), Förster Schreiber et al. (2018, SINS / zC-SINF), Curti et al. (2020a, KLEVER), Gillman et al. (2021, KROSS + KGES, median), Simons et al. (2021, CLEAR), (Li et al. 2022, MAMMOTH-Grism), Vallini et al. (2024), and Arribas et al. (2023, GA-NIFS). Hatched regions are reported in case of large samples. The predictions for the evolution of metallicity gradients from cosmological simulations are also shown, specifically for MUGS (enhanced feedback) and MaGICC (normal feedback) from Gibson et al. (2013), FIRE (Ma et al. 2017), Taylor & Kobayashi (2017), TNG50 (Hemler et al. 2021), and EAGLE (Tissera et al. 2022), together with the predicted evolution from the chemical evolution models from Mott et al. (2013, for both constant and variable SFE), Kubryk et al. (2015), Mollá et al. (2019), and Sharda et al. (2021). \n<!-- image --> \nquite di ff erent ( σ ∼ 0.2-0.4 kpc in our case versus ∼ 1.5 kpc in the case of Vallini et al. 2024). \nWe also report the predictions for the evolution of metallicity gradients with redshift from cosmological simulations. Some of them predict steeper negative gradients with increasing redshift (MUGS, with normal feedback, Gibson et al. 2013; TNG50, Hemler et al. 2021), in some cases extremely steep (Taylor & Kobayashi 2017, for which the case of a log( M ∗ / M ⊙ ) = 12 galaxy at the center of a cluster with no AGN is shown), while \nothers predict median flatter gradients at higher z though with increasing scatter (MAGICC, with enhanced feedback, Gibson et al. 2013; FIRE, Ma et al. 2017; EAGLE, Tissera et al. 2022). Predictions from chemical evolution models, also displayed in Fig. 7, are very heterogeneous as well, with little to no evolution with redshift (Mott et al. 2013 for the Milky Way, with variable SFE, and Sharda et al. 2021, for which the log( M ∗ / M ⊙ ) = 11.1 case is shown) and slightly to steeply increasing gradients with z (Mollá et al. 2019 and Kubryk et al. 2015, respectively, both \nfor the Milky Way), as well as gradients inverting at z ≳ 1 (Mott et al. 2013 with constant SFE). \nUnfortunately, predictions for the cosmic evolution of metallicity gradients from cosmological simulations and models are limited to z ≲ 3-4 so far. If we assumed that the trend with redshift follows a simple linear extrapolation to z ∼ 8 of the negative trends from TNG50 (Hemler et al. 2021) and MUGS (Gibson et al. 2013), as well as the model by Kubryk et al. (2015), we would get metallicity gradients that are incompatible with our estimates considering the uncertainties (Fig. 7), which would instead be more in line with flat gradient predictions at high z (MaGICC, Gibson et al. 2013; FIRE, Ma et al. 2017; EAGLE, Tissera et al. 2022; Mott et al. 2013 with variable SFE; Sharda et al. 2021). However, this very rough linear extrapolation is most likely wrong, given that metals are expected to start forming from uniformly distributed, pristine gas and therefore the gradients should converge to zero at some point in the past. The only case which extends beyond z ∼ 3-4 is the simulations of Taylor & Kobayashi (2017), which reach up to z ∼ 6, where gradients even steeper than -1.5 dex kpc -1 (not shown in the figure to avoid an excessive shrink of the y axis) are predicted, way steeper than those observed at z ∼ 6-8; however, even in this case, the redshifts up to 8 probed by the current observations are not explored. \nAll in all, we cannot draw any clear conclusion on what cosmological simulations and models best reproduce the observed gradients at high z . Instead, we stress the need for updated metallicity gradient predictions from cosmological simulations and chemical models at z ≳ 3 in order to match the redshifts reached by current observations. \nWe point out that the merger fraction increases with redshift (e.g. theoretically Fakhouri et al. 2010; Rodriguez-Gomez et al. 2015; O'Leary et al. 2021; observationally Duncan et al. 2019), therefore galaxies at high redshift often have disturbed morphologies due to interactions and do not have radially symmetric gas and chemical distributions (see e.g. Figs. 2, 3, and 4; also e.g. observations in de Graa ff et al. 2023; Arribas et al. 2023; Rodríguez Del Pino et al. 2023 and simulations in Pallottini et al. 2017, 2019). Specifically, the metallicity maps in our work show in some cases non-azimuthally symmetric structures which are averaged out when producing radial gradients (as it is e.g. the case of YD1, having the highest values to the W and the lowest to the E of its spatial emission peak). Therefore, radial gradients may not be the optimal means to describe the distribution of metals in highz galaxies and other alternative methods to quantify it (such as the metallicity scatter in the galaxy, as suggested by Maiolino & Mannucci 2019) should be sought and included in predictions from simulations. \nIn summary, the flat radial gradients and the asymmetric spatial distribution of metallicity observed in some cases seem to support the merging scenario, though ISM mixing due to past SN winds or re-accretion in the outer regions of metal-enriched gas from past galactic outflows (galactic fountains) are also a viable mechanism to explain the flatness of the gradients. Spatially asymmetric low-metallicity accretion from the circumgalactic medium (CGM) and intergalactic medium (IGM) could also contribute to produce non-azimuthally symmetric metallicities as suggested by Arribas et al. (2023).", '9. Conclusions': "In this work, we have presented new JWST NIRSpec IFU observations at low spectral resolution ( R ∼ 100; PRISM / CLEAR) of three highz systems, namely A2744-YD4 in the proto-cluster \nA2744-z7p9OD ( z ∼ 7.88), BDF-3299 ( z ∼ 7.11), and COSMOS24108 ( z ∼ 6.36). At these redshifts, the NIRSpec PRISM spectra (spanning ∼ 0.6-5.2 µ m) cover the rest-frame UV and optical spectral ranges and include the main optical emission lines from warm ( T ∼ 10 4 K) ionised gas, [O ii ], H γ , H β , [O iii ], and, for COSMOS24108, also H α and [S ii ]. We modelled the ionised gas lines in the spectra with Gaussian functions and mapped their emission. The main goal of this work is to study the spatially resolved gas-phase metallicity and investigate the shape of metallicity gradients at high redshift ( z ≳ 6). \nThe targets have very disturbed morphologies in both the stellar continuum and ionised gas line emission. We identify several spatial components concentrated within a few kiloparsecs in all the three systems. We obtained the main integrated properties of each spatial component in each target, specifically stellar mass from SED fitting, SFR from H α or H β , and gasphase metallicity by means of flux ratios of rest-frame optical emission lines which we modelled with Gaussian functions. We found log( M ∗ / M ⊙ ) ∼ 7.6-9.3, SFRs ∼ 1-15 M ⊙ yr -1 , and gasphase metallicities 12 + log(O / H) ∼ 7.7-8.3 by extracting integrated spectra from circular apertures centred on each spatial component. \nWe compared the stellar masses and gas-phase metallicities of the targets with the mass-metallicity relations (MZRs) inferred observationally at di ff erent redshifts and with those predicted by cosmological simulations. In general, the sources in the systems studied in this work are consistent with the distribution of the highestz galaxies to date ( z ∼ 3-10) from JADES, CEERS, and EROs samples in the mass-metallicity plane. Nevertheless, most of the sources in A2744-YD4 and COSMOS24108 lie at the upper end of this z ∼ 3-10 distribution, in terms of metallicity at a given stellar mass, and are closer to the best-fit MZRs measured at z ∼ 2-3 (e.g. MOSDEF and GLASS-JWST surveys) than to the MZR at z ∼ 3-10. Relative to the MZRs from cosmological simulations, the sources in the three systems studied in this work are in good agreement with the slopes predicted at z ∼ 6-8 by most simulations. \nWe inferred the gas-phase metallicity radial gradients by extracting integrated spectra from concentric radial annuli centred on the main sources of each system for which the S / N allowed for a robust metallicity estimate. The gas-phase metallicity gradients are flat within the uncertainties. Flat gradients can be associated with processes which mix the gas and the metals on galaxy scales and therefore wash out any gradient which may have formed in the galaxy, such as mergers, SN wind mixing, and re-accretion of metal-loaded galactic outflows in the outer regions. \nThese are among the very few measurements of spatially resolved gas metallicity at these high redshifts ( z ∼ 6-8). In particular, YD4 (at z ∼ 8) constitutes the highestz source in which a metallicity gradient has been probed so far. \nCosmological simulations and chemical evolution models make very di ff erent predictions regarding the cosmic evolution of metallicity gradients. Some of them predict steeper negative gradients with increasing redshift (e.g. MUGS and TNG50), while others predict median flatter gradients at higher z (e.g. MaGICC, FIRE, and EAGLE). Unfortunately, these predictions are generally limited to z ≲ 3, therefore no conclusions can be drawn on what simulations best reproduce the observed mainly flat gradients at z ∼ 6-8 found in this work. \nAll in all, the results of this work in terms of the MZR and gas-phase metallicity gradients at z ∼ 6-8 provide important constraints to guide future cosmological simulations and models. In particular, they urge for specific predictions on the cosmic evolu- \non of metallicity gradients and metallicity maps out to redshift of 8, given that most of such predictions are currently limited to z ≲ 3. \nMoreover, galaxies at high z tend to have more irregular morphologies and may not have azimuthally symmetric chemical distributions, as a result of frequent mergers and asymmetric low-mass gas infall. Radial gradients, which average out any azimuthal information, may then not be the optimal means to quantify the distribution of metals in highz galaxies. Therefore, other alternative quantitative tracers of the spatially resolved metallicity should be considered as part of predictions from simulations, to be compared with observations. \nAcknowledgements. We acknowledge support from European Union's HE ERC Starting Grant No. 101040227 - WINGS (G.V., S.C.), INAF Minigrant 'Reionization and fundamental cosmology with high-redshift galaxies' (M.C.), 'FirstGalaxies' Advanced Grant from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program, Grant agreement No. 789056 (A.J.B.), ERC Advanced Grant INTERSTELLAR H2020 / 740120 (A.F.), grant PID2021-127718NB-I00 funded by the Spanish Ministry of Science and Innovation / State Agency of Research, MICIN / AEI / 10.13039 / 501100011033 (S.A.). This work is based on observations made with the NASA / ESA / CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program #1893. 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Highest-resolution continuum map of A2744-YD4 from the original unsmoothed data cube in observed spectral range 1.2-2 µ m, where the spatial resolution is the highest (top-left). All the other maps are obtained from the (wavelength-dependent) spatially smoothed data cube, as in Fig. 2 (details in Sect. 2.2). First row: H β (centre) and [O ii ] (right) integrated fluxes. Second row: [O iii ] / H β (left) and [O iii ] / [O ii ] (right) line flux ratios. Contours mark the 2-3 µ m(observed-frame) continuum. \n<!-- image -->", 'Appendix B: Additional figures for BDF-3299': "Fig. B.1. NIRSpec PRISM / CLEAR spectra extracted from a circular aperture with radius of 0.1 '' centred at the location of the targets BDF3299-b and BDF-3299-c. \n<!-- image --> \nFig. B.2. Additional maps for BDF-3299. As in Fig. 3, the flux maps are obtained from the original data cube, while the line ratio maps from the spatially smoothed data cube (details in Sect. 2.2). Top row: maps of H β integrated flux (left), [O iii ] / H β (centre) and [O iii ] / [O ii ] (right) line flux ratios. \n<!-- image -->", 'Appendix C: Additional figures for COSMOS24108': "Fig. C.1. NIRSpec PRISM / CLEAR spectra extracted from a circular aperture with radius of 0.15 '' centred at the location of the targets COSMOS24108-b, COSMOS24108-c, COSMOS24108-[OIII]-Ea, and COSMOS24108-[OIII]-Eb. \n<!-- image --> \nFig. C.2. Additional maps for COSMOS24108. As in Fig. 4, the flux maps are obtained from the original data cube, while the line ratio maps from the spatially smoothed data cube (details in Sect. 2.2). Top row: median continuum map in the observed spectral range 3-3.5 µ m( ∼ 0.39-0.46 µ m rest-frame; left), with the contours of the continuum in the observed spectral range 1-2 µ m(same as in Fig. 4, left) superimposed; H β (centre-left), [O ii ] (centre-right), and H α (right) flux maps. Middle row: [O iii ] / H β (left) and [O iii ] / [O ii ] (centre-left) line ratio maps. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. C.3. Metallicity gradients for the two minor sources in COSMOS24108, [O iii ]-Ea and [O iii ]-Eb (top and mid panels, respectively), and for the main system (bottom panel), when centring the gradient around the peak of the [O iii ] emission instead of around the continuum peak as in Fig. 6. \n<!-- image -->"}
2024ARA&A..62...21M	Massive stars play a major role in the evolution of their host galaxies and serve as important probes of the distant Universe. It has been established that the majority of massive stars reside in close binaries and interact with their companion stars during their lifetimes. Such interactions drastically alter their life cycles and complicate our understanding of their evolution but are also responsible for the production of interesting and exotic interaction products. Extensive observation campaigns with wellunderstood detection sensitivities have enabled the conversion of observed properties into intrinsic characteristics facilitating a direct comparison to theory. Studies of large samples of massive stars in our Galaxy and the Magellanic Clouds have unveiled new types of interaction products providing critical constraints on the mass transfer phase and the formation of compact objects. The direct detection of gravitational waves has revolutionized the study of stellar mass compact objects providing a new window to study massive star evolution. Their formation processes are however still unclear. The known sample of compact object mergers will increase by orders of magnitude in the coming decade which is vastly outgrowing the number of stellarmass compact objects detected through electromagnetic radiation. Extensive observation campaigns with wellunderstood detection sensitivities have enabled the conversion of observed properties into intrinsic characteristics facilitating a direct comparison to theory. Studies of large samples of massive stars in our Galaxy and the Magellanic Clouds have unveiled new types of interaction products providing critical constraints on the mass transfer phase and the formation of compact objects. The direct detection of gravitational waves has revolutionized the study of stellar mass compact objects providing a new window to study massive star evolution. Their formation processes are however still unclear. The known sample of compact object mergers will increase by orders of magnitude in the coming decade which is vastly outgrowing the number of stellarmass compact objects detected through electromagnetic radiation.	2024-09-01T00:00:00Z	['10.1146/annurev-astro-052722-105936', '2024ARA&A..62...21M', 'arXiv:2311.01865', '10.48550/arXiv.2311.01865', '2023arXiv231101865M']	['massive stars', 'stellar evolution', 'rotation', 'compact objects', 'gravitational waves', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - High Energy Astrophysical Phenomena']	The Evolution of Massive Binary Stars	2,024	202	0.68	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	37	https://arxiv.org/pdf/2311.01865.pdf	{'No Header': 'Xxxx. Xxx. Xxx. Xxx. 2023. AA:1-42 \nhttps://doi.org/10.1146/((please add article doi)) \nCopyright © 2023 by the author(s). All rights reserved', 'Pablo Marchant, 1 and Julia Bodensteiner 2': '1 Institute of Astronomy, KU Leuven, Celestijnlaan 200D, 3001 Leuven, Belgium; email: [email protected]. \n2 ESO - European Organisation for Astronomical Research in the Southern Hemisphere, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany; email: [email protected].', 'Keywords': 'massive stars, binary star, stellar evolution, rotation, compact objects, gravitational waves', 'Abstract': 'Massive stars play a major role in the evolution of their host galaxies, and serve as important probes of the distant Universe. It has been established that the majority of massive stars reside in close binaries and will interact with their companion stars during their lifetime. Such interactions drastically alter their life cycles and complicate our understanding of their evolution, but are also responsible for the production of interesting and exotic interaction products. \n- · Extensive observation campaigns with well-understood detection sensitivities have allowed to convert the observed properties into intrinsic characteristics, facilitating a direct comparison to theory.\n- · Studies of large samples of massive stars in our Galaxy and the Magellanic Clouds have unveiled new types of interaction products, providing critical constraints on the mass transfer phase and the formation of compact objects.\n- · The direct detection of gravitational waves has revolutionized the study of stellar mass compact objects, providing a new window to study massive star evolution. Their formation processes are, however, still unclear. The known sample of compact object mergers will grow by orders of magnitude in the coming decade, turning into the best understood astrophysical population.', '1. INTRODUCTION': "Massive stars are powerful cosmic engines, capable of modifying their local, galactic and even extragalactic environments (Hopkins et al. 2014). Through their high luminosities, which can include a significant fraction of ionizing radiation, they are understood to play a critical role in the re-ionization of the Universe (Haiman & Loeb 1997). Radiation driven stellar winds can remove large fractions of the stellar birth mass (Vink 2022), providing kinetic feedback and chemically processed matter. At the end of their lives, energetic supernova explosions (SNe) further enrich their surroundings with matter that has undergone late nuclear burning stages (Nomoto et al. 2013), acting as one of the main drivers of the chemical evolution of galaxies. Although rare objects, massive stars dominate the integrated light of distant galaxies (e.g. Pettini et al. 2000). Yet, despite their broad astrophysical importance, various physical processes that dominate their evolution are still largely uncertain (see Langer 2012 for a recent review). \nOne critical aspect that undermines the understanding of massive stars is their scarcity. In the local Universe < 1% of stars born are expected to be massive (Kroupa 2002), which is further compounded with their orders of magnitude shorter lifetime compared to intermediate and low-mass stars. But another critical complication arises from the prevalence \nof close binaries in massive stars. For several decades multi-epoch spectroscopic observations have indicated that a large fraction of massive stars have close binary companions (Levato et al. 1987, Abt et al. 1990, Kobulnicky & Fryer 2007). By carefully accounting for observational biases it has been shown that the majority of massive stars undergo interaction phases that dominate their evolution, potentially with over half of them interacting before the end of the main-sequence (Sana et al. 2012). Detections made through photometry, astrometry or other techniques that cover different regimes not accessible by spectroscopy further indicate that a large number of stars have multiple companions (see Moe & Di Stefano 2017 for a recent compilation). Binary evolution adds on the complexity of massive star physics, providing a rich set of post-interaction products such as stripped stars (Shenar et al. 2020a, Drout et al. 2023), rapidly rotating accretors (de Mink et al. 2013, Renzo & Gotberg 2021), mergers (Schneider et al. 2019, Hirai et al. 2021), exotic supernovae (Chevalier 2012, Metzger 2022), X-ray binaries (Tauris & van den Heuvel 2006, Gilfanov et al. 2022) and double-degenerate binaries which may produce detectable gravitational wave (GW) emission (Tauris et al. 2017, Mandel & Farmer 2022). Observations of apparently single stars can hide a past of binary interaction, leading to an erroneous attribution of their physical properties to single-star physics. \nEven if theory can describe the intrinsic properties of a given stellar population, comparison to observations requires a clear understanding of the biases involved in sample selection and instrumental limitations. Large-scale surveys with clearly defined selection criteria play a critical role in this regard. This is the case for spectroscopic surveys such as the VLT-FLAMES Tarantula Survey (VFTS, see Evans et al. 2011, for an overview), photometric surveys such as Kepler (Borucki et al. 2010, Koch et al. 2010) and TESS (Ricker et al. 2015), and the astrometric measurements of the GAIA mission (Gaia Collaboration et al. 2016). The observation of transients through multi-band synoptic surveys has also provided breakthroughs on our understanding of SNe. For instance, the Zwicky Transient Facility (ZTF, Bellm et al. 2019) currently produces on the order of a million daily transient alerts, with dedicated infrastructure to perform spectroscopic follow-up and classification of the brightest ones (Fremling et al. 2020). Entire populations of high-mass X-ray binaries (HMXBs) can be probed in individual galaxies with X-ray observations (Gilfanov et al. 2022). However, the most significant development in the past decade did not involve electromagnetism, but rather the direct detection of gravitational waves (GWs) from merging compact objects (Abbott et al. 2016). Ground-based interferometers such as LIGO (Aasi et al. 2015) and Virgo (Acernese et al. 2015) have well-understood biases, making it straightforward to determine intrinsic population properties, or to apply biases synthetically to predicted populations. \nOwing to the large progress seen in the last decade, a review of the current state and standing problems in the field of massive binary evolution is of critical importance. However, a full comprehensive review is not possible within the scope of this document, and as such, our focus is on the progress that has been made since the review of Langer (2012) on single and binary massive-star evolution. Our scope is also limited to avoid overlap with the recent reviews of Eldridge & Stanway (2022) on the impact of massive binaries on the evolution of early galaxies, Kaaret et al. (2017) on ultraluminous X-ray sources and Murase & Bartos (2019) on multi-messenger astrophysics. We also exclude from the discussion the evolution of low-mass X-ray binaries, whose formation is potentially affected by stellar dynamics (eg. Ivanova et al. 2010). The field of transients and supernovae associated to binary evolution is also rapidly growing, and cannot be comprehensively covered in this review (for a recent \noverview, see chapter 13 of Tauris & van den Heuvel 2023). We do, however, soften the definition of 'massive star' 1 and include observations and findings for intermediate-mass stars when applicable to the massive star regime. \nThe outline of this review is as follows: In Section 2 we briefly mention the observational methods used to detect and characterize binaries, and discuss the constraints obtained on binary fractions and orbital parameters at different evolutionary stages. In Section 3 we review the main binary-interaction processes, and discuss the theoretical tools used to model them. We then discuss the properties of non-degenerate post-interaction products in Section 4, and how their observation constrains our theoretical models. In Section 5 we discuss the properties of single- and double-degenerate interaction products. In Section 6 we provide a brief review on the observations of gravitational-wave sources, and their potential formation through binary evolution. Finally, we conclude in Section 7.", '2. CONSTRAINTS ON BINARITY AT DIFFERENT EVOLUTIONARY STAGES': "The consensus that massive stars are predominantly part of binary or higher-order multiple systems has been established over the last decades using a multitude of different observations and detection techniques. Those probe different regions of the parameter space and suffer from different observational biases and limitations. \nPhotometric signatures such as eclipses and ellipsoidal variations can only be detected in the closest, shortest-period binaries, if their orientation towards the observer is favorable. Spectroscopic observations use radial velocity (RV) variations to detect binaries and constrain orbital parameters such as the period and eccentricity in the case of single-lined spectroscopic binaries (SB1s), where only one component, usually denoted as primary, is visible. Additionally, the mass ratio can be measured in double-lined spectroscopic binaries (SB2s), where the signature of both stars is discernible in the spectrum (it is usually defined as the mass of the less luminous star, the secondary, over the mass of the primary). Spectroscopic observations probe orbital periods that are of the order of the length of the observing campaign (usually up to a few years) but also suffer from severe observational biases depending on the system's parameters and orientation (e.g., Sana et al. 2012, and see Section 2.1). \nLong-baseline interferometry, which is still only technically feasible for brighter stars in our own Galaxy with current instrumentation, allows to probe binary systems that have angular separations between ∼ 1 and 100 mas, corresponding to orbital periods of the order of months to decades for Galactic distances (e.g., Le Bouquin et al. 2017, GRAVITY Collaboration et al. 2018, Bordier et al. 2022, Lanthermann et al. 2023). Similarly, highprecision long-term astrometric monitoring allows the detection of wider binaries. While barely any massive stars are included in the current Gaia data release 3 (DR3 Gaia Collaboration et al. 2021, see also Section 5.3), the fourth data release (DR4), expected towards the end of 2025, should improve this situation significantly and provide constraints on a large number of massive binaries. Binaries with even larger separations or more distant companions in higher-order multiples (with angular separations of ∼ 100 - 10 000 mas) can further be detected with high-angular resolution imaging techniques such as high-contrast \nT \nFigure 1 \n<!-- image --> \nBinary fractions and the index of the period distribution π for different classes of stars and evolutionary stages across the HRD. Stellar evolution tracks of different masses are shown, covering evolution from the zero-age MS until core-carbon depletion (or until just before the hydrogen envelope is removed in the AGB phase for the 5 M ⊙ and 7 M ⊙ models). The dashed line indicates the Humphreys-Davidson (HD) limit, above which there is a lack of observed stars (Humphreys & Davidson 1979). See text for references. \nor AO-supported imaging (e.g., Mason et al. 2009, Sana et al. 2014, Aldoretta et al. 2015, Kalari et al. 2022, Reggiani et al. 2022, Pauwels et al. 2023). \nGiven that binary interactions are thought to occur in close binaries with periods below ∼ 10 years (e.g., Podsiadlowski et al. 1992, and Section 3), we focus on spectroscopic surveys in the following, and only briefly mention complementary observations using other techniques. Thereby we go through different evolutionary stages, from the main sequence (MS) to further evolved stars. An overview of all reported binary properties is displayed in Figure 1, while references are provided in the text 2 .", '2.1. OB stars': 'Several spectroscopic studies targeting the multiplicity properties of OB stars at different metallicity were performed in recent years (for an overview, see Table 1). As most interactions occur after the MS when the primary expands (see Section 3), the properties of OB binaries are often considered as initial conditions and serve as input for other fields,', ':': "Binary fraction f bin The binary fraction is defined as f bin = N B /N with being N B the number of objects with at least one companion and N the number of objects. This can either be an observed or an intrinsic fraction. \nsuch as population synthesis computations (Eldridge et al. 2017) or GW progenitor studies (Belczynski et al. 2016). Multiplicity properties studied in the field as well as in cluster environments are described below. \nLarge magnitude-limited surveys of galactic OB stars in the field both in the Northern and Southern hemisphere such as IACOB or OWN showed that the observed binary fraction of O-type stars is around 55-65% (Abt et al. 1990, Sota et al. 2014, Barb'a et al. 2017, Britavskiy et al. 2019). For B-stars in the field, the observed binary fraction was reported to decrease down to 45-20 %, for early to late B-type stars, respectively (Chini et al. 2012). \nSpectroscopic surveys of OB stars in young star clusters in the Milky Way found similar observed binary fractions than the ones reported in the field (Sana et al. 2012, Kobulnicky et al. 2014, Banyard et al. 2022). While they target only a small peculiar region, given the similar distance to all stars, they can be considered volume-limited samples. The VLTFLAMES Tarantula Survey (VFTS), targeting the 30 Doradus star-forming complex in the LMC, (Evans et al. 2011, Sana et al. 2013, Dunstall et al. 2015) measured lower observed binary fractions of 25 ± 2% to 35 ± 3% for early-B and O-type stars. This is similar to the observed binary fraction measured in the young cluster NGC 346 in the SMC (Dufton et al. 2019). Using low-resolution spectroscopy, the observed binary fraction in NGC 330, a slightly older cluster in the SMC, was found to be even lower with only ∼ 13 ± 2% (Bodensteiner et al. 2020a). \nIn principle, the observed binary fractions give a lower limit on the intrinsic binary fractions. However, observed fractions are not directly comparable as different observing campaigns use different instruments and technical setups and their sensitivity to detect RV shifts thus varies. Moreover, there is a selection bias towards binary systems with similarly bright components rather than single stars at the faint end of magnitude-limited samples (Vanbeveren & Conti 1980). To obtain the true, intrinsic binary fraction, the observed binary fractions have to be bias-corrected using the probability of detecting binary systems with a given observing campaign. This not only requires an assumption on the underlying orbital parameter distributions, but also detailed knowledge of potential biases of the observing campaign (see e.g., Sana et al. 2012). Comparing the intrinsic binary fractions (where available) demonstrates that a vast majority of massive stars are members of close binary systems. Despite large error bars, there seems to be a trend between binary fraction and stellar mass. Whether this is indeed related to the mass, or potentially also to the metalliticy or the age of the cluster remains to be constrained by further work. \nMost surveys mentioned in Table 1 consider only the so-called SB1 bias (the chance to detect RV shifts of a given binary system larger than a given detection threshold, usually chosen to be 20 km s -1 ). An additional bias, the so-called SB2 bias, arises from unidentified SB2 systems that appear as rapidly rotating single stars if their RV shifts are too small for their spectral lines to effectively deblend (see Bodensteiner et al. 2021). This effect becomes more important for more rapidly rotating stars, for smaller RV variations, as well as for lower spectral-resolution data. Taking this bias into account will further increase some of the intrinsic binary fractions reported in Table 1. \nWhile some of the surveys mentioned in Table 1 are based on few epochs only (enough to measure RV variations and detect binaries), others had enough observations (i.e., ≳ 20 epochs) to fit binary orbits and constrain orbital parameters for SB1s and SB2s. Furthermore, additional observing campaigns were designed to follow-up previously detected binary systems, for example the MONOS survey for Galactic O-star binaries (Ma'ız Apell'aniz et al. 2019, Trigueros P'aez et al. 2021), or the TMBM (Almeida et al. 2017) and BBC programs \n(Villase˜nor et al. 2021) that followed O- and B-type binaries from the VFTS, respectively. \nThe exponent of the period distribution, which is defined as f (log P[days]) ∼ (log P) π , is reported to vary between π = -0 . 45 ± 0 . 39 for the Milky Way O stars (Sana et al. 2012) and π ∼ -0 . 2 for the LMC O stars (Almeida et al. 2017), which is close to flat on log P (see e.g., Banyard et al. 2022). The eccentricity distribution follows f ( e ) ∼ e η with η = -0 . 4 ± 0 . 2 (Sana et al. 2012). Covering a large range of mass ratios, Shenar et al. (2022b) reported that the mass-ratio distribution of VFTS binaries is also consistent with a flat distribution, that is f ( q ) ∼ q κ with κ = -0 . 2 ± 0 . 2. Comparing the observed orbital parameter distributions derived from detected binaries in different spectroscopic works implies them to be universal across metallicity and the considered stellar mass range (for a compilation, see figures 8 and 10 in Banyard et al. 2022). The orbital properties of close, eclipsing binaries in the Milky Way, LMC and SMC further corroborate that there are no statistically significant trends with metallicity (Moe & Di Stefano 2013). \nObservations have unambiguously shown that a large majority of O- and B-type MS stars are in binary or higher-order multiple systems. While the binary fraction seems to increase with stellar mass also outside the here considered mass range (for an overview, see e.g., Moe & Di Stefano 2017), the uncertainties are still large. The orbital parameter distributions seem universal across masses and metallicity, but more observations are required to further test this. The measured binary properties in young clusters and environments are thought to reflect the initial conditions, but their link to the outcome of star formation is not well understood (Duchˆene & Kraus 2013). Early processes such as inward migration or binary hardening seem to also play a role (e.g., Ram'ırez-Tannus et al. 2021, Bordier et al. 2022). The picture is further complicated by occurring interactions and post-interaction products polluting older populations and field stars (see e.g., Wang et al. 2020).", '2.2. OBe stars': 'OBe stars appear on average cooler and redder than their OB counterparts because of their rapid rotation and the IR excess from the disk (see Rivinius et al. 2013, for a recent review). While the formation mechanism of the OBe disk remains debated, different channels explaining the rapid rotation have been proposed. Those include their interpretation as single stars that approach critical rotation towards the end of the MS (e.g., Granada et al. 2013, Hastings et al. 2020b), or as mass gainers in previous binary interactions (e.g., Pols et al. 1991, Hastings et al. 2021). A potential way to distinguish those is to constrain OBe multiplicity properties: according to the single-star channel, OBe stars should have similar binary properties as OB stars. According to the binary channel, there should be a lack of MS companions and a significant number of stripped companions or compact objects. \nUnfortunately, multi-epoch spectroscopic surveys constraining close binary properties of large samples of classical OBe stars are lacking, in particular in comparison to works on OB stars (see Section 2.1). Additionally, OBe star spectra are complex and exhibit variability both in their disk emission and through stellar pulsations (e.g., Barnsley & Steele 2013, Baade et al. 2016, Labadie-Bartz et al. 2022). This complicates binary detections through spectroscopic and photometric techniques. \nAbt & Levy (1978) studied a small sample of B and Be stars, reporting similar binary properties, however based on low-quality data. High-angular resolution imaging and speckle interferometry studies reported a similar result, but their observations are not sensitive to short periods relevant for binary interactions (Oudmaijer & Parr 2010, Horch et al. 2020).', 'Classical OBe:': "Classical OBe stars are rapidly rotating, non-radially pulsating late-O and B-type stars whose spectra show, or have previously shown, emission lines that are formed in a gaseous circumstellar decretion disk. \nTable 1 Spectroscopic surveys of OB stars investigating binarity. The columns give the number of observed stars n ∗ , the age of the population for clusters and associations, the observed ( f obs bin ) and bias-corrected ( f intr bin ) spectroscopic binary fraction, the period P max up to which the observational biases are corrected for, the name of the survey, the environment (env.) and relevant references (Ref.). They are sorted by host galaxy. \nReferences: (1) Barb'a et al. (2017), (2) Sota et al. (2014), (3) Britavskiy et al. (2023), (4) Chini et al. (2012), (5) Abt et al. (1990), (6) Sana et al. (2012), (7) Banyard et al. (2022), (8) Kobulnicky et al. (2014), (9) Levato et al. (1987), (10) Sana et al. (2013), (11) Dunstall et al. (2015), (12) Dufton et al. (2019), (13) Bodensteiner et al. (2021) \nKlement et al. (2019) interpreted a detected turn-down in the spectral energy distributions (SEDs) of OBe stars as the presence of a companion truncating the disk. The nature of these companions remains unconstrained, however. Using multi-epoch low spectral resolution observations of the young SMC cluster NGC 330, Bodensteiner et al. (2021) reported a significant lower binary fraction of Be stars compared to B-type stars in the same cluster. \nConcerning Be star companions, based on a literature study, Bodensteiner et al. (2020c) pointed out a lack of close MS companions to massive Be stars 3 despite the fact that those companions are the easiest to detect and the most common companions to OB stars \n(Shenar et al. 2022b). In contrast, the sample of Be binaries with bloated stripped stars and subdwarf OB (sdOB) companions is continuously increasing (e.g., Wang et al. 2021, El-Badry & Quataert 2021, see Section 4.2). A well-studied class of Be binaries are Be X-ray binaries (BeXRBs) that are detected based on their X-ray emission (e.g., Raguzova & Popov 2005, Reig 2011, Coe & Kirk 2015, and see Section 5.1). \nOverall, the binary properties of OBe stars remain poorly constrained and the few measurements of orbital parameters are not enough to construct representative parameter distributions. The overall binary fraction of OBe stars seems to be lower than for OB stars while the companions are predominantly stripped stars or compact objects. This is in line with expectations of the binary channels of Be star formation, but further observations are required to consolidate or falsify this. On the lower-mass end (for late-type Be stars), the single-star channel might play a more important role (e.g., Kervella et al. 2008, Klement et al. 2021).", '2.3. Red supergiants': 'While the number of red supergiants (RSGs) in the Milky is low, they also show a high degree of photospheric variability (e.g., Kiss et al. 2006), hampering the detection of RV variability and the signature of companions. Given their large radii of up to 1500 R ⊙ , short-period binaries will not survive until the RSG phase and interact before, implying that there should be a lack of RSGs in close binaries, in contrast to their OB progenitors, and that several RSGs could actually be the rejuvenated products of previous interactions (dubbed red stragglers, Britavskiy et al. 2019). \nGiven the rarity of RSGs in our Galaxy, multiplicity studies mainly targeted RSGs in our neighboring galaxies. Compiling archival RV measurements of almost 1000 candidate RSGs in the LMC and SMC, Dorda & Patrick (2021) find a minimum binary fraction of 15 ± 3%. RSGs with OB companions can be detected based on signatures in the blue part of the spectrum (Neugent et al. 2018). Using this method and spectroscopic follow-up observations, Neugent et al. (2019) measured a bias-corrected binary fraction of 20 ± 7% for RSGs in the LMC. Patrick et al. (2019) investigated multi-epoch spectroscopy of 17 candidate RSGs in the 30 Dor region in the LMC observed in the context of the VFTS survey. Given that the detection probability for systems with periods up to 10 000 days and mass ratios > 0 . 3 is estimated to be almost 90%, the observed and intrinsic binary fractions are similar, namely ∼ 30%. Patrick et al. (2020) further investigated the binary properties of 15 RSGs in the 35-40 Myr old SMC clusters NGC 330 with multi-epoch spectroscopy. They report a bias-corrected binary fraction of 30 ± 10% for orbital periods between 2 . 3 < log P[days] < 4 . 3 and mass ratios > 0 . 1. \nBased on UV photometry, Patrick et al. (2022) measured an intrinsic binary fraction of 18 . 8 ± 1 . 5% for > 500 RSGs in the SMC for mass ratios > 0 . 3 and long periods above 1000 days. They find indications for a flat mass-ratio distribution, and a lack of high mass-ratio ( q > 0 . 5) systems above 15 M ⊙ (implying an absence of companions with similar masses to that of the RSG), which the authors interpreted as the most massive RSGs being merger products. Neugent (2021) extended the investigation to the Local Group galaxies M31 and M33. They reported a constant binary fraction of 33 . 5 +8 . 6 -5 . 0 % for M31, while the binary fraction in M33 drops from 41 . 2 +12 . 0 -7 . 3 % to 15 . 9 +12 . 4 -1 . 9 %, for inner and outer regions, respectively, which is attributed to a metallicity effect. \nOverall, the intrinsic binary fractions of RSGs measured in the SMC and LMC using', 'Red supergiants:': 'Red supergiants are the evolved products of massive star evolution; for many stars they mark the final stage of evolution before they end their lives with a SN explosion \nLuminous Blue Variable: Luminous blue variables (LBVs) are in a brief phase in the evolution of evolved massive stars. As the name implies, they populate the upper region of the HRD and are hot, can be above the HD limit, and are highly variable.', 'Wolf-Rayet stars:': 'WR stars are classified based on their spectra that are dominated by strong, broad emission lines, and comprise a number of objects with a different origin. \ndifferent detection techniques are in agreement with each other and amount to ∼ 30%. The role of metallicity, the nature of the companions of those RSGs as well as the orbital parameter distributions remain to be further constrained.', '2.4. Luminous Blue Variables': "While LBVs were for a long time considered to be the transitory phase between single-star BSGs and WR stars characterized by strong, eruptive mass loss (e.g., Conti 1975, Lamers & Nugis 2002), the link of LBV as direct progenitors of type-II SNe induced a shift in their interpretation, not as transitory phase but as potential end-point of stellar evolution (Kiewe et al. 2012, Groh et al. 2013). They were further proposed to not be a phase in single-star evolution, but to be binary interaction products (see see e.g., Gallagher 1989, Justham et al. 2014, Smith & Tombleson 2015, and Section 4). Constraining the binary properties of LBVs would thus also allow to put constraints on their evolutionary history. \nGiven the rarity of LBVs and their strong (spectroscopic and photometric) variability (see e.g., Humphreys & Davidson 1994)., binary detections are difficult. Targeting individual objects with different methods, several binary LBVs have been found so far (see e.g., Martayan et al. 2012, Boffin et al. 2016). Only few systematic studies of the binary properties of LBVs exist. Based on an X-ray survey using XMM-Newton, Naz'e et al. (2012) report X-ray properties for LBVs consistent with a binary a fraction between 2669% (detecting signatures of wind-wind collisions and potentially the O-type companions themselves). Targeting seven LBVs with high-angular resolution imaging, Martayan et al. (2016) find a binary fraction of ∼ 30%. Mahy et al. (2022a) combined spectroscopy and interferometry and reported a bias-corrected binary fraction for galactic LBVs of 62 +38 -24 % for periods below 1000 days and mass ratios between 0.1 and 1.0, higher than previously expected. The detected companions of LBVs are either OB MS stars or RSGs, and given the large radii of LBVs the periods of those systems are large. The full orbital parameter distributions such as the period distribution remain largely unconstrained.", '2.5. Wolf-Rayet stars': 'Different types of stars fall under the spectroscopic definition of Wolf-Rayet (WR) star, which is based on strong, broad emission lines in the spectrum. On the one hand there are classical WR (cWR) stars, sub-divided in nitrogen- (WN), carbon- (WC) or oxygen-rich (WO) depending on their composition. Most WR stars fall in the cWR category and are thought to be post-H-burning objects in an evolved stage of massive-star evolution after the RSG and potentially LBV phase. On the other hand, most of the rare WNh stars (nitrogenrich WR stars with hydrogen lines) are thought to be even more massive stars ( ≳ 100 M ⊙ at galactic and LMC metallicity) on the MS with strong stellar winds (see e.g., Crowther 2007, for a recent review). For those, Conti (1975) proposed an evolutionary sequence of O/WNh → LBV → WN → WC ( → WO). If this evolutionary connection indeed exists, their binary properties should also align in this sequence (i.e., remain the same or decrease in later stages). \nIn general, and using a multitude of techniques, similar observed binary fractions of ∼ 30-40% were reported for the Milky Way (e.g., van der Hucht 2001, Dsilva et al. 2020, 2022, 2023), lower-metallicity environments (such as the LMC and SMC, e.g., Foellmi et al. 2003a,b, Schnurr et al. 2008, and references therein) and higher-metallicity host galaxies \n(like the Triangulum or the Andromeda Galaxy, Neugent & Massey 2014). High-resolution spectroscopy of Galactic WN and WC stars showed that the bias-corrected binary fraction of WN stars is lower (0 . 52 +0 . 14 -0 . 12 %) than for WC stars (0 . 96 +0 . 04 -0 . 22 %). Also their period distributions differ, with a large number of short-period WN binaries (Dsilva et al. 2023, and references therein). This implies that while the evolutionary connection described above might hold for long-period systems, the short-period WN systems remain unexplained. Additionally, binary properties of WNh stars remain largely unconstrained.', '2.6. Blue and yellow supergiants': 'The class of blue supergiants (BSGs) comprises several types of objects: O supergiants are massive stars towards the end of their MS evolution. B supergiants are interpreted as a transitory post-MS phase, either in a brief traverse over the HRD or during a blue loop (Maeder & Meynet 2001, Vink 2022). Long-lived BSGs were suggested to be the product of stellar mergers. One particular case is that of SN1987A, which occurred in the LMC. Pre-explosion images indicated its progenitor was a BSG, which can be naturally explained as a merger product (Podsiadlowski et al. 1990, Menon & Heger 2017). Yellow supergiants (YSGs) are a heterogeneous group of stars crossing the HRD (Maeder & Meynet 2000). \nThe binary properties of BSGs and YSGs are not well constrained. In the context of the VFTS, Dunstall et al. (2015) reported an observed binary fraction ob B-type supergiants of 23 ± 6%, in agreement with the one observed for MS B-type stars targeted in the same survey. Observing OB giants to supergiants in the Galactic clusters Westerlund 1 with VLT-FLAMES, Ritchie et al. (2022) found an observed binary fraction of ≳ 40%, with an intrinsic binary fraction that might be significantly larger.', '3. BINARY INTERACTION PROCESSES': "The interactions that will happen (or have already happened) for the binary systems discussed in the previous section are mainly dictated by the radial evolution of both components. As a star evolves, it can eventually fill its Roche lobe, producing outflows through the first Lagrangian point L1 towards its companion. In binary evolution models, Roche lobes are approximated by a volume-equivalent radius, normally computed through the fit made by Eggleton (1983) in terms of the mass ratio 4 ( q ≡ m 1 /m 2 ): \nR RL , 1 ( q ) = f ( q ) a, f ( q ) = 0 . 49 q 2 / 3 0 . 6 q 2 / 3 +ln(1 + q 1 / 3 ) , 1. \nwhere R RL , 1 is the Roche lobe radius of the star with mass m 1 , and a is the orbital separation. Even if a star does not fill its Roche lobe, interactions can still happen through tidal effects, wind-orbit coupling and irradiation. On the opposite extreme, binary stars can extend well beyond the L1 point, either in stable contact systems or dynamically evolving common envelope (CE) configurations. \nRoche lobe overflow (RLOF) is the most transformative form of interaction in binaries. Following the work of Kippenhahn & Weigert (1967), phases of mass transfer are \nFigure 2 \n<!-- image --> \n(left) Radial evolution of massive stars at solar metallicity with different masses (see legend) as a function of time in units of the MS lifetime. The thick black line indicates the points at which the star develops a convective envelope containing more than 10% of the total mass. (right) Range of orbital periods (considering an arbitrary mass ratio) for which these stars would undergo Roche lobe overflow. \nnominated as 'Cases' (capitalized for historical reasons). Case A mass transfer refers to RLOF episodes during the MS of the donor star, while Case B mass transfer refers to RLOF before core-helium depletion. Case B is often split into early and late Case B, indicating interaction before or after the development of a convective envelope. Case C mass transfer corresponds to late interactions post core-helium depletion. In this review we make use of these definitions but note that they can be inconsistent in the literature, with Case B and Case C representing instead mass transfer before and after the ignition of helium, respectively (Podsiadlowski et al. 1992). Repeated interactions are marked with multiple letters, such that Case AB refers to a mass transfer event after the MS, which was preceded by Case A mass transfer. \nThe range in separations and orbital periods at which RLOF is expected is illustrated in Figure 2. As massive stars can expand by over two orders of magnitude, RLOF can occur at orbital separations ranging from tens of solar radii up to about ten AU (for mass ratios of order unity). For a given stellar radius R 1 , the period at which RLOF would occur is given by \nP RLOF = R 3 / 2 1 ( 4 π 2 Gm 1 ) 1 / 2 g ( q ) , g ( q ) ≡ ( f ( q )) -3 / 2 ( q 1 + q ) 1 / 2 . 2. \nThe function g ( q ) is bound between 3 . 3 and 1 . 35, resulting in a finite range of periods for which a star would undergo RLOF, independent of the mass ratio. As shown in the left panel of Figure 2, solar-metallicity massive stars are expected to undergo RLOF at the zero-age MS for periods ∼ 1 day, while at their maximum expansion they can undergo RLOF at periods of ∼ 10 years. As such, interacting binary systems operate over a large range of spatial and temporal timescales. \nOrbital evolution can be computed in terms of the orbital angular momentum, \nJ = m 1 m 2 √ Ga (1 -e 2 ) m 1 + m 2 . 3. \nIt is common to assume near-RLOF systems to be efficiently circularized (eg. Verbunt & Phinney 1995), in which case the time derivative of the orbital angular momentum is given by \n˙ J J = ˙ m 1 m 1 + ˙ m 1 m 2 -1 2 ˙ m 1 + ˙ m 2 m 1 + m 2 + 1 2 ˙ a a . 4. \nHaving a model for how the masses of a binary system evolve with time and how angular momentum is removed or added to the orbit allows for the integration of this equation to determine the evolution of the orbital separation. For some simple models the evolution of the separation as a function of the component masses can be determined analytically, serving as a useful probe to determine the future outcome of a binary observed pre-RLOF, or to understand the potential progenitors of a post-interaction system (see, eg., Soberman et al. 1997, Tauris & van den Heuvel 2023). The detailed dynamics of binary outflows and the angular momentum they remove remains an important uncertainty in evolutionary calculations (eg. Brookshaw & Tavani 1993, MacLeod & Loeb 2020, Willcox et al. 2023).", '3.1. Modeling tools for binary evolution': "Following the evolution of a binary system, including the properties of both its components, requires the use of computer simulations. Three different types of codes are used for this purpose. (Magneto)-hydrodynamical simulations can be used to probe short dynamical timescales, useful to model processes such as mergers and CE evolution (eg. Lombardi et al. 1995, Taam & Sandquist 2000, Schneider et al. 2019, Lau et al. 2022). Due to their computational cost 3D simulations can only explore a limited set of initial conditions, and cannot resolve nuclear- and thermal-timescale mass-transfer processes. To model longer evolutionary timescales, 1D stellar evolution codes are used, which at their core carry a significant resemblance to the models computed over half a century ago by Kippenhahn & Weigert (1967). 1D stellar evolution codes also serve as the source for initial conditions of 3D simulations. The third type of simulations are referred to as 'rapid' codes, and are based on semi-analytical models based on pre-computed 1D stellar models (eg. Hurley et al. 2002) which allows for sub-second calculation of full binary evolution models. \nIn contrast to rapid evolutionary codes, 1D simulations are referred to as 'detailed'. The majority of detailed binary models currently computed are done with either the MESA code (Paxton et al. 2011, 2015) or variations of the STARS code (Eggleton 1971, Eldridge et al. 2008). These codes are derived from the Henyey method (Henyey et al. 1959) to solve the equations of stellar structure in 1D, computing the evolution of two stars that are coupled through tidal interaction (see Section 3.2) and mass transfer (see Section 3.3). Detailed models can accurately follow phases of evolution on the nuclear and thermal timescales of its components, including phases of contact (see Section 3.4). Using 1D codes to model dynamical phases of evolution requires approximations that ignore (or parameterize) the uncertainties associated with 3D binary interactions (see Section 3.5). \nCompared to detailed evolutionary codes, a much broader set of tools are available to perform rapid calculations. The majority of these are based on the analytical fits of Hurley et al. (2000) to the single star evolution models of Pols et al. (1998), coupled with the semi-analytical approximations to binary evolution of Hurley et al. (2002). These include Startrack (Belczynski et al. 2002), binary c (Izzard et al. 2004), MOBSE (Giacobbo & Mapelli 2018), COSMIC (Breivik et al. 2020) and COMPAS (Riley et al. 2022). Even though these codes include multiple free parameters to adjust the physics of binary evolution, they", 'Detailed binary code:': '1D stellar evolution code that solves the differential equations of stellar structure and evolution for a binary system. Has a runtime of order a cpu hour.', 'Rapid binary code:': 'Code based on fits to single star evolution models and semi-analytical approximations to binary interactions. Has runtimes smaller than a cpu second. \nare limited to the physical assumptions of the stellar models of Pols et al. (1998). Some examples that break from this reliance on the fits of Hurley et al. (2002) are COMBINE (Kruckow et al. 2018), METISSE (Agrawal et al. 2020) and SEVN (Iorio et al. 2023). Additionally, various codes precede the work of Hurley et al. (2002), including SeBa (Portegies Zwart & Verbunt 1996) and the Brussels population synthesis code (Vanbeveren et al. 1998). One main weakness of rapid evolutionary codes is that, since they are based on single-star evolutionary models, they cannot capture the response of thermal timescale mass transfer on either component, which plays a critical role in determining the stability of mass transfer. \nOwing to their short runtimes, rapid codes have been the preferred tool to perform population synthesis calculations (see Han et al. 2020 for an overview), but detailed calculations are continuously taking a larger role in this area. Large grids of detailed models have been usually restricted to limited regions of the input parameters (eg. the grids of case A evolution of Nelson & Eggleton 2001, de Mink et al. 2007a and Sen et al. 2022). Currently, the BPASS (Eldridge et al. 2017) and POSYDON (Fragos et al. 2023) codes (which use STARS and MESA as their backends, respectively) provide openly available sets of detailed calculations covering the full range of parameters relevant to interacting massive binaries. Other large-scale population synthesis calculations done with detailed models have been performed (eg. Wang et al. 2020) but do not have an associated name. Although detailed population synthesis calculations are now feasible, further increasing the efficiency of calculations is critical for reproducibility and broad testing of theoretical uncertainties. One approach is the use of smart sampling of initial parameters rather than the use of a regular grid, coupled with interpolation (Rocha et al. 2022).', '3.2. Tidal Interaction': "Tidal torques in stars with radiative envelopes are attributed to the dynamical tide process (Zahn 1975, see Zahn 2008 for a recent review). In this process, gravity modes are excited by the tidal potential near the interface between the convective core and the radiative envelope, with these waves dissipating close to the stellar surface. The work of Zahn (1975) provides a straightforward method to compute the rate of tidal synchronization and circularisation, with the only non-trivial dependency being the computation of the structure constant E 2 . Many simulations (Hurley et al. 2002, Paxton et al. 2015) relied on an interpolation to the values of E 2 for zero-age MS models computed by (Zahn 1975), but more modern calculations accounting for evolved stages are available (Qin et al. 2018). Extensions to the Zahn (1975) model have also shown the potential for resonant interactions when the tidal frequency matches a natural oscillation frequency of the star, potentially leading to tidal locking (Witte & Savonije 1999). Direct calculations of the tidal torques are now possible on a timescale that allows for their integration in evolutionary calculations (eg. Sun et al. 2023). Computations of this type, relaxing some of the assumptions made by Zahn (1975), have shown that under some circumstances, tidal interaction arises from standing waves rather than traveling waves that are completely damped at the surface (Ma & Fuller 2023). \nAn associated growing field of study is the observation and theory of tidally excited oscillations (TEOs) in eccentric binaries (eg. Fuller 2017 and references within). The discovery of the prototypical system KOI-54 (Welsh et al. 2011) was enabled by the short cadence and high precision observations of the Kepler telescope. Owing to their characteristic lightcurves, such systems are often referred to as heartbeat stars. The two components of KOI-54 are only about twice the mass of the Sun, but despite not being massive they \nallow to probe tidal processes in stars with radiative envelopes. Further observations have pushed the observed heartbeat systems to the massive star regime, with TESS observations showing a system with a total mass of ∼ 150 M ⊙ (Koglyph[suppress]laczek-Szyma'nski et al. 2021). \nOne aspect that has received less attention is the effect of tidal deformation. Although tidally deformed stellar surfaces following the Roche potential are used to model observations (eg. Prˇsa & Zwitter 2005, Abdul-Masih et al. 2020), the impact of deformation on interior structure is seldom included in evolutionary models. By extending methods used to model centrifugal deformation in 1D stellar models, Fabry et al. (2022) has incorporated tidal deformation in binary evolution models, which can be applied to detached, semi-detached and contact binary systems. Another approach by Fellay & Dupret (2023) accounts for the full non-spherical mass distribution of each component in order to construct static structure models. Extended studies describing how tidal deformation modifies binary evolution are not available at the moment.", '3.3. Stable mass transfer': "When a star in a binary fills its Roche lobe, mass transfer will ensue and modify both the orbit and the structure of the donor. Whether or not the donor can remain in hydrostatic equilibrium in these conditions is normally described in terms of the mass radius exponents, \nζ ad ≡ d log R d log M ad , ζ RL ≡ d log dR RL log M , 5. \nwhich describe the adiabatic response of the donor radius and the change in the Roche lobe radius as mass is transferred (eg. Soberman et al. 1997). If ζ ad > ζ RL , the donor can adjust to mass transfer while remaining in hydrostatic equilibrium, producing either nuclear or thermal timescale events. In particular, donor stars with a significant fraction of their mass in a convective envelope are expected to expand in response to mass loss (Hjellming & Webbink 1987), favoring instability for late Case B and Case C mass transfer (see Figure 2). In detailed evolutionary calculations, this criteria is less useful, as ζ ad cannot directly be inferred from the structure of the star. In certain cases the pressure scale height at the photosphere can be an important fraction of the stellar radius, making the concept of a hard surface limited by its Roche lobe inapplicable. Such conditions require a different method to evaluate instability (Temmink et al. 2023). Rapid evolutionary codes rely on prescribed stability criteria, leading, for instance, to significant uncertainties in the predicted rates and formation processes of merging binary black holes (Olejak et al. 2021, Gallegos-Garcia et al. 2021). \nFigure 3 illustrates the typical evolution of two binaries undergoing stable Case A and early Case B mass transfer, closely resembling the evolution of the donor stars in Kippenhahn & Weigert (1967). Unless the evolution is modified by the secondary filling its own Roche lobe, Case A mass transfer is expected to be separated into a 'fast' and a 'slow' phase. As the more massive star in the Case A system initiates mass transfer, the orbit shrinks and leads to a thermal timescale mass transfer. After the mass ratio is inverted and the donor can thermally relax into the size of its Roche-lobe, mass transfer is driven by nuclear evolution. This evolutionary stage is referred to as Algol phase, and owing to the large ratio between thermal and nuclear timescales, a large majority of observable semidetached systems are expected to be in this phase. After the MS, a final phase of Case AB mass transfer strips most of the remaining envelope and leaves a star mostly composed of helium. The Case B system instead interacts while its expansion is driven by hydrogen \ṅ \ṅ \ṅ \nFigure 3 \n<!-- image --> \n(left) Mass transfer rate ˙ M mt and rate of change of mass of the accreting star ˙ M 2 as a function of the donor mass for a case A system (20 M ⊙ +16M ⊙ with an initial period of 2.5 days) and an early Case B system (same masses but with an orbital period of 15 days). Simulations include rotation and mass accretion is limited after the accreting star reaches critical rotation. (right) Evolution of the donor star in the HR diagram for the same two binaries. Different mass transfer phases are indicated with thick contours, and dots are placed in time intervals equal to 5% of the main-sequence lifetime of the donor star. \nshell burning and results in a single fast phase of Case B mass transfer removing most of the hydrogen envelope. \nThere are various processes that can modify this classical picture of the evolution of massive binary stars. Rapid rotation is argued to produce chemically homogeneous evolution in massive stars (Maeder 1987), in which case the MS evolution proceeds at almost constant radius. de Mink et al. (2009) argued that for massive binaries born near contact, tidal synchronization leads to rapid rotation and chemically homogeneous evolution of the primary, resulting in the first mass transfer phase being initiated by the initially less massive star (see also Marchant et al. 2017). Regarding the Algol phase, Sen et al. (2023) has shown that as more massive stars have a larger fraction of their mass in their convective cores, for sufficiently high donor masses ( ≳ 30M ⊙ ) the helium-enriched core can be exposed before the mass ratio inverts. If that happens, the star can thermally relax to its Roche lobe size before mass ratio inversion, leading to a slow case A phase with a more massive donor than accretor (which Sen et al. 2023 refer to as an inverse Algol). Regarding Case B mass transfer, Klencki et al. (2022) has suggested that at metallicities smaller or equal to that of the LMC, the thermal mass transfer phase can be interrupted, and followed by a long-lived nuclear timescale phase with blue or yellow supergiant donors. \nMass accretion onto the secondary results in increased rotational velocities as well as contamination with CNO processed material from the donor star. These are considered key indicators of past binary interaction, but unfortunately are also degenerate with expectations of rotational mixing. As post-interaction secondaries can become overluminous and evolve to long orbital periods (or become single stars in case the primary undergoes a SN), apparently single stars can actually be predominantly post-interaction products (de Mink et al. 2014). Our theoretical understanding of CNO enrichment and spin-up through accretion has not evolved significantly in the past decade, and the reader is referred to sections 3.1 and 3.2 of Langer (2012). \nOne critical aspect that remains unsolved is the interplay between mass transfer efficiency (ie. how much of the mass transferred by the donor is accreted onto the companion) and accretion spin-up. As shown by Packet (1981), an accreting star needs only to increase its mass by a few percent before reaching critical rotation. Whether or not accretion can proceed from that stage is uncertain. Langer et al. (2003) has pointed out that in the shortest period systems tidal interactions can prevent the accretor from reaching critical rotation, allowing for further accretion (see also Sen et al. 2023). It has also been argued that angular momentum could be transported outwards from the accretion disk while still allowing for an inwards mass flow (Paczynski 1991, Popham & Narayan 1991). Observational constraints are usually restricted to either the post-interaction or the Algol phase, while non-conservative phases are likely associated to thermal timescale mass transfer. Algol systems in the SMC appear to support a lowered efficiency with initial orbital period (de Mink et al. 2007b) but other post-interaction systems favor high accretion efficiencies even at long orbital periods (eg. Schootemeijer et al. 2018, Bodensteiner et al. 2020b, Vinciguerra et al. 2020). A keystone system to understand accretion efficiency is the massive binary β Lyrae (eg. Mourard et al. 2018), which is currently undergoing a rapid mass-transfer phase and has been resolved with the CHARA interferometer. \nAnother limitation of current detailed binary models is that mass transfer rates are determined from prescriptions that use the 1D model structure of the donor star. A commonly used prescription is that of Kolb & Ritter (1990), which treats separately the contributions from the extended atmosphere of the donor stars as well as from regions below the photosphere that are above the L1 equipotential. Lightening the assumptions of the Kolb & Ritter (1990) model (Marchant et al. 2021) or taking a different approach altogether to the computation of mass transfer rates (Cehula & Pejcha 2023) leads to qualitatively different evolution during fast mass transfer phases, with the potential to undergo overflow of the outer Lagrangian points.", '3.4. Contact Binaries': 'Contact binaries, where both components extend beyond the L1 equipotential, are precursors to stellar mergers and represent the most compact binary configurations possible. So long as material is contained within the equipotential of the second Lagrangian point L2, hydrostatic equilibrium is possible (Kuiper 1941). Just as with rotating stars, hydrostatic equilibrium in radiative layers requires the radiative flux to be proportional to gravity (eg. Fabry et al. 2022). However, the von Zeipel paradox (von Zeipel 1924) does not allow for both radiative and thermal equilibrium to hold simultaneously, such that large scale flows arise (Smith & Smith 1981, Tassoul & Tassoul 1982) and the surface flux deviates from a simple proportionality to effective gravity (eg. Espinosa Lara & Rieutord 2012). Contact binaries are expected to have similar temperatures at their surfaces, implying that the luminosity ratio between the two components is similar to the mass ratio ( L 1 /L 2 ≃ m 1 /m 2 , Lucy 1968). Owing to the steepness of the mass-luminosity relationship, this requires a significant redistribution of the luminosity beyond the L1 equipotential. \nThe internal structure of massive contact binaries, including tidal deformation and energy transport, is not well understood, with early work from the 70s not reaching a consensus (see Shu et al. 1980 and references within). Evolutionary models have characterized the conditions under which contact evolution happens, including the expansion of the accretor during thermal-timescale mass transfer as well the case where the more massive secondary in \nan Algol system expands due to nuclear evolution (Eggleton 1996, Nelson & Eggleton 2001, Wellstein et al. 2001). Modeling the evolution after that stage has remained uncertain, with models that either ignore the different physics of the contact stage, or use an ad-hoc mass transfer rate determined such that the surface of both stars remains in the same equipotential (de Mink et al. 2007a, Marchant et al. 2016). Population synthesis calculations of the LMC using this mass-transfer model indicate an overestimation of massive contact systems with mass ratios ≃ 1 when compared to observations (Menon et al. 2021), as most contact systems with mass ratios away from unity are predicted to evolve in a thermal timescale towards equalization of masses. Initial calculations by Fabry et al. (2022, 2023) have shown that the inclusion of energy transport in evolutionary models of massive contact binaries can extend long-lived phases of evolution with mass-ratios away from unity. \nAs the components of a short period contact binary rotate rapidly, rotational mixing is expected to make them overluminous and rich in nitrogen at their surface (and potentially operate more efficiently than in single star evolution, Hastings et al. 2020a). Observations of the contact binary VFTS 352 (Almeida et al. 2015), containing a ∼ 30 M ⊙ + 30M ⊙ binary with a period of 1.1 days, showed the system was indeed overluminous, providing potential support for rotational mixing and chemically homogeneous evolution. However, through detailed spectroscopic analysis accounting for the variable effective temperature across the surface, Abdul-Masih et al. (2019, 2021) did not find an indication of enrichment with CNO processed material. Whether or not rotational mixing is active in massive contact stars remains an open question. The systems analyzed by Abdul-Masih et al. (2019, 2021) show that both components share similar effective temperatures and luminosities, which is consistent with the scenario where energy is efficiencly transferred across the shared layers. Abdul-Masih et al. (2022) also showed that for a selection of contact binaries with data spanning more than a decade, the evolution of their orbital periods could be constrained to operate on their nuclear timescale, independent of their mass ratio.', '3.5. Mergers and common envelope evolution': "As a contact binary grows beyond its L2 equipotential, outflows removing large amounts of angular momentum are expected to lead to a quick coalescence (Pejcha et al. 2016). A stellar merger can naturally be confused with a single star, but current binaries could also be the result of a merger in a triple system as has been suggested for η Car (Hirai et al. 2021), HD 45166 (known as the quasi-Wolf Rayet star, Shenar et al. 2023) and even higher multiplicity systems (Vigna-G'omez et al. 2022). It has been argued that the origin of magnetic stars is associated to amplification processes during a stellar merger (Ferrario et al. 2009). Magneto-hydrodynamical merger simulations by Schneider et al. (2019) have shown that indeed sufficiently strong fields are produced, and identified the amplification process to be the magneto-rotational-instability. However, even if sufficiently strong magnetic fields are produced, they could be short-lived. Strong magnetic fields introduce an additional timescale, associated to Alfven waves travelling through the star, and there are significant restrictions to the field geometry to be stable in this timescale (Braithwaite & Spruit 2004). \nA merger can also be the outcome of unstable mass transfer leading to CE evolution. Whether or not a binary system undergoing CE will survive (ejecting its shared envelope) or merge is a long-standing problem in binary evolution (see Ivanova et al. 2013 for a review). Being one of the main formation channels proposed to form binary neutron stars (eg. Tauris et al. 2017, Vigna-G'omez et al. 2018), and merging binary black holes (eg. \nBelczynski et al. 2016, Bavera et al. 2021), uncertainties in CE evolution lead to order-ofmagnitude uncertainties on compact object merger rates (Mandel & Broekgaarden 2022). \nAs individual CE simulations of massive stars are expensive (eg. Lau et al. 2022), producing population predictions is reliant on simplified approximations. The most common approach is the use of the 'energy balance criterion', where the change in orbital separation is computed in terms of the binding energy of the envelope of the star inititating the CE phase, and a free efficiency parameter (Webbink 1984). Rapid population synthesis codes make use of fits to pre-computed binding energies as a function of the evolutionary stage of the star, but it has been recently pointed out that these could severely underestimate binding energies and overpredict the amount of stars surviving CE evolution (Klencki et al. 2021, Marchant et al. 2021). The computation of binding energies is also very uncertain, as it is not known a priori how much mass will be ejected before the CE. Recent results indicate that post-CE, the donor does not necessarily contract to become a hot stripped star, but rather undergoes a phase of stable mass transfer which completes the envelope stripping process (Fragos et al. 2019, Marchant et al. 2021, Hirai & Mandel 2022).", '4. NON-DEGENERATE POST-INTERACTION PRODUCTS': "As described in Section 2, a large number of stars in close binaries are predicted to interact at some point during their evolution. This not only drastically changes their evolution, but also implies that there should be a large number of post-interaction products (e.g., de Mink et al. 2014, Schneider et al. 2015, Wang et al. 2020). The different evolutionary channels described in Section 3 can lead to a multitude of interaction products with different properties, depending on the type of interaction. While there are several characteristics proposed to identify interaction products (see e.g., de Mink et al. 2014, for a list of observational indications), none of them are unambiguous, making their identification difficult. This implies that a large number of undetected interaction products might be interpreted as singleor non-interacting stars. As pointed out by de Mink et al. (2014), the 'best' single stars are most likely stars in pre-interaction binaries, which can usually be detected by large RV variations (see Section 2.1). \nIncluding interaction products in our understanding of stellar populations is crucial. For example, binary interactions were proposed to be responsible for the split MSs observed in young star clusters (Milone et al. 2018, Wang et al. 2020, 2022). Furthermore, stars that have undergone binary stripping during their evolution are predicted to significantly impact the integrated spectrum of stellar populations (Gotberg et al. 2019) and change their ionizing budget (Gotberg et al. 2020). Stripped stars are further thought to produce systematically different SN yields (Laplace et al. 2021). \nAn example of the evolution of a massive binary is schematically depicted in Figure 4. It demonstrates the multitude of different intermediate stages of interaction products, both non-degenerate (described here) and single- or double-degenerate (described in Section 5, which have different observational characteristics. It also illustrates potential end points of binary evolution. Those depend on the parameters of the system and the type of interaction that occurs. Figure 5 shows an observational overview of the mass donors in different mass regimes and evolutionary stages post-mass transfer in an HRD. We here include (candidate) objects detected by various techniques, which are described in the subsections below, and refrain from adding the mass gainers as often their parameters are not well constrained. \n<!-- image --> \nEvolution of a massive binary star from birth until the formation of a GW source, showing various intermediate post-interaction products. Phases in blue boxes represent end points for binary evolution.", '4.1. Potential merger products': 'Merger products are important in different aspects. As described in Sect. 2, the BSG progenitor of SN1987A can be explained as a merger product in a binary context, but not with a single-star solution (e.g., Podsiadlowski et al. 1990). Schneider et al. (2015) discuss how rejuvenation in mergers can shape the mass function of stellar populations, while Wang et al. (2022) proposed that the blue component of the split MSs observed in young star clusters are formed by pre-MS mergers. \nA handful of massive stars are potential stellar merger products. A famous example is the prototypical LBV η Car, which is currently in an eccentric binary system. Its giant eruption in the 19th century was proposed to be produced by the merging of the inner binary in a previous triple system (Portegies Zwart & van den Heuvel 2016, Smith et al. 2018, Hirai et al. 2021). If other LBVs could be the product of a stellar merger remains to be constrained. Another proposed merger product is the slowly-rotating early B star τ Sco (spectral type B0.2V Keszthelyi et al. 2021). Its spectrum shows a strong nitrogen excess (e.g., Martins et al. 2012), and the star is associated with a large-scale complex magnetic field (e.g., Petit et al. 2013). Schneider et al. (2019) reproduced the observed properties of τ Sco by a magnetohydrodynamic model of two merging MS stars, including its younger age in comparison to its parent association, the Upper Sco region. Based on a similar argument, Gies et al. (2022) interpreted the galactic runaway HD 93521 as merger product: it appears younger than the time it would have taken to travel from the galactic disk to its current location. Interestingly, their rotational properties are quite different: while HD 93521 is a \nrapid rotator, τ Sco is an extremely slowly rotating star (Nieva & Przybilla 2014) \nAnother proposed merger product is HD 45166, which was previously denoted quasi-WR due to its spectral appearance (it has similar, but narrower emission lines than normal WR stars, see e.g., Steiner & Oliveira 2005, and references therein). It was initially interpreted as first example of an intermediate-mass stripped star bridging the mass gap between sdOBs and WRs (see Section 4.2), but was recently reported to exhibit a strong magnetic field of 43 k G . This makes it a potential progenitor of a magnetar, a highly magnetized neutron star. To explain its properties, a merger of a hydrogen-rich star with a stripped star was proposed that expelled most of the H-rich envelope (Shenar et al. 2023). Given the proposed involvement of a stripped star, we include it in Figure 5.', '4.2. Observations of stripped stars': "It was long proposed that envelope stripping in cWRs occurs either because of their strong stellar winds (e.g., Grafener et al. 2011) or due to mass transfer in a binary system (e.g., Paczy'nski 1967, Vanbeveren et al. 1998). It remains uncertain which of the two channels is the dominant one, which most likely also varies as a function of metallicity (Shenar et al. 2020b). The viability of binary stripping is, however, demonstrated by the detection of lower-mass WR stars whose mass-loss rates are too low to strip their envelope (Schootemeijer & Langer 2018), in particular with rapidly-rotating companions (e.g., Shenar et al. 2016, 2019). While cWRs have current masses above ∼ 10 M ⊙ and launch strong, optically thick stellar winds, sdOB stars are their equivalent at the low-mass end (see Heber 2009, for a review). These core-helium burning objects with masses ≲ 1M ⊙ are thought to be stripped stars and are found in binaries with white-dwarf, low-mass MS or OBe star companions (e.g., Schaffenroth et al. 2022, and see Section 4.3). \nStripped stars with masses in between have only recently been reported based on UV photometry and followed up with spectroscopy (Drout et al. 2023, Gotberg et al. 2023), bridging sdOs with WRs and thus filling a long-standing gap in our understanding of postinteraction binaries. In particular, their current masses were estimated to be between 1 and 8M ⊙ , putting them in the intermediate mass range between WR and sdOB stars. They show a variety of composite spectra with different amounts of contribution from potential companions whose nature remains to be constrained. Furthermore, all systems with multiple epochs of observations showed significant RV variations indicative of the presence of a companion.", '4.3. Observational constraints on mass gainers': "Mass gainers from RLOF are expected to be rapidly rotating because of the accreted angular momentum (how rapidly still remains an open question, see Section 3). Based on a rapid population-synthesis calculation, de Mink et al. (2013) proposed that the rotational velocity distribution of massive stars is strongly shaped by binary interactions. \nIndeed, the observed rotational velocity distribution of single OB stars in the 30 Dor region is bimodal, with a majority of stars rotating with velocities around 100 km s -1 and a tail of high velocities, interpreted as signature of previous interactions (Ram'ırez-Agudelo et al. 2013, Dufton et al. 2013). A similar signature was reported for the young SMC cluster NGC346 (Dufton et al. 2019). Focusing on the O-type primaries of detected binaries, Ram'ırez-Agudelo et al. (2015) further found a lack of very rapidly rotating primaries compared to single O-type stars, implying that prior interaction is required to produce such \nT \nFigure 5 \n<!-- image --> \nObservational sample of stripped stars and post-interaction binaries. The HRD shows the relation between bloated, recently stripped stars (light pink, see text for references) and their contracted successors (Wang et al. 2021, and references therein). It further shows that the mass gap between sdO stars (grey) and WR stars (pink, Shenar et al. 2016, 2018, 2020b) was recently filled with intermediate-mass stripped stars (orange, Gotberg et al. 2023). Evolutionary tracks for donor stars of different masses undergoing case B mass transfer are shown, indicating their zero-age MS mass as well as their mass after stripping. Dots in the tracks are placed at intervals equal to 1% of the MS lifetime. See text for details and references. \nrapid rotators. Similar results were also reported for galactic O-type stars in the IACOB survey (Holgado et al. 2022, Britavskiy et al. 2023). The observed rotational velocities of stars in the slightly older cluster NGC 330 were interpreted as shaped by previous binary interactions (Bodensteiner et al. 2023). \nIf the stripped primary subsequently explodes as a SN, the system may be disrupted, potentially creating a rapidly-rotating runaway or walkaway star (e.g., Blaauw 1961, Renzo et al. 2019). The galactic runaway star ζ Oph was reported as such (with a space velocity of 30 km s -1 , see e.g., Renzo & Gotberg 2021). Britavskiy et al. (2023) found that almost 65% of apparently single fast-rotating O stars in IACOB are runaways. Investigating runaways in the VFTS, Sana et al. (2022) interpreted a population of rapidly rotating but slowly moving stars as results of binary ejections, in contrast to slowly rotating but rapidly moving stars interpreted as ejections by dynamical processes. Also the fraction of rapidly rotating OBe runaways matches binary population synthesis calculations (Boubert & Evans 2018). \nClassical OBe stars, which are in general rapid rotators, were proposed to be interaction products (see Section 2). Theoretical models agree on the feasibility of forming OBe stars according to this channel (e.g., van Bever & Vanbeveren 1997, Shao & Li 2014, Hastings et al. 2021), but the predicted number of OBe stars formed by this channel varies from a \nfew to basically 100%. The numbers depends strongly on model assumptions and uncertain interaction physics, such as the mass-transfer efficiency or the reaction of the accretor. Another remaining open question is how close to critical rotation OBe stars are (Rivinius et al. 2013). If the binary channel dominates Be star formation, their multiplicity properties would be fundamentally different from 'normal' OB stars. Firstly, there should be no Be+MS binaries (unless they were with a third companion in an initial triple system), and secondly their companions should be stripped stars or compact objects. In some cases, a massive enough companion might have exploded, potentially disrupting the system. \nSeveral well-known Be stars were proposed to be in binaries with an evolved companion. One example is the first Be star ever described, γ Cas, which is in a long-period binary system (e.g., Harmanec et al. 2000) 5 . Despite its brightness and various observational campaigns with different techniques, it remains debated if the companion is a white dwarf, a helium star or a neutron star (e.g. Langer et al. 2020a, and references therein). \nThe detection of stripped sdOB companions to Be stars further matches expectations of the binary channel. Given the temperatures and radii of those stars, they are faint in the optical and mostly detected in the UV (e.g., Gies et al. 1998, Peters et al. 2013, Wang et al. 2018, 2021, 2023). In a handful of cases, there is a direct signature of the sdO star in the optical spectrum (e.g. in ϕ Per or FY CMa, Poeckert 1981, Rivinius et al. 2004), or an observable change in the Be disk induced by the presence of the hot companion can be seen (e.g. in o Pup, Koubsk'y et al. 2012). A handful of such systems were also detected through interferometry (e.g., the first Be+sdB system κ Dra, Klement et al. 2022).", '4.4. Thermally contracting stripped stars': "Recently, a new type of OBe binary was reported in a brief evolutionary phase in between mass transfer and the OBe+sdOB phase, in which the stripped star has not yet contracted and is still similarly bright in the optical as the mass accretor, the Be star (see Figure 4). The first such systems reported were LB-1 (e.g., Irrgang et al. 2020, Shenar et al. 2020a) and HR6819 (e.g., Bodensteiner et al. 2020b, El-Badry & Quataert 2021). Initially they were reported as binary or triple system hosting a BH (Liu et al. 2019, Rivinius et al. 2020), mainly because of two observational characteristics: firstly, in contrast to sdOB systems, the optical spectrum is dominated by the narrow-lined stripped star showing large RV amplitudes indicative of a high mass ratio (in LB-1, the Be companion could only be revealed by spectral disentangling, Shenar et al. 2020a). Secondly, assuming the mass of the narrow-lined star, which appears like a 'normal' B-type star, from a comparison to single-star evolutionary tracks leads to a very high mass of the 'unseen' object. Given the proximity and brightness of HR 6819, the stripped star and Be companion with its disk could be resolved interferometrically (Frost et al. 2022). It was shown that the spectroscopic (and actual) masses of the stripped stars in LB-1 and HR 6819 are only ≲ 1 M ⊙ , and their companions are rapidly rotating Be stars with masses around 6 M ⊙ . \nOther systems were thereafter detected with a similar signature but in which the mass gainer currently shows no emission lines, for example the still uncertain, highly debated case of NGC1850BH1 (Saracino et al. 2022, 2023, El-Badry & Burdge 2022). Additional systems interpreted to have more massive stripped, bloated companions are VFTS 291 (Villase˜nor \net al. 2023), SMCSGS-FS 69 (Ramachandran et al. 2023) and AzV 476 (Pauli et al. 2022). \nThese thermally contracting systems provide a critical snapshot of a binary systems right after the interaction occurred (see Figure 5). A common feature is their mass ratios ( q = 5 for LB-1, and q ∼ 15 for HR6819), which were reported to require conservative mass transfer. While those systems are in a short evolutionary phase, they are more easily detectable with common observing techniques than their subsequent, longer-lived evolutionary stage (OB(e) + sdOB/WR systems) because of the higher optical brightness of the stripped star. El-Badry & Quataert (2021) also showed that the luminosity of the stripped stars is not purely powered by contraction, but also by shell-helium burning. This extends the lifetime of the contraction phase beyond a simple thermal timescale, making it more likely to observe binaries in this phase. \nAnother stripped star binary in a later evolutionary stage is the helium supergiant υ Sagittarius (Gilkis & Shenar 2023), which was reported to be in a currently interacting binary system during a second phase of mass-transfer. It was proposed that the more luminous primary is stripped of the remainder of its hydrogen envelope, while the accretor is a rapidly-rotating B-type star. The stripped primary of υ Sag is also included in Figure 1 (the evolutionary tracks do not include this second phase of mass transfer). \nThe aforementioned observed systems form an evolutionary sequence (see Figure 5) of recently stripped stars with OBe companions that later contract and appear as sdOB+OBe binaries when they are core-helium burning. Spectroscopically, the recently stripped stars in a contraction phase look like normal B-type MS stars. This is in contrast to expected surface abundances of stripped stars, which are thought to be dominated by helium (Gotberg et al. 2017, Schurmann et al. 2022). A potential explanation for the hydrogen-rich surface is that they have re-accreted hydrogen-rich material from the Be star decretion disk (Bodensteiner et al. 2020b). So far, a common characteristic of mass gainers is their rapid rotation. The fact that some of the potential mass-gainer companions do not show emission lines could simply be related to the transient nature of the Be phenomenon (e.g., Rivinius et al. 2013). The detection of additional such systems will allow to better constrain mass-transfer physics, the response of the accretor, and how close to the critical velocity it is spun up.", '5. SINGLE AND DOUBLE DEGENERATE BINARIES': "Once one of the components in a massive binary undergoes core-collapse, it can produce a neutron star (NS) or black hole (BH). The mapping between pre-explosion properties and the type and mass of the compact object depends on the physics of the supernova process (or lack thereof). A common method used to assess the post-explosion outcome is the compactness parameter, taken as a ratio between the mass and the radius at a specific mass coordinate (O'Connor & Ott 2011). Stars with higher compactness in their central regions are expected to collapse into a BH. Stellar evolution calculations done until corecollapse consistently show that, based on their compactness as well as other metrics, the boundary between NSs and BHs is not a simple mass threshold, but rather consists of 'islands' of explodability (Ugliano et al. 2012, Sukhbold & Woosley 2014, Sukhbold et al. 2016). Schneider et al. 2023 suggests that these variations in explodability lead to peaks in the mass distribution of BHs observed in gravitational wave sources. \nWhether or not the compact object remains bound to its companion depends on the total mass ejected and the kick imparted onto it. Observations of the proper motion of isolated NSs have been used to assess the distribution of kick velocities (Hobbs et al. 2005), \nFigure 6 \n<!-- image --> \n(left) Masses and distances to BH-HMXBs with measured dynamical masses, as well as detected inert BHs. Also shown are the masses of low mass X-ray transients with dynamical mass estimates from the BlackCAT catalogue (Corral-Santana et al. 2016) and the masses of binary neutron stars collected by Tauris et al. (2017). (right) masses of merging compact objects detected through GW emission. A representative band between 2 -5 M ⊙ is shown for the potential mass gap between NSs and BHs, while the upper mass gap due to pair-instability SNe is taken from (Marchant et al. 2019). \nand are often utilized in both rapid and detailed evolutionary calculations. Considering only isolated pulsars, however, is biased towards the strongest kicks (eg. O'Doherty et al. 2023), which can lead to an overestimate of unbound post-SN systems in population synthesis. Isolated BHs can be detected through lensing of their background stars, with a recent first detection (although data could also support a NS as the lens, Lam et al. 2022, Sahu et al. 2022). Data on isolated BHs is thus insufficient to estimate BH kick distributions, and instead BHs in X-ray binaries have been the standard method to determine BH kicks (e.g., Atri et al. 2019), showing support for both strong ( > 50 km s -1 ) and weak kicks at birth. One weakness of the methods that use low mass X-ray binaries to determine kicks is that they can only constrain the velocity imparted on the system at birth, and not the magnitude of the kick imparted onto the BH itself. \nIf the binary remains bound after the formation of the first or second compact object, and if the end result is not a binary BH, we can use electromagnetic observations to infer its orbital parameters, serving as a laboratory of the SN process. A detailed understanding of the intrinsic population of single-degenerate binaries is also crucial as an anchor point to constrain evolutionary models. Tauris et al. (2017) recently provided an extensive review on binary NSs, including their formation and intermediate stages. Thus, we only provide a brief overview of NSs in massive binaries and focus our discussion on the properties of BH-high mass X-ray binaries (BH-HMXBs) as well the recently identified category of inert BHs (corresponding to BH binaries wide enough that little mass is transferred and no accretion disk is formed). A compilation of masses and distances of NSs and BHs in \nmassive binaries is shown in Figure 6, including also GW observations. The detection of BHs through GWs has significantly outnumbered the electromagnetic sources for which we have mass constraints, but the population of inert BHs could grow by orders of magnitude in the coming decade (see Sect. 5.3).", '5.1. Neutron stars in massive binaries': 'Be X-ray binaries are composed of a compact object that is fed from the decretion disk of its companion Be star (Negueruela 1998, Okazaki & Negueruela 2001, see Reig 2011 for a review). Only one system has been claimed to contain a BH rather than a NS (Casares et al. 2014), but its status is currently contested with the compact object potentially being a stripped helium star instead (Rivinius et al. 2022, Janssens et al. 2023a). Assuming that a previous mass-transfer phase circularised the orbit before the SN, the eccentricity distribution of Be X-ray binaries is indicative of a subpopulation that underwent small kicks at NS formation (Pfahl et al. 2002). One explanation for this is the occurrence of electron capture SNe rather than iron-core collapse SNe (Podsiadlowski et al. 2004). The masses of Be stars in X-ray binaries have been used to argue that they underwent an efficient masstransfer process (Vinciguerra et al. 2020), but masses for Be stars are typically inferred from their spectral type, which carries significant uncertainty. \nIn OB+NS systems, after the OB star finishes its MS evolution, it will expand and fill its own Roche lobe (see Fig. 4). Owing to the large mass ratio, this most likely results is CE evolution, forming a compact stripped star+NS system, or instead merging. The outcome of a compact object merging with a stellar companion is uncertain, with possible outcomes being a stable Thorne-Zytkow object (Thorne & Zytkow 1975, see Farmer et al. 2023 for a recent picture) or a SN explosion (Chevalier 2012, Metzger 2022). Systems that survive CE and have stripped stars with masses ≲ 5 M ⊙ can undergo an additional phase of mass transfer after core helium depletion, which is referred to as Case BB mass transfer 6 (Tauris et al. 2015). This mass-transfer phase can leave an almost stripped helium core, producing SNe with ≲ 0 . 1 M ⊙ of ejecta and potentially small kick velocities (Moriya et al. 2017). \nIf the binary remains bound after the second SN, a double degenerate binary is produced. Neutron stars in binaries that are detected as radio pulsars provide an accurate probe of their radial motion through measurements of variations in the time of arrival of pulses. It was through timing measurement of the first radio pulsar (Hulse & Taylor 1975) that gravitational waves were first measured indirectly (Taylor et al. 1979, see Weisberg et al. 2010 for results with over three decades of timing data). Another remarkable case is the double pulsar (Kramer et al. 2006, 2021), where the detection of multiple post-Newtonian effects in the orbit allows the measurement of the individual pulsar masses to better than one part in ten thousand. Currently there are over 20 binary NSs detected (see Bernadich et al. 2023 for a recent collection), and the sample is expected to grow by up to an order of magnitude with the advent of the Square Kilometer Array (SKA, Keane et al. 2015). Even though binary NSs are usually discussed in the context of isolated binary evolution, it is important to consider alternative formation scenarios. Andrews & Mandel (2019) have suggested that the current sample contains a sub-population that is inconsistent with binary evolution and could have formed through dynamical processes.', '5.2. Black-hole high-mass X-ray binaries': "The X-ray flux of young star-forming galaxies without active galactic nuclei is dominated by HMXBs. The Chandra telescope has played a pivotal role in their study, allowing for the observation of X-ray binaries in galaxies beyond the local group (see Gilfanov et al. 2022 for a recent review). The brightest of these sources, with luminosities exceeding 10 39 erg s 1 , are referred to as ultraluminous X-ray sources (ULXs, Long & van Speybroeck 1983, see Kaaret et al. 2017 for a recent review). ULXs exceed the Eddington limit for a 10 M ⊙ BH, in some cases by over an order of magnitude, and as such they have been suggested to hold intermediate-mass BHs. This needs not be the case if there is significant beaming of radiation, which King et al. (2001) argues would be the case for rapid thermal-timescale mass transfer phases onto compact objects. Even more contrary to the idea that ULXs host intermediate mass BHs, the discovery of X-ray pulsations in some systems allowed a clear identification of their compact object accretors to be NSs (Bachetti et al. 2014). A clear identification of a BH accretor in a ULX is more elusive, but within our own galaxy it is suggested that the microquasar SS 433 (which hosts a BH) would appear as a ULX if observed from a different angle (Begelman et al. 2006). \nFor extragalactic X-ray binaries, spectroscopic follow-up that would allow the measurement of RVs and an estimate on the compact-object mass is challenging. Even when spectral features associated to the donor are detected, observed RV variations might not follow the orbital motion. Such is the case of IC10 X-1, which contains a WR star orbiting a compact object with a period of 35 hours. RV variations of the WR star indicated that IC10 X-1 hosts the most massive stellar-mass BH known (Prestwich et al. 2007, Silverman & Filippenko 2008). However, Laycock et al. (2015) showed that the phase of the RV variability was inconsistent with eclipses in the system and as such did not trace the orbital motion of the WR star, making the nature of the compact object unclear. Important adjustments have also been made to the measured masses of the BHs in Cyg X-1 and M33 X-7. A more accurate distance determination to Cyg X-1, placing it further away from us, has increased the mass estimate of its donor by ∼ 30% leading to a corresponding increase on its estimated BH mass (Miller-Jones et al. 2021). For M33 X-7, detailed spectroscopic modeling of its donor star has instead lowered the donor mass estimate by ∼ 30%, leading to a lower mass estimate for its BH (Ramachandran et al. 2022). \nIn the presence of an accretion disk around the BH, it is also possible to constrain its dimensionless spin (see Belczynski et al. 2021 for an overview of methods, as well as caveats). Table 2 lists the three known BH-HMXBs with spin estimates, all of which are high. For M33 X-7, Ramachandran et al. (2022) argued that the lowered mass estimate on the BH would lower its measured spin from 0 . 84 ± 0 . 05 (Liu et al. 2010) down to ∼ 0 . 6. For both Cyg X-1 and LMC X-1, the spin of the BH is near critical rotation. The origin of the high spin of these BH is unclear. It has been suggested that the source of the angular momentum in the BH progenitor came from tidal synchronization during an earlier stage of mass transfer (Valsecchi et al. 2010, Qin et al. 2018) or from a failed SN where some mass expanded and gained angular momentum from the orbit before falling back into the newly formed BH (Batta et al. 2017). Understanding the origin of spin in BH-HMXBs is crucial to understanding their potential link to GW sources, which are mostly observed to have low spins (see the discussion by Fishbach & Kalogera 2022). If BH-HMXBs inherit their spins from tidal coupling, longer-period systems could potentially exhibit lower spins, but it is unclear at which orbital period the BH would become X-ray inactive. \nTo get detectable X-ray fluxes in BH binaries, the BH not only has to accrete sufficient \nTable 2 Properties of BH-HMXBs and inert BHs, giving the distance d, orbital period P, mass of the compact object M CO , mass of the companion star M comp , and spin. Cyg X-3 could potentially have a neutron star instead of a BH. The error on the orbital period for all systems is below 1% . \nReferences: Cyg X-1 (Miller-Jones et al. 2021, Mahy et al. 2022b), Cyg X3 (Zdziarski et al. 2013, McCollough et al. 2016, Singh et al. 2002), NGC 330 X-1 (Binder et al. 2021, Rizzi et al. 2006, Crowther et al. 2010), M33 X-7 (Ramachandran et al. 2023, Gieren et al. 2013, Pietsch et al. 2006, Liu et al. 2010), LMCX-1 (Orosz et al. 2009, Gou et al. 2009, Pietrzy'nski et al. 2019), SS 433 (Hillwig & Gies 2008, Blundell & Bowler 2004), VFTS 243 (Shenar et al. 2022b, Pietrzy'nski et al. 2019), HD 130298 (Mahy et al. 2022b), Gaia BH1 (El-Badry et al. 2023b), Gaia BH2 (El-Badry et al. 2023a), NGC 3201 12560 (Giesers et al. 2018). \nmass, but also an accretion disk must form (Shapiro & Lightman 1976). Taking this into account, Vanbeveren et al. (2020) suggested that if the known WR+O binary systems in the solar neighborhood would evolve to become BH+O binaries, there should be over 100 BHHMXBs within a few kiloparsec of the Sun. A solution to that discrepancy would require most WR+O binaries to undergo a SNe and form a NS, or form a BH with a strong kick that unbinds the system. (Sen et al. 2021) improved upon the model for disk formation of Vanbeveren et al. (2020), and showed that owing to the fast winds of OB stars, only the closest OB+BH binaries would be observed as X-ray sources. Similarly, Hirai & Mandel (2021) suggested that tidal deformation in near-Roche-filling binaries leads to slow wind outflows through the vicinity of the L1 point, which can efficiently be captured by the BH and produce an accretion disk. If only near-Roche-filling systems can form discs around BHs, then WR+O star systems can still primarily evolve towards BH+O star binaries without being in conflict with the low number of observed BH-HMXBs. The vast majority of BHs with massive companions would then be inactive in X-rays.", '5.3. Inert black holes': "Recent work on the proposed population of long-period BHs which do not form accretion disks has referred to them as 'quiescent' or 'dormant'. However, this denomination can be confused with nomenclature used for X-ray active binaries, which can undergo periods of X-ray inactivity while still containing an accretion disk. We instead adopt the term 'inert' \nfor these BHs, to indicate that other than their gravitational influence on their companions, we do not expect them to become X-ray active in the near future. Using detailed population synthesis calculations, Langer et al. (2020b) has suggested that ∼ 3% of OB binaries contain BH companions, and almost all would be inert. \nThere are three main methods proposed to identify inert BHs in binaries. In low-mass BH binaries, ellipsoidal variability near Roche filling for the donor star can be identified through photometric measurements (eg. Gomel et al. 2021), although with a massive star donor it is expected that wind mass transfer would already make it active at this stage. Alternatively, the reflex motion of the companion star can be determined either through astrometry and/or by measuring its RV through spectroscopy. Currently it is only via spectroscopy and astrometry that confident detections have been made. \nIn the context of the Gaia mission, multiple studies pointed out the possibility to detect hundreds to thousands of inert galactic BHs in binaries (Breivik et al. 2017, Mashian & Loeb 2017, Yamaguchi et al. 2018, Yalinewich et al. 2018, Wiktorowicz et al. 2019, Janssens et al. 2022). Although this was expected to happen with the third data release of Gaia, the stringent criteria that were placed on the data in order to release orbital solutions excluded almost all massive stars (Janssens et al. 2023b). Current discoveries have instead been made in the low-mass regime, with (El-Badry et al. 2023b,a) reporting two confident detections of ∼ 9M ⊙ BHs orbiting ∼ 1M ⊙ stars. \nSpectroscopic measurements have allowed the first identification of inert BHs with massive companions. As part of the TMBM survey, Shenar et al. (2022b) followed up on the 51 apparent SB1 systems using spectral disentangling techniques to exclude the presence of a second luminous component. This allowed them to discover VFTS 243 (Shenar et al. 2022a), a binary consisting of a 25 ± 2 . 3M ⊙ O star and a 10 ± 2M ⊙ BH companion in a near-circular orbit ( e = 0 . 017 ± 0 . 012). Similarly, Mahy et al. (2022b) studied a sample of 32 Galactic SB1 stars and identified HD 130298 as an O+BH system, with masses similar to those of VFTS 243 but an eccentricity e = 0 . 457 ± 0 . 007. Contrary to BH-HMXBs, where large filling factors mean tidal forces are expected to lead to rapid circularisation, the eccentricity in inert BHs provides a direct constrain on the SN kick at their formation. Shenar et al. (2022a) has argued that the BH in VFTS 243 most likely formed with a weak kick and < 1M ⊙ of ejecta. Spectroscopic observations have also allowed for the detection of an inert BH with a low mass companion (Giesers et al. 2018) as well as two additional candidates (Giesers et al. 2019).", '6. GRAVITATIONAL WAVE SOURCES': 'Gravitational wave (GW) observations are emerging as a very fertile ground to study the evolution of massive and very massive stars. It is also very rapidly evolving, and as such, our objective here is not to provide a thorough review of the current state of this field. Rather, we aim to discuss the prospects of future GW observations as well as the binary processes that are thought to contribute to the observed sample.', '6.1. Observations of gravitational wave sources': 'Second generation GW observatories (including advanced LIGO, advanced Virgo and KAGRA) are currently on their third observing round (Abbott et al. 2020). The teams that develop each of these instruments are assembled into the LIGO-Virgo-KAGRA collabora- \ntion and up to the moment of writing have released almost a hundred detections, which are compiled in the Gravitational Wave Transient Catalogues (GWTCs, (The LIGO Scientific Collaboration et al. 2021)). For any given compact-object merger, the three bestconstrained quantities are the chirp mass, the mass ratio 7 and the effective spin, defined respectively as: \nM = ( m 1 m 2 ) 3 / 5 ( m 1 + m 2 ) 1 / 5 , q = m 2 m 1 , χ eff = m 1 χ 1 + m 2 χ 2 m 1 + m 2 , 6. \nwhere χ 1 and χ 2 are the components of the dimensionless spin of each compact object that are aligned with the orbital angular momentum. In practice, the frequency evolution of the source provides a measurement of the redshifted chirp mass (1+ z ) M , and to determine M one needs to make an assumption on the cosmology. Cases where an associated electromagnetic transient is detected are of particular importance, as a separate redshift measurement allows for the use of GW sources as standard candles, as was the case for GW170817 (Abbott et al. 2017). The chirp mass is directly derived by the frequency evolution of the binary, so for binaries that undergo many cycles in-band it can be accurately determined. For instance, the error on the chirp mass for GW170817 was smaller than 0.5%. Most measurements of q and χ eff show a significant degeneracy with current detector sensitivity, which can make it difficult to differentiate the nature of the merging compact objects (Hannam et al. 2013). In some sources additional information has been obtained by measuring precession (Hannam et al. 2022, Varma et al. 2022) as well as ringdown frequencies (Abbott et al. 2021). \nThe long-term prospects for GW astrophysics are certainly exciting. As the strain that is measured is proportional to the inverse of the luminosity distance (rather than an inverse square dependence as is the case for electromagnetic waves), detector improvements significantly increase the volume to which it is sensitive. This will possibly result in ∼ 10 4 detections by the end of the decade (Baibhav et al. 2019). In their expected sensitivity, third generation detectors will probe the Universe down to the redshift of the formation of the first stars (Hall & Evans 2019), making GW mergers the best characterized astrophysical population. Space-based missions such as the Laser Interferometer Space Antenna (LISA) will operate at lower frequencies, and allow for multi-band detections of stellar-mass merging binary BHs (Sesana 2016). \nBesides providing a probe into new astrophysical environments, two aspects of GW science make it very interesting to contrast observations against the theory of binary evolution. On the one hand, detector biases are very well understood, allowing for a transparent determination of intrinsic source properties. Owing to the limited amount of observations, at the moment most distribution properties are determined using parameterized models that contain expected features such as the lower and upper BH mass gaps (Abbott et al. 2023), but reliance on such models will lower as the number of detections increases. On the theoretical side, given a model that produces the rate of formation of double compact objects (including their masses, separations and spins), it is straightforward to provide expected rates of observation for specific detectors, as the time it takes for two compact objects to merge is easily calculable from general relativity (Peters 1964).', '6.2. Binary formation channels of gravitational wave sources': 'Although one could expect the natural formation scenario of a merging binary BH to involve the evolution of two stars in a binary system, a wide variety of processes that do not involve binary interaction are being considered (see Mandel & Farmer 2022 for an overview of some of the proposed channels). The relative contribution of different formation scenarios to the observed population is still uncertain, although comparison of predicted distributions to observations suggests that multiple processes contribute to it (e.g., Zevin et al. 2021). One important thing to keep in mind is that evolutionary paths towards a merging compact object are a rare outcome of massive-star evolution. Currently measured merger rates at redshift zero, in units of Gpc -3 yr -1, range between 10 -1700, 7 . 8 -130 and 16 -61 for NS+NS, NS+BH and BH+BH mergers (Abbott et al. 2023). In contrast, the rate for core-collapse SNe is on the order of 10 5 Gpc -3 yr -1 . Independent of the formation process, the main challenge to form a GW source is that the resulting binary BH needs to remain in a very compact orbit. For a circular compact object binary with a given orbital period and chirp mass, the time it takes for GWs to produce a merger is (Peters 1964): \nt delay = 11 . 9 Gyr ( P 5 [days] ) 8 / 3 ( M 30M ⊙ ) -5 / 3 . 7. \nThis implies that massive binaries that evolve to form merging binary BHs necessarily interact during their lifetime, and they need not only to survive to form a compact object binary but also finish their evolution in a short-period orbit. \nThe classical formation channel used to explain their formation is CE evolution. Shortly after Paczynski (1976) proposed CE as the formation scenario for cataclysmic variables, van den Heuvel (1976) suggested it could explain the formation of the Hulse-Taylor binary pulsar, while Tutukov & Yungelson (1993) argued it could also produce binary BHs (see Bavera et al. 2021 for a recent overview). An alternative scenario considers the case where both stars in a (near-)contact binary evolve chemically homogeneously (Mandel & de Mink 2016, Marchant et al. 2016) and form a binary BH. Chemically homogeneous evolution can also lead to the formation of merging BH+NS systems (Marchant et al. 2017). The third formation process that is commonly considered to form merging binary BHs is stable mass transfer (van den Heuvel et al. 2017), where a short-period binary is formed through non-conservative mass transfer from a star onto the first formed BH in the system. Picco et al. (2023) have suggested that stable mass transfer can also naturally produce NS+BH mergers, and that the mechanism is robust against uncertainties on the angular momentum budget of binary outflows. Current work suggests that population synthesis studies have systematically overestimated the contribution of the CE channel to the formation of merging binary BHs, owing to issues with adopted binding energies and criteria for unstable mass transfer (Klencki et al. 2021, Olejak et al. 2021, Marchant et al. 2021, Gallegos-Garcia et al. 2021). One channel that is not often discussed is that of pop III evolution, in which case the absence of metals is expected to lead to much smaller stellar radii, making it easier to form compact binaries by the end of their evolution (Kinugawa et al. 2014). \nA large number of studies have performed population synthesis (both using detailed and rapid codes) to compute the rates and distributions of compact object mergers. Mandel & Broekgaarden (2022) have made a compilation of published rate predictions, not only restricted to binary evolution, illustrating how rates for individual channels not only have uncertainties exceeding an order of magnitude, but that predictions from different groups can also have order of magnitude discrepancies. This is not entirely surprising, since pre- \ndicting merging compact objects requires an understanding of all phases of massive binary evolution coupled with the metallicity-dependent star-formation history. Even for individual systems, different evolutionary codes can produce wildly different outcomes both for single-star evolution (Romagnolo et al. 2023) as well as binary evolution (Belczynski et al. 2022). Moving forward will require collaboration within different research groups to identify the sources of these modeling discrepancies, as well as to define the different characteristic features of the distribution of merging compact objects that remain invariant independent of physical uncertainties (van Son et al. 2023).', '7. CONCLUSIONS': 'In the last decades, the field of massive binary evolution has grown significantly in terms of interest and scientific advances. This was driven first and foremost by large-scale spectroscopic observations, high-quality continuous photometric monitoring and high-precision astrometric survey. A consensus has emerged that binary evolution dominates the lives of massive stars, providing exciting new evolutionary pathways but also complicating our theoretical description of stellar evolution. In recent years, massive stars gained further visibility due to the detection of GWs, which opened a new window to constrain potential end products of massive star evolution. As a conclusion, we provide below summary points as well as future issues that we expect will shape the field in the coming decade.', 'SUMMARY POINTS': '- 1. A game changer in recent years have been large-scale surveys with well-understood biases, not only for electromagnetic observations in terms of spectroscopic, photometric or astrometric surveys, but also in terms of GW detections, for which detection biases are well constrained. Those have corroborated the finding that binary interactions dominate massive star evolution.\n- 2. Observations tentatively indicate that the multiplicity properties of OB stars, both binary fractions as well as orbital parameter distributions, are universal across the probed metallicity environments of the Milky Way, the LMC and SMC.\n- 3. Theoretical models of binary evolution rely on three distinct types of calculations, including 3D hydrodynamical codes, 1D stellar-evolution calculations and rapid semi-analytical approximations. Each of these play a critical role.\n- 4. Observational evidence is growing that a majority of rapidly rotating stars, in particular OBe stars, can be explained as accretors in previous binary interactions.\n- 5. In recent years, our picture of post-interaction products has been expanded to include massive stripped stars (closing the previous gap between sdOs and WRs) as well as thermally contracting objects right after mass stripping. These provide a novel perspective onto the mass transfer process.\n- 6. Inert BHs without accretion disks or detectable X-ray radiation have been identified in binary systems. These extend the known population of BHs to longer periods, and a much larger population will potentially be unveiled in the coming years.\n- 7. The number of GW observations has gone well beyond that of compact objects with dynamical masses measured from electromagnetic radiation. They provide a new window into massive star evolution, but their formation processes remain unclear.', 'FUTURE ISSUES': "- 1. The multiplicity fractions and orbital parameters of key objects such as OBe stars, WR stars or BSGs, remain poorly understood. Constraining those can provide us with important new insights on their evolutionary status and connection.\n- 2. Stellar evolution models still rely on simplified assumptions regarding how conservative mass transfer is, and the dynamics of outflows from binary systems. Narrowing down this uncertainty will require careful studies that combine detailed hydrodynamical simulations with long-term evolutionary calculations.\n- 3. Increased detections of inert black holes will provide a critical anchor point for binary evolution models. They will also give novel constraints on the supernova process and, in particular, potential kicks imparted onto the BH at birth. This will also require a clear understanding of the pre-SN binary properties, in particular the eccentricity and period distribution of WR+OB binaries.\n- 4. Recent observations indicate that triple- and higher-order multiple systems are common, even more so for more massive stars. Those could be important probes of the star-formation process and play a crucial role by inducing binary interactions through dynamical processes.\n- 5. It is critical to benchmark binary evolution models against each other, to clearly identify which physical assumptions lead to discrepant results.\n- 6. It is also important to keep in mind that significant uncertainty exists still in massive single-star evolution, which translates directly into binary evolution. In particular, the radial evolution of stars near the Eddington limit is poorly understood.\n- 7. New instrumentation on upcoming 40-m-class telescopes, such as HARMONI or MICADO at ESO's Extremely Large Telescope (ELT, first light expected before the end of the decade), will push our understanding of massive-star evolution to lower metallicity, for example by resolving individual stars in distant stellar populations.", 'DISCLOSURE STATEMENT': 'The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.', 'ACKNOWLEDGMENTS': 'The authors thank T. Shenar and A. Istrate for always insightful discussions and comments on the manuscript. PM acknowledges support from the FWO senior postdoctoral fellowship No. 12ZY523N.', 'LITERATURE CITED': "Aasi J, Abbott BP, Abbott R, Abbott T, Abernathy MR, et al. 2015. Classical and Quantum Gravity 32(7):074001 \n- Abbott BP, Abbott R, Abbott TD, Abernathy MR, Acernese F, et al. 2016. Phys. Rev. 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2024ApJ...975..207C	We simulate mergers between star clusters embedded within their natal giant molecular cloud. We extract initial conditions from cloudscale simulations of cluster formation and introduce different prescriptions for primordial binaries. We find that simulations that do not include primordial binaries result in a larger fraction of unbound stars than simulations that include a prescription for binaries based on observations. We also find a preferred direction of motion for stars that become unbound during the merger. Subcluster mergers within realistic gas environments promote binary disruption while mergers between idealized gasrich spherical clusters do not produce the same disruption. Binary systems with smaller semimajor axes are disrupted in simulations of subcluster mergers within their natal environment compared to simulations that do not include the realistic gas environment. We conclude that binary disruption and the production of an anisotropic distribution of unbound stars are the natural consequences of subcluster mergers during star cluster assembly.	2024-11-01T00:00:00Z	['10.3847/1538-4357/ad7f50', '2024ApJ...975..207C', '2024arXiv240913564C', '10.48550/arXiv.2409.13564', 'arXiv:2409.13564']	['Young star clusters', 'Star forming regions', 'Star clusters', 'Stellar dynamics', 'Binary stars', '1833', '1565', '1567', '1596', '154', 'Astrophysics - Astrophysics of Galaxies']	Binary Disruption and Ejected Stars from Hierarchical Star Cluster Assembly	2,024	202	0.54	['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']	1	https://arxiv.org/pdf/2409.13564.pdf	{'Binary Disruption and Ejected Stars from Hierarchical Star Cluster Assembly': 'Claude Cournoyer-Cloutier , 1, ∗ Jeremy Karam , 1, ∗ Alison Sills , 1 Simon Portegies Zwart , 2 and Maite J. C. Wilhelm 2 \n1 Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton, ON, L8S 4M1, Canada 2 Sterrewacht Leiden, Leiden University, Niels Bohrweg 2, 2333 Leiden, The Netherlands', 'ABSTRACT': 'We simulate mergers between star clusters embedded within their natal giant molecular cloud. We extract initial conditions from cloud-scale simulations of cluster formation and introduce different prescriptions for primordial binaries. We find that simulations that do not include primordial binaries result in a larger fraction of unbound stars than simulations which include a prescription for binaries based on observations. We also find a preferred direction of motion for stars that become unbound during the merger. Sub-cluster mergers within realistic gas environments promote binary disruption while mergers between idealized, gas-rich spherical clusters do not produce the same disruption. Binary systems with smaller semi-major axes are disrupted in simulations of sub-cluster mergers within their natal environment compared to simulations that do not include the realistic gas environment. We conclude that binary disruption and the production of an anisotropic distribution of unbound stars are the natural consequences of sub-cluster mergers during star cluster assembly. \nKeywords: Young star clusters (1833) - Star forming regions (1565) - Star clusters (1567) - Stellar dynamics (1596) - Binary stars (154)', '1. INTRODUCTION': "Star clusters form hierarchically within giant molecular clouds (GMCs, e.g. Lada & Lada 2003; Portegies Zwart et al. 2010). Simulations (e.g. Dobbs et al. 2022; Fujii et al. 2012, 2022; Grudi'c et al. 2018; Howard et al. 2018; Rieder et al. 2022) have shown that massive star clusters assemble through the merger of smaller subclusters. Observations also indicate that Orion (Fujii et al. 2022), Westerlund 2 (Sabbi et al. 2012; Zeidler et al. 2021) and R136 (Fahrion & De Marchi 2024; Fujii et al. 2012), among others, are assembling hierarchically. Simulations further suggest that sub-cluster mergers result in clusters that better match the density profiles (Fujii et al. 2012) of observed young massive clusters than isolated cluster formation, as well as providing a natural explanation for the light element abundance variations (Howard et al. 2019; Lah'en et al. 2024) observed in globular clusters. Sub-cluster mergers are also \nCorresponding author: Claude Cournoyer-Cloutier \[email protected] \n- ∗ Claude Cournoyer-Cloutier and Jeremy Karam are co-first authors \ndynamically rich processes, that can enhance the formation of new stars from compression of the surrounding gas (Fujii et al. 2022) and impart dynamical signatures in the stellar component of the clusters involved in the merger (e.g. Fujii et al. 2022; Karam & Sills 2024). Studying mergers between embedded star clusters therefore allows us to gain physical insights about the process of massive cluster formation. \nMost stars and protostars in star-forming regions are part of a binary (or higher-order) system, with a multiplicity fraction increasing strongly with stellar mass (see Moe & Di Stefano 2017; Offner et al. 2023, for recent reviews). The interplay between stellar multiplicity and stellar clustering is non-trivial. Observations reveal that - for solar-mass stars, at least - stellar multiplicity in young clusters depends on cluster density: higher-density environments (e.g. Orion, Duchˆene et al. 2018) have fewer wide binaries than low-density environments (e.g. Taurus, Kraus et al. 2011). Recent simulations of embedded (Cournoyer-Cloutier et al. 2021) and gas-free (Torniamenti et al. 2021) cluster assembly further suggest that changes to populations of binaries take place while the cluster is forming but that binary properties are then stable following gas expul- \nion. Similar trends with environment are observed in older clusters. Deacon & Kraus (2020) compare binaries in moving groups and open clusters, and find that stars formed in looser associations have more wide companions than stars formed in cluster-forming regions. For massive, dense globular clusters, stellar multiplicity is anti-correlated with cluster luminosity (and implicitly cluster mass, Milone et al. 2016), but not with central density. Taken together, those observations suggest that the differences in the populations of binaries are a result of a cluster's assembly process, rather than only its present-day properties. \nAccounting for the high multiplicity fraction of O and B stars, which are short-lived and rare, is simultaneously both more complicated and more important. Massive OB stars regulate the subsequent star formation in their natal clouds via feedback: they produce winds, radiation, and supernova explosions that heat the nearby gas and drive it away from the central, starforming regions, thus preventing the formation of other stars within the cluster. Massive stars routinely escape from their natal cluster; for example, recent observations show OB-stars moving away from the young massive clusters M16 (Stoop et al. 2023), M17 (Stoop et al. 2024), and NGC 3603 (Kalari et al. 2019). The removal of massive stars from their birth cluster can have consequences on galactic scales, due to the enrichment and supernova feedback from stars into the interstellar medium (Andersson et al. 2020). \nRunaway stars are often classified as stars with radial velocities v r ≳ 30 km/s. They can be produced through few-body dynamical interactions in a cluster, involving at least one binary (e.g. Poveda et al. 1967; Hoogerwerf et al. 2000; Fujii & Portegies Zwart 2011; Gvaramadze & Gualandris 2011). Through the interaction, the binary becomes more tightly bound and the energy lost is transferred to another star(s) as kinetic energy; if the amount of energy is large enough, the star's velocity may exceed the cluster's escape velocity and the star may become unbound from the cluster. Two populations of OB runaways are observed in 30 Doradus (Sana et al. 2022), attributed to binary disruption following a supernova (originally proposed by Blaauw 1961) and to few-body encounters within a star cluster. Simulations have shown that merger between sub-clusters may also enhance the production of runaway stars (Fujii et al. 2022; Polak et al. 2024). These simulations however did not include primordial binaries, and so could not compare the relative effects of sub-cluster mergers and few-body encounters on the production of runaway stars during cluster assembly. \nAny understanding of star cluster formation therefore requires an understanding of how a population of massive binaries evolves during cluster formation. Massive OB binaries are formed within embedded clusters, and the population of massive binaries in clusters is less dynamically processed than that of massive stars in the field, which are often runaways. There is however growing evidence supporting significant hardening of massive binaries during the process of cluster formation (e.g. Sana et al. 2017; Ram'ırez-Tannus et al. 2021; Bordier et al. 2022). It is clear that there is some highly nontrivial interplay between cluster mergers, binaries, and runaway stars. Some previous studies have started exploring this interplay (e.g. Fujii et al. 2012), using idealized initial conditions. \nIn this work, we utilize initial conditions drawn from a cluster formation simulation (Wilhelm 2024, Wilhelm et al. in prep.) 1 that includes a sub-cluster merger resulting in the production of runaway stars. The simulation only contains two sub-clusters, allowing us to cleanly isolate the merger. It was performed with Torch (Wall et al. 2019, 2020) and includes star formation and stellar feedback, but does not however have any population of primordial binaries 2 . We introduce primordial binaries to those initial conditions, and run a suite of simulations with different prescriptions for binaries. We run our simulations using only stellar dynamics and hydrodynamics. Those computational methods that are simpler and less expensive than those used in the Torch framework, allowing us to explore a larger range of parameters. In this paper, we study how sub-cluster mergers within giant molecular clouds influence a population of binary stars, and study the production of unbound stars during sub-cluster mergers. In Section 2, we describe our simulation methods and initial conditions. We present the results in Section 3 and discuss their implications for hierarchical cluster formation in Section 4. We summarize our key conclusions in Section 5.", '2. METHODS': 'We use initial conditions extracted from a simulation of star formation within a collapsing cloud with an initial mass of 10 4 M ⊙ , which naturally gives rises to a merger between two sub-clusters of stars with stellar masses of 1684.3 and 239.5 M ⊙ in 2603 and 444 stars. The merger takes place 2.13 Myr after the start of the simulation, and 0.66 Myr after the formation of the first star. The \n) \n( \nFigure 1. Initial conditions from the full Torch run, zoomed in to show the merging clusters. The full domain width is 17.5 pc, and the central 7 pc are shown. The surface density shown is calculated using the resolution of the gas cells, 0.136 pc. \n<!-- image --> \nsnapshot we use as our initial conditions is shown in Figure 1. In Section 2.1, we list the physics included in the full simulation. In Sections 2.3 and 2.2, we describe the numerical methods used in our simulations, while the initial conditions are outlined in Section 2.4.', '2.1. Torch simulation': 'The simulation from which we draw our initial conditions was done using Torch (Wall et al. 2019, 2020) and is presented in detail in Wilhelm (2024). In the text below, we describe the relevant physics implemented in the Torch simulation, and highlight how they lead to realistic initial conditions for the simulations presented in this work. \nThe Torch simulation uses magneto-hydrodynamics (MHD) coupled with stellar dynamics (including a treatment of dynamically-formed binaries), star formation, stellar evolution and feedback, and a sub-grid model for protoplanetary disks coupled through the Amuse framework (Portegies Zwart et al. 2009; Pelupessy et al. 2013; Portegies Zwart et al. 2013; Portegies Zwart & McMillan 2019). MHD is handled by the adaptive mesh refinement code Flash (Fryxell et al. 2000; Dubey et al. 2014), with a maximum spatial resolution of 0.136 pc. In the Torch simulation, star formation takes place in bound regions with high density gas ( ⩾ 3 . 819 x 10 -21 g cm -3 ) and converging flows. This results in the formation of subclusters with shapes consistent with observed embedded clusters (Cournoyer-Cloutier et al. 2023). The stars are sampled from a Kroupa (2001) initial mass function (IMF) from 0.08 M ⊙ to 100 M ⊙ . Stars more massive \nthan 7 M ⊙ provide feedback to the simulation in the form of momentum-driven winds, ionizing radiation and radiation pressure (Wall et al. 2020), resulting in a nonsmooth distribution of gas as seen in real star-forming regions.', '2.2. N-body dynamics': 'We use the stellar masses, positions and velocities from the full Torch run directly. We handle stellar dynamics with Petar (Wang et al. 2020a), which is optimized to handle large numbers of binaries and few-body encounters. Petar handles long-range interactions with a Barnes-Hut tree (Barnes & Hut 1986, as implemented by Iwasawa et al. 2016), short-range interactions with a fourth-order Hermite integrator (Makino & Aarseth 1992), and stable binaries and close encounters with the slow-down algorithmic regularization method (Wang et al. 2020b, sdar ). The longest timestep dt soft of the simulation, used for long-range interactions, must be chosen in conjunction with the changeover radii r in and r out between short- and long-range interactions. We set dt soft to 1/100 th of the shortest orbital period we could have for a binary in the long-range interaction regime. We set r out = 1 . 2120 x 10 -2 pc, r in = 1 . 2120 x 10 -3 pc, and dt soft = 27 . 948 years for our simulations. We also set r bin = 100 au for the radius under which we use slow-down algorithmic regularization.', '2.3. Hydrodynamics': 'We use the smoothed particles hydrodynamics (SPH) code Gadget-2 (Springel 2005) with a particle mass of 0.01 M ⊙ leading to 6.86 × 10 5 particles in total. We convert the gas from Flash to Gadget-2 using the method described in Karam & Sills (2024). The gravitational force from the gas on the stars and vice-versa is calculated with BHT ree (based on Barnes & Hut 1986, as implemented by Jun Makino) and the Bridge (Fujii et al. 2007) scheme. We use a bridge timestep of 2 5 dt soft = 894 . 336 years for our simulations. \nOur simulations do not include stellar feedback, stellar evolution, star formation or magnetic fields, in contrast with the Torch simulation from which we draw our initial conditions. It is however important to note that all stars in our simulations are younger than 3 Myr by the end of the runs, and that none of the stars would therefore have exploded in a supernova.', '2.4. Initial Conditions': "We summarize the different sets of initial conditions in Table 1 and in the text below. For our fiducial initial conditions (M0), we take the stars from the Torch simulation run at 2.13 Myr. The distribution of the \nTable 1. Overview of our simulations. Columns: the name of each simulation, whether the simulation has a primordial binary prescription, the prescription used for primordial binaries, and the initial conditions for the gas and cluster shape (see text for more details). Different numbers in the run names correspond to different random seeds. \nstars is shown in Figure 1. The fiducial initial conditions have 3047 stars for a total mass of 1923.8 M ⊙ . The stars are split into two clusters using a simple positional argument. For the runs in which we introduce binaries, each system (i.e. single star or binary) is placed so that its centre of mass is at the position of the star from M0 that has the mass closest to the system mass. This preserves the shape of the clusters, which is independent of the presence of binaries for embedded clusters (see Cournoyer-Cloutier et al. 2023). We also include two other runs without primordial binaries, M1 and M2 , where we randomly changed the mass of each star slightly from its mass in the Torch run but preserved each cluster's total mass. \nWe use three models for binaries, based on the observations presented in Moe & Di Stefano (2017, for stars above ∼ 1 M ⊙ ) and Winters et al. (2019, for M-dwarfs). In the all binaries models (AB1, AB2, AB3), we sample the full distribution of companions in the Galactic field using the technique presented in CournoyerCloutier et al. (2021). In the inner binaries models (IB1, IB2, IB3), we sample a distribution of inner companions; for intermediate- and high-mass stars, with a large triple fractions, this shifts the distribution to shorter periods. For each model, we vary the random seed to get the three different samplings. In the wide binaries models (WB1, WB2, WB3), we impose a minimum period cut- \nfollowing Ram'ırez-Tannus et al. (2021), \nP min = 10 (5 ± 1) exp ( -t/ 0 . 19 +0 . 06 -0 . 04 ) +1 . 40 (1) \nwhere P min is the minimum period in days and t is the cluster age in Myr. They derive this minimum period from the observed velocity dispersion of OB stars in young star clusters, which follows \nσ 1D = (13 . 3 ± 1 . 1) t +(2 . 0 ± 0 . 1) (2) \nWe pick three possible values for the age of the cluster: 0.66 Myr (for the formation time of the first star), 0.21 Myr (for the average age of the simulated stars at the start of our zoom-in simulations) and 0 Myr (since the mean formation time of the stars in the full simulation is slightly larger than 2.13 Myr). Using those ages, we modify the IB1 initial conditions, so that all systems made up of two OB stars (i.e. all systems where both stars have a mass of at least 2 M ⊙ ) have an orbital period of at least P min . We report the minimum period for an OB binary at the start of each simulation in Table 2, as P 0 OB , min . Although M0, M1 and M2 do not include any primordial binaries, they do include binaries formed through dynamics in the original Torch simulation. Those binaries have longer orbital periods than primordial binaries (see e.g Cournoyer-Cloutier et al. 2021, for a comparison between the properties of primordial and dynamically-formed binaries). \nTo investigate the effects of the background gas on the binaries (as it was shown to be important for the merger itself by Karam & Sills 2024), we also simulate an idealized merger and an isolated cluster of the same mass, both without background gas. We simulate a merger of two Plummer spheres of gas and stars, with the same stellar and gas masses as the merging clusters (PM). The relative velocity and impact parameter are also kept fixed, as well as the total gas mass, but the gas and stars are initialized as pairs of relaxed Plummer (1911) spheres with densities consistent with young massive clusters, following Karam & Sills (2022). We also simulate an isolated Plummer sphere with the total gas and stellar mass (PB). Both simulations use exactly the same stars and binaries as the IB1 simulation. A visual comparison of the gas between the merger inherited from the full initial conditions and the merger of two Plummer spheres is shown in Figure 2.", '3.1. Binary disruption': 'The binary fraction as a function of time is shown in the upper panel of Figure 3 for simulations with primor- \nFigure 2. Gas surface density for the merger inherited from the Torch simulation (left) and for the merger of two isolated Plummer spheres (right). The maximum surface density in the Plummer spheres simulation is about one order of magnitude higher than in the merger from the Torch simulation. \n<!-- image --> \ndial binaries. We calculate the binary fraction as \nF b = B S + B (3) \nwhere B is the number of stars with a bound companion within 10,000 au and S is the number of stars without a companion within 10,000 au. Binaries wider than 10,000 \nFigure 3. Binary fraction as defined in Equation 3 as a function of time for runs with primordial binaries. In merger runs with Torch initial conditions, about half of the total decrease in binary fraction takes place during the merger, which ends at 0.41 Myr. \n<!-- image --> \nau are not included in our statistics as they are very short-lived in dense stellar environments. \nThe initial binary fraction is ∼ 27% in all simulations with binaries. In the merger simulations with initial conditions inherited from the full Torch simulation (all the IB, AB and WB runs), the final binary fraction is between ∼ 19% and ∼ 21%, and the binary fraction at the end of the merger ( ∼ 0.40 Myr) is around 23-24%. For each of these simulations, about half of the decrease in the binary fraction takes place during the merger. The binary fraction therefore decreases by the same amount over the first 0.40 Myr and the last 1.60 Myr of the simulation, which indicates more rapid changes during the merger itself. The largest decrease takes place during the first ∼ 0.1 Myr of the simulation for all simulations realized with those initial conditions. We also note that the WB runs, which have all the same stars as IB1 and all the same binary systems for primaries less massive than 6 M ⊙ , follow almost exactly the same decrease in binary fraction as IB1 during the merger. When the sampling for the binaries is changed, some scatter is introduced but the overall trend remains. The AB runs show the largest amount of scatter. Since the binaries are sampled from the parameter space of all companions rather than inner companions only, there is a larger range in the distributions of semi-major axes and bind- \nFigure 4. Cumulative number of disrupted systems for IB1, PB and PM. The three simulations have the same initial population of binaries. \n<!-- image --> \nof the binaries, leading to more scatter in the fraction of disrupted systems. \nIn the Plummer sphere merger (PM), the binary fraction decreases from ∼ 27% to ∼ 23% over the course of the simulation, with a steeper decrease in the first 0.9 Myr. This is the simulation that has the highest binary fraction at the end of the simulation. The binary fraction in the isolated Plummer sphere (PB) decreases from ∼ 27% to ∼ 22%. The binary fractions in PM and PB follow the same decrease in the first 0.9 Myr of the simulation, but diverge in the second half of the simulation. \nWe plot the cumulative number of disrupted binaries as a function of time for all runs that use the IB prescription for primordial binaries, and present it in Figure 4. For merger simulations, most of the disruption takes place during the merger. It continues throughout the entire merger, but peaks in the early stages of the merger, during the first ∼ 0.1 Myr. We see a similar behaviour in runs with the other two binary prescriptions. More than half of the total disruptions take place during the merger, with roughly 100 binary systems being disrupted in the first 0.4 Myr. We also note that the distribution of disruptions as a function of time is almost identical for the IB runs during the merger, while they diverge at later times. Similar behaviour is observed in the AB and WB runs. The PM and PB runs also show \nFigure 5. Semi-major axes of binary systems disrupted in IB1, PB and PM during the first 0.41 Myr of the simulation, which corresponds to the time required for the merger to complete in IB1. The three simulations have the same initial population of binaries. \n<!-- image --> \ndisruption at early times, although not to the extent observed in the IB, AB, and WB runs. \nAll runs with primordial binaries and gas initial conditions inherited from a GMC-scale simulation show an excess of disruption during the merger. This excess is caused by the disruption of systems with smaller semi-major axes during the merger, as shown in Figure 5. We compare the distributions of semi-major axes for the binaries disrupted within the first 0.41 Myr of the simulation in IB1, PB and PM using a two-sample Kolomogorov-Smirnov (KS) test. We find that the semimajor axes of disrupted systems for PB and PM are consistent with being drawn from the same distributions. We are however confident at respectively 98.3% and 98.8% that the semi-major axes of the disrupted systems in IB1 are smaller than those in PB and PM.', '3.2. Unbound stars': "We present the unbound mass fraction for stars for all simulations in Figure 6. The fraction of unbound stars increases during the merger for all merger simulations (as expected from Karam & Sills 2022, 2024), while almost all stars remain bound to their host cluster in PB (the isolated Plummer sphere). A smaller fraction of stellar mass becomes unbound at early times in PM than in the other simulations with a primordial binary prescription, which behave very similarly to one \nFigure 6. Mass fraction of unbound stars as a function of time, for all simulations. \n<!-- image --> \nanother. By the time the merger ends in PM , at 0.60 Myr, the mass fraction of unbound stars has reached the same value as in the other simulations with primordial binaries. The most important difference is between the runs with primordial binaries, and those without primordial binaries. M0, M1 and M2 all have a similar mass fraction of unbound stars at the end of the merger, which is about 2.5 times higher than the mass fraction of unbound stars in the runs with Torch initial conditions and primordial binaries. This suggests that the underlying physical process responsible for the production of unbound stars during sub-clusters mergers is the same that leads to the disruption of binaries. In the absence of primordial binaries, the energy from the merger is converted to kinetic energy of the stars, leading to an increase in the fraction of unbound stars. In the presence of primordial binaries, however, the energy from the merger is used to disrupt binaries and is not sufficient to also unbind stars from the cluster. \nFor each unbound star, we calculate its threedimensional radial velocity v r relative to the cluster's center of mass. The distributions of velocities in the simulations with and without primordial binary distributions are similar. We label unbound stars with v r ≥ 30 km/s as runaways. We report the number of runaways N runaways in Table 2, along with the number of runaways more massive than 6 M ⊙ (N runaways , M ), the number of unbound stars more massive than 6 M ⊙ (N unbound , M ) and the number of runaways initially in \nFigure 7. Projected direction of motion in the xy plane of all unbound stars in simulations with Torch initial conditions, at the end of the merger. The length of the arrow is proportional to the projected velocity and the origin of the arrow corresponds to the location of the star. Runaway stars are color-coded by simulation while stars with v r < 30 km/s are shown in light grey. \n<!-- image --> \nFigure 8. Mollweide projection of the direction of motion of all unbound stars in simulations with Torch initial conditions, at the end of the merger. \n<!-- image --> \nbinaries (N runaways ,b ). The only merger simulation that does not produce any runaway stars is AB2 , which however has two unbound stars with v r ≥ 25 km/s, just below our runaway cut-off. PB, on the other hand, does not produce any runways, and the maximum radial velocity for an unbound star is 12.8 km/s. \nThe runaways produced during the merger for simulations with Torch initial conditions are shown in Figure 7, along with the unbound stars with v r ≤ 30 km/s. We also calculate the direction of motion of all unbound \nstars and the runaways stars after the mergers. We show the positions and directions of motion of all runaways produced during the merger in Figure 7. There are no runaways moving towards the (-x, +y) quadrant, which is the original location of the less massive cluster. If we consider all unbound stars produced during the merger for all Torch runs, however, the distribution of directions of motions peaks in the -x direction. Comparing the distribution of angles to uniform distributions in ϕ and θ with a KS test, we are confident at > 99.9% that the direction of motion of the unbound stars is not uniform. We present the three-dimensional direction of motion of unbound stars in Figure 8. The directions of motions are clustered around an angle ( ϕ, θ ) = (60 o , 0 o ). \nWe also report the binary fraction for stars that are unbound at the end of the simulation in Table 2. We calculate the binary fraction at the start of the simulation, at the end of the merger, and at the end of the simulation. The binary fraction for those stars decreases in all simulations, consistent with our expectation of disrupted binaries leading to unbound stars. We note that for simulations without primordial binaries, which only include wide, dynamically-formed bianries, the binary fraction of unbound stars at the end of the simulation is 0, while it ranges from 7 to 17% for all runs with primordial binaries (except PB).", '3.3. Outcomes for massive binaries': "We now consider specifically how massive OB binaries evolve throughout the simulations. We consider OB binaries to be systems with a primary mass above 6 M ⊙ (the lower mass threshold considered by Ram'ırezTannus et al. 2021) and companion mass above 2 M ⊙ (the lower mass limit for B-type main sequence stars). We are interested in two main features: can cluster mergers with background gas harden massive binaries, and can they disrupt them? \nWe first turn our attention to the OB binary with the shortest orbital period in each simulation. The minimum orbital period for an OB binary at the beginning and at the end of each simulation is reported in Table 2. There are no significant changes to the orbital period of the OB star with the shortest period in IB1, IB2, AB1, AB2, PB or PM. In IB3, the OB binary with the shortest period at the end of the simulation originally had a an orbital period of 123 days, and was hardened to an orbital period of 32.9 days through an exchange at 1.92 Myr. In AB3, the OB binary that had initially had the shortest orbital period, at 40.2 days, got hardened to an orbital period of 34.7 days, likely through a few-body encounter. \nIn simulations with initially wide massive binaries (WB1, WB2 and WB3), the minimum orbital period for OB binaries decreases significantly. In WB1, it decreases by a factor of > 2, going from ∼ 3100 days to ∼ 1440 days. The OB binary that has the shortest period at the end of the simulation however initially had an orbital period slightly longer than 13,000 days, and was hardened through an exchange at 1.23 Myr followed by a few-body interaction at 1.59 Myr. In WB2 and WB3, the OB binary that has the shortest orbital period at the end of the simulation also had the shortest possible period at the beginning of the simulation. Those periods were shorted by a factor of 7.4 and 25 in WB2 and WB3 respectively; we also note that the shortest period is shorter in WB3 than WB2 at the end of the simulation. This hardening is the expected behaviour from observations of OB binaries in young clusters (Ram'ırezTannus et al. 2021, orbital periods decrease during cluster formation). It however must be noted than none of the OB binaries in WB runs reaches an orbital period below 1000 days. Additionally, exchanges and interactions that significantly shorten the orbital period of OB binaries all take place well after the merger. \nWe look for disrupted OB binaries in all simulations that include primordial binaries. In the runs with Torch initial conditions, all the disruption of OB binaries takes place after the merger. There is one early disruption in PM. We stress that energetic binaries can be disrupted early if they encounter other energetic binaries. If we calculate the hard-soft limit for the resultant clusters, we find values around 2 x 10 44 erg, with a maximum value of 3.10 x 10 44 erg. Almost all the disrupted OB binaries were more energetic than this limit: this suggests that the short-period OB binaries observed in young star clusters might not be fully representative of the primordial population, despite their high binding energies. We also note that OB binaries were disrupted in all merger runs with primordial binaries, but not in the isolated Plummer sphere. \nOnly one massive star becomes unbound from its host cluster the during the merger: a star with mass 68 M ⊙ in M0 , the fiducial set of initial conditions. This star is the second most massive in the simulation. It forms a binary with semi-major axis ∼ 300 au with a star with mass 63 M ⊙ during the merger. It is then disrupted by an encounter with the most massive star in the simulation, which has a mass of 85 M ⊙ ; after the encounter, the 85 M ⊙ and 63 M ⊙ stars form a binary with semimajor axis ∼ 80 au. M0 , M1 and M2 produce the most massive unbound stars: by the end of the simulation, M0 has unbound stars with masses 22, 23, 50 and 68 M ⊙ , M1 has unbound stars with masses 49 and 66 M ⊙ , \nTable 2. Properties of unbound stars and OB binaries in the different simulations. Columns: run name, number of runaways, number of runways with mass above 6 M ⊙ , number of unbound stars with mass above 6 M ⊙ , number of runaways originally in binaries, fraction F b, i of stars unbound at the end of the simulation originally in binaries, fraction F b, m of stars unbound at the end of the simulation in binaries at the end of the merger, fraction F b, f of stars unbound at the end of the simulation in binaries at the end of the simulation, minimum initial period for an OB binary, minimum period for an OB binary at the end of the simulation. \nand M2 has an unbound stars with mass 29 M ⊙ . They have all been identified as members of a wide binary in at least one snapshot, but those did not persist between snapshots, indicating that a few-body interaction took place. This illustrates how wide, dynamically-formed binaries can results in the ejection of massive stars or large numbers of ejected stars, due to their larger dynamically cross-sections. Indeed, the other two simulations without primordial binaries, M1 and M2 , also have the largest number of runaway stars produced during the merger, and during the full simulation. On the other hand, the mass of the most massive unbound star in any simulation with primordial binaries is ∼ 11 M ⊙ .", '4.1. Causes of binary disruption during hierarchical cluster assembly': "There are several differences between the simulations using the initial conditions from Torch and those using idealized Plummer models. We discuss the effects of the stellar and gas distributions below. \nThe stellar density profiles for IB1, PB, and PM are very similar. All three simulations reach densities ≳ 10 4 M ⊙ pc -3 (and therefore number densities > 10 4 pc -3 ) in their central regions. If high stellar densities were the cause of the excess disruption we observe in the Torch runs, we would expect the mass or number density to be significantly higher in the Torch runs than in the idealized models, which is not the case. The gas distribution however differs strongly between the Torch runs \n- which were extracted from a GMC simulation - and the idealized Plummer models. The central gas density is about an order of magnitude higher in the Plummer models than in the Torch models, as illustrated in Figure 2. Conversely, the gas distribution is more extended in the Torch models, and includes background gas. \nThe more realistic gas distribution contributes in two ways to the excess disruption of binaries observed in the Torch models. First, the presence of background gas promotes the merger between the sub-clusters, as found by Karam & Sills (2024). This means that the stars from the smaller cluster are accreted unto the larger cluster much more quickly in the Torch simulations than in the Plummer spheres' merger. This rapid accretion means that binaries in either cluster are more likely to encounter other binaries and stars, and therefore have a larger chance to be disrupted. On the other hand, the Plummer spheres do not immediately merge, and the two clusters can be easily identified after the first passage. Second, the potential is shallower in the Torch runs, due to the more extended gas distribution. This allows binaries with higher binding energies (and therefore smaller semi-major axes) to be disrupted by allowing the stars to move apart more easily after an encounter. We note that the potential increases sharply during the first ∼ 0.2 Myr of the simulations, during which the disruption rate of binaries peaks in all Torch runs.", '4.2. Unbound stars in embedded star clusters': 'Recent observations have found a possible anisotropy in the distribution of runaway stars around the young massive star cluster R136 (Stoop et al. subm.). Using simulations of massive cluster formation, Polak et al. (2024) attribute this effect to the production of runaways from hierarchical cluster assembly inside a GMC. They find that an entire sub-cluster can become a system of runaway stars, due to the large tidal forces involved in the merger which completely destroys an incoming cluster. Our simulations however probe a lower cluster mass range, resulting in a smaller number of runaways, which is not sufficient to investigate whether those stars have a preferred direction. Considering the full population of unbound stars - of which runaways are only a subset - however allows us to conclude that stars ejected during sub-cluster mergers inside GMCs have a preferred direction of motion, even in a lowermass regime.', '4.3. Implications for massive cluster formation': "Our results offer several insights into the process of (massive) star cluster formation. First, the disruption of binaries during hierarchical cluster assembly within giant molecular clouds offers a natural explanation for the lower wide binary fraction found in the ONC than in other young star-forming regions (e.g. Duchˆene et al. 2018). As the ONC has been proposed to be the recent site of a merger between gas-rich sub-clusters (see e.g. Fujii et al. 2022), the disruption of the wide binaries during the merger process would be consistent with our findings. Howard et al. (2018) found that massive clusters acquire about half of their stellar mass via gas-rich mergers. Such a mode of cluster assembly could contribute to the low binary fractions observed in globular clusters (e.g. Milone et al. 2016). \nBinary disruption during gas-rich or gas-driven subcluster mergers leading to a shift towards smaller semimajor axes could also explain the apparent discrepancy between the results of Cournoyer-Cloutier et al. (2021) and Parker & Meyer (2014) or Torniamenti et al. (2021) regarding the evolution of field-like population of binaries in star clusters. In Cournoyer-Cloutier et al. (2021), we found that a field-like distribution of semi-major axes, like the one used in the AB runs, shifts to smaller values during hierarchical star cluster formation. On the other hand, for a similar distribution of binaries, Parker & Meyer (2014) use fractal initial conditions for their stars, while Torniamenti et al. (2021) use stellar positions inherited from the gas distribution. Neither, however, included gas along with the N-body dynamics while investigating the binaries. Both found the distribution of semi-major axes statistically unchanged by \nthe process of (gas-free) cluster assembly. Along with the results presented in this paper, this suggests that changes in the gas potential that coincide with subclusters merger play a critical role in disrupting binaries during cluster formation. \nAnother important implication of our results concerns the hardening of massive binary systems. The density in the inner 0.3 pc of IB1 is roughly 1.2 x 10 4 M ⊙ pc -3 at the start of the simulation, and 8.6 x 10 3 M ⊙ pc -3 after 0.9 Myr, around the typical central densities for young massive clusters (Portegies Zwart et al. 2010). Despite those high densities, however, simulations that started without any close massive binaries did not succeed in forming any OB binary with an orbital period shorter than 1000 days, while the most conservative choice proposed by Ram'ırez-Tannus et al. (2021) in Equation 1, using a cluster age of 2 Myr, gives a period of 337 days, about a factor of 4 shorter than the shortest period OB binary we get in WB1. Although our simulated clusters explore a lower mass range than the clusters studied by Ram'ırez-Tannus et al. (2021), our results suggest that the orbital parameters of OB binaries can be modified on timescales shorter than the cluster formation timescale.", '5. SUMMARY': 'We have conducted simulations of stellar sub-cluster mergers with different realistic prescriptions for the initial distribution of binaries. The shapes and masses of the sub-clusters, as well as the background gas, were taken from larger-scale simulations of cluster formation within a giant molecular cloud. We find that massive binaries can be disrupted or undergo significant changes to their orbital periods over timescales shorter than the cluster-formation timescale, in all of our merger simulations. The observed distributions of OB binaries is likely not the same as their formation distribution even in very young clusters and associations. \nWe also find that the merger process - and by extension, hierarchical cluster formation - lowers the binary fraction and disrupts a large number of binaries during the merger, in excess of what is observed for an isolated cluster or an idealized model of cluster merger. The disrupted systems also tend to have smaller semimajor axes than in idealized models. This excess disruption is attributed to the rapid merger driven by the background gas distribution, as well as the shallower potential. Sub-cluster mergers result in stars becoming unbound from their host cluster, with a larger mass fraction of unbound stars when primordial binaries are taken into account. Simulations without primordial binaries, in which the most massive stars tend to pair up \nin a wide, dynamically-formed binaries, are capable of ejection higher-mass stars for the cluster mass and density regime we consider. We further note that stars that become unbound during the merger tend to move in the same direction. \nWe conclude that the production of unbound stars with a preferred direction of motion is a natural consequence of sub-cluster mergers within a GMC, extending the results of Polak et al. (2024) to lower cluster masses. We further argue that viewing binary disruption as a by-product of sub-cluster mergers within a giant molecular cloud offers a natural explanation for the lower binary fraction observed in denser star-forming environments and in globular clusters, which form hierarchically through subsequent mergers. \nCCC is grateful for the hospitality of Leiden University, where this work was started during a visit in June 2023. The authors thank the referee for comments that improved the clarity of the paper. The authors warmly thank Eric Andersson, Sabrina Appel, Mordecai-Mark Mac Low, Stephen McMillan and Brooke Polak for ongoing discussions about Torch . The authors also thank Veronika Dornan, William Harris and Marta ReinaCampos for helpful discussions. CCC and JK are grateful to Gwendolyn Eadie for very valuable insights regarding the choice of initial stellar distributions. \nCCC is supported by a Canada Graduate Scholarship - Doctoral (CGS D) from the Natural Sciences and Engineering Research Council of Canada (NSERC). The visit to Leiden University was made possible by a Michael Smith Foreign Study Supplement (CGS MSFSS) held by CCC at the Max Planck Institute for Astrophysics in Summer 2023. JK and AS are supported by NSERC. This research was enabled in part by support provided by Compute Ontario (https://www. computeontario.ca/) and the Digital Research Alliance of Canada (alliancecan.ca) via the research allocation FT #2665: The Formation of Star Clusters in a Galactic Context. 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